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HEAT TRANSFER AND FLUID FLOW IN MINICHANNELS AND MICROCHANNELS Elsevier Internet Homepage http://www.elsevier.com Con... Author: Satish Kandlikar | Srinivas Garimella | Dongqing Li | Stephane Colin | Michael R. King

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Ana yt ca Methods for Heat Transfer and F u d F ow Prob ems

HEAT TRANSFER AND FLUID FLOW IN MINICHANNELS AND MICROCHANNELS Elsevier Internet Homepage http://www.elsevier.com Consult the Elsevier homepage for full catalogue information on all books, journals and electronic products and services Elsevier Titles of Related Interest INGHAM AND POP Transport Phenomena in Porous Media III 2005; ISBN: 0080444903 www.elsevier.com/locate/isbn/0080449031 W. RODI AND M. MULAS Engineering Turbulence Modelling and Experiments 6 2005; ISBN: 0080445446 www.elsevier.com/locate/isbn/0080445446 HARTNETT et al. Advances in Heat Transfer Volume 38 2004; ISBN: 0120200384 www.elsevier.com/locate/isbn/0120200384

Bernhard We gand Ana yt ca Methods for Heat Transfer and F u d F ow Prob ems Bernhard We gand Ana yt ca Methods fo...

Computat ona F u d Mechan cs and Heat Transfer

DONGQING LI Electrokinetics in Microfluidics 2004; ISBN: 0120884445 www.elsevier.com/locate/isbn/0120884445 Related Journals The following titles can all be found at: http://www.sciencedirect.com Applied Thermal Engineering Experimental Thermal and Fluid Science Fluid Abstracts: Process Engineering Fluid Dynamics Research International Communications in Heat and Mass Transfer International Journal of Heat and Fluid Flow International Journal of Heat and Mass Transfer International Journal of Multiphase Flow International Journal of Refrigeration International Journal of Thermal Sciences To Contact the Publisher Elsevier welcomes enquiries concerning publishing proposals: books, journal special issues, conference, proceedings, etc. All formats and media can be considered. Should you have a publishing proposal you wish to discuss, please contact, without obligation, the publishing editor responsible for Elsevier’s Mechanical Engineering publishing programme: Arno Schouwenburg Senior Publishing Editor Materials Science & Engineering Elsevier Limited The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, UK Tel.: +44 1865 84 3879 Fax: +44 1865 84 3987 E-mail: [email protected] General enquiries including placing orders, should be directed to Elsevier’s Regional Sales Offices – please access the Elsevier internet homepage for full contact details. HEAT TRANSFER AND FLUID FLOW IN MINICHANNELS AND MICROCHANNELS Contributing Authors Satish G. Kandlikar (Editor and Contributing Author) Mechanical Engineering Department Rochester Institute of Technology, NY, USA

F u d Dynam cs and Heat Transfer of Turbomach nery

Computat ona F u d Mechan cs and Heat Transfer, ~ ~ ~ ~~ ~ ~ ~ ~ ~ Ser es n Computat ona and Phys ca Processes n Mechan cs and Therma Sc ences W. J. M n...

Srinivas Garimella George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology, Atlanta, USA Computat ona F u d Mechan cs And Heat Transfer

Dongqing Li Department of Mechanical and Industrial Engineering University of Toronto, Ontario, Canada

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Stéphane Colin Department of General Mechanic National Institute of Applied Sciences of Toulouse Toulouse cedex, France Michael R. King Departments of Biomedical Engineering, Chemical Engineering and Surgery University of Rochester, NY, USA

Computat ona f u d mechan cs and heat transfer

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Opt ca methods and data process ng n heat and f u d f ow

ELSEVIER Inc. 525 B Street, Suite 1900 San Diego, CA 92101-4495, USA ELSEVIER Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, UK ELSEVIER Ltd 84 Theobalds Road London WC1X 8RR, UK © 2006 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier Ltd., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for nonprofit educational classroom use. Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2006 ISBN: 0-0804-4527-6 Typeset by Charon Tec Pvt. Ltd., Chennai, India www.charontec.com Printed in Great Britain

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Computat ona F u d Dynam cs and Heat Transfer: Emerg ng Top cs (Deve opments n Heat Transfer) (Deve opments n Heat Transfer Ob ect ves) Computat ona F u d Dynam cs and Heat T ransf er Transf ransfer WITPRESS WIT Press pub shes ead ng books n Sc ence ...

CONTENTS About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi viii x

Testing Laboratory

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satish G. Kandlikar and Michael R. King 1

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Chapter 2.

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Single-phase gas flow in microchannels . . . . . . . . . . . . . . . . . . . . . . Stéphane Colin 9 Chapter 3. Single-phase liquid flow in minichannels and microchannels . . Satish G. Kandlikar 87

OPEN

Chapter 4. Single-phase electrokinetic flow in microchannels . . . . . . . . . . . . Dongqing Li 137 Chapter 5. Flow boiling in minichannels and microchannels . . . . . . . . . . . . . Satish G. Kandlikar 175 Chapter 6. Condensation in minichannels and microchannels . . . . . . . . . . . . Srinivas Garimella 227 Chapter 7. Biomedical applications of microchannel flows . . . . . . . . . . . . . . . Michael R. King 409 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 v ABOUT THE AUTHORS Satish Kandlikar Dr. Satish Kandlikar is the Gleason Professor of Mechanical Engineering at Rochester Institute of Technology. He obtained his B.E. degree from Marathawada University and M.Tech. and Ph.D. degrees from I.I.T. Bombay. His research focuses on flow boiling and single phase heat transfer and fluid flow in microchannels, high flux cooling, and fundamentals of interfacial phenomena. He has published over 130 conference and journal papers, presented over 25 invited and keynote papers, has written contributed chapters in several handbooks, and has been editor-in-chief of a handbook on boiling and condensation. He is the recipient of the IBM Faculty award for the past three consecutive years. He received the Eisenhart Outstanding Teaching Award at RIT in 1997. He is an Associate Editor of several journals, including the Journal of Heat Transfer, Heat Transfer Engineering, Journal of Microfluidics and Nanofluidics, International Journal of Heat and Technology, and Microscale Thermophysical Engineering. He is a Fellow member of ASME. Srinivas Garimella Dr. Srinivas Garimella is an Associate Professor and Director of the Sustainable Thermal Systems Laboratory at Georgia Institute of Technology. He was previously a Research Scientist at Battelle Memorial Institute, Senior Engineer at General Motors, Associate Professor at Western Michigan University, and William and Virginia Binger Associate Professor at Iowa State University. Dr. Garimella received M.S. and Ph.D. degrees from The Ohio State University, and a Bachelors degree from the Indian Institute of Technology, Kanpur. He is Associate Editor of the ASME Journal of Energy Resources Technology and the International Journal of HVAC&R Research, and is Chair of the Advanced Energy Systems Division of ASME. He conducts research in the areas of vapor-compression and absorption heat pumps, phase-change in microchannels, heat and mass transfer in binary fluids, and supercritical heat transfer in natural refrigerants and blends. He has authored over 85 refereed journal and conference papers and several invited short courses, lectures and book chapters, and holds four patents. He received the NSF CAREER Award, the ASHRAE New Investigator Award, and the SAE Teetor Award for Engineering Educators. Dongqing Li Dr. Dongqing Li obtained his BA and MSc. degrees in Thermophysics Engineering in China, and his Ph.D. degree in Thermodynamics from the University of Toronto, Canada, in 1991. He was a professor at the University of Alberta and later in the University of Toronto from 1993 to 2005. Currently, Dr. Li is the H. Fort Flowers professor of Mechanical Engineering, Vanderbilt University. His research is in the areas of microfluidics and labon-a-chip. Dr. Li has published one book, 11 book chapters, and over 160 journal papers. He is the Editor-in-Chief of an international journal Microfluidics and Nanofluidics. vi About the Authors vii Stéphane Colin Dr. Stéphane Colin is a Professor of Mechanical Engineering at the National Institute of Applied Sciences (INSA) of Toulouse, France. He obtained his Engineering degree in 1987 and received his Ph.D. degree in Fluid Mechanics from the Polytechnic National Institute ofToulouse in 1992. In 1999, he created the Microfluidics Group of the Hydrotechnic Society of France, and he currently leads this group. He is the Assistant Director of the Mechanical Engineering Laboratory of Toulouse. His research is in the area of microfluidics. Dr. Colin is editor of the book Microfluidique, published by Hermes Science Publications. Michael King Dr. Michael King is an Assistant Professor of Biomedical Engineering and Chemical Engineering at the University of Rochester. He received a B.S. degree from the University of Rochester and a Ph.D. from the University of Notre Dame, both in chemical engineering. At the University of Pennsylvania, King received an Individual National Research Service Award from the NIH. King is a Whitaker Investigator, a James D. Watson Investigator of New York State, and is a recipient of the NSF CAREER Award. He is editor of the book Principles of Cellular Engineering: Understanding the Biomolecular Interface, published by Academic Press. His research interests include biofluid mechanics and cell adhesion. PREFACE In the last few decades, new frontiers have been opened up by advances in our ability to produce microscale devices and systems. The numerous advantages that can be realized by constructing devices with microscale features have, in many cases, been exploited without a complete understanding of the way the miniaturized geometry alters the physical processes. The augmentation of transport processes due to microscale dimensions is taken advantage of in nature by all biological systems. In the engineered systems that are the focus of this book, the challenge is to understand and quantify how utilizing microscale passages alters the fluid flow patterns and the resulting, momentum, heat, and mass transfer processes to maximize device performance while minimizing cost, size, and energy requirements. In this book, we are concerned with flow through passages with hydraulic diameters from about 1 µm to 3 mm, covering the range of microchannels and minichannels. Different phenomena are affected differently as we approach microscales depending on fluid properties and flow conditions; hence, classification schemes that identify a channel as macro, mini, or micro should be considered merely as guidelines. The main topics covered in this book are single-phase gas flow and heat transfer; singlephase liquid flow and heat transfer; electrokinetic effects on liquid flow; flow patterns, pressure drop, and heat transfer in convective boiling; flow patterns, pressure drop, and heat transfer during convective condensation, and finally biological applications. The coverage is intended to reflect the status of our current understanding in these areas. In each chapter, the fundamental physical phenomena related to the specific processes are introduced first. Then, the engineering analyses and quantitative methods derived from theoretical and experimental work conducted worldwide are presented. Areas requiring further research are clearly identified throughout as well as summarized within each chapter. There are two intended audiences for this book. First, it is intended as a basic textbook for graduate students in various engineering applications. The students will find the necessary foundation for the relevant transport processes in microchannels as well as summaries of the key models, results, and correlations that represent the state-of-the-art. To facilitate the development of the ability to use the new information presented, each chapter contains several solved example problems that are carefully selected to provide practical guidance for students as well as practitioners. Second, this book is also expected to serve as a source book for component and system designers and researchers. Wherever possible, the range of applicability and uncertainty of the analyses presented is provided so that analyzing new devices and configurations can be done with known levels of confidence. The comprehensive summary of the literature included in each chapter will also help the readers identify valuable source material relevant to their specific problem for further investigation. The authors would like to express appreciation toward the students and coworkers who have contributed significantly in their research and publication efforts in this field. The viii Preface ix first author would like to thank Nathan English in particular for his efforts in editing the entire manuscript and preparing the master nomenclature. The authors are thankful to the scientific community for their efforts in exploring microscale phenomena in microchannels and minichannels. We look forward to receiving information on continued developments in this field and feedback from the readers as we strive to improve this book in the future. We also recognize that in our zest to prepare this manuscript in a rapidly developing field, we may have inadvertently made errors or omissions. We humbly seek your forgiveness and request that you forward us any corrections or suggestions. Satish G. Kandlikar Srinivas Garimella Dongqing Li Stéphane Colin Michael King NOMENCLATURE A A,B,C,D,F A1 A2 A3 Ac Ap AT a a* a1 , a2 , a3 a1 . . . a5 B BB Bn b Bo Bo C, C C C* C0 C1 , C2 CC Cin Cf Section area, m2 (Chapters 2 and 6). Equation coefficients and exponents (Chapters 3 and 7). First-order slip coefficient, dimensionless (Chapter 2). Second-order slip coefficient, dimensionless (Chapter 2). High-order slip coefficient, dimensionless (Chapter 2). Cross-sectional area, m2 (Chapter 3). Total plenum cross-sectional area, m2 (Chapter 3). Total heat transfer surface area (Chapter 5). Speed of sound, m/s (Chapter 2); channel width, m (Chapter 3) equation constant in Eqs. (6.71), (6.83) and (6.87) (Chapter 6); coefficient in entrance length equations, dimensionless (Chapter 7). Aspect ratio of rectangular sections, dimensionless, a* = h/b (Chapter 2). Coefficients for the mass flow rate in a rectangular microchannel, dimensionless (Chapter 2). Coefficients in Eq. (6.107) (Chapter 6). Parameter used in Eqs. (6.9) and (6.41) (Chapter 6). Parameter used in Eq. (6.21) (Chapter 6). Coefficient in cell surface oxygen concentration equation (Chapter 7). Half-channel width, m (Chapter 2); channel height, m (Chapter 3); constant in Eqs. (6.71), (6.83) and (6.87) (Chapter 6). Boiling number, dimensionless, Bo = q /(GhLV ) (Chapter 5). Bond number, dimensionless, Bo = (L − V )gDh2 / (Chapter 5); Bo = g(L − G )((d/2)2 /) (Chapter 6). Constant, dimensionless (Chapter 1); coefficient in a Nusselt number correlation (Chapter 3); concentration, mol/m3 (Chapters 4 and 7); Chisholm’s parameter, dimensionless (Chapter 5); constant used in Eqs. (6.39), (6.40) and (6.157) (Chapter 6). Reference concentration, mol/m3 (Chapter 4). Ratio of experimental and theoretical apparent friction factors, dimensionless, C * = fapp,ex /fapp,th , (Chapter 3); non-dimensionalized concentration, C * = C(x, y)/Cin (Chapter 7). Oxygen concentration at the lower channel wall, mol/m3 (Chapter 7). Empirically derived constants in Eq. (6.35) (Chapter 6); parameter, used in Eq. (6.54) (Chapter 6). Coefficient of contraction, dimensionless (Chapter 6). Non-dimensionalized gas phase oxygen concentration, Cin = C˜ g /Cg (Chapter 7). Friction factor, dimensionless (Chapter 2). x Nomenclature Co Co Co Cp CS c c c c1 , c2 cp cv Ca CHF D D, D+ , D− Dcf Dh , DH Dle d dB E E1 , E2 Ex e Eo, Eö F, F Convection number, dimensionless, Co = [(1 − x)/x]0.9 [V /L ]0.5 (Chapter 5). √ /g( − ) v l Confinement number, dimensionless, Co = (Chapter 6). Dh Contraction coefficient, dimensionless (Chapter 5); distribution parameter in drift flux model, dimensionless, Co = j/( j) (Chapter 6). Specific heat capacity at a constant pressure, J/kg K (Chapter 6). Saturation oxygen concentration, mol/m3 (Chapter 7). Mean-square molecular speed, m/s (Chapter 2); constant in the thermal entry length equation, dimensionless (Chapter 3); constant in Eq. (6.57) (Chapter 6). Molecular thermal velocity vector m/s (Chapter 2). Mean thermal velocity, m/s (Chapter 2). Coefficients used in Eq. (6.133) (Chapter 6). Specific heat at a constant pressure, J/kg K (Chapters 2, 3 and 5). Specific heat at a constant volume, J/kg K (Chapter 2). Capillary number, dimensionless, Ca = µV / (Chapter 5). Critical heat flux, W/m2 (Chapter 5). Diameter, m (Chapters 1, 3, 5 and 6). Diffusion coefficient, diffusivity, m2 /s (Chapters 2 and 4). Diameter constricted by channel roughness, m, Dcf = D − 2 (Chapter 3). Hydraulic diameter, m (Chapters 1–5). Laminar equivalent diameter, m (Chapter 3). Mean molecular diameter, m (Chapter 2); diameter, m (Chapter 6). Departure bubble diameter, m (Chapter 5). Applied electrical field strength, V/m (Chapter 4); total energy per unit volume, J/m3 (Chapter 2); diode efficiency, dimensionless, (Chapter 2); parameter used in Eq. (6.141) (Chapter 6). Parameter used in Eq. (6.22) (Chapter 6). Electric field strength, V/m (Chapter 4). Internal specific energy, J/kg (Chapter 2); charge of a proton, e = 1.602 × 10−19 C (Chapter 4). Eötvös number, dimensionless, Eo = g (L − V ) L2 / in case of liquid gas contact (Chapters 5 and 6). Non-dimensional constant accounting for an electrokinetic body force (Chapter 4); general periodic function of unit magnitude (Chapter 4); force,N (Chapters 5 and 6); modified Froude number, dimensionless, F= F FM FS xi g UGS √ l −g Dg (Chapter 6); stress ratio, dimensionless, F = w / (L g) (Chapter 6); parameter used in Eq. (6.141) (Chapter 6); external force acting on a spherical cell, N (Chapter 7). External force per unit mass vector, N/kg (Chapter 2). Interfacial force created by evaporation momentum, N (Chapter 5). Interfacial force created by surface tension, N (Chapter 5). xii FFl Fg FT Fx f f fapp f fls Fp Frl Fr m Fr so Ft Nomenclature Fluid-surface parameter accounting for the nucleation characteristics of different fluid surface combinations, dimensionless (Chapter 5). Function of the liquid volume fraction and the vapor Reynolds number, used in Eq. (6.128) (Chapter 6). Dimensionless parameter of Eq. (6.112) (Chapter 6). Electrical force per unit volume of the liquid, N/m3 (Chapter 4). Volume force vector, N/m3 (Chapter 2). Fanning friction factor, dimensionless (Chapters 1, 3 and 5); single-phase friction factor, dimensionless (Chapter 6). Apparent friction factor accounting for developing flows, dimensionless (Chapter 3). Frequency, Hz (Chapter 2); velocity distribution function (Chapter 2). Superficial liquid phase friction factor, dimensionless (Chapter 6). Floor distance to mean line in roughness elements, m (Chapter 3). 2 Liquid Froude number, dimensionless, Fr 2l = V l /g (Chapter 6). Modified Froude number, dimensionless (Chapter 6). Soliman modified Froude number, dimensionless (Chapter 6). 0.5 G 2 x3 Froude rate, dimensionless Ft = (1−x) (Chapter 6). 2 gD g G Geq Gl Mass flux, kg/m2 s (Chapters 1, 5 and 6). Equivalent mass flux, kg/m2 s, Geq = Gl + Gl (Chapter 6). Mass flux that produces the same interfacial shear stress as a vapor core, fv l 2 kg/m -s, Gl = Gv v fl (Chapter 6). Gt g Gal H h hc hfg hG Total mass flux, kg/m2 s (Chapter 6). Acceleration due to gravity, m/s2 (Chapters 5 and 6). Liquid Galileo number, dimensionless, Gal = gD3 /l2 (Chapter 6). Maximum height, m (Chapter 4); distance between parallel plates or height, m (Chapter 7); parameter used in Eq. (6.141) (Chapter 6). Heat transfer coefficient, W/m2 -K (Chapters 1, 3, 5 and 6); channel half-depth, m (Chapter 2); specific enthalpy, J/kg (Chapters 2 and 5); wave height, m (Chapter 7). Average heat transfer coefficient, W/m2 K (Chapters 3 and 6). Film heat transfer coefficient, W/m2 K (Chapter 6). Latent heat of vaporization, J/kg (Chapter6). Gas-phase height in channel, m, hG ≤ 4 g (1− ) (Chapter 6). hLV hlv hlv I I i J J j Latent heat of vaporization at pL , J/kg (Chapter 5). Specific enthalpy of vaporization, J/kg (Chapter 6). Modified specific enthalpy of vaporization, J/kg (Chapter 6). Unit tensor, dimensionless (Chapter 2). Current, A (Chapter 3). Enthalpy, J/kg (Chapter 5). Mass flux vector, kg/m2 s (Chapter 2). Electrical current, A (Chapter 4). Superficial velocity, m/s (Chapters 5 and 6). h 4 Nomenclature * jg* , jG Ja Jal K K(x) K(∞) K1 K2 K90 Kc Ke Km k k1 k2 k B , b Kn Kn Ku L LG, LL Lent , Lh , Lhd Lt Leq l lSV lx,y,z LHS M M xiii Wallis dimensionless gas velocity, jg* = √ Gt x (Chapter 6). Dgv (l −v ) c T Jakob number, dimensionless, Ja = VL p,LhLV (Chapter 5). c (T −T ) Liquid Jakob number, dimensionless, Jal = pL hsatlv s (Chapter 6). Non-dimensional double layer thickness, K = Dh (Chapter 4); constant in Eqs. (6.56) and (6.95) (Chapter 6). Incremental pressure defect, dimensionless (Chapter 3). Hagenbach’s factor, dimensionless, K(x) when x > Lh (Chapter 3). Ratio of evaporation momentum to inertia forces at the liquid–vapor 2 interface, dimensionless, K1 = GqhLV VL (Chapter 5). Ratio of evaporation momentum to surface tension forces at the 2 liquid– vapor interface, dimensionless, K2 = hqLV VD (Chapter 5). Loss coefficient at a 90 bend, dimensionless (Chapter 3). Contraction loss coefficient due to an area change, dimensionless (Chapter 3). Expansion loss coefficient due to an area change, dimensionless (Chapter 3). Michaelis constant, mol/m3 (Chapter 7). Thermal conductivity, W/mK (Chapters 1–3, 5 and 6); constant, dimensionless (Chapter 7). Coefficient in the collision rate expression, dimensionless (Chapter 2). Coefficient in the mean free path expression, dimensionless (Chapter 2). Boltzmann constant, kB = 1.38065 J/K (Chapters 2 and 4). Knudsen number, Kn= /L, dimensionless (Chapters 1 and 2). Minimal representative length Knudsen number, Kn = /Lmin (Chapter 2). Kutateladze number, dimensionless Ku = Cp T /hfg (Chapter 6). Length or characteristic length in a given system, m (Chapters 1–3 and 5–7); Laplace constant, m, L = g(l−v ) (Chapter 6). Gas and liquid slug lengths in the slug flow regime, m (Chapter 6). Hydrodynamically developing entrance length, m, Lent = aHRe (Chapters 2, 3 and 7). Thermally developing entrance length, m, Lt = cRePrDh (Chapter 3). Total pipette length, m (Chapter 7). Microchannel length, m (Chapter 2). Characteristic length of a sampling volume, m (Chapter 2). Channel half height, m (Chapter 4). Left hand side (Chapter 7). Molecular weight, kg/mol (Chapter 2). Ratio of the electrical force to frictional force per unit volume, dimensionless, M = 2n∞ zeDh2 /µ UL (Chapter 4). xiv M, N MW m m ˙ m ˙* m ˙ ns Ma N N˙ N+ N− Nconf N0 n n1 , n2 , n3 ni nio nx Nu NuH Nui NuL Nuo NuT NuT Nux ONB P Pw p pR Pe PeF Nomenclature Equation exponents, dimensionless (Chapter 3). Molecular weight, g/mol (Chapter 7). Molecular mass, kg (Chapter 2); liquid volume fraction, dimensionless (Chapter 6); dimensionless constant in Eq. (6.57) (Chapter 6). Mass flow rate, kg/s (Chapters 2, 3 and 5). Mass flow rate, m/ ˙ m ˙ ns , dimensionless (Chapter 2). Mass flow rate for a no-slip flow, kg/s (Chapter 2). Mach number, dimensionless, Ma = u/a (Chapter 2). Avogadro’s number, 6.022137 · 1023 mol−1 (Chapter 2). Molecular flux, s−1 (Chapter 2). Non-dimensional positive species concentration (Chapter 4). Nondimensional negative species concentration √ (Chapter 4). /(g( − )) v l Confinement number, dimensionless, Nconf = (Chapter 6). Dh 2 Cellular uptake rate, mol/m s (Chapter 7). Number density, m−3 (Chapter 2); number or number of channels, dimensionless (Chapter 3); number of channels (Chapter 5); constant in Eqs. (6.41) and (6.57) (Chapter 6); number (Chapter 7). Constant in Eq. (6.21) (Chapter 6). Number concentration of type-i ion (Chapter 4). Bulk ionic concentration of type-i ions (Chapter 4). Normal vector in the x direction (Chapter 4). Nusselt number, dimensionless, Nu = hDh /k, (Chapters 1–3, 5 and 6). Nusselt number under a constant heat flux boundary condition, dimensionless (Chapter 3). Nusselt number for high interfacial shear condensation, dimensionless (Chapter 6). Average Nusselt number along a plate of length L, dimensionless (Chapter 6). Nusselt number for quiescent vapor condensation, dimensionless 1/n Nuo = (NunL1 ) + (NunT1 ) 1 (Chapter 6). Nusselt number for a turbulent film, dimensionless (Chapter 6). Nusselt number under a constant wall temperature boundary condition, dimensionless (Chapter 3). 1/n Combined Nusselt number, dimensionless, Nux = (Nuno2 ) + (Nuni 2 ) 2 (Chapter 6). Onset of nucleate boiling (Chapter 5). Wetted perimeter, m (Chapter 2); dimensionless pressure (Chapter 4); heated perimeter, m (Chapter 5); pressure, Pa (Chapter 6). Wetted perimeter, m (Chapter 3). Pressure, Pa (Chapters 1–3 and 5–7). Reduced pressure, dimensionless (Chapter 6). Peclét number, dimensionless, Pe = UH/D (Chapter 7). Peclét number of fluid, dimensionless (Chapter 4). Nomenclature Po Pr, Pr Q Q q q q q q qCHF R R R R+ R1, R2 Rp Rp,i Rpm r rb rc r1 r2 Ra Re, Re Re* Re+ ReDh Reg,si Rel,si Rel Rem Ret RSm S s xv Poiseuille number, dimensionless, Po = f Re (Chapters 2 and 3). Prandtl number, dimensionless, Pr = µcp /k (Chapters 2, 3, 5 and 6). Heat load, W (Chapter 3). Volumetric flow rate, m3 /s (Chapters 2, 3 and 7). Heat flux vector, W/m2 (Chapter 2). Heat flux, W/m2 , Chap. 2; dissipated power, W (Chapter 3); constant in Eq. (6.60) (Chapter 6). Volumetric flow rate per unit width, m2 /s (Chapter 7). Oxygen uptake rate on a per-cell basis, mol/s (Chapter 7). Heat flux, W/m2 (Chapters 5 and 6). Critical heat flux, W/m2 (Chapter 5). Gas constant (Chapter 1); upstream to downstream flow resistance, dimensionless (Chapter 5). Specific gas constant, J/kgK, R = cp − cv , (Chapter 2); radius, m (Chapter 6). Universal gas constant, 8.314511 J/molK (Chapter 2). Dimensionless pipe radius (Chapter 6). Radii of curvature of fluid–liquid interface, m. Mean profile peak height (Chapter 3); Pipette radius, m (Chapter 7). Maximum profile peak height of individual roughness elements, m (Chapter 3). Average maximum profile peak height of roughness elements, m (Chapter 3). Distance between two molecular centers, m (Chapter 2); radial coordinate, radius, radius of cavity, m (Chapters 2 and 4–7); constant in Eq. (6.60) (Chapter 6). Bubble radius, m (Chapter 5). Cavity radius, m (Chapter 5). Inner radius of an annular microtube, m (Chapter 2). Outer radius of annular microtube or a circular microtube radius, m (Chapter 2). Average surface roughness, m (Chapter 3). Reynolds number, dimensionless, Re = GD/µ (Chapters 1–5 and 7). Laminar equivalent Reynolds number, dimensionless, Re* = um Dle /µ (Chapter 3). Friction Reynolds number, dimensionless (Chapter 6). Reynolds number based on hydraulic diameter, dimensionless (Chapter 6). Reynolds number, based on superficial gas velocity at the inlet, dimensionless (Chapter 6). Reynolds number, based on superficial liquid velocity at the inlet, dimensionless (Chapter 6). Liquid film Reynolds number, dimensionless, Rel = G(1 − x)D/µl (Chapter 6). Mixture Reynolds number, dimensionless, Rem = GD/µm (Chapter 6). Transitional Reynolds number, dimensionless (Chapter 3). Mean spacing of profile irregularities in roughness elements, m (Chapter 3). Slip ratio, dimensionless, S = UG /UL (Chapter 6). Fin width or distance between channels, m (Chapter 3); constant in Eq. (6.60) (Chapter 6). xvi Sc Sh Sm St T Ts Tsat TSat TSub T+ t U USL , VL,S UGj , VGj UGS, VG,S u u uave uz uz * u* um ur uS uz0 UA V V Vl Vm v v v0 vLV W w We We X Nomenclature Schmidt number, dimensionless, Sc = µ/(D) (Chapter 2). Sherwood number, dimensionless, Sh = H/D (Chapter 7). Distance between two roughness element peaks, m (Chapter 3). Stanton number, dimensionless, St = h/cp G (Chapter 3). Temperature, K or C (Chapters 1–6). Liquid surface temperature, K or C (Chapters 3 and 5); surface temperature of tube wall, K or C (Chapter 6). Saturation temperature, K or C (Chapters 5 and 6). Wall superheat, K, TSat = TW − TSat (Chapter 5). Liquid subcooling, K, TSub = TSat − TB (Chapter 5). Dimensionless temperature in condensate film (Chapter 6). Time, s (Chapters 2 and 7). Uncertainty (Chapter 3); reference velocity, m/s (Chapter 4); potential, such as gravity (Chapter 7); average velocity, m/s (Chapter 7). Superficial liquid velocity, m/s (Chapter 6). Drift velocity in drift flux model, m/s, vG = jG / = Co j + VGj (Chapter 6). Superficial gas velocity, m/s (Chapter 6). Velocity, m/s (Chapters 2–4, 6 and 7). Velocity vector, m/s (Chapter 2). Average electroosmotic velocity, m/s (Chapter 4). Mean axial velocity, m/s (Chapter 2). Mean axial velocity, uz* = uz√ /uz0 , dimensionless (Chapter 2). Friction velocity, m/s, u* = i / l (Chapter 6). Mean flow velocity, m/s (Chapters 3 and 5). Relative velocity between a large gas bubble and liquid in the slug flow regime, m/s, ur = uS − (jG + jL ) (Chapter 6). Velocity of large gas bubble in slug flow regime, m/s (Chapter 6). Maximum axial velocity with no-slip conditions, m/s (Chapter 2). Overall heat transfer conductance, W/K (Chapter 6). Voltage, V, (Chapter 3); velocity, m/s (Chapters 5 and 6). Non-dimensional velocity, V = v/v0 (Chapter 4). Average velocity of a liquid film, m/s (Chapter 6). Zeroth-order uptake of oxygen by the hepatocytes (Chapter 7). Velocity, m/s (Chapter 4). Specific volume, v = 1/, m3 /kg (Chapters 2 and 5); velocity, m/s (Chapters 6 and 7). Reference velocity, m/s (Chapter 4). Difference between the specific volumes of the vapor and liquid phases, m3 /kg, vLG = vG − vL (Chapter 5). Maximum width, m (Chapters 3 and 4). Velocity, m/s (Chapter 7). Weber number, dimensionless, We = LG 2 / (Chapter 5). Weber number, dimensionless, We = VS2 D/ (Chapter 6). Cell density (Chapter 7). Nomenclature X X, Y, Z Xtt x x x, y, z x* , y* x* x+ Y y yb ys Z z z* zi xvii Martinelli parameter, dimensionless, X = {(dp/dz)L /(dp/dz)G }1/2 (Chapters 5 and 6). Coordinate axes (Chapter 4). Martinelli parameter for turbulent flow in the gas and liquid phases, dimensionless (Chapter 6). Mass quality, dimensionless (Chapters 5 and 6). Position vector, m (Chapter 2). Coordinate axes (Chapters 2–7); length (Chapter 6). Cross-sectional coordinates, dimensionless (Chapter 2). Dimensionless version of x, x* = x/RePrDh (Chapter 3). h Dimensionless version of x, x+ = x/D 3). Re (Chapter F /dz)GO Chisholm parameter, dimensionless, y = (dP (dPF /dz)LO (Chapter 6). Dimensionless parameter in Eq. 6.22 (Chapter 6). Bubble height, m (Chapter 5). Distance to bubble stagnation point from heated wall, m (Chapter 5). Ohnesorge number, dimensionless, Z = µ/(L)1/2 (Chapter 5). Heated length from the channel entrance, m (Chapter 5). Axial coordinate, dimensionless (Chapter 2). Valence of type-i ions (Chapter 4). Greek Symbols 1 ; 2 ; 3 c i r 1 , 2 , 3 A , B Convection heat transfer coefficient, W/m2 K (Chapter 2); coefficient in the VSS molecular model, dimensionless (Chapter 2); aspect ratio, dimensionless (Chapter 6); void fraction, dimensionless (Chapter 6). Coefficients for the pressure distribution along a plane microchannel, dimensionless (Chapter 2). Channel aspect ratio, dimensionless, c = a/b (Chapter 3). Eigenvalues for the velocity distribution in a rectangular microchannel, dimensionless (Chapter 2). 7.316 Radial void fraction, dimensionless, r = 0.8372 + 1 − rrw (Chapter 6). Coefficient in the VSS molecular model, dimensionless (Chapter 2); fin spacing ratio, dimensionless, = s/a (Chapter 3); angle with horizontal (Chapter 5); homogeneous void fraction, dimensionless (Chapter 6); velocity ratio, dimensionless (Chapter 6); multiplier to transition line, dimensionless, (F, X ) = constant, used by Sardesai et al. (1981) (Chapter 6). Coefficients for the pressure distribution along a circular microtube, dimensionless (Chapter 2). Empirically derived transition points for the Kariyasaki et al. void fraction correlation (Chapter 6). Euler or gamma function (Chapter 2). xviii

* f r b n n µ µ µn

P P2 / P1 p T TSat TSub t x + t ¯ h m w Nomenclature

Area ratio, dimensionless (Chapter 6); dimensionless length ratio, = L/H (Chapter 7); Specific heat ratio, dimensionless, = cp /cv (Chapter 2). Pressure drop, Pa (Chapter 6). Ratio of differential pressure between two system conditions (Chapter 6). Pressure drop, pressure difference, Pa (Chapters 1–3, 5 and 7). Temperature difference, K (Chapter 6). Wall superheat, K, TSub = TSat − TB (Chapter 5). Liquid subcooling, K, TSat = TW − TSat (Chapter 5). Elapsed time, s (Chapter 4). Quality change, dimensionless (Chapter 6). Mean molecular spacing, m (Chapter 2); film thickness, m (Chapter 6). Non-dimensional film thickness (Chapter 6). Thermal boundary layer thickness, m (Chapter 5). Average roughness, m (Chapter 3). Dielectric constant of a solution (Chapter 4). Dimensionless gap spacing (Chapter 7). Turbulent thermal diffusivity, m2 /s (Chapter 6). Momentum eddy diffusivity, m2 /s (Chapter 6). Electrical permittivity of the solution (Chapter 4). Zeta potential (Chapter 4). Dimensionless zeta potential, = ze/kb T (Chapter 4). Second coefficient of viscosity or Lamé coefficient, kg/m s, (Chapter 2). Temperature jump distance, m (Chapter 2). Temperature jump distance, dimensionless (Chapter 2). Exponent in the inverse power law model, dimensionless (Chapter 2). Exponent in the Lennard-Jones potential, dimensionless (Chapter 2). Fin efficiency, dimensionless (Chapter 3). Dimensionless surface charge density (Chapter 4); angle, degrees (Chapter 6). Dimensionless time (Chapter 4). Receding contact angle, degrees (Chapter 5). Dimensionless Michaelis constant, = Km /C * (Chapter 7). DebyeHuckel parameter, m−1 , = (2n∞ z 2 e2 /0 kb T )1/2 (Chapter 4). Constant in the inverse power law model, N m (Chapter 2). Constant in the Lennard-Jones potential, dimensionless (Chapter 2). Wavelength, m (Chapter 7); mean free path, m (Chapters 1 and 2); dimensionless parameter, = µ2L /(L Dh ) (Chapter 6). Bulk conductivity (Chapter 4). Roots of the transcendental equation tan (n ) = Sh/n (Chapter 7). Eigenvalues (Chapter 4). Dynamic viscosity, kg/ms (Chapters 1–7). Mobility (Chapter 4). Eigenvalues (Chapter 4). Nomenclature µH µm * e m TP sc se T t i i

* m W w

m i L h xix Homogeneous dynamic viscosity, kg/ms, µH = µG + (1 − )µL (Chapter 6). Mixture dynamic viscosity, kg/ms, 1/µm = x/µG + (1 − x)/µL (Chapter 6). Collision rate, s (Chapter 2). Coefficient of slip, m (Chapter 2). Coefficient of slip, dimensionless (Chapter 2). Inlet over outlet pressures ratio, dimensionless (Chapter 2). Density, kg/m3 (Chapters 1–7). Local net charge density per unit volume (Chapter 4). Mixture density, kg/m3 , 1/m = (1/l (1 − x) + (1/v )x) (Chapter 6). −1 Two-phase mixture density, kg/m3 , TP = xG + 1−x (Chapter 6). L Viscous stress tensor, Pa (Chapter 2); Area ratio, dimensionless (Chapter 3); surface charge density (Chapter 4); surface tension, N/m (Chapter 5); fractional saturation, s = C/C * (Chapter 7); cell membrane permeability to oxygen (Chapter 7). Tangential momentum accommodation coefficient, dimensionless (Chapter 2); surface tension, N/m (Chapter 6). Stress tensor, Pa (Chapter 2). Contraction area ratio (header to channel, >1), dimensionless (Chapter 5). Expansion area ratio (channel to header, <1), dimensionless (Chapter 5). Thermal accommodation coefficient, dimensionless (Chapter 2). Total collision cross-section, m2 (Chapter 2). Dimensionless time (Chapter 4); shear stress, Pa (Chapters 6 and 7); time scale, s (Chapter 7). Characteristic time for QGD and QHD equations, s (Chapter 2). Shear stress at vapor–liquid interface, Pa (Chapter 6). Dimensionless shear stress, Pa (Chapter 6). Shear stress due to momentum change, Pa (Chapter 6). Frictional wall shear stress, Pa (Chapters 2, 3 and 6). Average wall shear, Pa (Chapter 2). Intermolecular potential, J (Chapter 2); ratio of the characteristic diffusion time to the characteristic cellular oxygen uptake time, dimensionless (Chapter 7); velocity potential, m (Chapter 7). Angle, degrees (Chapter 6). Dimensionless electric field strength (Chapter 4). Eigenvalues for the velocity distribution in a rectangular microchannel, dimensionless (Chapter 2). Two-phase friction multiplier, dimensionless, L2 = pf ,TP / pf ,L , ratio of two-phase frictional pressure drop against frictional pressure drop of liquid flow (Chapters 5 and 6). Electrical potential (Chapter 4); dimensionless parameter, 1/3 = (w / ) µL /µW (W /L )2 (Chapter 6). Two-phase homogeneous flow multiplier, dimensionless (Chapter 5). xx j s , S Nomenclature Eigenvalues for the velocity distribution in a rectangular microchannel, dimensionless (Chapter 2). Two-phase separated flow multiplier, dimensionless (Chapters 5 and 6). Dimensionless double layer potential (Chapter 4). Dimensionless frequency (Chapter 4). Correction parameter used in Eqs. (6.52) and (6.53) (Chapter 6). Angular speed or vorticity, rad/s (Chapter 7); frequency, Hz (Chapter 4); temperature exponent of the coefficient of viscosity, dimensionless (Chapter 2). Subscripts 0 1-ph a AB an annu app av avg B B b BGKB c, cr, crit c cf CBD CHF circ const cp crit cst E e eo ep EQ eq ex F Lowest boundary condition (Chapters 4 and 7). Single phase (Chapter 5). Air, acceleration, ambient (Chapter 5); air (Chapter 6). Augmented Burnett equations (Chapter 2). Annular (Chapter 6). Flow in an annular microduct (Chapter 2). Apparent (Chapter 3). Average (Chapter 4). Average (Chapters 5 and 7). Bulk (Chapter 5); gas bubble (Chapter 6). Burnett equations, (Chapter 2). Bubble (Chapter 5); bulk (Chapter 3). Bhatnagar-Gross-Krook-Burnett equations (Chapter 2). Critical condition (Chapters 3, 5 and 6). Channel or in a single channel (Chapter 3); cavity mouth (Chapter 5); entrance contraction (Chapter 5). Calculated based on a flow diameter constricted by roughness elements (Dcf ) (Chapter 3). Convective boiling dominant (Chapter 5). Critical heat flux (Chapter 5). Flow in a circular microtube (Chapter 2). Constant (Chapter 4). Constant property (Chapter 3). Critical (Chapter 5). Constant (Chapter 7). Euler equations (Chapter 2). Outlet expansion, exit (Chapter 5). Electroosmostic (Chapter 4). Electrophoretic (Chapter 4). Set of equations (Chapter 2). Equivalent (Chapter 6). Experimental (Chapter 3). Frictional (Chapter 5). Nomenclature f f F/B f/d fd G g g GHS Gn H h H1 H2 hetero homo HS i in, i L l LG LS LO lv M M m max min MM n NBD NS ns o ONB plan QGD QHD r rect Fluid (Chapter 3). Fluid (Chapters 1, 3 and 4); frictional (Chapters 5 and 6); flooded (Chapter 6). Film-bubble region (Chapter 6). Film-bubble region (Chapter 6). Fully developed (Chapter 3). Gas (Chapter 6). Gas (Chapters 6 and 7). Gravitational (Chapters 5 and 6). Generalized hard sphere (Chapter 2). Refers to Gnielinski’s correlation (Chapter 3). Homogeneous (Chapter 6). Hydraulic (Chapter 6). Boundary condition with constant circumferential wall temperature and axial heat flux (Chapter 3). Boundary condition with constant wall heat flux, both circumferentially and axially (Chapter 3). Heterogeneous solution (Chapter 4). Homogeneous solution (Chapter 4). Hard spheres (Chapter 2). Species number (Chapters 4 and 7); vapor–liquid interface (Chapter 6). Inlet (Chapters 2, 3, 5 and 7). Liquid (Chapters 5 and 6). Liquid (Chapter 6). Gas-superficial (Chapter 6). Liquid-superficial (Chapter 6). Entire flow as liquid (Chapters 5 and 6). Liquid-vapor (Chapter 6). Momentum (Chapter 5). Maxwell model (Chapter 2). Mean (Chapter 3). Maximum (Chapters 4 and 5). Minimum (Chapter 5). Maxwell molecules (Chapter 2). Normal direction (Chapter 2). Nucleate boiling dominant (Chapter 5). Navier-Stokes equations (Chapter 2). No-slip (Chapter 2). Out, outlet (Chapters 2 and 3). Onset of nucleate boiling (Chapter 5). Flow between plane parallel plates (Chapter 2). Quasi-gasodynamic equations (Chapter 2). Quasi-hydrodynamic equations (Chapter 2). Radial coordinate, radius, m (Chapter 4). Flow in a rectangular Microchannel (Chapter 2). xxi xxii Nomenclature S S s s Sat, sat sh st str Sub SV T t th TP tp tr u UC V v VHS VSS W, w w wall x, y, z x, y z 0 1 2 ∞ Surface tension (Chapter 5). Stagnation (Chapter 5); superficial (Chapter 6); liquid slug (Chapter 6). Surface (Chapter 3); spherical (Chapter 7). Tangential direction (Chapter 2). Saturation at system or local pressure (Chapters 5 and 6). Shear (Chapter 6). Surface tension (Chapter 6). Stratified (Chapter 6). Subcooled, subcooling (Chapter 5). Sampling volume (Chapter 2). Two-phase mixture (Chapter 6). Total (Chapter 3); turbulent (Chapter 6). Theoretical (Chapter 3). Two-phase (Chapter 5). Two-phase (Chapter 5). Transition regime (Chapter 6). Unflooded (Chapter 6). Unit cell (Chapter 6). Vapor (Chapter 5). Vapor phase (Chapter 6). Variable hard spheres (Chapter 2). Variable soft spheres (Chapter 2). Wall, heated surface (Chapter 5). Fluid at the wall (Chapter 2); at the wall (Chapter 3); wall (Chapter 6). Wall (Chapter 2). Local value at a location or as a function of the co-ordinates (Chapters 3, 5 and 7). Cross-sectional coordinates (Chapter 2). Axial coordinate (Chapter 2). Standard conditions (Chapter 2); reference value (Chapter 2). First-order boundary conditions (Chapter 2). Second-order boundary conditions (Chapter 2). Infinity (Chapter 4). Superscripts +, − + * Charge designation (Chapter 4). Dimensionless parameters (Chapter 6). Dimensionless parameters (Chapters 2–4, 6 and 7). Operators Ñ Ñ˜ Nabla function (Chapter 2). Dimensionless gradient operator (Chapter 4). Averaged quantities (Chapter 7). Chapter 1 INTRODUCTION Satish G. Kandlikar Mechanical Engineering Department, Rochester Institute of Technology, Rochester, NY, USA Michael R. King Departments of Biomedical Engineering, Chemical Engineering and Surgery, University of Rochester, Rochester, NY, USA 1.1. Need for smaller flow passages Fluid flow inside channels is at the heart of many natural and man-made systems. Heat and mass transfer is accomplished across the channel walls in biological systems, such as the brain, lungs, kidneys, intestines, blood vessels, etc., as well as in many man-made systems, such as heat exchangers, nuclear reactors, desalination units, air separation units, etc. In general, the transport processes occur across the channel walls, whereas the bulk flow takes place through the cross-sectional area of the channel. The channel cross-section thus serves as a conduit to transport fluid to and away from the channel walls. A channel serves to accomplish two objectives: (i) bring a fluid into intimate contact with the channel walls and (ii) bring fresh fluid to the walls and remove fluid away from the walls as the transport process is accomplished. The rate of the transport process depends on the surface area, which varies with the diameter D for a circular tube, whereas the flow rate depends on the crosssectional area, which varies linearly with D2 . Thus, the tube surface area to volume ratio varies as 1/D. Clearly, as the diameter decreases, surface area to volume ratio increases. In the human body, two of the most efficient heat and mass transfer processes occur inside the lung and the kidney, with the flow channels approaching capillary dimensions of around 4 µm. Figure 1.1 shows the ranges of channel dimensions employed in various systems. Interestingly, the biological systems with mass transport processes employ much smaller dimensions, whereas larger channels are used for fluid transportation. From an engineering standpoint, there has been a steady shift from larger diameters, on the order of 10–20 mm, E-mail: [email protected]; [email protected] 1 2 Heat transfer and fluid flow in minichannels and microchannels Compact heat exchangers Boilers Power condensers Electronics cooling Henle’s loop Tubules I Aorta 25 mm Refrigeration evaporators/ condensers Large veins and arteries 2.5 mm II III Alveolar Alveolar ducts sacs 250 µm Capillaries 25 µm 2.5 µm Fig. 1.1. Ranges of channel diameters employed in various applications, Kandlikar and Steinke (2003). to smaller diameter channels. Since the dimensions of interest are in the range of a few tens or hundreds of micrometers, usage of the term “microscale” has become an accepted classifier for science and engineering associated with processes at this scale. As the channel size becomes smaller, some of the conventional theories for (bulk) fluid, energy, and mass transport need to be revisited for validation. There are two fundamental elements responsible for departure from the “conventional” theories at microscale. For example, differences in modeling fluid flow in small diameter channels may arise as a result of: (a) a change in the fundamental process, such as a deviation from the continuum assumption for gas flow, or an increased influence of some additional forces, such as electrokinetic forces, etc.; (b) uncertainty regarding the applicability of empirical factors derived from experiments conducted at larger scales, such as entrance and exit loss coefficients for fluid flow in pipes, etc., or (c) uncertainty in measurements at microscale, including geometrical dimensions and operating parameters. In this book, the potential changes in fundamental processes are discussed in detail, and the needs for experimental validation of empirical constants and correlations are identified if they are not available for small diameter channels. 1.2. Flow channel classification Channel classification based on hydraulic diameter is intended to serve as a simple guide for conveying the dimensional range under consideration. Channel size reduction has different effects on different processes. Deriving specific criteria based on the process parameters may seem to be an attractive option, but considering the number of processes and parameters that govern transitions from regular to microscale phenomena (if present), a simple dimensional classification is generally adopted in literature. The classification Chapter 1. Introduction 3 Table 1.1 Channel dimensions for different types of flow for gases at one atmospheric pressure. Channel dimensions (µm) Gas Continuum flow Slip flow Transition flow Free molecular flow Air Helium Hydrogen >67 >194 >123 0.67–67 1.94–194 1.23–123 0.0067–0.67 0.0194–1.94 0.0123–1.23 <0.0067 <0.0194 <0.0123 Table 1.2 Channel classification scheme. Conventional channels >3 mm Minichannels Microchannels Transitional Microchannels Transitional Nanochannels 3 mm ≥ D > 200 µm 200 µm ≥ D > 10 µm 10 µm ≥ D > 1 µm 1 µm ≥ D > 0.1 µm Nanochannels 0.1 µm ≥ D D: smallest channel dimension proposed by Mehendale et al. (2000) divided the range from 1 to 100 µm as microchannels, 100 µm to 1 mm as meso-channels, 1 to 6 mm as compact passages, and greater than 6 mm as conventional passages. Kandlikar and Grande (2003) considered the rarefaction effect of common gases at atmospheric pressure. Table 1.1 shows the ranges of channel dimensions that would fall under different flow types. In biological systems, the flow in capillaries occurs at very low Reynolds numbers. A different modeling approach is needed in such cases. Also, the influence of electrokinetic forces begins to play an important role. Two-phase flow in channels below 10 µm remains unexplored. The earlier channel classification scheme of Kandlikar and Grande (2003) is slightly modified, and a more general scheme based on the smallest channel dimension is presented in Table 1.2. In Table 1.2, D is the channel diameter. In the case of non-circular channels, it is recommended that the minimum channel dimension; for example, the short side of a rectangular cross-section should be used in place of the diameter D. We will use the above classification scheme for defining minichannels and microchannels. This classification scheme is essentially employed for ease in terminology; the applicability of continuum theory or slip flow conditions for gas flow needs to be checked for the actual operating conditions in any channel. 1.3. Basic heat transfer and pressure drop considerations The effect of hydraulic diameter on heat transfer and pressure drop is illustrated in Figs. 1.2 and 1.3 for water and air flowing in a square channel under constant heat flux and fully 4 Heat transfer and fluid flow in minichannels and microchannels Heat transfer coefficient (W/m2 -k) 1,000,000 100,000 Water 10,000 Air 1000 100 10 10 100 1000 10,000 Hydraulic diameter (side) of a square channel (µm) Fig. 1.2. Variation of the heat transfer coefficient with channel size for fully developed laminar flow of air and water. developed laminar flow conditions. The heat transfer coefficient h is unaffected by the flow Reynolds number (Re) in the fully developed laminar region. It is given by: h = Nu k Dh (1.1) where k is the thermal conductivity of the fluid and Dh is the hydraulic diameter of the channel. The Nusselt number (Nu) for fully developed laminar flow in a square channel under constant heat flux conditions is 3.61. Figure 1.2 shows the variation of h for flow of water and air with channel hydraulic diameter under these conditions. The dramatic enhancement in h with a reduction in channel size is clearly demonstrated. On the other hand, the friction factor f varies inversely with Re, since the product f · Re remains constant during fully developed laminar flow. The frictional pressure drop per unit length for the flow of an incompressible fluid is given by: pf 2 f G2 = L D (1.2) where pf /L is the frictional pressure gradient, f is the Fanning friction factor, G is the mass flux, and is the fluid density. For fully developed laminar flow, we can write: f · Re = C (1.3) where Re is the Reynolds number, Re = GDh /µ, and C is a constant, C = 14.23 for a square channel. Figure 1.3 shows the variation of pressure gradient with the channel size for a square channel with G = 200 kg/m2 s, and for air and water assuming incompressible flow conditions. These plots are for illustrative purposes only, as the above assumptions may not be Chapter 1. Introduction 5 Pressure gradient (Pa/m) 1.00E08 1.00E07 Water 1.00E06 Air 1.00E05 1.00E04 1.00E03 1.00E02 1.00E01 10 100 1000 10,000 Hydraulic diameter (side) of a square channel (µm) Fig. 1.3. Variation of pressure gradient with channel size for fully developed laminar flow of air and water. valid for the flow of air, especially in smaller diameter channels. It is seen from Fig. 1.3 that the pressure gradient increases dramatically with a reduction in the channel size. The balance between the heat transfer rate and pressure drop becomes an important issue in designing the coolant flow passages for the high-flux heat removal encountered in microprocessor chip cooling. These issues are addressed in detail in Chapter 3 under singlephase liquid cooling. 1.4. Special demands of microscale biological applications The use of minichannels, microchannels, and microfluidics in general is becoming increasingly important to the biomedical community. However, the transport and manipulation of living cells and biological macromolecules place increasingly critical demands on maintaining system conditions within acceptable ranges. For instance, human cells require an environment of 37 C and a pH = 7.4 to ensure their continued viability. If these parameter values stray more than 10% then cell death will result. All protein molecules themselves have preferred pH environments, and large variations from this can cause (sometimes irreversible) denaturation and loss of biological activity due to unfolding. High temperatures can also cause irreversible protein denaturation. However, in a polymerase chain reaction, or PCR, the rapid cycling (~1 min/cycle) of temperature between 94 C and 54 C for 30–40 cycles is necessary to induce repeated denaturation and annealing of DNA chains. Here, the microchannel geometry can be exploited due to the ease of changing the temperature of small liquid volumes, and in fact thermally driven natural convection can be used to achieve such temperature cycles (Krishnan et al., 2002). Additionally, the concentrations of solutes, such as dissolved gases, nutrients, and metabolic products, must be maintained within specified tolerances to ensure cell proliferation in microchannel bioreactors. Finally, local shear stresses can be critical in suspensions of biological particles. For instance, many cell types such as blood platelets 6 Heat transfer and fluid flow in minichannels and microchannels become activated to a highly adhesive state upon exposure to elevated shear stresses above 10 dynes/cm2 . Endothelial cells, which line the cardiovascular system, require a certain level of laminar shear stress on their luminal surface (≥0.5 dynes/cm2 ) or else they will not align themselves properly and can express surface receptor molecules which induce chronic inflammation. At much higher shear stresses such as 1500 dynes/cm2 , as can be produced around sharp corners for very high liquid flow rates, red blood cells are known to “lyse” or rupture. This is especially critical when designing artificial blood pumps or implantable replacement valves. All of these concerns must be separately addressed when developing new microscale flow applications for the transport and manipulation of biological materials. 1.5. Summary Microchannels are found in many biological systems where extremely efficient heat and mass transfer processes occur, such as lungs and kidneys. The channel size classification is based on observations of many different processes, but its use is mainly in arriving at a common terminology. The heat and mass transfer rates in small diameter channels are also associated with a high pressure drop penalty for fluid flow. The fundamentals of microchannels and minichannels in various applications are presented in this book. 1.6. Practice problems Problem 1.1 Calculate the heat and mass transfer coefficients for airflow in a human lung at various branches. Assume fully developed laminar flow conditions. Comment on the differences between this idealized case and the realistic flow conditions. Hint: Consult a basic anatomy book for the dimensions of the airflow passages and flow rates, and a heat and mass transfer book for laminar fully developed Nusselt number and Sherwood number in a circular tube. Problem 1.2 Calculate the Knudsen number for flow of helium flowing at a pressure of 1 millitorr (1 torr = 1 mmHg) in 0.1-, 1-, 10-, and 100-mm diameter tubes. What type of flow model is applicable for each case? Hint: Knudsen √ √ number is given by Kn = /Dh , and the mean free path is given by = (µ / 2RT ), where µ is the fluid viscosity, is the fluid density, T is the absolute temperature, and R is the gas constant. Problem 1.3 Calculate the pressure drop for flow of water in a 15-mm long 100-µm circular microchannel flowing at a temperature of 300 K and with a flow Reynolds number of (a) 10, (b) 100, and (c) 1000. Also calculate the corresponding water flow rates in kg/s and ml/min. Chapter 1. Introduction 7 References Kandlikar, S. G. and Grande, W. J., 2003, Evolution of microchannel flow passages – thermohydraulic performance and fabrication technology, Heat Transfer Eng., 24(1), 3–17, 2003. Kandlikar, S. G. and Steinke, M. E., 2003, Examples of microchannel mass transfer processes in biological systems, Proceedings of 1st International Conference on Minichannels and Microchannels, Rochester, NY, April 24–25, Paper ICMM20031124. ASME, pp. 933–943. Krishnan, M., Ugaz, V. M., and Burns, M. A., PCR in a Rayleigh–Benard convection cell, Science, 298, 793. Mehendale, S. S., Jacobi, A. M., and Shah, R. K., Fluid flow and heat transfer at micro- and meso-scales with applications to heat exchanger design, Appl. Mech. Rev., 53, 175–193. This page intentionally left blank Chapter 2 SINGLE-PHASE GAS FLOW IN MICROCHANNELS Stéphane Colin Department of Mechanical Engineering, National Institute of Applied Sciences of Toulouse, 31077 Toulouse, France Microfluidics is a rather young research field, born in the early eighties. Its older relative fluidics was in fashion in the sixties to seventies. Fluidics seems to have started in USSR in 1958, then developed in USA and Europe first for military purposes with civil applications appearing later. At that time, fluidics was mainly concerned with inner gas flows in devices involving millimetric or sub-millimetric sizes. These devices were designed to perform the same actions (amplification, logic operations, diode effects, etc.) as their electric counterparts. The idea was to design pneumatically, in place of electrically, supplied computers. The main applications were the concern of the spatial domain, for which electric power overload was indeed an issue due to the electric components of the time that generated excessive magnetic fields and dissipated too much thermal energy to be safe in a confined space. Most of the fluidic devices were etched in a substrate, by means of conventional machining techniques, or by insolation techniques applied on specific resins where masks protected the parts to preserve. The rapid development of microelectronics put a sudden end to pneumatic computers, but these two decades were particularly useful to enhance our knowledge about gas flows in minichannels or mini pneumatic devices. As microfluidics concerns smaller sizes – the inner sizes of Micro-Electro-MechanicalSystems (MEMS) – new issues have to be considered in order to accurately model gas microflows. These issues are mainly due to rarefaction effects, which typically must be taken into account when characteristic lengths are of the order of 1 µm, under usual temperature and pressure conditions. In this chapter, the role of rarefaction is explained and its consequences on the behavior of gas flows in microchannels are detailed. The main theoretical and experimental results from the literature about pressure-driven, steady or pulsed gas microflows are summarized. Heat transfer in microchannels and thermally driven gas microflows are also described. They are particularly interesting for vacuum generation, using microsystems without moving parts. E-mail: [email protected] 9 10 Heat transfer and fluid flow in minichannels and microchannels 2.1. Rarefaction and wall effects in microflows In microfluidics, theoretical knowledge for gas flows is currently more advanced than that for liquid flows (Colin, 2004). Concerning the gases, the issues are actually more clearly identified: the main micro-effect that results from shrinking down the devices size is rarefaction. This allows us to exploit the strong, although not complete analogy between microflows and low pressure flows that has been extensively studied for more than 50 years and particularly for aerospace applications. 2.1.1. Gas at the molecular level 2.1.1.1. Microscopic length scales Modeling gas microflows requires us to take into account several characteristic length scales. At the molecular level, we may consider the mean molecular diameter d, the mean molecular spacing and the mean free path (Fig. 2.1). The mean free path is the average distance traveled by a molecule between two consecutive collisions, in a frame moving with the stream speed of the gas. If we consider a simple gas, that is a gas consisting of a single chemical species with molecules having the same structure, then the mean free path depends on their mean diameter d and on the number density: n = −3 (2.1) and the inverse 1/n represents the mean volume available for one molecule. For example, a gas in standard conditions (i.e. for a temperature T0 = 273.15 K and a pressure p0 = 1.013 × 105 Pa), has about 27 million molecules in a cube of 1 µm in width (n0 = 2.687 × 1025 /m3 and 0 = 3.34 nm) and in the case of air for which d0 » 0.42 nm, the mean free path is 0 » 49 nm. When compared to the case of liquid water, the mean

d L d Fig. 2.1. Main characteristic length scales to take into account at the molecular level. Chapter 2. Single-phase gas flow in microchannels 11 molecular diameter of each are nearly equal, but the mean molecular spacing is about 10 times smaller and the mean free path is 105 times smaller than air! 2.1.1.2. Binary intermolecular collisions in dilute simple gases Gases that satisfy d 1 (2.2) are said to be dilute gases. In that case, most of the intermolecular interactions are binary collisions. Conversely, if Eq. (2.2) is not verified, the gas is said to be a dense gas. The dilute gas approximation, along with the equipartition of energy principle, leads to the classic kinetic theory and the Boltzmann transport equation. With this approximation, the mean free path of the molecules may be expressed as the ratio of the mean thermal velocity c to the collision rate : c = = √ 8RT / (2.3) The thermal velocity c of a molecule is the difference between its total velocity c and the local √ macroscopic velocity u of the flow (cf. Section 2.1.2), and its mean value c = 8RT /, which depends on the temperature T and on the specific gas constant R, is calculated from the Boltzmann equation. The estimation of the collision rate and consequently of the mean free path depends on the model chosen for describing the elastic binary collision between two molecules. Such a model also allows estimation of the total collision cross-section t = d 2 (2.4) in the collision plane and the mean molecular diameter d (Fig. 2.2), as well as the dynamic viscosity µ as a function of the temperature. A collision model generally requires the definition of the force F exerted between the two considered molecules. This force is actually repulsive at short distances and weakly attractive at large distances. Different approximated models are proposed to describe this force. The more classic one is the so-called inverse power law (IPL) model or point center st pd 2 d Fig. 2.2. Total collision cross-section in the collision plane. 12 Heat transfer and fluid flow in minichannels and microchannels of repulsion model (Bird, 1998). It only takes into account the repulsive part of the force and assumes that (2.5) F = |F| = r This derives from an intermolecular potential = ( − 1)r −1 (2.6) where , as well as the exponent have a constant value and r denotes the distance between the two molecules centers. Using the Chapman–Enskog theory, this model leads to a law of viscosity in the form T µ = µ0 (2.7) T0 with = +3 2( − 1) (2.8) The collision rate may be deduced as (Lengrand and Elizarova, 2004) 1 − − 1 2 T 2 RT 5 5 − − nt0 (2.9) =4 2 2 T0 ∞ where ( j) = 0 x j−1 exp (−x)dx is the Euler, or gamma function and T0 is a reference temperature for which the total collision cross-section t0 is calculated. From Eq. (2.3), we can see that the mean free path 1 = √ − 12 5 5 2 2 − 2 − nt0

T T0 − 1 2 (2.10) 1 is proportional to T − 2 and inversely proportional to the number density n. The simplest collision model is the hard sphere (HS) model, which assumes that the total collision cross-section t is constant and for which the viscosity is proportional to the square root of the temperature. Actually, the HS model may be described from the IPL with the exponent Õ ∞ and = 1/2. Bird, who proposed the variable hard sphere (VHS) model for applications to the Monte Carlo method (DSMC, cf. Section 2.2.4.2), has improved the HS model. The VHS model may be considered as a HS model with a diameter d that is a function of the relative velocity between the two colliding molecules. The Chapman–Enskog method leads to a viscosity µVHS √ − 1 2 T 15 m RT = 1 − 2 8 52 − 92 − t0 T0 (2.11) Chapter 2. Single-phase gas flow in microchannels 13 which verifies Eq. (2.7) and where m is the molecular mass. By eliminating t0 between Eqs. (2.9) or (2.10) and (2.11) and noting that ( j + 1) = j( j), the mean free path and the collision rate for a VHS model may be expressed as functions of the viscosity and the temperature: VHS = 30 RT µ (7 − 2)(5 − 2) (2.12) and µ 2(7 − 2)(5 − 2) VHS = √ 15 2RT (2.13) with = mn as the local density. Another classic model is the Maxwell molecules (MM) model. It is a special case of the IPL model with = 5 and = 1. Actually, the HS and the MM models may be considered as the limits of the more realistic VHS model, since real molecules generally have a behavior which corresponds to an intermediate value 1/2 ≤ ≤ 1. Another expression is frequently encountered in the literature: for example, √ Karniadakis √ and Beskok (2002) or Nguyen and Wereley (2002) use the formula M = /2µ/( RT ) proposed by Maxwell in 1879 (Eq. (2.57)). Other binary collision models based on the IPL assumption are described in the literature. Koura and Matsumoto (1991; 1992) introduced the variable soft sphere (VSS) model, which differs from the VHS model by a different expression of the deflection angle taken by the molecule after a collision. The VSS model leads to a correction of the mean free path and the collision rate values: VSS = VHS VSS = VHS (2.14) where = 6/[( + 1)( + 2)] and the value of is generally between 1 and 2 (Bird, 1998). Thus, the correction introduced by the VSS model in the mean free path and in the collision rate remains limited, less than 3%. For = 1, the VSS model reduces to the VHS model. Finally, some models can take into account the long-range attractive part of the force between two molecules by adding a uniform attractive potential (square-well model), or an IPL attractive component (Sutherland, Hassé and Cook or Lennard–Jones potential models) to the HS model. For example, the Lennard–Jones potential is = − ( − 1)r −1 ( − 1)r −1 (2.15) and the widely used Lennard–Jones 12-6 model corresponds to = 13 and = 7. The generalized hard sphere (GHS) model (Hassan and Hash, 1993) is a generalized model that combines the computational simplicity of the VHS model and the accuracy of complicated attractive–repulsive interaction potentials. More recently, the variable sphere (VS) molecular model proposed by Matsumoto (2002) provides consistency for diffusion and viscosity coefficients with those of any realistic intermolecular potential. 14 Heat transfer and fluid flow in minichannels and microchannels Table 2.1 Mean free path, dynamic viscosity, and collision rate for classic IPL collision models. r Model

k1 µ = k2 √ RT k2 HS ∞ 1 2 5 4 16 √ » 1.277 5 2 VHS

= 30 (7 − 2)(5 − 2) MM 5 1 2(7 − 2)(5 − 2) √ 15 2 2 » 0.798 VSS

= F= µ ∝ T = k1 +3 2( − 1) RT µ 2 +3 2( − 1) 5( + 1)( + 2) (7 − 2)(5 − 2) 4 (7 − 2)(5 − 2) √ 5( + 1)( + 2) 2 Table 2.1 resumes the relationships between the collision rate, the mean free path, the viscosity, the density and the temperature for classic IPL collision models. As the mean free path is an important parameter for the simulation of gas microflows (cf. Section 2.1.2), we should be careful when comparing some theoretical results from the literature and verify √ from which model has been calculated. Note √ that for the formula of Maxwell, k2,M = /2 is 2% lower than the value k2,HS = 16/(5 2) obtained from a HS model. Equation (2.11) is also interesting because it allows estimation of the mean molecular diameter d from viscosity data and a VHS model. Table 2.2 gives the value of d for different gases under standard conditions, obtained by Bird (1998), from a VHS or a VSS hypothesis. The ratio 0 /d is calculated for the mean value d of dVHS and dVSS . For any gas, we can see that the condition (2.2) is roughly verified and the different gases can reasonably be considered as dilute gases under standard conditions. However, the dilute gas approximation will be better for He than for SO2 . 2.1.2. Continuum assumption and thermodynamic equilibrium When applicable, the continuum assumption is very convenient since it erases the molecular discontinuities, by averaging the microscopic quantities on a small sampling volume. All macroscopic quantities of interest in classic fluid mechanics (density , velocity u, pressure p, temperature T , etc.) are assumed to vary continuously from point to point within the flow. For example, if we consider an air flow in a duct, for which the macroscopic velocity varies from 0 to 1 m/s and is parallel to the axis of the duct, the velocity of a molecule is of the order of 1 km/s and may take any direction. Similar considerations also concern the other mechanical and thermodynamic quantities. In order to respect the continuum assumption, the microscopic fluctuations should not generate significant fluctuations of the averaged quantities. Consequently, the size of a representative sampling volume must be large enough to erase the microscopic fluctuations, but it must also be small enough to point out the macroscopic variations, such as velocity or Chapter 2. Single-phase gas flow in microchannels 15 Table 2.2 Molecular weight, dynamic viscosity and mean molecular diameters under standard conditions (p0 = 1.013 × 105 Pa and T0 = 273.15 K) estimated from a VHS or a VSS model (data from Bird (1998)) with corresponding values of the ratio 0 /d. Gas M × 103 (kg/mol) µ0 × 107 (N-s/m2 )

dVHS (pm)

dVSS (pm) 0 /d Sea level air Ar CH4 Cl2 CO CO2 H2 HCl He Kr N2 N2 O Ne NH3 NO O2 SO2 Xe 28.97 39.948 16.043 70.905 28.010 44.010 2.0159 36.461 4.0026 83.80 28.013 44.013 20.180 17.031 30.006 31.999 64.065 131.29 171.9 211.7 102.4 123.3 163.5 138.0 84.5 132.8 186.5 232.8 165.6 135.1 297.5 92.3 177.4 191.9 116.4 210.7 0.77 0.81 0.84 1.01 0.73 0.93 0.67 1.00 0.66 0.80 0.74 0.94 0.66 1.10 0.79 0.77 1.05 0.85 419 417 483 698 419 562 292 576 233 476 417 571 277 594 420 407 716 574 – 1.40 1.60 – 1.49 1.61 1.35 1.59 1.26 1.32 1.36 – 1.31 – – 1.40 – 1.44 – 411 478 – 412 554 288 559 230 470 411 – 272 – – 401 – 565 7.97 8.06 6.95 4.78 8.04 5.98 11.51 5.88 14.42 7.06 8.06 5.85 12.16 5.62 7.95 8.26 4.66 5.86 Macroscopic quantity Macroscopic variations Sampling volume Microscopic fluctuations Representative sampling volume Control volume Fig. 2.3. The existence of a representative sampling volume (shaded area) is necessary for the continuum assumption to be valid. pressure gradients of interest in the control volume (Fig. 2.3). If the shaded area on Fig. 2.3 does not exist, the sampling volume is not representative and the continuum assumption is not valid. It may be considered that sampling a volume containing 10,000 molecules leads to 1% statistical fluctuations in the macroscopic quantities (Karniadakis and Beskok, 2002). Such a fluctuation level needs a sampling volume which characteristic length lSV verifies 16 Heat transfer and fluid flow in minichannels and microchannels lSV / = 104/3 » 22. Consequently, the control volume must have a much higher characteristic length L, that is L 104/3 (2.16) so that the statistical fluctuations can be neglected. For example, for air at standard conditions, the value of lSV corresponding to 1% statistical fluctuations is 72 nm. This is comparable to the value of the mean free path 0 = 49 nm. Moreover, the continuum approach requires that the sampling volume is in thermodynamic equilibrium. For the thermodynamic equilibrium to be respected, the frequency of the intermolecular collisions inside the sampling volume must be high enough. This implies that the mean free path = c / must be small compared with the characteristic length lSV of the sampling volume, itself being small compared with the characteristic length L of the control volume. As a consequence, the thermodynamic equilibrium requires that 1 L (2.17) This ratio Kn = L (2.18) is called the Knudsen number and it plays a very important role in gaseous microflows (see Section 2.1.3). If is obtained from an IPL collision model, √ = k2 µ/ RT (2.19) with k2 given by Table 2.1, and the Knudsen number can be related to the Reynolds number Re = uL µ (2.20) and the Mach number Ma = u a (2.21) by the relationship √ Ma Kn = k2 Re (2.22) Here, √ is the ratio of the specific heats of the gas and a, the local speed of sound, with a = RT for an ideal gas (Anderson, 1990), which is verified for a dilute gas (cf. Section 2.2.1). Equation (2.22) shows the link between rarefaction (characterized by Kn) and compressibility (characterized by Ma) effects, the latter having to be taken into account if Ma > 0.3. Chapter 2. Single-phase gas flow in microchannels 17 The limits that correspond to Eqs. (2.2), (2.16) and (2.17), with the indicative values (/d = 7, L/ = 100 and /L = 0.1) proposed by Bird (Bird, 1998), are shown in Fig. 2.4. In this figure, the mean free path has been estimated with an HS model, for which Eq. (2.10) can be written using Eq. (2.4) and = 1/2, in the form HS = √ 3 (2.23) 2d 2 √ Thus /L = (/d)3 (d/L)(1/ 2) and the thermodynamic equilibrium limit /L = 0.1 is a straight line with a slope equal to –3 in the log–log plot of Fig. 2.4. 2.1.3. Rarefaction and Knudsen analogy It is now clear that the similitude between low pressure and confined flows is not complete, since the Knudsen number is not the only parameter to take into account (cf. Fig. 2.4.). 107 Dilute gas Dense gas 106 Thermodynamic equilibrium 105 Thermodynamic disequilibrium Negligible statistical fluctuations 104 L/d 103 Significant statistical fluctuations L/d 100 l/L 0.1 d/d 7 102 101 102 101 d/d 100 100 Fig. 2.4. Limits of the main approximations for the modeling of gas microflows (from Bird (1998)). 18 Heat transfer and fluid flow in minichannels and microchannels However, it is convenient to differentiate the flow regimes in function of Kn, and the following classification, although tinged with empiricism, is usually accepted: l l l l For Kn < 10−3, the flow is a continuum flow (C) and it is accurately modeled by the compressible Navier–Stokes equations with classical no-slip boundary conditions. For 10−3 < Kn < 10−1, the flow is a slip flow (S) and the Navier–Stokes equations remain applicable, provided a velocity slip and a temperature jump are taken into account at the walls. These new boundary conditions point out that rarefaction effects become sensitive at the wall first. For 10−1 < Kn < 10, the flow is a transition flow (T) and the continuum approach of the Navier–Stokes equations is no longer valid. However, the intermolecular collisions are not yet negligible and should be taken into account. For Kn > 10, the flow is a free molecular flow (M) and the occurrence of intermolecular collisions is negligible compared with the one of collisions between the gas molecules and the walls. These different regimes will be detailed in the following sections. Their limits are only indicative and could vary from one case to another, partly because the choice of the characteristic length L is rarely unique. For flows in channels, L is generally the hydraulic diameter or the depth of the channel. In complex geometrical configurations, it is generally preferable to define L from local gradients (e.g. of the density : L = 1/| Ñ/|) rather than from simple geometrical considerations (Gad-el-Hak, 1999); the Knudsen number based on this characteristic length is the so-called local rarefaction number (Lengrand and Elizarova, 2004). Figure 2.5 locates these different regimes for air in standard conditions. For comparison, the cases or helium and sulphur dioxide are also represented. For SO2 , the dilute gas assumption is less valid than for air and He, and the binary intermolecular collision models described in Section 2.1.1.2 could be inaccurate. The shaded area roughly represents the domain of validity for classic gas flow models (C). The relationship with the characteristic length L expressed in micrometers is illustrated in Fig. 2.6, which shows the typical ranges covered by fluidic microsystems described in the literature. Typically, most of the microsystems which use gases work in the slip flow regime, or in the early transition regime. In simple configurations, such flows can be analytically or semi-analytically modeled. The core of the transition regime relates to more specific flows that involve lengths under 100 nanometers, as it is the case for a Couette flow between a hard disk and a read–write head. In that regime, the only theoretical models are molecular models that require numerical simulations. Finally, under the effect of both low pressures and small dimensions, more rarefied regimes can occur, notably inside microsystems dedicated to vacuum generation. 2.1.4. Wall effects We have seen that initial deviations from the classic continuum models appear when we shrink down the characteristic length L of the flow with respect to the boundary conditions. It is logical that the first sources of thermodynamic disequilibria appear on the boundary because at the wall there are fewer interactions between the gas molecules than in the core of the flow. Chapter 2. Single-phase gas flow in microchannels Standard conditions for He air 19 SO2 107 (C) 106 105 104 Kn (S) 3 10 L/d Kn 103 102 Kn 1 10 d/d 7 1 10 (T) 101 (M) 102 101 d/d 100 100 Fig. 2.5. Gas flow regimes as a function of the Knudsen number. 102 Free molecular flow (M) Hard disk drives Microchannels micropumps 101 Microvalves Transitional flow (T) Kn 100 Micronozzles Microflow sensors 101 Slip flow (S) 102 103 Continuum flow (C) 104 102 101 100 L (µm) 101 Fig. 2.6. Gas flow regimes for usual microsystems (from Karniadakis and Beskok (2002)). 102 20 Heat transfer and fluid flow in minichannels and microchannels Actually, the walls play a crucial role in gas microflows. First, shrinking system sizes results in an increase of surface over volume effects. It is interesting for heat transfer enhancement, but it also requires an excellent knowledge of the velocity, as well as the temperature and boundary conditions. The velocity slip and the temperature jump observed at the wall in the slip flow regime strongly depend on the nature and on the state of the wall, in relation to the nature of the gas. Thus, the roughness of the wall and the chemical affinity between the wall and the gas, although often not properly known, could highly affect the fluid flow and the heat transfer. 2.2. Gas flow regimes in microchannels Usually, the regime of gas flow in a macrochannel refers to viscous and compressibility effects, respectively quantified by the Reynolds number Re and the Mach number Ma. For low Reynolds numbers (typically for Re < 2000), the flow is laminar and it becomes turbulent for higher values of Re. For Ma < 1, the flow is subsonic and for Ma > 1, it is supersonic. Typically, if Ma > 0.3, compressibility effects should be taken into account. In microchannels on the other hand, the main effects are rarefaction effects, essentially quantified by the Knudsen number Kn. Consequently, the regime usually refers to these effects, according to the classification given in Section 2.1.3, and flows are considered as microflows for Kn > 10−3 . Actually, Eq. (2.22) shows that for a dilute gas, √ √ Re = k2 Ma/Kn, where k2 is of the order of unity, which implies that for subsonic microflows, viscous effects are predominant and the flows are laminar. The models used to describe gas flows in microchannels depend on the rarefaction regime. The main models are presented in Fig. 2.7, with the rough values of Kn for which DSMC or lattice Boltzmann Burnett equations with slip BC Euler equations QGD or QHD equations with first-order slip BC or Navier–Stokes equations with second-order slip BC Navier–Stokes equations with first-order slip BC Navier–Stokes equations with no-slip BC Kn 0 103 101 10 Fig. 2.7. Gas flow regimes and main models according to the Knudsen number. BC: boundary condition. Chapter 2. Single-phase gas flow in microchannels 21 each of them is applicable, that is valid and really interesting to use. The limits are indicative and may vary according to the considered configurations. These models are described in the following sections. Among the three main rarefied regimes (slip flow, transition flow and molecular flow), the slip flow regime is more specially detailed, since it is the most frequently encountered (cf. Fig. 2.6) and it allows analytic or semi-analytic developments. Moreover, the gas is considered as a dilute gas, which makes valid the ideal gas assumption. 2.2.1. Ideal gas model The classic equation of state for an ideal gas, p = RT = nkB T (2.24) is also called the Boyle–Mariotte’s law. Here, p is the pressure and = mn is the density, kB = R/N = (M/N)(R/M) = mR = 1.38065 J/K is the Boltzmann constant, R = 8.314511 J/mol/K being the universal gas constant, N = 6.022137 × 1023 mol−1 , the Avogadro’s number and M, the molecular weight. Equation (2.24) may be obtained from a molecular analysis, assuming that the gas is a dilute gas in thermodynamic equilibrium. If the thermodynamic equilibrium is not verified, Eq. (2.24) remains valid, providing the thermodynamic temperature T is replaced with the translational kinetic temperature Ttr , which measures the mean translational kinetic energy of the molecules. The specific ideal gas constant R = cp − c v (2.25) is the difference between the specific heats ∂h cp = ∂T p (2.26) and cv = ∂e ∂T v (2.27) where e is the internal specific energy, v = 1/ is the specific volume and h is the specific enthalpy. For an ideal gas, the specific heats are constant and their ratio is the index = cp cv (2.28) Therefore, for an ideal gas, cv = 1 R −1 cp = R −1 (2.29) 22 Heat transfer and fluid flow in minichannels and microchannels The specific heat ratio is constant for a monatomic gas ( = 5/3) but it is dependent of the temperature for a diatomic gas, with a value close to 7/5 near atmospheric conditions. Finally, we can note that for an ideal gas and an IPL collision model, the Knudsen number can be written as: √ k2 µ(T ) k2 µ(T ) RT Kn = √ = pL RT L (2.30) with the value of k2 given in Table 2.1. 2.2.2. Continuum flow regime The classic continuum flow regime (which corresponds to the shaded area in Fig. 2.5) may be accurately modeled by the compressible Navier–Stokes equations, the ideal gas equation of state, and classic boundary conditions that express the continuity of temperature and velocity between the fluid and the wall. 2.2.2.1. Compressible Navier–Stokes equations The compressible Navier–Stokes equations are the governing conservation laws for mass, momentum and energy. These laws are written assuming that the fluid is Newtonian, so that the stress tensor = −p I + (2.31) is a linear function of the velocity gradients. Thus, the viscous stress tensor has the form = µ[ Ñ Ä u + (Ñ Ä u)T ] + ( Ñ · u)I, where u is the velocity vector, I is the unit tensor, µ is the dynamic viscosity and is the second coefficient of viscosity, or Lamé coefficient. With Stokes’ hypothesis, 2µ + 3 = 0, which expresses that the changes of volume do not involve viscosity, the viscous stress tensor may be written as = µ[ Ñ Ä u + (Ñ Ä u)T − (2/3)(Ñ · u)I]. The validity of this assumption is discussed in Gad-el-Hak (1995). The Navier–Stokes equations also assume that the fluid follows the Fourier law of diffusion. Thus, the heat flux vector q is related to the temperature gradient by q = −kÑT , where the thermal conductivity k is a function of the temperature. With the above relations, the compressible Navier–Stokes equations may be written as: ∂ + Ñ · (u) = 0 ∂t (2.32) ∂(u) 2 + Ñ · u Ä u − µ Ñ Ä u + (Ñ Ä u)T − (Ñ · u)I + Ñp = f ∂t 3 (2.33) ∂E 2 T + Ñ · (E + p)u − µ Ñ Ä u + (Ñ Ä u) − (Ñ · u)I · u − kÑT = f · u ∂t 3 (2.34) Chapter 2. Single-phase gas flow in microchannels 23 These conservation equations are respectively the continuity (2.32), the momentum (2.33) and the energy (2.34) equations. The total energy per unit volume 1 E = u·u+e (2.35) 2 is a function of the internal specific energy e that may be written for an ideal gas, using Eqs. (2.24)–(2.26) and (2.28): e= p ( − 1) (2.36) The external volume forces f (gravitational forces, magnetic forces, etc.) are generally negligible in gas flow. It is even more the case in gas microflows, since the volume over surface ratio decreases with the characteristic length L and consequently, in microscale geometries, volume effects may be neglected when compared to surface effects. 2.2.2.2. Classic boundary conditions The previous set of Eqs. (2.24) and (2.32)–(2.34) must be completed with appropriate boundary conditions. In the continuum flow regime, they express the continuity of the velocity, u|w = uwall (2.37) and of the temperature, T |w = Twall (2.38) at the wall, the subscripts “wall” relating to the wall itself, and “w” to the conditions in the fluid at the wall. 2.2.3. Slip flow regime Actually, whatever the Knudsen number (even for very low values of Kn), there is in the neighborhood of the wall a domain in which the gas is out of equilibrium. This domain is called the Knudsen layer and has a thickness in the order of the mean free path. For very low Knudsen numbers (in the continuum flow regime), the effect of the Knudsen layer is negligible. In the slip flow regime, which is roughly in the range 10−3 < Kn < 10−1 , the Knudsen layer must be taken into account. Actually, the flow in the Knudsen layer cannot be analyzed from continuum by use of the Navier–Stokes equations, for example. But for Kn < 10−1, its thickness is small enough and the Knudsen layer can be neglected, providing the boundary conditions (2.37)–(2.38) are modified and express a velocity slip, as well as a temperature jump at the wall (Section 2.2.3.2). Thus, the Navier–Stokes equations remain applicable, but it could also be convenient to replace them by another set of conservation equations, such as the quasi-hydrodynamic (QHD) equations or the quasi-gasdynamic (QGD) equations (see Section 2.2.3.1). 24 Heat transfer and fluid flow in minichannels and microchannels 2.2.3.1. Continuum NS–QGD–QHD equations In a generic form, the conservation equations may be written in the following way: ∂ + Ñ · JEQ = 0 ∂t (2.39) for the continuity equation, ∂(u) + Ñ · (JEQ Ä u − EQ ) + Ñp = 0 ∂t (2.40) for the momentum equation and ∂E E+p +Ñ· JEQ − EQ · u + qEQ = 0 ∂t (2.41) for the energy equation. In these equations, JEQ is the mass flux vector, EQ , the viscous stress tensor and qEQ , the heat flux vector. The external volume forces f have been neglected. In the case of the Navier–Stokes equations (EQ º NS) these variables have the form: JNS = u (2.42) 2 NS = µ[ ÑÄu+(ÑÄu)T ]+( Ñ·u)I = µ Ñ Ä u + (Ñ Ä u)T − (Ñ · u)I 3 qNS = −k ÑT (2.43) (2.44) These lead to the previous Eqs. (2.32)–(2.34). In the Navier–Stokes equations, the density , the pressure p and the velocity u correspond to space averaged instantaneous quantities. From spatiotemporal considerations, the QGD and the QHD equations may be obtained (Lengrand and Elizarova, 2004). They are based on the same governing conservation laws (2.39)–(2.41), but these laws are closed with a different definition of JEQ , EQ and qEQ . Dissipative terms are added, which involve a parameter . This parameter = µ Sc a2 (2.45) is a time characteristic of the temporal averaging. Here, Sc = µ/(D) is the Schmidt number, which represents the ratio of the diffusive mass over the diffusive momentum fluxes. In the expression of Sc, D is the diffusion coefficient, which can be expressed as a function of and for a VSS molecule. Thus, for a VSS molecule, Sc = 5(2 + ) 3(7 − 2) (2.46) which reduces to 5/6 for a HS model. For an ideal gas, a2 = RT and may be written as: = 1 µ Sc p (2.47) Chapter 2. Single-phase gas flow in microchannels 25 The QGD equations are appropriate for dilute gases. For these equations (EQ º QGD): JQGD = JNS − [ Ñ · (u Ä u) + Ñp] (2.48) QGD = NS + [u Ä (u · ( Ñ Ä u) + Ñp) + I(u · Ñp + p Ñ · u)] (2.49) qQGD = qNS − u[u · Ñe + pu · Ñ(1/)] (2.50) From Eqs. (2.18), (2.19), (2.24) and (2.47), we can deduce that for a dilute gas and an IPL collision model, = Kn 1 L √ Sc k2 RT (2.51) Therefore, [ Ñ](KnÕ0) Õ 0, which shows that the above QGD equations degenerate into the classic NS equations when rarefaction is negligible. Another set of equations, the QHD equations, are appropriate to dense gases. In that case (EQ º QHD): JQHD = JNS − [u · ( Ñ Ä u) + Ñp] (2.52) QHD = NS + u Ä (u · ( Ñ Ä u) + Ñp) (2.53) qQHD = qNS (2.54) They also bring additional terms to the NS equations, which are proportional to the parameter . 2.2.3.2. First-order slip boundary conditions From a series of experiments at low pressures performed in 1875, Kundt and Warburg were probably the first who pointed out that for a flow of rarefied gas, slipping occurs at the walls (Kennard, 1938). For a gas flowing in the direction s parallel to the wall, the slip velocity may be written as: ∂us (2.55) uslip = us − uwall = ∂n w where is a length commonly called the coefficient of slip and n is the normal direction, exiting the wall. With appropriate assumptions, it is possible to calculate the magnitude of the coefficient of slip from the kinetic theory of gases. Following a paper initially dealing with stresses in rarefied gases due to temperature gradients (Maxwell, 1879), Maxwell added an appendix on the advise of a referee, with the aim of expressing the conditions which must be satisfied by a gas in contact with a solid body. Maxwell treated the surface as something intermediate between a perfectly reflecting and a perfectly absorbing surface. Therefore, he assumed that for every unit 26 Heat transfer and fluid flow in minichannels and microchannels n s (a) (b) Fig. 2.8. Maxwell hypothesis. (a) Perfect (specular) reflection of a fraction 1 − of the molecules. (b) Absorption and diffuse reemission of a fraction of the molecules. of area, a fraction of the molecules is absorbed by the surface (due to the roughness of the wall, or to a condensation–evaporation process (Kennard, 1938)) and is afterwards reemitted with velocities corresponding to those in still gas at the temperature of the wall. The other fraction, 1 − , of the molecules is perfectly reflected by the wall (Fig. 2.8). The dimensionless coefficient is called the tangential momentum accommodation coefficient. When = 0, the tangential momentum of the incident molecules equals that of the reflected molecules and no momentum is transmitted to the wall, as if the flow was inviscid. This kind of reflection is called specular reflection. Conversely, when = 1, the gas molecules transmit all their tangential momentum to the wall and the reflection is a diffuse reflection. From a momentum balance at the wall, Maxwell finally demonstrated that there is a slip at the wall, which takes the form: uslip = us − uwall = ∂us 3 µ ∂2 T 2− 3 µ ∂T M − + ∂n 2 T ∂s ∂n w 4 T ∂s w (2.56) with M = µ √ = 2 p

µ √ 2 RT (2.57) which is very close to the value of the mean free path HS deduced from an HS model. The equation of Maxwell is generally cited in the literature with the form uslip = us − uwall 2 − ∂us = + ∂n w 2 − ∂us = + ∂n w 3 µ ∂T 4 T ∂s w 3 R ∂T 4 k2 T ∂s w which omits the second term on the right-hand side of Eq. (2.56). (2.58) Chapter 2. Single-phase gas flow in microchannels 27 After non-dimensionalization with the characteristic length L, a reference velocity u0 and a reference temperature T0 , Eq. (2.58) is written as follows: * us* − uwall = 2− 3 Kn2 Re0 ∂T * ∂u* Kn s* + ∂n w 4k22 Ma20 ∂s * w (2.59) where us* = us /u0 , T * = T /T0 , n* = n/L and s* = s/L. Note that this non-dimensionalization is valid only if the reference lengths are the same in the s and n directions, which is generally not the case for a flow in a microchannel. In that case, it is preferable to keep the dimensional form of Eq. (2.58). The second term on the right-hand side, which is proportional to the square of the Knudsen number√Kn = /L, involves the Reynolds number Re0 = u0 L/µ and the Mach number Ma0 = u0 / RT0 defined at the reference conditions. The coefficient k2 depends on the intermolecular collision model and its value is given in Table 2.1. This second term is associated with the thermal creep, or transpiration phenomenon. It shows that a flow can be caused in the sole presence of a tangential temperature gradient, without any pressure gradient. In that case, the gas experiences a slip velocity and moves from the colder toward the warmer region. This can also result in pressure variation in a microchannel only submitted to tangential temperature gradients. The non-dimensional form of Eq. (2.59) clearly shows that slip is negligible for very low Knudsen numbers, but should be taken into account when Kn is no longer small, typically as soon as it becomes higher than 10−3 . In analogy with the slip phenomenon, Poisson suggested that there might also be a temperature jump at the wall, which could be described by an equation equivalent to T − Twall = ∂T ∂n w (2.60) where is called the temperature jump distance. Smoluchowski experimentally confirmed this hypothesis (Smoluchowski, 1898) and showed that was proportional to the mean free path . As for the slip, it can be assumed that a fraction T of the molecules have a long contact with the wall and that the wall adjusts their mean thermal energy. Therefore, these molecules are reemitted as if they were issuing from a gas at the temperature of the wall. The other fraction, 1 − T , is reflected keeping its incident thermal energy. The dimensionless coefficient T is called the energy accommodation coefficient. From an energy balance at the wall, it can be shown that (Kennard, 1938): T − Twall 2 − T 2 k ∂T = T + 1 µcv ∂n w (2.61) After non-dimensionalization with the characteristic length L and the reference temperature T0 , Eq. (2.61) is written as follows: T * * − Twall 2 − T 2 Kn ∂T * = T + 1 Pr ∂n * w (2.62) 28 Heat transfer and fluid flow in minichannels and microchannels where Pr = cp µcp = k k (2.63) is the Prandtl number. In the absence of tangential temperature gradients (∂T * /∂s* = 0), the boundary conditions (2.59) and (2.62) are called firstorder (i.e. (Kn)) boundary conditions. The velocity slip (respectively the temperature jump) is then proportional to the transverse velocity (respectively temperature) gradient and to the Knudsen number Kn. These boundary conditions are classically associated with the Navier–Stokes equations, but they can also lead to interesting results when used with the QGD or QHD equations. 2.2.3.3. Higher-order slip boundary conditions From a theoretical point of view, the slip flow regime is particularly interesting because it generally leads to analytical or semi-analytical models (cf. Section 2.3). For example, these analytical models allow us to calculate velocities and flow rates for isothermal and locally fully developed microflows between plane plates or in cylindrical ducts with simple sections: circular (Kennard, 1938), annular (Ebert and Sparrow, 1965), rectangular (Ebert and Sparrow, 1965; Morini and Spiga, 1998) … These models proved to be quite precise for moderate Knudsen numbers, typically up to about 0.1 (Harley et al., 1995; Liu et al., 1995; Shih et al., 1996; Arkilic et al., 2001). For Kn > 0.1, experimental studies (Sreekanth, 1969) or numerical studies (Piekos and Breuer, 1996) with the direct simulation Monte Carlo (DSMC) method show significant deviations with models based on first-order boundary conditions. Therefore, since 1947, several authors have proposed second-order boundary conditions, hoping to extend the validity of the slip flow regime to higher Knudsen numbers. Second-order boundary conditions take more or less complicated forms, which are difficult to group together in a sole equation. Actually, according to the assumptions, the second-order terms ((Kn2 )) may be dependent of (Chapman and Cowling, 1952; Karniadakis and Beskok, 2002) and involve tangential second derivatives ∂2 u* /∂s*2 (Deissler, 1964). In the simple case of a developed flow between plane plates, the tangential second derivatives are zero and one may compare most of the second-order models that take the generic form * us* − uwall = A1 Kn ∂ 2 u* ∂us* + A2 Kn2 *s2 * ∂n ∂n (2.64) In the particular case of a fully diffuse reflection ( = 1), the coefficients A1 and A2 proposed in the literature are compared in Table 2.3. We note significant differences, essentially for the second-order term. Moreover, some models based on a simple mathematical extension of Maxwell’s condition predict a decrease of the slip compared to the firstorder model, while other models predict an increase of the slip. This point is discussed in Section 2.3. The implementation of the boundary conditions (2.64) with the Navier–Stokes equations is not always easy and can lead to computational difficulties. The obvious interest in using Chapter 2. Single-phase gas flow in microchannels 29 Table 2.3 Coefficients of the main models of second-order boundary conditions proposed in the literature, for = 1. Author, year (reference) A1 A2 Maxwell (Maxwell, 1879) Schamberg, 1947 (Karniadakis and Beskok, 2002) Chapman and Cooling (Chapman and Cowling, 1952) Deissler (Deissler, 1964) Cercignani, 1964 (Sreekanth, 1969) Hsia et Domoto (Hsia and Domoto, 1983) Mitsuya (Mitsuya, 1993) Karniadakis et Beskok (Karniadakis and Beskok, 2002) 1 1 0 ( »1) 1 1.1466 1 1 1 0 −5/12 02 /2 ( »1/2) −9/8 −0.9756 −1/2 −2/9 1/2 higher-order boundary conditions led some authors to propose new conditions, which may be more appropriate for a treatment with the Navier–Stokes equations. Thus, Beskok and Karniadakis (1999) suggested a high-order form * us* − uwall = Kn 2− ∂us * 1 − A3 Kn ∂n * (2.65) that involves the sole first derivative of the velocity and an empirical parameter A3 (Kn). As for Xue and Fan (2000), they proposed * us* − uwall = 2− ∂u * tanh(Kn) s* ∂n (2.66) that leads to results close to that calculated by the DSCM method, up to high Knudsen numbers, of the order of 3. Other hybrid boundary conditions, such as * = us* − uwall 2− Kn ∂u * ∂p* Kn s* + Re * ∂n 2 ∂s (2.67) Jie et al. (2000), were also proposed in order to allow a more stable numerical solution, while giving results comparable to that from Eq. (2.65) for the cases that were tested. 2.2.3.4. Accommodation coefficients The accommodation coefficients depend on various parameters that affect surface interaction, such as the magnitude and the direction of the velocity. Fortunately, it seems that these 30 Heat transfer and fluid flow in minichannels and microchannels coefficients are reasonably constant for a given gas and surface combination (Schaaf, 1963). Many measurements of T and have been made, most of them from indirect macroscopic measurements, while a few of them are from direct measurements using molecular beam techniques. Schaaf (1963) reported values of ranging from 0.60 to 1.00, most of them being between 0.85 and 1.00. Recently, good correlations for microchannel flows were found in the slip flow regime, with values of between 0.80 and 1.00 (Section 2.3.5.2). Schaaf also reported values of T in a wide range, from 0.0109 to 0.990. 2.2.4. Transition flow and free molecular flow Typically, for Knudsen numbers higher than unity, the continuum Navier–Stokes or QGD/QHD equations become invalid. However, in the early transition regime (for Kn ~ 1), a slip-flow model may still be used providing the continuum model is corrected by replacing the Navier–Stokes equations with the Burnett equations. For higher Knudsen numbers, in the full transition and in the free molecular regimes, the Boltzmann equation must be directly treated by appropriate numerical techniques such as the DSMC or the lattice Boltzmann methods (LBM). 2.2.4.1. Burnett equations Higher-order fluid dynamic models, also called extended hydrodynamic equations (EHE), may be obtained from the conservation equations (2.39)–(2.41), deriving the form of the viscous stress tensor EQ and of the heat flux vector qEQ from a molecular approach and the Boltzmann equation. In a molecular description with the assumption of molecular chaos, a flow is entirely described if, for each time t and each position x, we know the number density n, the velocity and the internal energy distribution functions (Lengrand and Elizarova, 2004). For example, if we consider a monatomic gas, the velocity distribution function f (t, x, u) represents the number of particles in the six-dimensional phase space dx du at time t. It verifies the Boltzmann equation ∂f ∂f ∂f +u· +F· = Q( f , f * ) ∂t ∂x ∂u (2.68) which is valid in the entire Knudsen regime, that is for 0 ≤ Kn < ∞, since it does not require a local thermodynamic equilibrium hypothesis. In Eq. (2.68), F is an external body force per unit mass and Q( f , f * ) represents the intermolecular collisions. From a Chapman–Enskog expansion of the Boltzmann equation with the Knudsen number as a small parameter (i.e. f = f0 + Kn f1 + Kn2 f2 + · · ·), the form of the viscous stress tensor and of the heat flux vector may be obtained as follows: (0) (1) (2) (3) (i) EQ = EQ + EQ + EQ + EQ + · · · + EQ + (Kni+1 ) (2.69) Chapter 2. Single-phase gas flow in microchannels (0) (1) (2) (3) (i) qEQ = qEQ + qEQ + qEQ + qEQ + · · · + qEQ + (Kni+1 ) 31 (2.70) In the EHE, the mass flux vector JEQ = JNS = u is the same as for the Navier–Stokes equations, but according to the number of terms kept in Eqs. (2.69) and (2.70), we obtain the Euler, Navier–Stokes, Burnett or super-Burnett equations. Keeping only the first term in Eqs. (2.69) and (2.70), we obtain the zeroth-order approximation (i = 0) which results in the Euler (E) equations, with (0) (2.71) (0) (2.72) E = EQ = 0 qE = qEQ = 0 The Euler equations correspond to an inviscid flow (µ = 0) for which heat losses by thermal diffusion are neglected (k = 0). This system of equations is a hyperbolic system that can describe subsonic or supersonic flows, with the transition for Ma = 1 being modelized by a shock wave, which represents a discontinuity of the flow variables. The Euler equations are widely used for high speed flows in macrosystems such as nozzles and ejectors (Anderson, 1990). With the second terms in Eqs. (2.69) and (2.70), that is for a first-order approximation, we obtain the compressible NS equations, with 2 (0) (1) NS = EQ + EQ = 0 + µ Ñ Ä u + (Ñ Ä u)T − (Ñ · u)I 3 (0) (1) qNS = qEQ + qEQ = 0 − kÑT (2.73) (2.74) These classic equations are detailed in Section 2.2.2.1. According to the associated boundary conditions, they can be valid up to Knudsen numbers in the order of 10−1 or 1 (cf. Fig. 2.7). In the case of a plane flow in the (x, y) plane, Eq. (2.73) yields (1) xx (1) (1) xy = yx (1) yy ∂uy ∂ux = µ 1 + 2 ∂x ∂y ∂uy ∂ux =µ + ∂y ∂x ∂uy ∂ux + 2 = µ 1 ∂y ∂x with 1 = 4/3 and 2 = −2/3. (2.75) 32 Heat transfer and fluid flow in minichannels and microchannels As the Knudsen number becomes higher, additional higher-order terms from Eqs. (2.69) and (2.70) are required. The Burnett (B) equations correspond to a second-order approximation, with (0) (1) (2) (2.76) (0) (1) (2) (2.77) B = EQ + EQ + EQ qB = qEQ + qEQ + qEQ These new expressions for stress and heat-flux terms dramatically complicate the EHE systems. Yun et al. (1998) provide the expressions of these terms in the case of a plane flow: (2) xx

∂uy 2 ∂ux ∂uy ∂ux ∂uy ∂ux 2 ∂ux 2 µ2 = + 2 + 4 1 + 3 + 5 p ∂x ∂x ∂y ∂y ∂y ∂x ∂y ∂uy 2 RT ∂2 RT ∂2 ∂2 T ∂2 T + 6 + 7 R 2 + 8 R 2 + 9 + 10 ∂x ∂x ∂y ∂x2 ∂y2

RT ∂ 2 R ∂T ∂ R ∂T 2 RT ∂ 2 + 11 2 + 12 + 14 2 + 13 ∂x ∂x ∂x T ∂x ∂y R ∂T ∂ R ∂T 2 + 15 + 16 ∂y ∂y T ∂y (2) yy

∂uy 2 ∂uy 2 ∂ux ∂uy ∂ux ∂uy ∂ux 2 µ2 1 = + 2 + 4 + 3 + 5 p ∂x ∂y ∂x ∂y ∂x ∂y ∂x 2 ∂ux ∂2 T ∂2 T RT ∂2 RT ∂2 + 7 R 2 + 8 R 2 + 9 + (2.78) + 6 10 ∂y ∂y ∂x ∂y2 ∂x2

RT ∂ 2 R ∂T ∂ R ∂T 2 RT ∂ 2 + 11 2 + 12 + 14 2 + 13 ∂y ∂y ∂y T ∂y ∂x R ∂T ∂ R ∂T 2 + 15 + 16 ∂x ∂x T ∂x (2) (2) xy = yx = ∂uy ∂uy ∂ux ∂ux ∂ux ∂uy ∂ux ∂uy µ2 1 + + 2 + p ∂x ∂y ∂x ∂y ∂y ∂y ∂x ∂x RT ∂2 R ∂T ∂T RT ∂ ∂ ∂2 T + 4 + 5 + 6 2 ∂x∂y ∂x∂y T ∂x ∂y ∂x ∂y R ∂T ∂ ∂T ∂ + 7 + ∂x ∂y ∂y ∂x + 3 R Chapter 2. Single-phase gas flow in microchannels qx(2) = 33 ∂2 uy 1 ∂T ∂ux 1 ∂T ∂uy ∂2 ux ∂2 ux 1 ∂T ∂uy µ2 1 + 2 + 3 2 + 4 2 + 5 + 6 T ∂x ∂x T ∂x ∂y ∂x ∂y ∂x∂y T ∂y ∂x 1 ∂T ∂ux 1 ∂ ∂ux 1 ∂ ∂uy 1 ∂ ∂ux 1 ∂ ∂uy + 8 + 9 + 10 + 11 + 7 T ∂y ∂y ∂x ∂x ∂x ∂y ∂y ∂y ∂y ∂x qy(2)

∂2 uy ∂2 uy µ2 1 ∂T ∂uy 1 ∂T ∂ux ∂ 2 ux 1 ∂T ∂ux 1 + 2 + 3 2 + 4 2 + 5 + 6 = T ∂y ∂y T ∂y ∂x ∂y ∂x ∂x∂y T ∂x ∂y 1 ∂T ∂uy 1 ∂ ∂uy 1 ∂ ∂ux 1 ∂ ∂uy 1 ∂ ∂ux + 8 + 9 + 11 + 10 + 7 T ∂x ∂x ∂y ∂y ∂y ∂x ∂x ∂x ∂x ∂y

(2.79) Unfortunately, these Burnett equations experience stability problems for very fine computational grids. Zhong proposed in 1991 to add linear third-order terms from the super-Burnett equations (which corresponds to i = 3) in order to stabilize the conventional Burnett equations and maintain second-order accuracy for the stress and heat flux terms (Agarwal et al., 2001). This set of equations is called the augmented Burnett (AB) equations. Moreover, Welder et al. (1993) have shown that a linear stability analysis is not sufficient to explain the instability of the Burnett equations. This instability could be due to the fact that the conventional Burnett equations can violate the second law of thermodynamics at high Knudsen numbers (Comeaux et al., 1995). To overcome this difficulty, Balakrishnan and Agarwal have proposed another set of equations termed the Bhatnagar–Gross–Krook–Burnett or BGK–Burnett (BGKB) equations, which are consistent with the second principle and unconditionally stable for both monatomic and polyatomic gases. The BGKB equations also use third-order terms but the second-order terms {1 , . . . , 16 , 1 , . . . , 7 , 1 , . . . , 11 } are different than the ones of the conventional Burnett or AB equations, as well as the first-order terms (1 = −1.6 and 2 = −0.4 for = 1.4). The additional terms for the AB equations are (a) xx (a) yy 3 3 ∂ uy ∂ 3 uy ∂ ux µ3 ∂3 ux + 18 = 2 RT 17 + + p ∂x3 ∂x∂y2 ∂y3 ∂y∂x2 3 3 ∂ uy ∂ 3 uy ∂ ux µ3 ∂ 3 ux + 18 = 2 RT 17 + + p ∂y3 ∂y∂x2 ∂x3 ∂x∂y2 (a) (a) xy = yx = qx(a) qy(a) µ3 RT 8 p2

∂ 3 uy ∂ 3 uy ∂ 3 ux ∂ 3 ux + + + ∂y3 ∂x3 ∂y∂x2 ∂x∂y2 (2.80)

3 T ∂3 ∂ T µ3 ∂3 T ∂3 + 13 R 12 = + + p ∂x3 ∂x∂y2 ∂x3 ∂x∂y2 ∂3 T T ∂3 µ3 ∂3 T ∂3 + 13 R 12 = + + p ∂y3 ∂y∂x2 ∂y3 ∂y∂x2 (2.81) 34 Heat transfer and fluid flow in minichannels and microchannels and for the BGK–Burnett equations: (B) xx (B) yy (B) xy ∂3 uy ∂3 uy µ3 ∂3 ux ∂3 ux = 2 RT 1 3 + 2 + 3 2 + 4 3 p ∂x ∂x∂y2 ∂x ∂y ∂y ∂2 uy ∂2 ux ∂2 ux µ3 RT ∂ 1 2 + 5 + 6 2 − 2 p ∂x ∂x ∂x∂y ∂y ∂2 uy ∂2 uy ∂2 ux ∂ 7 2 + 8 + 4 2 + ∂y ∂x ∂x∂y ∂y

∂uy 3 ∂ux 2 ∂uy ∂ux ∂uy 2 ∂ux 3 µ3 + + 11 + 10 3 + 2 9 p ∂x ∂x ∂y ∂y ∂x ∂y 2 2 ∂uy ∂uy ∂ux ∂ux ∂ux ∂uy + 12 + − 4 +2 ∂x ∂y ∂y ∂y ∂x ∂x 2 ∂uy ∂ T ∂ux ∂2 T µ3 + 14 + + 2 R 13 p ∂x ∂y ∂x2 ∂y2 ∂3 uy ∂3 uy ∂3 ux ∂3 ux µ3 = 2 RT 1 3 + 2 + 3 2 + 4 3 p ∂y ∂y∂x2 ∂y ∂x ∂x ∂2 uy ∂2 uy ∂2 ux µ3 RT ∂ + 6 2 1 2 + 5 − 2 p ∂y ∂y ∂x∂y ∂x ∂2 uy ∂2 ux ∂2 ux ∂ + 4 2 7 2 + 8 (2.82) + ∂y ∂x∂y ∂x ∂x

∂uy 2 ∂ux ∂uy 3 ∂uy ∂ux 2 ∂ux 3 µ3 + + 10 3 + 11 + 2 9 p ∂y ∂y ∂x ∂x ∂y ∂x 2 2 ∂uy ∂uy ∂ux ∂ux ∂uy ∂ux + 12 + − 4 +2 ∂y ∂x ∂x ∂y ∂x ∂y 2 ∂uy ∂ T ∂ux ∂2 T µ3 + 14 + + 2 R 13 p ∂y ∂x ∂x2 ∂y2 ∂3 uy ∂3 uy µ3 ∂3 ux ∂3 ux = 2 RT 15 + 2 + + p ∂y∂x2 ∂y ∂x ∂y3 ∂x3 ∂2 uy ∂2 uy ∂2 ux ∂2 ux ∂2 ux ∂ ∂2 uy µ3 RT ∂ 6 2 + 16 + + 6 2 + 16 + − 2 p ∂y ∂x ∂x∂y ∂y2 ∂x ∂x2 ∂x∂y ∂y

∂uy 2 ∂uy ∂uy 2 ∂ux ∂uy ∂ux 2 µ3 ∂ux + 7 4 + + + 212 − 2 ∂x ∂y ∂x p ∂y ∂x ∂x ∂y 2 ∂uy ∂ux 2 ∂ T ∂ux ∂uy ∂2 T µ3 ∂ux + +2 + + + 2 R 17 ∂x ∂y ∂x ∂y p ∂y ∂x2 ∂y2 Chapter 2. Single-phase gas flow in microchannels qx(B) = 35 3 1 ∂ ∂2 T µ3 ∂3 T ∂2 T ∂ T R18 − + + p ∂x3 ∂y2 ∂x ∂x ∂x2 ∂y2 ∂2 uy ∂2 ux ∂2 ux µ3 ∂ux 19 2 + 20 + 6 2 + p ∂x ∂x ∂x∂y ∂y + + µ3 − p

∂uy ∂y 21 ∂2 uy ∂2 ux ∂2 ux + 22 + 7 2 2 ∂x ∂x∂y ∂y ∂uy ∂ux + ∂y ∂x 1 ∂ 1 ∂T + ∂x T ∂x

∂2 uy ∂2 uy ∂2 ux 23 2 + 24 + 6 2 ∂x ∂x∂y ∂y

∂uy 2 ∂ux 2 13 + ∂x ∂y ∂uy 2 ∂ux 2 ∂ux ∂uy ∂ux ∂uy + 17 2 + + + 214 ∂x ∂y ∂y ∂x ∂y ∂x + qy(B) = µ3 R ∂T 18 p T ∂x

∂2 T ∂2 T + ∂x2 ∂y2 (2.83) 3 1 ∂ ∂2 T ∂ T ∂3 T µ3 ∂2 T R18 − + p ∂y3 ∂x2 ∂y ∂y ∂x2 ∂y2 µ3 + p

∂uy ∂y

∂2 uy ∂2 uy ∂2 ux 19 2 + 20 + 6 2 ∂y ∂x ∂x∂y ∂ux + ∂x + µ3 − p

∂2 uy ∂2 uy ∂2 ux + 7 2 21 2 + 22 ∂y ∂x∂y ∂x ∂uy ∂ux + ∂y ∂x 1 ∂ 1 ∂T + ∂y T ∂y

∂2 uy ∂2 ux ∂2 ux 23 2 + 24 + 6 2 ∂y ∂x∂y ∂x ∂uy 2 ∂ux 2 + 13 ∂x ∂y ∂uy 2 ∂ux 2 ∂ux ∂uy ∂ux ∂uy + 17 2 + + 214 + ∂x ∂y ∂y ∂x ∂y ∂x + µ3 R ∂T 18 p T ∂y

∂2 T ∂2 T + 2 2 ∂x ∂y

The different coefficients for the above equations are given in Table 2.4. 36 Heat transfer and fluid flow in minichannels and microchannels Table 2.4 Coefficients for AB and BGKB equations (from Yun et al. (1998)); the coefficients for AB equations correspond to an HS collision model, and the coefficients for BGKB equations correspond to = 1.4. AB BGKB AB BGKB AB BGKB AB BGKB 1 2 3 4 5 6 7 8 9 10 1.199 −2.24 0.153 −0.48 −0.600 0.56 −0.115 −1.2 1.295 0.0 −0.733 0.0 0.260 −19.6 −0.130 −5.6 −1.352 −1.6 0.676 0.4 11 12 13 14 15 16 17 18 1 2 1.352 1.6 −0.898 −19.6 0.600 −18.0 −0.676 −0.4 0.449 −5.6 −0.300 −6.9 0.2222 – −0.1111 – −0.115 −1.4 1.913 −1.4 3 4 5 6 7 8 1 2 3 4 0.390 0.0 −2.028 −2.0 0.900 2.0 2.028 2.0 −0.676 0.0 0.1667 – 10.830 −25.241 0.407 −0.2 −2.269 −1.071 1.209 −2.0 5 6 7 8 9 10 11 12 13 −3.478 −2.8 −0.611 −7.5 11.033 −11.0 −2.060 −1.271 1.030 1.0 −1.545 −3.0 −1.545 −3.0 0.6875 – −0.625 – 1 2 3 4 5 6 7 8 9 10 BGKB 2.56 1.36 0.56 −0.64 0.96 1.6 −0.4 −0.24 1.024 −0.256 11 12 13 14 15 16 17 18 19 20 BGKB 1.152 0.16 2.24 −0.56 3.6 0.6 1.4 4.9 7.04 −0.16 21 22 23 24 1 2 −1.76 4.24 3.8 3.4 −1.6 −0.4 BGKB 2.2.4.2. DSMC method For high Knudsen numbers, the continuum approach (NS, QGD, QHD or B equations) is no longer valid, even with slip boundary conditions. A molecular approach is then required and the flow may be described from the Boltzmann equation (2.68). However, the integral formulation of Q( f , f * ), which represents the binary intermolecular collisions, is complex in the general case due to non-linearities and a great number of independent variables (Karniadakis and Beskok, 2002). Therefore, the numerical resolution of the Boltzmann equation is reserved to problems with a simple geometry or problems for which the rarefaction level allows simplifications. Thus, Q( f , f * ) is zero in the free molecular regime when Kn Õ ∞. Also, when Kn Õ 0, a semi-analytical resolution of the Boltzmann equation can be obtained using the method of moments of Grad, for which the distribution function is represented by a series of orthogonal Hermite polynomials (Grad, 1949) or the Chapman–Enskog method (Chapman and Cowling, 1952), for which the distribution function is expanded into a perturbation series with the Knudsen number being the small parameter (cf. Section 2.2.4.1). In the transition regime on the other hand, the numerical resolution of the Boltzmann equation requires approximate methods based on the simplification of Q( f , f * ). Sharipov and Selenev (1998) describe the different available methods – BGK equation (Bhatnagar et al., 1954), linearized Boltzmann equation (Cercignani et al., 1994)… – with their conditions of validity. Chapter 2. Single-phase gas flow in microchannels 37 The Direct Simulation Monte Carlo (DSMC) method is actually better suited to the transition regime. This method was developed by Bird (1978; 1998). Initially widely used for the simulation of low-pressure rarefied flows (Muntz, 1989; Cheng and Emmanuel, 1995), it is now often used for microfluidic applications (Stefanov and Cercignani, 1994; Piekos and Breuer, 1996; Mavriplis et al., 1997; Chen et al., 1998; Oran et al., 1998; Hudson and Bartel, 1999; Pan et al., 1999; Pan et al., 2000; Pan et al., 2001; Wu and Tseng, 2001; Wang and Li, 2004). As for the classical kinetic theory, the DSMC method assumes a molecular chaos and a dilute gas. The technique is equivalent to solving the Boltzmann equation for a monatomic gas with binary collisions. The DSMC technique consists in spitting off the simulations of the intermolecular collisions and of the molecular motion, with a time step smaller than the mean collision time 1/. The simulation is performed using a limited number of molecules, with each simulated molecule representing a great number W of real molecules. The control volume is divided into cells, whose size is of the order of /3, in order to correctly treat large gradient regions. The simulation involves four main steps (Oran et al., 1998): (1) Moving simulated molecules and modeling molecule–surface interactions, applying conservation laws. (2) Indexing and tracking molecules. (3) Simulating collisions with a probabilistic process, generally with a VHS model. (4) Sampling the macroscopic flow properties at the geometric center of the cells. The statistical error of the DSMC solution is inversely proportional to the square root of the total number of simulated molecules. The technique being explicit and timemarching, the simulation is unsteady. The solution of an unsteady problem is obtained from averaging an ensemble of many independent computations, to reduce statistical errors. For a steady problem, each independent computation is proceeded until a steady flow is established (Fig. 2.9). 2.2.4.3. Lattice Boltzmann method The lattice Boltzmann method (LBM) is appropriate for complex geometries and seems particularly effective for the simulation of flows in microsystems, where both mesoscopic dynamics and microscopic statistics are important (Karniadakis and Beskok, 2002). The method, whose detailed description is proposed by Chen and Doolen (1998), solves a simplified Boltzmann equation on a discrete lattice. The use of LBM for the simulation of gas microflows is recent, but LBM is destined to be more widely used for such applications in the future. For example, Nie et al. (2002) demonstrated that the LBM can capture the fundamental behaviors in microchannel flows, including velocity slip, nonlinear pressure drop along the channel and mass flow rate variation with Knudsen number. The numerical results found by Lim et al. (2002) are also in good agreement with the data obtained analytically and experimentally for a two-dimensional isothermal pressure-driven microchannel flow. 2.3. Pressure-driven steady slip flows in microchannels In this section, we consider the flow of an ideal gas through different microchannels in the slip-flow regime. The flow is steady, isothermal and the volume forces are neglected. 38 Heat transfer and fluid flow in minichannels and microchannels Start Read data Set constants Initialize molecules and boundaries Move molecules within t compute interactions with boundaries Reset molecular indexing Compute collisions Unsteady flow : repeat until required sample obtained No Interval t ? Sample flow properties No Time ttotal? Unsteady flow : average runs Steady flow : average samples after establishing steady flow Print final results Stop Fig. 2.9. DSMC flowchart (from Oran et al. (1998)). The flow is also assumed to be locally fully developed: the velocity profile in a section is the same as the one obtained for a fully developed incompressible flow, but the density is recalculated in each cross-section and depends on the pressure and the temperature via the equation of state (2.24). This assumption is no longer valid if the Mach number is greater Chapter 2. Single-phase gas flow in microchannels e y O x O h b h y er h 39 x r1 h b r2 (a) b (b) b (c) Fig. 2.10. Different microchannel sections. (a) Plane channel limited by parallel plates (b h); (b) circular or annular duct; (c) rectangular channel. than 0.3 (Harley et al., 1995; Guo and Wu, 1998). The main sections of interest are the rectangular and the circular cross-sections. Microchannels with rectangular cross-sections are easily etched in silicon wafers, for example by reactive ion etching (RIE) or by deep reactive ion etching (DRIE). If the aspect ratio of the microchannel is small enough, the flow may be considered as a plane flow between parallel plates and the modeling is much simpler. Gas flows in circular microtubes are also frequently encountered, for example in chromatography applications. Fused silica microtubes are proposed by different suppliers with a variety of internal diameters, from one micrometer to hundreds of micrometers. Due to their low cost, they are an interesting solution as connecting lines for a number of microfluidics devices or prototypes. The axial coordinate is z and the cross-section is in the (x, y) plane. The different sections considered in this section are represented in Fig. 2.10. Two different Knudsen numbers are usually defined for microchannel flows. Kn = = DH 4A/P (2.84) is defined from the hydraulic diameter as a reference length. The section area is noted A and the wetted perimeter is noted P. Kn = Lmin (2.85) is defined from the minimal representative length of the section. 2.3.1. Plane flow between parallel plates Here, h/b Õ 0 and the velocity only depends on the transverse coordinate y. With the previous assumptions, the momentum equation (2.33) reduces to d 2 uz 1 dp = dy2 µ dz (2.86) 40 Heat transfer and fluid flow in minichannels and microchannels 2.3.1.1. First-order solution The first-order boundary condition (2.58) for a fixed wall and an isothermal flow is written us |w = [(2 − )/] ∂us /∂n |w , that is for a flow between parallel plates: 2 − ∂uz (2.87) uz |y=h = − ∂y y=h to be associated with the condition of symmetry (cf. Fig. 2.10) duz =0 dy y=0 (2.88) The general solution of Eq. (2.86) is 1 dp uz = µ dz

y2 + a1 y + a2 2 (2.89) The condition of symmetry (2.88) yields a1 = 0 and the boundary condition (2.87) leads to a2 = −h(2 − )/ − h2 /2. Therefore, in a dimensionless form, the velocity distribution can be written as uz* = 1 − y*2 + 8 2− 2− Kn = 1 − y *2 + 4 Kn (2.90) where y* = y/h, uz* = uz /uz0 and uz0 = uz(y=0, Kn=0) represents the velocity at the center of the microchannel when rarefaction is not taken into account: uz0 = − h2 dp 2µ dz (2.91) The Knudsen number Kn = /L = /DH in Eq. (2.90) is defined from the hydraulic diameter DH = 4A/P = 4h, noting A = 16hb as the area and P = 4(b + h) as the wetted perimeter of the channel cross-section, with b h (Fig. 2.10(a)). Kn = /2h is defined from the depth of the microchannel. Equation (2.90) shows that the usual Poiseuille parabolic profile (uz* = 1 − y*2 ) is just globally translated with a quantity 2(2 − )/(h), due to the slip at the wall. The integration of the velocity distribution given by Eq. (2.90) over the cross-section leads to the mean velocity uz* 1 = 2 1 uz* dy* (2.92) −1 that is with Eq. (2.90): uz* = 2− 2 2− uz 2 Kn = + 4 Kn = +8 uz0 3 3 (2.93) Chapter 2. Single-phase gas flow in microchannels 41 We can now calculate the Poiseuille number Po, defined as: Po = Cf Re (2.94) where Cf = w 1 2 2 uz (2.95) is the friction factor that represents a dimensionless value of the average wall shear w and Re = uz DH µ (2.96) is the Reynolds number. The average wall shear is obtained from the force balance w P dz = −A dp (2.97) on a control volume that spans the channel and has an axial extension dz. Therefore, w = − DH dp 4 dz (2.98) Po = − 2 DH dp 1 2µ dz uz (2.99) and which leads with Eqs. (2.91) and (2.93) to: PoNS1,plan = 24 1 + 12 2− Kn = 24 1 + 6 2− Kn (2.100) Equation (2.100) shows that the Poiseuille number is less than its usual value of 24 for a Poiseuille flow between parallel plates (Shah, 1975), as soon as the Knudsen number is no longer negligible. This result shows that slip at the wall logically reduces friction and consequently, for a given pressure gradient, the flow rate is increased. Considering a long microchannel of length l with a pressure pi at the inlet and a pressure po at the outlet and neglecting the entrance effects, we can deduce from Eq. (2.93) the mass flow rate m ˙ = uz A = puz A/RT , which is independent of z. According to Eq. (2.30), the quantity (Kn p) is constant for an isothermal flow and it can be written as a function of the outlet conditions: Kn p = Kno po . Therefore, the integration of Eq. (2.93) yields po pi 2 2µmRTl ˙ 2− p+8 Kno po dp = − 3 h2A (2.101) 42 Heat transfer and fluid flow in minichannels and microchannels which leads to 2bh3 p2o µRTl m ˙ NS1,plan =

2 − 1 2− +8 Kno ( − 1) 3 (2.102) where = pi /po is the ratio of the inlet over outlet pressures and Kno is the outlet Knudsen number. The subscript NS1 refers to the Navier–Stokes equations with first-order slip flow boundary conditions. In dimensionless form that compares the actual mass flow rate m ˙ with the mass flow rate m ˙ ns obtained from a no-slip hypothesis, Eq. (2.102) becomes: m ˙ NS1,plan 2 − Kno 2 − Kno = 1 + 24 = 1 + 12 m ˙ ns,plan +1 +1 m ˙ *NS1,plan = (2.103) The pressure distribution along the z-axis can be found by expressing the conservation of mass flow rate through the channel: 2bh3 p2o 2 − 1 2− m ˙ NS1,plan = +8 Kno ( − 1) µRTl 3 3 2 2 2bh po 2− (p/po ) − 1 = (2.104) +8 Kno (p/po − 1) µRT (l − z) 3 Equation (2.104) shows that the pressure p * = p/po is the solution of the polynomial p*2 + 1 p * + 2 + 3 z * = 0 (2.105) where z * = z/l and 1 = 24 2− Kno 2 = −( + 1 ) 3 = ( − 1)( + 1 + 1 ) (2.106) Equation (2.105) is plotted in Fig. 2.11, for an inlet over outlet pressure ratio = 2. This figure shows that rarefaction and compressibility have opposite effects on the pressure distribution. Rarefaction effects (characterized by Kn) reduce the curvature in pressure distribution due to compressibility effects (characterized by Ma). The first-order mass-flow relation (2.102) accurately predicts the experimental data available in the literature, for moderate outlet Knudsen numbers and typically for Kno < 0.03. For higher values of Kno , some deviations appear (cf. Section 2.3.5). 2.3.1.2. Second-order solutions Therefore, the previous model can be improved to remain accurate for higher Knudsen numbers. For this purpose, secondorder – (Kn2 ) – models, based on second-order boundary conditions or the QHD equations, may be used. The use of second-order boundary conditions uz |y=h = − 2 2 − ∂uz 2 ∂ uz + A 2 ∂y y=h ∂y2 y=h (2.107) Chapter 2. Single-phase gas flow in microchannels 43 2 1.8 p* 1.6 1.4 1.2 1 0 0.2 0.4 0.6 0.8 1 z* Fig. 2.11. Pressure distribution along a plane microchannel. : incompressible flow; 2: Kno = 0; +: Kno = 0.05; #: Kno = 0.1. leads to a velocity distribution uz* = 1 − y*2 + 8 2− 2− Kn − 32A2 Kn2 = 1 − y *2 + 4 Kn − 8A2 Kn2 (2.108) and to a Poiseuille number PoNS2,plan = 24 2 1 + 12 2− Kn − 48 A2 Kn = 24 1 + 6 2− Kn − 12 A2 Kn2 (2.109) It is easy to verify that the mass flow rate is consequently corrected as: m ˙ *NS2,plan = m ˙ NS2,plan ln 2 − Kno − 96 A2 Kn2o 2 = 1 + 24 m ˙ ns,plan +1 −1 = 1 + 12 2 − Kno 2 ln − 24 A2 Kn o 2 +1 −1 (2.110) It is interesting to note that a close result can also be found keeping the first-order boundary conditions, but using the QHD equations in place of the NS equations (Elizarova and Sheretov, 2003). The solution of the momentum equation (2.40), with the expression 44 Heat transfer and fluid flow in minichannels and microchannels (2.53) of the viscous stress tensor and the boundary condition (2.87) yields m ˙ *QHD1,plan = m ˙ QHD1,plan ln 96 2 − Kno + 2 Kn2o 2 = 1 + 24 m ˙ ns,plan + 1 k2 Sc −1 = 1 + 12 24 2 − Kno 2 ln + 2 Kn o 2 + 1 k2 Sc −1 (2.111) Equations (2.110) and (2.111) only differ with their last term. For example, if we consider the second-order boundary conditions of Deissler, A2 = −9/8 and −96 A √2 = 108. This value must be compared with 96/(k22 Sc). For an HS model, k2 = 16/5 2 and Sc = 5/6. Therefore, 96/k22 Sc = 70, 7. The deviation on the second-order term between the two models is then around 42%. But, if we consider the more accurate VSS model, with = 0.74 for nitrogen under standard conditions (cf. Table 2.2), k2 = 1.064, Sc = 0.746 and 96/k22 Sc = 114, which corresponds to a deviation of about 5% between the second-order terms of NS2 and QDH1 models. Of course, this deviation is a function of the gas, the temperature, the choice of the second-order boundary condition and of the IPL model. 2.3.2. Gas flow in circular microtubes With the general assumptions of this section, the momentum equation (2.33) in polar coordinates reduces to 1 d duz 1 dp r = (2.112) r dr dr µ dz 2.3.2.1. First-order solution The first-order boundary condition (2.58) for a non-moving wall and an isothermal flow is now: 2 − ∂uz uz |r=r2 = − (2.113) ∂r r=r2 The general solution of Eq. (2.112) is 1 dp r 2 uz = + c1 ln r + c2 µ dz 4 (2.114) where c1 = 0 since the velocity has a finite value in r = 0. Using the boundary condition (2.113), we find c2 = − r2 2 − r2 − 2 2 4 (2.115) Chapter 2. Single-phase gas flow in microchannels 45 which leads to the dimensionless velocity profile uz* = 1 − r *2 + 4 2− Kn (2.116) where the Knudsen number Kn = /DH = /(2r2 ) = Kn = /Lmin is defined from a characteristic length L = 2r2 . Here, r * = r/r2 and uz* = uz /uz0 where uz0 = r22 dp − 4µ dz (2.117) is the velocity at the center of the microchannel when rarefaction is not taken into account. Equation (2.116) shows that the usual Hagen–Poiseuille velocity profile (uz* = 1 − r *2 ) is just globally translated with a quantity 2(2 − )/(r2 ) due to the slip at the wall. The integration of the velocity distribution given by Eq. (2.116) over the cross-section leads to the mean velocity uz* 1 = 1 uz* 2r * dr * (2.118) 0 that is with Eq. (2.116): uz* = 1 2− +4 Kn 2 (2.119) The Poiseuille number Po is still defined by Eq. (2.99), which leads with Eqs. (2.117) and (2.119) to: PoNS1,circ = 16 1 + 8 2− Kn (2.120) If Kn is not negligible, the Poiseuille number is less than its usual value of 16 for a circular cross-section (Shah, 1975). Following the same method as for a plane flow, the integration of Eq. (2.118) with m ˙ = puz A/RT yields m ˙ NS1,circ r24 p2o 2 − 1 2− = +4 Kno ( − 1) 4µRTl 4 (2.121) for a long microtube, neglecting entrance effects and m ˙ *NS1,circ = m ˙ NS1,circ 2 − Kno = 1 + 16 m ˙ ns,circ +1 (2.122) 46 Heat transfer and fluid flow in minichannels and microchannels The pressure distribution p* = p/po along the microtube is the solution of the polynomial p*2 + 1 p * + 2 + 3 z * = 0 (2.123) where z * = z/l and 1 = 16 2− Kno 3 = ( − 1) ( + 1 + 1 ) 2 = −1 − 1 − 3 (2.124) which qualitatively leads to the same conclusions as for a plane flow. 2.3.2.2. Second-order solution The first-order solution can easily be extended to second-order boundary conditions, following the same reasoning as in Section 2.3.1.2. We find uz* = 2− 1 +4 Kn − 8 A2 Kn2 2 (2.125) 16 PoNS2,circ = 2 1 + 8 2− Kn − 16 A2 Kn (2.126) and m ˙ *NS2,circ = ln m ˙ NS2,circ 2 − Kno − 32 A2 Kn2o 2 = 1 + 16 m ˙ ns,circ +1 −1 (2.127) The problem can also be solved from the QHD equations with first-order boundary conditions. Lengrand et al. (2004) showed that m ˙ *QHD1,circ = m ˙ QHD1,circ ln 2 − Kno 32 = 1 + 16 + 2 Kn2o 2 m ˙ ns,circ + 1 k2 Sc −1 (2.128) The deviation between the QHD1 and the NS2 models with respect to the second-order terms are exactly the same as in the case of a plane flow (Section 2.3.1.2). 2.3.3. Gas flow in annular ducts The solution for the slip-flow of a gas in a circular microduct is easily extensible to the case of an annular duct (Ebert and Sparrow, 1965). The inner radius r1 and the outer radius r2 are not necessarily small, but the distance r2 − r1 = DH /2 must be small enough so that the flow between the two cylinders is rarefied. The first-order boundary conditions are uz = 2 − ∂uz ∂r r=r1 uz = − 2 − ∂uz ∂r r=r2 (2.129) Chapter 2. Single-phase gas flow in microchannels 47 The integration of Eq. (2.112) leads to the velocity distribution 2− Kn uz * = 1 − r *2 + 4(1 − r1* ) * 2− * r1* (1 − r1*2 ) 1 + 4 2− Kn 2(1 − r1 ) Kn − ln r1 + r1* ln r1* − 2(1 − r1*2 ) 2− Kn (2.130) with Kn = Kn /2 = /(2(r2 − r1 )) and to its mean value uz * 1 2− = 1 + r1 *2 + 8(1 − r1* + r1*2 ) Kn 2 2 r1 * (1 − r1*2 )(1 + 4 2− Kn) + * r1 ln r1* − 2(1 − r1*2 ) 2− Kn (2.131) Thus, the Poiseuille number takes the form PoNS1,annu = 16(1 − r1* )2 1 + r1*2 + 8(1 − r1* + r1*2 ) 2− Kn + 2 r1* (1−r1*2 )(1+4 2− Kn) r1 * ln r1* −2(1−r1*2 ) 2− Kn (2.132) which generalizes the expression (2.120) obtained for a circular microtube. 2.3.4. Gas Flow in rectangular microchannels With the rectangular cross-section defined in Fig. 2.10(c) and the general assumptions of Section 2.3, the momentum equation takes the form d 2 uz 1 dp d 2 uz + = 2 2 dx dy µ dz (2.133) In a dimensionless form, the momentum equation can be written a*2 d 2 uz* d 2 uz* + = −1 dx*2 dy*2 (2.134) with x* = x b y* = y h uz* = uz h2 dp − µ dz and a* = h/b is the aspect ratio of the section with 0 < a* ≤ 1, if we assume h ≤ b. (2.135) 48 Heat transfer and fluid flow in minichannels and microchannels 2.3.4.1. First-order solution The first-order boundary conditions are * 2− * ∂uz uz y* =1 = − 2Kn ∂y * y* =1 uz* x* =1 (2.136) 2 − * ∂uz* 2a Kn =− ∂x * x* =1 with Kn = /(2h) and the conditions of symmetry duz * =0 dy* y* =0 duz* =0 dx* x* =0 (2.137) (2.138) (2.139) Ebert and Sparrow (1965) proposed a solution for the velocity distribution, compatible with the condition of symmetry (2.138): uz* = − ∞ i (x* ) cos (i y * ) (2.140) i=1 The eigenvalues i must verify i tan i = 1 2 − 2Kn (2.141) in order to be compatible with the boundary condition (2.136) without leading to a trivial velocity solution. The x* -dependent i functions are found by expanding the right-hand side of Eq. (2.134), that is unity, in terms of the orthogonal functions cos (i y * ): −1 = − ∞ i cos (i y * ) = − i=1 ∞ i=1 2 sin i cos (i y * ) i + sin i cos i (2.142) Therefore, to verify the momentum equation (2.134), the functions i must obey d 2 i i 2 i − * i − *2 = 0 dx*2 a a (2.143) The solution of Eq. (2.143), with the conditions (2.137)–(2.139) and using Eq. (2.141), is * æ ö cosh ai x * i è i = − 2 1 − (2.144) i ø i cosh a *i + 2Kn 2− i sinh a * Chapter 2. Single-phase gas flow in microchannels 49 Consequently, the velocity distribution is given by Eq. (2.140). This solution is given as a function of a Knudsen number Kn = /(2h) based on the channel depth, but it can also be expressed as a function of the previous Knudsen number Kn = /DH based on the hydraulic diameter by using the relation 2Kn = Kn (1 + a* ) (2.145) In the special case of a square section, for which a* = 1, both Knudsen numbers Kn and Kn are identical. 2.3.4.2. Second-order solution The above solution from Ebert et Sparrow was improved by Aubert and Colin (2001), in order to be more accurate for higher Knudsen numbers. This model is based on the second-order boundary conditions proposed by Deissler (1964): uz |y=h ∂2 uz 2 − ∂uz 9 2 ∂2 uz ∂2 uz (2.146) =− − 2 2 + + ∂y y=h 16 ∂y y=h ∂x2 y=h ∂z 2 y=h and uz |x=b 2 ∂ uz 2 − ∂uz 9 2 ∂2 uz ∂2 uz (2.147) =− − 2 2 + + ∂x x=b 16 ∂x x=b ∂y2 x=b ∂z 2 x=b This choice of Deissler boundary conditions, rather than other second-order formulations such as the ones proposed in Section 2.2.3.3, is due to the physical approach proposed by Deissler. The above equations are obtained from a three dimensional local momentum balance at the wall and the shear stress satisfies the definition of a Newtonian fluid used in the Navier–Stokes equations. In Eqs. (2.146) and (2.147), the term ∂2 uz /∂z 2 vanishes according to the continuity equation, which reduces to ∂uz /∂z = 0 for a locally fully developed flow. Finally, using the momentum equation (2.134), these equations may be written in a non-dimensional form as follows: uz* y* =1 * 2− 9 2 ∂2 uz * ∂uz 2Kn =− − Kn −1 2 ∂y * y* =1 4 ∂y* y* =1 (2.148) 2 * 2 − * ∂uz* 9 2 *2 ∂ uz a 2a Kn =− − Kn −1 ∂x * x* =1 4 ∂x*2 x* =1 (2.149) and uz* x* =1 50 Heat transfer and fluid flow in minichannels and microchannels The velocity distribution (2.140) proposed by Ebert and Sparrow, in the form of a single Fourier series, does not converge with second-order boundary conditions. Actually, the eigenvalues i obtained from the boundary condition (2.148) are such that the functions cos (i y * ) are no longer orthogonal on the interval [−1, 1]. Therefore, a new form uz* (x* , y* ) = ∞ i, j=1 * x 9 Aij Nij cos i * cos(j y * ) + Kn2 a 4 (2.150) is required, based on a double Fourier series. The functions ij (x * , y* ) = cos (i x * /a* ) cos (j y * ) are orthogonal over the domain [−1, 1] × [−1, 1]. They are solutions to the eigenvalues problem stemming from Eqs. (2.134), (2.148) and (2.149). The terms Nij are such that the functions Nij cos (i x* /a* ) cos (j y * ) are normed over the domain [−1, 1] × [−1, 1]. The conditions (2.148) and (2.149) require j and i to be solutions of Aubert (1999): 1 − 2 Kn (2 − ) 9 j tan j − Kn2 j2 = 0 4 (2.151) 1 − 2 Kn (2 − ) i 9 i tan * − Kn2 i2 = 0 a 4 (2.152) and In order to determine the terms Aij , the solution (2.150) is injected into the momentum equation (2.134). Furthermore, the right-hand side of Eq. (2.134) is expanded in a double Fourier series. Finally, it is found that 4 sin (i /a * ) sin j sin (i /a * ) cos j cos (i /a * ) sin j 9 + + Kn i j 2 2− i j * * sin j cos j 1 cos (i /a ) sin (i /a ) 1+ × + a

* i j

* ) sin ( /a * ) 9 1 cos ( /a i i + Kn cos2 j * + 4 2− a i −1 sin j cos j + cos2 (i /a

* ) 1 + (2.153) j

Aij Nij = i2 1 + j2

Whatever the aspect ratio a* , the convergence of the series (2.150) was verified. In every case, the asymptotic value of the sum is reached with a good precision from a number of roots j (Kn , Kn2 ) and i (a * , Kn , Kn2 ) from Eqs. (2.151) and (2.152) lower than 50. In Figure 2.12, the normalized velocity distribution uz* is plotted on the axis x* = 0. The solutions proposed by Morini and Spiga (1998) and by Ebert and Sparrow (1965) and the second-order solution of Eq. (2.150) are compared for = 1, Kn = 0.1 and for a square section (a * = 1). The slight deviation observed between the velocity distributions found by Morini and Spiga and by Ebert and Sparrow results from their two different ways of solving the same first-order problem. It is pointed out that the flow rate is underestimated (about Chapter 2. Single-phase gas flow in microchannels 51 0.5 1 2 0.4 0.3 uz* 3 0.2 0.1 0 0 0.2 0.4 y* 0.6 0.8 1 Fig. 2.12. Normalized velocity distribution in a square cross-section with diffuse reflection at the wall for Kn = 0.1. 1: second-order model; 2: first-order model (Ebert and Sparrow, 1965); 3: first-order model (Morini and Spiga, 1998). 10% in the case of Fig. 2.12) when the second-order terms are not taken into account. Moreover, it is verified that the velocity distribution obtained with a second-order model cannot be deduced from a simple translation of the first-order velocity distribution. The mass flow rate through the cross-section () is m ˙ = uz dx dy, that is with () Eqs. (2.135) and (2.150): m ˙ NS2,rect é ù ∞ a* sin (i /a * ) sin j h2 dp 4h2 ë 9 2 û =− Aij Nij + Kn µ dz a* i j 4 (2.154) i, j=1 In order to obtain m ˙ as a function of the inlet pressure pi and the outlet pressure po of a long rectangular microchannel of length l, Eq. (2.154) must be integrated along the microchannel. For an isothermal flow and known geometric dimensions, the Knudsen number only depends on the pressure p(z). Therefore, the solutions i and j of Eqs. (2.151) and (2.152) depend on the position along the z-axis. The bracketed term of Eq. (2.154) can be precisely fitted by a polynomial function a1 + a2 Kn + a3 Kn2 , where a1 , a2 and a3 are coefficients depending on the aspect ratio a* of the crosssection and on the momentum accommodation coefficient for a2 and a3 . Thus, m ˙ NS2,rect = − 4ph4 dp [a1 + a2 Kn + a3 Kn2 ] a* RT µ dz (2.155) Some numerical values of the coefficients a1 , a2 and a3 , obtained with a least squares fitting method from values calculated using Eq. (2.154) for Kn Î [0; 1], are given in 52 Heat transfer and fluid flow in minichannels and microchannels Table 2.5 Values of coefficients a1 , a2 and a3 for different values of and a * , for Deissler second-order boundary conditions. a* 0.5 0.6 0.7 0.8 0.9 1 0 a1 = 0.33333 a2 = 6.0000 a3 = 4.5000 a2 = 4.6667 a3 = 4.5000 a2 = 3.7143 a3 = 4.5000 a2 = 3.0000 a3 = 4.5000 a2 = 2.4444 a3 = 4.5000 a2 = 2.0000 a3 = 4.5000 0.01 a1 = 0.33123 a2 = 5.9694 a3 = 4.4528 a2 = 4.6459 a3 = 4.4542 a2 = 3.7007 a3 = 4.4548 a2 = 2.9917 a3 = 4.4550 a2 = 2.4405 a3 = 4.4549 a2 = 1.9993 a3 = 4.4549 0.025 a1 = 0.32808 a2 = 5.9228 a3 = 4.3840 a2 = 4.6141 a3 = 4.3875 a2 = 3.6797 a3 = 4.3887 a2 = 2.9788 a3 = 4.3895 a2 = 2.4337 a3 = 4.3896 a2 = 1.9977 a3 = 4.3892 0.05 a1 = 0.32283 a2 = 5.8420 a3 = 4.2761 a2 = 4.5591 a3 = 4.2813 a2 = 3.6423 a3 = 4.2843 a2 = 2.9547 a3 = 4.2858 a2 = 2.4201 a3 = 4.2858 a2 = 1.9925 a3 = 4.2853 0.1 a1 = 0.31233 a2 = 5.6774 a3 = 4.0697 a2 = 4.4431 a3 = 4.0832 a2 = 3.5613 a3 = 4.0906 a2 = 2.9000 a3 = 4.0943 a2 = 2.3860 a3 = 4.0952 a2 = 1.9751 a3 = 4.0944 0.25 a1 = 0.28081 a2 = 5.1332 a3 = 3.6073 a2 = 4.0464 a3 = 3.6287 a2 = 3.2675 a3 = 3.6449 a2 = 2.6823 a3 = 3.6560 a2 = 2.2267 a3 = 3.6626 a2 = 1.8626 a3 = 3.6647 0.5 a1 = 0.22868 a2 = 4.2443 a3 = 3.3489 a2 = 3.3519 a3 = 3.3588 a2 = 2.7120 a3 = 3.3708 a2 = 2.2289 a3 = 3.3856 a2 = 1.8487 a3 = 3.4048 a2 = 1.5395 a3 = 3.4292 1 a1 = 0.14058 a2 = 3.1207 a3 = 3.3104 a2 = 2.4552 a3 = 3.3176 a2 = 1.9782 a3 = 3.3287 a2 = 1.6187 a3 = 3.3438 a2 = 1.3364 a3 = 3.3644 a2 = 1.1077 a3 = 3.3908

Table 2.5. Note that the numerical values of the coefficient a1 correspond closely to the analytical expression a1 = 2 ∞ i=1 4i,ns

sin2 i,ns i,ns i,ns − a * tanh * a i,ns + cos i,ns sin i,ns (2.156) which can be deduced from the no-slip problem. In Eq. (2.156), i,ns = (2i − 1)/2. Since for an isothermal flow, the quantity Kn p is constant, the integration of Eq. (2.155) along the microchannel yields m ˙ NS2,rect = 4h4 p2o a1 2 2 − 1) + a Kn ( − 1) + a Kn ln ( 2 3 o o a* µRTl 2 (2.157) a3 1 ln m ˙ NS2,rect a2 + 2 Kn2 = 1 + 2 Kno m ˙ ns,rect a1 +1 a1 o 2 − 1 (2.158) and consequently m ˙ *NS2,rect = The mass flow rate is strongly dependent on the aspect ratio (cf. Fig. 2.13). If a* is less than 0.01, the simpler plane flow model of Section 2.3.1 is accurate enough, but for higher values of a* , it is no longer the case and the model presented in this section should be used. Chapter 2. Single-phase gas flow in microchannels 2.2 53 a* 1; second-order BC 2 a* 1; first-order BC m* 1.8 1.6 1.4 1.2 a* 0.1; second-order BC a* 0.1; first-order BC 1 1.1 1.5 1.9 2.3 2.7 3.1 3.5 Fig. 2.13. Influence of the aspect ratio and of the slip-flow model on the normalized mass flow rate in a rectangular microchannel, for Kno = 0.1 and = 1. BC: boundary condition. As an example, let us consider a pressure-driven flow in a long rectangular microchannel with an inlet over outlet pressure ratio = 2, an outlet Knudsen number Kno = 0.1 and an accommodation coefficient = 1. The flow rate deduced from Eq. (2.110) m ˙ NS2,plan = [(2bh3p2o )/(µRTl)][(2 − 1)/3 + 4((2 − )/)Kno ( − 1) + 9Kn2 o ln ] which was calculated from a plane model with Deissler boundary conditions, overestimates the flow rate m ˙ NS2,rect given by Eq. (2.157) and calculated from a rectangular model. The overestimation [(m ˙ plan − m ˙ rect )/m ˙ rect ]NS2 is about 0.5% for an aspect ratio a* = 0.01, it * rises to 5.3% for an aspect ratio a = 0.1 and reaches 112% for a square cross-section with a* = 1! The pressure distribution p(z) can also be deduced from the mass conservation. It is the solution of the equation a1 2 2 2 [(p/po ) − 1] + a2 Kno [(p/po ) − 1] + a3 Kno ln (p/po ) a1 2 2 2 ( − 1) + a2 Kno ( − 1) + a3 Kno ln = (1 − z * ) (2.159) which can be numerically solved. An example of the non-dimensional pressure distribution p* = p/pns as a function of z * = z/l is given in Fig. 2.14, for an outlet Knudsen number Kno = 0.1, an accommodation coefficient = 1 and for a * = 1 (square duct) or a* = 0.1. The maximum influence of the slip at the wall corresponds to z * = 0.77. A no-slip model overestimates the pressure by a few percents, as is the case for a plane microchannel (Fig. 2.11). Moreover, the greater the aspect ratio, the larger the pressure overestimation is. 2.3.5. Experimental data A discussion of the accuracy of the different models described in the previous section requires fine experimental data, notably to know which second-order boundary conditions are more appropriate. 54 Heat transfer and fluid flow in minichannels and microchannels 1 a* 0.1; first-order BC 0.99 a* 0.1; second-order BC p* 0.98 0.97 a* 1; first-order BC 0.96 a* 1; second-order BC 0.95 0 0.2 0.4 z* 0.6 0.8 1 Fig. 2.14. Influence of the aspect ratio and of the slip-flow model on the normalized pressure distribution along a rectangular microchannel, for Kno = 0.1 and = 1. BC: boundary condition. 2.3.5.1. Experimental setups for flow rate measurements Few experimental setups are described in the literature. The first techniques used to provide experimental data of gas flow rates in a microchannel were proposed by Arkilic et al. (1994) or by Pong et al. (1994). They both used accumulation techniques: the gas flowing through the microchannel accumulates in a reservoir, resulting in either a change in pressure (constant-volume technique) or a change in volume (constant-pressure technique). Actually, this method is limited because the tiny mass flows are very difficult to measure accurately using a single tank. In particular very small changes in temperature can overwhelm the mass flow measurement due to thermal expansion of the gas. To overcome this difficulty, a differential technique was used afterwards (Arkilic et al., 1998). The system is schematically represented in Fig. 2.15. Gas flows through the test microchannel and into the accumulation tank A. The mass flow through the system can be inferred from a measurement of the differential pressure between the accumulation tank A and a reference tank B. This arrangement is very insensitive to temperature fluctuations and allows accurate measurements of mass flow rates, with a sensitivity announced to be as low as 7 × 10−5 kg/s. Another experimental setup was designed by Lalonde (2001) for the measurement of gaseous microflow rates under controlled temperature and pressure conditions, in the range 10−7 to 10−13 m3 /s. Its principle is based on the tracking of a drop meniscus in a calibrated pipette connected in series with the microchannel. Its main advantage, compared to other similar setups presented in the literature (Shih et al., 1996; Zohar et al., 2002; Maurer et al., 2003), is the simultaneous measurement of the flow rate both upstream and downstream of the microchannel (Fig. 2.16). The inlet and the outlet pressures can both be independently tuned. The volume flow rate is measured by means of opto-electronical sensors that detect the passage of the two menisci of a liquid drop injected into the calibrated pipettes connected upstream and downstream of the microchannel. Chapter 2. Single-phase gas flow in microchannels 55 Differential pressure sensor Thermal shield p Tank B Microchannel Tank A T Insulation gate Copper casing High-pressure gas Vacuum pump Fig. 2.15. Schematic of an accumulation mass flow measurement system. Adapted from Arkilic et al. (1998). Pressurized gas Tank T Manometer Filter p Pressure regulator Manometer Calibrated pipettes and opto-electronic sensors Micro channel T p Vacuum Vacuum pump regulator Thermal shield Fig. 2.16. Schematic of a pointer tracking system for the measurement of gas microflow rates (Lalonde, 2001). 56 Heat transfer and fluid flow in minichannels and microchannels 1.4E09 1.2E09 Mean volume flow rates Q (m3/s) 1.0E09 8.0E10 6.0E10 4.0E10 0 2 4 6 8 10 12 Data label Fig. 2.17. Typical example of volumetric flow rate (Q) measurement with the setup of Lalonde (2001). A series of individual measurements of the volumetric flow rate can therefore be obtained, which allows the determination of the mean volumetric flow rates upstream and downstream of the microchannel and consequently, the mass flow rate through the microchannel. The comparison of the individual data given by each pipette, as well as the comparison of the two mean mass flow rates deduced from the two mean volumetric flow rates, must be consistent with the experimental uncertainties to allow the validation of the acquisition. A typical example is shown in Fig. 2.17. The upper data (circles) correspond to the downstream pipette and the lower ones (triangles) to the upstream pipette, through which the volumetric flow rate is lower, since the pressure is higher. The white symbols are relative to the first, the dark symbols to the second, of the two menisci of the drop in the pipette. In total, 44 individual data values are obtained for each test. Experimental uncertainties are reported by vertical bars. Other similar setups are described by Zohar et al. (2002) or by Maurer et al. (2003). They both provide a single but accurate measurement downstream of the microchannel. In the first case (Zohar et al., 2002), the outlet pressure is always the atmospheric pressure, whereas in the second case (Maurer et al., 2003), the outlet can be tuned as well as the upstream pressure. 2.3.5.2. Flow rate data Comparing the different models available to describe the slip flow regime in microchannels, we note significant differences, essentially for the second-order term. For example, in a Chapter 2. Single-phase gas flow in microchannels 57 plane flow between parallel plates, the mass flow rate predicted by Eq. (2.110) depends on the choice of the second-order boundary condition. According to the value of the coefficient A2 (see Table 2.3), the calculated mass flow rate may vary significantly. Moreover, some models generally based on a simple mathematical extension of Maxwell’s condition (2.58) and for which A2 > 0 predict a decrease of the slip compared to the first-order model. While other models, for which A2 < 0, predict an increase of the slip. It is also interesting to test the accuracy of a QHD model, which does not involve second-order boundary conditions, but is second-order Knudsen number dependent (cf. Eq. (2.111)). The increase of the slip due to secondorder terms was first confirmed by Lalonde (2001), who measured the flow rate of helium and nitrogen in rectangular microchannels and showed that for outlet Knudsen numbers Kno,M higher than 0.05, a first-order model underestimates the slip. For the comparison of the experimental flow rate data with the predictions from the different models presented above, the sources of error or uncertainty must be considered. They can globally be divided into three classes: (1) The uncertainties inherent to the fluid properties (density, viscosity, dissolved air, etc.) and to their dependence toward the experimental parameters (temperature, pressure). One can also note that some properties such as the momentum accommodation coefficient, , depend both on the fluid and the wall. (2) The uncertainties relative to the microchannel geometrical characteristics (dimensions of the section, roughness, etc.) and to their dispersion. (3) The uncertainties due to the flow rate measurement (metrology, leakage, operating conditions, etc.). Most of these uncertainties can be limited within an acceptable precision. This is generally not the case for the measurement of the cross-section dimensions. When these dimensions are of the order of 1 µm, the measures obtained by different means (profilometer, optical microscope, SEM) that require preliminary calibration can differ notably: up to about 5% for the width and about 10% for the depth (Anduze, 2000). Therefore, the data are hardly exploitable, since the hydraulic diameter plays a part to the power of 4 in the calculation of the flow rate. Also, an inaccuracy of only 10 nm when measuring a depth of 1 µm leads to an error of 4% in the estimation of the flow rate. The second parameter that can pose a problem is the accommodation coefficient , the value of which is not usually well known. In order to avoid interpretation errors due to these two mains sources of uncertainty, the idea is to compare among themselves data measured on a same microchannel, in a Knudsen number range that covers at least two of the following regimes: (1) In the little rarefied regime, differences between the flow rates predicted by the no-slip (ns), the NS1 and the NS2 models are negligible, of the order of the experimental uncertainties. The value of consequently does not play any significant role. (2) In the lightly rarefied regime, the difference between NS1 and NS2 models remains negligible, but the difference with the ns model is now significant. The influence of the accommodation coefficient may also become significant for the higher values of Kno . (3) In the moderately rarefied regime, the ns, NS1 and NS2 models predict significantly different flow rates, and is now a very sensitive parameter. 2h 0.545 µm Microchannel no.4 a* 0.011 2h 1.16 µm Microchannel no.3 a* 0.055 2h 1.88 µm Microchannel no.2 a* 0.089 2h 4.48 µm Microchannel no.1 a* 0.087 0.001 0.01 0.1 Moderately rarefied Lightly rarefied Heat transfer and fluid flow in minichannels and microchannels Little rarefied 58 Kno,VSS 1 Fig. 2.18. Knudsen number ranges covered by a series of rectangular microchannels tested by Lalonde (2001), the right part of the rectangles corresponds to the outlet Knudsen number values. Figure 2.18 represents these regimes with a series of microchannels experimentally studied by Lalonde (Colin et al., 2004). The microchannel no.1, with a measured depth 2h = 4.48 µm, allows a check of the transition from ns to NS1 models. First, the flow rate is measured for low values of the outlet Knudsen number Kno , that is for high values of Po . For Kno,VSS = 0.007, the differences between the flow rates predicted from the NS1 and NS2 models are negligible, whatever the value of the accommodation coefficient and the deviation with the ns model remains of the order of the experimental uncertainties. The depth 2h kept for the simulation is adjusted to 4.48 µm – exactly the measured value in that case – for a good correlation between the experimental data and the NS1 or NS2 model. Then, rarefaction is increased, by decreasing the outlet pressure; the difference between ns and NS1 or NS2 models becomes significant and the experimental data are in very good agreement with the slip flow models. The influence of is still negligible as is the deviation between NS1 and NS2 models. The microchannels nos.3 and 4 allow a check of the transition from NS1 to NS2 models. As an illustration, Fig. 2.19 shows some flow rate data relative to the microchannel no.3, whose measured depth is 2h = 1.15 µm. The depth for the simulation is adjusted to 1.16 µm, that is 0.01 µm more than the measured value, from data at low values of Kno , for which both NS1 and NS2 models accurately predict the experimental flow rates. An increase of the rarefaction leads to a deviation between these two models. The agreement between the experimental data and the NS2 model is good, whatever the gas – nitrogen (Fig. 2.19(b)) or helium (Fig. 2.19(c)) – with = 0.93. The rarefaction is increased even more and the difference between NS1 and NS2 models becomes very significant (Fig. 2.19(d)). The experiment shows the validity of the second-order slip flow model with Knudsen numbers higher than 0.2. This observation is confirmed with the microchannel no.4, up to Knudsen numbers around 0.25. Chapter 2. Single-phase gas flow in microchannels 59 2E11 1.6E11 m (kg/s) NS2 (s 0.93) 1.2E11 8E12 NS1 and NS2 (s 1) 4E12 ns 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

(a) 3.5E12 3.0E12 NS2 (s 0.93) m (kg/s) 2.5E12 2.0E12 NS2 (s 1) 1.5E12 ns NS1 (s 1) 1.0E12 5.0E13 1.3 (b) 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Fig. 2.19. Theoretical and experimental flow rates for a rectangular microchannel. 2 h = 1.16 µm, 2b = 21 µm, l = 5000 µm, T = 294.2 K. (a) Gas: N2 , po = 1.9 × 105 Pa, Kno,VSS = 2.5 × 10−2 ; (b) gas: N2 , po = 0.65 × 105 Pa, Kno,VSS = 7.3 × 10−2 . For nitrogen and helium flows, an attempt to summarize the results obtained with the microchannels nos.2, 3 and 4 is shown in Fig. 2.20. The dimensionless flow rate 1/m ˙ *, * little sensitive to the aspect ratio a , is plotted in function of Kno,M , for an inlet over outlet pressure ratio = 1.8. Up to Kno,M = 0.05, the first-order (NS1) and the second-order (NS2) slip flow models predict the same flow rate. For higher values of Kno , the deviation is significant and the experimental data are in very good agreement with the NS2 model, 60 Heat transfer and fluid flow in minichannels and microchannels 2.6E12 NS2 (s 0.93) m (kg/s) 2.1E12 1.6E12 NS2 (s 1) 1.1E12 NS1 (s 1) ns 6E13 1E13 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

(c) 1.8E12 1.6E12 1.4E12 NS2 (s 0.93) m (kg/s) 1.2E12 1.0E12 8.0E13 NS2 (s 1) 6.0E13 NS1 (s 1) ns 4.0E13 2.0E13 1.2 (d) 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Fig. 2.19. (Continued) (c) gas: He, po = 1.9 × 105 Pa, Kno,VSS = 7.9 × 10−2 ; (d) gas: He, po = 0.75 × 105 Pa, Kno,VSS = 2.0 × 10−1 . up to Kno,M = 0.25. Beyond that, the agreement is not so fine although the experimental data are closer to NS2 than to NS1 predictions. The experiments of Maurer et al. (2003) follow the same trend. The latest precise experimental data provided in the literature (Table 2.6) concern different gases and are relative to microchannels with cross-sections of comparable lengths l and depths 2h, but Chapter 2. Single-phase gas flow in microchannels 61 1.0 0.9 Encircled datum in Fig. 2.21(a) 0.8 Encircled datum in Fig. 2.21(b) 0.7 1/m* 0.6 NS1 (s 0.93) 0.5 0.4 0.3 NS2 (s 0.93) 0.2 0.1 0.0 0 0.1 0.2 0.3 0.4 0.5 Kno,M Fig. 2.20. Inverse reduced flow rate (1/m ˙ * ) in rectangular microchannels: comparison of experimental data with NS1 and NS2 slip flow models; = 1.8; T = 294.2 K; microchannels no.2 (white), no.3 (grey), no.4 (black); gas: N2 (circle) and He (square). Table 2.6 Recent experimental data of gas flows in rectangular microchannels. Microchannels dimensions Experimental conditions Theoretical comparison Reference l(µm) 2h (µm) a* (%) Gas Kno

Model Shih et al. (1996) 4000 1.2 3.0 He N2 7490 1.33 2.5 Ar N2 CO2 Lalonde (2001); Colin et al. (2004) 5000 0.54–4.48 1.1–8.7 N2 He 1.16 0.99 0.80 0.80 0.80 0.93; 1.00 0.93; 1.00 Plane, NS1 Arkilic et al. (2001) 0.16 0.055 0.05–0.41 0.05–0.34 0.03–0.44 0.004–0.16 0.029–0.47 Zohar et al. (2002) 4000 0.53; 0.97 1.3; 2.4 0.21; 0.38 0.11; 0.20 0.065; 0.12 1.00 1.00 1.00 Maurer et al. (2003) 10,000 1.14 0.6 He Ar N2 N2 He 0.054–1.1 0.17–1.46 0.87 0.91 Plane, NS1 Rectangular, NS2, k2 = k2,M , A2 = −9/8; plane, QHD1, k2 = k2,VSS Plane, NS1, k2 = k2,M Plane, NS2, A2 = −0.23(N2 ) , A2 = −0.26(He) manufactured with different processes. Arkilic et al. (2001) used two silicon wafers facing each other, Colin et al. (2004) studied microchannels etched by DRIE in silicon and covered with a glass sealed by anodic bonding, whereas Maurer et al. (2003) used microchannels etched in glass and covered with silicon. The lower and upper walls of the microchannels 62 Heat transfer and fluid flow in minichannels and microchannels fabricated by Zohar et al. (2002) were obtained by deposition of low-stress silicon nitride layers on a silicon wafer. A smart comparison of the data provided by these authors is also tricky, since the aspect ratios a* of the sections are different from one study to another. The two latter studies (Maurer et al., 2003; Colin et al., 2004) confirm that an adequate slip flow model based on second-order boundary conditions can be precise for high Knudsen numbers that usually apply to the transition regime. The experimental data from Lalonde have also been compared (Colin et al., 2003) to the mass flow predicted by a QHD model with first-order boundary conditions (QHD1). For the microchannel no.4, with a low aspect ratio a* = 0.011, the plane assumption is acceptable and Eq. (2.111) is valid. Figure 2.21 shows that the QHD1 model could be accurate for Knudsen numbers higher than the NS2 model, but with a different accommodation coefficient, = 1, both for nitrogen and helium. Actually, the analysis of this result should be qualified. For this comparison, the mean free path was calculated using the formula √ given by Maxwell (k2,M = /2) for the NS2 as well as for the QHD1 simulations, but the Schmidt value (0.74 for nitrogen and 0.77 for helium) used for the QHD1 simulation is very close to the one obtained from a VSS model, that is given by Eq. (2.46). Another comparison of the QHD1 model with the data of Lalonde is proposed in Elizarova and Sheretov (2003), using k2,VSS calculated from the VSS model (cf. Table 2.1). In this case, the better fit is found for = 1 for helium and = 0.93 for nitrogen. To conclude, it is clear that the NS2 or QHD1 models improve the accuracy of the NS1 model, but further studies that include smart experimental data are necessary to dissociate the roles of the accommodation coefficient, the collision model and the slip flow model. 2.3.5.3. Pressure data Pressure measurements along a microchannel have been successfully performed by different groups (Liu et al., 1995; Lee et al., 2002; Zohar et al., 2002; Jang and Wereley, 2004), using integrated pressure microsensors connected to the microchannels via capillary connections (Fig. 2.22). These experiments confirm the theoretical pressure distribution shown in Fig. 2.11. 2.3.5.4. Flow visualization Although micro-particle image velocimetry (micro-PIV) has been widely used to visualize liquid microflows for several years (Wereley et al., 1998), this technique has not been successfully applied to gaseous microflows as yet due to seeding and inertia issues (Wereley et al., 2002; Sinton, 2004). The molecular tagging velocimetry (MTV) could be an alternative and interesting technique for gas microflows visualization. This method was used to obtain the velocity profile of water in microtubes 705 µm in diameter, with a spatial resolution lower than 4% (Webb and Maynes, 1999). Recently, the technique was adapted to the analysis of gaseous subsonic or supersonic microjets generated by a 1 mm diameter nozzle. The gas was nitrogen, seeded with acetone and the spatial resolution was around 10 µm, with a maximal uncertainty of 5% for a Knudsen number between 0.002 and 0.035, that is in the slip flow regime. Chapter 2. Single-phase gas flow in microchannels Represented in Fig. 2.20 1.E12 m (kg/s) 1.E12 1.E12 63 NS2 (s 0.93) QHD1 (s 1) NS2 (s 1) 8.E13 NS1 (s 1) 6.E13 4.E13 ns0 2.E13 0.E00 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

(a) Represented in Fig. 2.20 4.E13 NS2 (s 1) NS2 (s 0.93) m (kg/s) 3.E13 QHD1 (s 1) 2.E13 NS1 (s 1) 1.E13 ns0 0.E00 1.3 (b) 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Fig. 2.21. Experimental data from Lalonde (outlet flow rate and inlet flow rate ), compared to NS2 and QHD1 models. Microchannel no.4. Po = 75 kPa, T = 294.2 K. (a) Gas: N2 ; Sc = 0.74; the encircled data correspond to Kno,M = 0.15. (b): Gas: He; Sc = 0.77; the encircled data correspond to Kno,M = 0.47. 2.3.6. Entrance effects In the previous sections, the microchannels have been considered as long microchannels and the entrance effects have been neglected. If the microchannel is short, such that the hydrodynamic development length Lhd is no longer negligible compared to the microchannel length, entrance effects have to be taken into account by a numerical approach. Rarefaction increases Lhd in the slip flow regime, due to slip at the walls. Barber and Emerson (2002) used a two-dimensional finite-volume Navier–Stokes solver to simulate the flow of a gas entering a plane microchannel in the slip-flow regime, with first-order boundary conditions. Their simulated data, for Kn Î [0; 0.1] and Re Î [0; 400] were fitted 64 Heat transfer and fluid flow in minichannels and microchannels Straight microchannel Inlet/outlet Integral pressure sensor Metal inter-connects Inlet/outlet (a) Pressure sensor Nitride membrane Piezoresistive elements Metal interconnects Capillary connection Microchannel Etching hole (b) Fig. 2.22. Microchannel with integrated pressure sensors (Zohar et al., 2002). (a) Microchannel with inlet and outlet; (b) close-up of the pressure sensor and its capillary connection with the microchannel. by a least-squares technique and led to the following expression for the hydrodynamic development length: Lhd 1 + 14.78 Kn 0.332 + 0.011 Re = DH 0.0271 Re + 1 1 + 9.78 Kn (2.160) which generalizes the expression Lhd /DH = 0.315/(0.0175 Re + 1) + 0.011Re found by Chen (1973) for a no-slip flow with a slight modification of the non-rarefied part of the equation. In Eq. (2.160), the Knudsen number Kn = /DH and the Reynolds number Re = u¯ z DH /µ are based on the hydraulic diameter DH = 4h of the microchannel, 2h in depth. The hydrodynamic development length is arbitrarily defined as the longitudinal distance required for the centerline velocity to reach 99% of its fully developed value. Equation (2.160) shows that for Kn = 0.1 and Re = 400, slip at the wall increases the entrance length of about 25%. Chapter 2. Single-phase gas flow in microchannels p * 1 65 1 0.9 2h 4.14 µm 0.8 0.7 1 2 0.6 2h 2.07 µm 2 0.5 0.4 3 0.3 2h 1.03 µm 0.2 3 0.1 0 0 20,000 40,000 60,000 f (Hz) 80,000 100,000 Fig. 2.23. Influence of microchannel depth 2h on the gain p* for a square microchannel closed at its outlet. l = 100 µm, T = 293 K, p = 0.11 MPa. 1, 2, 3: slip flow; 1 , 2 , 3 : no-slip flow. 2.4. Pulsed gas flows in microchannels Rarefaction affects steady microflows and modifies the behavior of unsteady microflows. The case of pulsed flows is of particular interest because such flows are encountered in many applications. For example, micropump actuators often generate sinusoidal pressure fluctuations inside a chamber in order to induce a flow in microchannels through microvalves or microdiodes. Detailed models for pulsed gaseous flows in the slip flow regime can be found in Caen et al. (1996) for circular microtubes or in Colin et al. (1998a) for rectangular microchannels. They are based on the Navier–Stokes equations with first-order slip and temperature jump conditions at the walls. If we consider a microchannel of length l, whose inlet is submitted to a sinusoidal pressure fluctuation with a small amplitude, the gain p* of the microchannel, that is the ratio of the outlet po over inlet pi fluctuating pressures amplitudes, can be calculated. For example, in the simple case of a microchannel closed at its outlet, this gain takes the form po 1 * p = (2.161) = pi cosh(l) # where ( f , Kn) » f (Kn) depends on the square root of the frequency f and on the Knudsen number. It can be shown that the band pass of the microchannel is underestimated when slip at the walls is not taken into account (Fig. 2.23). Obtaining experimental data to discuss these theoretical results is very hard, due to the very small size of the pressure sensors required for these experiments. However, 66 Heat transfer and fluid flow in minichannels and microchannels i o 2b1 l1 (a) 2b3 2b2 l2 2b1 i (b) l3 2b2 2b3 o Fig. 2.24. Two layouts of a microdiode placed in a microchannel. experimental data were obtained for microtubes down to 50 µm in diameter, using a commercially available pressure microsensor placed in a minichannel connected in series to one or several parallel microtubes (Colin et al., 1998b). The behavior of the micro- and minichannels association was simulated and compared to these data. The agreement was good, both for the gain and the phase of the association. The model for pulsed flows in constant-section microchannels can be easily extended to microchannels with slowly varying cross-sections (Aubert et al., 1998). It allows an understanding of the behavior of microdiffusers subjected to sinusoidal pressure fluctuations and to test the diode effect of a microdiffuser/nozzle placed in a microchannel that is subjected to sinusoidal pressure fluctuations. In Fig. 2.24, two layouts (a) and (b) are considered. In layout (a), the tested element is a microdiffuser, with an increasing section from inlet to outlet. In layout (b), the same element is used as a nozzle with a decreasing section from inlet to outlet. To obtain an exploitable comparison between the two layouts, the widths (2b1 , 2b2 , 2b3 ) and the lengths (l1 , l2 , l3 ) are the same in layouts (a) and (b). The angle 2 is also the same for the diffuser or the nozzle. An analysis of the frequency behavior for each layout shows significant differences, which is characteristic of a dynamic diode effect. Therefore, in order to characterize the dissymmetry of the transmission of the pressure fluctuations, an efficiency E of the diode is introduced. This efficiency E= p*(a) p*(b) = | po / pi |(a) | po / pi |(b) (2.162) defined as the ratio of the fluctuating pressure gain in layout (a) over the fluctuating pressure gain in layout (b), can be studied as a function of the frequency f . Some results are shown in Fig. 2.25, for different values of the depth 2 h with l1 = l2 = l3 = 3 mm, 2b1 = 2b3 = 467 µm, 2b2 = 100 µm and = 3.5 . Chapter 2. Single-phase gas flow in microchannels 67 1.05 1 3 1 0.95 2 0.9 E 0.85 0.8 0.75 0.7 0 5000 10000 15000 f (Hz) Fig. 2.25. Diode efficiency for different values of the depth. 1: 2h = 100 µm; 2: 2h = 62 µm, 3:2h = 6 µm. Figure 2.25 points out that with a microdiffuser (2h = 6 µm), E appears to be less than unity below a critical frequency. This denotes a reversed diode effect compared to the case of a diffuser with sub-millimetric dimensions (2h = 100 µm), for which E is less than unity beyond a critical frequency. Note also that the influence of slip (taken into account in Fig. 2.25) is not negligible when 2h = 6 µm. However, slip has little influence on the values of characteristic frequencies (i.e. the frequencies for which E is an extremum or equal to unity). Pulsed gaseous flows are also of great interest in short microchannels, or micro-orifices, for example in synthetic microjets. The synthetic microjet is a low power, highly compact microfluidic device that has potential for application in boundary layer control (Lee et al., 2003). The oscillation of an actuated membrane at the bottom of a cavity generates a pulsed flow through a micro-slit or orifice (Fig. 2.26). It results in a zero net-mass flux, but in a non-zero mean momentum flux. A series of such microjets can be used to control separation or transition in the boundary layer of the flow on a wing in order to reduce drag, to increase lift or to limit noise generated by the air flow. 2.5. Thermally driven gas microflows and vacuum generation Generating vacuum by means of microsystems concerns various applications such as the taking of biological or chemical samples, or the control of the vacuum level in the neighborhood of some specific microsystems during working. The properties of rarefied flows (due to both low pressure and small dimensions) allow unusual pumping techniques. For high Knudsen numbers, the flow may be generated without any moving mechanical component by only using thermal actuation, which is not possible with classical macropumps. 68 Heat transfer and fluid flow in minichannels and microchannels “Synthetic jet” Vortex ring ho do Orifice dc Cavity hc v Electrically oscillated membrane Fig. 2.26. Schematic of a micro-synthetic jet actuator (Mallinson et al., 2003). 2.5.1. Transpiration pumping Currently, the most studied technique is based on thermal transpiration. The basic principle requires two chambers filled with gas and linked with an orifice whose hydraulic diameter is small compared with the mean free path of the molecules. Chamber 1 is heated, for example with an element, so that T1 > T2 . By analyzing the probability that some molecules move from one chamber across the orifice, it can be shown that if the pressure is uniform ( p1 = p2 = p), a molecular flux N˙ 2Õ1 Ap =√ 2mkB √ √ T1 − T2 √ T1 T2 (2.163) from chamber 2 toward chamber 1 – from cold to hot temperatures – appears (Muntz and Vargo, 2002; Lengrand and Elizarova, 2004). If the net molecular flux is constant, the pressures necessarily verify that $ p1 = p2 T1 T2 (2.164) In Eqs. (2.163) √ and (2.164), m is the mass of a molecule and A is the area of the orifice. If p2 < p1 < p2 T1 /T2 , there is a net flow from 2 toward 1, which results in a decrease of p2 and/or an increase of p1 . This indicates that a basic microscale pump is working. Thus, a Knudsen compressor (Fig. 2.27) can be designed by connecting a series of chambers with very small orifices that have a cold region (temperature T2 ) on one side and a hot region (temperature T1 ) on the other side by means of an adequate local heater placed just downstream of the orifices. This multistage layout leads to important cumulated pressure drops, while keeping a satisfactory flow rate. In practice, the orifices must be Chapter 2. Single-phase gas flow in microchannels 69 T T1 T2 Flow n1 n1 n Element Fig. 2.27. Multistage Knudsen compressor. replaced by microchannels and this complicates the modeling (Vargo and Muntz, 1997). Moreover, it is generally difficult to maintain simultaneously a free molecular regime in the microchannels and a continuum regime in the chambers, which are the required conditions for the optimal efficiency corresponding to Eqs. (2.163) and (2.164). More elaborate models, based on the linearization of the Boltzmann equation (Loyalka and Hamoodi, 1990), are able to take into account the transition regime (0.05 < Kn < 10) both in the microchannels and in the chambers (Vargo and Muntz, 1999). Several design studies are proposed in the literature that show the theoretical feasibility of microscale thermal transpiration pumps. The first mesoscale prototypes, heated with an element at the wall (Vargo and Muntz, 1997; Vargo and Muntz, 1999; Vargo et al., 1999), were tested. An alternative solution suggesting heating within the gas was also proposed (Young, 1999), but experimental data are not available for this layout. Actually, there is yet much to do for the theoretical optimization and almost everything is to be done for the design of thermal transpiration micropumps using micro manufacturing techniques. The performances of these micropumps remain limited for technological reasons. A high vacuum level may become incompatible with the typical internal sizes of microsystems (Muntz and Vargo, 2002) because the regime in the chambers must be close to a continuum regime, which requires sizes too big for the lower pressures. 2.5.2. Accommodation pumping Accommodation pumping is another pumping technique. It exploits the property of gas molecules whose reflection on specular walls depends on their temperature. If the wall is warmer than the gas, the mean reflection angle is greater than the incident angle. Conversely, if the wall is colder than the molecules, they have a more tangential reflection. Consequently, in a microchannel with perfectly specular walls connecting two chambers at 70 Heat transfer and fluid flow in minichannels and microchannels T3 T1 T1 1 3 T2 2 Fig. 2.28. One stage of an accommodation micropump. different temperatures, a flow takes place from the hot toward the cold chambers (Hobson, 1970). If the walls of the microchannel give a diffuse reflection, this effect disappears. This property was confirmed with numerical simulations by the DSMC method (Hudson and Bartel, 1999). By linking two warm chambers (1 and 3) to a cold chamber (2) on one side with a smooth microchannel (1–2) and on the other side with a rough microchannel (3–2), a difference of pressure ( p1 < p3 ) appears between the two chambers at the same temperature (Fig. 2.28). Connecting in series several stages of that type, the pressure drops relative to each stage can be cumulated and allow the setup to reach high vacuum levels. The advantage of the accommodation pump is that it is operational without theoretical limitations concerning low pressures and contrary to the thermal transpiration pumping, the accommodation pumping does not require chambers with high dimensions. To compensate, the accommodation pump requires more stages to reach the same final pressure ratio. A pressure ratio = 100 requires 125 stages, whereas only 10 stages are enough for a transpiration pump with comparable temperatures. Lastly, no operational prototype was described in the literature so far, although the concept seems attractive. The highest-performance design was proposed by Hobson (1971; 1972); it requires a cold temperature T2 = 77 K for an atmospheric temperature T1 = T3 = 290 K. 2.6. Heat transfer in microchannels In the slip flow regime, heat transfer can be treated as well as mass transfer, providing the energy equation (2.41) and the temperature boundary conditions (2.61) are solved with the momentum equation and the slip-flow boundary conditions. Such heat transfer problems have an analytical solution only in the simplest cases. For example, if the compressibility of the flow can be neglected, the energy and the momentum equations are no longer Chapter 2. Single-phase gas flow in microchannels 71 coupled and if the flow is developed, longitudinal temperature gradients are constant and longitudinal velocity gradients are zero (see example in Section 2.6.1.1). In the general case, analytical or semi-analytical solutions are not possible and numerical simulations are required. Depending on the accommodation coefficients, heat transfer can be found to increase, decrease or remain unchanged, compared to non-slip-flow conditions. 2.6.1. Heat transfer in a plane microchannel 2.6.1.1. Heat transfer for a fully developed incompressible flow Let us consider the case of a plane microchannel (Fig. 2.10(a)), with a lower adiabatic wall (qw |y=−h = 0) and an upper wall with constant heat flux (qw |y=h = qh ). The flow is assumed to be hydrodynamically and thermally developed and incompressible. Fluid properties are assumed uniform. Noting y* = y/h, the momentum equation reduces to d 2 uz h2 dp = 2 µ dz dy* (2.165) and the energy equation to ∂T k ∂2 T = uz cp 2 2 * h ∂y ∂z (2.166) neglecting viscous dissipation. We introduce * = T − Twall 4 T − Twall = qh h/k Nu T − Twall (2.167) with qh = (T − Twall ) and Nu = DH /k = 4 h/k, noting as the convection heat transfer coefficient. The flow being thermally developed, ∂ * /∂z = 0, ∂Nu/∂z = 0 and ∂qh /∂z = 0, which implies ∂T /∂z = ∂T /∂z = ∂Twall /∂z. Consequently, the energy equation can be written as qh d 2 * dT = uz cp 2 * h dy dz (2.168) Moreover, the conservation of energy on a control volume of length dz and height 2h yields 2h cp uz dT = −qh dz and Eq. (2.168) reduces to uz d2 * =− 2 * 2 uz dy Neglecting thermal creep, the first-order boundary conditions are duz * duz * * = 0 u | = − y =0 z y =1 * * dy dy y* =1 (2.169) (2.170) 72 Heat transfer and fluid flow in minichannels and microchannels d * =0 dy* y* =−1 * |y* =1 = − * d * dy* y* =1 (2.171) with a dimensionless coefficient of slip * = /h = (/h)(2 − )/ = 4 Kn(2 − )/ and a dimensionless temperature jump distance * = /h = (/h)[(2 − T )/ T ][2/( + 1)]/ Pr = 8 [(2 − T )/T ][/( + 1)][Kn/Pr], where Pr is the Prandtl number. The resolution of this set of equations gives the velocity distribution uz = − h2 dp (1 − y*2 + 2 * ) 2 µ dz (2.172) and, with 1 uz = 2

1 −1 uz dy* uz 3(−y*2 + 1 + 2 * ) = uz 2(1 + 3 * ) (2.173) The integration of Eq. (2.169) with the boundary conditions (2.171) yields * = * + T and T = (1/2) Nu−1 = 1 1 + 3 * 1

1 *4 3 *2 1 y − y (1 + 2 * ) − y* (1 + 3 * ) 16 8 2 13 9 * + + 16 4 −1 (uz /uz )T (2.174) dy* are calculated and the Nusselt number is deduced as 26 + 147 * + 210 * * + 4 140 (1 + 3 * )2 2 (2.175) Equation (2.175) shows that heat transfer depends both on the dimensionless coefficient of slip * and on the dimensionless temperature jump distance * , that is it depends on the two accommodation coefficients and T as well as on the Knudsen number Kn. For classical values of the accommodation coefficients, the effect of the temperature jump is greater than the effect of velocity slip and the heat transfer decreases when rarefaction increases. The Nusselt number reduces to the classic value 70/13 (Rohsenow, 1998) when Kn = 0. Zhu et al. (2000) extended this solution to the problem of asymmetrically heated walls (0 = q0 = qh ) by combining the solutions of two sub-problems similar to the above problem. Inman (Karniadakis and Beskok, 2002) first studied the solution for a constant symmetric heat flux at the two walls. Chapter 2. Single-phase gas flow in microchannels 73 The problem with constant wall temperature was treated by Hadjiconstantinou and Simek (2002), who extended the study to the transition regime, using DSMC. They showed that the slip flow prediction (with first-order boundary conditions) is in good agreement with the DSMC results for Kn ≤ 0.2 and remains a good approximation beyond its expected range of applicability. 2.6.1.2. Heat transfer for a developing compressible flow This problem was studied by Kavehpour et al. (1997), using the same first-order slip flow and temperature jump boundary conditions as in Section 2.6.1.1. The flow was developing both hydrodynamically and thermally and two cases were considered: uniform wall temperature or uniform wall heat flux. Viscous dissipation was neglected, but compressibility was taken into account. The numerical methodology was based on the control volume finite difference scheme. It was found that the Nusselt number was substantially reduced for slip flow compared with continuum flow. 2.6.2. Heat transfer in a circular microtube Heat transfer in circular microtubes was simulated by Ameel et al. (1997), who considered a hydrodynamically fully developed incompressible flow, thermally developing with constant heat flux at the wall. Viscous dissipation and thermal creep were neglected and first-order velocity slip and temperature jump conditions were used, with accommodation coefficients taken equal to unity. They found the thermally fully developed solution Nu−1 = 24( − 1)(2 − 1)2 242 − 16 + 3 1 + 48(2 − 1)2 (242 − 16 + 3)( + 1) Pr (2.176) with = 1 + 4 Kn. Equation (2.176) reduces to the no-slip value Nu = 48/11 when Kn = 0. A semi-analytical solution is also given in Ameel et al. (1997) for the entrance region. For a Knudsen number Kn = 0.04, it is found that the Nusselt number for the fully developed flow is reduced 14% over that obtained with a no-slip boundary condition, with = 1.4 and Pr = 0.7. Larrodé et al. (2000) treated the case of constant temperature at the wall. 2.6.3. Heat transfer in a rectangular microchannel Yu and Ameel (2001) studied the heat transfer problem for a thermally developing flow in a rectangular microchannel. The flow was assumed to be incompressible and hydrodynamically fully developed, and a constant temperature was imposed at the walls. The velocity distribution was developed in series as discussed in Section 2.3.4.1 and the energy equation was solved by a modified generalized integral transform method. A competition between and was pointed out, with a transition value c of = / that depends on the aspect ratio a* (Table 2.7). 74 Heat transfer and fluid flow in minichannels and microchannels Table 2.7 Transition values of = / as a function of the aspect ratio, (from Yu and Ameel (2001)). a*−1 1 2 3 4 5 6 8 10 ∞ c 0.67 0.50 0.38 0.32 0.29 0.28 0.27 0.25 0.20 For < c , heat transfer is enhanced in comparison with the no-slip flow and if > c , heat transfer is reduced. Moreover, for < c , heat transfer increases with increasing * and it decreases if < c . According to the values available in the literature, it should be noted that may vary in a wide range from unity to 100 or more, for actual wall surface conditions. However, the value = 1.67, which corresponds to a diffuse reflection ( = 1) and a total thermal accommodation (T = 1) for air ( = 1.4; Pr = 0.7) is a representative value for many engineering applications. In this typical case, rarefaction effects result in a decrease of heat transfer, which can be as much as 40%. 2.7. Future research needs Theoretical knowledge is currently well advanced for gas flows in microchannels. However, there is yet a need for accurate experimental data, both for steady or unsteady gas microflows, with or without heat transfer. For example, in order to definitely validate the choice of the best boundary conditions in the slip flow regime, we need to isolate the influence of the accommodation coefficients, as it remains an open issue. Relationships between their values, the nature of the substrate and the micro-fabrication processes involved are currently not available. Theoretical investigations relative to unsteady or thermally driven microflows would also need to be supported by smart experiments. 2.8. Solved examples Example 2.1 A microchannel with constant rectangular cross-section is submitted to a pressure-driven flow of argon. The inlet and the outlet pressures are pi = 0.2 MPa and po = 25 kPa respectively. The geometry is given in Fig. 2.10(c), with a width 2b = 20 µm, a depth 2h = 1 µm and a length l = 5 mm. Assume a uniform temperature T = 350 K. (i) Calculate the Knudsen numbers Kn and Kn at the inlet and at the outlet of the microchannel, assuming that the intermolecular collisions are accurately described by a variable soft spheres model. (ii) Compare with the values obtained from a hard spheres model. (iii) Assuming a diffuse reflection at the wall, calculate the mass flow rate through the microchannel. Chapter 2. Single-phase gas flow in microchannels 75 (iv) Calculate the mass flow rate increase (in %) due to slip at the wall. (v) What would be the underestimation (in %) of the mass flow rate given by a no-slip model, in the same conditions but with an inlet pressure pi = 0.1 MPa? Solution: (i) Inlet and outlet Knudsen numbers with aVSS model (Answer: Kni = 3.48 × 10−2 ; Kni = 1.83 × 10−2 ; Kno = 2.78 × 10−1 ; Kno = 1.46 × 10−1 ) From Table 2.2, the temperature exponent of the coefficient of viscosity for argon (Ar) is = 0.81, the exponent for the VSS model is = 1.40 and the dynamic viscosity under standard conditions is µ0 = 211.7 × 10−7 Pa s. From Eq. (2.7), the viscosity at T = 350 K is: µ(T ) = µ0 (T /T0 ) = 2.117 × 10−5 (350/273.15)0.81 = 2.588 × 10−5 Pa s. The molecular weight of argon is given in Table 2.2: M = 39.948 × 10−3 kg/mol and its 2 gas constant is R = R/M = 8.314511/39.948 × 10−3 = 2.08 √ × 10 J/kg/K. The mean free path is given by Table 2.1: = k2 µ(T ) RT /p, with k2,VSS = 4(7 − 2)(5 − 2) = 0.996. √ 5( + 1)( + 2) 2 At the inlet, # √ i,VSS = k2 µ(T ) RT /pi = 0.996 × 2.588 × 10−5 × 2.08 × 102 × 350/2 × 105 = 34.8 nm and at the outlet, # √ o,VSS = k2 µ(T ) RT /po = 0.996 × 2.588 × 10−5 × 2.08 × 102 × 350/2.5 × 104 = 278 nm. The Knudsen number based on the hydraulic diameter DH = 4bh/(b + h) = 4 × 10 × 10−6 × 0.5 × 10−6 /((10 + 0.5) × 10−6 ) = 1.905 µm is Kn = /DH , that is Kni = i /DH = 3.48 × 10−8 /1.905 × 10−6 = 1.83 × 10−2 at the inlet and Kno = o /DH = 2.78 × 10−7 /1.905 × 10−6 = 1.46 × 10−1 at the outlet. The Knudsen number based on the microchannel depth is Kn = /(2h), that is Kni = i /(2 h) = 3.48 × 10−8 /1 × 10−6 = 3.48 × 10−2 at the inlet and 76 Heat transfer and fluid flow in minichannels and microchannels Kno = o /(2h) = 2.78 × 10−7 /1 × 10−6 = 2.78 × 10−1 at the outlet. (ii) Inlet and outlet Knudsen numbers with a HS model (Answer: Kni,HS = 4.13 × 10−2 ; Kni,HS = 2.17 × 10−2 ; Kno,HS = 3.30 × 10−1 ; Kno,HS = 1.73 × 10−1 ) With a HS model, = 0.5 and the viscosity µ(T ) = µ0 (T /T0 )0.5 = 2.117 × 10−5 (350/273.15)0.5 = 2.396 × 10−5 Pa s is underestimated of 7.4%. √ The coefficient k2,HS = 16/(5 2) = 1.277 (cf. Table 2.1). Consequently, # √ i,HS = k2,HS µ(T ) RT /pi = 1.277 × 2.588 × 10−5 × 2.08 × 102 × 350/2 × 105 = 41.3 nm and # √ o,HS = k2,HS µ(T ) RT /po = 1.277 × 2.588 × 10−5 × 2.08 × 102 × 350/2.5 × 104 = 330 nm which leads to: Kni,HS = i,HS /DH = 4.13 × 10−8 /1.905 × 10−6 = 2.17 × 10−2 Kno,HS = o,HS /DH = 3.30 × 10−7 /1.905 × 10−6 = 1.73 × 10−1 Kni,HS = i,HS /(2 h) = 4.13 × 10−8 /1 × 10−6 = 4.13 × 10−2 Kno,HS = o,HS /(2 h) = 3.30 × 10−7 /1 × 10−6 = 3.30 × 10−1 The overestimation of the Knudsen number calculated from an HS model, compared with the more accurate solution obtained with a VSS model, is 28.2%, which corresponds to the ratio (k2,HS − k2,VSS )/k2,VSS = 0.282. (iii) Mass flow rate (Answer: m ˙ = 4.89 × 10−12 kg/s) The rectangular cross-section has a nonnegligible aspect ratio a* = h/b = 1/20 = 0.05 > 0.01. Therefore, a rectangular model should be used. Due to the range covered by the Knudsen numbers, a second-order model (NS2) is required (cf. Fig. 2.7). The mass flow rate is obtained from Eq. (2.157): m ˙ NS2, rect = 4 h4 p2o a1 2 2 − 1) + a Kn ( − 1) + a Kn ln ( 2 3 o o a* µRTl 2 Chapter 2. Single-phase gas flow in microchannels 77 with a1 = 0.32283, a2 = 1.9925 and a3 = 4.2853 given by Table 2.5 for a diffuse reflection ( = 1). = pi /po = 200/25 = 8 and m ˙ NS2, rect = 4(10−6 )4 (2.5 × 10−4 )2 /(0.05 × 2.588 × 10−5 × 2.08 × 102 × 350 × 5 × 10−3 )× [0.32283(82 − 1)/2 + 1.9925 × 2.78 × 10−1 (8 − 1) + 4.2853 × (2.78 × 10−1 )2 ln8] = 4.89 × 10−12 kg/s (iv) Mass flow rate increase (Answer: 44.9%) From Eq. (2.158), m ˙ NS2, rect a2 a3 1 2 ln = 1 + 2 Kno + 2 Kno 2 m ˙ *NS2, rect = m ˙ ns, rect a1 +1 a1 −1 = 1+2 1.9925 1 ln 8 4.2853 2.78 × 10−1 +2 (2.78 ×10−1)2 2 = 1.449 0.32283 8+1 0.32283 8 −1 which means that slip at the wall increases the flow rate of 44.9%. (v) Underestimation of the mass flow rate (Answer: 46.7%) If the inlet pressure is decreased to pi = 0.1 MPa, the pressure ratio is = 4 and m ˙ *NS2, rect = 1 + 2 1 ln 4 1.9925 4.2853 2.78 × 10−1 +2 (2.78 × 10−1 )2 2 = 1.877. 0.32283 4+1 0.32283 4 −1 Therefore, although the outlet Knudsen number is unchanged, the inlet pressure number is increased and rarefaction effects are enhanced. Slip at the wall now increases the flow rate of 87.7%. In others words, the underestimation of the mass flow rate with a no-slip model is 46.7%, since (m ˙ NS2 − m ˙ ns )/m ˙ NS2 = 1 − 1/m ˙ *NS2 = 1 − 1/1.877 = 0.467. Example 2.2 In a hard disk drive, the flow of gas between the hard disk and the read–write head can locally be comparable to a fully developed micro-Couette flow between parallel plane plates. The shield of the hard disk drive is sealed and the gas inside is pure nitrogen. The distance between the two plates is 2h = 0.2 µm, the read-write head is at r = 3 cm from the center of the disk, whose speed of rotation is ˙ = 7200 rpm. One of the plates (y = −h) is fixed and the other ( y = h) moves with a constant velocity U in the z-direction. Assume a uniform temperature T = 300 K, a uniform pressure p = 90 kPa and an accommodation coefficient = 0.9. (i) Write the velocity distribution as a function of the Knudsen number. (ii) Calculate the mass flow rate per unit width and compare it to the case of a no-slip Couette flow. (iii) Calculate the friction factor and compare its value to the one obtained with a no-slip assumption. 78 Heat transfer and fluid flow in minichannels and microchannels Solution y* (i) Velocity distribution Answer: uz = U2 4 Kn(2−)/ +1 + 1 The momentum equation reduces to d 2 uz /dy2 = 0, and the boundary conditions are uz y=−h = ((2 − )/) duz /dyy=−h and uz y=+h = U − ((2 − )/) duz /dyy=+h. Thus, the velocity profile is linear: uz = a1 y + a2 . The boundary conditions yields −a1 h + a2 = ((2 − )/) a1 and a1 h + a2 = U − ((2 − )/) a1 , which leads to 2 a1 = U /(h + (2 − )/) and 2 a2 = U . Then uz = U /2(y/((2 − )/ + h) + 1). With y * = y/h and Kn = y/DH = y/(4 h), the velocity distribution is written as uz = U /2(y* /(4 Kn(2 − )/ + 1) + 1). The centerline velocity is independent of the slip at the walls and the velocity transverse gradient is reduced as rarefaction is increased. (ii) Mass flow rate (Answer: m ˙ = (p M/RT )(˙ r 2 /60)2h = 4.57 × 10−6 kg/s/m) 1 The mass flow rate is m ˙ = uz A = (p/RT ) uz A with uz = 1/2 −1 uz dy * = U /2. Therefore, the mass flow rate per unit width is: m ˙ = = p U pM r2 ˙ 4h = 2h RT 2 RT 60 9 × 104 × 28.013 × 10−3 × 7200 × 3 × 10−2 × 2 × 2 × 10−7 = 4.57 × 10−6 kg/m 8.3145 × 300 × 60 The mass flow rate is the same as the one for a Couette flow without slip at the wall. (iii) Friction factor (Answer: Cf* = Cf /Cf ,ns = 0.57) From Eq. (2.95), Cf = 2 w /(uz 2 ) and w = w = µ duz /dy is uniform since the velocity U 1 z profile is linear. With du dy = 2 h 1 + 4 Kn(2 − )/ , we obtain: Cf = 4 µRT hpU (1 + 4 Kn(2 − )/) From Table 2.2, the temperature exponent of the coefficient of viscosity for nitrogen (N2 ) is = 0.74, the exponent for the VSS model is = 1.36 and the dynamic viscosity under standard conditions is µ0 = 1.656 × 10−5 Pa s. From Eq. (2.7), the viscosity at T = 300 K is: µ(T ) = µ0 (T /T0 ) = 1.656 × 10−5 (300/273.15)0.74 = 1.775×10−5 Pa s. The molecular weight of nitrogen is given in Table 2.2: M = 28.013 × 10−3 kg/mol and its gas constant is R = R/M = 8.314511/28.013 × 10−3 = 2.97 × 102 J/kg/K. √ The mean free path is given by √ Table 2.1: = k2 µ(T ) RT /p, with √ k2 = [4 (7 − 2)(5 − 2)]/[5( + 1)( + 2) 2 ] = 1.06. Therefore, = k µ(T ) RT /p = 2 √ 1.06 × 1.775 × 10−5 × 2.97 × 102 × 300/9 × 104 = 62.6 nm and the Knudsen number based on the hydraulic diameter is: Kn = /(4h) = 6.26 × 10−8 /(4 × 10−7 ) = 1.56 × 10−1 . Chapter 2. Single-phase gas flow in microchannels 79 The influence of rarefaction on the friction is then given by: Cf* = Cf Cf ,ns = Cf /Cf (Kn=0) = = 1 1 + 4Kn(2 − )/ 1 = 0.57 1 + 4 × 0.156(2 − 0.9)/0.9 The friction factor is reduced of 43%, due to slip at the walls. 2.9. Problems Problem 2.1 A flow of helium is generated in a circular microtube by axial pressure and temperature gradients. The microtube has a length l = 5 mm and a uniform cross-section with a diameter 2 r2 = 10 µm. In a section where the flow may be considered as locally fully developed, the gradients are dp/dz = −5 Pa/µm and dT /dz = 0.5 K/µm, for a pressure p = 100 kPa and a temperature T = 400 K. Assume a tangential momentum accommodation coefficient = 0.9. (i) Write the velocity distribution as a function of the Knudsen number, in a dimensionless form: uz* (r * , Kn). (ii) Calculate the part of the velocity increase due to slip at the wall. (iii) Calculate the part of the velocity increase due to thermal creep. Problem 2.2 A flow of air in a rectangular microchannel is generated by a pressure gradient. The inlet pressure is pi = 3.5 bar and the outlet pressure is po = 150 mbar. The temperature T = 350 K is uniform and the tangential momentum accommodation coefficient is = 0.8. The microchannel length is l = 10 mm and its uniform section has a depth 2h = 1.2 µm and a width 2b = 4.8 µm. (i) Calculate the mass flow rate with a plane flow assumption and a no-slip model. (ii) Calculate the mass flow rate with a plane flow assumption and a Navier–Stokes model with first-order slip boundary conditions. (iii) Calculate the mass flow rate with a plane flow assumption and a Navier–Stokes model with Deissler second-order slip boundary conditions. (iv) Calculate the mass flow rate with a plane flow assumption and a quasi-hydrodynamic model with first-order slip boundary conditions. (v) Calculate the mass flow rate with a no-slip model in a rectangular section. (vi) Calculate the mass flow rate with a Navier–Stokes model in a rectangular section with Deissler second-order slip boundary conditions. (vii) Calculate the deviations of the above different models, compared to the most accurate among them. 80 Heat transfer and fluid flow in minichannels and microchannels Problem 2.3 Consider a microchannel with a uniform cross-section whose width is very large compared with its depth 2h = 5 µm. A pressure-driven flow of helium is generated, the lower wall ( y = −h) is adiabatic and the upper wall ( y = h) has a constant heat flux qh = −200 W/cm2 . Consider a section far from the inlet, such that the flow in this section is thermally and hydrodynamically developed. The flow conditions in this section are such that the Knudsen number based on the hydraulic diameter is Kn = 2 × 10−2 and the accommodation coefficients are = 0.9 and T = 0.6. The wall temperature Twall = 400 K is measured with a micro-sensor. (i) Calculate the mean temperature T of the gas in the section. (ii) Calculate the temperature T0 at the center of the section ( y = 0). (iii) Compare with the case of a similar non rarefied flow. Problem 2.4 Consider a microchannel, with a length l = 5 mm and whose uniform rectangular crosssection has a low aspect ratio: its width is 2b = 200 µm and its depth is 2h = 1.5 µm. A flow of nitrogen is generated with a pressure gradient. The inlet pressure is pi = 3 × 105 Pa and the outlet pressure is po = 1 × 105 Pa. The temperature T = 300 K is uniform and the tangential momentum accommodation coefficient is = 0.95. (i) Calculate the mean velocity in a crosssection of the microchannel, as a function of the local pressure gradient and Knudsen number. (ii) Calculate the mass flow rate. (iii) Calculate the pressure value at the middle of the microchannel (at z = l/2) and compare it to the no-slip value. (iv) Calculate the maximal velocity reached in the microchannel outside the entrance region. (v) Calculate the maximal Mach number and justify the locally fully developed assumption. Problem 2.5 A multistage Knudsen compressor is designed to locally tune the pressure level or air inside a microsystem. The geometry of this Knudsen compressor is given in Fig. 2.27. Ten chambers are connected via 9 series of 100 short microchannels. These microchannels have rectangular cross-sections with a depth 2h = 0.5 µm and a width 2b = 20 µm. Each chamber is assumed large enough for the flow inside to be considered as a continuum flow. The temperature in these chambers is T2 = 300 K and each element is able to rise the temperature at the outlet of the microchannels to the value T1 = 600 K. Chamber 10 is connected to a region, whose pressure p10 = 10 mbar is maintained constant by means of a primary pump. The initial temperature Tt=0 = T2 = 300 K is uniform before the elements heating. The initial pressure in each chamber is also uniform and is equal to the pressure Chapter 2. Single-phase gas flow in microchannels 81 in chamber 10: pt=0 = p10 = 10 mbar. For t > 0, heating of the elements can be regulated in order to tune the pressure in chamber 1 in the range 1 to 10 mbar. (i) Calculate the possible range covered by the Knudsen number in the microchannels during the pumping process and check that the free molecular regime assumption in the microchannel is valid. (ii) Calculate the maximal value of the mass flow rate between two adjacent chambers at the beginning of the pumping progress. 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D., and Liao, Q., An analysis for heat transfer between two unsymmetrically heated parallel plates with micro spacing in slip flow regime, in Proceedings of the International Conference on Heat Transfer and Transport Phenomena in Microscale, G. P. Celata, ed. Banff, Canada, pp. 114–120, 2000. Zohar, Y., Lee, S. Y. K., Lee, Y. L., Jiang, L., and Tong, P., Subsonic gas flow in a straight and uniform microchannel, J. Fluid Mech., 472, 125–151, 2002. Chapter 3 SINGLE-PHASE LIQUID FLOW IN MINICHANNELS AND MICROCHANNELS Satish G. Kandlikar Mechanical Engineering Department, Rochester Institute of Technology, Rochester, NY, USA 3.1. Introduction 3.1.1. Fundamental issues in liquid flow at microscale Microchannels are used in a variety of devices incorporating single-phase liquid flow. The early applications involved micromachined devices such as micropumps, microvalves, and microsensors. This was followed by a thrust in the biological and life sciences with a need for analyzing biological materials, such as proteins, DNA, cells, embryos, and chemical reagents. The field of micromixers further received attention with developments in microreactors, where two chemical species are mixed prior to introducing them into a reaction chamber. The high flux heat dissipation from high-speed microprocessors provided the impetus for studies on heat transfer in microchannels. The developments in the microelectromechanical devices naturally require heat removal systems that are equally small. Cooling of mirrors employed in high-power laser systems involves cooling systems that cover very small footprints. Advances in biomedical and genetic engineering require controlled fluid transport and its precise thermal control in passages of several micrometer dimensions. A proper understanding of fluid flow and heat transfer in these microscale systems is therefore essential for their design and operation. In dealing with liquid flows in minichannels and microchannels in the absence of any wall surface effects, such as the electrokinetic or electroosmotic forces that are covered in Chapter 4, the flow is not expected to experience any fundamental changes from the continuum approximation employed in macrofluidic applications. Gad-el-Hak (1999) argued that liquids such as water should be treated as continuous media with the results obtained E-mail: [email protected] 87 88 Heat transfer and fluid flow in minichannels and microchannels from classical theory being applicable in channels larger than 1 µm. However, there remain a number of unresolved issues that require further study. The main areas of current research are summarized below: (a) Experimental validation of the laminar and turbulent flow transport equations – the laminar flow friction factor and heat transfer equations derived from theoretical considerations are expected to hold in microchannel applications in the absence of any changes in the transport processes or any new physical phenomenon. Although explicit equations and experiments for mass transfer are not covered, the conclusions reached for the momentum and heat transfer are expected to be applicable to mass transport processes as well. (b) Verification of the laminar-to-turbulent flow transition at microscale – experimental evidence in this regard needs to be critically evaluated. (c) The effect of large relative roughness values on the flow – large values of relative roughness are more commonly encountered in microchannels. Their effect on the laminarto-turbulent transition, friction factors, and heat transfer needs to be investigated. (d) Verification of empirical constants derived from macroscale experiments – a number of constants (such as for the losses due to flow area changes, bends, etc.), whose values are derived from macroscale fluid flow experiments need to be verified for microscale applications. This chapter will be devoted to answering the above questions on the basis of the evidence available in literature. 3.1.2. Need for smaller flow passages The flow passage dimensions in convective heat transfer applications have been shifting towards smaller dimensions for the following three main reasons: (a) Heat transfer enhancement. (b) Increased heat flux dissipation in microelectronic devices. (c) Emergence of microscale devices that require cooling. Employing smaller channel dimensions results in higher heat transfer performance, although it is accompanied by a higher pressure drop per unit length. The higher volumetric heat transfer densities require advanced manufacturing techniques and lead to more complex manifold designs. An optimum balance for each application leads to different channel dimensions. For example, in the refrigeration industry, the use of microfin tubes of 6–8 mm diameter have replaced the plain tubes of larger diameters. In automotive applications, the passage dimensions for radiators and evaporators have approached a 1 mm threshold as a balance between the pumping power, heat transfer, and cleanliness constraints imposed by the overall system. Microelectronic devices, which include a variety of applications such as PCs, servers, laser diodes, and RF devices, are constantly pushing the heat flux density requirements to higher levels. What seemed to be an impossibly high limit of 200 W/cm2 of heat dissipation in 1993 now seems to be a feasible target. The new challenge for the coming decade is on the Chapter 3. Single-phase liquid flow in microchannels and minichannels 89 Cooling water from building HVAC system Water-to-water HX Pump In-line filter High heat flux chips cooled with microchannel heat sinks Water cooled cold plate Auxiliary localized air cooling HX Fig. 3.1. Schematic of a cluster of servers with high heat flux chips cooled with microchannel heat sinks, cold plates, and localized air cooling integrated with a secondary chilled water loop from the building HVAC system, Kandlikar (2005). order of 600–1000 W/cm2 . The available temperature differences are becoming smaller, in some cases as low as only a few C with external copper heat sinks. These high levels of heat dissipation require a dramatic reduction in the channel dimensions, matched with suitable coolant loop systems to facilitate the fluid movement away from the heat source. A cooling system for a microscale device might require cooling channels of a few tens of micrometers as compared to more conventional sized channels with 1–3 mm flow passage dimensions. In addition, several such units may be clustered together and a secondary cooling loop may be employed to remove the heat with a conventional cooling system. Figure 3.1 shows a schematic of a microchannel cooling system configuration for cooling a server application. The combination of the microchannel heat exchangers, mounted directly on the chip or in the heat sink that is bonded to the chip, water cooled cold plates with minichannel or microchannel flow passages, and auxiliary localized cooling systems will be able to address the complex cooling needs of the high send servers. The cooling system is integrated with the building HVAC system as described by Kandlikar (2005). A schematic of direct liquid cooling of a multichip module or a heat sink is shown in Fig. 3.2. The liquid flows through the cold plates that are attached to a substrate cap. In advanced designs, direct cooling of chips is accomplished by circulating water, a waterantifreeze mixture, oil, or a dielectric fluid, such as FC-72, FC-77 or FC-87 through microchannels that are fabricated on the chip surface. Copper heat sinks with integrated microchannels and minichannels are expected to dominate heat sink applications. 90 Heat transfer and fluid flow in minichannels and microchannels Microchannels Alternate supply and return manifolds separated by header bars Fig. 3.2. Schematic of a microchannel cooling arrangement on a chip or a heat sink with alternate supply and return manifolds created by header bars, design originally proposed by Tuckerman (1984). 3.2. Pressure drop in single-phase liquid flow 3.2.1. Basic pressure drop relations One-dimensional flow of an incompressible fluid in a smooth circular pipe forms the basis for the pressure drop analysis in internal flows. The following equations are readily derived based on the continuum assumption for Newtonian liquid flows in minichannels and microchannels. Considering the equilibrium of a fluid element of length dx in a pipe of diameter D, the force due to pressure difference dp is balanced by the frictional force due to shear stress w at the wall. D2 dp = (D dx)w (3.1) 4 The pressure gradient and the wall shear stress are thus related by the following equation: dp 4w = dx D (3.2) For Newtonian fluids, the wall shear stress w is expressed in terms of the velocity gradient at the wall: du w = µ (3.3) dy w where µ is the dynamic viscosity. The Fanning friction factor f is used in heat transfer literature because of its ability to represent the momentum transfer process of fluid flow Chapter 3. Single-phase liquid flow in microchannels and minichannels 91 in a manner consistent with the heat and mass transfer process representations: f = w 2 (1/2)um (3.4) where um is the mean flow velocity in the channel. The frictional pressure drop p over a length L is obtained from Eqs. (3.2) and (3.4) respectively: p = 2L 2fum D (3.5) The Fanning friction factor f in Eq. (3.5) depends on the flow conditions, the channel wall geometry and surface conditions: (a) (b) (c) (d) laminar or turbulent flow, flow-channel geometry, fully developed or developing flow, smooth or rough walls. For non-circular flow channels, the D in Eq. (3.5) is replaced by the hydraulic diameter Dh represented by the following equation. Dh = 4Ac Pw (3.6) where Ac is the flow-channel cross-sectional area and Pw is the wetted perimeter. For a rectangular channel of sides a and b, Dh is given by Dh = 4ab 2ab = 2(a + b) (a + b) (3.7) 3.2.2. Fully developed laminar flow The velocity gradient at the channel wall can be readily calculated from the well-known Hagen–Poiseuille parabolic velocity profile for the fully developed laminar flow in a circular pipe. Using this velocity profile, w and f are obtained from Eqs. (3.3) and (3.4). The result for friction factor f is presented in the following form: f = Po Re (3.8) where Po is the Poiseuille number, (Po = f Re), which depends on the flow-channel geometry. Table 3.1 gives the f Re product and the constant Nusselt number in the fully developed laminar flow region for different duct shapes as derived from Kakac et al. (1987). It can be seen that for a circular pipe, Po = f Re = 16 (3.9) Shah and London (1978) provide the following equation for a rectangular channel with short side a and long side b, and a channel aspect ratio defined as c = a/b. 92 Heat transfer and fluid flow in minichannels and microchannels Table 3.1 Fanning friction factor and Nusselt number for fully developed laminar flow in ducts, derived from Kakac et al. (1987). NuH NuT Po = f Re Circular 4.36 3.66 16 Flat channel 8.24 7.54 24 3.61 4.13 4.79 5.33 6.05 6.49 8.24 2.98 3.39 3.96 4.44 5.14 5.60 7.54 14.23 15.55 17.09 18.23 19.70 20.58 24.00 4.00 3.34 15.05 10 30 60 90 120 2.45 2.91 3.11 2.98 2.68 1.61 2.26 2.47 2.34 2.00 12.47 13.07 13.33 13.15 12.74 1 2 4 8 16 4.36 4.56 4.88 5.09 5.18 3.66 3.74 3.79 3.72 3.65 16.00 16.82 18.24 19.15 19.54 Duct shape b a Rectangular, aspect ratio, b/a = b 1 2 3 4 6 8 ∞ Hexagon Isosceles Triangle, Apex angle = u Ellipse, Major/Minor axis a/b = b a Nu = hDh /k; Re = um Dh /µ; NuH – Nu under a constant heat flux boundary condition, constant axial heat flux, and uniform circumferential temperature; NuT – Nu under a constant wall temperature boundary condition; f – friction factor. Po = f Re = 24(1 − 1.3553c + 1.94672c − 1.70123c + 0.95644c − 0.25375c ) (3.10) 3.2.3. Developing laminar flow As flow enters a duct, the velocity profile begins to develop along its length, ultimately reaching the fully developed Hagen–Poiseuille velocity profile. Almost all the analyses available in literature consider a uniform velocity condition at the inlet. The Chapter 3. Single-phase liquid flow in microchannels and minichannels 93 length of the hydrodynamic developing region Lh is given by the following well-accepted equation: Lh = 0.05 Re Dh (3.11) Since the pressure gradients found in small diameter channels are quite high, the flow lengths are generally kept low. In many applications, the length of channel in the developing region therefore forms a major portion of the flow length through a microchannel. To account for the developing region, the pressure drop equations are presented in terms of an apparent friction factor. Apparent friction factor fapp accounts for the pressure drop due to friction and the developing region effects. It represents an average value of the friction factor over the flow length between the entrance section and the location under consideration. Thus the pressure drop in a channel of hydraulic diameter Dh over a length x from the entrance is expressed as: p = 2x 2fapp um Dh (3.12) The difference between the apparent friction factor over a length x and the fully developed friction factor f is expressed in terms of an incremental pressure defect K(x): K(x) = ( fapp − f ) 4x Dh (3.13) For x > Lh the incremental pressure defect attains a constant value K(∞), known as Hagenbach’s factor. Combining Eqs. (3.12) and (3.13), the pressure drop can be expressed in terms of the incremental pressure drop: p = 2( fapp Re)µum x 2( f Re)µum x u2 = + K(x) m 2 2 2 Dh Dh (3.14) For a circular tube, Hornbeck (1964) obtained the axial velocity distribution and pressure drop in a non-dimensional form. He estimated the fully developed region to begin at x+ = 0.0565, with a value of K(∞) = 1.28 for a circular duct. Chen (1972) proposed the following equation for K(∞) for the circular geometry: K(∞) = 1.20 + 38 Re (3.15) The non-dimensionalized length x+ given by: x+ = x/Dh Re (3.16) 94 Heat transfer and fluid flow in minichannels and microchannels fapp Re 102 ac 0.1 ac 0.2 ac 0.5 ac 1.0 101 103 101 102 x (x/Dh)/Re 100 Fig. 3.3. Apparent friction factors for rectangular ducts in the developing region for different aspect ratios ( = 1/c ), Phillips (1987). Shah and London (1978) showed that the frictional pressure drop in the developing region of a circular duct obtained by Hornbeck (1964) can be accurately described by the following equation: p 1.25 + 64x+ − 13.74(x+ )1/2 = 13.74(x+ )1/2 + 2 (1/2)um 1 + 0.00021(x+ )−2 (3.17) Rectangular geometries are of particular interest in microfluidics applications. Shah and London (1978) and Kakac et al. (1987) present comprehensive summaries of the available literature. Phillips (1987) reviewed the available information, including that by Curr et al. (1972) and compiled the results for the apparent friction factor in a rectangular duct as shown in Fig. 3.3. It can be seen that fully developed flow is attained at different x+ values, with low aspect ratio ducts reaching it earlier. The constant in Eq. (3.11) is around 1 (x+ = 100 ) for rectangular ducts as seen from Fig. 3.3. Table 3.2 derived from Phillips (1987) gives the values of the apparent friction factor in a tabular form. For other channel aspect ratios, a linear interpolation is suggested. Alternatively, use the curve-fit equations provided in Appendix A. The results for the fully developed friction factors and Hagenbach’s factors in trapezoidal channels are reported by Kakac et al. (1987). By considering the rectangular channels as a subset of the trapezoidal geometry, Steinke and Kandlikar (2005a) obtained the following curve-fit equation for the Hagenbach’s factor for rectangular channels: K(∞) = 0.6796 + 1.2197c + 3.30892c − 9.59213c + 8.90894c − 2.99595c (3.18) Chapter 3. Single-phase liquid flow in microchannels and minichannels 95 Table 3.2 Laminar flow friction factor in the entrance region of rectangular ducts. fapp Re x+ = (x/Dh )/Re c = 1.0 c = 0.5 c = 0.2 c ≤ 0.1 c ≥ 10 0 0.001 0.003 0.005 0.007 0.009 0.01 0.015 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.20 >1.0 142.0 111.0 66.0 51.8 44.6 39.9 38.0 32.1 28.6 24.6 22.4 21.0 20.0 19.3 18.7 18.2 17.8 15.8 14.2 142.0 111.0 66.0 51.8 44.6 40.0 38.2 32.5 29.1 25.3 23.2 21.8 20.8 20.1 19.6 19.1 18.8 17.0 15.5 142.0 111.0 66.1 52.5 45.3 40.6 38.9 33.3 30.2 26.7 24.9 23.7 22.9 22.4 22.0 21.7 21.4 20.1 19.1 287.0 112.0 67.5 53.0 46.2 42.1 40.4 35.6 32.4 29.7 28.2 27.4 26.8 26.4 26.1 25.8 25.6 24.7 24.0 For intermediate values use the curve-fit equations provided in Appendix A at the end of the chapter. The analysis presented in this section assumes a uniform velocity profile at the entrance of the channel. In many microfluidic applications, the channels have the manifold surfaces flush with the two opposing channel surfaces. The effect of such an arrangement was investigated numerically by Gamrat et al. (2004). They showed that the resulting apparent friction factors in the entrance region could be up to 50% lower than the theoretical predictions. 3.2.4. Fully developed and developing turbulent flow A number of correlations with comparable accuracies are available in literature for fully developed turbulent flow in smooth channels. The following equation by Blasius is used extensively: f = 0.0791 Re−0.25 (3.19) A more accurate equation was presented by Phillips (1987) to cover both the developing and fully developed flow regions. He presented the fanning friction factor for a circular tube in terms of the following equation: fapp = AReB (3.20) 96 Heat transfer and fluid flow in minichannels and microchannels where A = 0.09290 + 1.01612 x/Dh B = −0.26800 − (3.21) 0.32930 x/Dh (3.22) For rectangular channel geometries, Re is replaced with the laminar-equivalent Reynolds number (Jones, 1976) given by: Re* = um [(2/3) + (11/24)(1/c )(2 − 1/c )]Dh um Dle = µ µ (3.23) where Dle is the laminar-equivalent diameter given by the term in the square parenthesis in Eq. (3.23). 3.3. Total pressure drop in a microchannel heat exchanger 3.3.1. Friction factor An earnest interest in microchannel flows began with the pioneering work on direct chip cooling with water by Tuckerman and Pease (1981). Recently, a number of investigators Δp Microchannel Inlet plenum Ki Frictional losses Developing region Outlet plenum Ko Fig. 3.4. Schematic representation of the experiments employed by researchers for pressure drop measurements in microchannels, Steinke and Kandlikar (2005a). Chapter 3. Single-phase liquid flow in microchannels and minichannels 97 including Li et al. (2000), Celata et al. (2002) and Steinke and Kandlikar (2005a) critically evaluated the available literature and presented explanations for the large deviations from the classical theory reported by some of the researchers. A schematic representation of the pressure drop experiments conducted by the researchers is shown in Fig. 3.4. Since measuring the local pressure along the flow is difficult in microchannels, researchers have generally measured the pressure drop across the inlet and outlet manifolds. The resulting pressure drop measurement represents the combined effect of the losses in the bends, entrance and exit losses, developing region effects, and the core frictional losses. Thus, the measured pressure drop is the sum of these components (Phillips, 1987): 2 4fapp L um 2 p = (Ac /Ap ) (2K90 ) + (Kc + Ke ) + 2 Dh (3.24) where Ac and Ap are the total channel area and the total plenum cross-sectional area, K90 is the loss coefficient at the 90 degree bends, Kc and Ke represent the contraction and expansion loss coefficients due to area changes, and fapp includes the combined effects of frictional losses and the additional losses in the developing flow region. Phillips (1987) studied these losses and recommends K90 to be approximately 1.2. The contraction and exit losses can be read from Fig. 3.5 derived from Kays and London (1984) and Phillips (1990). (a) Kc Laminar 4(L/D)/Re

0.20 0.10 0.05 Kc Turbulent Re 3000 5000 10,000 Ke Turbulent Re 10,000 5000 3000 Ke Laminar 4(L/D)/Re 0.05 0.10 0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Area ratio, s 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Kc and Ke Kc and Ke 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Kc Laminar Re 2000 10,000 ∞ Ke Re 10,000 2000 Laminar 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (b) Area ratio, s Fig. 3.5. Contraction and expansion loss coefficients for flow between inlet and outlet manifolds and the microchannels, Kays and London (1984). 98 Heat transfer and fluid flow in minichannels and microchannels 4.0 Uncorrected Corrected C * 1.0 3.5 * C app 3.0 2.5 2.0 1.5 1.0 0.5 0 200 400 600 Reynolds number, Re 800 * Fig. 3.6. Variation of Capp = fapp,ex /fapp,th for a 200 µm square microchannel array after correcting for entrance and exit losses. Figure 3.5(a) is applicable to the low aspect ratio channels, c < 0.1 and Fig. 3.5(b) is for 0.1 ≤ c ≤ 1.0. Equation (3.24) can also be written in terms of the fully developed friction factor f and the pressure drop defect K(x): 2 um 4fL 2 p = (Ac /Ap ) (2K90 ) + (Kc + Ke ) + + K(x) 2 Dh (3.25) For L > Lh , K(x) is replaced by the Hagenbach’s factor K(∞) as discussed earlier in Section 3.2.3. Steinke and Kandlikar (2005a) conducted experiments on square silicon microchannels with 200 µm sides and 10 mm in length, and analyzed the data in detail. A parameter C * is introduced to represent the ratio of the experimental and theoretical apparent friction factors. Figure 3.6 shows the data before and after applying the corrections in C * = fapp,ex /fapp,th , where the subscripts ex and th refer to experimental and theoretical values respectively. The uncorrected data converges to within 30% of the theoretical predictions after all corrections are applied. It is suspected that the high errors are due to slight variations in the channel cross-sectional area over its length, different entrance conditions, and errors associated with interpolations during data reduction. Figure 3.7 shows the excellent agreement between the laminar flow theory and experimental results obtained by Judy et al. (2002) in 15–150 µm round and square microchannels made of fused silica and stainless steel with distilled water, methanol and isopropanol. Similar agreement is seen in Fig. 3.8 by Bucci et al. (2004) in 172, 290 and 520 µm diameter stainless steel circular tubes with water in the fully developed laminar flow region. Chapter 3. Single-phase liquid flow in microchannels and minichannels 99 70 60 50 D 21.7 m, L/D 2341 D 40.1 m, L/D 2532 f Re 40 0 200 D 30.5 m, L/D 2019 f Re 64 400 600 800 70 60 D 51.9 m, L /D 2192 D 97.9 m, L /D 2983 f Re 64 50 40 0 400 800 D 77.6 m, L/D 3926 D 148.6 m, L/D 2030 1200 Re 1600 2000 Fig. 3.7. Comparison between theory and experimental data of Judy et al. (2002) in 50–150 µm circular and square microchannels in the fully developed laminar region. (Note: f in this figure represents the Darcy friction factor, which is four times the Fanning friction factor used elsewhere in this text.) 1 f [] D 172 m D 290 m D 520 m 64/Re 0.1 100 1000 Re[] Fig. 3.8. Comparison between theory and experimental data of Bucci et al. (2004) in 171–520 µm diameter circular tubes in the fully developed laminar region. (Note: f in this figure represents the Darcy friction factor, which is four times the Fanning friction factor used elsewhere in this text.) Niklas and Favre-Marinet (2003) analyzed the flow of water in a network of triangular microchannels with Dh = 110 µm. The contributions due to various losses were carefully analyzed in both their numerical as well as experimental work. They concluded that the classical theory is applicable to modeling the flow through the entire system. 100 Heat transfer and fluid flow in minichannels and microchannels Another point that needs to be emphasized is the large errors that are associated with microscale experiments as pointed out by Judy et al. (2002). A detailed analysis was conducted by Steinke and Kandlikar (2005a) for rectangular microchannels using actual measured parameters. Their final expression for uncertainty in estimating the product f Re for rectangular channels is given by:

2 Uf Re U p 2 UQ 2 Uµ 2 U 2 UL + + + +3 = 2 f Re µ p L Q 2 2

1/2 Ua Ub Ua 2 Ub 2 +5 +5 +2 +2 (3.26) a b a+b a+b The details of the derivation are given by Steinke (2005). Note that the uncertainties in the height a and width b measurements have a major influence on the overall uncertainty, followed by the uncertainties in the density and volumetric flow rate measurements. The low flow Reynolds number data in Fig. 3.6 exhibits extremely large uncertainties due to the errors associated with the flow rate measurement. The large discrepancy reported in literature by some of the earlier investigators is the result of the uncertainties in channel dimensions and flow rate measurements, entrance and exit losses, and the developing region effects. Figure 3.9 shows a picture of a microchannel cross-section obtained with a scanning electron microscope by Steinke and Kandlikar (2005a). Note that the actual channel profile G01-008 a 200 m b 250 m s 100 m 93.9 m 193.6 m 244.2 m 33.9 m 54.3 m 85.1 Deg 20 m EHT 20.00 kV WD 6 mm l Probe 50 pA Mag 670 x Stage at T 90.0° Date: 1 Apr 2005 Time: 17:02:39 File Name G01-008_003.tif Fig. 3.9. Cross-sectional view of a rectangular microchannels etched in silicon showing distortion in channel geometry from ideal rectangular profile, Steinke and Kandlikar (2005a). Chapter 3. Single-phase liquid flow in microchannels and minichannels 101 deviates significantly from the intended rectangular profile. The side walls are undercut by about 20 µm and the corners are rounded. This illustrates the need to accurately measure the flow area and the for estimating the pressure drop characteristics of microchannels. Similar conclusions were reached by other investigators, for example, Li et al. (2000), Celata (2002), Judy et al. (2002), Baviere et al. (2004), and Tu and Hrnjak (2003). The increasing deviations from the theoretical values at higher Reynolds numbers in many data sets reported in literature is believed to be due to the increased length of the developing region. Baviere and Ayela (2004) measured the local pressures along the flow in a microchannel of height 7.5 µm. The range of Reynolds numbers considered was between 0.1 and 15. Their results were found to agree with the classical theory. They also identified the errors associated with the channel height measurement as being the largest source of uncertainty. Baviere et al. (2004) conducted experiments with bronze, altuglas, and silicon coated microchannels with channel heights between 7.1 and 300 µm and channel widths between 1 and 25 mm. Their results were also in excellent agreement with the classical theory. Other effects that may influence the flow in microchannels are (i) high viscous dissipation causing a change in the fluid viscosity at the wall, and (ii) electrokinetic effects, which are covered in Chapter 4. The effect of viscous dissipation is reviewed by Shen et al. (2004). Xu et al. (2000) modeled this effect and found that the velocity profile was modified due to the viscous dissipation. The resulting friction factors were predicted to be lower due to a reduction in the viscosity at higher liquid temperatures. The viscous effect was studied experimentally by Judy et al. (2002). Their pressure drop data correlated well with the theory when they used the average of the inlet and outlet fluid temperatures from their stainless steel microtubes, which were 15–100 µm in diameter. Koo and Kleinstreuer (2004) concluded that viscous dissipation effects increase rapidly with a decrease in the channel dimensions and hence should be considered along with the imposed heat sources at the channel walls. Figure 3.10 shows a plot of the friction factor versus Reynolds number using the experimental data points reported in literature as compiled by Steinke and Kandlikar (2005a). The lines represent the theoretical values for different channel aspect ratios. These data points were screened to eliminate the data that were obtained with large uncertainties and those which did not account for the entrance and exit losses. It is seen that the general trends are followed, though the errors are still quite large mainly due to the uncertainties associated with channel size measurement. 3.3.2. Laminar-to-turbulent transition The laminar-to-turbulent flow transition is another topic that was analyzed by a number of investigators. The laminar-to-turbulent transition in abrupt entrance rectangular ducts was found to occur at a transition Reynolds number of Ret = 2200 for c = 1 and at Ret = 2500 (Hartnett et al., 1962) for parallel plates with c = 0. For other aspect ratios, a linear interpolation between these two values is recommended. Some of the initial studies indicated an early transition to turbulent flow in microchannels. However, a number of recent studies showed that the laminar-to-turbulent transition 102 Heat transfer and fluid flow in minichannels and microchannels 1E05 1E04 Friction factor, f 1E03 1E02 1E01 1E00 1E01 1E02 1E03 1E04 1E03 f Re 24 f Re 16 f Re 14 Data 1E01 1E01 1E03 Reynolds number, Re 1E05 Fig. 3.10. Comparison of friction factors reported in literature with the theoretical values for different microchannel geometries. Entrance and exit losses and the developing region at the entrance were not accounted for in a number of data sets reviewed by Steinke and Kandlikar (2005a). remains unchanged. Figure 3.8 by Bucci et al. (2004) shows that the transition occurred around Ret = 2000 for circular microtubes 171–520 µm in diameter. The results of Baviere et al. (2004) also indicate that the laminar-to-turbulent transition in smooth microchannels is not influenced by the channel dimensions and occurs around 2300. Similar results were reported by a number of investigators, including Bucci et al. (2004), Schmitt and Kandlikar (2005), Kandlikar et al. (2005) for minichannels with Dh < 1 mm; and Li et al. (2000) for 80 µm ≤ Dh ≤ 166.3 µm. The transition from the laminar-to-turbulent region is influenced by the channel surface roughness. Further details of this effect are discussed in the next section on roughness. 3.4. Roughness effects 3.4.1. Roughness representation Darcy (1857) investigated the effects of surface roughness on the turbulent flow of water in rough pipes made of cast iron, lead, wrought iron, and asphalt-covered cast iron. The pipes were 12 to 500 mm in diameter and 100 m long. Fanning (1886) proposed a correlation for the pressure drop as a function of roughness. Mises (1914) is credited with introducing the term relative roughness, which was originally defined as the ratio of absolute roughness to the pipe radius (we now use pipe diameter to normalize roughness parameters, such as average roughness Ra ). Nikuradse (1933) presented a comprehensive review of the literature covering uniform roughness and roughness structures such as corrugations, and Chapter 3. Single-phase liquid flow in microchannels and minichannels Sm 1 Rp 1 Sm 2 Rp 2 103 Sm 3 Rp Main profile Mean line 3 Fp Floor profile Mean line Fig. 3.11. Maximum profile peak height (Rp ), Mean spacing of profile irregularities (RSm ), and Floor distance to mean line (Fp ). some of the available correlations relating friction factor to relative roughness. He also conducted a systematic study on friction factor by applying uniform diameter sand grain particles with Japanese lacquer to the inner pipe surface and measuring the pressure drop. He identified three regions which form the basis of the current Moody diagram that is used for friction factor estimation. The term relative roughness is used for the ratio /Dh where is the average roughness. Nikuradse used the diameter of the uniform sand grain particles to represent the roughness . The range of relative roughness in Nikuradse’s experiments on circular pipes was 0.001 ≤ /D ≤ 0.033. Furthermore, Nikuradse identified three ranges to describe his laws of resistances. In Range I, for low Reynolds numbers corresponding to laminar flow, the friction factor was independent of surface roughness and was given by the now established classical equation f = 16/Re for circular pipes. In Range II, he described the roughness effects on the friction factor in turbulent flow by f = 0.079/Re1/4 , and finally in Range III, the friction factor is independent of the Reynolds number but depends on the relative roughness, f = 0.25/[1.74 + 2 log(2/D)]2 . Note that the original equations by Nikuradse have been modified to express them in terms of the Fanning friction factor (which is one fourth of Darcy friction factor, f = 0.25 fDarcy ). Moody (1944) used the available data on the friction factor and developed the wellknown Moody diagram covering the relative roughness range of 0 ≤ /D ≤ 0.05. Following the earlier investigators, he used the tube root diameter of a circular pipe, similar to Nikuradse’s work, in expressing the relative roughness /D. A number of different parameters were studied for their suitability in representing the roughness effects. Kandlikar et al. (2005) investigated parameters based on various roughness characterization schemes and proposed a set of three parameters: Mean Profile Peak Height (Rp ), Mean Spacing of Profile Irregularities (RSm ), and the Floor Distance to Mean Line (Fp ), as shown in Fig. 3.11. Two of these parameters (Rp and RSm ) are defined in the ASME B46.1-2002 and the other (Fp ) is proposed by Kandlikar et al. (2005). l Average maximum profile peak height ( Rpm ): The distance between the average of the individual highest points of the profile (Rp,i ) and the mean line within the evaluation length. The mean line represents the conventional average roughness value (Ra ). 104 l Heat transfer and fluid flow in minichannels and microchannels Mean spacing of profile irregularities ( RSm ): Consists of the mean value of the spacing between profile irregularities within the evaluation length. The irregularities of interest are the peaks, so this is equivalent to the Pitch. 1 Sm i n n RSm = (3.27) i=1 l Floor distance to mean line (Fp ): Consists of the distance between the main profile mean line (determined by Ra ) and the floor profile mean line. The floor profile is the portion of the main profile that lies below the main profile mean line. The three parameters described above allow the characterization of the peak height, peak spacing and the distance from the floor to the mean line. These parameters will define the characteristics of the surface roughness that influence the location and shape of the fluid flow streamlines (as described by Webb et al., 1971 and Kandlikar et al., 2005) and, consequently, the size of the recirculation flow zones between roughness elements. From the above parameters, the equivalent roughness can be estimated by the following relationship: = Rpm + Fp (3.28) For the sand grain roughness employed by Nikuradse, the new definition of roughness yields the same value, and hence no correction is needed for any of the friction factor correlations and charts. The use of mean spacing between profile irregularities is expected to represent the structured roughness surfaces. Such surfaces may be designed in the future to obtain specific pressure drop and heat transfer performance characteristics. 3.4.2. Roughness effect on friction factor For microchannels, the relative roughness values are expected to be higher than the limit of 0.05 used in the Moody diagram. Kandlikar et al. (2005) considered the effect of crosssectional area reduction due to protruding roughness elements and recommended using the constricted flow area in calculating the friction factor. Using a constricted diameter Dcf = D − 2, a modified Moody diagram was presented as shown in Fig. 3.12. In the turbulent region, it was found that such a representation yielded a constant value of friction factor above /Dcf > 0.03. In the turbulent fully rough region, 0.03 ≤ /Dcf ≤ 0.05, the friction factor based on the constricted flow diameter is given by: f Darcy,cf = 0.042 (3.29a) In terms of the Fanning friction factor, we get: fcf = fDarcy,cf /4 = 0.042/4 = 0.0105 (3.29b) Since experimental data is not available beyond /Dcf > 0.05, using Eq. (3.29) for higher relative roughness values than 0.05 is not recommended. Note that the friction factor and Chapter 3. Single-phase liquid flow in microchannels and minichannels Transition zone 0.10 0.09 0.08 0.07 0.06 Laminar flow fcf = 64/Recf 0.05 0.04 Plateau 0.030 0.020 0.014 0.010 0.008 0.006 0.004 0.03 0.002 0.02 0.001 0.0005 Relative roughness, /Dcf Friction factor based on constricted flow, fDarcy,cf 105 0.00025 Smooth pipe 0.01 1E02 1E03 1E04 1E05 1E06 1E07 1E08 Reynolds number based upon constricted flow, Recf Fig. 3.12. Darcy friction factor plot based on a constricted flow diameter, Kandlikar et al. (2005). the geometrical and flow parameters are based on the constricted flow diameter as given by the following equations: Dcf = D − 2 p = 2 2fcfum,cf L Dh,cf um,cf = m/A ˙ cf Recf = um,cf Dh,cf µ (3.30) (3.31) (3.32) (3.33) In the fully developed laminar flow region, the constricted friction factor is given by the following equation, Kandlikar et al. (2005): Laminar region, 0 ≤ /Dh,cf ≤ 0.15; fcf = Po Recf (3.34) where the Poiseuille number Po is given by either Eq. (3.9) or (3.10) depending on the channel geometry. In the turbulent region, use the modified Miller equation as recommended by Kandlikar et al. (2005). 106 Heat transfer and fluid flow in minichannels and microchannels Fully developed turbulent region, 0 ≤ /Dh,cf < 0.03; −2 /Dh,cf 5.74 0.25 log10 + 0.9 3.7 Recf fcf = 4 Fully developed turbulent region, 0.03 ≤ /Dh,cf ≤ 0.05 fcf = 0.0105 (3.35) (3.36) No recommendation can be made for the turbulent region with /Dh,cf > 0.05 because of the lack of detailed experimental data in literature for this region. Further experimental work in this area is strongly recommended. For rectangular channels with all sides having a roughness , the mean flow velocity is calculated using the constricted flow area and the constricted channel dimensions of acf = a − 2 and bcf = b − 2. The hydraulic diameter is calculated using the constricted channel dimensions. Schmitt and Kandlikar (2005) conducted experiments to study the effect of a large relative roughness of /Dh,cf up to 0.15 in smooth and artificially roughened rectangular minichannels. They introduced sawtooth roughness elements in a 10.3 mm wide and 100 mm long test channel. Cross-sectional views of the smooth channels and channels with sawtooth roughness elements are shown in Fig. 3.13. The channel gap bcf was varied to produce values of /Dh,cf from 0.03 to 0.15. Differential pressure taps for measuring b (a) Smooth channel b bcf 72.9 m 500 m (b) Aligned sawtooth b bcf 72.9 m 500 m 250 m (c) Offset sawtooth Fig. 3.13. Roughness elements used by Schmitt and Kandlikar (2005) and Kandlikar et al. (2005). Chapter 3. Single-phase liquid flow in microchannels and minichannels 107 the local pressure drops were located at several locations along the flow length in the fully developed region. The smooth channel results reported by Kandlikar et al. (2005) closely follow the classical laminar flow theory for friction factor. The experimental data from the two sawtooth structures are represented in Fig. 3.14 for flow of water with b = 500 µm, resulting in a Dh of 953 µm. This plot uses the height b in calculating Dh and Re. The dashed line represents the f Re = Constant line corresponding to the channel aspect ratio c = a/b. It is seen that the agreement in the low Reynolds number region corresponding to laminar flow is considerably off. Figure 3.15 shows the same data used in Fig. 3.14, but plotted with the constricted flow areas and using the constricted flow parameters given by Eqs. (3.30)–(3.33). The f 1.00 0.10 0.01 100 1000 Re Lam., theory Aligned sawtooth 10,000 Turb., theory-extended Offset sawtooth Fig. 3.14. Fully developed friction factor versus Reynolds number, both based on hydraulic diameter for water flow. Dh = 953 µm, b = 500 µm, bcf = 354 µm, w = 10.03 mm, /Dh = 0.0735, Kandlikar et al. (2005). 1.000 fcf 0.100 0.010 0.001 100 Lam., theory Constricted aligned sawtooth 1000 Recf 10,000 Turb., theory-extended Constricted offset sawtooth Fig. 3.15. Fully developed friction factor versus Reynolds number, both based on constricted flow hydraulic diameter; water flow. Dh,cf = 684 µm, b = 500 µm, bcf = 354 µm, w = 10.03 mm, /Dh,cf = 0.1108, Kandlikar et al. (2005). 108 Heat transfer and fluid flow in minichannels and microchannels results shown in Fig. 3.15 in the laminar region show good agreement with the accepted laminar flow theory ( fcf Recf = Constant). Similar results were obtained for different gap sizes (yielding different relative roughness values) for both the offset sawtooth and aligned sawtooth geometries. 3.4.3. Roughness effect on the laminar-to-turbulent flow transition The transition from laminar-to-turbulent flow has been reported to occur in microchannels at Reynolds numbers considerably below 2300. In many experiments, the transition has been mistakenly identified to occur early based on experimental data uncorrected for the developing length (Steinke and Kandlikar, 2005a). Kandlikar et al. (2003) conducted experiments with stainless steel tubes and noted that early transition occurred for a 0.62 mm ID stainless steel tube with a surface roughness /Dh of 0.355%. Schmitt and Kandlikar (2005) conducted careful experiments with plain and sawtooth roughened channels with air and water. Their results for smooth rectangular channels showed a transition Reynolds number between 2000 and 2300, but for increasing relative roughness values there was a decreasing transition Reynolds numbers as seen in Figs. 3.14 and 3.15. Figure 3.16 shows the transition Reynolds number as a function of the relative roughness (based on the constricted hydraulic diameter). The following equations are used to describe the roughness effects based on their experimental data. Laminarto-turbulent transition criteria: For 0 < /Dh,cf ≤ 0.08: Ret,cf = 2300 − 18,750(/Dh,cf ) (3.37) For 0.08 < /Dh,cf ≤ 0.15: Ret,cf = 800 − 3270(/Dh,cf − 0.08) (3.38) 2500 Air Water 2000 Recf 1500 1000 500 0 0 0.05 0.1 0.15 /Dh,cf Fig. 3.16. Transition Reynolds number variation with relative roughness based on constricted flow diameter, Equations (3.37) and (3.38) plotted along with data from Schmitt and Kandlikar (2005). Chapter 3. Single-phase liquid flow in microchannels and minichannels 109 Friction factor in the transition region for rough tubes with /Dh,cf ≤ 0.05 at a given Recf can be obtained by a linear interpolation of (i) the laminar friction factor obtained from Eq. (3.34) at the transition Reynolds number, and (ii) the turbulent friction factor at Recf = 2300 given by Eq. (3.35) or (3.36) depending on the value of /Dh,cf . For /Dh,cf > 0.05 in the transition region, additional experimental data is needed before any recommendations can be made. 3.4.4. Developing flow in rough tubes Developing flow in rough tubes has not been explored in literature. As a preliminary estimate, the methods described in Section 3.2.3 for smooth tubes are recommended by introducing the constricted flow diameter in calculating the flow velocity, Reynolds number and friction factor. Further research in this area is needed. 3.5. Heat transfer in microchannels 3.5.1. Fully developed laminar flow The Nusselt number in fully developed laminar flow is expected to be constant as predicted by the classical theory. However, there are a number of investigations reported in literature that show a trend increasing with Reynolds number in this range. This results from the experimental uncertainties as discussed in Section 3.5.3. The Nusselt number in the fully developed laminar flow is constant and depends on the channel geometry and the wall heat transfer boundary condition. Table 3.1 presents the Nusselt numbers for commonly used geometries under constant heat flux and constant wall temperature boundary conditions. For a rectangular channel, the Nusselt number depends on the channel aspect ratio c = a/b, and the wall boundary conditions. Three boundary conditions are identified in literature and the Nusselt number for each one is given below. Constant wall temperature, T-boundary condition: NuT = 7.541(1 − 2.610c + 4.9702c − 5.1193c + 2.7024c − 0.5485c ) (3.39) Constant circumferential wall temperature, uniform axial heat flux, H1 boundary condition: NuH1 = 8.235(1 − 2.0421c + 3.08532c − 2.47653c + 1.05784c − 0.18615c ) (3.40) Constant wall heat flux, both circumferentially and axially: NuH2 = 8.235(1 − 10.6044c + 61.17552c − 155.18033c + 176.92034c − 72.92365c ) (3.41) 110 Heat transfer and fluid flow in minichannels and microchannels Table 3.3 Fully developed laminar flow Nusselt numbers. c = a/b Nufd,3 Nufd,4 0 0.10 0.20 0.30 0.40 0.50 0.70 1.00 1.43 2.00 2.50 3.33 5.00 10.00 >10.00 8.235 6.939 6.072 5.393 4.885 4.505 3.991 3.556 3.195 3.146 3.169 3.306 3.636 4.252 5.385 8.235 6.700 5.704 4.969 4.457 4.111 3.740 3.599 3.740 4.111 4.457 4.969 5.704 6.700 8.235 a – unheated side in three-side heated case. For intermediate values, use the curve-fit equations provided in Appendix A at the end of the chapter. In reality, all practical situations fall somewhere in the middle of these three boundary conditions. This becomes an especially important issue in the case of microchannels because of the difficulty in identifying a correct boundary condition with discretly spaced heat sources, and two-dimensional effects in the base and the fins. The heating in microchannel geometries generally comes from three sides, as a cover of glass or some other material is bonded on top of the microchannels to form the flow passages. The fully developed Nusselt numbers for both three- and four-side heated ducts have been compiled from various sources, including Wibulswas (1966), and Phillips (1987) and are given in Table 3.3. Note that the side with dimension a is not heated, and the channel aspect ratio is defined as c = a/b. 3.5.2. Thermally developing flow The thermal entry length is expressed by the following form for flow in ducts: Lt = c Re Pr Dh (3.42) For circular channels, the leading constant c in Eq. (3.42) is found to be 0.05, while for rectangular channels, the plots presented by Phillips (1987) suggest c = 0.1. The local heat transfer in the developing region of a circular tube is given by the following equations (Shah and London, 1978): Nux = 4.363 + 8.68(103 x* )−0.506 e−41x * (3.43) Chapter 3. Single-phase liquid flow in microchannels and minichannels 111 Table 3.4 Thermal entry region Nusselt numbers. Nux,4 x* c ≤ 0.1* c = 0.25 c = 0.333 c = 0.5 c = 1.0 c ≥ 10** 0.0001 0.0025 0.005 0.00556 0.00625 0.00714 0.00833 0.01 0.0125 0.0167 0.025 0.033 0.05 0.1 1 31.4 11.9 10 9.8 9.5 9.3 9.1 8.8 8.6 8.5 8.4 8.3 8.25 8.24 8.23 26.7 10.4 8.44 8.18 7.92 7.63 7.32 7 6.63 6.26 5.87 5.77 5.62 5.45 5.35 27.0 9.9 8.02 7.76 7.5 7.22 6.92 6.57 6.21 5.82 5.39 5.17 5.00 4.85 4.77 23.7 9.2 7.46 7.23 6.96 6.68 6.37 6.05 5.7 5.28 4.84 4.61 4.38 4.22 4.11 25.2 8.9 7.1 6.86 6.6 6.32 6.02 5.69 5.33 4.91 4.45 4.18 3.91 3.71 3.6 31.6 11.2 9.0 8.8 8.5 8.2 7.9 7.49 7.2 6.7 6.2 5.9 5.55 5.4 5.38 x* = x/(Re PrDh ); * – parallel plates, both sides heated; ** – parallel plates, one side heated. For intermediate values, use the curve-fit equations provided in Appendix A. x* = x/Dh Re Pr (3.44) For rectangular channels with the four-side heating configuration, Nusselt numbers in the thermally developing region presented in Table 3.4 are derived from Phillips’ (1987) work. For the three-side heating configuration the following scheme is suggested by Phillips (1990). Three-side heating, c ≤ 0.1 and c ≥ 10, use four-side heating table without any modification. Three-side heating, 0.1 < c < 10: Nux,3 (x * , c ) = Nux,4 (x * , c ) * , ) Nufd,3 (x * = xfd c * , ) * Nufd,4 (x = xfd c (3.45) The subscripts x, 3 and x, 4 refer to the location at a distance x in the heated length for the three-sided and four-sided heating cases respectively. The Nusselt numbers in the fully developed region for both heating configurations are obtained from Table 3.3, and in the developing region Nusselt numbers for the four-sided heating are obtained from Table 3.4. The use of fully developed hydrodynamic conditions in heat transfer analysis is reasonable for water. Garimella and Singhal (2004) noted that assuming fully developed hydrodynamic conditions and thermally developing conditions resulted in a satisfactory agreement with their data for microchannels. 112 Heat transfer and fluid flow in minichannels and microchannels 3.5.3. Agreement between theory and available experimental data on laminar flow heat transfer Laminar flow heat transfer in microchannels has been studied by a number of researchers. Agreement with the classical laminar flow theory is expected to hold, but the reported data show significant scatter due to difficulties encountered in making accurate local heat flux and temperature measurements. Steinke and Kandlikar (2005b) reviewed the available data and presented a comprehensive table showing the range of parameters employed. The laminar flow heat transfer in the fully developed region is expected to be constant, but the data taken from literature show a generally linear increase in Nusselt number with flow Reynolds number as seen in Fig. 3.17. The main reasons for this discrepancy have been attributed to the following factors: (i) Entrance region effects: The researchers have used the fully developed theoretical values for Nusselt numbers in their comparison with the experimental data. Due to the relatively short lengths employed in microchannels, the influence of the entrance region cannot be neglected. The entrance region effects become more significant at higher Reynolds numbers, in part explaining the increasing Nusselt number trend with Reynolds number as seen in Fig. 3.17. (ii) Uncertainties in experimental measurements: The uncertainties in heat transfer measurements have been analyzed by a number of investigators, including Judy et al. (2002) and Steinke and Kandlikar (2005b). The following equation for the uncertainty in the Nusselt number calculation from the uncertainties in the experimental measurements is presented by Steinke and Kandlikar for a rectangular channel of base 1E04 Nusselt number, Nu 1E03 1E02 1E01 1E00 1E01 1E00 1E01 1E02 1E03 1E04 1E05 Reynolds number, Re Fig. 3.17. Selected experimental data for single-phase liquid flow in microchannels and minichannels, Dh = 50 µm to 600 µm. Chapter 3. Single-phase liquid flow in microchannels and minichannels 113 width a and height b:

2

U kf 2 UI UV 2 U Ts 2 U Ti 2 + + +4 +2 UNu = Nu · kf I V Ts Ti 1/2 2 2 2 U f 2 UTo 2 UL Ua Ub +2 +3 +4 +5 +2 To L a b f (3.46) where I – current, V – voltage, Ts – surface temperature, Ti – fluid inlet temperature, To – fluid outlet temperature, L – flow length, a and b – cross-section dimensions, and f – fin efficiency. The uncertainties in the measurement of the surface temperature and flow channel dimensions play a critical role in the overall uncertainty. The importance of accurate geometrical measurements was emphasized in the friction factor estimation as well. Accurate surface temperature measurement poses a significant challenge due to the small dimensions of the test section. Use of silicon chips with integrated circuits to measure the temperatures is recommended. Another factor that makes the temperature measurements critical in the overall uncertainty is the small temperature difference between the surface and the fluid at the outlet. Since the heat transfer coefficients are very high in microchannels, and the flow is relatively low (low Reynolds number), the outlet temperature in many experiments approaches the surface temperature. Proper experiments need to be designed to account for this effect. (iii) Ambiguity in the determination of the thermal boundary condition: Several experiments reported in literature were conducted with microchannels or minichannels fabricated on copper or silicon substrates. The actual boundary conditions for these test sections are difficult to ascertain as they fall in between the constant temperature and constant heat flux boundary conditions. Furthermore, in many cases the heating is three-sided with the side walls acting as fins. The fin efficiency effects also alter the heat flux and temperature distributions. A clear comparison is only possible after the conjugate heat transfer effects are incorporated into a detailed numerical simulation of the test section. 3.5.4. Roughness effects on laminar flow heat transfer Kandlikar et al. (2003) studied the heat transfer and pressure drop of laminar flow in smooth and rough stainless steel tubes of 1.067 and 0.62 mm ID. The surface roughness of the inner tube wall was changed by treating it with two different acid mixtures. The surface roughness actually went down after the acid treatment as the protruding peaks in the surface profile were smoothed out. The effect of changes in the relative roughness on pressure drop was minimal, but the heat transfer in the entry region showed a distinct dependence on roughness. Figure 3.18 shows the local Nusselt number at a location 52 mm from the start of the heated length. The flow was hydrodynamically fully developed prior to entering the heated section. Heat transfer and fluid flow in minichannels and microchannels Nu 114 20 18 16 14 12 10 8 6 4 2 0 Nu(Twmid, e/d 0.00355) Nu(Twmid, e/d 0.0029) Nu(Twmid, e/d 0.00161) 0 500 1000 1500 Re 2000 2500 3000 Fig. 3.18. The effect of roughness in the entrance region (x = 52 mm) of a 0.62 mm inner diameter tube with a fully developed velocity profile and a developing temperature profile, Kandlikar et al. (2003). It is seen from Fig. 3.18 that the Nusselt number increases even for a modest increase in the relative roughness from 0.161% to 0.355%. Also note that the location where the heat transfer measurements were taken was preceded by a tube length longer than the hydraulic entry length, which was 380 mm at Re = 2300. A number of researchers did not correctly identify the developing region and have mistakenly reported the increasing trend in the Nu versus Re plot as being in the fully developed region. It is not possible to present any equations for heat transfer enhancement with roughness because of the lack of systematic data covering the entire region. Further research in this area is strongly recommended. With structured roughness surfaces, the enhancement results from the combined effects of area enhancement and that due to the flow modification caused by the roughness elements. 3.5.5. Heat transfer in the transition and turbulent flow regions A detailed discussion was presented earlier on the critical Reynolds number for the laminarto-turbulent transition. For smooth channels, the well-established criterion of Rec = 2300 is expected to hold. The effect of roughness is described by Eqs. (3.32) and (3.33). Since the transition region is encountered in many minichannel and microchannel heat exchangers, there is a need for generating accurate experimental data for both smooth and rough tubes in this region. Phillips (1990) suggests using the following equations in the developing turbulent region. For larger values of x, the influence of the term [1 + (Dh /x)2/3 ] reduces asymptotically to 1: For 0.5 < Pr < 1.5: Nu = 0.0214[1.0 + (Dh /x)2/3 ] [Re0.8 − 100]Pr 0.4 (3.47) Nu = 0.012[1.0 + (Dh /x)2/3 ] [Re0.87 − 280]Pr 0.4 (3.48) For 1.5 < Pr < 500: Further validation of these equations in microchannels is warranted. Adams et al. (1997) conducted experimental work in the turbulent region with flow of water in 0.76 and Chapter 3. Single-phase liquid flow in microchannels and minichannels 115 0.109 mm diameter circular channels. Based on their data, they proposed the following equation that also matches the data by Yu et al. (1995) within ±18.6%: Nu = NuGn (1 + F) (3.49) where NuGn = ( f /8)(Re − 1000)Pr 1 + 12.7( f /8)1/2 (Pr 2/3 − 1) (3.50) f = (1.82 log(Re) − 1.64)−2 (3.51) F = CRe(1 − (D/Do )2 ) (3.52) NuGn represents the Nusselt number predicted by Gnielinski (1976) correlation. The least-squares fit to all the data sets studied byAdams et al. (1997) resulted in C = 7.6 × 10−5 and Do = 1.164 mm. Heat transfer coefficients in microchannels are very high due to their small hydraulic diameters. The high pressure gradients have led researchers to employ low flow rates. However, with the reduced flow rate, the ability of the fluid stream to carry the heat away for a given temperature rise becomes limited. In order to improve the overall cooling performance, the following two options are available. (a) Reduce the flow length of the channels. (b) Increase the liquid flow rate. As a result, employing multiple streams with short paths in a microchannel heat exchanger is recommended, similar to a split flow arrangement providing two streams as will be discussed in Section 3.7. The reduced flow length will then enable the designer to employ higher flow rates under a given pressure drop limit. This scheme offers several advantages over a single-pass arrangement where the fluid traverses the entire length of the heat exchanger: (i) The reduced flow length reduces the pressure drop: The short flow length effectively reduces the overall pressure drop. (ii) Larger developing region: The multiple inlets result in a larger channel area being under developing conditions where the heat transfer is higher. (iii) Higher flow velocities: Some of the pressure drop reduction could be used to increase the flow velocity of individual streams. The possibility of employing turbulent flow should also be explored as the heat transfer coefficient is higher in this region. 3.5.6. Variable property effects The property ratio method is usually recommended in accounting for the property variations due to temperature changes in the heat exchanger flow passages. The following equations are recommended for liquids: f /fcp = [µw /µb ]M (3.53) 116 Heat transfer and fluid flow in minichannels and microchannels Nu/Nucp = [µw /µb ]N (3.54) where the subscript cp refers to the constant property solution obtained from appropriate equations or correlations. For laminar flow, M = 0.58 and N = −0.14, and for turbulent flow, M = 0.25 and N = −0.11 (Kays and London, 1984). 3.6. Microchannel and minichannel geometry optimization The first practical implementation of microchannels in silicon devices was demonstrated by Tuckerman and Pease (1981). They were able to dissipate 7.9 MW/m2 with a maximum substrate temperature rise of 71 C and a pressure drop of 186 kPa. The allowable temperature differences and the pressure drops for cooling today’s microprocessor chips have become significantly smaller, and many investigators have been working towards optimization of the channel geometrical configuration. The application of microchannels to electronics cooling imposes severe design constraints on the system design. For a given heat dissipation rate, the flow rate, pressure drop, fluid temperature rise, and fluid inlet to surface temperature difference requirements necessitate optimization of the channel geometry. A number of investigators studied the geometrical optimization of microchannel heat exchangers, For example, Phillips (1987), Harpole and Eninger (1991), Knight et al. (1992), Ryu et al. (2002), Bergles et al. (2003), and Kandlikar and Upadhye (2005). The results found by Kandlikar and Upadhye are presented below. Kandlikar and Upadhye (2005) considered a microchannel system as shown in Fig. 3.19. For a chip of width W = 10 mm and length L = 10 mm, they presented an analysis scheme for heat transfer and pressure drop by incorporating the entrance region effects. The number of channels was used as a parameter in developing the optimization scheme. The maximum chip temperature was set at 360 K while the fluid inlet temperature was set equal to 300 K. The channel depth was assumed to be 200 µm. Figures 3.20 and 3.21 show the parametric plots resulting from the optimization program. The fin spacing ratio, defined as = s/a, is plotted as a function of the number of channels on a 10 mm wide chip with pressure drop, water flow rate and fin thickness as parameters. A lower limiting value of approximately 40 µm was considered achievable for Cover plate b a s L W Fig. 3.19. Microchannel geometry used in microprocessor heat sink channel size optimization, Kandlikar and Upadhye (2005). Chapter 3. Single-phase liquid flow in microchannels and minichannels 117 fin thickness using silicon microfabrication techniques. For a given pressure drop limit, the optimal number of channels and other channel geometrical parameters can be determined from this plot. Such plots could be generated for other system configurations as well. The effect of introducing offset strip-fins in the microchannel flow passages and a split flow arrangement, as shown in Figs. 3.22 and 3.23 respectively, was also analyzed by 1 80 55 0.8 45 0.7 80 40 40 50 35 60 70 35 0.6 60 70 45 Fin spacing ratio 45 50 0.9 40 30 40 50 0.5 25 40 30 35 30 0.4 0.3 25 30 0.2 50 60 70 80 Number of channels 90 100 Fig. 3.20. Contour plot of fin spacing ratio versus the number of channels with pressure drop across them in kPa (dash-dot lines) and fin thickness in µm (solid lines) as parameters for water flow in plain rectangular microchannels in a single-pass arrangement at a heat flux of 3 MW/m2 , Kandlikar and Upadhye (2005). 60 5 70 5 1.1 50 5 70 1.7 0.7 60 2 2.5 1.5 0.8 5 1.2 0.6 40 50 1.5 60 2 0.5 2.5 Fin spacing ratio 50 1.2 0.9 3 2 1. 5 0.3 0.2 50 30 5 40 1.7 0.4 0 80 1.7 1. 1 30 60 70 80 Number of channels 90 100 Fig. 3.21. Contour plot of fin spacing ratio versus the number of channels with water flow rate in 10−3 kg/s (dash-dot lines) and fin thickness in µm (solid lines) as parameters for water flow in plain rectangular microchannels in a single-pass arrangement at a heat flux of 3 MW/m2 , Kandlikar and Upadhye (2005). 118 Heat transfer and fluid flow in minichannels and microchannels Fins with length l Fig. 3.22. Offset strip-fins shown in the top view, with individual fin length l along the flow length, Kandlikar and Upadhye (2005). (a) Single-pass arrangement (b) Split-flow arrangement Fig. 3.23. Schematics of single-pass and split-flow arrangements for fluid flow through microchannels, Kandlikar and Upadhye (2005). Kandlikar and Upadhye (2005). The resulting pressure drop versus the dissipated heat flux is shown in Fig. 3.24. It can be seen that the pressure drop is reduced considerably in the split flow arrangement. 3.7. Enhanced microchannels The use of enhanced microchannels was proposed by Kishimoto and Sasaki (1987). The need for higher heat transfer coefficients than those attainable with plain microchannels was identified by Kandlikar and Grande (2004), and some specific enhancement geometries were suggested by Steinke and Kandlikar (2004). Kandlikar and Upadhye (2005) analyzed the enhanced offset strip-fin geometry and the results were presented in the previous section. Colgan et al. (2005) presented detailed experimental results comparing various offset fin geometries. A three-dimensional rendition is shown in Fig. 3.25. There are several inlet and outlet manifolds, with short flow lengths through the enhanced structures. The apparent friction factors and Nusselt numbers for plain and several enhanced configurations are shown in Figs. 3.26 and 3.27. A plot of fapp versus Stanton number is shown in Fig. 3.28. The relationship deviates somewhat from a linear behavior because the manifold effects are included in the data. Chapter 3. Single-phase liquid flow in microchannels and minichannels 119 50 Pressure drop, p (kPa) Enhanced microchannels 40 Single-pass arrangement Split-flow arrangement 30 20 10 100 150 200 250 Heat load, Q (W) 300 350 Fig. 3.24. Comparison of pressure drops for the enhanced microchannels with offset strip-fins (l = 0.5 mm) in single-pass and split-flow arrangements on a 10 mm × 10 mm chip, Kandlikar and Upadhye (2005). Fig. 3.25. Three-dimensional rendition of the IBM enhanced silicon chip with short multiple fluid streams. (Colgan et al., 2005). Microfabrication technology opens up a whole new set of possibilities for incorporating enhancement structures derived from past experience in compact heat exchanger development. Different manufacturing constraints enter into play for microchannels on a copper substrate as compared to the silicon microfabrication technology. Novel developments are expected to arise as we continue forward in integrating the microfabrication technology using creative ways to achieve high-performance heat transfer devices. Steinke and Kandlikar (2004) present a good overview of singlephase enhancement techniques and illustrate various enhancement configurations that may be considered especially with silicon microfabrication technology. 120 Heat transfer and fluid flow in minichannels and microchannels 150 Continuous 60 m pitch (2) Stag., 100 m pitch, wide chan. and no gap (5) Stag., 100 m pitch, wide chan. and 40 m gap (3) Stag., 100 m pitch, narrow chan. and no gap (2) Stag., 100 m pitch, narrow chan. and 40 m gap (3) Model, y 6.3x 32.3 100 90 * Re f app 80 70 60 50 40 30 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 9 10 Re*(Dh /L) Fig. 3.26. Apparent friction factor comparison of a plain and five enhanced microchannel configurations, Colgan et al. (2005). 30 Nu 20 10 9 8 7 6 Continuous 75 m pitch Continuous 60 m pitch (2) Stag., 100 m pitch, wide chan. and no gap (5) 5 Stag., 100 m pitch, wide chan. and 40 m gap (3) Stag., 100 m pitch, narrow chan. and no gap (2) Stag., 100 m pitch, narrow chan. and 40 m gap (3) laminar flow, theory (aspect ratio 3) Model, y 2.97(x^0.50) 4 3 2.5 4 5 6 7 8 10 20 Re Pr Dh/L 30 40 50 60 70 80 Fig. 3.27. Nusselt number comparison of a plain and five enhanced microchannel configurations, Colgan et al. (2005). Chapter 3. Single-phase liquid flow in microchannels and minichannels 121 0.09 0.08 0.07 0.06 model => St 0.08 fapp St 0.05 0.04 0.03 0.02 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 fapp 1.5 Fig. 3.28. Stanton number and friction coefficient relationship for microchannels with different fin geometries, Colgan et al. (2005). 3.8. Solved examples Example 3.1 Microchannels are directly etched into silicon in order to dissipate 100 Watts from a computer chip over an active surface area of 10 mm × 10 mm. The geometry may be assumed similar to Fig. 3.19. Each of the parallel microchannels has a width a = 50 µm, depth b = 350 µm, and a spacing s = 40 µm. The silicon thermal conductivity may be assumed to be k = 180 W/m-K. Assume a uniform heat load over the chip base surface and a maximum water inlet temperature of 35 C. Assume one-dimensional steady state conduction in the chip substrate. (i) Calculate the number of flow channels available for cooling. (ii) Assuming the temperature rise of the water to be limited to 10 C, calculate the required mass flow rate of the water. (iii) Calculate the flow Reynolds number using the mean water temperature for fluid properties. (iv) Check whether the fully developed flow assumption is valid. (v) Calculate the average heat transfer coefficient in the channels. (vi) Calculate fin efficiency. (vii) Assuming the heat transfer coefficient to be uniform over the microchannel surface, calculate the surface temperature at the base of the fin at the fluid inlet and fluid outlet sections. (viii) Calculate the pressure drop in the core of the microchannel. (ix) If two large reservoirs are used as the inlet and outlet manifolds, calculate the total pressure drop between the inlet and the outlet manifolds. 122 Heat transfer and fluid flow in minichannels and microchannels Solution Properties: Water at 40 C (Incropera and DeWitt (2002)) = 991.8 kg/m3 ; µ = 655 × 10−6 N-s/m2 ; cp = 4179 J/kg-K; k = 0.632 W/m-K; Pr = 4.33 (i) Number of flow channels (Answer: 111 channels) Assumptions (1) The first and the last channels are half the channel width from the chip edge. The remaining width left for channels is: 10 mm – 50 µm = 0.00995 m. The pitch is: 50 µm + 40 µm = 90 µm = 90 × 10−6 m. Assumption 1 requires that the pitch repeat up to the last channel, but not include the last channel. Therefore, the number of channels is found by: (0.00995 m – 50 × 10−6 m)/(90 × 10−6 m) = 110 channels. But recall that that the above calculation was for a repeating pitch up to but not including the last channel so the number of channels available for cooling is n = 111. (ii) Required flow rate of water (Answer: 0.00239 kg/s or 143.6 ml/min, total) The inlet water temperature is given as 35 C, and the stated assumption is that the water temperature rise is limited to 10 C.The mass flow rate can be calculated using the properties ˙ p T where q is of water at the average temperature of 40 C by using the equation q = mc the power dissipated from the chip, cp is the specific heat of water at constant pressure and T is the temperature change of the water. The total mass flow rate is m ˙t = q = 100 W/(4179 J/kg-K)(10 K) = 0.00239 kg/s = 143.6 ml/min, cp T and the mass flow rate in one channel is given by m ˙c= q = 21.6 × 10−6 kg/s = 1.3 ml/min ncp T (iii) Flow Reynolds number (Answer: Re = 164.7) The flow Reynolds number using the mean water temperature (40 C) for fluid properties can be found using the hydraulic diameter Dh given by Eq. (3.7) Dh = 2ab = 2(50 × 10−6 m)(350 × 10−6 m)/(50 × 10−6 m + 350 × 10−6 m) (a + b) = 87.5 × 10−6 m in the Reynolds number equation given by Re = um Ac Dh m ˙ c Dh um Dh = = µ Ac µ Ac µ = (21.6 × 10−6 kg/s)(87.5 × 10−6 m) = 164.7 (17.5 × 10−9 m2 )(655 × 10−6 N-s/m2 ) This low Reynolds number clearly indicates laminar flow regime. Chapter 3. Single-phase liquid flow in microchannels and minichannels 123 (iv) Fully developed assumption (Answer: fully developed assumption is valid – hydrodynamic entrance length = 0.72 mm and thermal entrance length = 6.25 mm) The fully developed flow assumption is valid if the thermal and hydrodynamic entrance lengths are less than the channel length. For laminar flow, the hydrodynamic entrance length is given by Eq. (3.11) as Lh = 0.05 Re Dh = 0.05(164.7)(87.5 × 10−6 m) = 720 × 10−6 m = 0.72 mm, and the thermal entrance length is given by Eq. (3.42) as Lt = 0.1Re Pr Dh = 0.1(164.7)(4.33)(87.5 × 10−6 m) = 6.24 × 10−3 m = 6.24 mm Since the thermal and hydrodynamic entrance lengths are less than the channel length, the fully developed flow assumption is valid. (v) Average heat transfer coefficient (Answer: 47.4 × 103 W/m2 -K) Assumptions (1) Constant heat flux boundary condition. (2) Three-sided heating condition. The Nusselt number using the thermal conductivity of water at the average temperature (40 C) is given by Nu = h¯ Dh /k. A value for the fully developed Nusselt number can be obtained from Table 3.3 using the aspect ratio of the channel c = a/b = 1/7. By linear interpolation, the Nusselt number is Nufd,3 = 6.567. The average heat transfer coefficient is kNu h¯ = = (0.632 W/m-K)(6.567)/87.5 × 10−6 m = 47.4 × 103 W/m2 -K Dh (vi) Fin efficiency (Answer: 67%) Assumptions (1) Adiabatic tip condition. For an adiabatic tip, the fin efficiency equation is given by f = tanh(mb)/mb where b is the fin height 1/2 ¯ (350 × 10−6 m) and m is defined as m = (hP/kA c) In the above equation, k is thermal conductivity of the fin material (given as k = 180 W/m-K), Ac is the cross-sectional area of the fin L × s = (0.01 m) (40 × 10−6 m), and P is the perimeter of the fin which can be defined as 2L since the width of the fin is much smaller than its length. The term mb can be written as 1/2 1/2 ¯ h2L 2h¯ mb = b= b ksL ks 1/2 2(47.4 × 103 W/m2 -K) = (350 × 10−6 m) = 1.270 (180 W/m-K)(40 × 10−6 m) 124 Heat transfer and fluid flow in minichannels and microchannels Therefore, the fin efficiency can be calculated as f = tanh(1.270)/1.270 = 0.672 = 67%. (vii) Inlet and outlet surface temperatures at the base of the fin (Answer: T s,i = 36.3 C and T s,o = 48.6 C) Assumptions (1) The heat flux is constant over the chip surface. The surface heat flux considering the fin efficiency is given by q = q/(2bf + a)nL 100 W = 2(350 × 10−6 m)(0.672) + 50 × 10−6 m

(111)(0.01 m) = 173 × 103 W/m2 The relationship between heat flux, heat transfer and temperature difference is given by q = h(Ts − Tf ) where the subscripts s and f refer to the surface and fluid, respectively. The local heat transfer coefficients at the inlet and outlet of the microchannels are needed in order to compute the surface temperatures. Since the flow is developing at the entrance of the microchannel, Tables 3.3 and 3.4 are used with the channel aspect ratio and x* for the inlet, where x* is defined by Eq. (3.44). The entrance of the channel is assumed to begin at x = 0.1 mm from the edge of the chip so x* is * xin = 0.1 × 10−3 m = 1.603 × 10−3 (87.5 × 10−6 m)(164.7)(4.33) Using linear interpolation in Table 3.3, the fully developed Nusselt numbers for threesided and four-sided heating are found to be 6.567 and 6.273, respectively. The thermal entry region Nusselt number for four-sided heating is found using Table 3.4 to be 18.42 for the inlet. Equation (3.45) is used to obtain the local Nusselt number at the entrance as Nux,3 (x* , c ) = 18.42 6.567 = 19.28 6.273 Alternatively, equations given in Appendix A may be used for interpolating values from Tables 3.2 to 3.4. Using the thermal conductivity of water at the given inlet temperature (35 C), the local heat transfer coefficient at the inlet is found to be h= kNu = (0.625 W/m-K)(19.28)/87.5 × 10−6 m = 137.7 × 103 W/m2 -K Dh Since the flow is fully developed at the outlet, the three-sided Nusselt number fromTable 3.3 is used with the thermal conductivity of water at the outlet temperature (45 C). The local heat transfer coefficient at the exit is found to be h= kNu = (0.638 W/m-K)(6.567)/87.5 × 10−6 m = 47.9 × 103 W/m2 -K Dh Chapter 3. Single-phase liquid flow in microchannels and minichannels 125 Using the value of the heat flux obtained above, the local heat transfer coefficients, and the given inlet and outlet water temperatures, the surface temperatures at the base of the fin at the fluid inlet and outlet are Ts = q + Tf => Ts,i = 36.3 C h and Ts,o = 48.6 C (viii) Pressure drop in microchannel core (Answer: 43.6 kPa) Assumptions (1) The core of the microchannel includes pressure drop due to only frictional losses in the fully developed region and the loss due to the developing region. Since the flow is fully developed at the exit, the pressure drop is defined by p = 2 2( f Re)µum L um + K(∞) 2 Dh2 where K(∞) is the Hagenbach factor, which is defined by Eq. (3.18) as K(∞) = 0.6796 + 1.2197c + 3.30892c − 9.59213c + 8.90894c − 2.99595c = 0.8969 The f Re term is given by Eq. (3.10) as f Re = 24(1 − 1.3553c + 1.94672c − 1.70123c + 0.95644c − 0.25375c ) = 20.2 The pressure drop in the core of the microchannel is given by Eq. (3.14) as p = 2(20.2)(655 × 10−6 N-s/m2 )(1.24 m/s)(0.01 m) (87.5 × 10−6 m)2 + (0.8969) (991.8 kg/m3 )(1.24 m/s)2 = 43.6 kPa 2 (ix) Total pressure drop (Answer: 45.0 kPa) Assumptions (1) Reservoirs are large so the Area of the reservoir >> the Area of microchannel. The total pressure drop between the inlet an outlet manifolds would include the pressure drop calculated above (43.6 kPa) plus the minor losses at the entrance and exit. 2 /2) where K is a loss coefficient related to The minor loss is defined by p = K(um area changes at the entrance or exit. Taking the area ratio as zero, based on Assumption 1, for a laminar flow regime, the contraction and expansion loss coefficients Kc and Ke can be obtained from Fig. 3.5(b) as Kc = 0.8 and Ke = 1.0. 126 Heat transfer and fluid flow in minichannels and microchannels The total pressure drop is obtained by p = u2 u2 u2 2( f Re)µum L + K(∞) m + Kc m + Ke m = 45.0 kPa 2 2 2 2 Dh The contraction and expansion coefficients were taken as the largest values on the chart in Fig. 3.5(b) in order to design for the maximum expected pressure drop. Example 3.2 A microchannel is etched in silicon. The microchannel surface is intentionally etched to provide an average roughness of 12 µm. The microchannel dimensions measured from the root of the roughness elements are: width – 200 µm, height – 200 µm, length – 10 mm. Water flows through the microchannels at a temperature of 300 K. Calculate the core ˙ = 180 × 10−6 kg/s. frictional pressure drop when, (i) m ˙ = 90 × 10−6 kg/s, and (ii) m Schematic b a e = 12 m Solution Properties: From Incropera and DeWitt (2002), saturated water at 300 K: µf = 0.855 × 10−3 N-s/m2 , = 997 kg/m3 , cp,f = 4179 J/kg-K, kf = 0.613 W/m-K. (i) Core frictional pressure drop when m ˙ = 90 × 10−6 kg/s (Answer: 29,365 Pa) Calculate the hydraulic diameter using the constricted width (acf ) and constricted height (bcf ) in Eq. (3.6) Dh,cf = 4Ac = acf = bcf = 0.000176 m Pw Calculate the Reynolds number of the fluid in a microchannel using Eq. (3.33) Recf = um,cf Dh,cf (997.01 kg/m3 )(2.91 m/s)(176 × 10−6 m) = 598 = µ (0.855 × 10−3 Ns/m2 ) Chapter 3. Single-phase liquid flow in microchannels and minichannels 127 For fully developed laminar flow, the hydrodynamic entry length may be obtained using Eq. (3.11) Lh,cf = 0.05 Recf Dh,cf = 0.05(598)(0.176 mm) = 5.26 mm Since L > Lh,cf , the fully developed flow assumption is valid. The f Re term can be obtained using Eq. (3.10) f Re = 24(1 − 1.3553c + 1.94672c − 1.70123c + 0.95644c − 0.25375c ) = 14.23 The core frictional pressure drop can be calculated using Eq. (3.14) 2 um,cf 2( f Re)µm,cf L p = + K(∞) 2 Dh,cf where K(∞) is given by Eq. (3.18) K(∞) = (0.6796 + 1.2197c + 3.30892c − 9.59213c + 8.90894c − 2.99595c ) = 1.53 Hence p = 2(14.23)(0.855 × 10−3 Ns/m2 )(2.91 m/s)(0.01 m) (176 × 10−6 m)2 + (1.53) (997.01 kg/m3 )(2.91 m/s)2 = 29,365 Pa 2 (ii) Core frictional pressure drop when m ˙ = 180 × 10−6 kg/s (Answer: 68,694 Pa) Calculate the Reynolds number of the fluid in a microchannel using Eq. (3.33) Recf = (997.01 kg/m3 )(5.83 m/s)(176 × 10−6 m) um,cf Dh,cf = = 1200 µ (0.855 × 10−3 Ns/m2 ) For fully developed laminar flow, the hydrodynamic entry length may be obtained using Eq. (3.11) Lh,cf = 0.05 Recf Dh,cf = 0.05(1200)(0.176 mm) = 10.5 mm Since L < Lh,cf , the fully developed flow assumption is not valid. Calculate the core frictional pressure drop with the developing laminar flow assumption. The apparent friction factor can be obtained from Table 3.2 using x+ calculated from Eq. 3.16 x+ = x/Dh 0.01 m = 0.0475 = Recf (176 × 10−6 m)(1200) 128 Heat transfer and fluid flow in minichannels and microchannels From Table 3.2 fapp Re = 21.35 so the core frictional pressure drop can be calculated using the following equation 2( fapp Re)µm,cf L p = 2 Dh,cf = 2(21.35)(0.855 × 10−3 Ns/m2 )(5.83 m/s)(0.01 m) = 68,694 Pa (176 × 10−6 m)2 Comments (1) The total frictional pressure drop has to consider the minor losses because the actual p is higher than the core frictional pressure drop. Example 3.3 Consider a copper minichannel heat sink with an area of 30 mm × 30 mm, and relevant dimensions in Fig. 3.14 with a = 1 mm, b = 3 mm, and s = 1.5 mm. The heat dissipation is 100 W and the water inlet temperature is 30 C. The maximum surface temperature in the heat sink is limited to 80 C. Calculate the water flow rate under these conditions. Also calculate the frictional pressure drop in the core. Assume a constant heat transfer coefficient corresponding to fully developed conditions and take the thermal conductivity of copper to be 400 W/m-K. Assumptions Laminar flow, constant heat flux equally distributed over the surface area, one-dimensional steady state conduction, and three-sided heated condition with an adiabatic tip for fin efficiency a = 1 mm = 10−3 m, b = 3 mm = 3 × 10−3 m, s = 1.5 mm = 1.5 × 10−3 m, w = L = 30 × 10−3 m, Tf ,i = 30 C, Ts,o = 80 C, q = 100 W, kCu = 400 W/m-K, Ac = a×b = 3 × 10−6 m2 , c = a/b = 1/3 b a s Solution Equation 3.7 gives the hydraulic diameter, Dh = 2(10−3 )(3 × 10−3 ) 2ab = = 1.5 × 10−3 m (a + b) (10−3 + 3 × 10−3 ) Chapter 3. Single-phase liquid flow in microchannels and minichannels 129 The Nusselt number can be obtained using Table 3.3 for a three-sided heated channel. Through interpolation this gives, Nu = Nufd,3 = 5.224 For calculating water properties, let’s assume the water exit temperature is equal to the maximum surface temperature of the heat sink. Then, the average water temperature is, Tavg = (30 C + 80 C)/2 = 55 C Properties of water at 55 C (from Incropera and DeWitt, 2002): = 985 kg/m3 , cp = 4.183 kJ/kg-K, µ = 505 × 10−6 kg/m-s, k = 648 × 10−3 W/m-K The average heat transfer coefficient is calculated using the Nusselt number which is rearranged to give, kNu (0.648)(5.224) h¯ = = = 2.26 × 103 W/m2 -K Dh (1.5 × 10−3 ) By relating the conductive and convective heat transfer for a constant heat flux, the average temperature difference between the surface and the fluid is given as q T = h(2bf + a)nL overall width w 0.030 = = = 12 channel + fin a+s 0.001 + 0.0015 tanh(mb) The fin efficiency is, f = mb $ $ 2h¯ 2(2.26 × 103 ) mb = b= × (3 × 10−3 ) = 0.260 kCu s (400)(1.5 × 10−3 ) The number of channels is, n = f = tanh(0.260) = 0.978 (0.260) T s,o T(x) Ts,i ΔT q h T f,o Tf,i x 130 Heat transfer and fluid flow in minichannels and microchannels T = 100 (2.26 × 103 )[2(3 × 10−3 )(0.978) + 10−3 ](12)(30 × 10−3 ) = 17.9 C The fluid outlet temperature is, Tf ,o = Ts,o − T = 80 C − 17.9 C = 62.1 C The average water temperature is, Tavg = (30 C + 62.1 C)/2 = 46.1 C Tavg is off by 9 degrees. It is necessary to iterate again with the updated average temperature of 46 C Properties of water at 46 C (from Incropera and De Witt, 2002): = 990 kg/m3 , cp = 4.180 kJ/kg-K, µ = 588 × 10−6 kg/m-s, k = 639 × 10−3 W/m-K Following identical steps the new values are, h¯ = 2.23 × 103 W/m2 -K, mb = 0.258, f = 0.978, T = 18.2 C, Tf ,o = 61.8 C Tavg = (30 + 61.8)/2 = 45.9 C The calculated average temperature is approximately the same as the assumed value. No more iterations are necessary. Ts,i = Tf ,i + T = 30 C + 18.2 C = 48.2 C q = mc ˙ p Tfluid m ˙t = q (100) = 752 × 10−6 kg/s = cp Tfluid (4180)(61.8 − 30) Qc = m ˙ t (1/n) (752 × 10−6 )(1/12) = = 63.3 × 10−9 m3 /s (990) um = Qc (63.3 × 10−9 ) = = 0.0211 m/s Ac (3 × 10−6 ) Re = Dh um (1.5 × 10−3 )(0.0211)(990) = = 53.3 µ (588 × 10−6 ) or 3.80 ml/min Flow is laminar so original assumption is correct. 2( f Re)µum L u2 Equation 3.14 gives the pressure drop, p = + K(∞) m 2 2 Dh f Re can be determined with Eq. 3.10, f Re = 24(1 − 1.3553c + 1.94672c − 1.70123c + 0.95644c − 0.25375c ) = 17.1 Hagenbach’s factor comes from Eq. (3.22), K(∞) = 0.6796 − 1.2197c + 3.30892c − 9.59213c + 8.90894c − 2.99595c = 1.20 Chapter 3. Single-phase liquid flow in microchannels and minichannels 131 The pressure drop in the core is therefore, p = 2(17.1)(588 × 10−6 )(0.0211)(30 × 10−3 ) (990)(0.0211)2 = 5.92 Pa + (1.20) (1.5 × 10−3 )2 2 We need to calculate the hydrodynamic and thermal entrance lengths to justify fully developed conditions. Using Eq. (3.11) we have, Lh = 0.05 Dh Re = 0.05(1.5 × 10−3 )(53.3) = 0.00400 m = 4.00 mm Similarly Eq. (3.42) gives, Lt = 0.05 Dh Re Pr = 0.05(1.5 × 10−3 )(53.3)(3.85) = 0.0154 m = 15.4 mm This validates the fully developed assumption as given in the problem statement. Comments If the heat dissipation is increased to 200 W the flow rate will be 18.4 ml/min. This increases the pressure drop to 38.8 Pa. This illustrates the performance hit for increased heat dissipation and the need to develop microchannel cooling under turbulent conditions where the heat transfer coefficient will be larger. For turbulent flow more heat can be dissipated with minimal performance loss in pressure drop. 3.9. Practice problems Problem 3.1 Solve Example 1 with the substrate as copper, with a thermal conductivity of 380 W/m-K. Problem 3.2 For Example 3.1, plot the outlet surface temperature and pressure drop as a function of the channel depth b over a range of 200 to 600 µm. Problem 3.3 For Example 3.1, plot the outlet surface temperature and pressure drop as a function of the channel width a over a range of 50 to 200 µm. Problem 3.4 Solve Example 3.1 with a split flow arrangement. Neglect the area reduction caused by the central manifold. Problem 3.5 Solve Example 3.1 with copper substrate and a split flow arrangement. Assume the channel width to be 100 µm and a depth of 800 µm. Neglect the area reduction caused by the central manifold. Problem 3.6 For Example 3.3 with copper, plot the outlet surface temperature and the pressure drop as a function of the channel depth over a range of 200 to 2000 µm. Problem 3.7 In Example 3.3, redesign the heat sink if the design heat load increases to 500 W. You may have to rework the channel geometry and flow arrangements. Problem 3.8 Design a microchannel heat exchanger to dissipate 800 Watts from a copper heat sink of 20 mm × 20 mm heated surface area. The inlet water temperature is 40 C, and 132 Heat transfer and fluid flow in minichannels and microchannels the maximum surface temperature in the heat sink is desired to be below 60 C. Check your channel dimensions from manufacturing standpoint (provide the manufacturing technique you will be implementing). Do not neglect the temperature drop occurring in the copper between the base of the channels and the bottom surface of the heat sink receiving heat. Show the details of the manifold design. Problem 3.9 A copper minichannel heat sink with an area of 30 mm × 30 mm, and relevant dimensions in Fig. 3.14 are a = 1 mm, b = 3 mm, and s = 1.5 mm. Calculate the maximum heat dissipation possible with an inlet water temperature of 30 C and the maximum surface temperature in the heat sink is limited to 80 C. Calculate the water flow rate under these conditions. Also calculate the frictional pressure drop in the core under these conditions. Problem 3.10 Design a minichannel heat exchanger to dissipate 5 kW of heat from a copper plate with a footprint of 10 cm × 12 cm. The plate surface temperature should not exceed 60 C and the inlet design temperature for water is 35 C. Calculate the water flow rate, outlet water temperature, and core frictional pressure drop. Compare the performance of (a) straight once-through flow passages and (b) split flow passages. Problem 3.11 Design the heat exchanger for Problem 3.10 in aluminum, and compare its performance with a copper heat sink. Appendix A Table A.1 Curve-fit equations for Tables 3.2, 3.3, and 3.4 Constants Equations a b c d e f 1 2 3 4 5 6 7 8 9 10 11 12 141.97 142.05 142.1 286.65 8.2321 8.2313 36.736 30.354 31.297 28.315 6.7702 9.1319 −7.0603 −5.4166 −7.3374 25.701 2.0263 1.9349 2254 1875.4 2131.3 3049 −3.1702 −3.7531 2603 1481 376.69 337.81 1.2771 −2.295 17559 13842 14867 27038 0.4187 0.48222 1431.7 1067.8 800.92 1091.5 0.29805 0.92381 66172 154970 144550 472520 2.1555 2.5622 14364 13177 14010 26415 2.2389 7.928 555480 783440 622440 1783300 2.76 × 10−6 5.16 × 10−6 −220.77 −108.52 −33.894 8.4098 0.0065322 0.0033937 1212.6 −8015.1 −13297 −35714 NA NA Equation form: a + cx0.5 + ex a + cx + ex2 ; Equations 5–10: y = 0.5 1.5 1 + bx + dx + fx 1 + bx + dx2 + fx3 2 −1.5 Equations 11–12: y = a + bx + c(ln x) + d ln x + ex Equations 1–4: y = Chapter 3. Single-phase liquid flow in microchannels and minichannels 133 Equation variables Laminar flow friction factor in entrance region of rectangular ducts, Table 3.2: y-variable – friction factor x-variable – Equation 1: c = 1.0 Equation 2: c = 0.5 Equation 3: c = 0.2 Equation 4: 0.1 = c = 10 Fully developed Nusselt number for three-side heating, Table 3.3: y-variable – Nusselt number for three-sided heating x-variable – Equation 5: c Fully developed Nusselt number for four-side heating, Table 3.3: y-variable – Nusselt number for four-sided heating x-variable – Equation 6: c Thermal entry region Nusselt numbers for four-sided heating, Table 3.4: y-variable – Nusselt number for four-sided heating in the entry region x-variable – Equation 7: c = 0.1 Equation 8: c = 0.25 Equation 9: c = 0.333 Equation 10: c = 0.5 Equation 11: c = 1.0 Equation 12: c = 10 References Adams, T. M., Abdel-Khalik, S. I., Jeter, M., and Qureshi, Z. 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Wibulswas, P., Laminar flow heat transfer in non-circular ducts, PhD Thesis, London, UK, London University, 1966. Xu, B., Ooi, K. T., Wong, N. T., and Choi, W. K., Experimental investigation of flow friction for liquid flow in microchannels, Int. Com. Heat Mass Trans., 27(8), 1165–1176, 2000. Yu, D., Warrington, R., Barron, R., and Ameel, T., An experimental investigation of fluid flow and heat transfer in microtubes, Proceedings of the ASME/JSME Thermal Engineering Conference, 1, ASME, 523–530, 1995. Chapter 4 SINGLE PHASE ELECTROKINETIC FLOW IN MICROCHANNELS Dongqing Li Department of Mechanical Engineering, Vanderbilt University, Nashville, TN, USA 4.1. Introduction It is important to understand electrokinetic-driven liquid flow in microchannels when designing and controlling lab-on-a-chip devices. Lab-on-a-chip devices are miniaturized bio-medical or chemistry laboratories on a small glass or plastic chip. Generally, a lab-ona-chip device has a network of microchannels, electrodes, sensors and electrical circuits. Electrodes are placed at strategic locations on the chip. Applying electrical fields along microchannels controls the liquid flow and other operations in the chip. These labs on a chip can duplicate the specialized functions as their room-sized counterparts, such as clinical diagnostics, DNA scanning and electrophoretic separation. The advantages of these labs on a chip include dramatically reduced sample size, much shorter reaction and analysis time, high throughput, automation and portability. The key microfluidic functions required in various lab-on-a-chip devices include pumping, mixing, thermal cycling, dispensing and separating. Most of these processes are electrokinetic processes. Basic understanding, modeling and controlling of these key microfluidic functions/processes are essential to systematic design and operational control of the lab-on-a-chip systems. Because all solid–liquid (aqueous solutions) interfaces carry electrostatic charge, there is an electrical double layer (EDL) field in the region close to the solid–liquid interface on the liquid side. Such an EDL field is responsible for at least two basic electrokinetic phenomena: electroosmosis and electrophoresis. Briefly, electroosmosis is the liquid motion in a microchannel caused by the interaction between the EDL at the liquid-channel wall interface with an electrical field applied tangentially to the wall. Electrophoresis is the motion of a charged particle relative to the surrounding liquid under an applied electrical field. Essentially all on-chip microfluidic processes are realized by using these two phenomena. This chapter will review basics of the EDL field, E-mail: [email protected] 137 138 Heat transfer and fluid flow in minichannels and microchannels and discuss some key on-chip microfluidic processes. A more comprehensive review of the electrokinetic-based microfluidic processes for lab-on-achip applications can be found elsewhere (Li, 2004). 4.2. EDL field It is well-known that most solid surfaces obtain a surface electric charge when they are brought into contact with a polar medium (e.g. aqueous solutions). This may be due to ionization, ion adsorption or ion dissolution. If the liquid contains a certain amount of ions (for instance, an electrolyte solution or a liquid with impurities), the electrostatic charges on the solid surface will attract the counterions in the liquid. The rearrangement of the charges on the solid surface and the balancing charges in the liquid is called the EDL (Hunter, 1981; Lyklema, 1995). Immediately next to the solid surface, there is a layer of ions that are strongly attracted to the solid surface and are immobile. This layer is called the compact layer, normally about several Angstroms thick. Because of the electrostatic attraction, the counterions concentration near the solid surface is higher than that in the bulk liquid far away from the solid surface. The coions concentration near the surface, however, is lower than that in the bulk liquid far away from the solid surface, due to the electrical repulsion. So there is a net charge in the region close to the surface. From the compact layer to the uniform bulk liquid, the net charge density gradually reduces to zero. Ions in this region are affected less by the electrostatic interaction and are mobile. This region is called the diffuse layer of the EDL. The thickness of the diffuse layer is dependent on the bulk ionic concentration and electrical properties of the liquid, usually ranging from several nanometers for high ionic concentration solutions up to several microns for pure water and pure organic liquids. The boundary between the compact layer and the diffuse layer is usually referred to as the shear plane. The electrical potential at the solid–liquid surface is difficult to measure directly. The electrical potential at the shear plane is called the zeta potential and can be measured experimentally (Hunter, 1981; Lyklema, 1995). In practice, the zeta potential is used as an approximation to the potential at the solid–liquid interface. The ion and electrical potential distributions in the EDL can be determined by solving the Poisson–Boltzmann equation (Hunter, 1981; Lyklema, 1995). According to the theory of electrostatics, the relationship between the electrical potential and the local net charge density per unit volume e at any point in the solution is described by the Poisson equation: Ñ 2 = − e (4.1) where is the dielectric constant of the solution. Assuming the equilibrium Boltzmann distribution equation is applicable, which implies uniform dielectric constant, the number concentration of the type-i ion is of the form zi e ni = nio exp − (4.2) kb T where nio and zi are the bulk ionic concentration and the valence of type-i ions, respectively, e is the charge of a proton, kb is the Boltzmann constant, and T is the absolute temperature. Chapter 4. Single phase electrokinetic flow in microchannels 139 For a symmetric electrolyte (z− = z+ = z) solution, the net volume charge density e is proportional to the concentration difference between symmetric cations and anions, via.: ze (4.3) e = ze(n+ − n− ) = −2zeno sinh kb T Substituting Eq. (4.3) into the Poisson equation leads to the well-known Poisson– Boltzmann equation: 2zeno ze 2 sinh (4.4) Ñ = kb T Solving the Poisson–Boltzmann equation with proper boundary conditions will determine the local EDL potential field and hence, via Eq. (4.3), the local net charge density distribution. 4.3. Electroosmotic flow in microchannels Consider a microchannel filled with an aqueous solution. There is an EDL field near the interface of the channel wall and the liquid. If an electric field is applied along the length of the channel, an electrical body force is exerted on the ions in the diffuse layer. In the diffuse layer of the EDL field, the net charge density, e is not zero. The net transport of ions is the excess counterions. If the solid surface is negatively charged, the counterions are the positive ions. These excess counterions will move under the influence of the applied electrical field, pulling the liquid with them and resulting in electroosmotic flow. The liquid movement is carried through to the rest of the liquid in the channel by viscous forces. This electrokinetic process is called electroosmosis and was first introduced by Reuss in 1809 (Reuss, 1809). In most lab-on-a-chip applications, electroosmotic flow is preferred over the pressuredriven flow. One of the reasons is the plug-like velocity profile of electroosmotic flow. This means that fluid samples can be transported without dispersion caused by flow shear. Generally, pumping a liquid through a small microchannel requires applying very large pressure difference depending on the flow rate. This is often impossible because of the limitations of the size and the mechanical strength of the microfluidic devices. Electroosmotic flow can generate the required flow rate even in very small microchannels without any applied pressure difference across the channel. Additionally, using electroosmotic flow to transport liquids in complicated microchannel networks does not require any external mechanical pump or moving parts, it can be easily realized by controlling the electrical fields via electrodes. Although high voltages are often required in electroosmotic flow, the required electrical power is very small due to the very low current involved, and it is generally safe. However, the heat produced in electroosmotic flow often presents problems to many applications where solutions of high electrolyte concentrations and long operation time are required. Most channels in modern microfluidic devices and micro-electro-mechanical systems (MEMS) are made by micromachining technologies. The cross-section of these channels 140 Heat transfer and fluid flow in minichannels and microchannels y x 2H z L 2W Fig. 4.1. Geometry of microchannel. The shaded region indicates the computational domain. is close to a rectangular shape. In such a situation, the EDL field is two-dimensional and will affect the two-dimensional flow field in the rectangular microchannel. In order to control electroosmotic pumping as a means of transporting liquids in microstructures, we must understand the characteristics of electroosmotic flow in rectangular microchannels. In the following, we will review the modeling and the numerical simulation results of electroosmotic flow in rectangular microchannels (Arulanandam and Li, 2000a). The EDL field, the flow field and the volumetric flow rate will be studied as functions of the zeta potential, the liquid properties, the channel geometry and the applied electrical field. Consider a rectangular microchannel of width 2W , height 2H and length L, as illustrated in Fig. 4.1. Because of the symmetry in the potential and velocity fields, the solution domain can be reduced to a quarter section of the channel (as shown by the shaded area in Fig. 4.1). The 2D EDL field can be described by the Poisson–Boltzmann equation: ∂2 ∂2 2n∞ ze ze + 2 = sinh − ∂y2 ∂z 0 kb T Along the planes of symmetry, the following symmetry boundary conditions apply: at y = 0 ∂ =0 ∂y at z = 0 ∂ =0 ∂z Along the surfaces of the solution domain, the potential is the zeta potential: at y = H = at z = W = The Poisson–Boltzmann equation and the boundary conditions can be transformed into non-dimensional equations by introducing the following dimensionless variables: y z ze 4Across-sectional 4HW y * = z* = * = = Dh = Dh Dh kb T Pwetted H +W The non-dimensional form of the Poisson–Boltzmann equation is given by: ∂2 * ∂2 * + = (Dh )2 sinh * ∂y*2 ∂z *2 (4.5) Chapter 4. Single phase electrokinetic flow in microchannels 141 where , the Debye–Huckle parameter, is defined as follows: = 2n∞ z 2 e2 0 kb T 1/2 (4.6) and 1/ is the characteristic thickness of the EDL. The non-dimensional parameter Dh is a measure of the relative channel diameter, compared to the EDL thickness. Dh is often referred to as the electrokinetic diameter. The corresponding non-dimensional boundary conditions as follows: ∂ * =0 ∂y* at y* = 0 at y* = at z * = 0 H ze * = * = Dh kb T ∂ * =0 ∂z * at z * = W ze * = * = Dh kb T If we consider that the flow is steady, two-dimensional, and fully developed, and there is no pressure gradient in the microchannel, the general equation of motion is given by a balance between the viscous or shear stresses in the fluid and the externally imposed electrical field force: 2 ∂ u ∂2 u = Fx + µ ∂y2 ∂z 2 where Fx is the electrical force per unit volume of the liquid, which is related to the electric field strength Ex and the local net charge density as follows: Fx = e Ex Therefore, the equation for the electroosmotic flow can be written as:

2 2n∞ ze ∂ u ∂2 u + 2 = sinh Ex µ ∂y2 ∂z 0 (4.7) The following boundary conditions apply along the planes of symmetry: at y = 0 ∂u =0 ∂y at z = 0 ∂u =0 ∂z Along the surface of shear (the surface of the solution domain), the velocity boundary conditions are given by: at y = H u = 0 at z = W u = 0 The equation of motion can be non-dimensionalized using the following additional transformations: u* = u U Ex* = Ex L 142 Heat transfer and fluid flow in minichannels and microchannels where U is a reference velocity and L is the distance between the two electrodes. The non-dimensionalized equation of motion becomes: ∂ 2 u* ∂ 2 u* + = MEx* sinh * ∂y*2 ∂z *2 (4.8) M is a new dimensionless group, which is a ratio of the electrical force to the frictional force per unit volume, given by: M= 2n∞ zeDh2 µUL The corresponding non-dimensional boundary conditions are given by: at y* = 0 at y* = ∂u* =0 ∂y* H * u =0 Dh at z * = 0 at z * = ∂u* =0 ∂z * W * u =0 Dh Numerically solving Eqs. (4.5) and (4.8) with the boundary conditions will allow us to determine the EDL field and the electroosmotic flow field in such a rectangular microchannel. As an example, Let’s consider a KCl aqueous solution. At a concentration of 1 × 10−6 M, = 80 and µ = 0.90 × 10−3 kg/ms. An arbitrary reference velocity of U = 1 mm/s was used to non-dimensionalize the velocity. According to experimental results (Mala et al., 1997), zeta potential values changes from 100 to 200 mV, corresponding to three concentrations of the KCl solution, 1 × 10−6 , 1 × 10−5 and 1 × 10−4 M. The hydraulic diameter of the channel varied from 12 to 250 µm, while the aspect ratio (H /W ) varied from 1:4 to 1:1. Finally, the applied voltage difference ranged from 10 V to 10 kV. The EDL potential distribution in the diffuse double layer region is shown in Fig. 4.2. The nondimensional EDL potential profile across a quarter section of the rectangular channel exhibits characteristic behavior. The potential field drops off sharply very close to the wall. The region where the net charge density is not zero is limited to a small region close to the channel surface. Figure 4.3 shows the non-dimensional electroosmotic velocity field for an applied potential difference of 1 kV/cm. The velocity field exhibits a profile similar to plug flow, however, in electroosmotic flow, the velocity increases rapidly from zero at the wall (shear plane) to a maximum velocity near the wall, and then gradually drops off to a slightly lower constant velocity that is maintained through most of the channel. This unique profile can be attributed to the fact that the externally imposed electrical field is driving the flow. In the mobile part of the EDL region very close to the wall, the larger electrical field force exerts a greater driving force on the fluid because of the presence of the net charge in the EDL region. Variation of Dh affects the following nondimensional parameters: the electrokinetic diameter, and the strength of the viscous forces in the ratio of electrical to viscous forces. Chapter 4. Single phase electrokinetic flow in microchannels 143 6 1.0 0.8 0.6 0.4 0.2 0.0 (z/ W 2 ) 4 na ndim en sio 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Non-dim 0.1 0.0 ensional height (y /H ) No 0 lw idt h ntial Non-dimensional EDL pote 8 Fig. 4.2. Non-dimensional EDL potential profile in a quarter section of a rectangular microchannel with Dh = 79, * = 8 and H :W = 2:3. 14 12 10 l ve Non-dimensiona locity (u/U) 16 8 6 4 2 1.0 0.8 0.6 0.4 0.2 0.0 00 .9 0 .8 0 Non .7 0.6 -dim 0 ens .5 0.4 iona 0 l he .3 0.2 ight 0 (y/H .1 0.0 ) h idt W (z/ ) lw na io ns e dim on N Fig. 4.3. Non-dimensional velocity field in a quarter section of a rectangular microchannel with Dh = 79, * = 8, H /W = 2/3, Ex* = 5000 and M = 2.22. 144 Heat transfer and fluid flow in minichannels and microchannels 55 C 106 M, z 0.200 V 50 C 105 M, z 0.150 V Volumetric flow rate (l/min) 45 C 104 M, z 0.100 V 40 35 30 25 20 15 10 5 0 0 50 100 150 200 250 Hydraulic diameter (m) Fig. 4.4. Variation of volumetric flow rate with hydraulic diameter for three different combinations of concentration and zeta potential, with H /W = 2/3, and Ex = 1 kV/cm. The volumetric flow rate increased with approximately Dh2 as seen in Fig. 4.4. This is expected, since the cross-sectional area of the channel also increases proportionately to Dh2 . When larger pumping flow rates are desired, larger diameter channels would seem to be a better choice. However, there is no corresponding increase in the average velocity with increased hydraulic diameter. This is because the nature of electroosmotic flow – the flow is generated by the motion of the net charge in the EDL region driven by an applied electrical field. When the double layer thickness (1/) is small, an analytical solution of the electroosmotic velocity can be derived from an one-dimensional channel system such as a cylindrical capillary with a circular cross-section, given by vav = − Ez r 0 µ (4.9) Equation (4.9) indicates that the electroosmotic flow velocity is linearly proportional to the applied electrical field strength and linearly proportional to the zeta potential. The negative sign indicates the flow direction and has to do with the sign of the potential. If potential is negative (i.e. a negatively charged wall surface), the excess counterions in the diffuse layer are positive, therefore the electroosmotic flow in the microchannel is towards the negative electrode. With a rectangular microchannel not only the hydraulic diameter but also the channel shape will influence the velocity profile. This is because of the impact of the channel Chapter 4. Single phase electrokinetic flow in microchannels 145 0.75 Volumetric flow rate (l/min) 0.60 0.45 0.30 0.15 C 106 M, z 0.200 V C 105 M, z 0.150 V C 104 M, z 0.100 V 0.00 0.0 0.2 0.4 0.6 Aspect ratio (H/W ) 0.8 1.0 Fig. 4.5. Variation of volumetric flow rate with aspect ratio for three different combinations of concentration and zeta potential, with Dh = 24 µm and Ex = 1 kV/cm. In this case, z/W = 1.0 represents the channel wall, and z/W = 0 represents the center of the channel. geometry on the EDL. Figure 4.5 shows the relationship between the aspect ratio (H /W ) and the volumetric flow rate for a fixed hydraulic diameter. As the ratio of H :W approaches 1:1 (for a square channel), the flow rate decreases. This is because of the larger role that corner effects have on the development of the EDL and the velocity profile in square channels. From Eq. (4.6), it is clear that increasing the bulk ion concentration in the liquid results in an increase in or a decrease in the EDL thickness 1/. Correspondingly, the EDL potential field falls off to zero more rapidly with distance, that is, the region influenced by the EDL is smaller. The ionic concentration effect on the velocity or the flow rate can be understood as follows. Since ionic concentration influences the zeta potential, as the ionic concentration is increased, the zeta potential decreases in value. As the zeta potential decreases, so does the electroosmotic flow velocity (Eq. (4.9)) and the volumetric flow rate. 4.4. Experimental techniques for studying electroosmotic flow In most electroosmotic flows in microchannels, the flow rates are very small (e.g. 0.1 µL/min) and the size of the microchannels is very small (e.g. 10–100 µm), it is extremely difficult to measure directly the flow rate or velocity of the electroosmotic flow in microchannels. To study liquid flow in microchannels, various microflow visualization methods have evolved. Micro particle image velocimetry (microPIV) is a method that was 146 Heat transfer and fluid flow in minichannels and microchannels adapted from well-developed PIV techniques for flows in macro-sized systems (Talyor and Yeung, 1993; Herr et al., 2000; Wereley and Meinhart, 2001; Singh et al., 2001; Selvaganapathy et al., 2002). In the microPIV technique, the fluid motion is inferred from the motion of sub-micron tracer particles. To eliminate the effect of Brownian motion, temporal or spatial averaging must be employed. Particle affinities for other particles, channel walls, and free surfaces must also be considered. In electrokinetic flows, the electrophoretic motion of the tracer particles (relative to the bulk flow) is an additional consideration that must be taken. These are the disadvantages of the microPIV technique. Dye-based microflow visualization methods have also evolved from their macro-sized counterparts. However, traditional mechanical dye injection techniques are difficult to apply to the microchannel flow systems. Specialized caged fluorescent dyes have been employed to facilitate the dye injection using selective light exposure (i.e. the photoinjection of the dye). The photo-injection is accomplished by exposing an initially nonfluorescent solution seeded with caged fluorescent dye to a beam or a sheet of ultraviolet (UV) light. As a result of the UV exposure, caging groups are broken and fluorescent dye is released. Since the caged fluorescent dye method was first employed for flow tagging velocimetry in macro-sized flows in 1995 (Lempert et al., 1995), this technique has since been used to study a variety of liquid flow phenomena in microstructures (Dahm et al., 1992; Paul et al., 1998; Herr et al., 2000; Johnson et al., 2001; Molho et al., 2001; Sinton et al., 2002; 2003a; Sinton and Li, 2003a, b). The disadvantages of this technique are that it requires expensive specialized caged dye, and extensive infrastructure to facilitate the photo-injection. Recently, Sinton and Li (Sinton and Li, 2003a) developed a microchannel flow visualization system and complimentary analysis technique by using caged fluorescent dyes. Both pressure-driven and electrokinetically driven velocity profiles determined by this technique compare well with analytical results and those of previous experimental studies. Particularly, this method achieved a high degree of near-wall resolution. Generally, in the experiment, a caged fluorescent dye is dissolved in an aqueous solution in a capillary or microchannel. It should be noted that the caged dye cannot emit fluorescent light at this stage. The UV laser light is focused into a sheet crossing the capillary (perpendicular to the flow direction). The caged fluorescent dye molecules exposed to the UV light are uncaged and thus are able to shine. The resulting fluorescent dye is continuously excited by an argon laser and the emission light is transmitted through a laser-powered epi-illumination microscope. Full frame images of the dye transport are recorded by a progressive scan CCD camera and saved automatically on the computer. In the numerical analysis, the images are processed and cross-stream velocity profiles are calculated based on tracking the dye concentration maxima through a sequence of several consecutive images. Several sequential images are used to improve the signal to noise ratio. Points of concentration maxima make convenient velocimetry markers as they are resistant to diffusion. In many ways, the presence of clearly definable, zero-concentration-gradient markers is a luxury afforded by the photo-injection process. The details of this technique can be found elsewhere (Sinton and Li, 2003a, b). In an experimental study (Sinton and Li, 2003a), the CMNB-caged fluorescein with the sodium carbonate buffer and 102 µm i.d. glass capillaries were used. Images of the uncaged dye transport in four different electroosmotic flows are displayed in vertical sequence in Chapter 4. Single phase electrokinetic flow in microchannels (a) (b) (c) 147 (d) Fig. 4.6. Images of the uncaged dye in electroosmotic flows through a 102 µm i.d. capillary at 133 ms intervals with applied electric field strength: (a) 0 V/0.14 m; (b) 1000 V/0.14 m; (c) 1500 V/0.14 m; and (d) 2000 V/0.14 m. Fig. 4.6. The dye diffused symmetrically as shown in Fig. 4.6(a). Image sequences given in Fig. 4.6(b–d) were taken with voltages of 1000 V, 1500 V, and 2000 V, respectively (over the 14 cm length of capillary). The field was applied with the positive electrode at left and the negative electrode at right. The resulting plug-like motion of the dye is characteristic of electroosmotic flow in the presence of a negatively charged surface at high ionic concentration. The cup-shape of the dye profile was observed in cases Fig. 4.6(b–d) within the first 50 ms following the UV light exposure. This period corresponded to the uncaging time scale in which the most significant rise in uncaged dye concentration occurs. Although the exact reason for the formation of this shape is unknown, it is likely that it was an artifact of the uncaging process in the presence of the electric field. Fortunately, however, the method is relatively insensitive to the shape of the dye concentration profile. Once formed, it is the transport of the maximum concentration profile that provides the velocity data. This also makes the method relatively insensitive to beam geometry and power intensity distribution. Figure 4.7 shows velocity data for the four flows corresponding to the image sequences in Fig. 4.6. Each velocity profile was calculated using an 8-image sequence and the numerical analysis technique described in reference (Sinton and Li, 2003a). The velocity profile resulting from no applied field, Fig. 4.6(a), corresponds closely to stagnation as expected. This run also serves to illustrate that, despite significant transport of dye due to diffusion, the analysis method is able to recover the underlying stagnant flow velocity. Although the other velocity profiles resemble that of classical electroosmotic flow (Wereley and Meinhart, 2001), a slight parabolic velocity deficit of approximately 4% was detected in all three flows. This was caused by a small back-pressure induced by the electroosmotic fluid motion (e.g. caused by the not-perfectly leveled capillary along the length direction). In additional to these PIV and dye-based techniques, the electroosmotic flow velocity can be estimated indirectly by monitoring the electrical current change while one solution is replaced by another similar solution during electroosmosis (Arulanandam and Li, 2000b; Ren et al., 2002; Sinton et al., 2002). In this method, a capillary tube is filled with an electrolyte solution, then brought into contact with another solution of the same electrolyte 148 Heat transfer and fluid flow in minichannels and microchannels R (m) 50 0 50 0 0.2 0.4 0.6 0.8 1 Velocity (mm/s) Fig. 4.7. Plots of velocity data from four electroosmotic flow experiments through a 102 µm i.d. capillary with applied electric field strengths of 0 V/0.14 m; 1000 V/0.14 m; 1500 V/0.14 m and 2000 V/0.14 m (from left to right). but with a slightly different ionic concentration. Once the two solutions are in contact, an electrical field is applied along the capillary in such a way that the second solution is pumped into the capillary and the 1st solution flows out of the capillary from the other end. As more and more of the second solution is pumped into the capillary and the first solution flows out of the capillary, the overall liquid conductivity in the capillary is changed, hence the electrical current through the capillary is changed. When the second solution completely replaces the first solution, the current will reach a constant value. Knowing the time required for this current change and the length of the capillary tube, the average electroosmotic flow velocity can be calculated by: uav,exp = L t (4.10) where L is the length of the capillary and t is the time required for the higher (or lower) concentration electrolyte solution to completely displace the lower (or higher) concentration electrolyte solution in the capillary tube. Figure 4.8 shows an example of the measured current–time relationship in a 10-cm-long glass capillary of 100 µm in diameter with a KCl solution under an applied electrical field of 350 V/cm. An example of measured electroosmotic velocity as a function of the applied electrical field is given in Fig. 4.9. As seen in this figure, a linear relationship between the applied voltage and the average velocity is evident. In addition, it is clear that the average velocity is independent of the diameter of the capillary tubes. However, in these experiments it is often difficult to determine the exact time required for one solution completely replacing another solution. This is because the gradual changing of the current with time and the small current fluctuations exist at both the beginning 149 6 5 5.5 4 5 3 4.5 2 Current 4 Voltage 3.5 0 50 100 Voltage (kV) Current (mA) Chapter 4. Single phase electrokinetic flow in microchannels 1 0 150 Time (s) Fig. 4.8. A typical result of current versus time. For the specific case of capillary diameter D = 100 µm, Ex = 3500 V/10 cm. KCl concentration in Reservoir 1 is C95% = 0.95 × 10−4 M. KCl concentration in Reservoir 2 is C100% = 1 × 10−4 M. 4 C 1 103 M KCl Average velocity (mm/s) 3.5 3 2.5 2 1.5 R 50 µm 1 R 75 µm 0.5 R 100 µm 0 0 2000 4000 6000 Voltage (V) Fig. 4.9. Average velocity versus the applied voltage and the capillary diameter for a 10-cm-long polyamide coated silica capillary tube. and the end of the replacing process. These make it very difficult to determine the exact beginning and ending time of the current change from the experimental results. Consequently, significant errors could be introduced in the average velocity determined in this way. It has been observed that, despite the curved beginning and ending sections, the measured current–time relationship is linear in most part of the process, as long as the concentration difference is small. Therefore, a new method was developed to determine the average electroosmotic velocity by using the slope of the current–time relationship (Ren et al., 2002). When a high concentration electrolyte solution gradually replaces another solution of the same electrolyte with a slightly lower concentration in a microchannel under an applied electric field, the current increases until the high concentration solution completely replaces the low concentration solution, at which time the current reaches a 150 Heat transfer and fluid flow in minichannels and microchannels constant value. If the concentration difference is small, a linear relationship between the current and the time is observed. This may be understood in the following way. In such a process, because the concentration difference between these two solutions is small, for example, 5%, the zeta potential, which determines the net charge density and hence the liquid flow, can be considered as constant along the capillary. Consequently, the average velocity can be considered as a constant during the replacing process. As one solution flows into the capillary and the other flows out of the capillary at essentially the same speed, and the two solutions have different electrical conductivities, the overall electrical resistance of the liquid in the capillary will change linearly, and hence the slope of current–time relationship is constant during this process. The slope of the linear current–time relationship can be described as: slope = I EA(b2 − b1 ) (EAb ) Ltotal = u¯ ave = · t t Ltotal Ltotal (4.11) where Ltotal is the total length of the capillary and u¯ ave is the average electroosmotic velocity, E is the applied electrical field strength, A is the cross-section area of the capillary, (b2 − b1 ) is the bulk conductivity difference between the high-concentration solution and the low-concentration solution. Using this equation, the average electroosmotic flow velocity can be evaluated from the slope of the measured current–time relationship. Figure 4.10 shows an example of the measured average electroosmotic velocity by the slope method. In the straightforward current–time method, the total time required for one solution completely replacing another solution has to be found, which often involves some inaccuracy of choosing the beginning and the ending points of the replacing process. In this slope method, instead of identifying the total period of time, only a middle section of the current–time data is required to determine the slope of the current–time relationship. This 3 Average velocity (mm/s) 100 m 200 m 2 1 0 0 10 20 30 Electrical field strength (kV/m) Fig. 4.10. The average velocity versus the applied voltage and the capillary diameter for 1 × 10−4 M KCl solution in a 10-cm-long capillary tube. Chapter 4. Single phase electrokinetic flow in microchannels 151 method is easier and more accurate than finding the exact beginning and the ending points of the replacing process. 4.5. Electroosmotic flow in heterogeneous microchannels Many microchannels do not have uniform surface properties. The surface heterogeneity may be caused by the impurities of the materials, by adhesion of solutes (e.g. surfactants, proteins and cells) of the solution, and by desired chemical surface modification. In the majority of lab-on-a-chip systems, electrokinetic means (i.e. electroosmotic flow, electrophoresis) are the preferred method of species transport. Although the surface heterogeneity has been long recognized as a problem leading to irregular flow patterns and nonuniform species transport, only recently have researchers begun to investigate the potential benefits the presence of surface heterogeneity (nonuniform surface -potential or charge density) may have to offer. Early studies examining these effects were conducted by Anderson (1985), Ajdari (1995; 1996; 2001) and Ghosal (2002). In Ajdari’s works it was predicted that the presence of surface heterogeneity could result in regions of bulk flow circulation, referred to as “tumbling” regions. This behavior was later observed in slit microchannels experimentally by Stroock et. al. (Stroock et al., 2000), who found excellent agreement with their flow model. In another study Herr et al. (2000) used a caged dye velocimetry technique to study electroosmotic flow in a capillary in the presence of heterogeneous surface properties and observed significant deviations from the classical plug type velocity profile, an effect which was predicted by Keely et al. (1994). In another clever application Johnson et al. (2001) used a UV excimer laser to introduce surface heterogeneity to the side wall of a polymeric microchannel and demonstrated how this could reduce sample band broadening around turns. Recently the use of surface heterogeneity to increase the mixing efficiency of a T-shaped micromixer was proposed (Erickson and Li, 2002b). Here we wish to review a general theoretical analysis of 3D electroosmotic flow structures in a slit microchannel with periodic, patchwise heterogeneous surface patterns (Erickson and Li, 2003a). Let’s consider the electroosmotically driven flow through a slit microchannel (i.e. a channel formed between two parallel plates) exhibiting a periodically repeating heterogeneous surface patterns shown in Fig. 4.11. Since the pattern is repeating, the computational domain is reduced to that over a single periodic cell, demonstrated by the dashed lines in Fig. 4.11. To further minimize the size of the solution domain it has been assumed that the heterogeneous surface pattern is symmetric about the channel mid-plane, resulting in the computational domain shown in Fig. 4.12. As a result of these two simplifications, the inflow and outflow boundaries, surfaces 2 and 4 (as labeled in Fig. 4.12 and referred by the symbol ), represent periodic boundaries on the computational domain, while surface 3 at the channel mid-plane represents a symmetry boundary. From Fig. 4.11 it can be seen that in all cases the surface pattern is symmetric about 5 and 6 and thus these surfaces also represent symmetry boundaries. For a general discussion of periodic boundary conditions, however, the reader is referred to a paper by Patankar et al. (1977). In order to model the flow through this periodic unit, it requires a description of the ionic species distribution, the double layer potential, the flow field and the applied potential. The 152 Heat transfer and fluid flow in minichannels and microchannels Fig. 4.11. Periodic patchwise surface heterogeneity patterns. Dark regions represent heterogeneous patches, light regions are the homogeneous surface. The percent heterogeneous coverage is 50%. 3

Iy y x z Iz

Ix Direction of applied electric field Fig. 4.12. Domain for the periodical computational cell showing location of computational boundaries. divergence of the ion species flux (for simplicity here we consider a monovalent, symmetric electrolyte as our model species), often referred to as the Nernst–Planck conservation equations, is used to describe the positive and negative ion densities (given below in non-dimensional form), ˜ + Pe+ N + V = 0 ˜ · −ÑN ˜ + − N + Ñ (4.12a) Ñ ˜ + Pe− N − V = 0 ˜ · −ÑN ˜ − + N − Ñ Ñ (4.12b) where N + and N − are the non-dimensional positive and negative species concentrations (N + = n+ /n∞ , N − = n− /n∞ where n∞ is the bulk ionic concentration), is the Chapter 4. Single phase electrokinetic flow in microchannels 153 non-dimensional electric field strength ( = e/kb T where e is the elemental charge, kb is the Boltzmann constant and T is the temperature in Kelvin), V is the non-dimensional velocity (V = v/vo where vo is a reference velocity) and the ~ sign signifies that the gradient operator has been nondimensionalised with respect to the channel half height (ly in Fig. 4.12). The Peclet number for the two species are given by Pe+ = vo ly /D+ and Pe− = vo ly /D− where D+ and D− are the diffusion coefficients for the positively charged and negatively charged species respectively. Along the heterogeneous surface (1 in Fig. 4.12) and symmetry boundaries of the computational domain (3 , 5 and 6 ), zero flux boundary conditions are applied to both Eqs. (4.12a) and (b) (see reference (Erickson and Li, 2002a, b) for explicit equations). As mentioned earlier, along faces 2 and 4 , periodic conditions are applied which take the form shown below: N2+ = N4+ , N2− = N4− , on 2 , 4 , on 2 , 4 The velocity field is described by the Navier–Stokes equations for momentum, modified to account for the electrokinetic body force, and the continuity equation as shown below: 2 ˜ ˜ +Ñ ˜ V − F(N + − N − )Ñ ˜ Re(V · ÑV) = −ÑP (4.13a) ˜ ·V =0 Ñ (4.13b) where F is a non-dimensional constant (F = n∞ ly kb T /µvo ) which accounts for the electrokinetic body force, Re is the Reynolds Number (Re = vo ly /µ, where is the fluid density and µ is the viscosity) and P is the non-dimensional pressure. Along the heterogeneous surface, 1 , a no-slip boundary condition is applied. At the upper symmetry surface, 3 , we enforce a zero penetration condition for the y-component of velocity and zero gradient conditions for the x and z terms, respectively. Similarly a zero penetration condition for the z-component of velocity is applied along 5 and 6 while zero gradient conditions are enforced for the x and y components. As with the Nernst–Planck equations, periodic boundary conditions are applied along 2 and 4 . For the potential field, we choose to separate, without loss of generality, the total non-dimensional electric field strength, , into two components (X , Y , Z) = (X , Y , Z) + E(X ) (4.14) where the first component, (X , Y , Z), describes the EDL field and the second component E(X ) represents the applied electric field. In this case the gradient of E(X ) is of similar order of magnitude to the gradient of (X , Y , Z) and thus cannot be decoupled to simplify the solution to the Poisson equation. As a result for this case the Poisson equation has the form shown below: ˜ 2 + d 2 E / d X 2 + K 2 (N + − N − ) = 0 Ñ (4.15a) where K is the non-dimensional double layer thickness (K 2 = 2 ly2 /2 where = (2n∞ e2 /w kb T )1/2 is the Debye–Huckel parameter). Two boundary conditions commonly 154 Heat transfer and fluid flow in minichannels and microchannels applied along the solid surface for the above equation are either an enforced potential gradient proportional to the surface charge density, or a fixed potential condition equivalent to the surface -potential. In this case we choose the former resulting in the following boundary condition applied along the heterogeneous surface ∂/∂Y = −(X , Z) on 1 (4.15b) where is the non-dimensional surface charge density ( = ly e/w kb T , where is the surface charge density). Along the upper and side boundaries of the computational domain, 3 , 5 and 6 , symmetry conditions are applied while periodic conditions are again are applied at the inlet and outlet, 2 and 4 . In order to determine the applied electric field, E(X ), we enforce a constant current condition at each cross-section along the x-axis as below: Jconst Ymax Zmax = [J + (X , Y , Z) − J − (X , Y , Z)]nx dY dZ 0 (4.16a) 0 where nx in the above is a normal vector in the x-direction and J + and J − are the nondimensional positive and negative current densities. Assuming monovalent ions for both positive and negative species (as was discussed earlier) and using the ionic species flux terms from Eqs. (4.12a) and (b), condition 4.16(a) reduces to the following: Jconst Ymax Zmax ∂N + Pe+ − ∂( + E) Pe+ ∂N − + − = − − − N + −N dY dZ ∂X Pe ∂X Pe ∂X 0 0 Ymax Zmax [Pe+ (N + − N − )Vx ]dY dZ + 0 (4.16b) 0 which is a balance between the conduction (term 2) and convection (term 3) currents with an additional term to account for any induced current due to concentration gradients. Solving Eq. (4.16b) for the applied potential gradient yields: Z max Y max dE = dX − 0 0

∂N + ∂X − Pe+ ∂N − Pe− ∂X Z max Y max

N+ + 0 0 − N+ + Pe+ − Pe− N Pe+ − Pe− N

∂ ∂X dY dZ dY dZ Z max Y max Pe+ N + − N − Vx dY dZ − Jconst + 0 0 Z max Y max N+ 0 0 + Pe+ − Pe− N (4.16c) dY dZ Chapter 4. Single phase electrokinetic flow in microchannels 155 As an example, let us consider a lx = 50 µm, ly = lz = 25 µm computational domain, as shown in Fig. 4.12, containing a 10−5 M KCl solution and with an applied driving voltage of 500 V/cm and a homo = −4 × 10−4 C/m2 . The numerical simulations of the above described model revealed three distinct flow patterns, depending on the degree of surface heterogeneity, as shown in Fig. 4.13. The grey scales in these figures represent the magnitude of the velocity perpendicular to the direction of the applied electric field (v2 = vy2 + vz2 ) scaled by the maximum velocity in a homogeneous channel, which in this example is 1.7 mm/s. At low degrees of surface heterogeneity (hetero = −2 × 10−4 C/m2 ) the streamline pattern shown in Fig. 4.13(a) was obtained. As can be seen a net counter-clockwise flow perpendicular to the applied electric field is present at the first transition plane (i.e. at the initial discontinuity in the heterogeneous surface pattern) and a clockwise flow at the second transition plane. This flow circulation is a pressure-induced effect. It results from the transition from the higher local fluid velocity (particularly in the double layer) region over the homogeneous surface on the right-hand side at the entrance to the left-hand side after the first transition plane (and vice versa at the second transition plane). To satisfy continuity then there must be a net flow from right to left at this point, which in this case takes the form of the circulation discussed above. The relatively straight streamlines parallel with the applied electric field indicate that at this level the heterogeneity is too weak to significantly disrupt the main flow. While a similar circulation pattern at the transition planes was observed as the degree of surface heterogeneity was increased into the intermediate range (−2 × 10−4 C/m2 = hetero = +2 × 10−4 C/m2 ), it is apparent from the darker contours shown in Fig. 4.13(b) that the strength of the flow perpendicular to the applied electric field is significantly stronger reaching nearly 50% of the velocity in the homogeneous channel. Unlike in the previous case it is now apparent that the streamlines parallel with the applied electric field are significantly distorted due to the much slower or even oppositely directed velocity over the heterogeneous patch. At even higher degrees of heterogeneity (+2 × 10−4 C/m2 = hetero = +4 × 10−4 C/m2 ) a third flow structure is observed in which a dominant circulatory flow pattern exists along all three coordinate axes, as shown in Fig. 4.13(c). This results in a negligible, or even nonexistent, bulk flow in the direction of the applied electric field (which is to be expected since the average surface charge density for these cases is very near zero). As indicated by the grey scales, the velocity perpendicular to the flow axis has again increased in magnitude, reaching a maximum at the edge of the double layer near the symmetry planes at the location where a step change in the surface charge density has been imposed. It is also shown (Erickson and Li, 2003a) that the induced 3D flow patterns are limited to a layer near the surface with a thickness equivalent to the length scale of the heterogeneous patch. Additionally, the effect becomes smaller in magnitude and more localized as the average size of the heterogeneous region decreases. Moreover, it was demonstrated that while convective effects are small, the electrophoretic influence of the applied electric field could distort the net charge density field near the surface, resulting in a significant deviation from the traditional Poisson–Boltzmann double layer distribution. That is the reason why the Nernst–Planck conservation equations, instead of the Boltzmann distribution equation, were used in this study. 156 Heat transfer and fluid flow in minichannels and microchannels Channel height (microns) 0.5 0.4 20 0.3 10 0.2 0 40 20 0.1 20 10 0 0 0 (a) Channel height (microns) 0.5 0.4 20 0.3 10 0.2 40 0 20 0.1 20 10 0 0 0 (b) Channel height (microns) 0.5 0.4 20 0.3 10 0.2 0 40 20 0.1 20 10 0 0 0 (c) Fig. 4.13. Electroosmotic flow streamlines over a patchwise heterogeneous surface pattern with homo = −4 × 10−4 C/m2 and (a) hetero = −2 × 10−4 C/m2 (b) hetero = +2 × 10−4 C/m2 (c) hetero = +4 × 10−4 C/m2 . The grey scale represents magnitude of velocity perpendicular to the applied potential field, scaled by the maximum velocity in a homogeneous channel. Arrow represents direction of applied electric field. Chapter 4. Single phase electrokinetic flow in microchannels 157 4.6. AC electroosmotic flow Electroosmotic flow induced by unsteady applied electric fields or electroosmotic flow under an alternating electrical (AC) field has its unique features and applications. Oddy et al. (2001) proposed and experimentally demonstrated a series of schemes for enhanced species mixing in microfluidic devices using AC electric fields. In addition Oddy et al. (2001) also presented an analytical flow field model, based on a surface slip condition approach, for an axially applied (i.e. along the flow axis) AC electric field in an infinitely wide microchannel. Comprehensive models for such a slit channel have also been presented by Dutta and Beskok (2001), who developed an analytical model for an applied sinusoidal electric field, and Söderman and Jönsson (1996), who examined the transient flow field caused by a series of different pulse designs. As an alternative to traditional DC electroosmosis, a series of novel techniques have been developed to generate bulk flow using AC fields. For example Green et al. (2000) experimentally observed peak flow velocities on the order of hundreds of micrometers per second near a set of parallel electrodes subject to two AC fields, 180 out-of-phase with each other. The effect was subsequently modeled using a linear double layer analysis by Gonzàlez et al. (2000). Using a similar principal, both Brown et al. (2002) and Studer et al. (2002) presented microfluidic devices that incorporated arrays of nonuniformly sized embedded electrodes which, when subject to an AC field, were able to generate a bulk fluid motion. Also using embedded electrodes, Selvaganapathy et al. (2002) demonstrated what they termed fr-EOF or bubble-free electroosmotic flow in which a creative periodic waveform was used to yield a net bulk flow while electrolytic bubble formation was theoretically eliminated. Recently, Erickson and Li presented a combined theoretical and numerical approach to investigate the time periodic electroosmotic flow in a rectangular microchannel (Erickson and Li, 2003b). This work considers the time-periodic electroosmotic flow in a straight, rectangular microchannel. Assuming a fully developed flow field and considering the geometric symmetry, the analytical domain can be reduced to the upper left-hand quadrant of the channel cross-section as shown in Fig. 4.14. For pure electroosmotic flows (i.e. absent z2 Analytical domain Y ly z1 O X lx Fig. 4.14. Illustration of the analytical domain in AC electroosmotic flow in a rectangular microchannel. 158 Heat transfer and fluid flow in minichannels and microchannels of any pressure gradients) of incompressible liquids, the Navier–Stokes equations may take the following form: f ∂v = µÑ 2 v + e E(t) ∂t (4.17) where v is the flow velocity, t is time, f is the fluid density, e is the net charge density, µ is the fluid viscosity and E(t) is a general time-periodic function with a frequency = 2f and describes the applied electric field strength. To solve this flow equation, we must know the local net charge density e . This requires solving the EDL field. Introducing the non-dimensional double layer potential, = ze/kb T , and non-dimensional double layer thickness K = Dh (where Dh is the hydraulic diameter of the channel, Dh = 4lx ly /2(lx + ly ), and is the Deybe–Hückel parameter). We obtain the non-linear Poisson–Boltzmann distribution equation: ј 2 − K 2 sinh = 0 (4.18) where the ~ signifies that the spatial variables in the gradient operator have been nondimensionalized with respect to Dh (i.e. X = x/Dh , Y = y/Dh ). Eq. (4.18) is subject to the symmetry conditions along the channel center axes, that is, ∂/∂X = 0 at X = 0 ∂/∂Y = 0 at Y = 0, and the appropriate conditions at the channel walls, = Z1 at X = Lx /2 = Z2 at Y = Ly /2 where Z1,2 = ze1,2 /kb T , Lx = lx /Dh and Ly = ly /Dh . Generally, for multi-dimensional space and complex geometry Eq. (4.18) must be solved numerically. However, under the condition that the double layer potential, = ze/kb T , is small, Eq. (4.18) can be linearized (the so-called the Deybe– Hückel approximation), yielding, ј 2 − K 2 = 0 (4.19) The Deybe–Hückel approximation is considered valid only when |ze| < kb T , which in principal is limited to cases where || ≤ 25 mV at room temperature. For the rectangular geometry shown in Fig. 4.14 an analytical solution to Eq. (4.19), subject to the aforementioned boundary conditions, can be obtained using a separation of variables technique: ∞ 4(−1)n+1 cosh([2n + K 2 ]1/2 Y ) Z1 cos(n X ) (X , Y ) = (2n − 1) cosh([2n + K 2 ]1/2 LY ) n=1 ∞ 4(−1)n+1 cosh([µ2n + K 2 ]1/2 X ) (4.20) Z2 cos(µn Y ) + (2n − 1) cosh([µ2n + K 2 ]1/2 LX ) n=1 Chapter 4. Single phase electrokinetic flow in microchannels 159 where n and µn are the eigenvalues given by n = (2n − 1)/2LX and µn = (2n − 1)/ 2LY , respectively. Introducing the non-dimensional time, = µt/f Dh2 , and the nondimensional frequency, = f Dh2 /µ, scaling the flow velocity by the electroosmotic slip velocity, V = vzeµ/Ez kb T , where Ez is a constant equivalent to the strength of the applied electric field, and combining Eq. (4.17) with the Poisson equation: Ñ 2 = e and Eq. (4.18) yield the non-dimensional flow equation, ∂V = ј 2 V − K 2 sinh F() ∂ (4.21) where F is a general periodic function of unit magnitude such that E() = Ez F(). Along the channel axes Eq. (4.21) is subject to a symmetry boundary conditions, ∂V /∂X = 0 at X = 0 ∂V /∂Y = 0 at Y = 0 while no-slip conditions are applied along the solid channel walls, V = 0 at X = Lx /2 and Y = Ly /2 In order to obtain an analytical solution to Eq. (4.21), the Deybe–Hückel approximation is implemented and F() is chosen to be a sinusoid function, resulting in the following form of the equation: ∂V = ј 2 V − K 2 (X , Y )sin (4.22) ∂ Using a Green’s Function approach, an analytical solution to Eq. (4.22), subject to the homogeneous boundary conditions discussed above, can be obtained ∞ V (X , Y , ) = ∞ ∞ −16K 2 (−1)n+1 cos(l X )cos(µm Y ) LX LY (2n − 1) l=1 m=1 n=1 2l + µ2m sin() − cos() + exp − 2l + µ2m [f1mn + f2ln ] × 2 2 l + µ2m + 2 (4.23) where f1mn /f2ln are given by: ì ï í0 f1mn = Z1 µm (−1)m+1 LX ï î 2 n + µ2m + K 2 2 ì ï í0 f2ln = Z2 l (−1)l+1 LY ï î 2 µn + 2l + K 2 2 ü n = m ï ý n=m ï þ ü n = l ï ý n=lï þ 160 Heat transfer and fluid flow in minichannels and microchannels Equation (4.23) represents the full solution to the transient flow problem. The equation can be somewhat simplified in cases where the quasi-steady state time periodic solution (i.e. after the influence of the initial conditions has dissipated) is of interest. As these initial effects are represented by the exponential term in Eq. (4.23), the quasi-steady state time periodic solution has the form shown below: V (X , Y , ) = ∞ ∞ ∞ −16K 2 (−1)n+1 cos(l X )cos(µm Y ) LX LY (2n − 1) l=1 m=1 n=1 2l + µ2m sin() − cos() [ f1mn + f2ln ] × 2 2 l + µ2m + 2 (4.24) In the above analytical description of the uniaxial electroosmotic flow in a rectangular microchannel, the governing parameter is which represents the ratio of the diffusion time scale (tdiff = f Dh2 /µ) to the period of the applied electric field (tE = 1/). Figure 4.15 compares the time-periodic velocity profiles (as computed from Eq. (4.24)) in the upper left-hand quadrant of a square channel for two cases: (a) = 31 and (b) = 625. These two values correspond to frequencies of 500 Hz and 10 kHz in a 100 µm square channel or equivalently a 100 µm and a 450 µm square channel at 500 Hz. To illustrate the essential features of the velocity profile a relatively large double layer thickness has been used, = 3 × 106 m−1 (corresponding to a bulk ionic concentration n∞ = 10−6 M), and a uniform surface potential of = −25 mV was selected (within the bounds imposed by the Deybe–Hückel linearization). From Fig. 4.15, it is apparent that the application of the electrical body force results in a rapid acceleration of the fluid within the double layer. In the case where the diffusion time scale is much greater than the oscillation period (high , Fig. 4.15(b)) there is insufficient time for fluid momentum to diffuse far into the bulk flow and thus while the fluid within the double layer oscillates rapidly the bulk fluid remains almost stationary. At = 31 there is more time for momentum diffusion from the double layer, however the bulk fluid still lags behind the flow in the double layer (this out-ofphase behavior will be discussed shortly). Extrapolating from these results, when < 1, such that momentum diffusion is faster than the period of oscillation, the plug type velocity profile characteristic of steady state electroosmotic flow would be expected at all times. Another interesting feature of the velocity profiles shown in Fig. 4.15 is the local velocity maximum observed near the corner (most clearly visible in the = 625 case at t = /2 and t = /). The intersection of the two walls results in a region of double layer overlap and thus an increased net charge density over that region in the double layer. This peak in the net charge density corresponds to a larger electrical body force and hence a local maximum electroosmotic flow velocity, it also increases the ratio of the electrical body force to the viscous retardation allowing it to respond more rapidly to changes in the applied electric field. The finite time required for momentum diffusion will inevitably result in some degree of phase shift between the applied electric field and the flow response in the channel. From Fig. 4.15, however, it is apparent that within the limit of > 1, this phase shift is significantly different in the double layer region than in the bulk flow. It is apparent that the response of the fluid within the double layer to the AC field is essentially immediate, Non-dimensional velocity Non-dimensional velocity Chapter 4. Single phase electrokinetic flow in microchannels 1 0.5 0 0.5 1 1 0.5 0 p/2 t v 0.5 1 0.4 0.4 0.4 0.2 0 0 Non-dimensional velocity Non-dimensional velocity Y 1 0.5 0 0.5 1 X 1 0.5 p tv 0 0.5 1 0.4 0.4 0.2 0 0 X 1 0 0.5 1 0.2 0 0 Y 0.5 0.4 0.2 0.2 Non-dimensional velocity Y Non-dimensional velocity 0.2 0 0 X 0.4 X 1 0.5 3p/2 t v 0 0.5 1 0.4 0.4 0.4 0.2 0.4 0.2 0.2 0 0 0.2 0 0 X Y Non-dimensional velocity Y Non-dimensional velocity 0.4 0.2 0.2 Y 1 0.5 0 0.5 1 X 1 0.5 2p t v 0 0.5 1 0.4 0.4 0.4 0.2 0.4 0.2 0.2 0.2 0 0 0 0 Y (a) 161 X Y X (b) Fig. 4.15. Steady-state time periodic electroosmotic velocity profiles in a square microchannel at (a) = 30 and (b) = 625 (equivalent to an applied electric field frequency of (a) 500 Hz and (b) 10 kHz in a 100 µm square channel). 162 Heat transfer and fluid flow in minichannels and microchannels however, the bulk liquid lags behind the applied field by a phase shift depending on the value. Additionally while the velocity in the double layer reaches its steady state oscillation almost immediately, the bulk flow requires a period before the transient effects are dissipated. In Eq. (4.24) the out-of-phase cosine term ( cos()) is proportionally scaled by , thus as expected when is increased, the phase shift for both the double layer and bulk flow velocities is increased as is the number of cycles required to reach the steady state. It is interesting to note the net positive velocity at the channel midpoint within the transient period before decaying into the steady state behavior. This is a result of the initial positive impulse given to the system when the electric field is first applied and is reflected by the exponential term in Eq. (4.23). The transient oscillations were observed to decay at an exponential rate, as expected from this transient term in Eq. (4.23). Similar to the out-of-phase cosine term, this exponential term is also proportionally scaled by the non-dimensional frequency, suggesting that the effect of the initial impulse becomes more significant with increasing . Non-dimensional velocity (V ) 0.6 0.4 0.2 0 0.2 0.4 0.6 Non-dimensional velocity (V ) (a) 0.03 0.02 0.01 0 0.01 0.02 0.03 (b) 0 10 20 30 40 vt (rads) 50 60 Fig. 4.16. Transient stage velocity at channel midpoint for impulsively started flows using different waveforms at (a) = 30 and (b) = 625. Solid line represents sin wave, long dashed line represents triangular wave and doted line represents square wave. Chapter 4. Single phase electrokinetic flow in microchannels 163 The results presented above are limited to sinusoidal waveforms. To examine how the flow field will respond to different forms of periodic excitation, such as a square (step) or triangular waveforms, numerical solutions to Eqs. (4.17) and (4.18) are required. Figure 4.15 shows that the particular waveform also has a significant effect on the transient response of the bulk fluid (here the channel mid point is chosen as a representative point). As can be seen in Fig. 4.16(a) for the = 31 case, a square wave excitation tends to produce higher velocities whereas the triangular wave exhibits slightly smaller bulk velocities when compared with the sinusoidal waveform. As is increased, the initial positive impulsive velocity is observed for both additional waveforms as shown in Fig. 4.16(b). As expected the fluid excited by the square waveform exhibits higher instantaneous velocities, which lead to an increase in the number of cycles required to reach the time periodic quasi-steady state oscillation. 4.7. Electrokinetic mixing Let us consider a simple T-shaped microfluidic mixing system (Erickson and Li, 2002b). Without loss of the generality, we will consider that two electrolyte solutions of the same flow rate enter a T-junction separately from two horizontal microchannels, and then start mixing while flowing along the vertical microchannel, as illustrated in Fig. 4.17. The flow is generated by the applied electrical field via electrodes at the upstream and the downstream positions. This simple arrangement has been used for numerous applications including the dilution of a sample in a buffer (Harrison et al., 1993), the development of complex species gradients (Jeon et al., 2000; Dertinger et al., 2001), and measurement of the diffusion coefficient (Kamholz et al., 2001). Generally most microfluidic mixing systems are limited to the low Reynolds number regime and thus species mixing is strongly diffusion dominated, as opposed to convection or turbulence dominated at higher Reynolds numbers. Consequently mixing tends to be slow and occur over relatively long distances and time. As an example the concentration gradient generator presented by Dertinger et al. (2001) required a mixing channel length on the order of 9.25 mm for a 45 µm × 45 µm crosssectional channel or approximately 200 times the channel width to achieve nearly complete mixing. Enhanced microfluidic mixing over a short flow distance is highly desirable for lab-on-a-chip applications. One possibility of doing so is to utilize the local circulation flow caused by the surface heterogeneous patches. To model such an electroosmotic flow and mixing process, we need the following equations. The flow field is described by the Navier–Stokes Equations and the continuity Equation (given below in non-dimensional form): Re ∂V ˜ ˜ + ј 2 V + (V · Ñ)V = −ÑP ∂ ˜ =0 ÑV (4.25) (4.26) where V is the non-dimensional velocity (V = v/veo , where veo is calculated using Eq. (4.27) given below), P is the non-dimensional pressure, is the non-dimensional time and Re is 164 Heat transfer and fluid flow in minichannels and microchannels w Inlet stream 1 Inlet stream 2 w Larm Mixing channel Y Lmix X Mixed stream Fig. 4.17. T-Shaped micromixer formed by the intersection of two microchannels, showing a schematic of the mixing or dilution process. the Reynolds number given by Re = veo L/ where L is a length scale taken as the channel width (w from Fig. 4.17) in this case. The ~ symbol over the Ñ operator indicates the gradient with respect to the non-dimensional coordinates (X = x/w, Y = y/w and Z = z/w). It should be noted that in order to simplify the numerical solution to the problem, we have treated the electroosmotic flow in the thin EDL as a slip flow velocity boundary condition, given by: veo = µeo Ñ = w Ñ µ (4.27) where µeo = (w /µ) is the electroosmotic mobility, w is the electrical permittivity of the solution, µ is the viscosity, is the zeta potential of the channel wall, and is the applied electric field strength. In general the high voltage requirements limit most practical electroosmotically driven flows in microchannels to small Reynolds numbers, therefore to simplify Eq. (4.25) we ignore transient and convective terms and limit ourselves to cases where Re < 0.1. We consider the mixing of equal portions of two buffer solutions, one of which contains a species of interest at a concentration, co . Species transport by electrokinetic means is accomplished by three mechanisms: convection, diffusion and electrophoresis, and can be described by:

∂C ˜ Pe + Ñ · (C(V + Vep )) = ј 2 C ∂ (4.28) 0 0 100 100 200 300 Downstream distance (microns) Downstream distance (microns) Chapter 4. Single phase electrokinetic flow in microchannels 400 200 300 400 500 (a) 165 500 (b) Fig. 4.18. Electroosmotic streamlines at the mid-plane of a 50 µm T-shaped micromixer for the (a) homogeneous case with = −42 mV, (b) heterogeneous case with six offset patches on the leftand right-channel walls. All heterogeneous patches are represented by the crosshatched regions and have a = + 42 mV. The applied voltage is app = 500 V/cm. where C is the non-dimensional species concentration (C = c/co , where co is original concentration of the interested species in the buffer solution.), Pe is the Péclet number (Pe = veo w/D, where D is the diffusion coefficient), and Vep is the non-dimensional electrophoretic velocity equal to vep /veo where vep is given by: vep = µep Ñ (4.29) and µep = (w p /) is the electrophoretic mobility (w is the electrical permittivity of the solution, µ is the viscosity, p is the zeta potential of the to-bemixed charged molecules or particles) (Hunter, 1981). As we are interested in the steady state solution, the transient term in Eq. (4.28) can be ignored. The above described model was solved numerically to investigate the formation of electroosmotically induced flow circulation regions near surface heterogeneities in a T-shaped micromixer and to determine the influence of these regions on the mixing effectiveness. In Fig. 4.18 we compare the mid-plane flow field near the T-intersection of a homogeneous mixing channel with that of a mixing channel having a series of 6 asymmetrically 166 Heat transfer and fluid flow in minichannels and microchannels (a) (b) Fig. 4.19. 3D species concentration field for a 50 µm × 50 µm T-shaped micromixer resulting from the flow fields shown in Fig. 4.23. (a) homogeneous case and (b) heterogeneous case with offset patches. Species diffusivity is 3 × 10−10 m2 /s and zero electrophoretic mobility are assumed. distributed heterogeneous patches on the left- and right-channel walls. For clarity the heterogeneous regions are marked as the crosshatched regions in this figure. The homogeneous channel surface has a potential of −42 mV. A -potential of = +42 mV was assumed for the heterogeneous patches. Apparently, the channel with heterogeneous patches generates local flow circulations near the patches. These flow circulation zones are expected to enhance the mixing of the two streams. Figure 4.19 compares the 3D concentration fields of the homogeneous and heterogeneous mixing channel shown in Fig. 4.18. In these figures a neutral mixing species (i.e. µep = 0, thereby ignoring any electrophoretic transport) with a diffusion coefficient D = 3 × 10−10 m2 /s is considered. While mixing in the homogeneous case is purely diffusive in nature, the presence of the asymmetric circulation regions, Fig. 4.19(b), enables enhanced mixing by convection. Recently a passive electrokinetic micromixer based on the use of surface charge heterogeneity was developed (Biddiss et al., 2004). The micromixer is a T-shaped microchannel structure (200 µm in width and approximately 8 µm in depth) made from Polydimethylsiloxane (PDMS) and is sealed with a glass slide. Microchannels were fabricated using a rapid prototyping/soft-lithography technique. The glass surface was covered by a PDMS mask with the desired heterogeneous pattern, then treated with a Polybrene solution. After removing the mask, the glass surface will have selective regions of positive surface charge while leaving the majority of the glass slide with its native negative charge (Biddiss et al., 2004). Finally the PDMS plate (with the microchannel structure) will be bonded to the glass slide to form the sealed, T-shaped microchannel with heterogeneous patches on the mixing channel surface. A micromixer consisting of 6 offset staggered patches (in the mixing channel) spanning 1.8 mm downstream and offset 10 µm from the channel centerline with a width of 90 µm Chapter 4. Single phase electrokinetic flow in microchannels (a) 167 (b) Fig. 4.20. Images of steady-state species concentration fields under an applied potential of 280 V/cm for (a) the homogeneous microchannel and (b) the heterogeneous microchannel with six offset staggered patches. and a length of 300 µm, was analyzed experimentally. Mixing experiments were conducted at applied voltage potentials ranging between 70 and 555 V/cm and the corresponding Reynolds numbers range from 0.08 to 0.7 and Péclet numbers from 190 to 1500. The liquid is a 25 mM sodium carbonate/bicarbonate buffer. To visualize the mixing effects, 100 µM fluorescein was introduced through one inlet channel. As an example, Fig. 4.20 shows the experimental images of the steady state flow for the homogenous and heterogeneous cases at 280 V/cm. The enhanced mixing effect is obvious. This experimental study shows that the passive electrokinetic micromixer with an optimized arrangement of surface charge heterogeneities can increase flow narrowing and circulation, thereby increasing the diffusive flux and introducing an advective component of mixing. Mixing efficiencies were improved by 22–68% for voltages ranging from 70 to 555 V/cm. For producing a 95% mixture, this technology can reduce the required mixing channel length of up to 88% for flows with Péclet numbers between 190 and 1500, and Reynolds numbers between 0.08 and 0.7. In terms of required channel lengths, at 280 V/cm, a homogeneous microchannel would require a channel mixing length of 22 mm for reaching a 95% mixture. By implementing the developed micromixer, an 88% reduction in required channel length to 2.6 mm was experimentally demonstrated. Practical applications of reductions in required channel lengths include improvements in portability and shorter retention times, both of which are valuable advancements applicable to many microfluidic devices. 168 Heat transfer and fluid flow in minichannels and microchannels 4.8. Electrokinetic sample dispensing An important component of many bio- or chemical lab-chips is the microfluidic dispenser, which employs electroosmotic flow to dispense minute quantities (e.g. 300 pico-liters) of samples for chemical and biomedical analysis. The precise control of the dispensed sample in microfluidic dispensers is key to the performance of these lab-on-a-chip devices. Let us consider a microfluidic dispenser formed by two crossing microchannels as shown in Fig. 4.21 (Ren and Li, 2002). The depth and the width of all the channels are chosen to be 20 µm and 50 µm, respectively. There are four reservoirs connected to the four ends of the microchannels. Electrodes are inserted into these reservoirs to set up the electrical field across the channels. Initially, a sample solution (a buffer solution with sample species) is filled in Reservoir 1, the other reservoirs and the microchannels are filled with the pure buffer solution. When the chosen electrical potentials are applied to the four reservoirs, the sample solution in Reservoir 1 will be driven to flow toward Reservoir 3 passing through the intersection of the cross channels. This is the so-called loading process. After the loading process reaches a steady state, the sample solution loaded in the intersection will be “cut” or dispensed into the dispensing channel by the dispensing solution flowing from Reservoir 2 to 4. This can be realized by adjusting the electrical potentials applied to these four reservoirs. This is the so called the dispensing process. The volume and the concentration of the dispensed sample are the key parameters of this dispensing process, and they depend on the applied electrical field, the flow field and the concentration field during the loading and the dispensing processes. To model such a dispensing process, we must model the applied electrical field, the flow field and the concentration field. To simplify the analysis, we consider this as a 2D problem, y Reservoir 4 z x Reservoir 1 Wy Reservoir 3 Wx Reservoir 2 Fig. 4.21. The schematic diagram of a crossing microchannel dispenser. Wx and Wy indicate the width of the microchannels. Chapter 4. Single phase electrokinetic flow in microchannels 169 that is, ignoring the variation in the z-direction. The 2D applied electrical potential in the liquid can be described by: ∂2 * ∂2 * + =0 ∂x*2 ∂y*2 Here the non-dimensional parameters are defined by: * = , x* = x , h y* = y h where is a reference electrical potential and h is the channel width chosen as 50 µm. Boundary conditions are required to solve this equation. We impose the insulation condition to all the walls of microchannels, and the specific non-dimensional potential values to all the reservoirs. Once the electrical field in the dispenser is known, the local electric field strength can be calculated by = −Ñ E The electroosmotic flow field reaches steady state in milli-seconds, much shorter than the characteristic time scales of the sample loading and sample dispensing. Therefore, the electroosmotic flow here is approximated as steady state. Furthermore, we consider a thin EDL, and use the slip flow boundary condition to represent the electroosmotic flow. The liquid flow field can thus be described by the following non-dimensional momentum equation and the continuity equation: * ueo * * * ∂ueo ∂u* ∂P * ∂2 ueo ∂2 ueo + v*eo eo = − * + *2 + *2 * * ∂x ∂y ∂x ∂x ∂y * ueo * ∂v*eo ∂P * ∂2 v*eo ∂2 v*eo * ∂veo + v = − + + eo ∂x* ∂y* ∂y* ∂x*2 ∂y*2 * ∂ueo ∂v* + eo =0 * ∂x ∂y* where ueo , veo are the electroosmotic velocity component in x- and y-direction, respectively, and non-dimensionalized as follows: P* = P − Pa , (v/h)2 * ueo = ueo h , v v*eo = veo h v The slip velocity conditions are applied to the walls of the microchannels, the fully developed velocity profile is applied to all the interfaces between the microchannels and the reservoirs, and the pressures in the four reservoirs are assumed to be the atmospheric pressure. The distribution of the sample concentration can be described by the conservation law of mass, taking the following form: ∂C * ∂C * ∂Ci* * ∂2 Ci* Di ∂2 Ci* * * * i i + ueo + uep + veo + vep = + *2 ∂ ∂x * ∂y* v ∂x*2 ∂y 170 Heat transfer and fluid flow in minichannels and microchannels R4 R3 Y (mm) 0.465 0.47 0.474 0.485 R1 0.476 10.05 0.48 10.05 0.478 10.1 0.49 Y (mm) R4 10.1 R3 R1 10 10 9.95 2.45 2.5 R2 2.55 9.95 2.45 2.6 2.5 R2 2.55 2.6 X (mm) X (mm) R4 R4 10.1 10.1 0.281 0.282 0.284 R3 0.285 10 Y (mm) 25 0.28 R1 10.05 0.283 0.2825 Y (mm) 10.05 R3 R1 10 0.287 0.2895 0.291 9.95 2.45 2.5 2.55 R2 X (mm) 2.6 9.95 2.45 2.5 R2 2.55 2.6 X (mm) Fig. 4.22. Examples of the applied electrical field (left) and the flow field (right) at the intersection of the microchannels in a loading process (top) and in a dispensing process (bottom). where Ci is the concentration of the ith species, ueo and veo are the components of the electroosmotic velocity of the ith species, Di is the diffusion coefficient of the ith species, and uepi and vepi are the components of the electrophoretic velocity of the ith species given by uepi = Eµepi , where µepi is the electrophoretic mobility. The non-dimensional and = t/(h2 /v), where C is a parameters in the above equation are defined by C * = C/C, reference concentration. Figure 4.22 shows the typical electrical field and flow field (computer simulated) for the loading and dispensing process, respectively. In this figure, the non-dimensional applied electrical potentials are: * (1) = 1.0, * (2) = 1.0, * (3) = 0.0, * (4) =1.0 for the loading process, and * (1) = 0.2, * (2) = 2.0, * (3) = 0.2, * (4) = 0.0 for the dispensing process, where * (i) represents the non-dimensional applied electrical potential to ith reservoir. For this specific case, the electrical field and the flow field for the loading process are symmetric to the middle line of the horizontal channel, and the electrical field and the flow field for the dispensing process are symmetric to the middle line of the vertical channel. Chapter 4. Single phase electrokinetic flow in microchannels Loading (steady state) 1 (a) 171 dispensing 4 3 (b) (c) Fig. 4.23. The loading and dispensing of a focused fluorescein sample: (a) Processed images; (b) iso-concentration profiles at 0.1Co , 0.3Co , 0.5Co , 0.7Co , and 0.9Co , calculated from the images and (c) corresponding iso-concentration profiles calculated through numerical simulation. The electrokinetic dispensing processes of fluorescent dye samples were investigated experimentally (Sinton et al., 2003b, c, d). The measurements were conducted by using a fluorescent dye-based microfluidic visualization system. Fig. 4.23 shows a sample dispensing process and the comparison of the dispensed sample concentration profile with the numerically simulated results. Both the theoretical studies and the experimental studies have demonstrated that the loading and dispensing of sub-nanolitre samples using a microfluidic crossing microchannel chip can be controlled electrokinetically. The ability to inject and transport large axial extent, concentration-dense samples was demonstrated. Both experimental and numerical results indicate the shape, cross-stream uniformity, and axial extent of the samples were very sensitive to changes in the electric fields applied in the loading channel. In the dispensing process, larger samples were shown to disperse less than focused samples, maintaining more solution with the original sample concentration. 4.9. Practice problems 1. Model and simulate 1D steady state electroosmotic flow in a circular capillary under an applied DC field. Plot and discuss the EDL field and flow field, effects of zeta potential, bulk ionic concentration, applied voltage and diameter. 2. Model and simulate a 1D steady state electroosmotic flow in a microchannel under an applied DC field and a small externally applied pressure difference along the channel. Plot and discuss the flow field in terms of the value and the direction of the applied pressure gradient, zeta potential and the applied electrical field. 172 Heat transfer and fluid flow in minichannels and microchannels 3. Model and simulate 1D electroosmotic flow in a circular capillary under an applied AC field. Plot and discuss the flow field, effects of the zeta potential, the bulk ionic concentration, the AC voltage and frequency, and diameter of the capillary. 4. Model and simulate a steady state electroosmotic flow in a microchannel with a 90 degree turn under an applied DC field. Plot and discuss the flow field in terms of the zeta potential, the channel size and the applied electrical field. 5. 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Söderman, O. and Jönsson, B., Electro-osmosis: velocity profiles in different geometries with both temporal and spatial resolution, J. Chem. Phys., 105, 10300–10311, 1996. Stroock, A. D., Weck, M., Chiu, D. T., Huck, W. T. S., Kenis, P. J. A., Ismagilov, R. F., and Whitesides, G. M., Patterning electro-osmotic flow with patterned surface charge, Phys. Rev. Lett., 84, 3314–3317, 2000. Studer, V., Pépin, A., Chen, Y., and Ajdari, A., Fabrication of microfluidic devices for AC electrokinetic fluid pumping, Microelectronic Eng., 61–62, 915–920, 2002. Taylor, J. A. and Yeung, E. S., Imaging of hydrodynamic and electrokinetic flow profiles in capillaries, Anal. Chem., 65, 2928–2932, 1993. Wereley, S. T. and Meinhart, C. D., Second-order accurate particle image velocimetry, Exp. Fluids, 31, 258–268, 2001. Chapter 5 FLOW BOILING IN MINICHANNELS AND MICROCHANNELS Satish G. Kandlikar Mechanical Engineering Department, Rochester Institute of Technology, Rochester, NY, USA 5.1. Introduction Flow boiling in minichannels is of great interest in compact evaporator applications. Automotive air-conditioning evaporators use small passages with plate-fin heat exchangers. Extruded channels with passage diameters smaller than 1 mm are already being applied in compact condenser applications. Developments in evaporators to this end are needed to overcome the practical barriers associated with flow boiling in narrow channels. Another application where flow boiling research is actively being pursued is in the heat removal from high heat flux devices, such as computer chips, laser diodes and other electronics devices and components. Flow boiling is attractive over single-phase liquid cooling from two main considerations: (i) High heat transfer coefficient during flow boiling. (ii) Higher heat removal capability for a given mass flow rate of the coolant. Although the heat transfer coefficients are quite high in single phase flow with small diameter channels, flow boiling yields much higher values. For example, the single-phase heat transfer coefficient under laminar flow of water in a 200 µm square channel is around 10,000 W/m2 C (Fig. 1.2), whereas the flow boiling heat transfer coefficients can exceed 100,000 W/m2 C (Steinke and Kandlikar, 2004). In other words, larger channel diameters can be implemented with flow boiling at comparable or even higher heat transfer coefficients than single-phase systems. This feature becomes especially important in view of the filtration requirements to keep the channels clean. Another major advantage of flow boiling systems is the ability of the fluid to carry larger amounts of thermal energy through the latent heat of vaporization. With water, the latent heat is significantly higher (2257 kJ/kg) than its specific heat of 4.2 kJ/kg C E-mail: [email protected] 175 176 Heat transfer and fluid flow in minichannels and microchannels at 100 C. This feature is especially important for refrigerant systems. Although the latent heat of many potential refrigerants is around 150–300 kJ/kg at temperatures around 30–50 C, it still compares favorably with the single-phase cooling ability of water. However, the biggest advantage is that a suitable refrigerant can be chosen to provide desirable evaporation temperatures, typically below 50 C, without employing a deep vacuum as would be needed for flow boiling with water. Further research in identifying/developing specific refrigerants is needed. The specific desirable properties of a refrigerant for flow boiling application are identified by Kandlikar (2005) and are discussed in Section 5.9. 5.2. Nucleation in minichannels and microchannels There are two ways in which flow boiling in small diameter channels is expected to be implemented. They are: (i) two-phase entry after a throttle valve, (ii) subcooled liquid entry into the channels. The first mode is applicable to the evaporators used in refrigeration cycles. The throttle valve prior to the evaporator can be designed to provide subcooled liquid entry, but more commonly, a two-phase entry with quality between 0 and 0.1 is employed. Although this represents a more practical system, the difficulties encountered with proper liquid distribution in two-phase inlet headers have been a major obstacle to achieving stable operation. Even liquid distribution in the header provides a more uniform liquid flow through each channel in a parallel channel arrangement. Subcooled liquid entry is an attractive option, since the higher heat transfer coefficients associated with subcooled flow boiling can be utilized. Such systems are essentially extensions of single-phase systems and rely largely on the temperature rise of the coolant in carrying the heat away. In either of the systems, bubble nucleation is an important consideration. Even with a two-phase entry, it is expected that a slug flow pattern will prevail, and nucleation in liquid slugs will be important. With subcooled liquid entry, early nucleation is desirable for preventing the rapid bubble growth that has been observed by many investigators. An exhaustive review of this topic is provided by Kandlikar (2002a, b). A number of researchers have studied the flow boiling phenomena (e.g. Lazarek and Black, 1982; Cornwell and Kew, 1992; Kandlikar et al., 1995; Kuznetsov and Vitovsky, 1995; Cuta et al., 1996; Kew and Cornwell, 1996; Kandlikar and Spiesman, 1997; Kandlikar et al., 1997; Kasza et al., 1997; Lin et al., 1998; 1999; Jiang et al., 1999; Kamidis and Ravigururajan, 1999; Lakshminarasimhan et al., 2000; Kandlikar et al., 2001; Hetsroni et al., 2002). The inception of nucleation plays an important role in flow boiling stability, as will be discussed in a later section. The nucleation criteria in narrow channels have been studied by a number of investigators, and it is generally believed that there are no significant differences from the conventional theories for large diameter tubes proposed by Bergles and Rohsenow (1962; 1964), Sato and Matsumura (1964), and Davis and Anderson (1966). These theories are extensions of the pool boiling nucleation models proposed by Hsu and Graham (1961), and Hsu (1962). Kenning and Cooper (1965), and Kandlikar et al. (1997) Chapter 5. Flow boiling in microchannels and minichannels 177 have suggested further modifications based on the local temperature field in the vicinity of a nucleating bubble under various flow conditions. Consider subcooled liquid entering a small hydraulic diameter channel at an inlet temperature TB,i . Assuming (i) constant properties, (ii) uniform heat flux, and (iii) steady conditions, the bulk temperature TB, z along the flow length z is given by the following equation: ˙ p) TB, z = TB,i + (q Pz)/(mc (5.1) where the q : heat flux, P: heated perimeter, z: heated length from the channel entrance, m: ˙ mass flow rate through the channel, and cp : specific heat. The wall temperature TW, z along the flow direction is related to the local bulk fluid temperature through the local heat transfer coefficient hz . TW, z = TB, z + q/hz (5.2a) The local heat transfer coefficient hz is calculated with the single-phase liquid flow equations given in Chapter 3. For simplicity, the subscript z is not used in the subsequent equations. Considering the complexity in formulation (corner effects in rectangular channels, flow maldistribution in parallel channels, variation in local conditions, etc.), the equations for fully developed flow conditions are employed. For more accurate results, the equations presented in Chapter 3 should be employed in the developing region. Small cavities on the heater surface trap vapor or gases and serve as nucleation sites. As the heater surface temperature exceeds the saturation temperature, a bubble may grow inside the cavity and appear at its mouth as shown in Fig. 5.1(a). The force resulting from the difference in pressures between the outside liquid pL and the inside vapor pV is balanced by the surface tension forces. A force balance along a diametric plane through the bubble yields the following equation: pV − pL = 2/rb (5.2b) TB y = dt y pL pV TL,yb y = yb ur rb yS rb Tw T (a) 2rc (b) Fig. 5.1. Schematic representation of (a) temperature and pressure around a nucleating bubble and (b) stagnation region in front of a bubble in the flow (Kandlikar et al., 1997). 178 Heat transfer and fluid flow in minichannels and microchannels where : surface tension, and rb : bubble radius. Whether the bubble is able to nucleate and the cavity is able to act as a nucleation site depends on the local temperature field around the bubble. The local temperature in the liquid is evaluated by assuming a linear temperature gradient in a liquid sublayer of thickness y = t from the temperature at the wall to the temperature in the bulk liquid. Equating the heat transfer rates obtained from the equivalent conduction and convection equations, the thickness t is given by: t = kL /h (5.3) where kL is the thermal conductivity of the liquid and h is the single-phase heat transfer coefficient in the liquid prior to nucleation. The heat transfer coefficient can be obtained from equations given in Chapter 3. At a given location z, the temperature in the liquid at y = y b is obtained from the linear temperature profile as shown in Fig. 5.1: TL,yb = TW − (yb /t )(TW − TB ) (5.4) where TL,yb : liquid temperature at y = yb , and TW : wall temperature. Neglecting the effect of interface curvature on the change in saturation temperature, and introducing the Clausius–Clapeyron equation, dp/dT = hLV /[TSat (vV − vL )], into Eq. (5.2b) to relate the pressure difference to the corresponding difference in saturation temperatures, the excess temperature needed to sustain a vapor bubble is given by: (pV − pL ) = [TL,Sat (pV ) − TSat ]hLV TSat (vV − vL ) (5.5) where TL,Sat ( pV ): saturation temperature in K corresponding to the pressure pV , TSat : saturation temperature in K corresponding to the system pressure pL , hLV : latent heat of vaporization at pL , and vV and vL : vapor and liquid specific volumes. Combining Eqs. (5.2a) and (5.5), and assuming vV >> vL , we get: TL,Sat (pV ) = TSat + 2 TSat rb V hLV (5.6) As a condition for nucleation, the liquid temperature TL,yb in Eq. (5.4) should be greater than TL,Sat (pV ), which represents the minimum temperature required at any point on the liquid–vapor interface to sustain the vapor bubble as given by Eq. (5.6). Combining Eqs. (5.4) and (5.6) yields the condition for nucleating cavities of specific radii: (yb /t )(TW − TB ) − (TW − TSat ) + 2 TSat =0 rb V hLV (5.7) The liquid subcooling and wall superheat are defined as follows: TSub = TSat − TB (5.8) TSat = TW − TSat (5.9) Chapter 5. Flow boiling in microchannels and minichannels 179 The bubble radius rb and height yb are related to the cavity mouth radius rc through the receding contact angle r as follows: rb = rc /sin r (5.10) yb = rb (1 + cos r ) = rc (1 + cos r )/sin r (5.11) Substituting Eqs. (5.10) and (5.11) into Eq. (5.7), and solving the resulting quadratic equation for rc , Davis and Anderson (1966) obtained the range of nucleation cavities given by: t sin r {rc,min , rc,max } = 2(1 + cos r ) $ × 1

TSat TSat + TSub

8TSat ( TSat + TSub )(1 + cos r ) 1− 2 V hLV t TSat (5.12) The minimum and maximum cavity radii rc,min and rc,max are obtained from the negative and positive signs of the radical in Eq. (5.12), respectively. Different investigators have used different models to relate the bubble radius to the cavity radius and to the location where the liquid temperature TL is determined. Hsu (1962) assumed yb = 1.6rb , which effectively translates into a receding contact angle of r = 53.1 . Substituting this value into Eq. (5.12), the range of cavities nucleating from Hsu’s criterion is given by: TSat TSat + TSub $ 12.8TSat ( TSat + TSub ) × 1 1− 2 V hLV t TSat t {rc,min , rc,max } = 4

(5.13) Bergles and Rohsenow (1964), and Sato and Matsumura (1964) considered a hemispherical bubble at the nucleation inception with yb = rb = rc . The resulting range of nucleating cavities is given by: {rc,min , rc,max } = TSat TSat + TSub $ 8TSat ( TSat + TSub ) × 1 1− 2 V hLV t TSat t 2

(5.14) Kandlikar et al. (1997) analyzed the flow around a bubble and found that a stagnation point occurred at a certain distance yS from the bubble base as shown in Fig. 5.1(b). For receding contact angles in the range of 20–60 , the location of the stagnation point was given by: yS = 1.1rb = 1.1(rc /sin r ) (5.15) 180 Heat transfer and fluid flow in minichannels and microchannels Since a streamline farther away from the wall at this location would sweep over the bubble as seen in Fig. 5.1(b), the temperature at y = yS was taken as the liquid temperature at y = yb . The resulting range of nucleation cavities is then given by: t sin r TSat {rc,min , rc,max } = 2.2 TSat + TSub $ 8.8TSat ( TSat + TSub ) × 1 1− (5.16) 2 V hLV t TSat Note that there was a typographical error in the original publication by Kandlikar et al. (1997). The correct value for the constant is 8.8 as given in Eq. (5.16), though the actual graphs in the original publication were plotted using the correct value of 8.8 by Kandlikar et al. (1997). The onset of nucleate boiling (ONB) is of particular interest in flow boiling. The radius rc,crit of the first cavity that will nucleate (if present) is obtained by setting the radical term in Eq. (5.16) to zero: t sin r TSat rc,crit = (5.17) 2.2 TSat + TSub For a given heat flux, the wall superheat at the ONB, TSat,ONB is given by: TSat,ONB = # 8.8TSat q /(V hLV kL ) (5.18) If the local wall superheat at a given section is lower than that given by Eq. (5.18), nucleation will not occur. The local subcooling at the ONB can be determined from the following equation: TSub,ONB = q − TSat,ONB h (5.19) In a channel with subcooled liquid entering, the local subcooling at the section where nucleation occurs is given by Eq. (5.19). If the subcooling is negative, it means that the local liquid is superheated and will cause extremely high bubble growth rates. Later it will be shown that such high rates result in reverse flow that leads to severe pressure drop fluctuations. Figure 5.2 shows the comparison of different nucleation models with the experimental data by Kandlikar et al. (1997). A high-power microscope and a high-speed camera were used to visualize the nucleation activity and measure the underlying cavity dimensions. Cavities were largely rectangular in shape and the larger side of the opening was used in determining the cavity radius. The cavities nucleate at a certain minimum wall superheat and continue to nucleate for higher wall superheats. A majority of the data points shown in Fig. 5.2 correspond to higher values of wall superheat than the minimum required for nucleation. It is seen that all data points fall very close to or above the criterion given by Eq. (5.14). As seen from Fig. 5.2, the criterion by Davis and Anderson predicts higher wall superheats for larger cavities, whereas Bergles and Rohsenow’s criterion allows the larger cavities Chapter 5. Flow boiling in microchannels and minichannels 181 16 Bergles and Rohsenow Hsu Davis and Anderson Truncated/stagnation Re 1997 Re 1664 14 TWall TSat (°C) 12 10 8 TBulk 80°C 6 4 2 0 0 2 4 6 8 10 Cavity radius (µm) 12 14 16 Fig. 5.2. Comparison of different nucleation criteria against the experimental data taken with water at 1 atm pressure in a 3 × 40 mm channel, r = 40 , truncated/stagnation model by Kandlikar et al. (1997). to nucleate at lower wall superheats, and Hsu’s predictions are also quite close to the data. The criterion by Kandlikar et al. (1997) includes the contact angle effect. The above analysis assumes the availability of cavities of all sizes on the heater surface. If the cavities of radius rc,crit are not available, then higher superheats may be required to initiate nucleation on the existing cavities. The nucleation criteria plotted in Fig. 5.2 indicate the superheat needed to activate cavities of specific radii. Alternatively, Eq. (5.7) along with Eqs. (5.3) and (5.10) may be employed to determine the local conditions required to initiate nucleation at a given cavity. The resulting equation for the wall superheat required to nucleate a given size cavity of radius rc is given by: TSat |ONB at rc = 1.1rc q 2 sin r TSat + kL sin r rc V hLV (5.20) The cavity size used in Eq. (5.20) may be smaller or larger than rc,crit given by Eq. (5.20). Among the available cavities, the cavity with the smallest superheat requirement will nucleate first. Equation (5.19) can be employed to find the local bulk temperature at the nucleation location for a given heat flux. Again, if the local subcooling is negative, it means that the bulk liquid at this condition is superheated and will lead to severe instabilities as will be discussed later. The applicability of the above nucleation criteria to minichannels and microchannels is an open area. These equations were applied to design nucleation cavities in microchannels by Kandlikar et al. (2005) as will be discussed later in Section 5.6. 5.3. Non-dimensional numbers during flow boiling in microchannels Table 5.1 shows an overview of the non-dimensional numbers relevant to two-phase flow and flow boiling as presented by Kandlikar (2004). The groups are classified as empirical and theoretical based. 182 Heat transfer and fluid flow in minichannels and microchannels Table 5.1 Non-dimensional numbers in flow boiling. Non-dimensional number Significance Relevance to microchannels Groups based on empirical considerations Martinelli parameter, X , dp dp X2 = dz F L dz F V Ratio of frictional pressure drops with liquid and gas flow, successfully employed in two-phase pressure drop models It is expected to be a useful parameter in microchannels as well Convection number, Co Co = [(1 − x)/x]0.8 [V/L ]0.5 Co is a modified Martinelli parameter, used in correlating flow boiling heat transfer data Its direct usage beyond flow boiling correlations may be limited Boiling number, Bo q Bo = G hLV Heat flux is non-dimensionalized with mass flux and latent heat, not based on fundamental considerations Since it combines two important flow parameters, q and G, it is used in empirical treatment of flow boiling Groups based on Fundamental Considerations q G hLV 2 K 1 represents the ratio of evaporation momentum to inertia forces at the liquid–vapor interface Kandlikar (2004) derived this number that is applicable to flow boiling systems where surface tension forces are important K 2 represents the ratio of evaporation momentum to surface tension forces at the liquid–vapor interface Kandlikar (2004) derived this number that is applicable in modeling interface motion, such as in critical heat flux Bo represents the ratio of buoyancy force to surface tension force. Used in droplet and spray applications Since the effect of the gravitational force is expected to be small, Bo is not expected to play an important role in microchannels Eo is similar to Bond number, except that the characteristic dimension L could be Dh or any other suitable parameter Similar to Bo, Eo is not expected to be important in microchannels except at very low-flow velocities and vapor fractions Capillary number, Ca µV Ca = Ca represents the ratio of viscous to surface tension forces, and is useful in bubble removal analysis Ca is expected to play a critical role as both surface tension and viscous forces are important in microchannel flows Ohnesorge number, Z Z represents the ratio of viscous to the square root of inertia and surface tension forces, and is used in atomization studies The combination of the three forces masks the individual forces, it may not be suitable in microchannel research K1 = K2 = q hLV 2 L V D V Bond number, Bo Bo = g(L − V )D2 Eötvös number, Eo Eo = g(L − V ) L2 µ Z= (L)1/2 (Continued ) Chapter 5. Flow boiling in microchannels and minichannels 183 Table 5.1 (Continued) Non-dimensional number Significance Relevance to microchannels Weber number, We We represents the ratio of the inertia to the surface tension forces. For flow in channels, Dh is used in place of L We is useful in studying the relative effects of surface tension and inertia forces on flow patterns in microchannels Ja represents the ratio of the sensible heat required for reaching a saturation temperature to the latent heat Ja may be used in studying liquid superheat prior to nucleation in microchannels and effect of subcooling LG 2 We = Jakob number, Ja L cp,L T Ja = V hLV Adapted from Kandlikar (2004). F S Flow Liquid FM Vapor plug Channel wall Evaporation at the interface Fig. 5.3. Schematic representation of evaporation momentum and surface tension forces on an evaporating interface in a microchannel or a minichannel (Kandlikar, 2004). With the exception of the empirically derived Boiling number, the non-dimensional numbers used in flow boiling applications have not incorporated the effect of heat flux. This effect was recognized by Kandlikar (2004) as causing rapid interface movement during the highly efficient evaporation process occurring in microchannels. Figure 5.3 shows an evaporating interface occupying the entire channel. The change of phase from liquid to vapor is associated with a large momentum change due to the higher specific volume of the vapor phase. The resulting force is used in deriving two nondimensional groups K1 and K2 as shown in Table 5.1. The evaporation momentum force, the inertia force and the surface tension forces are primarily responsible for the two-phase flow characteristics and the interface shape and its motion during flow boiling in microchannels and minichannels. K1 represents a modified boiling number with the incorporation of the liquid to vapor density ratio, while K2 relates the evaporation momentum and surface tension forces. Future research work in this area is needed to utilize these numbers in modeling of the flow patterns and critical heat flux (CHF) phenomenon. 184 Heat transfer and fluid flow in minichannels and microchannels The Bond number compares the surface tension forces and the gravitational forces. Under flow boiling conditions in narrow channels, the influence of gravity is expected to be quite low. The Weber number and Capillary numbers account for the surface tension, inertia, and viscous forces. These numbers are expected to be useful parameters in representing some of the complex features of the flow boiling phenomena. 5.4. Flow patterns, instabilities and heat transfer mechanisms during flow boiling in minichannels and microchannels A number of investigators have studied the flow patterns, and pressure drop and heat transfer characteristics of flow boiling in minichannels and microchannels. Kandlikar (2002a) presented a comprehensive summary in a tabular form. An abridged version of the table is reproduced in Table 5.2. The ranges of parameters investigated along with some key results are included in this table. Some of the researchers focused on obtaining the heat transfer coefficient data. Table 5.3, derived from Steinke and Kandlikar (2004), gives the details of the experimental conditions of some of the studies reported in the literature. It may be noted that there are very few local data available that report local heat transfer coefficients. The influence of surface tension forces becomes more dominant in small diameter channels, and this is reflected in the flow patterns observed in these channels. A comprehensive summary of adiabatic flow pattern studies is presented by Hewitt (2000). Kandlikar (2002a, b) presented an extensive summary of flow patterns and associated heat transfer during flow boiling in microchannels and minichannels. Although a number of investigators have conducted extensive studies on adiabatic two-phase flows with air–water mixtures, there are relatively fewer studies available on evaporating flows. Cornwell and Kew (1992) conducted experiments with R-113 flowing in 1.2 mm × 0.9 mm rectangular channels. They mainly observed three flow patterns as shown in Fig. 5.4 (isolated bubbles, confined bubbles and high quality annular flow). They noted the heat transfer to be strongly influenced by the heat flux, indicating the dominance of nucleate boiling in the isolated bubble region. The convective effects became important during other flow patterns. These flow patterns are observed in minichannels at relatively low heat flux conditions. Mertz et al. (1996) conducted experiments in single and multiple channels with water and R-141b boiling in rectangular channels 1, 2, and 3 mm wide. They observed the presence of nucleate boiling, confined bubble flow and annular flow. The bubble generation process was not a continuous process, and large pressure fluctuations were observed. Kasza et al. (1997) observed the presence of nucleate boiling on the channel wall similar to the pool boiling case. They also observed nucleation in the thin films surrounding a vapor core. Bonjour and Lallemand (1998) reported flow patterns of R-113 boiling in a narrow space between two vertical surfaces. They noted that the Bond number effectively identifies the transition of flow patterns from conventional diameter tubes to minichannels. For smaller diameter channels, however, the gravitational forces become less important and Bond number is not useful in modeling the flow characteristics. The presence of nucleation followed by bubble growth was visually observed by Kandlikar and Stumm (1995), and Kandlikar and Spiesman (1997). Chapter 5. Flow boiling in microchannels and minichannels 185 Table 5.2 Summary of investigations on evaporation in minichannels and Microchannels. Fluid and ranges of parameters G(kg/m2 -s), q -(kW/m2 ) Channel size, Dh (mm) horizontal (unless otherwise stated) Lazarek and Black (1982) R-113, G = 125–750. q = 14–380 Cornwell and Kew (1992) Author (year) Flow patterns Remarks Circular, D = 3.1, L = 123 and 246 Not observed Subcooled and saturated data, h almost constant in the two-phase region, dependent on q . Behavior similar to large diameter tubes R-113, G = 124–627, q = 3–33 Rectangular, 75 channel 1.2 × 0.9 36 channel 3.25 × 1.1 Isolated bubble, confined bubble, annular slug h was dependent on the flow pattern. Isolated bubble region, h ~ q0.7 , lower q effect in confined bubble region, convection dominant in annular slug region Moriyama and Inoue (1992) R-113, G = 200–1000, q = 4–30 Rectangular, 0.035–0.11 gap, w = 30, L = 265 Flattened bubbles, with coalescence, liquid strips/film Data in narrow gaps obtained and correlated with an annular film flow model. Nucleate boiling ignored, although h varied with q Wambsganss et al. (1993) R-113 G = 50–100, q = 8.8–90.7 Circular, D = 2.92 mm Not reported Except at the lowest heat and mass fluxes, both nucleate boiling and convective boiling components were present Bowers and Mudawar (1994) R-113, 0.28–1.1 ml/s, q = 1000–2000 Minichannels and, microchannels D = 2.54 and 0.51 Not studied Minichannels and microchannels compared. Minichannels are preferable unless liquid inventory or weight constraints are severe Mertz et al. (1996) Water/R-141b G = 50–300 q = 3–227 Rectangular, 1, 2, and 3 mm wide, aspect ratio up to 3 Nucleate boiling, confined bubble and annular Single- and multi-channel test sections. Flow boiling pulsations in multichannels, reverse flow, nucleate boiling dominant Ravigururajan et al. (1996) R-124, 0.6–5 ml/s, 20–400 W 270 µm wide, 1 mm deep, and 20.52 mm long, Not studied Experiments were conducted over 0–0.9 quality and 5 C inlet subcooling. Wall superheat from 0 C to 80 C (Continued ) 186 Heat transfer and fluid flow in minichannels and microchannels Table 5.2 (Continued ) Author (year) Fluid and ranges of parameters G(kg/m2 -s), q -(kW/m2 ) Channel size, Dh (mm) horizontal (unless otherwise stated) Flow patterns Remarks Tran et al. (1996) R-12, G = 44–832, q = 3.6–129 Circular, D = 2.46, Rectangular, Dh = 2.4 Not studied Local h obtained up to X = 0.94. Heat transfer in nucleate boiling dominant and convective dominant regions obtained Kasza et al. (1997) Water, G = 21, q = 110 Rectangular, 2.5 × 6.0 × 500 Bubbly, slug Increased bubble activity on wall at nucleation sites in the thin liquid film responsible for high heat transfer Tong et al. (1997) Water, Circular, G = 25–45 ×103 , D = 1.05–2.44 CHF 50– 80 MW/m2 Not studied Pressure drop measured in highly subcooled flow boiling, correlations presented for both single-phase and two-phase Bonjour and Lallemand (1998) R-113, q = 0–20 Rectangular, vertical 0.5–2 mm gap, 60 mm wide, and 120 mm long Three flow patterns with nucleate boiling Nucleate boiling with isolated bubbles, nucleate boiling with coalesced bubbles and partial dryout, transition criteria proposed Peng and Wang (1998) Water, ethanol, and mixtures Rectangular, a = 0.2–0.4, b = 0.1–0.3, L = 50; triangular, Dh = 0.2–0.6, L = 120 Not observed No bubbles observed, propose a fictitious boiling. Microscope/high-speed video not used resulting in erroneous conclusions Kamidis and Ravigururajan (1999) R-113, Re = 190–1250; 25–700 W Circular, D = 1.59, 2.78, 3.97, 4.62 Not studied Extremely high h, up to 11 kW/m2 C were observed. Fully developed sub-cooled boiling and CHF were obtained Kuznetsov and R-318C, Shamirzaev G = 200–900, (1999) q = 2–110 Annulus, 0.9 gap × 500 Confined bubble, cell, annular Capillary forces important in flow patterns, thin film suppresses nucleation, leads to convective boiling Lin et al. (1999) Circular, D=1 Not studied Heat transfer coefficient obtained as a function of quality and heat flux. Trends are similar to large tube data Downing et al. R-113, (2000) Ranges not clearly stated Circular coils, Dh = 0.23–1.86, helix diameter = 2.8–7.9 Not studied As the helical coil radius became smaller, pressure drop reduced – possibly due to rearrangement in flow patterns Hetsroni et al. (2001) Triangular, = 55 , Periodic n = 21, 26, Dh = 0.129– annular 0.103, L = 15 R-141b, G = 300–2000 q = 10–150 Water, Re = 20–70, q = 80–360 Periodic annular flow observed in microchannels. Significant enhancement noted in h during flow boiling (Continued ) Chapter 5. Flow boiling in microchannels and minichannels 187 Table 5.2 (Continued ) Fluid and ranges of parameters G(kg/m2 -s), q -(kW/m2 ) Channel size, Dh (mm) horizontal (unless otherwise stated) Kennedy et al. (2000) Water, G = 800–4500, q = 0–4000 Lakshminarasimhan, et al. (2000) Flow patterns Remarks Circular, D = 1.17 and 1.45, L = 160 Not studied q at the OFI was 0.9 of q required for saturated vapor at exit Similarly, G at OFI was 1.1 times G for saturated exit vapor condition R-11, G = 60–4586, Rectangular, 1 × 20 × 357 mm, Boiling incipience observed through LCD Boiling front observed in laminar flow, not visible in turbulent flow due to comparable h before and after, flow boiling data correlated by Kandlikar (1990) correlation Kandlikar et al. (2001) Water, G = 80–560 Rectangular, 1/6 channels, 1 × 1, L = 60 High-speed photography Flow oscillations and flow reversal linked to the severe pressure drop fluctuations, leading to flow reversal during boiling Khodabandeh and Palm (2001) R-134a/ R-600a, G not measured, q = 28–424 Circular tube, 1.5 mm diameter Not studied h compared with 11 correlations. Mass flow rate not measured, assumed constant in all experiments – perhaps causing large discrepancies with correlations at higher h Kim and Bang (2001) R-22, G = 384–570, q = 2–10 Square tube, 1.66 mm, rectangular 1.32 ×1.78 Flow pattern observed in rectangular chain Heat transfer coefficient somewhat higher than correlation predictions. Slug flow seen as the dominant flow pattern Koo et al. (2001) Water, 200 W heat sink Parallel rectangular microchannels 50 µm × 25 µm Thermal profile predicted on the chip and compared with experiments Pressure drop using homogeneous flow model in good agreement with data. Kandlikar (1990) correlation predictions in good agreement with data Lee and Lee (2001a, b) R-113 G = 50–200, q = 3–15 Rectangular, 0.4–2 mm high, 20 mm wide Not reported Pressure drop correlated using Martinelli-Nelson parameter, Heat transfer predicted will by Kandlikar (1990) Author (year) (Continued ) 188 Heat transfer and fluid flow in minichannels and microchannels Table 5.2 (Continued) Author (year) Fluid and ranges of parameters G(kg/m2 -s), q -(kW/m2 ) Channel size, Dh (mm) horizontal (unless otherwise stated) Flow patterns Remarks correlation for film Re > 200, new correlation developed using film flow model for film Re < 200 Serizawa and Feng (2001) Air–water, jL = 0.003– 17.52 m/s, jG = 0.0012– 295.3 m/s. Circular tubes, diameters of 50 µm for air–water and and 25 µm for steam–water Flow patterns identified over the ranges of of flow rates studied Two new flow patterns identified: liquid ring flow and liquid lump flow. Steam– water ranges not given Warrier et al. (2001) FC-84, G = 557–1600, q = 59.9 Rectangular, dimensions not available, hydraulic diameter = 0.75 mm Not studied Overall pressure drop and local heat transfer coefficient determined. A constant value of C = 38 used in Eq. (1). Heat transfer coefficient correlated as a function of Boiling number alone OFI: Onset of flow instability. Adapted from Kandlikar (2002a). Table 5.3 Available literature for evaporation of pure liquid flows in parallel minichannel and microchannel passages. Author (year) Fluid Lazarek and Black (1982) Moriyama and Inoue (1992) Wambsganss et al. (1993) Bowers and Mudawar (1994) R-113 R-113 R-113 R-113 Peng et al. (1994) Cuta et al. (1996) Mertz et al. (1996) Ravigururajan et al. (1996) Tran et al. (1996) Kew and Cornwell (1997) Ravigururajan (1998) Yan and Lin (1998) Kamidis and Ravigururajan (1999) Dh (mm) 3.150 0.140–0.438 2.920 2.540 and 0.510 Water 0.133–0.343 R-124 0.850 Water 3.100 R-124 0.425 R-12 2.400–2.460 R-141b 1.390–3.690 R-124 0.850 Re G (kg/m2 s) (kW/m2 ) Type* Vis.** 57–340 107–854 313–2906 14–1714 125–750 200–1000 50–300 20–500 14–380 4.0–30 8.8–90.75 30–2000 O O/L L O N Y N N 500–1626 32–184 50–300 142–411 44–832 188–1480 3583–10,369 – 1.0–400 10–110 5.0–25 3.6–129 9.7–90 20–700 O O O L L L L N N N N N Y N 50–900 90–200 5.0–20 50–300 L L N N 200–2000 100–570 57–210 217–626 345–2906 1373–5236 11,115– 32,167 R-134a 2.000 506–2025 R-113 1.540–4.620 190–1250 (Continued ) Chapter 5. Flow boiling in microchannels and minichannels 189 Table 5.3 (Continued) Author (year) Fluid Dh (mm) Re Lin et al. (1999) Mudawar and Bowers (1999) Bao et al. (2000) Lakshminarasimhan et al. (2000) Jiang et al. (2001) R-141b Water 1.100 0.902 1591 16–49 Water 0.026 Kandlikar et al. (2001) Kim and Bang (2001) Koizumi et al. (2001) Lee and Lee (2001c) Lin et al. (2001) Hetsroni et al. (2002) Qu and Mudawar (2002) Warrier et al. (2002) Yen et al. (2002) Yu et al. (2002) Zhang et al. (2002) Faulkner and Shekarriz (2003) Hetsroni et al. (2003) Kuznetsov et al. (2003) Lee et al. (2003) Lee and Garimella (2003) Molki et al. (2003) Park et al. (2003) Qu and Mudawar (2003) Wu and Cheng (2003) Steinke and Kandlikar (2004) Water R-22 R-113 R-113 R-141b Vertrel XF Water FC-84 R-123 Water Water Water 1.000 1.660 0.500–5.000 1.569–7.273 1.100 0.158 0.698 0.750 0.190 2.980 0.060 1.846–3.428 Water R-21 Water Water R-134a R-22 Water Water Water 0.103–0.161 1.810 0.036–0.041 0.318–0.903 1.930 1.660 0.349 0.186 0.207 R-11/R-123 1.950 R-11 3.810 G (kg/m2 s) (kW/m2 ) 568 2 × 104 – 1 × 105 1200–4229 50–1800 1311– 60–4586 11,227 1541–4811 2 × 104 – 5 × 104 100–556 28–48 1883–2796 384–570 67–5398 100– 800 220–1786 52–209 536–2955 50–3500 35–68 148–290 338–1001 135–402 440–1552 557–1600 65–355 50–300 534–1612 50–200 127 590 20,551– 3106–6225 41,188 8.0–42 51–500 148–247 30–50 22–51 170–341 300–3500 260–1080 717–1614 100–225 1473–2947 300–600 338–1001 135–402 75– 97 112 116–1218 157–1782 Type* Vis.** – 1 × 103 – 2 × 105 5–200 7.34–37.9 L O N N L O N Y – O Y 1–150 2.0–10 1.0–110 2.98–15.77 1–300 22.6–36 1–1750 1.0–50 5.46–26.9 50–200 2.2 ×104 250–2750 O L O L L L N L L L O O Y Y Y N N Y N N Y N N N 80–220 3.0–25 0.2–301 – 14 10.0–20 10.0–1300 226 5–930 O L O O L L L O L Y N Y N N N N Y Y Adapted from Steinke and Kandlikar (2004). * O = overall, L = local; ** Visualization, Y = yes, N = no. Low x Isolated bubble High x Confined bubble Fig. 5.4. Flow patterns observed by Cornwell and Kew (1992) during flow boiling of R-113 in 1.2 mm × 0.9 mm rectangular channel. The contribution of nucleate boiling to flow boiling heat transfer was clearly confirmed in the above studies, as well as a number of other studies reported in the literature, (e.g. Lazarek and Black, 1982; Wambsganss et al., 1993; Tran et al., 1996; Yan and Lin, 1998; Bao et al., 2000; Steinke and Kandlikar, 2004). However, for microchannels, it was 190 (a) Heat transfer and fluid flow in minichannels and microchannels (b) Fig. 5.5. Flow patterns in 200 µm × 1054 µm parallel channels (a) expanding bubble flow pattern (Kandlikar et al., 2005) each frame shows the progression of boiling in the same channel at 1 ms time intervals, and (b) a snapshot of interface movements in six parallel channels connected by common headers (Kandlikar and Balasubramanian, 2005). also noted that the rapid evaporation and growth of a vapor bubble following nucleation caused major flow excursions, often resulting in reversed flow. Figure 5.5(a) shows a sequence of frames obtained by Kandlikar et al. (2005) for water boiling in 200 µm square parallel channels. The images are taken at intervals of 1 ms and the flow direction is upward. The growth of bubbles and their expansion against the flow direction is clearly seen. Figure 5.5(b) shows the movement of the liquid–vapor interface in both directions as obtained by Kandlikar and Balasubramanian (2005). Similar observations have been made by a number of investigators, including Mertz et al. (1996), Kennedy et al. (2000), Kandlikar et al. (2001), Peles et al. (2001), Kandlikar (2002a, b), Zhang et al. (2002), Hetsroni et al. (2001; 2003), Peles (2003), Brutin andTadrist (2003), Steinke and Kandlikar (2004), and Balasubramanian and Kandlikar (2005). Flow instability poses a major concern for flow boiling in minichannels and microchannels. A detailed description of the flow boiling instabilities is provided by Kandlikar (2002a, b; 2005). Instabilities during flow boiling have been studied extensively in large diameter tubes. The excursive or Ledinegg and the parallel channel instabilities have been studied extensively in the literature. These instabilities are also present in small diameter channels as discussed by Bergles and Kandlikar (2005). Nucleation followed by an increase in the flow resistance due to two-phase flow in channels leads to a minimum in the pressure drop demand curve leading to instabilities. Parallel channel instabilities also occur at the minimum in the pressure drop demand curve. In a microchannel, in addition to these two instabilities, there is a phenomenon that comes into play due to rapid bubble growth rates that causes instability and significant flow reversal problems. Large pressure fluctuations at high frequencies have been reported by a number of investigators including Kew and Cornwell (1996), Peles (2003), and Balasubramanian and Kandlikar (2005) among others. The stability of the flow boiling process has been studied analytically using linear and non-linear stability analyses. Some of the semi-empirical methods have yielded limited success (e.g. Peles et al., 2001; Brutin et al., 2002; Stoddard et al., 2002; Brutin and Pressure drop (kPa) Chapter 5. Flow boiling in microchannels and minichannels 191 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Time (s) Fig. 5.6. Pressure fluctuations observed during flow boiling of water in 1054 µm × 197 µm parallel minichannels (Balasubramanian and Kandlikar, 2005). Bubble/slug growth rate (m/s) 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 Length of bubble/slug (mm) Fig. 5.7. Bubble/slug growth rates during flow boiling of water in 1054 µm × 197 µm parallel minichannels (Balasubramanian and Kandlikar, 2005). Tadrist, 2003). However, these models cannot be tested because of a lack of extensive experimental data under stable boiling conditions. It is expected that with the availability of such data sets, more rigorous models, similar to those available for flow boiling in conventional sized tubes, will be developed in the near future. The pressure fluctuations associated with the flow boiling process were observed by many researchers as described earlier. Figure 5.6 shows a typical plot of the pressure drop fluctuations obtained during flow boiling in minichannels by Balasubramanian and Kandlikar (2005). A fast fourier transform analysis of the instantaneous pressure drop signal revealed that the dominant frequency was dependent on the wall temperature and was related to the nucleation activity within the channels. The growth rate of the growing bubbles was measured to be as high as 3.5 m/s. Figure 5.7 shows the growth rate of a slug in 1054 µm × 197 µm parallel channels. As the bubbles reach the opposite channel wall, the growth rate stabilizes until the rapid evaporation from the walls causes the growth rate to 192 Heat transfer and fluid flow in minichannels and microchannels (1) (2a) (2b) (2c) Fig. 5.8. Schematic representation of bubble growth in (1) large diameter tubes, and (2a–c) microchannels and minichannels. Flowing and evaporating liquid film Annular flow pattern in macrochannels Liquid slug Evaporating liquid film Vapor core Expanding-bubble flow pattern in microchannel Fig. 5.9. Comparison of the annular flow pattern in macrochannels and the expanding-bubble flow pattern in microchannels and minichannels. increase again. The numerical simulation by Mukherjee and Kandlikar (2004) confirmed the bubble growth observed in Fig. 5.7. Nucleation followed by rapid bubble growth is believed to be the cause of instabilities. As a bubble grows under pool boiling conditions, the bubble growth rate is much higher immediately following its inception. The growth rate is initially proportional to time t and is controlled by the inertial forces. As the bubble grows, the growth rate slows down and follows a t 1/2 trend in the thermally controlled region. In large diameter channels (above 1–3 mm, depending on heat flux), the bubbles grow to sizes that are smaller than the channel diameter and leave the heater surface under inertia forces. In flow boiling in macrochannels, the bubble growth is similar to that in pool boiling, except that the flow causes the bubbles to depart early as shown in Fig. 5.8. These departing bubbles contribute to the bubbly flow. As more bubbles are formed, they coalesce and develop into slug and annular flows as shown in Fig. 5.9. Chapter 5. Flow boiling in microchannels and minichannels 193 (a) (b) (c) (d) (e) (f) Fig. 5.10. The expanding-bubble flow pattern observed in a 197 µm × 1054 µm channel, using water at atmospheric pressure, the time duration between successive images is 0.16 ms, G = 120 kg/m2 s, q = 317 kW/m2 , Ts = 110.9 C, flow is from left to right (Kandlikar and Balasubramanian, 2005). In microchannels and minichannels, as a bubble nucleates and initially grows in the inertia controlled region, it encounters the channel walls prior to entering the thermally controlled region found in the conventional channels. The large surface area to fluid volume ratio in the channel causes the liquid to heat up rapidly. Thus, the bubble encounters a superheated liquid as it continues to grow and spreads over the other areas of the channel wall. The availability of heat from the superheated layer and from the channel walls causes a rapid expansion of the bubble, leading to the expandingbubble flow pattern shown in Fig. 5.8. The bubble occupies the entire channel cross-section and continues to grow as shown in Fig. 5.9. The expanding-bubble flow pattern differs from the annular flow pattern mainly in that the liquid on the wall acts similar to the film under a growing vapor bubble, rather than as a flowing film. This makes the heat transfer mechanism very similar to the nucleate boiling mechanism. A number of investigators have confirmed the strong heat flux dependence of the heat transfer coefficient during flow boiling in microchannels, which indicates the dominance of nucleate boiling. Figure 5.10 shows the expanding-bubble flow pattern observed during flow boiling in a rectangular minichannel by Kandlikar and Balasubramanian (2005). Note the rapid interface movement on the right side of the bubbles (downstream). The liquid film is essentially stationary and occasionally dries out before the upstream liquid slug flows through and rewets the surface. At other times, severe flow reversal was observed at the same site, and the bubble interface moved upstream rapidly. Mukherjee and Kandlikar (2004) performed a numerical simulation of the bubble growth process in a microchannel evaporator. The bubble initially nucleated from one of the walls and then grew to occupy the entire channel cross-section. The bubble shapes were compared with the experimental observations by Balasubramanian and Kandlikar (2005) for the same geometry of 197 µm × 1054 µm with water in the expanding-bubble flow pattern region (Fig. 5.11). Their agreement validates numerical simulation as a powerful tool for analyzing the flow boiling phenomenon. 194 Heat transfer and fluid flow in minichannels and microchannels 0.082 ms 0.165 ms 0.247 ms 0.325 ms 0.354 ms Fig. 5.11. Comparison of bubble shapes obtained from numerical simulation by Mukherjee and Kandlikar (2004) and experimental observations (Balasubramanian and Kandlikar, 2005). 5.5. CHF in microchannels The two-phase flow and local wall interactions during flow boiling set a limit for the maximum heat flux that can be dissipated in microchannels and minichannels. In the electronics cooling application, the inlet liquid is generally subcooled, while a two-phase mixture under saturated conditions may be introduced if a refrigeration loop is used to lower the temperature of the evaporating liquid. Thus the CHF under both subcooled and saturated conditions is of interest. Kandlikar (2001) modeled the CHF in pool boiling on the basis of the motion of the liquid– vapor–solid contact line on the heater surface. CHF was identified as the result of the interface motion caused by the evaporation momentum force at the evaporating interface near the heater surface. This force causes the “vapor-cutback” phenomenon separating liquid from the heater surface by a film of vapor. High-speed images of the interface motion under such conditions were obtained by Kandlikar and Steinke (2002). This phenomenon is also responsible in restoring the CHF conditions on subsequent rewetting of the heater surface. Extension of this pool boiling CHF model to microchannels and minichannels is expected to provide useful results. Bergles (1963) and Bergles and Rohsenew (1964) provide extensive coverage on subcooled CHF in macrochannels. Under subcooled flow boiling conditions, the primary concern is the explosive growth of vapor bubbles upon nucleation. At the location where nucleation is initiated, the bulk liquid may have a low value of liquid subcooling. In some cases, the liquid may even be under superheated conditions. This situation arises when the proper nucleation sites are not available, and nucleation is initiated toward the exit end of the microchannels. This would result in an explosive bubble growth following nucleation as shown in Fig. 5.12(a). Since the bubble growth is quite rapid, it often results in a reverse flow in the entire channel. When such bubble growth occurs near the channel entrance, the vapor is pushed back into the inlet plenum as shown in Fig. 5.12(b). The vapor bubbles growing near the entrance region find the path of least resistance into the inlet plenum. The resulting instability leads to a CHF condition. Chapter 5. Flow boiling in microchannels and minichannels 195 Inlet manifold (a) Outlet manifold (b) Fig. 5.12. Reverse flow of vapor into the inlet manifold leading to early CHF. The unstable operating conditions are largely responsible for the low values of CHF reported in literature. It may be recalled that the CHF under pool boiling conditions is around 1.6 MW/m2 , while the values reported for narrow channels are significantly lower than this value. For example, Qu and Mudawar (2004) reported CHF data with a liquid subcooling between 70 C and 40 C, and their CHF values (based on the channel area) are only 316.2 to 519.7 kW/m2 . The main reason for such low values is believed to be the instabilities, which were reported by Qu and Mudawar in the same paper (see Fig. 5.12). Bergles and Kandlikar (2005) reviewed the available CHF data and concluded that all the available data in the literature on microchannels suffers from this instability. The effect of mass flux on the CHF is seen to be quite significant. For example, Roach et al. (1999) obtained CHF data in 1.17- and 1.45-mm diameter tubes and noted that the CHF increased from 860 kW/m2 (for a 1.15-mm diameter tube with G = 246.6 kg/m2 s), to 3.699 MW/m2 (for a 1.45-mm diameter tube with G = 1036.9 kg/m2 s). It is suspected that the higher flow rate results in a higher inertia force and induces a stabilization effect. This trend is supported by the experimental results obtained by Kamidis and Ravigururajan (1999) with R-113 in 1.59, 2.78, 3.97, and 4.62 mm tubes, and by Yu et al. (2002) with water in a 2.98 mm diameter tube. The influence of tube diameter is somewhat confusing in light of the instabilities. In general, the CHF decreased with the tube diameter, and in many cases the reduction was rather dramatic (e.g. Qu and Mudawar, 2004). The presence of flow instability, especially in small diameter tubes, needs to be addressed in obtaining reliable experimental data. Further research in this area is warranted. 5.6. Stabilization of flow boiling in microchannels The reversed flow that leads to unstable operation poses a major concern in implementing flow boiling in practical applications. The rapid growth of a vapor bubble in a superheated liquid environment leads to flow reversal, which is identified as a major cause of instability. The flow instability results from the reversed flow occurring in the parallel channels. Two methods for reducing the instabilities are discussed in this section. 5.6.1. Pressure drop element at the inlet to each channel The parallel channels provide an effect similar to the upstream compressibility for each of the channels. Therefore, placing a flow restrictor in the flow loop prior to the inlet Heat transfer and fluid flow in minichannels and microchannels 0.00 ms 0.75 Y 1 0.75 0.5 X 0 1 2 4 3 X 0.22 ms 0.05 ms 1 1 0.75 0.5 Y 0.75 0.5 0.25 0.25 0 0 1 2 3 0 0.5 0.25 4 Z Z 0.5 0.25 0 0.25 0.5 X 0 0.25 0.5 0 1 2 3 0.29 ms 1 0.75 0.75 X 3 4 0.5 0.25 0.25 0 0 0.5 0.25 0 0.25 0.5 Z Z 0 2 Y 1 0.5 1 4 X 0.11 ms 0.5 0.25 0 0.25 0.5 Y 0 0.5 0.25 0 0.25 0.5 4 3 2 1 0.25 0 Z Z 0.5 0.25 0 0.25 0.5 0.5 0.25 0 Y 0.16 ms 1 0 1 2 X 3 Y 196 4 Fig. 5.13. Simulation of bubble growth in a microchannel with upstream to downstream flow resistance ratio, R = 1 (Mukherjee and Kandlikar, 2005). manifold will not reduce the instability arising in each channel. To reduce the intensity and occurrence of reverse flow, a pressure drop element (essentially a flow constrictor or a length of reduced cross-sectional area) is placed at the entrance of each channel. This introduces an added resistance to fluid flow, but provides an effective way to reduce the flow instabilities arising from the reverse flow. Mukherjee and Kandlikar (2004; 2005) conducted an extensive numerical analysis of bubble growth in a microchannel in an effort to study the effect of the pressure drop elements on flow stabilization. They utilized the level set method to define the interface and track its movement by applying conservation equations. Figure 5.13 shows the consecutive frames of a bubble growing in a microchannel. The resistance to flow in both the upstream and downstream directions was the same. The bubble growth is seen to be slow at the beginning, but becomes more rapid after the bubble touches the other heated channel walls. This leads to an extension of the inertia-controlled region, where the heat transfer is more efficient since it does not depend on the diffusion of heat across a thin layer surrounding the liquid–vapor interface. Mukherjee and Kandlikar (2005) introduced a new parameter R to represent the upstream to downstream flow resistance ratio. In their simulation, they showed that increasing the upstream resistance reduced the intensity of the backflow. The results are shown in Fig. 5.14. In Fig. 5.14(a), the resistance in both directions is the same, while in Fig. 5.14(b), the inlet to outlet ratio is 0.25 indicating a four-fold higher flow resistance in the backward direction. The resulting bubble growth in Fig. 5.14(b) shows that reverse flow is completely eliminated in this case. As expected, the backflow characteristics were also found to be dependent on the heat flux and the local liquid superheat at the nucleation site. Chapter 5. Flow boiling in microchannels and minichannels Unit vector 1 0.75 Y 0.5 0.25 0 0 1 2 3 4 X (a) Unit vector 1 0.75 Y 0.5 0.25 0 0 (b) 197 1 2 3 4 X Fig. 5.14. Bubble growth with unequal flow resistances on the upstream and downstream flow directions, upstream to downstream flow resistance ratio (a) R = 1 and (b) R = 0.25 (Mukherjee and Kandlikar, 2005). Channel details, 200 µm square channel, X : channel length direction; Y : channel height direction. 5.6.2. Flow stabilization with nucleation cavities Introducing artificial nucleation sites on the channel wall is another method for reducing the instability. Introducing nucleation cavities of the right size would initiate nucleation before the liquid attains a high degree of superheat. Kandlikar et al. (2005) experimentally observed nucleation behavior with the introduction of artificial cavities into the microchannel. Figure 5.15 shows nucleation on these cavities much earlier with significant reduction in the reverse flow and instabilities. The flow pattern present after nucleation is initiated is an important feature of flow boiling in minichannels and microchannels. The bubbles expand to occupy the entire crosssection, with intermittent liquid slugs between successive expanding bubbles. Additional nucleation within the slugs further divides them. Figures 5.4, 5.5, and 5.10 show the two-phase structure with a vapor core surrounded by a film of liquid. This flow pattern is similar to the annular flow pattern, but with an important distinction. In an annular flow pattern, liquid flows in the thin film surrounding the vapor core. The velocity profile and flow rate in the film of the classical annular flow are determined from the well-known triangular relationship between the wall shear stress, pressure drop, and the liquid film flow rate. The sizes of the nucleation cavities required to initiate nucleation are functions of local wall and bulk fluid temperatures, heat transfer coefficient, and heat flux. Examples 5.1 and 5.2 illustrate the size ranges of nucleating cavities under given flow and heat flux conditions. Implementing cavities and pressure drop elements together was found to be most effective in reducing the instabilities. The placement of nucleation cavities and the area reduction at the inlet section are some of the design variables that need to be taken into account while designing a flow boiling system. Table 5.4 shows the effect of various configurations in reducing the flow boiling instabilities. Heat transfer and fluid flow in minichannels and microchannels Flow direction 198 (a) (b) (c) (d) (e) (f) Fig. 5.15. Stabilized flow with large fabricated nucleation sites; successive frames from (a) to (f) taken at 0.83 ms time intervals illustrate stabilized flow in a single channel from a set of six parallel vertical microchannels; water, G = 120 kg/m2 s, q = 308 kW/m2 , Ts = 114 C (Kandlikar et al., 2005). Table 5.4 Effect of artificial nucleation sites and pressure drop elements on flow boiling stability in 1054 µm × 197 µm channel (Kandlikar et al., 2005). Case Open header, 5–30 µm nucleation sites 51% area PDEs, 5–30 µm nucleation sites 4% area PDEs, 5–30 µm nucleation sites Average surface temperature ( C) Pressure drop (kPa) Pressure fluctuation (± kPa) Stability 113.4 12.8 2.3 Unstable 113.0 13.0 1.0 111.5 39.5 0.3 Partially stable Completely stable PDE: pressure drop element. PDE, refers to pressure drop elements, placed at the inlet of each channel providing the specified cross-sectional area for flow. For the cases investigated, the pressure drop elements or artificial nucleation sites alone did not eliminate the instabilities as noted from the pressure drop fluctuations. The 4% PDEs are able to eliminate the instabilities completely, but they introduce a very large pressure drop. The effect of 51% area reduction Chapter 5. Flow boiling in microchannels and minichannels 199 in conjunction with the artificial nucleation sites is interesting to note. There was little pressure drop increase due to area reduction as compared to the fully open channel. It is therefore possible to arrive at an appropriate area reduction in conjunction with the artificial nucleation sites with marginal increase in the pressure drop. 5.7. Predicting heat transfer in microchannels Flow boiling heat transfer data was obtained by a number of investigators. Many of the researchers reported unstable operating conditions during their tests. Therefore, this data and the matching correlations should be used with some degree of caution. At the same time, it is recommended that experimental data under stable operating conditions be obtained by providing artificial nucleation sites and pressure drop elements near the inlet inside each channel. A good survey of available experimental data and correlations is provided by Qu and Mudawar (2004), and Steinke and Kandlikar (2004). Table 5.3 is adapted from Steinke and Kandlikar (2004), and lists the ranges of experimental data available in the literature. It can be seen that the fluids investigated include: water, R-21, R-22, R-113, R-123, R-124, R141b, FC-84, and Vertrel XF. The mass fluxes, liquid Reynolds numbers, and heat fluxes range from 20 to 6225 kg/m2 s, 14 to 5236, and up to 2 MW/m2 , respectively. As the tube diameter decreases, the Reynolds number shifts toward a lower value. In order to establish the parametric trends, the local flow boiling heat transfer data are needed. In the papers listed in Table 5.3, only those by Hetsroni et al. (2002), Yen et al. (2002), and Steinke and Kandlikar (2004) deal with tubes around 200 µm and report local heat transfer data. Qu and Mudawar (2003) also report local data for parallel rectangular channels of 349 µm hydraulic diameter. The experimental data of Bao et al. (2000) for a 1.95 mm diameter tube indicates a strong presence of the nucleate boiling term, while Qu and Mudawar’s data exhibits a strong influence of mass flux, indicating the presence of convective boiling. The decreasing trends in heat transfer coefficient with quality seen in Qu and Mudawar’s data was also seen in Steinke and Kandlikar’s (2004) data. A number of issues arise that make it difficult to assess the available experimental data accurately. Some of the factors are: (i) the presence of instabilities during the experiments, (ii) different ranges of parameters, especially heat flux and mass flux. The influence of instabilities was discussed in the earlier sections. The ranges of mass fluxes employed in small diameter channels typically fall in the laminar range. The correlations developed for large diameter tubes are in a large part based on the turbulent flow conditions for Reynolds numbers based on all-liquid flow. Some of the data available in literature was seen to be correlated by the all-nucleate boiling-type correlations, such as Lazarek and Black (1982), and Tran et al. (1996). However, recent data reported by Qu and Mudawar has very large errors (36.2% and 98.8%). Similar observations can be made with the Steinke and Kandlikar (2004) data. Some of the more recent correlations by Yu et al. (2002) and Warrier et al. (2002) employ only the heat flux dependent terms (similar to Lazarek and Black (1982), and Tran et al. (1996)) and correlate Qu and Mudawar’s (2003) data well. However, the presence of both nucleate 200 Heat transfer and fluid flow in minichannels and microchannels boiling and convective boiling terms to a varying degree is reported by Kandlikar and Balasubramanian (2004). They recommend using the laminar flow equations for the allliquid flow heat transfer coefficients in the Kandlikar (1990) correlation and cover a wide range of data sets. The flow boiling correlation by Kandlikar (1990) utilizes the all-liquid flow, singlephase correlation. Since most of the available data was in the turbulent region, use of the Gnielisnki correlation was recommended. However, later, Kandlikar and Steinke (2003), and Kandlikar and Balasubramanian (2004) introduced the laminar flow equation for laminar flow conditions based on ReLO . The correlation based on the available data is given below: For ReLO > 100: h hTP = larger of TP,NBD (5.21) hTP,CBD hTP,NBD = 0.6683Co−0.2 (1 − x)0.8 hLO + 1058.0Bo0.7 (1 − x)0.8 FFl hLO (5.22) hTP,CBD = 1.136Co−0.9 (1 − x)0.8 hLO + 667.2Bo0.7 (1 − x)0.8 FFl hLO (5.23) where Co = [(1 − x)/x]0.8 (V /L )0.5 and Bo = q /GhLV . The single-phase all-liquid flow heat transfer coefficient hLO is given by: for 104 ≤ ReLO ≤ 5 × 106 for 3000 ≤ ReLO ≤ 104 hLO = hLO = for 100 ≤ ReLO ≤ 1600 hLO = ReLO Pr L ( f/2)(kL /D) 0.5 1 + 12.7(Pr 2/3 L − 1)( f/2) (ReLO − 1000) Pr L ( f/2)(kL /D) 0.5 1 + 12.7(Pr 2/3 L − 1)( f/2) NuLO k Dh (5.24) (5.25) (5.26) In the transition region between Reynolds numbers of 1600 and 3000, a linear interpolation is suggested for hLO . For Reynolds number below and equal to 100 (Re ≤ 100) the nucleate boiling mechanism governs, and the following Kandlikar Correlation is proposed, For ReLO ≤ 100, hTP = hTP,NBD = 0.6683Co−0.2 (1 − x)0.8 hLO + 1058.0Bo0.7 (1 − x)0.8 FFl hLO (5.27) The single-phase all-liquid flow heat transfer coefficient hLO in Eq. (5.27) is found from Eq. (5.26). The fluid surface parameter FFL in Eqs. (5.22), (5.23), and (5.27) for different fluid surface combinations is given in Table 5.5. These values are for copper or brass surfaces. For stainless steel surfaces, use FFL = 1.0 for all fluids. For silicon surfaces, no data is currently available. Use of the values listed in Table 5.5 for copper is suggested. Chapter 5. Flow boiling in microchannels and minichannels 201 Table 5.5 Recommended FFl (fluid surface parameter) values in flow boiling correlation by Kandlikar (1990; 1991). Fluid FFl Water R-11 R-12 R-13B1 R-22 R-113 R-114 R-134a R-152a R-32/R-132 R-141b R-124 Kerosene 1.00 1.30 1.50 1.31 2.20 1.30 1.24 1.63 1.10 3.30 1.80 1.00 0.488 q1 = 5 kW/m2 q2 = 15 kW/m2 q1 – Kandlikar (90) q2 – Kandlikar (90) 7000 6000 h TP ( W/m2 K ) 5000 4000 3000 2000 1000 0 0.0 0.2 0.4 0.6 0.8 1.0 Quality (x) Fig. 5.16. Yan and Lin (1998) data points for R-134a compared to the correlation by Kandlikar and Balasubramanian (2004) using the laminar single-phase equation; Dh = 2 mm, G = 50 kg/m2 s, q = 5 and 15 kW/m2 , ReLO = 506. The above correlation scheme is based on the data available in the literature. It has to be recognized that all of the data suffer from the instability condition to some extent. It is expected that the correlation will undergo some changes as new data under stabilized flow conditions become available. A comparison of the correlation scheme described in Eqs. (5.21)–(5.27) with some of the experimental data available in the literature is shown in Figs 5.16–5.18. The decreasing 202 Heat transfer and fluid flow in minichannels and microchannels 10000 h TP (W/m2 K) 8000 6000 4000 2000 0 0 0.2 0.4 0.6 Quality (x) 0.8 1 Fig. 5.17. Yen et al. (2002) data points for HCFC 123 compared to the correlation by Kandlikar and Balasubramanian (2004) using the laminar singlephase flow equation; Dh = 0.19 mm, G = 145 kg/m2 -s, q = 6.91 kW/m2 , pavg = 151.8 kPa, ReLO = 86. q14 G 157 kg/m2 s 100 q15 119 kW/m2 q14 q16 151 kW/m2 q15 h TP( kW/m2 K ) Correlation q14 182 kW/m2 q16 80 Correlation q15 Correlation q16 60 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 Quality (x) Fig. 5.18. Steinke and Kandlikar (2004) data points compared to the correlation by Kandlikar and Balasubramanian (2004) using the laminar flow equation with the nucleate boiling dominant term only; Dh = 207µm, G = 157 kg/m2 -s, q = 119, 151 and 182 kW/m2 , ReLO = 116. trend in the heat transfer coefficient with quality is evident in the data, indicating the dominance of the nucleate boiling mechanism. The complex nature of flow boiling in small diameter channels, including liquid–vapor interactions, presence of expanding bubbles with thin evaporating film, nucleation of bubbles in the flow as well as in the thin film, etc. make it difficult to present a comprehensive analytical model to account for the heat Chapter 5. Flow boiling in microchannels and minichannels 203 transfer mechanisms during flow boiling. Further efforts with high-speed flow visualization techniques are recommended to provide the fundamental information on this topic. 5.8. Pressure drop during flow boiling in microchannels and minichannels The pressure drop in a microchannel or a minichannel heat exchanger is the sum of the following components: p = pc + pf ,1-ph + pf ,tp + pa + pg + pe (5.28) where subscripts c: contraction at the entrance; f,1-ph: single-phase pressure loss due to friction, including entrance region effects; f,tp: two-phase frictional pressure drop; a: acceleration associated with evaporation; g: gravitational; and e: expansion at the outlet. Equations for calculating each of these terms are presented in the following sections. 5.8.1. Entrance and exit losses The inlet to the microchannel may be single-phase liquid or a twophase mixture. It is common to have liquid at the inlet when a liquid pump and a condenser are employed in the cooling system. When the refrigeration system forms an integral part of the cooling system, the refrigerant is throttled prior to entry into the microchannels. With the need to incorporate pressure drop elements in each channel, liquid at the inlet in such cases is also possible. The contraction losses in the single-phase liquid are covered in Chapter 3. The nature of the liquid’s entry into the channels is another factor that needs to be taken into consideration. The channel floor may be flush with the manifold, or may be shallower or deeper than the manifold. Lee and Kim (2003) used a micro-PIV system to identify the entrance losses with sharp and smooth channel entrances. For the two-phase entry and exit losses, Coleman (2003) recommends the following scheme proposed by Hewitt (2000). The following equation is used to calculate the pressure loss due to a sudden contraction of a two-phase mixture using a separated flow model: 2 1 G2 1 pc = − 1 + 1 − 2 h (5.29) 2L Co c where G: mass flux, c : contraction area ratio (header to channel > 1), Co – contraction coefficient given by: Co = 1 0.639(1 − 1/c )0.5 + 1 (5.30) and h : two-phase homogeneous flow multiplier given by: h = [1 + x(L /V − 1)] with x: local quality. (5.31) 204 Heat transfer and fluid flow in minichannels and microchannels The exit pressure loss is calculated from the homogeneous model: pe = G 2 e (1 − e )s (5.32) where e : area expansion ratio (channel to header < 1), and s : the separated flow multiplier given by: L s = 1 + − 1 [0.25x(1 − x) + x2 ] (5.33) V The frictional pressure drop in the single-phase region prior to nucleation is calculated from the equations presented in Chapter 3. In the two-phase region, the following equations may be used to calculate the frictional, acceleration and gravity components of the pressure drop. The local friction pressure gradient at any section is calculated with the following equation:

dpF dpF = 2 (5.34) dz dz L L The two-phase multiplier L2 is given by the following equation by Chisholm (1983): L2 = 1 + 1 C + 2 X X (5.35) The value of the constant C depends on whether the individual phases are in the laminar or turbulent region. Chisholm recommended the following values of C: Both phases turbulent C = 21 (5.36a) Laminar liquid, turbulent vapor C = 12 (5.36b) Turbulent liquid, laminar vapor C = 10 (5.36c) Both phases laminar C=5 (5.36d) The Martinelli parameter X is given by the following equation:

dpF - dpF X2 = dz L dz V (5.37) Mishima and Hibiki (1996) found that the constant C depends on the tube diameter, and recommended the following equation for C: C = 21(1 − e −319Dh ) (5.38) where Dh is in m. Mishima and Hibiki’s correlation is used extensively and is recommended. English and Kandlikar (2005) found that their adiabatic air–water data in a 1-mm square channel was overpredicted by using Eq. (5.38). Upon further investigation, they found that Chapter 5. Flow boiling in microchannels and minichannels 205 12 Mishima-Hibiki Experimental data English and Kandlikar (2005) Pressure drop (kPa/m) 10 8 6 4 2 0 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 Mass quality (x) Fig. 5.19. Comparison of experimental data from English and Kandlikar (2005) with their correlation, and Mishima and Hibiki’s correlation (1996). the diameter correction recommended by Mishima and Hibiki (1996) should be applied to C in Eq. (5.35). Accordingly, the value of C will change depending on the laminar or turbulent flow of individual phases. Thus the following modified equation is recommended for frictional pressure drop calculation in microchannels and minichannels: L2 = 1 + C(1 − e−319Dh ) 1 + 2 X X (5.39) The value of C is obtained from Eqs. (5.36a)–(5.36d). Figure 5.19 shows the agreement of experimental data obtained by English and Kandlikar (2005) during air–water flow in a 1-mm square channel. The average absolute deviation was found to be 3.5%, which was within the experimental uncertainties. Further validation of Eq. (5.39) is recommended. The acceleration pressure drop is calculated from the following equation assuming homogeneous flow: pa = G 2 vLV xe (5.40) where vLV : difference between the specific volumes of vapor and liquid phases = vV − vL . The above equation assumes the inlet flow to be liquid only and exit quality to be xe . For a two-phase inlet flow, the xe should be replaced with the change in quality between the exit and inlet sections. Heat transfer and fluid flow in minichannels and microchannels 10 Superficial liquid velocity (m/s) Superficial liquid velocity (m/s) 206 D 2.05 mm 1 0.1 Bubbly Slug 0.01 0.01 0.1 Churn Annular 1 10 100 10 D 4.08 mm 1 0.1 Bubbly 0.01 0.01 Superficial gas velocity (m/s) Slug 0.1 Churn 1 Annular 10 100 Superficial gas velocity (m/s) Fig. 5.20. Flow pattern map for minichannels derived from Mishima and Hibiki (1996). The gravitational pressure drop will be a very small component. It can be calculated from the following equation based on the homogeneous flow model: vLV g( sin )L ln 1 + xe pg = vLV xe vL (5.41) Equation (5.41) also assumes a liquid inlet condition. In case of two-phase inlet flow, the difference between the exit and inlet quality should be used in place of xe . 5.9. Adiabatic two-phase flow Adiabatic two-phase flow has been studied extensively in literature, especially with air– water flow. For small diameter channels, the work of Mishima and Hibiki (1996) is particularly noteworthy. Figure 5.20 shows their flow pattern map. The effect of surfactants on the adiabatic two-phase flow pressure drop was studied by English and Kandlikar (2005) in a 1-mm square minichannel with air and water with superficial gas and liquid velocities of 3.19–10 m/s and 0.001–0.02 m/s, respectively. The resulting flow pattern was stratified (annular) in all cases, which is near the bottom right corner in Fig. 5.20. The pressure drop was not affected by variation in surface tension from 0.034 to 0.073 N/m. The reason for this is believed to be the absence of a three-phase line (such as would exist with discrete droplets sliding on the wall). The Mishima and Hibiki (1996) correlation for pressure drop was modified as shown in Fig. 5.19 using Eqs. (5.39) and (5.36). 5.10. Practical cooling systems with microchannels Use of microchannels in a high heat flux cooling application using flow boiling systems was reviewed by Pokharna et al. (2004) for notebook computers and for server applications by Kandlikar (2005). The major issues that need to be addressed before implementation of flow boiling becomes practical are listed as follows (Kandlikar, 2005). Chapter 5. Flow boiling in microchannels and minichannels 207 High heat flux cooling systems with flow boiling have lagged behind the single-phase liquid cooled systems because of some of the operational challenges that still remain to be resolved. These are listed below: (i) Need for low-pressure water or a suitable refrigerant to match the saturation temperature requirement for electronics cooling. (ii) Unstable operation due to rapid bubble expansion and occasional flow reversal. (iii) Unavailability of CHF data and a lack of fundamental understanding of the flow boiling phenomenon in microchannel passages. The use of water is very attractive from a heat transfer performance perspective, but the positive pressure requirement may lead to a vacuum system, which is generally not desirable due to possible air leakage, which raises the saturation temperature in the system. Development of a suitable refrigerant suitable for this application is recommended. The desirable characteristics of an ideal refrigerant in a flow boiling system may be listed as (Kandlikar, 2005): (a) saturation pressure slightly above the atmospheric pressure at operating temperatures; (b) high latent heat of vaporization; (c) good heat transfer and pressure drop related properties (high liquid thermal conductivity, low liquid viscosity, low hysteresis for ONB); (d) high dielectric constant if applied directly into the chip; (e) compatible with silicon (for direct chip cooling) and copper (for heat sink applications), and other system components; (f) low leakage rates through pump seals; (g) chemical stability under system operating conditions; (h) low cost; (i) safe for human and material exposure under accidental leakages. The fundamental understanding of the flow boiling phenomenon is slowly emerging, and efforts to stabilize the flow using nucleating cavities and pressure drop elements is expected to help in making flow boiling systems as a viable candidate for high heat flux cooling application. Another application for flow boiling in narrow channels is in compact evaporators. The evaporators used in automotive application currently do not employ some of the small channel configurations used in condenser application due to stability problems. Stabilized flow boiling, as described in Section 5.6, will enable practical implementation of microchannels and minichannels in a variety of compact evaporator applications. 5.11. Solved examples Example 5.1 Water is used as the cooling liquid in a microchannel heat sink. The dimensions of one channel are a = 1054 µm × b = 50 µm, where a is the unheated length in the three-sided heating case. The inlet temperature of the water is 70 C, and the Reynolds number in the channel is 600. 208 Heat transfer and fluid flow in minichannels and microchannels (1) Calculate the insipient boiling location and cavity radius and plot the wall superheat and liquid subcooling at the ONB versus the cavity radius rc . Also plot the predicted heat transfer coefficient as a function of quality for the following values of heat flux: (i) q = 50 kW/m2 , (ii) q = 340 kW/m2 , (iii) q = 1 MW/m2 . (2) Calculate the pressure drop in the heat exchanger core for a heat flux of 1 MW/m2 and a channel length of 20 mm. Assumptions Fully developed laminar flow Nusselt number; three-sided heating; properties of water are at saturation temperature at 1 atm; Receding contact angle is 40 . Solution Properties of water at 100 C: hLV = 2.26 × 106 J/kg, iL = 4.19 × 105 J/kg, V = 0.596 kg/m3 , vL = 0.001044 m3 /kg, vg = 1.679 m3/kg, cp,L = 4217 J/kg-K, µV = 1.20 × 10−5 N-s/m2 , µL = 2.79 × 10−4 N-s/m2 , kL = 0.68 W/m-K, = 0.0589 N/m, Pr = 1.76 (Incropera and DeWitt, 2002). Properties of water at 70 C: iL = 2.93 × 105 J/kg. Part 1 (i) Heat flux is 50 kW/m2 The fully developed Nusselt number is found using Table 3.3 with aspect ratio c = a/b = 1054 µm/50 µm = 21.08. Since the aspect ratio is greater than 10, the Nusselt number for three-sided heating is 5.385. The hydraulic diameter and heated perimeter are: Dh = 4Ac 2ab 2 × 1054 × 10−6 × 50 × 10−6 = = 9.547 × 10−5 m = Pw (a + b) (1054 + 50) × 10−6 = 95.47 µm P = a + 2b = (2 × 50 + 1054) × 10−6 = 1.154 × 10−3 m = 1154 µm The heat transfer coefficient is h = kL Nu/Dh = (0.68 W/m-K)(5.385)/(9.55 × 10−5 m) = 38,355 W/m2 -K. Under the fully developed assumption, a value for the temperature difference between the wall temperature and the bulk liquid temperature can be found using the equation q = h( T ) = h(TWall − TBulk ) = h(TWall − TSat + TSat − TBulk ) = h( TSat + TSub ). Therefore, the sum of TSat and TSub can be found for a given heat flux as q /h = ( TSat + TSub ) = (50,000 W/m2 )/(38,355 W/m2 -K) = 1.30 C. The wall superheat at the critical cavity radius is found by setting the expression under the radical in Eq. (5.16) equal to zero. 2 )=0 1 − [8.8TSat ( TSat + TSub )]/(V hLV t TSat Noting that t = kL /h, the equation q = h( TSat + TSub ) can be written as q = kL t ( TSat + TSub ). Chapter 5. Flow boiling in microchannels and minichannels 209 Solving this equation for t : t = kL ( TSat + TSub )/q = (0.68 × 1.3)/50,000 = 1.77 × 10−5 m = 17.7 µm and substituting it into the expression under the radical in Eq. (5.16), the value for TSat at the critical rc can be obtained as: $ 8.8TSat q 8.8 × 0.0589 × 373.15 × 50,000 TSat = = = 3.25 C V hLV kL 0.596 × 2.26 × 106 × 0.68 The cavity radius can be found by using Eq. (5.17): rc,crit = t sin r 2.2

TSat TSat + TSub

= 1.77 × 10−5 × sin 40 2.2

3.25 1.3 = 12.9 × 10−6 m = 12.9 µm Using Eq. (5.1) and the definitions of wall superheat and liquid subcooling given by Eqs. (5.8) and (5.9), the location where ONB occurs is found: ˙ p (TB,z − TB,i )mc (375.1 − 343.15) × 92.4 × 10−6 × 4217 = q P 50,000 × 1.15 × 10−3 = 0.216 m = 21.6 cm z= Realistically, this length is too large for a microchannel heat exchanger and the actual length is expected to be much shorter. Also, the heat flux is very low for a microchannel heat exchanger. Therefore, it will be operating under single-phase liquid flow conditions throughout. Plotting Eq. (5.16) as a function of TSat illustrates the nucleation criteria for the given heat flux. The wall superheat at ONB is plotted as a function of the nucleation cavity radius ( T ONB versus rc ) by using Eq. (5.20). The liquid subcooling is also presented on this plot. Note that the negative values of TSub indicate that the bulk is superheated at the point of nucleation. This will lead to significant flow instabilities as discussed in the Section 5.4. TONB versus rc Nucleation criteria rc,min rc,max TONB (°C) Cavity radius (µm) 100 20 15 10 TSat,ONB TSub,ONB 5 0 5 10 15 20 10 1 0 10 20 TSat (°C) 30 0.1 1 10 rc (µm) 100 210 Heat transfer and fluid flow in minichannels and microchannels The Boiling number is calculated from Table 5.1: Bo = q Dh 50,000 × 9.547 × 10−5 q = = = 1.262 × 10−5 GhLV hLV Re µL 2.26 × 106 × 600 × 2.79 × 10−4 For a flow of higher Reynolds number, the value of hLO would not be the same as the single-phase value of h, as can be seen with Eqs. (5.24)–(5.26). As the Reynolds number is 600, the heat transfer coefficient hLO is equal to the single-phase value of h by Eq. (5.26), hLO = h = 38,355 W/m2 K. The Convection number is a function of quality: Co = [(1 − x)/x]0.8 [V /L ]0.5 = [(1 − x/x)]0.8 (0.596/957.9) = [(1 − x)/x]0.8 × 6.22 × 10−4 From Eq. (5.21), the flow boiling hTP is the larger of Eqs. (5.22) and (5.23), which can be plotted as functions of x as seen below using an FFl of 1: h hTP = larger of TP,NBD hTP,CBD hTP,NBD = hLO [0.6683Co−0.2 (1 − x)0.8 + 1058.0Bo0.7 (1 − x)0.8 FFl ] = 38,355{0.6683([(1 − x)/x]0.8 × 6.22 × 10−4 ) −0.2 (1 − x)0.8 + 1058.0(1.262 × 10−5 )0.7 (1 − x)0.8 × 1} hTP,CBD = hLO [1.136Co−0.9 (1 − x)0.8 + 667.2Bo0.7 (1 − x)0.8 FFl ] = 38,355{1.136([(1 − x)/x]0.8 × 6.22 × 10−4 )−0.9 (1 − x)0.8 + 667.2(1.262 × 10−4 )0.7 (1 − x)0.8 × 1} Predicted heat transfer coefficient versus quality 1.2E06 NBD CBD hTP (W/m2 K) 1.0E06 8.0E05 6.0E05 4.0E05 2.0E05 0.0E00 0 0.2 0.4 0.6 Quality (x ) 0.8 1 Since hTP,NBD yields higher values than hTP,CBD , it represents the flow boiling thermal conditions and should be used for the present case. Note that hTP increases with quality for this case. Chapter 5. Flow boiling in microchannels and minichannels 211 (ii) Heat flux is 340 kW/m2 a = 1054 µm; b = 50 µm; Ti = 70 C; Re = 600; c = 21.08; h = 38,355 W/m2 K ( TSat + TSub ) = 8.86 C; TSat = 8.48 C; rc,crit = 5.0 µm; z = 2.9 cm 20 rc,min rc,max 15 TONB (°C) Cavity radius (µm) TONB versus rc Nucleation criteria 100 10 10 5 0 5 10 TSat,ONB TSub,ONB 15 20 1 0 10 20 30 0.1 1 TSat (°C) 10 100 rc (µm) Predicted heat transfer coefficient versus quality hTP (W/m2 K) 1.2E06 NBD CBD 1.0E06 8.0E05 6.0E05 4.0E05 2.0E05 0.0E00 0 0.2 0.4 0.6 Quality (x) 0.8 1 In plotting the heat transfer coefficient versus quality, it is seen that the nucleate boiling dominant prediction is actually higher than the convective dominant prediction for very low qualities (<0.1) and should be used in those cases. There is an increase in the nucleate boiling behavior in comparison to the lower heat flux case. (iii) Heat flux is 1 MW/m2 a = 1054 µm; b = 50 µm; Ti = 70 C; Re = 600; c = 21.08; h = 38,355 W/m2 K ( TSat + TSub ) = 26.1 C; TSat = 14.5 C; rc,crit = 2.9 µm; z = 0.6 cm The nucleate boiling dominant prediction is higher than the convective boiling dominant for qualities lower than 0.4 and should be used for those cases. Note that a high wall superheat is needed to initiate nucleation (14.5 C). Also, the cavity sizes for nucleation become smaller as the heat flux increases. Thus a variety of different cavity sizes are needed on a surface to operate it at different heat flux conditions. Another point to note is that the saturation temperature of water at 1 atm is quite high for electronics cooling applications. Two options can be pursued: one is to use low-pressure steam, and the other is to use refrigerants. The next example illustrates the use of refrigerants. 212 Heat transfer and fluid flow in minichannels and microchannels Nucleation criteria rc,min rc,max TONB versus rc 30 25 20 15 10 5 0 5 10 15 20 0.1 TONB (°C) Cavity radius (µm) 100 10 1 0 10 20 TSat (°C) 40 1 10 100 rc (µm) Predicted heat transfer coefficient versus quality 1.4E06 NBD CBD 1.2E06 hTP (W/m2 K) 30 TSat,ONB TSub,ONB 1.0E06 8.0E05 6.0E05 4.0E05 2.0E05 0.0E00 0 0.2 0.4 0.6 Quality (x) 0.8 1 Part 2 From Part 1, (iii), it is found that the onset of nucleate boiling occurs at 0.6 cm from the entrance end. Therefore, the flow is single-phase up to 0.6 cm at which point it becomes two-phase flow. However, the bulk is subcooled at this location and so Eq. (5.1) is used to determine the location where the saturation temperature is reached: z= ˙ p (TB,z − TB,i )mc (100 − 70) × 92.4 × 10−6 × 4217 = 0.0101 m = 1.01 cm = 1 × 106 × 1.15 × 10−3 q P Using this approach simplifies the problem, but keep in mind that since ONB actually occurs before this location in the channel, the actual pressure drop in the core of the channel will be slightly higher due to the longer two-phase flow length. The exit quality can be solved for by using the following equation to get xe = 0.055, where AT is the heated perimeter multiplied by the channel length: q AT = m[(i ˙ L,@TSat + xe hLV,@TSat ) − iL,@TB,in ] Chapter 5. Flow boiling in microchannels and minichannels xe = = 213 q AT + iL,@TB,in − iL,@TSat hLV,@TSat m ˙ 10,00,000 × 1.15 × 10−3 × 0.02 5 5 2.26 × 106 + 2.93 × 10 − 4.19 × 10 9.24 × 10−5 = 0.055 The total mass flux can be calculated from the Reynolds number and channel geometry: G = ReµL /Dh = (600 × 2.79 × 10−4 )/(9.547 × 10−5 ) = 1753 kg/m2 s The superficial Reynolds numbers of the liquid and vapor phases in the two-phase flow region are found by using the following equations. The average quality over the channel length is taken as half the exit quality. ReV = −5 G x2e Dh 1753 × 0.055 2 × 9.547 × 10 = = 384 µV 1.2 × 10−5 9.547 × 10−5 G 1 − x2e Dh 1753 1 − 0.055 2 ReL = = = 583 µL 2.79 × 10−4 The single-phase friction factors fL and fV are obtained by using Eq. (3.10) with the Reynolds numbers calculated above. Po = f Re = 24(1 − 1.3553c + 1.94672c − 1.70123c + 0.95644c − 0.25375c ) = 24(1 − 1.3553 × 21.08 + 1.9467 × 21.082 − 1.7012 × 21.083 + 0.9564 × 21.084 − 0.2537 × 21.085 ) = 22.56 The friction pressure gradients given in Eq. (5.37) are defined as: 2 2fL G 2 1 − x2e = D h L L 2 22.56 2 583 17532 1 − 0.055 2 = 2,458,697 Pa/m = 2.46 MPa/m = 9.547 × 10−5 × 957.9 2 2 0.055 2 2fV G 2 x2e 2 22.56 dpF 384 1753 2 − = = dz V Dh V 9.547 × 10−5 × 0.5956 dpF − dz

= 4,800,308 Pa/m = 4.80 MPa/m

dpF - dpF 2.46 2 X = = 0.5125 = dz L dz V 4.80 214 Heat transfer and fluid flow in minichannels and microchannels The two-phase multiplier is calculated with Eq. (5.39): −5 −319Dh 5 1 − e−319×9.547×10 C 1 − e 1 1 + 2 =1+ = 3.16 L2 = 1 + + √ X X 0.5125 0.5125 and the two-phase pressure drop per unit length is calculated with Eq. (5.34):

dpF dpF = 2 = 24,58,697 × 3.16 = 7,769,482 Pa/m = 7.77 MPa/m dz TP dz L L The frictional two-phase pressure drop is found by multiplying the frictional pressure gradient by the two-phase flow length (0.02 m – 0.0101 m = 0.0099 m) to get 76.7 kPa. The total pressure drop in the core is found by adding the pressure drop in the singlephase flow section length (0.0101 m) and the two-phase pressure drop: p = pf ,1-ph + pf ,tp = 26,800 + 76,700 = 103,500 Pa = 103.5 kPa (15.0 psi) Example 5.2 Microchannels are directly etched into the silicon chips to dissipate a heat flux of 13,000 W/m2 from a computer chip. The geometry may be assumed similar to Fig. 3.19. Each of the parallel microchannels has a width a = 200 µm, height b = 200 µm, and length L = 10 mm. Refrigerant-123 flows through the horizontal microchannels at an inlet temperature of TB,i = 293.15 K. The heated perimeter P = b + a + b = 600 × 10−6 m, and the cross-sectional area Ac = a × b = 40 × 10−9 m. Assume r from Fig. 5.1 is 20 , and Re = 100. Solution Properties of R-123 at TSat = 300.9 K and 1 atm: µL = 404.2 × 10−6 N-s/m2 , µV = 10.8 × 10−6 N-s/m2 , L = 1456.6 kg/m3 , V = 6.5 kg/m3 , cp,L = 1023 J/kg K, kL = 75.6 × 10−3 W/m K, kV = 9.35 × 10−3 W/m K, L = 14.8 × 10−3 N/m, hLV = 170.19 × 103 J/kg, iL = 228 × 103 J/kg, iV = 398 × 103 J/kg. (i) Calculate the incipient boiling location and cavity radius (Answers: z = 0.584 × 10−3 m, and r c,crit = 2.23 × 10−6 m) From Eq. (5.18): # TSat,ONB = 8.8TSat q /(V hLV kL ) = 8.8(14.8 × 10−3 )(300.9)(13,000)/(6.5)(170.19 × 103 )(75.6 × 10−3 ) = 2.47 K Calculate the hydraulic diameter: Dh = 4Ac = a = b = 200 × 10−6 m Pw Chapter 5. Flow boiling in microchannels and minichannels 215 From Table 3.3, Nufd,3 = 3.556, and note that: h= (3.556)(0.0756) NukL = 1344 W/m2 K = Dh (200 × 10−6 ) From Eq. (5.19): TSub,ONB = q 13,000 − TSat,ONB = − 2.47 = 7.2 K h 1344 From Eq. (5.8), TB at the onset of nucleate boiling is: TB,ONB = TSat − TSub,ONB = 300.9 − 7.2 = 293.7 K Calculate the flow velocity using the Reynolds number: V = (100)(404.2 × 10−6 ) ReµL = 0.139 m/s = Dh (1456.6)(200 × 10−6 ) Calculate the mass flow rate: m ˙ = VAc = (1456.6)(0.139)(40 × 10−9 ) = 8.09 × 10−6 kg/s Calculate the mass flux: G= 8.09 × 10−6 m ˙ = = 202 kg/m2 s Ac 40 × 10−9 The incipient boiling location can be calculated by rearranging Eq. (5.1): mc ˙ p,L z = (TB,ONB − TB,i ) q P (8.09 × 10−6 )(1023) = (293.7 − 293.15) (13,000)(600 × 10−6 ) = 0.584 × 10−3 m To find the cavity radius, substitute Eqs. (5.3) and (5.19) into Eq. (5.17) we get rc,crit = kL sin r TSat,ONB (75.6 × 10−3 )( sin 20 )(2.47) = 2.23 × 10−6 m = 2.2q 2.2(13,000) (ii) Plot the wall superheat and liquid subcooling versus nucleating cavity radius for q = 5, 13, and 30 kW/m2 . Equations (5.19) and (5.20) are used to plot the following figures. 216 Heat transfer and fluid flow in minichannels and microchannels q 5000 W/m2 TSat TSub 20 10 0 0.1 1.0 10.0 100.0 10 20 30 TSat TSub 30 Tsat and Tsub (K) TSat and TSub (K) 30 q 13,000 W/m2 40 40 20 10 0 . 0.1 1.0 10.0 100.0 10 20 30 rc (µm) q 30,000 W/m2 100 TSat and TSub (K) 80 60 rc (µm) Tsat Tsub 40 20 0 0.1 20 1.0 10.0 100.0 40 60 80 rc (µm) (iii) Calculate the pressure drop in the test section (Answer: 426 Pa) For fully developed laminar flow, the hydrodynamic entry length may be obtained using Eq. (3.11): Lh = 0.05ReDh = 0.05(100)(0.200 ) = 1.0 mm Since L > Lh , the fully developed flow assumption is valid. From Eq. (5.1):

mc ˙ p,L (8.09 × 10−6 )(1023) z = (TB,z − TB,i ) = (300.97 − 293.15) q P (13,000)(600 × 10−6 ) = 8.29 × 10−3 m Note that z is the location where two-phase boiling begins. So, we need to find the singlephase pressure drop until z and then add to that the value of the two-phase pressure drop from z to L. The total pressure drop can be found using Eq. (5.28): p = pc + pf ,1-ph + pf ,tp + pa + pg + pe Chapter 5. Flow boiling in microchannels and minichannels 217 To find the single-phase pressure drop, the f Re term can be obtained using Eq. (3.10) and using an aspect ratio of one: c = a/b = 200/200 = 1 f Re = 24(1 − 1.3553c + 1.94672c − 1.70123c + 0.95644c − 0.25375c ) = 24(1 − 1.3553 × 1 + 1.9467 × 12 − 1.7012 × 13 + 0.9564 × 14 − 0.2537 × 15 ) = 14.23 The single-phase core frictional pressure drop can be calculated using Eq. (3.14): 2( f Re)µL Um z L Um2 + K(∞) · 2 2 Dh pf ,1-ph = where K(∞) is given by Eq. (3.18) K(∞) = (0.6796 + 1.2197c + 3.30892c − 9.59213c + 8.90894c − 2.99595c ) = 1.53 2(14.23)(0.404 × 10−3 )(0.139 )(0.00829 ) (200 × 10−6 )2 pf ,1-ph = + (1.53) (1456.6 )(0.139 )2 2 = 353 Pa Note that pc , pg and pe can be neglected because we are calculating the core pressure drop in horizontal microchannels. The following Eqs. (9.1-12, 14, 35, 36, 37, 39 and 40) are from Chapter 9 of Handbook of Phase Change (Kandlikar et al., 1999). Assuming that the liquid pressure is 1 atm, then iIN = 228 × 103 J/kg. The heated area in the two-phase region is: Ah,tp = (b + a + b)(L − z) = (200 × 10−6 + 200 × 10−6 + 200 × 10−6 ) × (10 × 10−3 − 8.29 × 10−3 ) = 1.026 × 10−6 m2 From Eq. (9.1-12): iTP = iIN + q Ah,tp (13,000)(1.026 × 10−6 ) = 229.6 × 103 J/kg = 228 × 103 + GAc (8.09 × 10−6 ) From Eq. (9.1-14): xe = iTP − iL (229.6 × 103 − 228 × 103 ) = 0.0094 = hLV (170.19 × 103 ) 218 Heat transfer and fluid flow in minichannels and microchannels Use the average thermodynamic quality between 0 and xe , xavg = 0.0047. From Eq. (9.1-39): ReV = GxDh (202)(0.0047)(200 × 10−6 ) = 17.58 = µV (10.8 × 10−6 ) From earlier the vapor friction factor is: fV = 14.23 14.23 = 0.809 = ReV 17.58 From Eq. (9.1-35): dpF − dz = V 2fV G 2 x2 2(0.809)(202)2 (0.0047)2 = 1122 Pa/m = Dh V (200 × 10−6 )(6.5) From Eq. (9.1-40): ReL = fL = G(1 − x)Dh (202)(1 − 0.0047)(200 × 10−6 ) = 99.48 = µL (0.4042 × 10−3 ) 14.23 14.23 = = 0.143 ReL 99.48 From Eq. (9.1-36): dpF − dz = L 2fL G 2 (1 − x)2 2(0.143)(202)2 (1 − 0.0047)2 = 39,683 Pa/m = Dh L (200 × 10−6 )(1456.6) From Eq. (5.37): X = 2 dpF dz 39,683 dpF = = 35.37 dz V 1122 L Assuming both phases are laminar, from Eq. (5.36d), C = 5. Using Eq. (5.39): L2 1 C(1 − e−319Dh ) 5(1 − e−319(200×10 + 2 =1+ =1+ √ X X 35.37 −6 ) ) + 1 = 1.08 35.37 2 F F From Eq. (5.34) pf ,tp = dp = dp dz dz L L = (39,683)(1.08) = 42,858 Pa/m The acceleration pressure drop is calculated from Eq. (5.40): pa = G 2 vLV xe = (202)2 (0.00069)(0.0094) = 0.26 Pa Chapter 5. Flow boiling in microchannels and minichannels 219 The value of the acceleration pressure drop is negligible, and hence the total pressure drop is p = pf ,1-ph + pf ,tp = (353) + (42,858)(0.01 − 0.00829) = 426 Pa (iv) Plot the predicted heat transfer coefficient as a function of quality From Table 5.1, the Boiling number is: q (13,000) = = 0.378 × 10−3 GhLV (202)(170.19 × 103 ) Bo = From Table 5.1, the convection number is: Co = [(1 − x)/x]0.8 [V /L ]0.5 = [(1 − x)/x]0.8 (6.5/1456.6)0.5 = 0.0668[(1 − x)/x]0.8 For ReLO ≤ 100, from Eqs. (5.26) and (5.27): hLO = NukL (3.556)(0.0756) = = 1344 W/m2 K Dh (200 × 10−6 ) From Table 3 in page 391 of Handbook of Phase Change (Kandlikar et al., 1999), assume FFl = 1.3. For ReLO ≤ 100, hTP = hTP,NBD and we can plot the predicted heat transfer coefficient as a function of quality using Eq. (5.27). This is very similar to the method followed in the previous example, but only the nucleate boiling dominant prediction is used from Eq. (5.27): hTP = hTP,NBD = hLO {0.6683Co−0.2 (1 − x)0.8 + 1058.0Bo0.7 (1 − x)0.8 FFl } = 1344{0.6683(0.0668[(1 − x)/x]0.8 )−0.2 (1 − x)0.8 + 1058.0(0.378 × 10−3 )0.7 (1 − x)0.8 FFl } h TP (W/m2 K) 16,000 14,000 hTP, NBD 12,000 10,000 8000 6000 4000 hTP,NBD 2000 0 0 0.2 0.4 0.6 Quality (x) 0.8 1 220 Heat transfer and fluid flow in minichannels and microchannels The total frictional pressure drop has to consider the minor losses because the actual p is higher than the core frictional pressure drop. Due to the laminar conditions and the dominance of nucleate boiling effects, note the decreasing trend in hTP as a function of quality. 5.12. Practice problems 1. A microchannel with dimensions of 120 µm × 400 µm and 3 mm length operates at an inlet pressure of 60 kPa. The inlet liquid is at saturation temperature. A heat flux of 600 kW/m2 is applied to the three sides of the channel walls: (i) Calculate the mass flux to provide an exit quality of 0.1 from this channel. Assuming that cavities of all sizes are available, calculate the distance at which ONB occurs. What is the nucleating cavity diameter at this location? (ii) Calculate the pressure drop in the channel. (iii) Plot the variation of heat transfer coefficient with channel length assuming an evaporator pressure to be mean of the inlet and exit pressures. 2. R-123 is used to cool a microchannel chip dissipating 400 W from a chip 12 mm × 12 mm surface area. Assuming an allowable depth of 300 µm, design a cooling system to keep the channel wall temperature below 70 C. Calculate the associated pressure drop in the channel. 3. A copper heat sink has a 60 mm × 120 mm base. 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R., Pan, T., and Dhir, V. K., Heat transfer and pressure drop in narrow rectangular channels, in Proceedings of the Fourth International Conference on Multiphase Flow, May 27–June 1, New Orleans: ASME, 2001. Warrier, G. R., Pan, T., and Dhir, V. K., 2002, Heat transfer and pressure drop in narrow rectangular channels, Exp. Thermal Fluid Sci., 26, 53–64, 2002. Wu, H. Y. and Cheng, P., Liquid/twophase/vapor alternating flow during boiling in microchannels at high heat flux. Int. Communi. Heat Mass Transfer 30(3), 295–302, 2003. Yan, Y. and Lin, T., Evaporation heat transfer and pressure drop of refrigerant R-134a in a small pipe, Int. J. heat Mass Transfer, 42, 4183–4194, 1998. Yen, T. H., Kasagi, N., and Suzuki, Y., Forced convective boiling heat transfer in microtubes at low mass and heat fluxes, in Symposium on Compact Heat Exchangers on the 60th Birthday of Ramesh K. Shah, August 24, Grenoble France, pp. 401–406, 2002. Yu, W., France, D. M., Wambsganss, M. W., and Hull, J. R., Two-phase pressure drop, boiling heat transfer, and critical heat flux to water in a small-diameter horizontal tube. Int. J. Multiphase Flow 28, 927–941, 2002. Zhang, L., Koo, J.- M., Jiang, L., and Asheghi. M., Measurements and modeling of two-phase flow in microchannels with nearly constant heat flux boundary conditions, J. Microelectromech. Syst., 11(1), 12–19, 2002. Chapter 6 CONDENSATION IN MINICHANNELS AND MICROCHANNELS Srinivas Garimella George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA 6.1. Introduction 6.1.1. Microchannel applications Microchannels are increasingly being used in industry to yield compact geometries for heat transfer in a wide variety of applications. Considerable literature exists on single-phase flow, pressure drop, and heat transfer in such channels, as has been documented elsewhere in this book and can be seen in some recent reviews of the literature (Sobhan and Garimella, 2001; Garimella and Sobhan, 2003; Garimella and Singhal, 2004; Liu and Garimella, 2004). Similarly, boiling and evaporation in microchannels have also been studied due to the electronics cooling industry’s need to remove high heat fluxes through vaporization from compact devices that must be maintained at relatively low temperatures while being not readily accessible or conducive to the installation of large and complex cooling systems. In addition to high heat flux applications like electronics cooling, microchannels have also seen increasing use for the purpose of reducing inventories of hazardous fluids in components and for reducing material consumption for component fabrication. Ultimately, in these applications, microchannels are viewed as being responsible for the mitigation of ozone depletion by enabling the use of smaller amounts of environmentally harmful fluids, and also reducing greenhouse gas emissions by improving component and system energy efficiencies. Utilization in micro-sensors and micro-actuators, and in biological diagnostic devices are other candidate applications. The art of the utilization of microchannels for achieving high heat transfer rates is in fact much farther ahead than the science of obtaining a comprehensive understanding of phase change in these channels. For example, the large heat transfer coefficients in microchannels, and the large surface area-to-volume ratios they offer, has led to their use in compact E-mail: [email protected] 227 228 Heat transfer and fluid flow in minichannels and microchannels Air flow Refrigerant flow Fig. 6.1. Schematic of a microchannel tube, multilouver fin condenser. condensers for the air-conditioning systems in automobiles for many years. These condensers consist of rectangular tubes with multiple parallel microchannels (Fig. 6.1) cooled by air flowing across multilouver fins. The microchannels used in these condenser designs typically have hydraulic diameters in the 0.4–0.7 mm range, although an understanding of the fundamental phase-change phenomena in these heat exchangers is just beginning to emerge. Garimella and Wicht (1995) demonstrated that substantial reductions in condenser sizes can be achieved for small (~18 kW) condensers for ammonia–water absorption spaceconditioning systems for residential application by using microchannel condensers instead of conventional round-tube, flat-fin designs. This compactness was attributed to large condensation heat transfer coefficients, high surface-to-volume ratios, and reduced air-side pressure drops due to the use of multilouver fins. Noting the potential enhancements due to the use of microchannels, Webb and Ermis (2001) investigated condensation of R134a in extruded aluminum tubes with multiple parallel microchannels with and without microfins, and with hydraulic diameter 0.44 < Dh < 1.564 mm. Their work documented that heat transfer coefficients and pressure gradients increase with decreasing hydraulic diameter. Webb and Lee (2001) investigated the use of these microchannel condensers as replacements for 2-row, roundtube condensers with 7-mm diameter tubes. They noted material cost reductions of up to 55% over the conventional geometries due to a combination of air-side and tube-side improvements. Other advantages such as air-side pressure drop (in part, due to form drag reduction over flat tubes) and material weight reduction were also noted due the use of microchannel condensers, depending on the specific tube and fin geometry parameters. Jiang and Garimella (2001) demonstrated the implementation of microchannel tubes for residential air-conditioning systems. They compared conventional round-tube, flat-fin geometries to systems with air-coupled microchannel evaporators and condensers as well as systems that coupled refrigerant heat exchange to intermediate glycol–water solution loops. Their work showed that indoor and outdoor units of air-coupled microchannel systems can be packaged in only one-half and one-third the space required for a conventional system, respectively. Furthermore, they showed that significant additional compactness can be achieved through hydronic coupling, because of the high heat transfer coefficients on both sides of the liquid-coupled heat exchangers, and the counterflow orientation, which is more suitable in the presence of a temperature glide. Chapter 6. Condensation in minichannels and microchannels 229 It was also shown that the refrigerant charge of the microchannel air-coupled system is 20% less than that of the round-tube heat pump. For the hydronically coupled system, the refrigerant charge is only 10% of the charge in the round-tube heat pump, yielding significant benefits pertaining to the reduction of the environmental impact of air-conditioning systems. These representative studies clearly demonstrate the wide-ranging benefits that can be obtained by the use of microchannels in phase-change applications. 6.1.2. Microchannel phenomena Researchers have discussed the term “microchannel” at some length in the literature and elsewhere in this book. One extreme of the size range (sub-micron size channels) is where continuum phenomena do not apply, and properties such as viscosity, ice point, and others change due to molecular and electrokinetic forces, but not much information is available for that size range, and there are few practical phase-change applications at that range. However, such non-continuum phenomena do not have to be present for there to be differences in flow and heat transfer phenomena from conventional, larger diameter channels. One simple way of looking at microchannels is the existence of deviations in pressure gradients and heat transfer coefficients from widely available models, which is sufficient to warrant studies that understand and account for the different flow phenomena. In boiling applications, for example, the microchannel designation may be applied when the channel size approaches the bubble departure diameter, which would lead to differences from macrochannel phenomena. Essentially, when channel size affects flow, pressure gradients and heat transfer in a manner that are not accounted for by models developed for the larger channels, it warrants a different treatment. Thus, microchannel phase change differs from its counterpart in macrochannels due to differences in the relative influences of gravity (whose effects are less prominent at small scales), shear, and surface tension, the latter playing a dominant role. The microchannel behavior, therefore, depends on phenomena that do not exist or are not prominent at the larger scales, or phenomena that are suppressed at the small scales. As these phenomena would depend on the liquid and vapor phase properties of the fluid under consideration, a rigorous definition of the term microchannel would necessarily depend on the fluid in question. Serizawa et al. (2002), who conducted air–water and steam–water flow visualization studies in 50-µm channels, recommend the Laplace constant L > Dh (also referred to as the capillary length scale) as the criterion for determining whether a channel should be considered to be a microchannel: L= (6.1) g (l − v ) The Laplace constant represents the ratio of surface tension and gravity forces, and has been interpreted as the dimension at which influence of surface tension becomes more important than the stratifying effect of gravity (Fukano and Kariyasaki, 1993). This criterion implies that for a high surface tension fluid pair like air–water at standard temperature and pressure, a channel with Dh < 2.7 mm would be a microchannel, while for R-134a at 1500 kPa, the corresponding value is Dh < 0.66 mm. Another interpretation of a different form of this same quantity is provided in terms of the confinement of bubbles in channels (Kew 230 Heat transfer and fluid flow in minichannels and microchannels and Cornwell, 1997). Thus, typically, microchannel two-phase flows are characterized by “confined” single elongated bubbles occupying much of the channel (as will be discussed at length in a subsequent section), whereas multiple bubbles may occupy the cross-section in macrochannels. Also, typical bubble sizes decrease as the pressure increases, affecting how the channel may be viewed in relation to the bubble size. Based on this interpretation, Kew and Cornwell (1997) recommend the use of the confinement number (Co > 0.5) as the transition criterion (for boiling applications) beyond which microchannel effects are present: # /(g(l − v )) Co = Dh (6.2) Thus, Co = L/Dh , and the transition Dh value is 1.32 mm, that is double the value recommended by Serizawa et al. (2002). However, the practical value of such a categorization is somewhat limited – even a change in operating temperature would affect whether the channel should be treated as a microchannel through the influence of temperature on surface tension. What is more important is to recognize and account for the respective flow phenomena in the development of pressure drop and heat transfer models. For example, substantial differences due to channel size have been documented by several researchers at hydraulic diameters of a few millimeters, as noted by Palm (2001). It is also clear that the increase in surface area associated with the use of microchannels increases the importance of surface forces over that of body forces, as noted by Serizawa and Feng (2004). Thus, surface phenomena, and in some cases, surface characteristics, become more prominent in these small channels, and the interactions between the fluid and wall increase in importance. This is another rationale for the commonly noted observation that in microchannels, surface tension and viscous forces dominate over gravitational forces. As will also become clear in the following sections, neither the same measurement techniques nor the same modeling approaches used for the larger scale channels are adequate for addressing these phenomena that are specific to microchannels. 6.1.3. Chapter organization and contents The focus of this chapter is condensation in microchannels. Compared to single-phase flow and boiling in such channels, condensation has received less attention. Part of this is because researchers have concentrated primarily on the removal of heat from small and inaccessible surfaces such as high-flux electronics components, using single-phase and boiling processes. Condensation has thus far been generally coupled with air: the dominant air-side resistance in such situations may be one reason for the relatively smaller attention to condensation. However, liquid coupling, condensation in heat pipes, fluid charge minimization, and other factors are making this process more prominent. The heat removed from the sources must also be rejected, requiring innovative compact designs and a better understanding of condensation in microchannels. This chapter presents the available information in an emerging field for which directly relevant literature is limited; therefore, it is by no means a comprehensive treatment of all aspects of condensation in microchannels. The chapter is written with the primary Chapter 6. Condensation in minichannels and microchannels 231 application of refrigerant condensation and condensers as the basis, but can be extended as necessary to other related applications. Heat transfer coefficients and pressure drops for condensation inside microchannels are strongly dependent on the different flow patterns that are established as the fluid undergoes a transition from vapor to liquid. Methods to predict flow regimes as a function of flow-related parameters such as mass flux, quality, and fluid properties are therefore presented first. Where possible, the differences between the flow patterns in large channels and microchannels are discussed. This is followed by a brief discussion of void fraction in condensing flows. The void fraction is almost invariably required for the modeling of pressure drop and heat transfer in two-phase flow. (Once again, it is noted that void fractions in microchannels have proved to be difficult to measure due to issues such as optical access, and the dynamic and sometimes indeterminate nature of the vapor–liquid interface.) Models for the prediction of pressure drops in the different flow regimes are then presented along with a comparison of the predicted pressure drops from the different models. This is followed by a presentation of the corresponding models for condensation heat transfer in microchannels. Finally, areas of critical research needs for furthering the understanding of condensation in microchannels are pointed out. At the time of this writing, there is ongoing research in this field at several laboratories around the world, and the information presented here will no doubt need to be updated on a continuous basis. 6.2. Flow regimes The first step in understanding condensation in microchannels is the characterization of how the condensing fluid flows through the channels, because this forms the basis for the associated pressure drop and heat transfer. As the fluid condenses in the channel and progresses from the vapor phase toward the liquid phase over a range of qualities, different flow patterns are established at different regions of the condenser. The progression of flow regimes, and in particular, the transitions between the different flow mechanisms, is different in microchannels than those in larger diameter tubes, primarily due to differences in relative magnitudes of gravity, shear, viscous, and surface tension forces. Similar differences exist between flow regimes and transitions in circular and non-circular microchannels. Therefore, extrapolation of large round-tube correlations to smaller diameters and non-circular geometries could introduce errors into pressure drop and heat transfer predictions. Also, the design of optimal condensers often requires the provision of varying tube-side flow area. If a constant tube-side cross-sectional area is provided, the high velocities in the initial portion of the condenser would result in excessive pressure drops with a corresponding decrease in saturation temperature and driving temperature difference, while the low velocities toward the end of the condenser would result in very low heat transfer coefficients. A gradually decreasing flow cross-section through the use of varying numbers of tubes per pass is therefore desirable for optimal condenser design. The selection of tube sizes and pass arrangements for a given application would require not just an average condensation coefficient, but rather an understanding of the variation in heat transfer coefficient and pressure drop with quality, phase velocities, and flow patterns. Unfortunately, the bulk of the literature on two-phase flow regimes is on adiabatic flows, in which air–water, nitrogen–water, and air–oil mixtures, typically in the absence of heat 232 Heat transfer and fluid flow in minichannels and microchannels transfer, are used to simulate condensing flows. The obvious reason for this approach is the substantial simplification in experimental facilities that it offers. Thus, the requisite vapor–liquid “quality” can simply be “dialed in” by independently controlling the air and liquid flow rates entering the test section, whereas this must be achieved indirectly though upstream and downstream conditioning of a condensing fluid through pre- and postheating/cooling, as appropriate. In addition, experiments on refrigerants require closed loops, whereas a nitrogen–water or air–water experiment can be conducted in a oncethrough manner, with the exiting fluid streams simply being exhausted and drained. Finally, several fluids (most refrigerants), at condensing temperatures of interest, are at pressures much higher than atmospheric, which presents challenges in providing optical access during the condensation process. On the other hand, air–water experiments are typically conducted at or near atmospheric pressure. The properties of the air–water fluid pair are significantly different from those of refrigerant vapor–liquid mixtures; therefore, application of the results from air–water studies and extrapolation to condensing flows must be done with appropriate caution. Also, the absence of the heat of condensation during air–water experiments must be recognized while applying these studies to condensation in microchannels. Nevertheless, due to the preponderance of adiabatic air–water studies in the literature, and the relative scarcity of condensation studies, a discussion of the significant studies on adiabatic twophase flow is provided here first, followed by studies on condensation (Table 6.1). 6.2.1. Adiabatic flow 6.2.1.1. Conventional tubes Early attempts at flow regime mapping were conducted on relatively large tubes using air– water or air–oil mixtures (Alves, 1954; Baker, 1954; Govier and Omer, 1962). Alves (1954) recognized several different flow patterns for air–water and air–oil flows corresponding to different gas and liquid superficial velocities in a 25.4-mm tube. Through comparisons of pressure drop data with the Lockhart– Martinelli (1949) correlation, he suggested that it was necessary to take flow regimes into account while computing two-phase pressure drops. Baker (1954) developed a flow regime map with flow pattern transitions based on functions of the mass flux of the gas phase and the liquid-to-gas ratio. Flow patterns such as bubble, plug, stratified, wavy, slug, annular, and spray were observed for oil and gas flow in 25.4–102-mm tubes, and combined with the data of Jenkins (1947), Alves (1954), and others to develop the map. He found that pressure drops for the large pipes (>200 mm) were 40–60% less than those predicted by the Lockhart–Martinelli (1949) correlation. He also found that the onset of slug flow occurred at lower values of the Lockhart–Martinelli parameter, X , in the larger pipes, and that the pipe diameter affects the two-phase multiplier. Thus, the influence of pipe diameter on flow patterns has been acknowledged for quite some time. The importance of tube diameter was also recognized and quantified by Govier et al. (1957) and Govier and Short (1958), particularly for upward flow of air–water mixtures, even though their work was limited to a relatively larger diameter range (16.00– 63.50 mm). Mandhane et al. (1974) tested the available flow maps with data representing a wide range of flow conditions for pipe diameters ranging from 12.7 to 165.1 mm. They Table 6.1 Summary of flow regime studies. Investigators Hydraulic diameter Adiabatic flow: conventional tubes Taitel and Dukler (1976) Fluids Orientation/ conditions Range/ applicability Techniques, basis, observations • Devised theoretical approach to flow regime mapping using a momentum balance on a stratified flow pattern Air–water Air–water Horizontal 0.043 < jG < 170.7 m/s 0 < jL < 7.315 m/s • “Best fit” map from available flow regime maps • Introduced physical property corrections Adiabatic flow: small circular channels Suo and Griffith 1–6 mm Air–water (1964) Horizontal adiabatic l /g >> 1 µl /µg > 25 • Transition from elongated bubble to annular and bubbly flow • Criterion to determine when buoyancy effects can be neglected Mandhane et al. (1974) 12.7–165.1 mm Barnea et al. (1983) 4–12 mm Air–water Horizontal T = 25 C Pexit = 1 atm 0.001 < jL < 10 m/s 0.01 < jG < 100 m/s • Recommended Taitel and Dukler (1976) model for all transitions except stratified to non-stratified • Modified stratified-to-intermittent transition boundary of Taitel and Dukler (1976) Damianides and Westwater (1988) 1–5 mm Air–water Horizontal P ~ 5 atm T ~ 15–20 C 0.001 < jL < 10 m/s 0.01 < jG < 100 m/s • Documented effect of hydraulic diameter • Some agreement for 5 mm with Taitel and Dukler (1976), but poor agreement for 1 mm • 1-mm tube showed large intermittent flow region • Increasingly small stratified flow region in small diameter tubes Fukano et al. (1989) 1–4.9 mm Air–water Horizontal adiabatic Pexit = 1 atm 0.04 < jG < 40 m/s 0.2 < jL < 4 m/s • Results agreed with Barnea et al. (1983) but not with Damianides and Westwater (1988) • Mandhane et al. (1974) cannot predict transitions in small tubes • Taitel and Dukler (1976) and Weisman et al. (1979) not reliable for small diameters Brauner and Maron (1992) 4–25.2 mm Air–water Horizontal 0.01 < jG < 40 m/s 0.001 < jL < 0.3 m/s • Incorporated the stabilizing effect of surface tension associated with practically finite wavelengths • Transition criteria based on Eotvos number (Continued ) Table 6.1 (Continued ) Investigators Hydraulic diameter Fluids Ide et al. (1995) 0.5–6 mm Air–water Vertical upwards Mishima and co-workers (Mishima and Hibiki, 1996; Mishima et al., 1997; Mishima and Hibiki, 1998) Coleman and Garimella (1999) 1–4 mm Air–Water Vertical upwards 0.0896 < jG < 79.3 m/s 0.0116 < jL < 1.67 m/s • Neutron radiography for flow visualization and void fractions • Recommended transition criteria of Mishima and Ishii (1984) 5.5-, 2.6-, 1.75-, 1.3-mm circular Dh = 5.36 mm ( = 0.725) rectangular 1.1- and 1.45-mm circular; semi-triangular 1.09 and 1.49 mm Air–water Horizontal 0.1 < jG < 100 m/s 0.01 < jL < 10 m/s • Diameter has significant effect on flow regime transitions • Intermittent flow regime increases in small diameter tubes • Comparisons with the criteria of Damianides and Westwater (1988), Fukano et al. (1989) and Weisman et al. (1979) Air–water Horizontal adiabatic 0.02 < jG < 80 m/s 0.02 < jL < 8 m/s • Observed bubbly, churn, slug, slug-annular and annular flow • No stratified flow • Maps agreed with Damianides and Westwater (1988), Fukano et al. (1989) and Fukano and Kariyasaki (1993) • Poor agreement with analytically derived criteria (Suo and Griffith, 1964; Taitel and Dukler, 1976) • Dispersed bubbly flow not observed in smallest channel, but capillary bubble flow pattern seen • Trends agreed well with several investigators (Barnea et al., 1983; Galbiati and Andreini, 1992a; Mishima and Hibiki, 1996; Ide et al., 1997), but not with Taitel and Dukler (1976) and Mishima and Ishii (1984) • Transitions difficult to distinguish for air–water, but clear for R-134a • Lower of R-134a compared to air–water led to differences • R-134a transitions agreed with Hashizume (1983) and Wang et al. (1997a) Triplett et al. (1999b) Orientation/ conditions Range/ applicability • Studied “liquid lumps” formed in vertically upward flow • Velocity characteristics of liquid lumps used to interpret flow pattern with corresponding maps by Fukano et al. (1989) Zhao and Bi (2001a) Triangular channels Dh = 2.886, 1.443, and 0.866 mm Air–water Vertical upward 0.1 < jG < 100 m/s; For Dh = 2.886, 1.443 0.08 < jL < 6 m/s For Dh = 0.866 0.1 < jL < 10 m/s Yang and Shieh (2001) 1, 2, and 3 mm Air–water Horizontal adiabatic T = 25–30 C 0.016 < jG < 91.5 m/s 0.006 < jL < 2.1 m/s 300 < G < 1600 kg/m2 -s T = 30 C R-134a Techniques, basis, observations Tabatabai and Faghri (2001) 4.8–15.88-mm condensation 12.3–1-mm air–water Water R-12 R-113 R-22 R-134a • Flow regime maps based on relative effects of surface tension, shear, and buoyancy forces • Better agreement with available data in sheardominated regions • Predicts movement of boundaries with channel size Horizontal Adiabatic flow: narrow, high-aspect ratio, rectangular channels Troniewski and Rectangular Air–water Horizontal Ulbrich (1984) 0.09 < < 10.10 7.45 < Dh < 13.1 Vertical T = 25–35 C 0.6 < jG < 43 m/s 200 < jL < 1600 m/s T = 25–35 C 0.7 < µL < 40 × 10−3 Pa s 995 < L < 1150 kg/m3 • Corrections to axes of Baker (1954) map based on single-phase velocity profiles Wambsganss et al. (1991; 1994) Rectangular = 6, 0.167 Dh = 5.44 mm Air–water Horizontal adiabatic 0.0001 < x < 1 50 < G < 2000 kg/m2 -s • Flow visualization and dynamic pressure measurements • Qualitative agreement with large circular channels but quantitative differences • Transition criteria for bubble or plug flow to slug flow based on root-mean-square pressure changes Xu and co-workers (Xu, 1999; Xu et al., 1999) Rectangular gap = 0.3, 0.6, and 1 mm Air–water Vertical adiabatic Pexit = 1 atm 0.01 < jG < 10 m/s 0.01 < jL < 10 m/s • Bubbly, slug, churn, and annular flow in 0.6- and 1-mm channels • Bubbly flow absent in 0.3-mm channel • Extended criteria of Mishima and Ishii (1984) for application to narrow rectangular geometry Hibiki and Mishima (2001) Rectangular channels with gap 0.3–17 mm Air–water, steam–water Vertical upward Air–water data (from Sadatomi et al., 1982; Lowry and Kawaji, 1988; Ali et al., 1993; Mishima et al., 1993; Wilmarth and Ishii, 1994; Xu, 1999) Steam– water data (from Hosler, 1968) • Criteria specifically for narrow rectangular channels based on Mishima and Ishii (1984) criteria • Collision/coalescence of bubbles increase when maximum distance between bubbles < projected bubble diameter (Continued ) Table 6.1 (Continued ) Investigators Hydraulic diameter Adiabatic flow: microgravity Rezkallah et al. 9.525 mm (Rezkallah, 1995; Rezkallah and Zhao, 1995; Lowe and Rezkallah, 1999) Adiabatic flow: D << 1 mm Kawahara 100 µm et al. (2002) Fluids Orientation/ conditions Range/ applicability Techniques, basis, observations • • • • Slug at µg different from “Taylor bubble” profile seen at 1-g Annular flow regime similar to 1-g Little effect of µL on transitions Later verified and modified based on 6–40-mm tubes Air–water Air–glycerin/ water Microgravity 0.09 < jG < 30 m/s 0.015 < jL < 3.5 m/s 0.6 < jG < 12 m/s 0.7 < jL < 2 m/s Nitrogen–water Horizontal 0.1 < jG < 60 m/s 0.02 < jL < 4 m/s • Low Re at small Dh implies greater effects of wall shear and surface tension • Observed liquid slug, gas core with smooth-thin liquid film, gas core with smooth-thick liquid film, gas core with ring-shaped liquid film, and gas core with deformed interface • No gravitational effects, absence of bubbly flow (no bubble breakup due to absence of liquid phase turbulence at low Re) • Flow conditions determine likely, but not unique patterns • Map based on probability of occurrence of patterns • Four flow regimes (slug-ring, ring-slug, semi-annular, and multiple), which are combinations of flow mechanisms Chung and Kawaji (2004) 530, 250, 100, and 50 µm Nitrogen–water Horizontal 0.02 < jG < 73 m/s 0.01 < jL < 5.77 m/s • Extended work of Kawahara et al. (2002) for effect of diameter • 530- and 250-µm flow similar to ~1-mm flow • Slug flow in small channels; absence of bubbly, churn, slug-annular, and annular flow Chung et al. (2004) 96-µm square Nitrogen–water 100-µm circular Horizontal Circular data (from Chung and Kawaji, 2004) • Negligible effect of channel shape Serizawa et al. (2002) 20-, 25-, and 100-µm circular Horizontal 0.0012 < jG < 295.3 m/s • Results different from other investigators of D << 1-mm channels • Numerous terms to define flow patterns 0.003 < jL < 17.52 m/s • Bubbly flow in 25-µm channel Air–water, steam (only 50 µm) • High capillary pressure at small bubble interface keeps bubble spherical and prevents coalescence • Slug flow due to entrance phenomenon, not bubble coalescence • Liquid ring flow due to rupture of liquid bridges between bubbles at high gas velocities • Surface tension prevents liquid slugs from spreading as film, leads to dry patches, liquid “lumps” • State that air–water and steam–water patterns in 25-µm channel similar; photographs presented seem dissimilar • Surprisingly state that flow map agrees with large channel Mandhane et al. (1974) map, but graphs indicate otherwise Condensing flow Traviss and Rohsenow (1973) Breber et al. (1980) 8 mm R-12 Horizontal condensing 100 < G < 990 kg/m2 -s 10 < T < 40.6 C 4.8–50.8 mm R-11, R-12, R-113, steam, n-pentane Horizontal condensing 108.2 < P < 1250 kPa 17.6 < G < 990 kg/m2 -s R-113, steam, propane, methanol, n-pentane R-12, R-113, steam Horizontal condensing Sardesai et al. (1981) 24.4 mm Soliman (1982; 1986) 4.8 < D < 15.9 mm Tandon et al. (1982; 1985a) Hashizume et al. (Hashizume, 1983; Hashizume et al., 1985; Hashizume and Ogawa, 1987) 4.8 < D < 15.9 mm 10 mm R-12, R-113, R-22 R-12, R-22 • • • • • Condensing Condensing Horizontal condensing • • • • 0.01 < jg* < 20 0.001 < 1 − / < 3 570 < p < 1960 kPa Disperse, annular, semi-annular, and slug flow observed Froude number as basis for condensation mode von Karman velocity profile to describe film velocity Taitel and Dukler (1976) map to develop simple criteria for condensation in horizontal tubes Criteria based on ratio of shear to gravity forces All map boundaries vertical and horizontal lines – easy to use Defined transition regions, rather than abrupt transitions Investigated annular– stratified/wavy transition Taitel and Dukler (1976) annular–stratified/wavy transition criterion as basis to develop transition criterion from data • Annular–wavy Froude number transition criterion based on data from different sources • Mist–annular Weber number transition criterion • Simple transition criteria, good agreement with data for annular, semi-annular, and wavy flows • Refrigerant flow patterns different from air–water • Modified property corrections of Baker (1954) map proposed by Weisman et al. (1979) to make them applicable to R-12 and R-22 (Continued ) Table 6.1 (Continued ) Investigators Hydraulic diameter Fluids Orientation/ conditions Range/ applicability Techniques, basis, observations Wang et al. (1997a, b) 6.5 mm R-22, R-134a, R-407C Horizontal 50 < G < 700 kg/m2 -s Tsat = 2, 6, and 20 C • At low G, intermittent and stratified flow; at high G, annular • Agreed with modified Baker (1954) map of Hashizume et al. (Hashizume, 1983; Hashizume et al., 1985; Hashizume and Ogawa, 1987) Dobson and co-workers (Dobson, 1994; Dobson et al., 1994; Dobson and Chato, 1998) 3.14, 4.6, and 7.04 mm R-12, R-134a, R-22, R32/R125 Horizontal Condensing 25 < G < 800 kg/m2 -s T = 35–60 C • Good agreement with Mandhane et al. (1974) map after axis transformation • Primary distinction between gravity and shear-controlled flows • Criteria based on modified Soliman number and mass flux Coleman and Garimella (2000 a, b; 2003), Garimella (2004) Round: 4.91 mm; Square: 4, 3, 2, 1 mm; Rectangular: 4 × 6; 6 × 4; 2 × 4; 4 × 2 mm R-134a Horizontal condensing P = 1379–1724 kPa 150 < G < 750 kg/m2 -s • Flow regimes subdivided into flow patterns, G–x based transition criteria • Dh more important than shape, aspect ratio for transition criteria • Intermittent and annular flow increases as Dh decreases • Wavy flow non-existent in 1-mm channel Cavallini et al. (2001; 2002a, b) 8 mm R-22, R-134a, R-125, R-32, R-236ea, R-407C, and R-410A Condensation, horizontal tubes 30 < Tsat < 50 C 100 < G < 750 kg/m2 -s • Criteria similar to Breber et al. (1979; 1980) for primary flow regimes (annular, stratified, wavy, and slug) • Recommended Kosky and Staub (1971) model for annular flow El Hajal et al. (2003) 3.14 < D < 21.4 mm R-22, R-134a, R-125, R-32, R-236ea, R-410a Horizontal condensing 16 < G < 1532 kg/m2 -s 0.02 < pr < 0.8 76 < (We/Fr)L < 884 • Transitions between stratified, wavy, intermittent, annular, mist and bubbly flow on G–x coordinates, adapting boiling map of Kattan et al. (1998a, b; 1998c) with updates by Zurcher et al. (1999) • Void fraction for map construction deduced from heat transfer database, subsequently averaged with homogeneous value Chapter 6. Condensation in minichannels and microchannels 239 developed a “best fit” map from the different flow regime maps available at the time, and also introduced physical property corrections for gas and liquid density, gas and liquid viscosity, and surface tension to be applied to the transition lines. However, these corrections did not yield a significant improvement in flow pattern prediction. A theoretical model of pressure drop for slug flows in horizontal and near-horizontal tubes was developed by Dukler and Hubbard (1975) with the two main contributions to pressure drop being the acceleration of the liquid to the slug velocity, and the pressure drop due to wall shear at the back of the slug. This model was validated using pressure–time traces and individual phase velocity measurements. Nicholson et al. (1978) extended this model to include elongated bubble flows by using a constant slug length, and also used the model to predict flow regime transitions. Taitel and Dukler (1976) devised a theoretical approach to flow regime mapping for air–water mixtures using a momentum balance on a stratified flow pattern. A set of four non-dimensional parameters was used to define instability-driven transitions from the stratified to annular or intermittent flows, and to distinguish between stratified smooth and stratified wavy flows. This theoretical model showed good agreement with the experimentally developed flow map of Mandhane et al. (1974). The effects of fluid properties (liquid viscosity, surface tension, gas density) and pipe diameter on two-phase horizontal flow patterns were investigated by Weisman et al. (1979) for pipe diameters ranging from 11.5 mm to 51 mm. Using data from other investigators as well as new flow visualization and pressure fluctuation data, modifications to existing correlations were developed for the flow-regime transitions. It was concluded that for the range of diameters tested, the pipe diameters and fluid properties have only moderate influences on the transition lines. Some investigators (Moore and Turley, 1983; Cai et al., 1996) have used statistical approaches to better define the transitions between flow regimes (plug or bubble to slug flow, annular to annular–mist flow) that are difficult to distinguish visually. This approach utilizes the root mean square and fluctuations in the pressure/time signals, frictional pressure gradients, and chaos theory to provide an objective means to interpret flow pattern transitions. Annunziato and Girardi (1987) used differential pressure measurement and local void fraction probes to measure the temporal fluctuations in a 89-mm diameter tube. 6.2.1.2. Small circular channels Among the studies on flow regime maps for microchannels using adiabatic air–water mixtures, Suo and Griffith (1964) correlated the transition from elongated bubble to annular and bubbly flow in capillary tubes (1.0 < D < 1.6 mm) by using the average volumetric flows of the liquid and gas phases, and the velocity of the bubbles. They also provided a criterion which determines when buoyancy effects can be neglected in terms of the parameter l gD2 /. However, it must be noted that they restrict their analysis to situations where l /g 1 and µl /µg > 25. While they explicitly state these restrictions in their paper, several other air–water adiabatic flow papers do not establish such explicit restrictions; therefore, caution is advised when applying these air–water based criteria to refrigerant condensation. For example, for refrigerant R-134a condensing at 1500 kPa, the density ratio is 14, while the viscosity ratio is 9.7. For the air–water pair at standard conditions, the corresponding ratios are l /g = 839 and µl /µg = 49. Similarly, the surface tension of the air–water pair is 17 times higher than that of R-134a. 240 Heat transfer and fluid flow in minichannels and microchannels Barnea et al. (1983) classified two-phase flow patterns in small horizontal tubes (4 < D < 12 mm) into to four major regimes (dispersed, annular, intermittent and stratified). They found that all transitions except the stratified to non-stratified transition were satisfactorily described by the Taitel– Dukler (1976) model. Therefore, they modified the stratified-to-intermittent flow transition boundary predicted by Taitel and Dukler (1976) by accounting for surface tension in terms of the gas-phase height in the channel as follows: hG ≤ 4 $ g 1 − 4 (6.3) Even though the above equation includes surface tension and gravitational terms, they commented that for small channels, this condition is always satisfied, which leads to a further simplification of the criterion, namely, hG ≤ 4 D. They also provided different interpretations for the formation of intermittent flow from stratified flows. In large channels, they attribute the transition to the Kelvin–Helmholtz instability, whereas in small tubes with wetting liquid, wetting of the wall due to capillary forces causes the film to climb and form a bridge, which leads to intermittent flow. Damianides and Westwater (1988) developed individual flow regime maps for air– water mixtures for 1 < Dh < 5 mm to document the effect of diameter on the flow regime transitions. They found some agreement between their results and the transition criteria of Taitel and Dukler (1976) for the 5-mm case, but the agreement was poor for 1-mm tubes. The 1-mm tube showed a large intermittent flow region, with the transition from annular flow attributed to the generation of roll waves. For the larger diameter tubes, they reported the existence of an increasingly large region of stratified wavy flow, with the waves creeping up to form annular flow at larger gas velocities. The agglomeration of plugs of gas led to slug flow, rather than the growth of a finite amplitude wave, which meant that the Kelvin– Helmholtz instability could not be used to predict the onset of slugging. As observed by almost all investigators, they also stated increasingly important role of surface tension at the smaller channel diameters. They noted that the intermittent-todisperse flow transition occurred at larger liquid flow rates in the larger tubes, and that the intermittent-to-annular transition occurred at larger gas flow rates in smaller tubes. The stratified flow region was essentially non-existent in the 1-mm tube. Fukano et al. (1989) also investigated air–water flow patterns in tubes with 1 < Dh < 4.9 mm and recorded bubbly, annular, plug, and slug flows, the latter two being combined into one intermittent region for pressure drop analyses. Their results agreed with the work of Barnea et al. (1983), and while there is some disagreement between the results reported by Fukano et al. (1989) and Damianides and Westwater (1988), these studies do point out that the flow regime map presented by Mandhane et al. (1974) cannot sufficiently predict the flow regime transitions in small diameter tubes. Also, it appears that the theoretical predictions of Taitel and Dukler (1976) and the correlations presented by Weisman et al. (1979) are not reliable for small diameters. Brauner and Maron (1992) noted that several previous researchers had based their flow regime transition analyses on single stability criteria of infinitely long waves. Therefore, they incorporated the stabilizing effects of surface tension associated with practically finite Chapter 6. Condensation in minichannels and microchannels 241 wavelengths, and were able to predict the transition from stratified flow over a wide range of channel sizes. This resulted in a transition criterion based on the Eotvos number: E¨o = (2)2 >1 l − g D2 g (6.4) Ide et al. (1995) studied the “liquid lumps” formed during vertical upward air–water flow in six tubes with 0.5 < D < 6.0 mm, including the velocity of the lumps, the corresponding disturbance wave and the base film. A momentum-averaged mean velocity of liquid lumps was used to characterize the flow for cases with large liquid lumps and many small waves. The liquid lump velocity characteristics were then used to interpret the flow patterns with corresponding maps (Fukano et al., 1989) for horizontal capillary tubes. Mishima and Hibiki (1996) investigated flow patterns for upward flow of air– water mixtures in vertical tubes (an orientation that is not particularly relevant for condensation applications) with 1 < D < 4 mm. Some variants of flow patterns typically seen in larger tubes were observed, but the transitions were in reasonable agreement with Mishima and Ishii’s (1984) transition criteria. Mishima et al. (1997; 1998) used neutron radiography, which uses the fact that thermal neutrons easily penetrate heavy materials such as metals, but are attenuated by light materials containing hydrogen, that is water, to non-intrusively investigate qualitative and quantitative two-phase flow phenomena. Thus, the incident neutron beam is attenuated in proportion to the liquid phase thickness, allowing projection of the image of the twophase flow, which is then converted to an optical image by the scintillator. The image is intensified and enlarged, and detected by a high-speed video camera. Air–water vertical upward flow was visualized to identify the prevailing flow regime, rising velocity of bubbles, and wave height and interfacial area in annular flow. Representative images of slug flow before and after processing are shown in Fig. 6.2. Void profiles and void fraction variations were also recorded as discussed in a subsequent section. By utilizing attenuation characteristics of neutrons in materials, measurements of void profile and average void fraction were performed. This technique enables flow visualization in situations without optical access and where X-ray radiography is not applicable. Coleman and Garimella (1999) investigated the effect of tube diameter and shape on flow patterns and flow regime transitions for air–water flow in circular tubes of 5.5-, 2.6-, 1.75-, and 1.3-mm diameter, and a rectangular tube with Dh = 5.36 mm and aspect ratio = 0.725. Gas and liquid superficial velocities were varied from 0.1 m/s < USG < 100 m/s, and 0.01 m/s < USL < 10.0 m/s, respectively. Bubble, dispersed, elongated bubble, slug, stratified, wavy, annular–wavy, and annular flow patterns were observed (Fig. 6.3). This figure also shows their attempt to bring some uniformity to the terminology used to describe these flow mechanisms by the various investigators cited above. Thus, they divided the mechanisms into four major regimes (stratified, intermittent, annular, and dispersed) and subdivided these major regimes further into the applicable patterns. Thus, when comparing with the results of various investigators, the nomenclature for at least the major regimes would be somewhat uniform. The tube diameter had a significant effect on the transitions. As the tube diameter is decreased, the transition from the intermittent regime to the dispersed or bubbly regime occurred at progressively higher USL . Also, the transition from intermittent to annular flow occurs at a nearly constant USL , for a given tube, with the 242 Heat transfer and fluid flow in minichannels and microchannels (a) (b) Fig. 6.2. Raw and processed images of slug flow in vertical upward air–water flow. Reprinted from Mishima, K., Hibiki, T. and Nishihara, H., Visualization and measurement of two-phase flow by using neutron radiography, Nuclear Engineering and Design, 175(1–2), pp. 25–35 (1997) with permission from Elsevier. transition USL increasing as the diameter first decreases from 5.5 mm, but approaching a limiting value as the diameter decreases further to 1.75 and 1.30 mm. The size of the intermittent regime increases in small diameter tubes. The primary difference in their results for round and rectangular channels of approximately the same Dh was that the rectangular tube showed a transition to disperse flows at higher USL , presumably because it is more difficult to dislodge the liquid from the corners of the rectangular tube due to the surface tension forces. As with other investigators, they found that very few points for the 5.5-mm tube were in the stratified regime, whereas the Taitel–Dukler (1976) criteria predicted the existence of a stratified region for these conditions. The agreement with the intermittentto-disperse and the annular-to-disperse transitions was somewhat better. However, the assumptions inherent in the Taitel–Dukler analyses were not considered applicable to the small diameter channels. The agreement with the data of Damianides and Westwater (1988) Chapter 6. Condensation in minichannels and microchannels 243 Stratified regime: wavy flow pattern Intermittent regime: elongated bubble pattern Intermittent regime: slug flow pattern Annular regime: wavy–annular pattern Annular regime: annular pattern Dispersed regime: bubble flow pattern Dispersed regime: dispersed flow pattern Fig. 6.3. Air–water flow patterns in circular and rectangular tubes. Reprinted from Coleman, J. W. and Garimella, S., Characterization of two-phase flow patterns in small diameter round and rectangular tubes, International Journal of Heat & Mass Transfer, 42(15), pp. 2869–2881 (1999) with permission from Elsevier. was much better. The transition to the dispersed region (including the effect of diameter) was also in good agreement with the results of Fukano et al. (1989). The transition correlations presented by Weisman et al. (1979) were only in agreement for the largest tube studied by Coleman and Garimella, with the agreement deteriorating as Dh decreased. 244 Heat transfer and fluid flow in minichannels and microchannels Triplett et al. (1999b) conducted an investigation similar to that of Coleman and Garimella (1999) of flow regimes in adiabatic air–water flows through circular microchannels of 1.1- and 1.45-mm diameter, and in semi-triangular microchannels with Dh = 1.09 and 1.49 mm. They identified bubbly, churn, slug, slug–annular, and annular flow patterns (stratified flow was not observed as was the case with many other investigators), and plotted the data on gas and liquid superficial velocity axes. The progression between the respective regimes as the gas and liquid velocities are changed is shown in Fig. 6.4. Bubbly flow undergoes a transition to slug flow when the gas superficial velocity UGS increases due to the corresponding rise in the void fraction and the coalescence of bubbles. At high superficial liquid velocities ULS , as the overall mass flux is increased, churn flow is established due to the disruption of slugs. Churn flow also occurred when large waves disrupted wavy– annular flows. Slug–annular flow is established at low UGS and ULS , and changes to annular flow when UGS increases. The flow regime maps for all channels tested were substantially similar. Their maps agreed in general with the results of Damianides and Westwater (1988) and Fukano and Kariyasaki (1993), but the agreement with almost all analytically derived transition criteria (Suo and Griffith, 1964; Taitel and Dukler, 1976) was poor. Zhao and Bi (2001a) investigated flow patterns for cocurrent upward air–water flow in vertical equilateral triangular channels with Dh = 2.886, 1.443, and 0.866 mm. The flow patterns encountered included dispersed bubbly flow, slug flow, churn flow and annular flow for Dh = 2.886 and 1.443 mm. For Dh = 0.866 mm, dispersed bubbly flow was not observed, but a capillary bubbly flow pattern, consisting of a succession of ellipsoida bubb es, spanned the cross-sect on of the channe (F g. 6.5) at ow gas f ow rates. Long s ugs were seen at h gh superf c a gas ve oc t es n the s ug f ow reg me. Churn and annu ar f ows occurred at h gher superf c a gas ve oc t es as the channe s ze decreased. These trends were n agreement w th the resu ts of other nvest gators (Barnea et a ., 1983; Ga b at and Andre n , 1992a; M sh ma and H b k , 1996; Ide et a ., 1997) for sma c rcu ar tubes. However, occurrence of the churn–annu ar trans t on ne at h gher gas ve oc t es w th decreas ng channe d ameter d d not agree w th the f nd ngs of Ta te et a . (1980) and M sh ma and Ish (1984), who pred cted the oppos te trend for convent ona tubes. The s gn f cant d screpanc es between the data and the Ta te et a . and the M sh ma–Ish trans t on cr ter a were attr buted to the arge d fference n channe s ze and the men scus effect due to the tr angu ar shape. Yang and Sh eh (2001) observed bubb y, p ug, wavy, s ug, annu ar, and d spersed reg mes dur ng a r–water and R-134a (ad abat c) f ow through 1-, 2-, and 3-mm hor zonta tubes. Trans t ons between the reg mes, espec a y s ug-to-annu ar, were d ff cu t to d st ngu sh n a r–water f ow, wh e a the R-134a trans t ons were c ear. In a manner much ke the study by Co eman and Gar me a (1999), they p otted the r f ow reg me maps for the d fferent tubes versus the trans t on nes of Ta te and Duk er (1976), Barnea et a . (1983), and Dam an des and Westwater (1988). The resu ts were n genera y good agreement w th those of the atter two nvest gators, but as po nted out by numerous other nvest gators, the strat f ed– nterm ttent trans t on pred cted by Ta te and Duk er (1976) was not seen, due to the napp cab ty of the bas s for th s trans t on to the sma er tubes. The s ug–annu ar trans t on for R-134a occurred at ower gas ve oc t es, wh e the nterm ttent–bubb y trans t on occurred at h gher qu d ve oc t es, both attr buted to the ower surface tens on of R-134a compared to the a r–water pa r. The R-134a trans t ons were n good agreement w th the Chapter 6. Condensat on n m n channe s and m crochanne s (a) ULS 3.021 m/s, UGS 0.083 m/s (b) ULS 5.997 m/s, UGS 0.396 m/s (c) ULS 0.213 m/s, UGS 0.154 m/s (d) ULS 0.608 m/s, UGS 0.498 m/s (e) ULS 0.661 m/s, UGS 6.183 m/s (f) ULS 1.205 m/s, UGS 4.631 m/s (g) ULS 0.043 m/s, UGS 4.040 m/s (h) ULS 0.082 m/s, UGS 6.163 m/s ( ) ULS 0.082 m/s, UGS 73.30 m/s ( ) ULS 0.271 m/s, UGS 70.42 m/s F g. 6.4. F ow reg mes for ad abat c a r–water f ow through 1.09-mm c rcu ar channe s. Repr nted from Tr p ett, K. A., Gh aas aan, S. M., Abde -Kha k, S. I. and Sadowsk , D. L., Gas– qu d two-phase f ow n m crochanne s, Part I: Two-phase f ow patterns, Internat ona Journa of Mu t phase F ow, 25(3), pp. 377–394 (1999) w th perm ss on from E sev er. 245 246 Heat transfer and f u d f ow n m n channe s and m crochanne s Channe nner wa A A Gas–wa nterface Gas– qu d nterface Gas L qu d A–A v ew F g. 6.5. Cap ary bubb e n tr angu ar channe . Repr nted from Zhao, T. S. and B , Q. C. Co-current a r–water two-phase f ow patterns n vert ca tr angu ar m crochanne s, Internat ona Journa of Mu t phase F ow, 27(5), pp. 765–782 (2001) w th perm ss on from E sev er. resu ts of Wang et a . (1997a) and Hash zume (1983), a though these two nvest gat ons focused on ower mass f uxes than those stud ed by Yang and Sh eh (2001). Tabataba and Faghr (2001) deve oped a f ow reg me map for m crochanne s w th part cu ar attent on to the nf uence of surface tens on. They noted that n two-phase f ow, r pp es are generated on the annu ar ayer w th an ncrease n gas-phase ve oc ty, wh ch n turn eads to the format on of co ars and br dges. They d st ngu sh s ug, p ug, and bubb e reg mes based on the s ze and gap between the br dges. As w th other nvest gators, they state that n arge tubes, grav tat ona forces pu the qu d f m down, prevent ng br dge format on. The r map s based on the re at ve effects of surface tens on, shear, and buoyancy forces. The map uses the rat o of pressure d fference due to surface tens on and that due to shear forces as one ax s, wh e the rat o of superf c a ve oc t es VG,S /V ,S forms the other ax s. The vo d fract on mode of Sm th (1969) s used to compute the requ red vapor or gas-core f ow area and nterfac a surface area. The three terms n the force ba ance per un t ength of the reg on of nterest are g ven by: Fsurface tens on = 2(1 − )0.5 L (6.5) Chapter 6. Condensat on n m n channe s and m crochanne s u 2 Fshear = Df L 4 Fbuoyancy = g − g Ag L 247 (6.6) (6.7) where the vo d fract on appears n each of the three equat ons above ( nd rect y through the gas phase area n the buoyancy term). Th s estab shes a cond t on for the surface tens on forces be ng dom nant: |Fsurface tens on | > |Fshear | + |Fbuoyancy | (6.8) wh ch then y e ds a cr ter on for the trans t on from surface-tens on-dom nated reg ons to shear-dom nated reg ons: é ù0.5 2(1 − )0.5 − g − g Ag (6.9) u < B = ë D f 4

û

Re at ng the actua phase ve oc t es to the correspond ng superf c a ve oc t es resu ts n the fo ow ng form of the trans t on cr ter on: VSG VSG > (1 − )B VSL (6.10) For the annu ar-to- nterm ttent trans t on, they used the fo ow ng cr ter on for the qu d vo ume fract on m proposed by Gr ff th and Lee (1964), wh ch s based on co ars form ng n annu ar f ow due to nstab t es at a wave ength of 5.5 × D, w th the co ar u t mate y form ng br dges and s ugs: 2 r 2r − 43 r 3 m= = 0.06 (6.11) r 2 (5.5)2r where the numerator s the tota qu d vo ume per s ug and the denom nator s the tota vo ume of the s ug at the most unstab e wave ength of 5.5. Tabataba and Faghr compared th s f ow reg me map w th data n the terature for condensat on of steam and refr gerants n tubes w th D > 4.8 mm, and for a r–water f ow n sma er d ameter tubes, and stated that better agreement was obta ned n the surfacetens on-dom nated reg ons, and a so that the map s ab e to pred ct the movement of the boundar es w th channe s ze. 6.2.1.3. Narrow, h gh aspect rat o, rectangu ar channe s Most of the research on two-phase f ow n sma hydrau c d ameter rectangu ar channe s uses tubes of e ther sma ( < 0.50) or arge ( > 2.0) aspect rat os (Hos er, 1968; Jones Jr. and Zuber, 1975; Lowry and Kawa , 1988; W marth and Ish , 1994). In a study on f ow reg me maps for 0.125 < < 0.50 and 11.30 < Dh < 33.90 mm, R chardson (1959) showed that the sma er aspect rat o suppressed the strat f ed and wavy f ow reg mes and promoted the onset of e ongated bubb e and s ug f ows. Tron ewsk and U br ch (1984) proposed 248 Heat transfer and f u d f ow n m n channe s and m crochanne s correct ons to the axes of the Baker (1954) map based on the s ng e-phase ve oc ty prof es n rectangu ar channe s for hor zonta and vert ca channe s w th 0.09 < < 10.10 and 7.45 < Dh < 13.10 mm. Lowry and Kawa (1988) stud ed rectangu ar geometr es w th Dh < 2.0 mm and 40 < < 60 n vert ca upward f ows and conc uded that the Ta te and Duk er (1976) mode was not va d for narrow channe f ow. Wambsganss et a . (1991) reported f ow patterns and f ow reg me trans t ons n a s ng e rectangu ar channe w th aspect rat os of 6.0 and 0.167 and Dh = 5.45 mm through f ow v sua zat on and dynam c pressure measurements. They found qua tat ve agreement w th the correspond ng f ow reg me maps for c rcu ar channe s and arger rectangu ar channe s, but noted quant tat ve d fferences between those maps and the r resu ts for the re at ve y sma er rectangu ar channe s. Wambsganss et a . (1994) extended th s work to deve op cr ter a for trans t on from bubb e or p ug f ow to s ug f ow based on root-meansquare pressure changes. The trans t on was dent f ed from a d st nct change n the s ope or a oca peak n the pressure drop versus Mart ne parameter X or qua ty x-graph. Xu (1999) nvest gated ad abat c a r/water f ow n vert ca rectangu ar channe s w th gaps of 0.3, 0.6, and 1.0 mm. The f ow reg mes (bubb y, s ug, churn, and annu ar) n the 1 and 0.6 mm channe s were found to be s m ar to those reported n other stud es. At sma er gaps, the bubb y–s ug, s ug–churn and churn–annu ar f ow trans t ons occurred at ower superf c a gas ve oc t es, due to the squeez ng of the gas phase a ong the w dth of the narrow channe s, wh ch was be eved to fac tate these trans t ons. In the 0.3-mm channe , bubb y f ow was absent, but cap-bubb y f ow was seen, character zed by two-d mens ona (2-D) sem -c rcu ar tops and f at bottoms. In add t on, the presence of drop ets together w th s ug and annu ar f ows (attr buted to the ncreased nf uence of surface tens on and shear) ed h m to add s ug-drop et and annu ar-drop et reg mes to the others found n the b gger channe s. Xu et a . (1999) ater extended the trans t on cr ter a deve oped by M sh ma and Ish (1984) for app cat on to the narrow rectangu ar geometry. A though th s or entat on and the correspond ng phys ca cons derat ons are not part cu ar y re evant to condensat on, the r approach can be mod f ed to y e d ana ogous trans t on cr ter a for some of the trans t ons encountered n condens ng f ows. They based the bubb y-to-s ug trans t on on the ncreas ng probab ty of bubb e co s on and coa escence, wh ch wou d ead to s ug f ow, w th th s trans t on occurr ng n the range of vo d fract ons 0.1 < < 0.3, as suggested by M sh ma and Ish (1984). The s ug–churn trans t on was a so adapted from the work of M sh ma and Ish (1984), based on the f ow becom ng unstab e when two ne ghbor ng s ugs start to touch each other as the channe mean vo d fract on ncreases. The trans t on to annu ar f ow was assumed to occur somewhat arb trar y for vo d fract ons >0.75. They ust f ed th s based on the f nd ngs of Armand (1946), who found that the vo d fract on s re ated to the vo umetr c qua ty through a s mp e near re at onsh p = 0.833 for < 0.9, but ncreased sharp y when > 0.9, s gn fy ng annu ar f ow due to the un nterrupted gas core. A though these nvest gators app ed some reasonab e phys ca ns ghts to the deve opment of the trans t on cr ter a, the agreement was on y acceptab e w th the 1-mm channe data, y e d ng poor agreement w th the 0.6- and 0.3-mm data. Us ng much the same approach as Xu et a . (1999), H b k and M sh ma (2001) adapted M sh ma and Ish s (1984) mode for round tubes to deve op a f ow reg me map and trans t on cr ter a spec f ca y for narrow rectangu ar cross-sect ons w th vert ca upf ow. The r map cons sted of bubb y, s ug, churn, and annu ar f ows. The bubb y-to-s ug Chapter 6. Condensat on n m n channe s and m crochanne s 249 trans t on was based on f uctuat ons of bubb es d str buted n a square att ce pattern. They postu ated that co s ons and coa escences of bubb es ncrease s gn f cant y when the max mum d stance between bubb es s ess than the pro ected d ameter of a f at bubb e. The bubb e spac ng was re ated to vo d fract on, wh ch y e ded the trans t on cr ter on n terms of the vo d fract on. The s ug-to-churn trans t on was based on the pressure grad ents around the bubb e and n the qu d f m surround ng t, wh e the churn f ow-to-annu ar f ow cr ter on was based on f ow reversa cons derat ons or the destruct on of qu d s ugs due to deformat on or entra nment. The r trans t on cr ter a were n reasonab e agreement w th data for a r–water f ows n rectangu ar channe s w th gaps of 0.3–17 mm, and w th some h gh-pressure water bo ng data. 6.2.1.4. Ins ghts from re evant m crograv ty work Based on the re at ve y sma nf uence of grav ty on two-phase f ow n m crochanne s, ns ghts nto these f ows can a so be obta ned from re ated work on s m ar f ows n reduced grav ty env ronments. Rezka ah et a . (Rezka ah, 1995; Rezka ah and Zhao, 1995) recorded two-phase f ow patterns aboard the NASA KC-135 a rcraft us ng a r– water and a r–g ycer n/water (to obta n d fferent v scos t es) n a 9.525-mm tube. They c ass f ed the f ows nto three reg mes (F g. 6.6): surface-tens on-dom nated bubb y and s ug f ows, an ntermed ate frothy s ug–annu ar reg on, and nert a-dom nated annu ar f ow and p otted the f ow reg mes on qu d- and gas-phase Weber number (WeSG , WeSL ) (where We = VS2 D/) coord nates based on the respect ve superf c a ve oc t es. They noted that s ug f ow under these cond t ons was d fferent from the “Tay or bubb e” prof e typ ca y seen at earth norma grav ty: the fore and aft port ons of the s ug were smooth and ha fspher ca , w th tt e entra nment of sma bubb es – s m ar observat ons w be descr bed Wave Bubb e Tay or bubb e L qu d f m Frothy s ug Gas S ug Gas core Bubb y S ug Trans t ona Annu ar F g. 6.6. M crograv ty f ow patterns documented by Lowe and Rezka ah (1999). Repr nted from Lowe, D. C. and Rezka ah, K. S., F ow reg me dent f cat on n m crograv ty two-phase f ows us ng vo d fract on s gna s, Internat ona Journa of Mu t phase F ow, 25(3), pp. 433–457 (1999) w th perm ss on from E sev er. 250 Heat transfer and f u d f ow n m n channe s and m crochanne s for m crochanne f ows n a subsequent part of th s sect on. The annu ar f ow reg me was s m ar to what s observed at 1-g, except that the f m th ckness was arger. The trans t on from bubb y/s ug f ow to ntermed ate f ow occurred at Weber number (based on the gas superf c a ve oc ty) of WeSG ~ = 1, wh e the trans t on to annu ar f ow started at WeSG ~ = 20. The bubb y-to-s ug f ow trans t on occurred at the rat o of superf c a qu d and gas ve oc t es VSL = C · VSG (1.2 < C < 4.6). Based on the r own add t ona work, vo d fract on nformat on, and data from other nvest gators compr s ng data for 6 < D < 40 mm, they ater (Rezka ah, 1996) mod f ed the f rst trans t on to We = 2, and a so recommended that the Weber numbers based on actua phase ve oc t es, rather than superf c a ve oc t es be used as coord nates for the f ow reg me map. L qu d v scos ty was found to have tt e effect on the trans t ons. These trans t on cr ter a were a so subsequent y (Lowe and Rezka ah, 1999) va dated through very deta ed measurements and ana yses of the vo d fract on s gna s n the var ous reg mes. Ga b at et a . (1992a, b; 1994) nvest gated f ow patterns n hor zonta and vert ca two-phase f ow of a r–water m xtures n cap ary tubes, and deve oped trans t on cr ter a between these patterns. The Ta te – Duk er (1976) strat f ed-toannu ar f ow trans t on cr ter a were mod f ed to account for the nf uence of surface tens on. They a so po nted out that n et m x ng effects are s gn f cant n estab sh ng fu y deve oped f ow. Two-phase f ow n cap ary tubes was treated as ana ogous to m crograv ty s tuat ons. 6.2.1.5. Channe s w th D << 1 mm F ow reg mes n two-phase f ow through m crochanne s w th D << 1 mm have on y recent y been nvest gated. One of the f rst groups of nvest gators of such f ows nc ude Feng and Ser zawa (1999; 2000) and Ser zawa and Feng (2001), who stud ed a r–water and steam–water f ow through 25- and 50-µm channe s. The r use of the term “ qu d r ng f ow” for such sma m crochanne s has s nce been used by other nvest gators n severa d fferent forms, as d scussed throughout th s sect on. Essent a y, as the gas f ow rate ncreases, the qu d br dge between ad acent gas s ugs n s ug f ow becomes unstab e, and trans t ons to th s so-ca ed qu d r ng f ow pattern. Kawahara et a . (2002) then conducted a comprehens ve study of f ow reg mes, vo d fract on and pressure drop for hor zonta n trogen–water f ow n 100-µm c rcu ar channe s. These authors contend that, n compar son w th f ow n channe s n the 1-mm d ameter range, the order of magn tude reduct on n channe d ameter resu ts n ower Reyno ds numbers and greater effects of wa shear and surface tens on, wh ch are man fested n d fferent types of f ow patterns. A though they stated the need for refract ve ndex match ng to remove the effects of opt ca d stort on ar s ng due to the sma rad us of curvature, they chose to conduct the exper ments w thout such apparatus. Th s s because they fe t that the near-wa phenomena, a though d storted, cou d be v ewed n more deta w thout opt ca correct on, and the effects of d stort on cou d be accounted for dur ng the ana ys s phase. They observed f ve d fferent f ow mechan sms: qu d a one ( qu d s ug), gas core w th smooth-th n qu d f m, gas core w th smoothth ck qu d f m, gas core w th r ng-shaped qu d f m, and gas core w th a deformed nterface. These f ow reg mes are shown n F g. 6.7 for var ous comb nat ons of qu d and gas ve oc t es. The ax symmetr c (non-strat f ed) f ow patterns c ear y demonstrated the absence of grav tat ona effects. S m ar y, the absence of sma , d spersed bubb es (bubb y f ow) was attr buted to the qu d phase Re, be ng very ow; thus, no bubb e breakup nduced Chapter 6. Condensat on n m n channe s and m crochanne s 251 L qu d a one L qu d a one L qu d a one Gas core f ow w th a smooth-th n qu d f m Gas core f ow w th a smooth-th n qu d f m Gas core f ow w th a smooth-th ck qu d f m Gas core f ow w th a smooth-th ck qu d f m Gas core f ow w th a r ng-shaped qu d f m Gas core f ow w th a r ng-shaped qu d f m Gas core f ow w th a deformed nterface (b) F ow d rect on Gas core f ow w th a deformed qu d f m (c) F ow d rect on Gas core f ow w th a smooth-th n qu d f m Gas core f ow w th a r ng-shaped qu d f m Gas core f ow w th a r ng-shaped qu d f m Nose of a gas s ug Ta of a gas s ug Gas bubb e (a) F ow d rect on F g. 6.7. A r–water f ow reg mes n 100-µm channe s. (a) UL = 0.15 m/s, UG = 6.8 m/s; (b) UL = 0.56 m/s, UG = 20.3 m/s; (c) UL = 3.96 m/s, UG = 19.0 m/s. Repr nted from Kawahara, A., Chung, P.M.-Y. and Kawa , M. Invest gat on of two-phase f ow pattern, vo d fract on and pressure drop n a m crochanne , Internat ona Journa of Mu t phase F ow, 28(9), pp. 1411–1435 (2002) w th perm ss on from E sev er. by qu d phase turbu ence cou d occur. They a so recogn zed that a g ven comb nat on of qu d and gas phase ve oc t es does not a ways resu t n a un que f ow pattern – t s mp y estab shes the more ke y patterns. Other nvest gators such as Co eman and Gar me a (2000a; 2003) have addressed th s ssue by def n ng “over ap zones” n the r f ow reg me maps, n wh ch the f ow cou d trans t on from one reg me to another at the same cond t ons. Therefore, they deve oped a f ow reg me map based on the probab ty of occurrence of the d fferent patterns at the d fferent cond t ons. These probab ty ana yses ed them to def ne four f ow reg mes (s ug-r ng, r ng-s ug, sem -annu ar, and mu t p e), wh ch were comb nat ons of the f ow mechan sms ment oned above. At ow qu d f ow rates, f ows w th a gas core surrounded by a smooth-th n qu d f m occurred at ow gas f ow rates, wh e the gas core was accompan ed w th a r ng-shaped qu d f m at h gh gas f ow rates. At h gh qu d f ow rates, the gas core was surrounded by a th ck qu d f m, or the gas core moved through a deformed f m n serpent ne manner. At a comb nat on of h gh gas and qu d f ow rates, severa comb nat ons of f ows occurred to y e d the “mu t p e” f ow pattern. Chung and Kawa (2004) extended the work of Kawahara et a . (2002) to nvest gate the effect of channe d ameter, by conduct ng exper ments on n trogen–water f ow through 530-, 250-, 100-, and 50-µm channe s. They found that the two-phase f ow character st cs n the 530- and 250-µm channe s were s m ar to those reported for channe s of ~1-mm d ameter, for examp e by Tr p ett et a . (1999b). For channe s sma er than these, on y s ug f ow was observed, w th the absence of bubb y, churn, s ug–annu ar and annu ar f ow attr buted to the greater v scous and surface tens on effects. From a comb nat on of f ow v sua zat on and vo d fract on and pressure drop ana yses, they nterpreted s ug f ow as the rap d mot on of a gas s ug through a re at ve y qu escent qu d f m retarded by the wa . Based on the va ues of the pert nent d mens on ess parameters, they observed that the re evant forces for these channe s were nert a, surface tens on, v scous force, and grav ty, n decreas ng order of mportance. Furthermore, w th decreas ng channe s ze, the Bond number, the superf c a Reyno ds numbers, the Weber number, and the cap ary number 252 Heat transfer and f u d f ow n m n channe s and m crochanne s a decrease, wh ch mp es that the nf uence of grav tat ona and nert a forces decreases, wh e the mportance of surface tens on and v scous forces ncreases. Chung et a . (2004) a so nvest gated the effect of channe shape on f ow patterns, vo d fract on, and pressure drop by conduct ng s m ar tests on a 96-µm square, and a 100-µm c rcu ar, m crochanne . The f ow reg me maps for the two channe s were substant a y s m ar, except that the square channe d d not exh b t the r ng-s ug f ow pattern that was present n the c rcu ar channe . They reasoned that whereas n the c rcu ar channe , the qu d f m must ncrease n th ckness or deve op nto th cker r ngs per od ca y, n the square channe , t accumu ates n the corners of the square channe , wh ch was a so supported by the ana yses of Ko b and Cerro (1991; 1993). Kawahara et a . (2005) subsequent y a so nvest gated water/n trogen and ethano –water/n trogen m xtures n 50-, 75-, 100-, and 251-µm c rcu ar channe s to demonstrate that f u d propert es have tt e effect on f ow patterns n these channe s. Ser zawa et a . (2002) a so conducted a study s m ar to that of Kawahara et a . (2002) on a r–water f ow n c rcu ar tubes of 20-, 25- and 100-µm d ameter, and steam–water f ow n a 50-µm c rcu ar tube. Representat ve f ow patterns recorded by them are shown n F g. 6.8. It must be noted that they deve op a part cu ar y mag nat ve and var ed set of terms that ead to further pro ferat on (and perhaps confus on) about the descr ptors for (a) Bubb y f ow (b) S ug f ow (c) L qu d r ng f ow (d) L qu d ump f ow F g. 6.8. A r–water f ow patterns n 25-µm c rcu ar channe . Repr nted from Ser zawa, A., Feng, Z. and Kawara, Z. Two-phase f ow n m crochanne s, Exper menta Therma & F u d Sc ence, 26(6–7), pp. 703–714 (2002) w th perm ss on from E sev er. Chapter 6. Condensat on n m n channe s and m crochanne s 253 the observed f ow mechan sms. Thus, the terms they use nc ude: d spersed bubb y f ow, gas s ug f ow, qu d r ng f ow, qu d ump f ow, skewed barbecue (Yak tor ) shaped f ow, annu ar f ow, frothy or w spy annu ar f ow, r vu et f ow, and qu d drop ets f ow. Un ke the f nd ngs of Kawahara et a . (2002), they observed bubb y f ow n the 25-µm channe , often bubb es a most the s ze of the channe c ose y fo owed by much sma er, f ne y d spersed bubb es. They noted that the cap ary pressure at the sma 5-µm bubb e nterface was as h gh as 0.3 bar, wh ch kept the bubb e from d stort ng from a spher ca shape or from coa esc ng. They exp a ned that s ug f ow occurs not due to bubb e coa escence, but rather because of an entrance phenomenon, whereby gas at a h gh f ow rate enters the channe , but the gas bubb e ve oc ty s not h gh enough to overcome the surface tens on of the qu d br dge. Th s s an exp anat on qu te d fferent from most other nvest gators, and requ res further study. They a so state that surface tens on prevents the qu d s ug from spread ng as a f m, and coup ed w th the h gh pressure of the gas bubb e, wh ch pushes t to expand across the ent re cross-sect on, eads to dry spots and patches. L qu d r ng f ow or g nates from qu d br dges between s ugs that are ruptured when the gas ve oc ty s very h gh. L qu d r ng mot on s governed by a ba ance between v scous forces at the wa and shear at the gas nterface: v scous forces dom nate up to a certa n thresho d th ckness, beyond wh ch the shear from the gas core pushes the qu d r ng. L qu d umps are formed at even h gher gas ve oc t es, when qu d s entra ned and osc ates, but surface tens on prevents t from spread ng as a f m. L qu d umps are cons dered by them to be “part a y cont nuous f ms” rather than the term r vu et that s often used n the terature. They a so present steam–water f ow reg mes that nc ude bubb y, s ug, qu d r ng and qu d drop et f ows, and state that these f ow mechan sms are s m ar to those shown for a r–water n a 25-µm tube (F g. 6.8). However, g ven the degree of d st nct on they make n th s work between sma var at ons n f ow mechan sms, t does not appear that the steam–water and a r–water f ow patterns are a that s m ar. Severa of the r photographs show arge gas bubb es appear ng together w th ad o n ng sma bubb es, and deta ed exp anat ons are prov ded based on the prom nence of surface effects n sma channe s. G ven the above emphas s on the un queness of m crochanne two-phase f ow, one of the r more surpr s ng f nd ngs s that the r f ow reg me map s deemed to agree w th the map of Mandhane et a . (1974), wh ch s one of the more w de y quoted f ow maps for much arger convent ona channe s. No rea exp anat on s g ven for th s stated agreement w th such a map, a though a qu ck ook at the map and the r data revea s that the agreement s not as good as they state. 6.2.1.6. Dependence on contact ang e and surface propert es Bara as and Panton (1993) stud ed the effect of contact ang e on a r–water f ow by us ng 1.6-mm d ameter hor zonta pyrex, po yethy ene, po yurethane, and f uoropo ymer res n tubes. These tubes const tuted three part a y wett ng and one non-wett ng comb nat on w th contact ang es of 34 , 61 , 74 , and 106 . S m ar f ow maps were found for the part a y wett ng tubes, except for the ex stence of r vu et f ow, wh ch rep aces wavy f ow when the contact ang e becomes arge. The trans t ons for the non-wett ng system (except for the p ug–s ug trans t on) were cons derab y d fferent. The trans t ons between s ug and r vu et f ow and those between s ug and annu ar f ow sh ft to 254 Heat transfer and f u d f ow n m n channe s and m crochanne s ower gas ve oc t es. The non-wett ng systems fac tate movement of qu d due to shear forces, caus ng the trans t on to r vu et or annu ar f ow at ower gas ve oc t es. Ser zawa and Kawahara (2001) po nted out that surface wettab ty effects on f ow patterns are more s gn f cant n sma er tubes due to the proport onate y arger nf uence of surface forces n these tubes. Th s was conf rmed by conduct ng tests w th very carefu y c eaned tubes, fo owed by tests us ng on y c ean ng w th ethano , as a so ment oned n Ser zawa et a . (2002). 6.2.2. Condens ng f ow Compared to the number of nvest gat ons on ad abat c two-phase f ows, there are few part cu ar y re evant stud es on condens ng f ows n sma d ameter channe s. A few of the key nvest gat ons toward the ow end of the hydrau c d ameter range from among the ava ab e stud es are summar zed here. One of the ear er, we c ted nvest gat ons of f ow patterns dur ng condensat on of refr gerants was conducted by Trav ss and Rohsenow (1973), who stud ed condensat on of R-12 n an 8-mm d ameter tube. A w de range of qua t es, mass f uxes (100 < G < 990 kg/m2 -s), and saturat on temperatures (10 C < T < 40.6 C) was tested. D sperse, annu ar, sem -annu ar (referred to by many as wavy, w th a th n qu d f m at the top and a th cker f m at the bottom of the tube), and s ug f ows were observed. The r ma n focus, as s the case w th most of the stud es on tubes w th D > ~5 mm, was the trans t on between shear-dom nated annu ar f ow and grav ty-dom nated strat f ed f ow. They used the Froude number Fr2 = V¯ 2 /g, where V¯ s the average ve oc ty of the qu d f m and s the f m th ckness, as the bas s for determ n ng the app cab e mode of condensat on. They used the von Karman un versa ve oc ty prof e to descr be the f m ve oc ty. Us ng th s ve oc ty prof e, and express ng the wa shear stress n terms of the two-phase fr ct ona pressure drop (wh ch was computed us ng a two-phase mu t p er to the vapor-phase pressure drop), they deve oped express ons for the f m Reyno ds number Re = G(1 − x)D/µ n terms of Fr , the Ga eo number Ga = gD3 / 2 , and the Mart ne parameter Xtt . By p ott ng the r data on a Re versus Xtt graph, they proposed that the annu ar to sem -annu ar trans t on occurs at a constant Fr of 45. They further state that th s boundary may be used to d st ngu sh between non-strat f ed d sperse and annu ar f ows and “strat f ed” sem -annu ar, wavy, s ug and p ug f ows. Th s deve opment s w de y ava ab e n the re evant textbooks, and the deta s are not presented here. Breber et a . (1980) used the Ta te –Duk er (1976) map d scussed n the prev ous sect on to deve op s mp e trans t on cr ter a for condensat on n hor zonta tubes, once aga n bas ng them on the rat o of shear forces to grav ty forces on the qu d f m, wh ch are quant f ed through the Wa s d mens on ess gas ve oc ty g* : g* = √ Gt x Dgv ( − v ) (6.12) n add t on to the rat o of qu d vo ume to vapor vo ume. In the absence of a re ab e way to pred ct the s p between the phases, the Mart ne parameter X s used to quant fy th s: $ P 1 − x 0.9 v 0.5 µ 0.1 X = = (6.13) Pv x µv Chapter 6. Condensat on n m n channe s and m crochanne s 255 They app ed these cr ter a to data from a var ety of stud es on f u ds such as R-12, R-113, steam, and n-pentane f ow ng through 4.8 < D < 22.0-mm tubes, at 108 < p < 1249 kPa, and 18 < G < 990 kg/m2 -s. On the bas s of the compar sons between these data and the aforement oned coord nate axes, they recommended the fo ow ng s mp f ed trans t on cr ter a: Annu ar f ow Wavy/strat f ed f ow S ug ( nterm ttent) f ow D sperse (bubb y) f ow g* > 1.5; g* < 0.5; g* < 1.5; g* > 1.5; X X X X < 1.0 < 1.0 > 1.5 > 1.5 One advantage of these cr ter a s the s mp c ty they afford: the trans t on nes appear as hor zonta or vert ca nes, he p ng n the c ear bracket ng of the var ous reg mes. The ex stence of a trans t on band between the respect ve reg mes a so a ows for the more rea st c account ng of over ap zones through near nterpo at on of heat transfer coeff c ents and other performance parameters. The authors do note, however, that ts pred ct ve capab t es are the weakest for the wavy–s ug trans t on (wh ch happens to be of cons derab e nterest n sma d ameter and m crochanne s). Sardesa et a . (1981) a so nvest gated the annu ar-to-strat f ed/wavy trans t on for a var ety of f u ds (R-113, steam, propano , methano , and n-pentane) condens ng n hor zonta 24.4-mm d ameter tubes. They nstrumented the r test sect on w th thermocoup es around the c rcumference on the tube at severa ax a ocat ons to determ ne the c rcumferent a var at on of heat transfer coeff c ents, wh ch enab ed them to compute the rat o of the heat transfer coeff c ent at the top and bottom of the tube. They used the Ta te –Duk er (1976) annu ar–strat f ed/wavy trans t on cr ter on (represent ng a rat o of the ax a shear force to the c rcumferent a grav ty force n the qu d f m) as the bas s for deve op ng the r exper menta y der ved trans t on cr ter on. Thus, they used the data to f nd a mu t p er to th s trans t on ne (F, X ) = constant, wh ch wou d resu t n a ne para e to the Ta te – Duk er trans t on ne. The data showed that the heat transfer coeff c ent rat o stayed c ose to un ty (s gn fy ng ax -symmetr c annu ar f ow) down to = 1.75, be ow wh ch t decreased rap d y, s gn fy ng grav ty-dr ven condensat on. An emp r ca f t der ved from the square root of the two-phase fr ct ona mu t p er of Ch sho m and Suther and (1969–70) was used such that g2 F = 1 s used as the upper m t of grav ty-contro ed condensat on, wh e g2 F = 1.75 s used as the ower m t for annu ar f ow, w th g2 = 0.7X 2 + 2X + 0.85. Here, F s the mod f ed Froude number g ven by: F= g UGS √ − g Dg (6.14) and UGS s the superf c a gas phase ve oc ty. So man (1982) a so proposed an annu ar-wavy trans t on cr ter on based on data from d fferent sources that nc uded R-12, R-113, and steam f ow ng through 4.8 < D < 15.9-mm tubes for 28 C < T < 110 C. Dur ng the progress on from vapor to qu d, the ncrease n the qu d vo ume and the s gn f cance of grav ty, coup ed w th the decreas ng ve oc ty, and thus the nert a effects, were quant f ed based on the decrease n the Froude number. Us ng the qu d f m th ckness express ons of Kosky (1971), the Mart ne parameter Xtt , and the 256 Heat transfer and f u d f ow n m n channe s and m crochanne s Azer et a . (1972) equat on G = 1 + 1.09Xtt0.039 , he deve oped the fo ow ng express ons for the f m Reyno ds number: Re = 10.18Fr 0.625 Ga0.313 (G /Xtt )−0.938 for Re ≤ 1250 Re = 0.79Fr 0.962 Ga0.481 (G /Xtt )−1.442 for Re > 1250 (6.15) The data from the var ous sources were then used to determ ne that the annu ar-to-wavy and nterm ttent trans t on occurs when Fr = 7 n the above express ons, rrespect ve of the d ameter or f u d under cons derat on. Subsequent y, So man (1986) a so deve oped a corre at on for the m st–annu ar trans t on. M st f ow occurs n the entrance reg on of the condenser, w th the qu d phase f ow ng as entra ned drop ets w thout any v s b e qu d f m at the wa . He used the arge d screpanc es between exper menta y determ ned heat transfer and pressure drop va ues and pred ct ons typ ca y us ng annu ar f ow mode s as the rat ona e to propose that m st f ow must be treated separate y. He reasoned that entra nment occurs due to the nert a of the vapor phase (G VG2 ) shear ng drop ets from the surface of the qu d f m, wh e v scous (µL VL /) and surface tens on (/D) forces tend to stab ze the qu d f m, w th the ba ance between them be ng represented by the Weber number, for wh ch the fo ow ng express ons were der ved: 0.3 2 µ −0.4 G 0.64 G for ReLS ≤ 1250 We = 2.45ReGS G D 0.3 0.084 (6.16) µ2G µG 2 L 0.79 2.55 0.157 We = 0.85ReGS Xtt /G G D µL G for ReLS > 1250 Based on the above def n t ons and compar son w th the database ment oned above, the fo ow ng cr ter a were estab shed: We < 20 We > 30 a ways annu ar a ways m st (6.17) Tandon et a . (1982) a so used data for R-12 and R-113 from severa sources for the range * as 4.8 < D < 15.9 mm, and proposed a f ow reg me map w th d mens on ess gas ve oc ty G the ord nate and (1 − )/ as the absc ssa, wh ch resu ted n s mp e trans t on equat ons and good agreement w th the data for annu ar, sem -annu ar, and wavy f ows. The corre at on by Sm th (1969) was used to compute the vo d fract on . S m ar to the Breber et a . (1980) cr ter a, the r trans t on cr ter a, g ven be ow, a so resu t n hor zonta and vert ca trans t on nes on the map. Thus: Spray * 6 < G Annu ar/sem -annu ar * ≤ 6 and 1 ≤ G Wavy * ≤1 G and and 1− ≤ 0.5 1− ≤ 0.5 1− ≤ 0.5 (6.18) Chapter 6. Condensat on n m n channe s and m crochanne s S ug P ug * ≤ 0.5 and 0.01 ≤ G * ≤ 0.01 G and 257 1− ≥ 0.5 1− ≥ 0.5 Subsequent y, Tandon et a . (1985a) a so demonstrated good agreement w th the r own data on condensat on of R-12, R-22, and m xtures of the two f u ds f ow ng through 10-mm tubes. Hash zume and co-workers (Hash zume, 1983; Hash zume et a ., 1985; Hash zume and Ogawa, 1987) conducted a three-part study on f ow patterns, vo d fract on, and pressure drop for refr gerant (R-12 and R-22) f ows n 10-mm d ameter hor zonta tubes. Tests were conducted at saturat on pressures n the range 570 < p < 1960 kPa, correspond ng to saturat on temperatures n the range 20 C < T < 72 C for R12 and 4 C < T < 50 C for R-22, and t was shown that these f ow patterns for refr gerants are qu te d fferent from those for the a r–water system. The f ows were ana yzed us ng s mp f ed mode s for annu ar and strat f ed f ow, and the ve oc ty prof es n the qu d and gas phases were descr bed us ng the Prandt m x ng ength. They mod f ed the property correct ons to the Baker map (1954) proposed by We sman et a . (1979) (spec f ca y, they decreased the surface tens on exponent n = (W /)1 [(µL /µW )(W /L )2 ]1/3 from 1 to 0.25) to make those maps app cab e for R-12 and R-22. Wang et a . (1997a,b) nvest gated two-phase f ow patterns for refr gerants R-22, R-134a, and R-407C n a 6.5-mm c rcu ar tube over the mass f ux range 50 < G < 700 kg/m2 -s, for saturat on temperatures of 2 C, 6 C and 20 C. They noted, as d d Wambsganss et a . (1991), that pr or f ow pattern stud es had focused on d ameters n the range 9.5 < D < 75.0 mm, and had been conducted at h gh mass f uxes (G > 300 kg/m2 -s), wh ch are not part cu ar y app cab e to refr gerat on and a r-cond t on ng app cat ons. At G = 100 kg/m2 -s, p ug, s ug, and strat f ed f ow patterns were observed, w th no occurrence of annu ar f ow. As the mass f ux ncreased to 200 and 400 kg/m2 -s, annu ar f ow appeared, and dom nated the map at h gher mass f uxes. They found the trans t ons n the r study to be n good agreement w th the mod f ed Baker map of Hash zume (Hash zume, 1983; Hash zume et a ., 1985; Hash zume and Ogawa, 1987), w th the trans t ons occurr ng at somewhat ower mass f uxes, wh ch they attr buted to the sma er d ameter used n the r study. The s ug-to-wavy and wavy-to-annu ar f ow reg me trans t ons for the refr gerant b end R-407C occurred at h gher G and x than for the pure f u ds. The h gher phase dens t es and the consequent ower gas- and qu d-phase ve oc t es for R-407C compared to R-22, and the h gher qu d-phase v scos ty of the const tuents of R-407C were assumed to be the cause for th s agg ng behav or of the refr gerant f ow trans t ons. One of the more exhaust ve stud es of refr gerant condensat on n sma d ameter (D = 3.14, 4.6, and 7.04 mm) tubes was conducted by Dobson and Chato (1998) us ng refr gerants R-12, R-22 R-134a, and two d fferent compos t ons of a near-azeotrop c b end of R-32/R-125 over the mass f ux range 25 < G < 800 kg/m2 -s. The authors prov de a good overv ew of the var ous stud es on condensat on, nc ud ng a step-by-step descr pt on of the progress on of f ows from one reg me to another as the qua ty changes at d fferent mass f uxes. At the owest mass f ux of 25 kg/m2 -s, the f ow was ent re y smooth strat f ed for a qua t es, wh e wavy f ow was observed at 75 kg/m2 -s. As the mass f ux was ncreased to 150–300 kg/m2 -s, condensat on occurred over annu ar, wavy–annu ar, wavy and s ug f ow 258 Heat transfer and f u d f ow n m n channe s and m crochanne s as the qua ty decreased. F u d propert es and tube d ameter affected the f ow trans t ons the most at these mass f uxes. At the h ghest mass f uxes, (500, 650, and 800 kg/m2 -s), the f ow progressed from annu ar-m st ( nd cat ng entra nment) at h gh qua t es to annu ar, wavy–annu ar and s ug f ow w th decreas ng qua ty. W th tests conducted at T = 35 and 45 C, they commented that the decreas ng d fference n vapor and qu d propert es w th an approach toward the cr t ca pressure ( ncreas ng reduced pressure) exp a ned the changes n the f ow patterns. Thus, annu ar f ow was seen over a arger port on of the map at ower reduced pressures; for examp e, annu ar f ow was estab shed n R-134a at 35 C at qua t es as ow as 15% at G = 600 kg/m2 -s. A so, the R-32/R-125 b end showed trans t ons to annu ar f ow at much h gher qua t es than the other f u ds, w th a s gn f cant amount of strat f cat on seen at the h gh temperature case even at G = 600 kg/m2 -s. It was a so shown that the s ug f ow reg on ncreases at h gher reduced pressures. As the tube d ameter decreased, the trans t on from wavy f ow to wavy–annu ar f ow, and from wavy– annu ar to annu ar f ow moved to ower qua t es. They found very good agreement w th the Mandhane et a . (1974) map after correct ng the superf c a vapor ve oc ty w th the factor # g /a to account for the fact that the Mandhane et a . map was based pr mar y on a r– water data, where the gas phase dens ty s drast ca y ower than refr gerant vapor dens ty. Th s correct on accounts for the d fferent gas-phase k net c energ es better than the or g na map. S nce the r stated ob ect ve was to pr mar y deve op f ow-reg me-based heat transfer corre at ons, they d v ded the above-ment oned f ow reg mes nto grav ty-dom nated and shear-contro ed reg mes (as done by the many other researchers c ted above), and d d not prov de deta ed trans t on cr ter a between each of the f ow reg mes observed. Instead, n the r pr or work (Dobson, 1994; Dobson et a ., 1994), they estab shed that the wavy– annu ar to annu ar f ow trans t on wou d occur at a So man-mod f ed Froude number Frso of 18, nstead of Frso = 7 recommended by So man (1982). In the heat transfer corre at ons a so deve oped n th s work, and to be d scussed n a subsequent sect on of th s chapter, they recommend that the shear-contro ed annu ar f ow corre at on be used for G ≥ 500 kg/m2 -s for a x, wh e for G < 500 kg/m2 -s, the annu ar f ow corre at on shou d be used for Frso > 20, and the grav ty-dr ven corre at on shou d be used for Frso < 20. The stud es by Co eman and Gar me a (2000a, b; 2003) and Gar me a (2004) are perhaps the on y nvest gat ons to date that address spec f ca y a the aspects of the sub ect at hand. Thus, severa stud es c ted above e ther report deta ed f ow reg mes n m crochanne s as sma as 25 µm, but for ad abat c a r–water f ow, or those that study f ow reg mes n condens ng f ows are typ ca y restr cted to D > ~5 mm, w th the except on of the one 3.14-mm tube stud ed by Dobson and Chato (1998) and d scussed above. The work of Co eman and Gar me a s mu taneous y addressed: (a) sma d ameters, (b) condens ng nstead of ad abat c f ow, (c) f ow v sua zat on dur ng condensat on, (d) temperatures representat ve of pract ca condensat on processes, and (e) a w de range of mass f uxes and qua t es. They conducted f ow v sua zat on stud es for refr gerant R-134a n n ne d fferent tubes of round, square, and rectangu ar cross-sect on tubes w th 1 < Dh < 4.91 mm over the mass f ux range 150 < G < 750 kg/m2 -s, and qua ty range 0 < x < 1. The channe s stud ed by them are shown n F g. 6.9. In the r stud es, refr gerant of a des red qua ty was supp ed to the test sect on by pre-condens ng a superheated vapor. V sua zat on was conducted n a counterf ow heat exchanger, w th refr gerant f ow ng through an nner g ass tube of the crosssect on of nterest, and a r f ow ng through the space between th s nner tube and another transparent outer P ex g as tube, thus enab ng v sua zat on of the respect ve f ow reg mes. Chapter 6. Condensat on n m n channe s and m crochanne s 259 5 mm 4 4 4 6 6 4 Tube shape (C rcu ar–Square–Rectangu ar) and aspect rat o I 4 2 4 4 4 4 Aspect rat o II 3 3 2 2 Tube m n atur zat on 2 4 1 1 F g. 6.9. F ow v sua zat on geometr es of Co eman and Gar me a (2000a, b; 2003). Heat transfer between the co d a r and refr gerant causes condensat on. Compressed a r f ow ng n the annu us prov ded a ow d fferent a pressure for the g ass m crochanne , mak ng t poss b e to conduct tests at saturat on pressures as h gh as 1379–1724 kPa. Post condensers/subcoo ers comp eted the c osed oop fac ty. They recorded f ow patterns for each mass f ux at nom na y 5–10% qua ty ncrements for a tota of about 50 data per tube, to prov de a f ne reso ut on n the var at on of f ow patterns a ong the condensat on process. Four ma or f ow reg mes, nc ud ng annu ar, nterm ttent, wavy, and d spersed f ow were dent f ed, w th the reg mes further subd v ded nto f ow patterns (F g. 6.10). In the annu ar f ow reg me, the vapor f ows n the core of the tube w th a few entra ned qu d drop ets, wh e qu d f ows a ong the c rcumference of the tube wa . The f ow patterns w th n th s reg me (m st, annu ar r ng, wave r ng, wave packet, and annu ar f m) show the vary ng nf uences of grav ty and shear forces as the mass f ux and qua ty changes. F ow patterns w th a s gn f cant nf uence of grav ty (vapor f ow ng above the qu d, or a not ceab e d fference n f m th ckness at the top and bottom of the tube) and a wavy nterface were ass gned to the wavy f ow reg me. The waves at the qu d–vapor nterface are caused by nterfac a shear between the two phases mov ng at d fferent ve oc t es. Thus, th s reg me was subd v ded nto d screte waves of arger structure mov ng a ong the phase nterface, and d sperse waves w th a arge range of amp tudes and wave engths super mposed upon one another, as shown n F g. 6.10. The other f ow reg mes shown n the f gure have a ready been descr bed n deta n th s chapter. In some cases, the f ow mechan sms corresponded to more than one f ow reg me, typ ca y nd cat ng a trans t on between the respect ve reg mes. A typ ca f ow reg me map for the 4.91-mm c rcu ar tube p otted us ng the mass f ux G, and qua ty x-coord nates s shown n F g. 6.11. A ma or port on of th s map s occup ed by the wavy f ow reg me w th a sma reg on where the p ug, s ug, and d screte wave f ow patterns coex st. The waves become ncreas ng y d sperse as the qua ty and mass f ux s ncreased (shown by the arrow n F g. 6.11). The approx mate demarcat on 260 Heat transfer and f u d f ow n m n channe s and m crochanne s F ow patterns F ow reg mes Annu ar Wavy Interm ttent D spersed M st f ow D screte wave (0) S ug f ow Bubb y f ow Annu ar r ng D screte wave (1) S ug f ow Bubb y f ow Wave r ng D screte wave (2) P ug f ow Bubb y f ow D sperse wave (3) P ug f ow Wave packet Note: Numbers above denote ntens ty of secondary waves Annu ar f m F g. 6.10. Condensat on f ow reg mes and patterns (1 < Dh < 4.9 mm). Repr nted from Co eman, J. W. and Gar me a, S., Two-phase f ow reg mes n round, square and rectangu ar tubes dur ng condensat on of Refr gerant R134a, Internat ona Journa of Refr gerat on, 26(1), pp. 117–128 (2003) w th perm ss on from E sev er and Internat ona Inst tute of Refr gerat on ( f r@ f r.org or www. f r.org). 800 P ug and s ug f ow D screte waves 700 D sperse waves Annu ar r ng pattern rse M st f ow D sperse waves M st ds pe 500 Wave packet pattern cre te to 300 Ds 400 P ug/S ug f ow Mass f ux (kg/m2-s) 600 Annu ar f m D screte waves D screte waves and p ug/s ug Annu ar f m 200 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Qua ty F g. 6.11. Condensat on f ow reg me map (4.91 mm). Repr nted from Co eman, J. W. and Gar me a, S., Two-phase f ow reg mes n round, square and rectangu ar tubes dur ng condensat on of Refr gerant R134a, Internat ona Journa of Refr gerat on, 26(1), pp. 117–128 (2003) w th perm ss on from E sev er and Internat ona Inst tute of Refr gerat on ( f r@ f r.org or www. f r.org). Chapter 6. Condensat on n m n channe s and m crochanne s 800 1 mm 2 mm 3 mm 4 mm 700 Mass f ux (kg/m2-s) 261 600 500 400 300 200 Interm ttent 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Qua ty F g. 6.12. Effect of Dh on nterm ttent f ow reg me. From Gar me a, S., Condensat on f ow mechan sms n m crochanne s: Bas s for pressure drop and heat transfer mode s, Heat Transfer Eng neer ng, 25(3), pp. 104–116 (2004). between d screte and d sperse waves s shown by the dashed ne n th s f gure, a though th s trans t on occurs gradua y. The effect of hydrau c d ameter on the f ow reg me maps s shown n F gs. 6.12 and 6.13. F gure 6.12, wh ch dep cts the trans t on from the nterm ttent reg me for the four square tubes nvest gated, shows that the s ze of the nterm ttent reg me ncreases as Dh decreases, w th th s effect be ng greater at the ower mass f uxes. The arge ncrease n the s ze of the nterm ttent reg me n the sma er hydrau c d ameter tubes s because surface tens on ach eves a greater s gn f cance n compar son w th grav tat ona forces at these d mens ons. Th s a so occurs because n square channe s, t s eas er for the qu d to be he d n the sharp corners, counteract ng to some extent, the effects of grav ty. Th s fac tates p ug and s ug f ow at h gher qua t es as the hydrau c d ameter s decreased. F gure 6.13 shows that the 4-mm tube map s dom nated by the wavy f ow reg me (w th an absence of the annu ar f m f ow pattern). As Dh s decreased, the annu ar f ow reg me appears and occup es an ncreas ng port on of the map. Thus, for the 4-mm tube, the effects of grav ty dom nate, resu t ng n most of the f ow reg me map be ng covered by the wavy f ow reg me. As the hydrau c d ameter decreases, the effects of surface tens on ncreas ng y counteract the effects of grav ty, promot ng and extend ng the s ze of the annu ar f m f ow pattern reg on nstead of the more strat f ed wavy f ow reg me. Thus, as Dh decreases, the wavy f ow reg me s ncreas ng y rep aced by the annu ar f ow reg me, and s nonex stent n the Dh = 1-mm tube. Furthermore, surface tens on stab zes the waves, wh ch eads to more d screte waves at sma d ameters. The effect of tube shape was nvest gated us ng maps for c rcu ar, square, and rectangu ar tubes of approx mate y the same Dh ~ 4–4.9 mm (F g. 6.14). Th s f gure shows that the 262 Heat transfer and f u d f ow n m n channe s and m crochanne s 800 1 1 mm 2 2 mm 3 3 mm Mass f ux (kg/m2-s) 700 ar nu An mm) (1 600 500 ar nu An mm) (2 r u a Ann m) (3 m 400 300 200 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Qua ty F g. 6.13. Effect of Dh on annu ar f ow reg me. From Gar me a, S., Condensat on f ow mechan sms n m crochanne s: Bas s for pressure drop and heat transfer mode s, Heat Transfer Eng neer ng, 25(3), pp. 104–116 (2004). 800 Wavy 600 Annu ar D sperse wave 500 400 300 200 100 0.0 Interm ttent Mass f ux (kg/m2-s) 700 D screte wave 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Qua ty Ma or trans t ons D screte-to-d sperse wave Round (4.91 mm) Square (4 4 mm) Rectangu ar 4 (H) 6 (W) mm Rectangu ar 6 (H) 4 (W) mm F g. 6.14. Effect of tube shape on f ow reg me trans t ons. From Gar me a, S., Condensat on f ow mechan sms n m crochanne s: Bas s for pressure drop and heat transfer mode s, Heat Transfer Eng neer ng, 25(3), pp. 104–116 (2004). Chapter 6. Condensat on n m n channe s and m crochanne s 263 nterm ttent reg me s arger n the round tube than n the square tube at ower mass f uxes and approx mate y the same at h gher mass f uxes. The extent of the nterm ttent reg me for the rectangu ar tubes s n between that of the c rcu ar and square tubes. The wavy f ow reg me s a so arger n the round tube. The square and rectangu ar channe s he p qu d retent on n the corners and a ong the ent re c rcumference of the tube ead ng to annu ar f ow, rather than preferent a y at the bottom of the tube as wou d be the case n the wavy f ow reg me. In the 4 × 6-mm and 6 × 4-mm tubes, the arger aspect rat o resu ts n a s ght ncrease n the s ze of the nterm ttent reg me at the ower mass f uxes, and a sma reduct on n the s ze of th s reg me at the h gher mass f uxes. However, these effects are sma , and t can be conc uded that th s trans t on ne s on y weak y dependent on the aspect rat o. The sma er aspect rat o a so resu ts n a arger annu ar f m f ow pattern reg on, wh ch s to be expected because of the reduced nf uence of grav ty for the tubes w th the sma er he ght. D screte waves are more preva ent n the round tube compared to the square and rectangu ar tubes. It was a so found that the sma er aspect rat o resu ts n a sma er wavy f ow reg me but w th a arger fract on of d screte waves. At the h gher mass f uxes, the effect of the aspect rat o s neg g b e. The ncrease n the annu ar f m f ow pattern reg on was more pronounced n the sma er hydrau c d ameter tubes tested by them, perhaps due to the greater nf uence of surface tens on. Thus, wh e tube shape had some effect on the trans t ons between the var ous f ow reg mes, the nf uence of hydrau c d ameter was found to be far more s gn f cant. A two-part study on f ow reg mes and heat transfer dur ng condens ng f ows n channe s w th 3.14 < D < 21 mm has a so appeared recent y (E Ha a et a ., 2003; Thome et a ., 2003). A though th s work s pr mar y on arge channe s, the comprehens ve nature of the database used for mode deve opment warrants d scuss on n th s chapter. However, due to the h gh y coup ed nature of the f ow reg me (E Ha a et a ., 2003) and heat transfer parts (Thome et a ., 2003), t s presented n a subsequent sect on on heat transfer. 6.2.3. Summary observat ons and recommendat ons The above d scuss on has shown that numerous stud es have been conducted to understand two-phase f ow n sma d ameter channe s. The preponderance of stud es s on ad abat c a r– water f ows. Wh e these f ows have h stor ca y prov ded some understand ng of condensat on phenomena, many of the d screpanc es between the behav or observed n these stud es and the actua f ow reg mes dur ng the condensat on process can be attr buted to the substant a y d fferent propert es of the a r–water pa r compared to refr gerant vapor/ qu d pa r. It a so appears that v deo record ng techno ogy s afford ng nvest gators the ab ty to capture every nuance of two-phase f ow n great deta . H gh-speed v deo presents the nvest gator w th a p ethora of mages separated n t me by very sma ncrements. Wh e th s s certa n y advantageous n understand ng the ntr cac es of two-phase f ow, t has ed to the further pro ferat on of creat ve, confus ng, and vague def n t ons of f ow patterns. Just the number of f ow patterns c ted n th s chapter a one wou d approach about 50! There are many def n t ons w th re at ve y ess d st nct on. F ow patterns from d fferent nvest gators can be as eas y thought to agree as not, depend ng on the nterpretat on of these terms. As one who has h mse f pored over numerous frames of h gh-speed v deo, t s easy to understand how each researcher attempts to capture every nuance of a dynam c 264 Heat transfer and f u d f ow n m n channe s and m crochanne s bubb e or a wave n a descr pt on of the pattern. It s not c ear whether such deta ed descr pt ons have ed to a s gn f cant advancement n the quant f cat on of the re evant phys ca phenomena that are respons b e for these mechan sms n the f rst p ace. These m nute d st nct ons between f ow patterns wou d on y be usefu f they ead to quant tat ve mode s of essent a features such as nterfac a area, shear, s p, vo d fract on, momentum coup ng, and u t mate y, heat transfer and pressure drop. The summary eva uat on of the above-ment oned stud es s as fo ows: (1) Ad abat c f ow patterns are now ava ab e for a w de range of m crochanne s, most y us ng a r or n trogen/water m xtures, a the way down to 25 µm. Extrapo at on of a r–water f ow patterns to condens ng refr gerant app cat ons must be done w th caut on, however. (2) Many of the stud es on sma c rcu ar and narrow rectangu ar channe s were conducted on co- or countercurrent vert ca upf ow, wh ch s an or entat on se dom used n condensat on. (3) It has now been very we documented that grav tat ona forces have tt e s gn f cance n the sma channe s (D < ~1 mm) of nterest n th s book. Th s has been va dated through observat ons of symmetr c f ms, bubb es and s ugs, and a so f ow patterns ndependent of channe or entat on. (4) In the channe s w th 1 < D < 10 mm, the f ow patterns can broad y be c ass f ed as d sperse, annu ar, wavy, and nterm ttent. Most of the phase-change stud es n th s s ze range have focused on channe s w th D > 3 mm, and the pr mary focus of most of these nvest gat ons has been to understand grav ty-dom nated versus shear-contro ed condensat on. (5) In channe s w th D < 1 mm, t appears that the pr mary patterns are some k nd of s ug/p ug f ow, a trans t on reg on des gnated as qu d r ng by some nvest gators, and annu ar f ow (w th some m st f ow at h gh qua t es and mass f uxes). (6) Maps and trans t on cr ter a such as those by Ta te –Duk er (1976) are tempt ng to use because of the r ana yt ca bas s; however, rare y do they pred ct condensat on phenomena n m crochanne s – the or g na nvest gators perhaps never ntended for them to be app ed n th s manner. (7) For the ~5-mm channe s, s mp e cr ter a such as those by Breber et a . (1980), Tandon et a . (1982), and So man (1982; 1986) may suff ce, w th m nor mod f cat ons to match the app cat on under cons derat on to d st ngu sh between grav ty and shear-contro ed condensat on. The more recent descr pt ve report ng of f ow patterns by Dobson and Chato (1998) may a so be used to obta n qua tat ve ns ghts. (8) Wh e f ow pattern descr pt ons are ava ab e for channe s w th D < ~5 mm, and even for channe s n the 50–250 µm range, exp c t equat ons that quant fy the trans t on cr ter a are st m ss ng. A comb nat on of the ana yt ca approaches of Tabataba and Faghr (2001), some of the ana yses of M sh ma and Ish (1996), and the ongo ng work of Kawahara, Kawa , and others (Kawahara et a ., 2002; Chung and Kawa , 2004) can be brought to bear to obta n such cr ter a. (9) In add t on, trans t on cr ter a that are app cab e over a w de range of f u d propert es, operat ng cond t ons, and channe s zes must be deve oped based on the under y ng forces and va dated by the deta ed exper menta y obta ned phase-change maps such as those of Co eman and Gar me a (2000a, b; 2003) and Gar me a (2004). (10) S m ar y, the nvest gators of m crochanne s w th D << 1 mm shou d a so test condensat on processes to supp ement the nformat on gathered from ad abat c f ow v sua zat on stud es. The ass gnment of f ow reg mes for n ne representat ve cases (comb nat ons of mass f uxes and qua t es) for condensat on of refr gerant R-134a n a 1-mm d ameter channe us ng severa of the prom nent trans t on cr ter a s ustrated n Examp e 6.1. 6.3. Vo d fract on 6.3.1. Convent ona tubes An mportant parameter for the ca cu at on of heat transfer coeff c ents, and more so pressure drops, s the vo d fract on n two-phase f ow. The vo d fract on a so determ nes the amount of f u d to be charged nto a c osed oop system such as a refr gerat on cyc e. Vo d fract on s s mp y the rat o of the cross-sect ona area Ag occup ed by the gas, to the tota cross-sect ona area of the channe (see Tab e 6.2): = Ag A (6.19) The ca cu at on of th s parameter s an essent a step and prov des c osure to the set of equat ons that must be so ved to est mate the pressure drop and heat transfer n two-phase f ow. Wh e the vapor qua ty x s a thermodynam c quant ty determ ned by the state of the two-phase m xture, the vo d fract on depends on the character st cs of the f ow, and has been the sub ect of severa nvest gat ons. The s p, that s, the rat o of the ve oc t es of the two phases, s requ red to enab e computat on of the vo d fract on. The s mp est assumpt on that can be made s of homogeneous f ow, where t s assumed that the vaporand qu d-phase ve oc t es are equa , and that the two phases behave ke a s ng e, un form m xture that has representat ve average propert es, wh ch y e ds the fo ow ng express on: = xv (1 − x)/ + x/v (6.20) Wh e the homogeneous mode s n fact qu te arb trary, and serves as a start ng po nt for the more representat ve mode s of vo d fract on, t has ga ned some acceptance for the mode ng of m crochanne s, as w be d scussed ater. Vo d fract on mode s for convent ona channe s are read y ava ab e n textbooks such as those by Carey (1992), Co er and Thome (1994), and Hew tt et a . (1994); therefore, the more prom nent corre at ons are s mp y presented here w thout d scuss on. Wh e the deta s of the f ow patterns must be taken nto account to obta n mode s based on the phys ca phenomena, severa of the more w de y used corre at ons often assume annu ar f ow and re ate the vo d fract on to a parameter that resemb es the Mart ne parameter X , often Xtt (wh ch assumes turbu ent f ow n both phases). Incorporat ng the appropr ate express ons for the pressure drops n the two phases, these express ons can be represented by the fo ow ng gener c express on (Butterworth, 1975): −1 1 − x n1 v n2 µ n3 = 1 + BB (6.21) x µv 266 Tab e 6.2 Summary of vo d fract on stud es. Invest gator Hydrau c d ameter F u ds Or entat on 6 mm Bo ng heavy water Vert ca 38 mm Bo ng water Hor zonta , vert ca 11 mm A r–water Vert ca Z v (1964) Severa Steam–water Hor zonta , vert ca Tandon et a . (1985b) 6.1 mm Bo ng heavy water Vert ca 22 mm Steam–water Range/ app cab ty Convent ona tubes Premo et a . (1971) • S p rat o-based corre at on 0.7 < p < 5.9 MPa 0 < x < 0.38 650 < G < 2050 kg/m2 -s 380 < q < 1200 kW/m2 1.725 < p < 14.5 MPa 0.003 < x < 0.17 750 < G < 1950 kg/m2 -s 20 < q < 140 kW/m2 Atmospher c pressure 0.005 < x < 0.525 50 < G < 1330 kg/m2 -s • Sem -emp r ca mode • Assumed strat f ed annu ar f ow ( qu d-phase) w th qu d entra nment n vapor-phase (homogeneous m xture phase), equa ve oc ty head n both phases • States that s mode ndependent of f ow reg me, p, G, x; app cab e to hor zonta and vert ca co-current f ow • For K = 0.4, app cab ty assumed on y for c rcu ar tubes • Not recommended for x < 0.01 n bo ng f ow (due to therma nequ br um) • Ana yt ca vo d fract on mode (assum ng annu ar f ow), m n m zes rate of energy d ss pat on • Three mode s: (I) w thout qu d entra nment and wa fr ct on, (II) w th wa fr ct on, and (III) w th qu d entra nment • Data show wa fr ct on effect << effect of qu d entra nment • As p ncreases, mode (I) approaches homogeneous mode • S mp est mode (I) w de y used, does not re y on data 0.7 < p < 6 MPa, x < 0.41 650 < G < 2050 kg/m2 -s 380 < q < 1200 kW/m2 0.24 < < 0.92 Atmospher c pressure x < 0.04 • Sem -emp r ca vo d fract on mode • Assumed annu ar f ow w thout qu d entra nment n vapor-phase and turbu ent f ow n both phases • State that mode performs as we as Sm th s mode (1969) and better than Wa s (1969) and Z v s (1964) corre at ons Heat transfer and f u d f ow n m n channe s and m crochanne s Sm th (1969) Techn ques, bas s, observat ons Yashar et a . (2001) • M crof n tubes: 0.2-mm f n he ght • 7.3, 8.9 mm (evaporat on) • 8.9 mm (condensat on) R-134a, R-410A Hor zonta 75 < G < 700 kg/m2 -s 0.05 < x < 0.8 • • • • E Ha a et a . (2003) 8 mm R-22, R-134a, R-236ea, R-125, R-32, R-410A Hor zonta (condens ng) 65 < G < 750 kg/m2 -s 0.22 < pS < 3.15 MPa 0.15 < x < 0.88 • Used ear er bo ng maps (Kattan et a ., 1998a, b, c; Zurcher et a ., 1999) as bas s • Deduced vo d fract ons for annu ar f ow from heat transfer database (Cava n et a ., 2001; 2002a, b) and f m th ckness cons derat ons • Logar thm c mean of homogeneous and Ste ner (1993) mode s Koyama et a . (2004) 7.52 mm (smooth) 8.86 mm (m crof n) R-134a Hor zonta (ad abat c) 0.01 < x < 0.96 p = 0.8, 1.2 MPa G = 125, 250 kg/m2 -s (smooth) G = 90, 180 kg/m2 -s (m crof n) • Vo d fract on ncreases as pressure decreases (smooth and m crof n) • Smooth-tube data agree w th Sm th s (1969) and Baroczy s (1965) mode s • M crof n tube data ower than smooth tube va ues • Resu ts d fferent from Yashar et a . (2001), who reported s m ar va ues for both tube types • Yashar et a . (2001) corre at on overpred cts m crof n tube data • n m crof n tubes ncreases w th ncreas ng G, at ow p, x • Mode for strat f ed– annu ar and annu ar f ow; reasonab e agreement w th data, espec a y at h gh x A r–water Hor zonta < 0.9 atmospher c pressure • Emp r ca corre at on Sma channe s Armand (1946) Measurements n condens ng m crof n tubes (uncerta nty 10%) S m ar trends for m crof n and smooth tubes Lower vo d fract ons compared to evaporat on, for same G, x No effect of f n arrangement (he x ang e) Kar yasak et a . (1991) 1, 2.4, 4.9 mm A r–water Hor zonta 0.1 < G < 25 m/s 0.03 < L < 2 m/s • Numerous curve-f ts to match data w thout phys ca cons derat ons M sh ma and H b k (1996) 1–4 mm A r–water Vert ca 0.0896 < G < 79.3 m/s 0.0116 < L < 1.67 m/s • Image process ng from neutron rad ography for non- ntrus ve y measur ng vo d fract on • Corre ated vo d fract on us ng dr ft-f ux mode w th emp r ca parameters for vert ca bubb y and s ug f ow (Cont nued ) Tab e 6.2 (Cont nued) Invest gator Hydrau c d ameter F u ds Or entat on Range/ app cab ty Techn ques, bas s, observat ons • D str but on parameter funct on of D, arger va ues for sma channe s than reported by Ish (1977) for arger tubes • Agreed w th Kar yasak et a . (1992) corre at on H b k and M sh ma (1996) 1–4 mm A r–water Vert ca < 19 m/s • Best agreement w th 1-mm data • Techn que a so works for annu ar f ow at h gh n arger tubes H b k et a . (1997) 3.9 mm A r–water Vert ca upward G = 0.131 m/s L = 0.0707 m/s • Corre at on for rad a vo d fract on prof e for vert ca s ug f ow M sh ma et a . (1997) 2.4 mm A r–water Vert ca upward < 8 m/s • Neutron rad ography mage process ng for var at on of vo d fract on a ong channe ength for s ug f ow • R se of bubb e tracked us ng h gh vo d fract on reg on • Merg ng of vo d fract on peaks nd cated bubb e coa escence Kureta et a . (2001; 2003) 3, 5 mm Steam–water Vert ca upward 240 < G < 2000 kg/m2 -s • Instantaneous and t me-averaged 2-D for subcoo ed bo ng (at ow ) Tr p ett et a . (1999a) 1.1, 1.45 mm A r–water Hor zonta (ad abat c) 0.02 < G < 80 m/s 0.02 < L < 8 m/s • Vo umetr c vo d fract on from st photographs (rotat ona symmetry assumed, uncerta nty ~15%) • Vo d fract on ncreases w th ncreas ng gas ve oc ty and decreases w th ncreas ng superf c a qu d ve oc ty • Homogeneous mode best for s ug and bubb e f ow, Premo et a . (1971), Chexa et a . (Chexa and Le ouche, 1986; Chexa et a ., 1992; Chexa et a ., 1997), Lockhart–Mart ne (1949), Baroczy (1965) overpred ct for annu ar and churn f ow 9.525 mm A r–water Vert ca upward (ad abat c) 0.1 < G < 18 m/s 0.07 < L < 2.5 m/s 0.1 < < 0.9 • • • • M crograv ty E kow and Rezka ah (1997a, b) PDF of vo d f uctuat ons at 1-g, µ-g w th capac tance sensor Average n s ug f ow ~10% h gher at µ-g Larger f uctuat ons n bubb e and s ug f ow at 1-g F uctuat ons decreased n trans t ona and annu ar f ow, due to ncreas ng dom nance of nert a forces • Compared w th dr ft-f ux mode , at 1-g: power aw d str but on n s ug f ow, sadd e-shaped d str but on n bubb y f ow; at µ-g, power aw d str but on for both f ows Lowe and Rezka ah (1999) 9.525 mm M crochanne s Ser zawa 20, 25, 100, et a . (2002) 50 µm A r–water Vert ca upward 0.09 < G < 20.94 m/s 0.04 < L < 3.03 m/s p = 89 kPa, µ-g, 1-g • Capac tance sensor to measure , dent fy f ow reg mes us ng t me trace and PDF A r–water (steam–water) Hor zonta 0.0012 < G < 295.3 m/s 0.003 < L < 17.52 m/s • n bubb y and s ug f ow from v deo ana ys s • Fo ows Armand (1946) and Kar yasak et a . (1991) corre at ons Kawahara et a . (2002) 100 µm N trogen–water Hor zonta 0.1 < G < 60 m/s 0.02 < L < 4 m/s • T me-averaged vo d fract on and curve-f t from v deo ana ys s • ncreases w th ncreas ng homogenous vo d fract on. Very ow for < 0.8, rap d ncrease for > 0.8 • D fferent from Ser zawa et a . (2002) who found near corre at on to homogeneous for s m ar d ameters • Low at h gh gas f ow rates nd cates arge s p rat os and weak momentum coup ng between phases Chung and Kawa (2004) 50, 100, 250, 530 µm N trogen–water Hor zonta T = 22.9–29.9 C 0.02 < G < 72.98 m/s 0.01 < L < 5.77 m/s • 530 µm: homogeneous mode , 250 µm: Armand (1946) corre at on, 50, 100 µm: Kawahara et a . (2002) corre at on • S ght y d fferent constants for 50-µm channe Chung et a . (2004) 96-µm square N trogen–water Hor zonta (ad abat c) 0.000310 < Bo < 0.000313 0.00010 < WeLS < 25 0.000019 < WeGS < 9 1 < ReLS < 438 1 < ReGS < 612 • Kawahara et a . (2002) corre at on ho ds for 96 µm square channe Kawahara et a . (2005) 50, 75, 100, 251 µm Water–N trogen, Ethano –water/ n trogen Hor zonta 0.08 < G < 70 m/s 0.02 < L < 4.4 m/s • No effects of f u d propert es • 50-, 75-, 100 µm data fo ow Kawahara et a . (2002) corre at on, 251 µm data fo ow near Armand (1946) corre at on • Used to demarcate m cro-/m n channe s (100–251 µm) 270 Heat transfer and f u d f ow n m n channe s and m crochanne s A conven ent tabu at on of the ead ng coeff c ent BB and the exponents n1 . . . n3 for th s mode appears n Carey (1992) for the Z v (1964) entropy m n m zat on mode , the Wa s (1969) separated cy nder mode , the Lockhart–Mart ne (1949) mode , the Thom (1964) corre at on, and the Baroczy (1965) corre at on. Hew tt et a . (1994) a so prov de the fo ow ng corre at on deve oped by Premo et a . (1971) based on the s p rat o uG /uL : S = 1 + E1 y 1 + yE2 − yE2 1/2 (6.22) where y = /1 − , and s the vo ume rat o (a so equ va ent to the homogeneous vo d fract on): = xL xL + (1 − x) G (6.23) The parameters E1 and E2 are g ven by: L G 0.22 E1 = 1.578Re−0.19 LO E2 = 0.0273WeLO Re−0.51 LO

L G −0.08 (6.24) where ReLO = GD/µL and WeLO = G 2 D/L . Once the s p rat o s ca cu ated us ng the above equat ons, the vo d fract on s obta ned as fo ows: = x x + S (1 − x) G /L (6.25) Severa nvest gators have a so used the vo d fract on by Sm th (1969), wh ch was based on an annu ar f ow assumpt on, and a so assumed equa ve oc ty heads of the core and annu ar f m f ows. Sm th stated that th s corre at on, g ven be ow, can be used rrespect ve of f ow reg me, pressure, mass f ux, qua ty, and other parameters: / ì é −1 1−x ùü 0 L ï ï

0 í ý + 0.4 x G 1−x 1 G ú ê = 1+ × ë0.4 + 0.6 1−x û ï ï L x 1 + 0.4 x î þ (6.26) Tandon et a . (1985b) deve oped an ana yt ca mode for the vo d fract on n annu ar f ows. They assumed an ax -symmetr c annu ar f m and a core w th no qu d entra nment, and turbu ent f ow n both phases, wh ch was ust f ed based on Carpenter and Co burn s (1951) f nd ng that at h gh gas ve oc t es, the qu d phase am nar–turbu ent trans t on cou d occur at ReL ~ 240. Fr ct ona pressure drops were computed us ng the Lockhart– Mart ne (1949) method, and the von Karman un versa ve oc ty prof e was used to represent the ve oc ty d str but on n the qu d f m. They found good agreement w th data from Isb n et a . (1957) for steam–water f ow through a 22-mm tube, and Rouhan and Becker (1963) for bo ng of steam n 6.1-mm vert ca tubes, and a so w th the corre at on of Sm th et a . Chapter 6. Condensat on n m n channe s and m crochanne s 271 (1969). The vo d fract on s re ated to the qu d f m Reyno ds number and the Mart ne parameter as fo ows: 4 3 −0.63 −1 −2 [F [F )] )] (X (X + 0.9293Re 50 < ReL < 1125 = 1 − 1.928Re−0.315 tt tt L L 3 4 [F (Xtt )]−1 + 0.0361Re−0.176 [F (Xtt )]−2 = 1 − 0.38Re−0.088 ReL > 1125 L L (6.27) where F(Xtt ) = 0.15[Xtt−1 + 2.85Xtt−0.476 ]. A rev ew of the app cab ty of severa d fferent vo d fract on mode s appears n R ce (1987), who po nted out that the cho ce of the vo d fract on mode affects refr gerant charge pred ct on to a s gn f cant degree, and that mode s based on non-refr gerant f u ds, geometr es, and f ow cond t ons y e d h gh eve s of uncerta nty when app ed to refr gerants. Improper charge pred ct on can ead to prob ems n off-des gn and trans ent performance. Yashar et a . (2001) po nted out that common y used vo d fract on mode s (Sm th, 1969; Wa s, 1969; Premo et a ., 1971) are best su ted for evaporat on app cat ons due to the r bas s n a r–water f ows or n water bo ng app cat ons. As stated prev ous y n th s chapter, a r–water and steam–water m xtures are character zed by ow gas-phase dens t es, whereas n condensat on, part cu ar y of refr gerants, the vapor-phase dens ty s much h gher due to the h gher pressures nvo ved. Due to these h gher vapor-phase dens t es, the vapor ve oc t es are ower, wh ch favors strat f ed f ows, and eads to a dependence of the vo d fract on on mass f ux – a feature that s often not nc uded n severa of the w de y used mode s. Not ng that trans t ons between strat f ed and annu ar reg ons are n fact gradua due to the movement of the qu d f m up around the c rcumference of the tube w th ncreas ng ve oc t es, Hur burt and Newe (1999) deve oped a mode for the non-un form f m th ckness dur ng th s strat f ed–annu ar trans t on. (It was po nted out n the prev ous sect on that other nvest gators have refereed to th s reg on as sem -annu ar or wavy f ow.) Graham et a . (1999) attempted to account for the dependence of the vo d fract on on the non-un form f m th ckness by us ng the Froude rate (Yashar et a . 2001), a parameter that compares the vapor k net c energy to the energy requ red to move the qu d phase upward a ong the c rcumference. Thus, the Froude rate parameter s the Froude number mu t p ed by the square root of the vapor-to- qu d mass f ow rate rat o: 0.5 G 2 x3 Ft = (6.28) (1 − x) g2 gD They reasoned that var at on of the vo d fract on dur ng the trans t on from strat f ed to annu ar f ow wou d be accounted for by th s parameter (vapor k net c energy/grav tat ona drag) comb ned w th the Lockhart–Mart ne parameter (v scous drag/vapor k net c energy), and proposed the fo ow ng express on: −0.321 1 + Xtt (6.29) = 1+ Ft In the above express on, both phenomena are assumed to act ndependent y. Yashar et a . (2001) measured vo d fract ons dur ng condensat on n m crof n (0.2-mm f n he ght) tubes 272 Heat transfer and f u d f ow n m n channe s and m crochanne s of 8.9-mm d ameter. Essent a y, nked shutoff va ves were actuated dur ng tests to so ate the nventory of the two-phase m xture n the test sect on, wh ch prov ded the vo umeaveraged vo d fract on for the g ven test cond t on. (It shou d be noted that th s techn que, used by many nvest gators, does not prov de the area fract on at a cross-sect on of the tube, wh ch s the more re evant parameter, but rather a vo umetr c average.) These tests showed that the trends for smooth and m crof n tubes were s m ar and that for evaporat on, the vo d fract on was re at ve y nsens t ve to mass f ux. However, mass f ux effects were seen n condensat on, part cu ar y at ow mass f uxes (G < 200 kg/m2 -s), presumab y due to the arger vapor dens t es at condens ng cond t ons. Koyama et a . (2004) nvest gated vo d fract ons for the ad abat c f ow of R-134a n hor zonta 7.52-mm smooth and 8.86-mm m crof n tubes over the qua ty range 0.01 < x < 0.96 us ng qu ck c os ng va ves, an approach s m ar to that used by Yashar et a . (2001). The vo d fract ons n the smooth tube agreed we w th the Sm th (1969) and Baroczy (1965) corre at ons. The vo d fract ons were ower at the h gher pressures, whereas the effect of mass f ux was m n ma . They a so found that the vo d fract ons n the m crof n tube were ower than those n the smooth tube; thus the smooth-tube vo d fract on mode s were unab e to pred ct the exper menta va ues for the m crof n tube. They stated that because of the h gher pressure drop n the m crof n tubes, the mean qu d ve oc ty s ower and the qu d occup es a arger port on of the crosssect on, wh ch eads to ower vo d fract ons n the m crof n tubes. The vo d fract on n m crof n tubes was found to ncrease w th an ncrease n mass f ux, part cu ar y at ow pressures and qua t es. They a so deve oped mode s for the vo d fract on n m crof n tubes for strat f ed–annu ar and annu ar f ow. They assumed that the qu d f ows at the bottom of the tube and a so f s the grooves around the who e c rcumference n strat f ed–annu ar f ow. For annu ar f ow, they assumed that the grooves were f ed un form y. Us ng these assumpt ons, they terat ve y so ved s mp e momentum ba ance equat ons between the vapor and qu d reg ons to obta n the vo d fract on for the exper menta cond t ons. They found reasonab y good agreement between the exper menta and numer ca y pred cted va ues, espec a y at h gh vapor qua t es. 6.3.2. Sma channe s For the sma d ameter range, Kar yasak et a . (1991) measured vo d fract on n 1-, 2.4-, and 4.9-mm tubes and proposed severa emp r ca curve-f ts to the measured vo d fract ons n sma reg ons as a funct on of the homogeneous vo ume fract on (see Tab e 6.2): < A : B < < 0.6 ( nterm ttent f ow): B < , 0.6 < < 0.95 ( nterm ttent f ow): B < , 0.95 < (annu ar and nterm ttent f ow): = = 0.833 (6.30) = 0.69 + 0.0858 = 0.83 og (1 − ) + 0.633 where A and B are a so emp r ca y der ved trans t on po nts. Wh e the above corre at on matched the r data, t appears to have been der ved s mp y based on stat st ca curve-f ts to fo ow the data c ose y, and the trends exh b ted show numerous unrea st c d scont nu t es, changes n s ope and nf ect ons for wh ch no phys ca rat ona e s prov ded. The f rst nterm ttent f ow corre at on above s the so-ca ed Armand (1946) corre at on. Chapter 6. Condensat on n m n channe s and m crochanne s 273 M sh ma and H b k (1996) used neutron rad ography to conduct f ow v sua zat on stud es on a r–water f ows n 1–4-mm d ameter tubes. The vo d fract on was corre ated we by the dr ft-f ux mode w th a new equat on for the d str but on parameter as a funct on of nner d ameter. The dr ft-f ux mode was used to corre ate the vo d fract on as fo ows: uG = G / = Co + UG (6.31) where uG s the gas ve oc ty, s the m xture vo umetr c f ux, = G + L , Co s the d str but on parameter, and VG s the dr ft ve oc ty. For each reg me (bubb y, s ug, annu ar and churn), the requ red d str but on parameter was taken from Ish (1977). However, n accordance w th the f nd ngs of severa other researchers (G bson, 1913; Zukosk , 1966; Tung and Par ange, 1976; Kar yasak et a ., 1992), the dr ft ve oc ty n s ug and bubb y f ows was assumed to be zero, and the d str but on parameter was f t to the fo ow ng equat on based on the data: Co = 1.2 + 0.510e−0.691D (6.32) where the tube d ameter D s n mm. These f nd ngs were n good agreement w th the resu ts of Kar yasak et a . (1992), who mode ed the r data n terms of the vo umetr c vo d fract on = G /( G + L ). The d str but on parameters resu t ng from the data and the above equat on for the sma d ameter channe s are arger than those reported by Ish (1977) for arger channe s. M sh ma et a . (1993) a so found th s ncrease n Co for narrow rectangu ar channe s, and attr buted th s trend to the centra zed vo d prof e and the am nar zat on of f ow n these sma er and narrower channe s. H b k and M sh ma (1996) deve oped an approx mate method for the process ng of mages obta ned from neutron rad ography, wh ch was then used for non- ntrus ve y measur ng vo d fract ons n a r–water f ows n tubes w th 1 < D < 4 mm. The agreement between the data and the pred ct ons of the rad ography techn que was best at 1 mm, but the techn que was deemed to be acceptab e for annu ar f ows w th arge vo d fract ons n the arger tubes a so. H b k et a . (1997) then used the neutron rad ography method to measure rad a vo d fract on d str but ons n vert ca upward a r–water f ow n a 3.9-mm channe . An assumed power aw or sadd e-shaped vo d fract on prof e was subsequent y ver f ed us ng measured ntegrated var at ons n the vo d fract on prof e a ong the channe ax s. A tr p e-concentr c-tube apparatus w th some channe s f ed w th water or gas n d fferent comb nat ons prov ded base ne attenuat ons and ensured ca brat on of the exper menta resu ts. The method was demonstrated by deve op ng the fo ow ng equat on for vo d fract on n s ug f ow: 7.316 r r = 0.8372 + 1 − (6.33) rw The var at on n vo d fract on a ong channe ength n s ug f ow was determ ned by M sh ma et a . (1997), who used the mages shown n the prev ous sect on on f ow patterns (F g 6.2) to compute the vo d prof e across the channe at nterva s of 6.45 mm (F g. 6.15). Th s f gure shows that the h gh vo d fract on reg on moves upward w th the r se of the s ug bubb e. A so, the merg ng of two peaks nto one peak s ustrated, correspond ng to the coa escence of two bubb es. Unesak et a . (1998) ater conducted Monte Car o s mu at ons for the tr p e-concentr ctube apparatus descr bed above to further va date the techn que. 274 Heat transfer and f u d f ow n m n channe s and m crochanne s F ow d rect on S ug f ow t 80 ms t 64 ms t 48 ms Vo d fract on (%) t 32 ms t 16 ms 60 40 20 0 t 0 ms 0 40 80 100 20 60 Long tud na pos t on (mm) 120 F g. 6.15. Progress on of vo d fract on prof es w th t me. Repr nted from M sh ma, K., H b k , T. and N sh hara, H., V sua zat on and measurement of two-phase f ow by us ng neutron rad ography, Nuc ear Eng neer ng and Des gn, 175(1–2), pp. 25–35 (1997) w th perm ss on from E sev er. Th s group (Kureta et a ., 2001; 2003) a so measured nstantaneous and t me-averaged vo d fract ons for subcoo ed bo ng of water (at ow vo d fract ons) n narrow rectangu ar channe s (3- and 5-mm gap) us ng neutron rad ography n the mass f ux range 240 < G < 2000 kg/m2 -s. A though the r pr mary ob ect ve was to determ ne the po nt of net vapor generat on and ts effect on cr t ca heat f ux, the r techn que and resu ts demonstrate the feas b ty of non- ntrus ve y obta n ng tempora y (0.89 ms) and spat a y (2-D) reso ved vo d fract ons n two-phase f ows n m n channe s. A representat ve d str but on of nstantaneous vo d fract ons obta ned by them s shown n F g. 6.16. Such tempora reso ut on s part cu ar y he pfu for f ows more preva ent n m n channe s (such as s ug f ows) w th arge tempora var at ons n vo d fract on. F g. 6.17 shows vo d fract on prof es computed by averag ng nstantaneous va ues over 2 s (2250 mages). Such techn ques cou d a so be app ed to ga n deta ed ns ghts nto condens ng f ows. Tr p ett et a . (1999a, b) used st photographs of f ow patterns n 1.1- and 1.45-mm hor zonta c rcu ar channe s (a though, as noted n the prev ous sect on, f ow patterns were a so recorded for sem -tr angu ar channe s) to obta n the vo umetr c vo d fract on. Bubb es n the bubb y f ow pattern were assumed to be spheres or e pso ds, wh e the bubb es n s ug f ow were assumed to cons st of cy nders and spher ca segments. Annu ar f ow vo d fract ons were est mated by averag ng cy ndr ca segments of the vapor core. The est mated uncerta nt es n these vo d fract ons were re at ve y h gh, 15%. A so, they were unab e to re ab y measure vo d fract ons n s ug-annu ar and churn f ow, and ass gned a vo d fract on of 0.5 to the port on of churn f ow n wh ch the gas phase was d spersed. In genera they found that the vo d fract on ncreases w th ncreas ng gas superf c a ve oc ty, UGS at a constant qu d superf c a ve oc ty, ULS , and decreases w th ncreas ng ULS for constant UGS . They compared the r resu ts w th the vo d fract on corre at ons of Butterworth (1975), Premo et a . (1971), and the corre at on of Chexa et a . (Chexa and Le ouche, 1986; Chexa et a ., 1992; Chexa et a ., 1997), wh ch s based on the dr ft-f ux mode of Zuber Chapter 6. Condensat on n m n channe s and m crochanne s Vo d fract on 1 () 0 100 275 Vo d fract on 1 100 mm Vapor bubb e 50 F ow-d rect ona d stance (mm) 0.5 0 15 0 0 15 Transverse pos t on (mm) 30 mm (a) 0 ms (b) 10.7 ms (c) 21.3 ms F g. 6.16. Instantaneous 2-D vo d fract on prof es n narrow rectangu ar channe s. Repr nted from Kureta, M., H b k , T., M sh ma, K. and Ak moto, H., Study on po nt of net vapor generat on by neutron rad ography n sub-coo ed bo ng f ow a ong narrow rectangu ar channe s w th short heated ength, Internat ona Journa of Heat and Mass Transfer, 46(7), pp. 1171–1181 (2003) w th perm ss on from E sev er. s 5 mm, G 600 kg/(m2s), T n 90°C Vo d fract on, () 1 0.9 MW/m2 0.5 0.7 MW/m2 0 100 15 50 F ow-d rect ona d stance, z (mm) 0 0 15 Transverse pos t on, d (mm) F g. 6.17. T me-averaged 2-D vo d fract on n narrow rectangu ar channe s. Repr nted from Kureta, M., H b k , T., M sh ma, K. and Ak moto, H., Study on po nt of net vapor generat on by neutron rad ography n sub-coo ed bo ng f ow a ong narrow rectangu ar channe s w th short heated ength, Internat ona Journa of Heat and Mass Transfer, 46(7), pp. 1171–1181 (2003) w th perm ss on from E sev er. and F nd ey (1965). They found that the homogeneous f ow mode was the best pred ctor of the measured vo d fract ons n bubb y and s ug f ow, that s, at ow UGS va ues. For annu ar and churn f ow, however, the homogeneous mode and other emp r ca mode s s gn f cant y overpred cted the exper menta va ues. They be eve that th s overpred ct on by the emp r ca corre at ons, typ ca y based on annu ar f ow n arge channe s, s due to the s p be ng greater n the arger channe s than n m crochanne s. 276 Heat transfer and f u d f ow n m n channe s and m crochanne s 6.3.3. M crograv ty As ment oned n the prev ous sect on on f ow reg mes, Rezka ah and co-workers have conducted nvest gat ons on two-phase f ow n m crograv ty env ronments that perta ns to f ows n sma d ameter channe s due to the decreased nf uence of grav ty. E kow and Rezka ah (1997a, b) conducted deta ed measurements of vo d fract on f uctuat ons n earth norma (vert ca upward) and m crograv ty cond t ons for a r–water f ow n a 9.53-mm tube. Such f uctuat ons were a so measured by Jones and Zuber (1975) us ng s gna s from gamma dens tometers, by We sman et a . (1979) (d scussed n the prev ous sect on) us ng pressure s gna s, by Das and Pattanayak (1993; 1994a, b; 1995a, b; 1996) and Matuszk ew cz et a . (1987) us ng mpedance and conduct v ty probes, respect ve y, and by Geraets and Borst (1988) and Keska et a . (1992) us ng capac tance probes. Probab ty dens ty funct ons (PDFs) of the vo d fract on f uctuat ons were arger n the bubb e and s ug f ow reg mes at 1-g compared to those at m crograv ty. Th s d fference decreased as nert a forces became more predom nant n trans t ona and annu ar f ows. They nferred that at m crograv ty cond t ons, the s p ve oc ty n bubb e f ow s a most zero – an observat on made by severa others c ted n th s chapter, and the bubb e movement and qu d turbu ence are h gh y suppressed. In s ug f ow, the average vo d fract on va ues at 1-g were found to be ~10% ower than those at m crograv ty. The vo d fract on s gna f uctuated from h gh to ow va ues, based on whether the channe was occup ed by the qu d or the gas phase. At 1-g, bubb es coa esce read y due to contact between Tay or bubb es, ead ng to f uctuat ons n the vo d fract on (0–0.75), wh e at m crograv ty, Tay or bubb es d d not nteract w th each other, resu t ng n much sma er vo d fract on f uctuat ons (0.3–0.55). The re at onsh p between the vo umetr c vo d fract on and the area-based vo d fract on can be expressed as fo ows (see Tab e 6.2): 6 5 UG = Co + (6.34) where UG s the prev ous y def ned dr ft ve oc ty and s the vo umetr c f ux, ( = USG + USL ). The d str but on coeff c ent Co = /( ) accounts for the nonun form d str but on of the vo d fract on over the cross-sect on of the tube, and depends on the f ow reg me and the vo d fract on prof e. In m crograv ty f ow, the second term above (dr ft ve oc ty) was neg ected (as was the case w th other nvest gators d scussed above), wh ch y e ded a d str but on coeff c ent of 1.25 n bubb y and s ug f ow (center ne vo d fract on > wa vo d fract on). The d str but on coeff c ent be ng the same for bubb y and s ug f ows at zero grav ty s because of the s p between the two phases be ng neg g b e n both types of f ow. Wh e Co at 1-g n s ug f ow was c ose to th s va ue (Co = 1.17), n bubb y f ow, t was found to be 0.61, nd cat ng a sadd e-shaped vo d fract on prof e, w th a h gher vo d fract on at the wa than at the center. Lowe and Rezka ah (1999) deve oped th s techn que further, us ng a concave para e p ate capac tance sensor to measure vo d fract ons for dent fy ng f ow reg mes. They were ab e to def ne ob ect ve cr ter a for f ow reg me dent f cat on based on the vo d fract on PDF n con unct on w th h gh-speed v deo mages. An ustrat on of the d fferent s gna s and PDFs for the three pr mary f ow reg mes of nterest n m crochanne s s shown n Chapter 6. Condensat on n m n channe s and m crochanne s

Probab ty 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0 0.2 0.4 (a1) 0.8 1 0.06 <> 0.605 Probab ty 0.05 0.04 0.03 0.02 0.01 0 0.2 0.4 (a2) 0.6 0.8 1 T me (s) 0 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b2) 1 1.1 0.30 Probab ty 0.25

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.10 0.09 <> 0.067 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 (b1)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.6 T me (s) 277 <> 0.879 0.20 0.15 0.10 0.05 0 0.2 0.4 (a3) T me (s) 0.6 0.8 1 0.00 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b3) 1 1.1

F g. 6.18. Vo d fract on s gna s and PDF. Repr nted from Lowe, D. C. and Rezka ah, K. S., F ow reg me dent f cat on n m crograv ty two-phase f ows us ng vo d fract on s gna s, Internat ona Journa of Mu t phase F ow, 25(3), pp. 433–457 (1999) w th perm ss on from E sev er. F g. 6.18. A ow average vo d fract on s gna w th sma f uctuat ons correspond ng to the passage of the bubb es, and a PDF w th a s ng e narrow peak at the ow end of the vo d fract on range character zes bubb y f ow. A vo d fract on s gna w th arge f uctuat ons between h gh and ow va ues correspond ng to the passage of gas bubb es and qu d s ugs character zes s ug f ow. The correspond ng PDF has two peaks. Annu ar f ow exh b ts a re at ve y constant vo d fract on of about 0.80–0.90, w th the PDF show ng a very narrow peak. Sma d ps n the vo d fract on trace correspond to the passage of d sturbance waves. Such traces and PDFs were a so used to dent fy trans t ons between the respect ve reg mes. Pack ng dens t es of spher ca and e ongated bubb es were used to nterpret the ocat ons of the trans t ons from s ug-annu ar f ow. 6.3.4. M crochanne s Ser zawa et a . (2002) stud ed a r–water f ow n m crochanne s (20, 25, and 100 µm) and steam–water f ow n a 50-µm tube. A though a var ety of f ow reg mes (d scussed n 278 Heat transfer and f u d f ow n m n channe s and m crochanne s the prev ous sect on) were dent f ed by them, cross-sect ona -averaged vo d fract ons were ca cu ated from h gh-speed v deo p ctures on y for the bubb y and s ug f ow reg mes ( qu d r ng f ow was not nc uded for vo d fract on ana ys s). The Armand (1946) corre at on = 0.833, d scussed above n connect on w th the work of Kar yasak et a . (1991), was recommended for these reg mes (Tab e 6.2). Kawahara et a . (2002) measured vo d fract ons for n trogen–water f ow n a 100-µm tube by ana yz ng v deo frames and ass gn ng a vo d fract on of 0 to the qu d s ug, and a vo d fract on of 1 to a frame w th a gas core surrounded by a smooth-th n qu d f m or r ng-shaped qu d f m. T me-averaged vo d fract ons were then ca cu ated over severa frames. For h gh qu d f ow rates, gas-core f ow w th a th ck qu d f m was ass gned a vo d fract on (0 < < 1), thus account ng for a f n te f m th ckness surround ng the core n such f ows. They found that the vo d fract on s not strong y dependent on L , and deve oped the fo ow ng emp r ca f t for the vo d fract on n terms of the homogeneous vo d fract on: = 0.030.5 1 − 0.970.5 (6.35) Th s var at on of vo d fract on (F g. 6.19) s s gn f cant y d fferent from the near re at onsh p w th the homogeneous vo d fract on descr bed above. Thus, vo d fract ons rema n 1.0 Vo d fract on () 0.8 Emp r ca corre at on, Eq. (6.35) b, Homogeneous f ow 0.8b, A et a . (1993) L 0.02–0.06 m/s L 0.11–0.20 ms/s L 0.42–0.56 m/s L 1.34–1.51 ms/s L 3.50–4.40 m/s 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Homogeneous vo d fract on (b) F g. 6.19. Vo d fract ons for a r–water f ow n 100-µm channe s. Repr nted from Kawahara, A., Chung, P. M. -Y. and Kawa , M. Invest gat on of two-phase f ow pattern, vo d fract on and pressure drop n a m crochanne , Internat ona Journa of Mu t phase F ow, 28(9), pp. 1411–1435 (2002) w th perm ss on from E sev er. Chapter 6. Condensat on n m n channe s and m crochanne s 279 ow even at h gh va ues of ( < 0.8), and ncrease steep y for 0.8 < < 1. They attr buted th s phenomenon to the observat on that s ng e-phase qu d f ow occurred most frequent y at h gh qu d f ow rates for a gas f ow rates, and a so at ow qu d and gas f ow rates. Wa shear and surface tens on resu ted n channe br dg ng under such cond t ons, and the assoc ated ow average vo d fract ons. At ow qu d and h gh gas f ow rates, gas-core f ow surrounded by a qu d f m w th weak momentum coup ng was seen. Wh e the qu d f ow was governed by wa shear and surface tens on, the gas phase f owed w thout s gn f cant res stance due to the undeformed shape character st c of the f ow n such sma channe s observed by them. The non near dependence of the area-based vo d fract on on the vo umetr c vo d fract on mp es that s p rat os are much h gher (as h gh as 16) n these channe s compared to the much ower va ues seen n channe s w th Dh > 1 mm. It shou d be noted that these resu ts contrad ct the resu ts of Ser zawa et a . (2002), who reported the adequacy of the Armand corre at on for a s m ar range of Dh . Chung and Kawa (2004) nvest gated the effect of channe d ameter on vo d fract on for the f ow of n trogen–water m xtures n c rcu ar 530-, 250-, 100-, and 50-µm channe s. For the arger channe s (530 and 250 µm), mu t p e mages at each data po nt were ana yzed by est mat ng the gas fract on vo ume as a comb nat on of symmetr ca shapes. For the sma er channe s, the vo d fract on was est mated as descr bed above n Kawahara et a . (2002). They found that the vo d fract on for the 530-µm channe was pred cted we by the homogeneous f ow mode , and the vo d fract on for the 250-µm channe agreed we w th the Armandtype corre at on. However, the 100- and 50-µm channe vo d fract ons fo owed the s ow near ncrease up to = 0.8, fo owed by a steep exponent a ncrease captured by Eq. (6.35) above that was deve oped by them ear er (Kawahara et a ., 2002). (Based on th s atter work, however, they mod f ed the ead ng constants for the 50-µm channe to 0.02 and 0.98 n the numerator and denom nator, respect ve y.) They attr buted the departure from homogeneous f ow at h gh vo umetr c vo d fract ons to a arger s p between the phases. Th s mod f ed equat on nd cates a further decrease from the homogeneous mode , even compared to the va ues for the 100-µm channe . They argued that n the arger (m n ) channe s, the wavy and deformed gas– qu d nterface eads to more momentum exchange between the gas and qu d phases, wh ch s ows the f ow of the gas bubb es, keep ng the vo d fract on h gh. On the other hand, accord ng to them, the ack of th s momentum exchange n the m crochanne s eads to a ower vo d fract on. Chung et a . (2004) ater conf rmed that the coeff c ents n the vo d fract on corre at on for c rcu ar 100-µm channe s (C1 = 0.03, and C2 = 0.97) wou d app y to square channe s w th Dh = 96 µm. Kawahara et a . (2005) further nvest gated the effect of f u d propert es and d ameter on vo d fract on. They tested the two-phase f ow of water–n trogen and ethano –water/n trogen gas m xtures (of vary ng ethano concentrat ons) n 50-, 75-, 100-, and 251-µm c rcu ar channe s. The r resu ts showed that the vo d fract on was not sens t ve to f u d propert es over the range of d ameters stud ed. The vo d fract on data for the 50-, 75-, and 100-µm channe s agreed w th the corre at on of Kawahara et a . (2002), but the data for the 251-µm d ameter channe fo owed the homogeneous f ow type mode of Armand (1946). They nterpreted these resu ts to mp y that the boundary between “m crochanne s” and “m n channe s” wou d e between 100 and 251 µm. 280 Heat transfer and f u d f ow n m n channe s and m crochanne s 6.3.5. Summary observat ons and recommendat ons The above d scuss on has shown that vo d fract on mode s for convent ona tubes have been deve oped on the bas s of an assumpt on of homogeneous f ow, or as a funct on of s p rat o, the Lockhart–Mart ne parameter, and those that ncorporate some dependence on the mass f ux. Many of these mode s assume annu ar f ow regard ess of the f ow cond t ons. A so, severa of these mode s were deve oped us ng data from a r–water ad abat c f ow, or evaporat ng f ows, a though n pract ce, they are rout ne y used for condensat on a so. Yashar et a . (2001) acknow edged the h gher vapor phase dens t es n condens ng f ows compared to a r–water or evaporat ng f ows, and po nted out that th s eads to ower vapor ve oc t es, more strat f ed f ows, and a dependence of the vo d fract on on mass f ux. Th s group (Graham et a ., 1999) a so used a Froude Rate parameter and deve oped a s mp e mode for the chang ng vo d fract on from strat f ed to annu ar f ows by account ng for the rat o of the vapor k net c energy to the grav tat ona drag, comb ned w th the Lockhart– Mart ne parameter, wh ch accounts for the rat o of the v scous drag to the vapor k net c energy. L nked shutoff va ves were used by th s group as we as by Koyama et a . (2004) to measure the vapor trapped n the test sect on, wh ch y e ds the vo umetr c vo d fract on, but not the cross-sect ona vo d fract on. M sh ma, H b k , and co-workers (M sh ma et a ., 1993; H b k and M sh ma, 1996; M sh ma and H b k , 1996; H b k et a ., 1997; M sh ma et a ., 1997; Kureta et a ., 2001; Kureta et a ., 2003) have used neutron rad ography to measure the d str but on of vo d fract ons n 2-D n sma channe s, as we as the tempora evo ut on of the vo d fract on, for examp e, as the vapor and qu d phases n s ug f ow pass through the reg on of nterest. Tr p ett et a . (1999a, b) have used mage ana ys s to measure vo d fract ons n these channe s, wh e Rezka ah and co-workers (E kow and Rezka ah, 1997a, b; C arke and Rezka ah, 2001) have focused on non- ntrus ve measurement of vo d fract on us ng capac tance sensors n m crograv ty env ronments. The r work has a so dent f ed the use of t me traces and PDF of the vo d fract on as s gnatures for the d fferent k nds of f ow reg mes and trans t ons between them. Kawa , Kawahara, and co-workers (Kawahara et a ., 2002; Chung and Kawa , 2004; Chung et a ., 2004; Kawahara et a ., 2005) have used mage ana ys s to obta n vo d fract on mode s for the <1-mm range. The r work shows that the vo d fract on for channe s of ~1-mm d ameter fo ows the homogeneous f ow mode , wh e sma er channe s (D < 250 µm) show much ower vo d fract ons at vo umetr c vo d fract on < 0.8, beyond wh ch the vo d fract on sharp y r ses to the homogeneous va ue. Ser zawa et a . (2002), on the other hand, found a var ant of the homogeneous f ow mode to app y to sma er geometr es. As was the case for the dent f cat on of f ow reg mes and trans t on cr ter a, much of the work on vo d fract ons has been conducted us ng ad abat c a r–water f ow. The few stud es on vo d fract ons n condensat on are for arger d ameters (~7–8 mm). C ear y, much add t ona research on vo d fract ons n d fferent f ow reg mes dur ng condensat on n m crochanne s s requ red. Vo d fract ons computed for representat ve cond t ons for R134a condens ng n a 1-mm tube at a pressure of 1500 kPa, at d fferent mass f uxes and qua t es us ng a w de var ety of corre at ons are ustrated n Examp e 6.2. A compar son of the resu t ng vo d fract ons s shown n F g. 6.20. Chapter 6. Condensat on n m n channe s and m crochanne s x 0.2 x 0.5 281 x 0.8 0.9 0.8 0.7 0.6 G 750 kg/m2-s 1.0 0.9 0.8 0.7 G 400 kg/m2-s Vo d fract on (a) 0.5 1.0 0.6 0.9 0.8 0.7 0.6 G 150 kg/m2-s 0.5 1.0 0.5 ) ) ) ) ) e 4) 3) 5) 1) 1) od (196 1964 1949 (196 1993 1985 (200 1969 197 00 2 m ( ( ( ( ( ( ( . . . . us zy Z v ne hom ner t a m th et a ta ta eo aroc T te on e a e S rt re en a S a o B g h a M nd H em Yas mo nd Pr E Ta Ho rt a a h ck Lo F g. 6.20. Vo d fract ons for R-134a condens ng n a 1-mm tube. 6.4. Pressure drop As was noted n the preced ng sect ons, there s re at ve y tt e nformat on on f ow reg mes and vo d fract on n condens ng f ows through m crochanne s. Th s s part cu ar y true for pressure drops dur ng condensat on. Therefore, the ava ab e re evant nformat on on pressure drops n condens ng f ows through re at ve y sma channe s and pr mar y ad abat c f ows through m crochanne s s presented here (Tab e 6.3). Tab e 6.3 Summary of pressure drop stud es. Invest gator Hydrau c d ameter (mm) F u ds C ass ca corre at ons Lockhart and 1.5–26 mm Mart ne (1949) Ch sho m (1973) Fr ede (1979; 1980) Or entat on/ cond t ons Range/ app cab ty Techn ques, bas s, observat ons A r and benzene, Ad abat c kerosene, water, and var ous o s • P corre ated based on whether gas and qu d phases are am nar or turbu ent • P re ated n terms of two-phase mu t p er 2 to correspond ng qu d-or gas-phase pressure drop • Ch sho m (1967) corre ated L2 n terms of Xtt Steam • Mod f ed procedure and equat on deve oped by Baroczy (Baroczy and Sanders, 1961; Baroczy, 1966) to deve op corre at on based on qu d on y P D > 1 mm Ad abat c µ /µv < 1000 Condensat on or ad abat c qu d–vapor stud es (~2 < Dh < ~10 mm) Hash zume 10 mm R-12, R-22 Hor zonta 570 < p < 1960 kPa et a . (1985) condens ng 2 • Deve oped LO corre at on from database of 25,000 po nts • Inc udes surface tens on effects • P express ons for annu ar and strat f ed f ow, qu d-and gas-phase ve oc ty n terms of Prandt m x ng ength • Strat f ed f ow mode ed as f ow between para e p ates Wang 6.5 mm et a . (1997b) R-22, R-134a, R-407C Hor zonta ad abat c • Mod f ed Ch sho m (1967) C for L2 for G > 200 kg/m2 -s 50 < G < 700 kg/m2 -s T = 2 C, 6 C, and 20 C • Emp r ca corre at on for G2 , for G < 200 kg/m2 -s Hur burt and Newe (1999) 3–10 mm R-22, R-134a (Dobson, 1994) R-11, R-12, R-22 (Sacks, 1975) Hor zonta 0.2 < x < 0.9 200 < G < 650 kg/m2 -s • Sca ng equat on to pred ct P at new cond t on from ava ab e nformat on at a d fferent cond t on • Assumed bu k of condensat on occurs n annu ar f ow • Turbu ent vapor and qu d f ms, aw of wa assumed Chen et a . (2001) 1.02, 3.17, 5.05, 7.02 mm A r–water Hor zonta 3.17, 5.05, 7.02, 9 mm R-410A Room temperature 50 < G < 3000 kg/m2 -s 0.001 < x < 0.9 5 C and 15 C 50 < G < 600 kg/m2 -s 0.1 < x < 0.9 • Accounted for ncreased nf uence, decreased g nf uence • Poor agreement w th Ch sho m (1967), Fr ede (1979; 1980), and homogeneous f ow mode s • Mod f ed homogeneous mode to nc ude Bo and We to account for effect of surface tens on and mass f ux W son et a . (2003) Dh = 7.79, 6.37, 4.40, and 1.84 R-134a R-410A Hor zonta condens ng T = 35 C 75 < G < 400 kg/m2 -s • F attened round smooth, ax a , and he ca m crof n tubes • P ncreases as tube approaches rectangu ar shape Souza et a . (1993) 10.9 mm R-12 R-134a Hor zonta Pr = 0.07–0.12 200 < G < 600 kg/m2 -s 0.1 < x < 0.9 2 • New corre at on for LO • Demonstrated effect of o n refr gerant on pressure drop Cava n et a . (2001; 2002a, b) 8 mm R-22, R-134a, R-125, R-32, R-236ea, R-407C, and R-410A Hor zonta tubes 30 < Tsat < 50 C 100 < G < 750 kg/m2 -s • Mod f ed Fr ede (1979) corre at on to app y on y to annu ar f ow, whereas t had or g na y been ntended for annu ar and strat f ed reg mes A r–water stud es (~1 < Dh < ~10 mm) Zhao and 9.525 mm A r–water Rezka ah (1995) Vert ca upward 1-g, µ-g 0.1 < G < 18 m/s 0.1 < L < 2.5 m/s • P about the same for 1-g and µ-g, espec a y n annu ar f ow • Re at ve y good agreement w th homogeneous, Lockhart and Mart ne (1949) and Fr ede (1979; 1980) corre at ons Fukano et a . (1989) 1, 2.4, and 4.9 mm A r–water Hor zonta sotherma Pex t = 1 atm 0.04 < G < 40 0.2 < L < 4 • Corre at on for P n bubb y, s ug, p ug, and annu ar f ow • Proposed a new s ug f ow mode account ng for s ug/bubb e ve oc ty rat os for d fferent gasand qu d-phase reg mes, and for expans on osses as qu d f owed from the annu ar f m surround ng the gas bubb e nto qu d s ug reg on M sh ma and H b k (1996) 1–4 mm A r–water Vert ca upwards 0.0896 < G < 79.3 m/s 0.0116 < L < 1.67 m/s • C n Ch sho m (1967) corre at on decreases for sma D, proposed new corre at on for C based on Dh Tr p ett et a . (1999a) C rcu ar 1.1 and 1.45 mm, sem -tr angu ar 1.09 and 1.49 mm A r–water Hor zonta ad abat c 0.02 < G < 80 m/s 0.02 < L < 8 m/s • Homogeneous mode pred cted bubb y and s ug f ow data we at h gh ReL • Poor agreement n s ug–annu ar and annu ar reg on, a so n s ug f ow at very ow ReL • Fr ede (1979; 1980) corre at on resu ted n arge dev at ons Tran et a . (2000) C rcu ar 2.46 and 2.92 mm, rectangu ar 4.06 × 1.7 mm R-134a, R-12, R-113 Bo ng 138 < P < 856 kPa 33 < G < 832 kg/m2 -s • Deve oped corre at on for P dur ng bo ng of refr gerants • Trends s m ar to arge-d ameter tubes • Proposed mod f ed vers on of Ch sho m (1973) corre at on to account for surface tens on effects (Cont nued ) Tab e 6.3 (Cont nued ) Invest gator Hydrau c d ameter (mm) F u ds Or entat on/ cond t ons Range/ app cab ty Techn ques, bas s, observat ons Zhao and B (2001b) Equ atera tr angu ar channe s Dh = 0.866, 1.443, and 2.886 mm A r–water Vert ca upward co-current 0.1 < G < 100 m/s For Dh = 2.886, 1.443 mm 0.08 < L < 6 m/s For Dh = 0.866 mm 0.1 < L < 10 m/s • Fr ct ona P corre ated us ng Lockhart and Mart ne (1949) approach Lee and Lee (2001) Rectangu ar 20 mm w de Gap 0.4–4 mm A r–water Hor zonta 175 < ReLO < 17700 0.303 < X < 79.4 • Commented on weaker coup ng between phases n sma er channe s, corre ated parameter C n the two-phase mu t p er (Ch sho m, 1967) n terms of non-d mens ona parameters to nc ude surface tens on effects Hor zonta ad abat c 400 < G < 1400 kg/m2 -s 0.1 < x < 0.9 • P n m crof n tube h gher than n p a n tube • Equ va ent Re-based mode for P • Conc uded surface tens on does not affect P Hor zonta T = 40–50 C • Used Akers et a . (1959) equ va ent mass ve oc ty concept to corre ate condensat on P Condensat on stud es (~0.4 < Dh < ~5 mm) Yang and Rectangu ar R-12 Webb (1996b) p a n: Dh = 2.64, M crof n: Dh = 1.56 Yan and L n (1999) 2 mm R-134a Zhang and Webb (2001) 3.25 and 6.25 mm Mu t port extruded A Dh = 2.13 mm R-134a, R-22 R-404A Ad abat c T = 20–65 C 200 < G < 1000 kg/m2 -s • Fr ede (1979; 1980) corre at on d d not pred ct data we • P funct on of reduced pressure rather than dens ty or 2 v scos ty rat o; proposed new corre at on for LO Gar me a et a . (2002; 2003a, b; 2005) 0.5–4.91 mm R-134a Hor zonta T ~ 52 C 0 • F ow reg me based mode s for nterm ttent and annu ar/m st/d sperse f ow reg mes • Interm ttent f ow mode treats P as comb nat on of P due to qu d s ug, f m–bubb e nterface, and trans t ons between s ug and bubb e, s ug frequency deduced from P data • Annu ar f ow mode : nterfac a fr ct on factor der ved from measured pressure drops corre ated w th ReL , fL , X , and surface tens on parameter • Comb ned mode ensures smooth trans t ons across reg mes N trogen–water Hor zonta 0.1 < G < 60 m/s 0.02 < L < 4 m/s • Homogeneous corre at ons d d not pred ct data we • Lockhart and Mart ne (1949) approach w th Ch sho m (1967) coeff c ent s gn f cant y overpred cted data • Proposed new va ues for parameter C based on data N trogen–water Hor zonta T = 22.9–29.9 C 0.02 < G < 72.98 m/s 0.01 < L < 5.77 m/s • Effect of d ameter nvest gated • New s ug f ow mode based on work of Gar me a et a . (2002; 2003a, b; 2005) for 50- and 100-µm tubes A r–Water Stud es (Dh < ~500 µm) Kawahara et a . 100 µm (2002) Chung and Kawa (2004) 530, 250, 100, and 50 µm 286 Heat transfer and f u d f ow n m n channe s and m crochanne s 6.4.1. C ass ca corre at ons Pressure drops n convent ona channe s have ong been ca cu ated us ng three we -known corre at ons by Lockhart and Mart ne (1949), Ch sho m (1973), and Fr ede (1979; 1980), somet mes w th mod f cat ons to account for the spec f c geometry or f ow cond t ons under nvest gat on. The Lockhart–Mart ne (1949) corre at on was based on ad abat c f ow of a r and benzene, kerosene, water, and var ous o s f ow ng through 1.5– 26-mm p pes, and the pressure drops were corre ated based on whether the nd v dua qu d and gas phases were cons dered to be n am nar or turbu ent f ow. As many of the separated f ow corre at ons for sma channe s are a so based on th s bas c methodo ogy, a br ef overv ew s presented here. The two-phase pressure drop s expressed n terms of two-phase mu t p ers to the correspond ng s ng e-phase qu d or gas-phase pressure drop: L2 = (dPF /dz) (dPF /dz)L (6.36) 2 = G (dPF /dz) (dPF /dz)G (6.37) 2 LO = (dPF /dz) (dPF /dz)LO (6.38) where 2 s the two-phase mu t p er, the subscr pts L and G refer to the f ow of the qu d and gas phases f ow ng through the who e channe , and LO refers to the ent re f u d f ow occurr ng as a qu d phase through the channe . These mu t p ers are n turn most often 0.5 (dPF /dz)L corre ated n terms of the Mart ne parameter X = (dP (wh ch has been referred /dz) F G to n the prev ous sect ons). Ch sho m (1967) deve oped the fo ow ng corre at ons for the two-phase mu t p ers of Lockhart and Mart ne : L2 = 1 + 1 C + 2 X X 2 = 1 + CX + X 2 G (6.39) (6.40) where C depends on the f ow reg me of the qu d and gas phases. Wh e these (Lockhart and Mart ne , 1949; Ch sho m, 1967) corre at ons have shown cons derab e dev at ons from the data for sma channe s w th phase-change f ows, they cont nue to be the bas s for many of the more recent corre at ons, as w be seen ater n th s sect on. Ch sho m (1973) mod f ed the procedure and equat ons deve oped by Baroczy (Baroczy and Sanders, 1961; Baroczy, 1966) that account for f u d propert es, qua ty, and mass f ux based on steam, water/a r, and mercury/n trogen data to deve op the fo ow ng corre at on: 2 LO = 1 + Y 2 − 1 [Bx(2−n)/2 (1 − x)(2−n)/2 + x2−n ] (6.41) Chapter 6. Condensat on n m n channe s and m crochanne s 287 where n s the exponent for Reyno ds number n the turbu ent s ng e-phase fr ct on factor corre at on, for examp e, n = 0.5 for the B as us equat on. The parameter Y s the Ch sho m parameter: (dPF /dz)GO Y = (dPF /dz)LO 0.5 (6.42) and B s g ven by: 55 G 0.5 520 = 0.5 YG 15,000 = 2 0.5 Y G B = 0 < Y < 9.5 9.5 < Y < 28 (6.43) 28 < Y Fr ede (1979; 1980) deve oped the fo ow ng corre at on based on a database of 25,000 po nts for ad abat c f ow through channe s w th d > 1 mm: 2 LO =E+ 0.324FH (6.44) Fr 0.045 We0.035 where E = (1 − x)2 + x2 L fGO G fLO (6.45) F = x0.78 (1 − x)0.24 H= L G 0.91 µG µL (6.46) 0.19 1− µG µL 0.7 (6.47) 2 and We = G 2 D/ , and f and Fr = G 2 /gDTP TP LO and fGO are the s ng e-phase fr ct on factors for the tota f u d f ow occurr ng as qu d and gas respect ve y. The two-phase m xture dens ty s ca cu ated as fo ows: TP = x 1−x + G L −1 (6.48) The above corre at on s the more w de y used corre at on for vert ca upward and hor zonta f ow, and s recommended by Hew tt et a . (1994) for s tuat ons where surface tens on data are ava ab e, and by Hetsron (1982) for µ /µv < 1000. Wh e these corre at ons have been used w de y, often the r pred ct ve capab t es are not part cu ar y good, pr mar y because nvest gators have tr ed to use them beyond the r or g na y ntended range of app cab ty, and a so because these mode s do not account for the f ow reg mes that are estab shed over these d verse ranges of cond t ons and geometr es. 288 Heat transfer and f u d f ow n m n channe s and m crochanne s 6.4.2. Condensat on or ad abat c qu d–vapor stud es (~2 < Dh < ~10 mm) Hash zume et a . (1985) conducted f ow v sua zat on and pressure drop exper ments on refr gerants R-12 and R-22 f ow ng through hor zonta 10-mm tubes over the range 20 C < T < 50 C. They deve oped pressure drop express ons for annu ar and strat f ed f ow, descr b ng the qu d- and gas-phase ve oc ty prof es n terms of the Prandt m x ng ength. The pressure drop was mode ed as dP/dL = (4(Re+ )2 µ2L /L D3 ) where Re+ s the fr ct on Reyno ds number. Strat f ed f ow was mode ed as f ow between para e p ates w th the qu d poo form ng one of the para e p ates, w th an equ va ent gap between the p ates der ved n terms of the tube d ameter, qua ty, and mass f ux. Trans t on (wavy) reg on pressure drops were ca cu ated by nterpo at ng between the annu ar and strat f ed reg ons, w th the correspond ng trans t ons estab shed by the mod f ed Baker map deve oped by them (Hash zume, 1983). They ater (Hash zume and Ogawa, 1987) tr ed to extend th s work to a r–water and steam–water f ows, but found that the best agreement was w th refr gerant data, and not w th a r–water or steam–water m xtures. Wang et a . (1997b) measured pressure drops dur ng the ad abat c f ow of refr gerants R-22, R-134a, and R-407C n a 6.5-mm tube for the mass f ux range 50 < G < 700 kg/m2 -s. 2 ana ogous to the Lockhart–Mart ne (1949) They computed two-phase mu t p ers G corre at on from the measured pressure drops, and tr ed to obta n the best f t for the C parameter n Ch sho m s (1967) equat on for the mu t p er, because they found that C was strong y dependent on the f ow pattern. For R-22 and R-134a, the nterm ttent f ow data were pred cted we w th C = 5. For G > 200 kg/m2 -s, wavy-annu ar f ow was observed w th the two-phases be ng turbu ent, and the mu t p ers d d not depend on the mass f ux. For G = 50 and 100 kg/m2 -s, there was a pronounced nf uence of the mass f ux. They stated that for G = 100 kg/m2 -s, the f ow s not shear dom nated, thus the waves do not reach the top of the tubes, and the pressure drop wou d therefore depend on the wetted per meter, wh ch n turn depends on the mass f ux. However, for G > 200 kg/m2 -s, even though the f ow cont nues to be wavy (w th a th cker f m at the bottom), the waves do reach the top of the tube and render the tube comp ete y wet, wh ch makes the wetted per meter re at ve y constant as the mass f ux var es. Based on these cons derat ons, they deve oped the fo ow ng equat ons for the two-phase mu t p er: For G > 200 kg/m2 -s: 2 G = 1 + 9.4X 0.62 + 0.564X 2.45 (6.49) and for G < 200 kg/m2 -s, the C parameter n the Ch sho m (1967) corre at on for the Lockhart–Mart ne (1949) mu t p er was mod f ed to nc ude a mass f ux dependence as fo ows: 0.938 C = 4.566 × 10−6 X 0.128 ReLO

L G −2.15 µL µG 5.1 (6.50) Hur burt and Newe (1999) deve oped sca ng equat ons for condens ng f ows that wou d enab e the pred ct on of vo d fract on, pressure drop, and heat transfer for a refr gerant at a g ven cond t on and tube d ameter from the ava ab e resu ts for another s m ar f u d (R-11, Chapter 6. Condensat on n m n channe s and m crochanne s 289 R-12, R-22, and R-134a) operat ng at a d fferent cond t on n the d ameter range 3–10 mm. To ach eve th s, they assumed that the bu k of the condensat on process occurs under annu ar f ow cond t ons, and that even though n rea ty, the condens ng f m s non-un form around the c rcumference, an equ va ent average f m th ckness wou d suff ce for the pred ct on of shear stress. They further cons dered that the pr mary res stance to heat transfer occurs n the v scous and buffer ayers of the f m, regard ess of the actua th ckness. The approach used for mode ng the f m and core f ows was s m ar to that of Rohsenow et a . (Rohsenow et a ., 1957; Bae et a ., 1971; Trav ss et a ., 1973), assum ng turbu ent vapor and qu d f ms and aw-of-the-wa un versa ve oc ty and temperature prof es. (The qu d f m was assumed to be turbu ent even at ow Re (Re > 240) due to nteract ons w th the turbu ent vapor core, based on the work of Carpenter and Co burn (1951).) They used the nterfac a shear stress, rather than the vo d fract on, as an nput to determ ne the qu d f m th ckness, wh ch enab ed them to capture the dependence of vo d fract on on mass f ux. They used vo d fract on, pressure drop, and heat transfer data of Sacks (1975) for R-11, R-12, and R-22, and the R-22 and R-134a data of Dobson (1994) to va date the r mode . Us ng these cons derat ons, they deve oped the fo ow ng sca ng equat on for pressure drop pred ct on: −0.75 −0.25 −0.15 0.125 P2 / P1 = g2 /g1 L2 /L1 µg2 /µg1 µL2 /µL1 × (D2 /D1 ) −1.0 2 (G2 /G1 ) (x2 /x1 ) 1.65 1 − x2 1 − x1 0.38 (6.51) Th s equat on pred cts a f ve-fo d ncrease n pressure drop between R-22 and R-11, pr mar y due to the drast c d fference n saturat on pressure at 40 C (959 kPa for R-22, 175 kPa for R-11), wh ch eads to a much h gher vapor-phase dens ty for R-22 (65.7 kg/m3 ) than for R-11 (9.7 kg/m3 ). Th s approach was cons dered to be va d for the qua ty range 0.2 < x < 0.8, where annu ar f ow preva s n genera . A though there were some d sagreements between the trends pred cted by th s equat on and the exper menta data, part cu ar y n the effect of mass f ux, the approach prov des some genera gu dance for the mode ng of annu ar f ows. Chen et a . (2001) attempted to account for the ncreased nf uence of surface tens on and the decreased nf uence of grav ty n tubes w th D < 10 mm for f u ds encompass ng a w de range of propert es, a r–water ( n 1.02-, 3.17-, 5.05-, 7.02 mm tubes) and R-410A ( n 3.17-, 5.05-, 7.02-, 9.0 mm tubes). The a r–water tests were conducted at room temperature, wh e the R-410A tests were conducted at 5 C and 15 C, somewhat ower than typ ca condensat on temperatures of nterest. Poor agreement was found between the r pressure drop data and the pred ct ons of the Ch sho m (1967), Fr ede (1979), and homogeneous f ow (w th average v scos ty (Beatt e and Wha ey, 1982)) mode s. Therefore, they mod f ed the homogeneous f ow mode by nc ud ng the Bond number and the Weber number to account for the effects of surface tens on and mass f ux: dP dP hom = dz dz hom (6.52) 1 + (0.2 − 0.9 exp(−Bo)) Bo < 2.5 hom = 1 + We0.2 /exp Bo0.3 − 0.9 exp(−Bo) Bo ≥ 2.5 290 Heat transfer and f u d f ow n m n channe s and m crochanne s where We = G 2 D/m and Bo = g(L − G ) (d/2)2 / . They a so deve oped a s m ar mod f cat on to the Fr ede (1979) corre at on us ng the rat ona e that, when used for sma tubes, th s corre at on does not emphas ze surface tens on (We) enough, and may emphas ze grav ty (Fr) too much. The resu t ng mod f cat on s as fo ows: dP dP = dz dz Fr ede ì 0.45 0.0333ReLO ï ï (6.53) ï Bo < 2.5 í 0.09 ReG (1 + 0.4exp(−Bo)) = ï We0.2 ï ï î Bo ≥ 2.5 (2.5 + 0.06Bo) These mod f cat ons mproved the pred ct ve capab t es of the homogeneous and Fr ede corre at ons from dev at ons of 53.7% and 218.0% to 30.9% and 19.8%, respect ve y. In add t on, these equat ons a so agreed reasonab y we w th the pressure drop data of Hash zume (1983) for R-12 and R-22 n 10-mm tubes. W son et a . (2003) stud ed the effect of progress ve y f atten ng 8.91-mm round smooth tubes and tubes w th ax a and he ca m crof n tubes on vo d fract on, pressure drop, and heat transfer dur ng condensat on of refr gerants R-134a and R-410A at 35 C. The hydrau c d ameters of the tubes tested were 7.79, 6.37, 4.40, and 1.84 mm, w th the m crof n tubes hav ng 60 f ns of 0.2 mm he ght for a surface area ncrease of 60% over a smooth tube. The pressure drop at a g ven mass f ux and qua ty ncreased as the tube approached a rectangu ar shape. They found that the fo ow ng c rcu ar tube qu d-on y two-phase mu t p er corre at on of Jung and Radermacher (1989) pred cted the r resu ts w th n 40%: 2 LO = 12.82Xtt−1.47 (1 − x)1.8 (6.54) S m ar y, the fo ow ng annu ar and strat f ed f ow corre at on of Souza et a . (1993) w th the Dh of the f attened tube as the character st c d mens on, a so pred cted the r data w th n 40%: L2 = 1.376 + C1 Xtt−C2 Fr < 0.7: C1 = 4.172 + 5.48Fr − 1.564Fr 2 C2 = 1.773 − 0.169Fr Fr > 0.7: C1 = 7.242; C2 = 1.655 (6.55) √ where Fr = G/ L gD . The correspond ng s ng e-phase fr ct on factor was ca cu ated us ng the Co ebrook (1939) equat on and an express on for the re at ve roughness of the m crof n tubes deve oped by Cava n et a . (2000). W son et a . (2003) deemed th s 40% agreement to be adequate and d d not deve oped a new corre at on spec f ca y for f at (f attened) tubes. Chapter 6. Condensat on n m n channe s and m crochanne s 291 6.4.3. A r–water stud es (~1 < Dh < ~10 mm) Zhao and Rezka ah (1995) measured pressure drops for upward a r–water f ow data n a 9.5-mm tube at 1-g and m crograv ty cond t ons and found that the pressure drops were about the same at both env ronments. In add t on, the data were n re at ve y good agreement w th the homogeneous, Lockhart–Mart ne (1949) and Fr ede (1979) corre at ons, a though Fr ede s corre at on was found to overest mate the data. They reasoned that at reduced grav ty, annu ar f ow pressure drops wou d not be affected due to the dom nance of nert a forces. In bubb e and s ug f ows, the s p between the two phases decreases due to a decrease n gas-phase ve oc ty, ead ng to a h gher vo d fract on, and therefore ncreas ng the qu d phase pressure drop due to the reduced qu d f ow area. However, n bubb y f ow, the decrease n turbu ence at reduced grav ty due to the s m ar ve oc t es of the gas and qu d phases wou d ead to ower pressure drops. The d fference n pressure drop at 1-g and µ-g wou d be due to a trade-off between these compet ng nf uences. However, at the re at ve y h gh ve oc t es and fu y turbu ent f ows they tested, sma changes n bubb e ve oc ty due to the change n grav ty were not deemed to be very s gn f cant n a ter ng the pressure drop. Fukano et a . (1989) proposed corre at ons for pressure drop n bubb y, s ug, p ug, and annu ar f ow based on exper ments w th a r–water f ow n 1-, 2.4-, and 4.9mm tubes. In s ug f ow, sequent a photographs showed that the ve oc ty of the arge gas bubb e can be corre ated as uS = 1.2 ( G + L ) where G + L s the qu d s ug ve oc ty. From th s, the re at ve ve oc ty between the gas bubb e and the qu d n the s ug s g ven by ur = 0.2 ( G + L ). From photograph c observat ons, t was a so estab shed that the qu d s ug ength s g ven by: LL /(LL + LG ) = K L /( L + G ) (6.56) where the constant of proport ona ty K var ed from 0.9 at 1 mm to 0.72 at 4.9 mm. Assum ng that n s ug and p ug f ows, the pressure drop occurs n the qu d s ug on y, wh e n annu ar and bubb y reg ons, t occurs over the ent re ength of the channe , they deve oped the fo ow ng equat ons for the two-phase mu t p er: n L G + L n+c+1 n−m T L2 ∝ K ReL (6.57) L T L where c = 0 for nterm ttent f ow and c = 1 for annu ar and bubb y f ows, subscr pt T refers to the two-phase m xture, and m and n refer to the Re exponent n the s ng e-phase fr ct on factor for the qu d and two-phase f ows, respect ve y, dependent on whether the f ow s am nar or turbu ent. As the exper menta L2 va ues were arger than those pred cted by th s treatment, they proposed a new s ug f ow mode that accounts for the expans on osses ( oss coeff c ent = 1) as the qu d f owed from the annu ar f m surround ng the gas bubb e nto the qu d s ug reg on. Th s add t ona contr but on of the expans on osses was shown to be s gn f cant, and ed to much better agreement between the measured and pred cted va ues. The ana ys s a so showed that for a 2.4-mm channe , the expans on osses cou d n fact be arger than the fr ct ona osses n the s ug. The expans on osses were found to f rst ncrease w th ncreas ng G because the qu d s ug ength decreases, presumab y 292 Heat transfer and f u d f ow n m n channe s and m crochanne s ead ng to more expans on events; but further ncreases n G make the expans on osses ess s gn f cant because at very h gh G , the fr ct ona osses ncrease rap d y. M sh ma and H b k (1996) (whose work was d scussed n the prev ous sect ons on f ow reg mes and vo d fract on) measured fr ct ona pressure drops n a r–water f ows through 1–4-mm tubes. By compar ng the r resu ts w th the Lockhart–Mart ne (1949) corre at on, they not ced that the parameter C n Ch sho m s (1967) curve-f t (Eq. (6.39) above) to the mu t p er decreased w th a decrease n tube d ameter. S m ar trends were a so observed by Sugawara et a . (1967) for a r–water f ow n hor zonta 0.7–9.1-mm round tubes, Ungar and Cornwe (1992) for the f ow of ammon a n hor zonta 1.46–3.15-mm round tubes, M sh ma et a . (1993) for vert ca upward f ow of a r–water n rectangu ar 1.07 × 40-, 2.45 × 40-, and 5.00 × 40-mm channe s, Sadatom et a . (1982) for vert ca upward f ow of a r–water through rectangu ar (7–17) × 50- and 7 × 20.6-mm channe s, and by Mor yama et a . (1992a,b) for the f ow of R113-n trogen through hor zonta rectangu ar (0.007– 0.098) × 30-mm channe s. Inc ud ng the data from these nvest gators and the r own data, they deve oped the fo ow ng equat on for the parameter C: C = 21(1 − exp(−0.319Dh )) (6.58) where Dh s n mm. They state that th s equat on s va d for vert ca and hor zonta round tubes as we as rectangu ar ducts (a though the pred ct ons d d not agree we w th the ammon a vapor data of Ungar and Cornwe (1992)). It shou d be noted that severa nvest gators have shown that the homogeneous f ow mode s reasonab y successfu n pred ct ng pressure drop dur ng ad abat c f ow and bo ng n channe s w th a d ameter of a few mm. These nc ude the prev ous y ment oned work of Ungar and Cornwe (1992) on the f ow of ammon a n hor zonta 1.46–3.15-mm round tubes, and Kureta et a . (1998) on the bo ng of water n 2–6-mm channe s at atmospher c pressure over a w de range of mass f uxes. S m ar y, Tr p ett et a . (1999a) found that th s mode was ab e to pred ct some of the r data on ad abat c f ow of a r–water m xtures through 1.1- and 1.45-mm c rcu ar, and 1.09- and 1.49-mm sem -tr angu ar channe s. In part cu ar, the homogeneous mode pred cted the data n the bubb y and s ug f ow reg mes we at h gh ReL . The agreement n the s ug-annu ar and annu ar f ow reg ons was poor, as was the agreement w th the s ug f ow data at very ow ReL . A so, the Fr ede (1979) corre at on resu ted n arger dev at ons from the data, espec a y at ow ReL va ues. Th s same group (Ekberg et a ., 1999) a so nvest gated a r–water f ow n two narrow, concentr c annu w th nner and outer d ameters of 6.6 and 8.6 mm for one annu us, and 33.2 and 35.2 mm for the second annu us, respect ve y. The measured pressure drops d d not show part cu ar y good agreement w th any of the ava ab e mode s, and even the Fr ede corre at on, wh ch they state prov ded the best agreement, had dev at ons of over 50%. Tran et a . (2000) deve oped a corre at on for pressure drop dur ng bo ng of refr gerants n c rcu ar (2.46 and 2.92 mm) and rectangu ar (4.06 × 1.7 mm) channe s. The r tests d d not record oca pressure drops – the pressure drop was measured from the n et at nom na y zero qua ty to a vary ng ex t qua ty. For the tests w th arge ex t qua t es, therefore, the measured pressure drops cou d be the resu t of cons derab y vary ng oca pressure drops as the f ow reg mes change a ong the test sect on. A though they found that the trends n the data ( n terms of the effect of mass f ux, qua ty and saturat on pressure) were s m ar Chapter 6. Condensat on n m n channe s and m crochanne s 293 to those found n arge tubes (Ecke s et a ., 1994), the arge tube corre at ons (Ch sho m, 1967, 1973; Fr ede , 1979; Jung and Radermacher, 1989; Souza and P menta, 1995) typ ca y underpred cted the data, w th the d screpancy ncreas ng at arge qua t es and mass f uxes. The reason for the h gher pressure drop n sma tubes was thought to be the fact that coa esced bubb es n sma channe s are conf ned, e ongated, and s de over a th n qu d f m, whereas n arge tubes, the bubb es may grow and f ow unrestr cted through the tubes. In v ew of th s, they proposed a mod f ed vers on of the Ch sho m (1973) corre at on that accounted for the ro e of surface tens on through the conf nement number (def ned n a prev ous sect on of th s chapter) ntroduced by Cornwe and Kew (1993) as fo ows: 7 8 Pf = pfLO 1 + 4.3Y 2 − 1 Nconf x0.875 (1 − x)0.875 + x1.75 (6.59) where Y s the rat o of the gas- and qu d-on y pressure drops def ned n Eq. (6.42) above. Zhao and B (2001b) measured pressure drops for upward co-current a r–water f ow through equ atera tr angu ar channe s w th Dh = 0.866, 1.443, and 2.886 mm. They found that the vo d fract on cou d be s mp y represented by = 0.838, wh ch s very s m ar to the Armand (1946) corre at on and was used to compute acce erat on osses n two-phase f ow. The fr ct ona component was then corre ated us ng the Lockhart–Mart ne (1949) approach. For th s, they computed the requ red s ng e-phase fr ct on factor by rep ac ng the am nar and turbu ent fr ct on factors n the comprehens ve am nar-trans t on-turbu ent corre at on of Church (1977) w th the correspond ng express ons for tr angu ar channe s. Th s mod f cat on y e ded two-phase mu t p ers that were between the va ues pred cted by the Ch sho m (1967) express on us ng C = 5 and C = 20. Lee and Lee (2001) nvest gated pressure drop for a r–water f ow through 20-mm w de hor zonta rectangu ar channe s w th gaps of 0.4–4 mm. They reca ed that the or g na bas s for the Ch sho m (1967) corre at on of the two-phase mu t p er n the Lockhart– Mart ne (1949) corre at on cons sted of account ng for the pressure drop due to (a) gas phase, (b) qu d phase, and (c) nteract ons between the two phases. The parameter C n Ch sho m s corre at on represents th s nteract on term; wh ch n turn depends on the f ow reg mes of the two phases. Therefore, they proposed d fferent va ues for C, account ng for the gap s ze as we as the phase f ow rates. They reasoned that as the gap s ze decreases, the f ow tends more and more to p ug and s ug f ow, w th an ncreas ng effect of surface tens on due to the curved gas/ qu d nterface at the edge of the bubb e. In such surfacetens on-dom nated f ows, they stated that of the severa d mens on ess parameters dent f ed by Suo and Gr ff th (1964) for cap ary tubes, the Reyno ds number of the qu d s ug, Re = (L Dh )/µL , the rat o of v scous and surface tens on effects, = µL /, and a comb nat on of parameters ndependent of the qu d s ug ve oc ty, = µ2L /(L Dh ), were the re evant ones. Lee and Lee then used the r data to obta n nd v dua va ues of the constant A and exponents q, r, and s for each comb nat on of qu d and gas f ow reg mes n the fo ow ng equat on for the parameter C n the two-phase mu t p er: C = Aq r ResLO (6.60) Regress on ana ys s showed that the dependence on parameters and above, that s, surface tens on effects, was on y s gn f cant when both phases were am nar, w th C be ng 294 Heat transfer and f u d f ow n m n channe s and m crochanne s s mp y a funct on of ReLO when e ther or both phases were n the turbu ent reg me. Th s approach pred cted the r data w th n ±10%, and a so pred cted the data of Wambsganss et a . (1992) and M sh ma et a . (1993) w th n 15% and 20%, respect ve y. Th s corre at on s va d for 175 < ReLO < 17700 and 0.303 < X < 79.4. 6.4.4. Condensat on stud es (~0.4 < Dh < 5 mm) Yang and Webb (1996b) measured pressure drops n s ng e- and ad abat c twophase f ows of refr gerant R-12 n rectangu ar p a n and m crof n tubes w th Dh = 2.64 and 1.56 mm, respect ve y. The measurements were conducted at somewhat h gher mass f uxes (400 < G < 1400 kg/m2 -s) than those of nterest for refr gerat on and a r-cond t on ng app cat ons. The r s ng e-phase fr ct on factors were corre ated by fsmooth = 0.0676Re−0.22 Dh and fm crof n = 0.0814Re−0.22 . The pressure grad ent n the m crof n tube was h gher than Dh that of the p a n tube. They tr ed to mode the two-phase pressure drops us ng the Lockhart– Mart ne (1949) approach and curve-f tt ng the two-phase mu t p er v2 , but th s d d not ead to a sat sfactory corre at on. The equ va ent mass ve oc ty concept of Akers et a . (1959) was then attempted to deve op a corre at on. Thus, the s ng e-phase fr ct on factor s f rst ca cu ated us ng the aforement oned equat ons at the Reyno ds number ReDh for the qu d phase f ow ng a one. The equ va ent Reyno ds number for two-phase f ow s then ca cu ated as fo ows: 1/2 Geq Dh Reeq = (6.61) ; Geq = G (1 − x) + x µ v The equ va ent fr ct on factor s therefore based on an equ va ent a - qu d f ow that y e ds the same fr ct ona pressure drop as the two-phase f ow. Based on the r data, the two-phase fr ct on factor at th s equ va ent Reyno ds number was f t to the fo ow ng equat on: f 0.12 = 0.435Reeq f (6.62) F na y, the two-phase pressure drop s ca cu ated as fo ows: Pf = f Re2eq µ2 4L 2 Dh3 (6.63) Th s approach was ab e to pred ct both p a n and m crof n tube data w th n ±20%. It shou d be noted that th s approach co apses both the s ng e- and two-phase pressure grad ents for each tube on the same ne, when p otted aga nst the Reeq . From the r data, Yang and Webb (1996b) a so conc uded that surface tens on d d not p ay a ro e n determ n ng the pressure drop n these tubes, a though n re ated work (Yang and Webb, 1996a), they found surface tens on to be s gn f cant for the heat transfer coeff c ent, espec a y at ow G and h gh x. Subsequent y, Yan and L n (1999) used the same Akers et a . (1959) equ va ent mass ve oc ty concept that was used by Yang and Webb (1996b) to corre ate condensat on Chapter 6. Condensat on n m n channe s and m crochanne s 295 pressure drop of R-134a n a 2-mm c rcu ar tube. They arranged 28 such copper tubes n para e he d between copper b ocks, to wh ch were so dered s m ar coo ng channe s to form a crossf ow heat exchanger that served as the test sect on. The resu t ng fr ct on factor corre at on s as fo ows: f = 498.3Re−1.074 eq (6.64) It shou d be noted that th s s cons derab y d fferent from the resu ts of Yang and Webb c ted above; however, Yan and L n s s ng e-phase fr ct on factors were s gn f cant y h gher than the pred ct ons of the B as us equat on. They attr buted th s to the nf uence of entrance engths and tube roughness, but th s d screpancy may exp a n the substant a d fferences between the two stud es. Zhang and Webb (2001) measured ad abat c twophase pressure drops for R-134a, R-22, and R-404A f ow ng through 3.25- and 6.25-mm c rcu ar tubes and through a mu t -port extruded a um num tube w th a hydrau c d ameter of 2.13 mm. They noted that the Fr ede (1979) corre at on, a though recommended w de y for arger tubes, was deve oped from a database w th D > 4 mm, and therefore d d not pred ct the r data we , espec a y at the h gher reduced pressures (Tsat = 65 C). S nce the dependence of the Fr ede corre at on on Weber number and Froude number was weak, they dec ded not to nc ude them as var ab es wh e corre at ng the r data. A so, based on the work of Wadekar (1990) for convect ve bo ng, they postu ated that the pressure drop wou d be a funct on of the reduced pressure rather than the dens ty and v scos ty rat os. Us ng these cons derat ons, and add t ona data from a prev ous study on mu t -port tubes w th round 1.45- and 0.96-mm ports, and 1.33-mm rectangu ar ports, they deve oped the fo ow ng mod f ed vers on of the Fr ede corre at on: 2 LO = (1 − x)2 + 2.87x2 ( p/pcr t )−1 + 1.68x0.8 (1 − x)0.25 (p/pcr t )−1.64 (6.65) Gar me a et a . deve oped exper menta y va dated mode s for pressure drops dur ng condensat on of refr gerant R-134a n nterm ttent f ow through c rcu ar (Gar me a et a ., 2002) and non-c rcu ar (Gar me a et a ., 2003b) m crochanne s (F g. 6.21) w th 0.4 < Dh < 4.9 mm. In add t on, they deve oped a mode for condensat on pressure drop n annu ar f ow (Gar me a et a ., 2003a), and further extended t to a comprehens ve mu t -reg me pressure drop mode (Gar me a et a ., 2005) for m crochanne s for the mass f ux range 150 < G < 750 kg/m2 -s. For Dh < 3.05 mm, they used f at tubes w th mu t p e extruded para e channe s to ensure accurate y measurab e f ow rates and heat ba ances. Three such tubes were brazed together (F g. 6.21), w th refr gerant f ow ng through the center tube, and coo ant (a r) f ow ng n counterf ow through the top and bottom tubes. The ow therma capac ty and heat transfer coeff c ents of a r ma nta ned sma changes n qua ty n the test sect on, wh ch n turn enab ed the measurement of the pressure drop var at on as a funct on of qua ty w th h gh reso ut on. The measurement techn ques were f rst ver f ed by conduct ng s ng e-phase pressure drop measurements for each tube n the am nar and turbu ent reg mes for both the superheated vapor and subcoo ed qu d cases. The s ng e-phase pressure drops were n exce ent agreement w th the va ues pred cted by the Church (1977) corre at on. 296 Heat transfer and f u d f ow n m n channe s and m crochanne s C193: Dh 4.91 mm C120: Dh 3.05 mm C60: 10 channe s, Dh 1.52 mm C30: 17 channe s, Dh 0.761 mm C20: 23 channe s, Dh 0.506 mm C rcu ar channe s S30: 17 channe s, Dh 0.762 mm B32: 14 channe s, Dh 0.799 mm T33: 19 channe s, Dh 0.839 mm RK15: 20 channe s, Dh 0.424 mm W29: 19 channe s, Dh 0.732 mm N21: 19 channe s, Dh 0.536 mm Non-c rcu ar channe s Pout Refr gerant out et A r n et Pn A r out et Test sect on schemat c Refr gerant n et F g. 6.21. Refr gerant condensat on geometr es of Gar me a. From Gar me a, S., Agarwa , A. and K on, J. D., Condensat on Pressure Drop n C rcu ar M crochanne s, Heat transfer Eng neer ng, 26(3), pp. 1–8 (2005). The measured pressure drops nc uded expans on and contract on osses due to the headers at both ends of the test sect on, and the pressure change due to dece erat on caused by the chang ng vapor fract on as condensat on takes p ace, and was represented as fo ows: Pmeasured = Pfr ct + Pexp+contr + Pdece (6.66) Chapter 6. Condensat on n m n channe s and m crochanne s 297 The pressure drop due to contract on was est mated us ng a homogeneous f ow mode recommended by Hew tt et a . (1994): 2 1 L G2 2 Pcontr = 1+x −1 +1− −1 (6.67) Cc G 2L where s the area rat o (Atest−sect on /Aheader ) and Cc s the coeff c ent of contract on, wh ch s n turn a funct on of th s area rat o as g ven by Ch sho m (1983): Cc = 1 0.639(1 − )0.5 + 1 (6.68) For the expans on nto the header from the test sect on, the fo ow ng separated f ow mode recommended by Hew tt et a . (1994) was used: Pexpans on = G 2 (1 − ) S L (6.69) where S , the separated f ow mu t p er, s a so a funct on of the phase dens t es and the qua ty. These est mates were va dated us ng pressure drop measurements on a “nearzero” ength test sect on, n wh ch the measured pressure drops (w th sma fr ct ona pressure drops) showed exce ent agreement w th the contract on/expans on contr but ons ca cu ated as shown above. Contract on and expans on oss contr but ons were ess than 5% of the tota measured pressure drop for a the r data. The pressure change due to acce erat on (dece erat on) of the f u d (due to the change n qua ty across the test sect on) was est mated as fo ows (Carey, 1992): Pacce erat on G 2 x2 G 2 (1 − x)2 = + V L (1 − )

x=xout G 2 x2 G 2 (1 − x)2 − + V L (1 − ) (6.70) x=x n where the vo d fract on was eva uated us ng the Baroczy (1965) corre at on. For a most a the r data, the dece erat on term was extreme y sma compared to the overa pressure drop. These est mates were a so corroborated by ad abat c f ow and condensat on tests at the same nom na cond t ons. The res dua fr ct ona component of the two-phase pressure drop, wh ch genera y was at east an order of magn tude arger than these m nor osses, was used for deve op ng condensat on pressure drop mode s for the respect ve f ow reg mes. To deve op f ow reg me-based pressure drop mode s, trans t on from the nterm ttent to the other f ow reg mes (based on the r ear er f ow v sua zat on work) was determ ned as fo ows: x≤ a G+b (6.71) where a and b are geometry-dependent constants g ven by: a = 69.57 + 22.60 × exp(0.259 × Dh ) (6.72) 298 Heat transfer and f u d f ow n m n channe s and m crochanne s b = −59.99 + 176.8 × exp(0.383 × Dh ) (6.73) where Dh s n mm. On a mass-f ux versus qua ty map, these trans t on nes appear n the ower eft corner, as shown n F g. 6.22. For pressure drop mode deve opment, they broad y categor zed the f ow nto the pr mary reg mes, nterm ttent ( ower eft) and annu ar f m/ m st/d sperse patterns of the annu ar f ow reg me (upper r ght). The very few data that were n the wavy reg me n these sma channe s were ncorporated nto the annu ar or nterm ttent reg mes, as appropr ate. The over ap reg on n F g. 6.22 accounts for the gradua trans t on between the pr mary reg mes, dur ng wh ch the f ow sw tches back and forth between the respect ve reg mes. Pressure drop mode s for these reg mes are descr bed be ow. The un t ce for the deve opment of the nterm ttent mode s shown n F g. 6.23, n wh ch the vapor-phase trave s as ong so tary bubb es surrounded by an annu ar qu d f m 400 M st/annu ar/d sperse f ow mode ap e n zo 200 Interm ttent f ow mode 600 0.5 mm 0.76 mm 1.5 mm 3.0 mm 4.91 mm er Ov Mass f ux G, (kg/m2-s) 800 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Vapor qua ty (x) 0.9 1.0 F g. 6.22. F ow reg me ass gnment. From Gar me a, S., Agarwa , A. and K on, J. D., Condensat on Pressure Drop n C rcu ar M crochanne s, Heat transfer Eng neer ng, 26(3), pp. 1–8 (2005). F m–s ug trans t on reg on L qu d Vapor U f m Dh D bubb e U bubb e U s ug U f m Lbubb e Ls ug F g. 6.23. Un t ce for nterm ttent f ow. Repr nted from Gar me a, S., K on, J. D. and Co eman, J. W. An exper menta y va dated mode for twophase pressure drop n the nterm ttent f ow reg me for c rcu ar m crochanne s, Journa of F u ds Eng neer ng, 124(1), pp. 205–214 (2002) w th perm ss on from the Amer can Soc ety of Mechan ca Eng neers. Chapter 6. Condensat on n m n channe s and m crochanne s 299 and separated by qu d s ugs. Based on the recorded f ow patterns, t was assumed that the bubb e s cy ndr ca , and that there s no entra nment of vapor n the s ug, or qu d n the bubb e. Further, for any g ven cond t on, t was assumed that the ength/frequency/speed of bubb es/s ugs s constant, w th no bubb e coa escence, and a smooth bubb e/f m nterface. The tota pressure drop for th s f ow pattern nc udes contr but ons from: the qu d s ug, the vapor bubb e, and the f ow of qu d between the f m and s ug as fo ows: Ptota = Ps ug + Pf/b + Ptrans t ons (6.74) A s mp e contro vo ume ana ys s (Gar me a et a ., 2002; 2003b) s m ar to that performed by Suo and Gr ff th (1964) was used to ca cu ate the qu d s ug ve oc ty from the superf c a phase ve oc t es, n turn ca cu ated from the overa mass f ux and the qua ty. Thus, Us ug = V + L , wh ch y e ded the s ug Reyno ds number Res ug = (L Us ug Dh )/µL . Th s Res ug was used to ca cu ate the pressure grad ent n the s ug from an appropr ate s ng ephase fr ct on factor corre at on from the terature. The resu ts of severa nvest gat ons (Suo and Gr ff th, 1964; Duk er and Hubbard, 1975; Fukano et a ., 1989) d scussed n th s chapter suggested that Ububb e = 1.2Us ug . Un ke a r–water stud es, where many of these nvest gators have neg ected the pressure drop n the f m–bubb e reg on, s nce the rat o of the qu d to vapor dens t es for R-134a s much ower (approx mate y 16:1 versus 850:1 for a r–water) as s the rat o of qu d and vapor v scos t es (approx mate y 10:1 versus 50:1), the contr but on of the f m/bubb e reg on was not neg ected. Based on the data, the f m was assumed to be am nar, wh e the gas bubb e was turbu ent. The respect ve Reyno ds numbers were def ned as fo ows: Ref m = L Uf m (Dh − Dbubb e ) µL Rebubb e = V (Ububb e − U nterface ) Dbubb e µV (6.75) The f m f ow was assumed to be dr ven by the comb nat on of the pressure grad ent n the f m/bubb e reg on and shear at the f m/bubb e nterface. The ve oc ty prof e for comb ned Couette–Po seu e f ow through an annu us where the nner surface moves at the nterface ve oc ty, U nterface , s represented by the superpos t on of the pressure-dr ven and the shear-dr ven components: − dP n(Rtube /r) dx f/b uf m (r) = R2tube − r 2 − R2tube − R2bubb e 4µL n(Rtube /Rbubb e ) + U nterface n(Rtube /r) n(Rtube /Rbubb e ) The bubb e ve oc ty, assum ng a power aw prof e (n = 7) s g ven by: 1/n (n + 1)(2n + 1) r (U ) ububb e (r) = − U +U nterface 1 − bubb e nterface 2n2 Rbubb e (6.76) (6.77) A shear ba ance at the f m–bubb e nterface y e ds the nterface ve oc ty: U nterface = −(dP/dx)f/b 2 (Rtube − R2bubb e ) 4µL (6.78) 300 Heat transfer and f u d f ow n m n channe s and m crochanne s W th a parabo c ve oc ty prof e n the f m, the average f m ve oc ty can a so be ca cu ated. A so, w th the f m pressure grad ent ca cu ated us ng a s ng e-phase fr ct on factor for the gas phase, cont nu ty y e ds:

Rbubb e 2 Rbubb e 2 Us ug = Ububb e (6.79) + Uf m 1 − Rtube Rtube These ca cu at ons y e d 0.899 < Rbubb e /Rtube < 0.911; that s, the bubb e d ameter s ~90% of the tube d ameter. W th the pressure drop/ ength known n the s ug and the f m/bubb e reg ons, the fr ct ona pressure drop was ca cu ated us ng the re at onsh p proposed by Fukano et a . (1989) for the ength of the s ug (Eq. (6.56)): Ls ug dP Pfr ct on on y = Ltube 1− dx f/b Ls ug + Lbubb e

Ls ug dP + (6.80) dx s ug Ls ug + Lbubb e The rema n ng contr but on to the tota pressure drop s the oss assoc ated w th the f ow of qu d between the f m and the s ug. As the f m moves s ow y, the front of the qu d s ug s constant y tak ng up f u d from the f m. A pressure oss s assoc ated w th the acce erat on and subsequent m x ng of th s qu d, w th the tota pressure oss from these trans t ons g ven by Pf m/s ug trans t ons = NUC × Pone trans t on , where NUC s the number of ce s per un t ength. The pressure drop for one trans t on s based on the force to acce erate the port on of the qu d f m overtaken by the rear of the bubb e from the average f m ve oc ty to the average s ug ve oc ty (Duk er and Hubbard, 1975): Rbubb e 2 Pone trans t on = L 1 − (Us ug − Uf m )(Ububb e − Uf m ) (6.81) Rtube These components of the tota pressure drop are shown be ow:

Ls ug P Lbubb e dP NUC dP one + + P trans t on = L dx f m LUC dx s ug LUC L (6.82) bubb e A dep ct on of these var ous contr but ons to the measured pressure drop for c rcu ar channe s s shown n F g. 6.24. The number of un t ce s per un t ength requ red for c osure of the mode was computed from the s ug frequency , wh ch was corre ated based on s ug Re and Dh :

b NUC Dh Dh a Res ug = = (6.83) = Dh Ububb e Ltube LUC The coeff c ents a and b were f t us ng the d fference between the measured pressure drop and the pressure drop ca cu ated as descr bed above for the s ug and bubb e/f m reg ons; that s, the net pressure drop due to trans t ons. The corre at on y e ded a = 2.437, b = −0.560 for both c rcu ar and non-c rcu ar (except tr angu ar, for wh ch d fferent Chapter 6. Condensat on n m n channe s and m crochanne s 301 35 4.91 mm 3.05 mm 15 1.52 mm 20 0.76 mm ΔP (kPa) 25 0.51 mm 30 ΔPs ug ΔPbubb e ΔP trans t ons *Note: ΔPs ug and ΔPbubb e are pred cted va ues whereas the tota ΔP s measured 10 5 0 Increas ng mass f ux and qua ty F g. 6.24. Contr but on of each pressure drop mechan sm to tota pressure drop. Repr nted from Gar me a, S., K on, J. D. and Co eman, J. W. An exper menta y va dated mode for two-phase pressure drop n the nterm ttent f ow reg me for c rcu ar m crochanne s, Journa of F u ds Eng neer ng, 124(1), pp. 205–214 (2002) w th perm ss on from the Amer can Soc ety of Mechan ca Eng neers. coeff c ents were necessary, as d scussed n Gar me a et a . (2003b)) channe s. For c rcu ar channe s (0.5–4.91 mm), the pred cted pressure drops were on average w th n ±13.5% of the measured va ues. The nterm ttent mode was a so extended to the d screte wave f ow reg on by Gar me a et a . (2005). Th s s because as the progress on from the nterm ttent reg on to the d screte wave reg on occurs, the gas bubb es start d sappear ng, to be rep aced by strat f ed, we def ned qu d and vapor ayers. W th n the d screte wave reg on, travers ng from the nterm ttent f ow boundary toward the annu ar f ow boundary, the bubb es d sappear comp ete y, w th the number of un t ce s per un t ength approach ng zero. Based on th s conceptua zat on of the nterm ttent and d screte f ow reg ons, the s ug frequency mode deve oped by Gar me a et a . (2002; 2003b) for nterm ttent f ow was mod f ed to nc ude data from the d screte wave f ow reg on, resu t ng n the fo ow ng comb ned mode for the two reg ons:

−0.507 Dh Dh NUC = = 1.573 Res ug (6.84) Ltube LUC Gar me a et a . (2003a; 2005) a so deve oped pressure drop mode s for annu ar f ow condensat on n m crochanne s, assum ng equa pressure grad ents n the qu d and gas core at any cross-sect on, un form f m th ckness, and no entra nment of the qu d n the gas core. The measured pressure drops were used to compute the Darcy form of the nterfac a fr ct on factor to represent the nterfac a shear stress as fo ows: 1 1 P = × f g Vg2 × L 2 D (6.85) 302 Heat transfer and f u d f ow n m n channe s and m crochanne s The above equat on uses the nterface d ameter, D . Th s same express on can be represented n terms of the more conven ent tube d ameter, D, by us ng the Baroczy (1965) vo d fract on mode : P 1 G 2 × x2 1 = × f × 2.5 L 2 g × D (6.86) The rat o of th s nterfac a fr ct on factor (obta ned from the exper menta data) to the correspond ng qu d-phase Darcy fr ct on factor was then computed and corre ated n terms of the Mart ne parameter and the surface tens on parameter = L µL / ntroduced by Lee and Lee (2001): f = A × X a Reb c f (6.87) where L = G(1−x) s the qu d superf c a ve oc ty. The phase Reyno ds numbers (1−) requ red above and for the Mart ne parameter were def ned n terms of the f ow areas of each phase: GD (1 − x) Re = √ 1 + µ Reg = GxD √ µg (6.88) The fr ct on factors were computed us ng f = 64/Re for Re < 2100 and the B as us express on f = 0.316 × Re−0.25 for Re > 3400. Regress on ana ys s on data grouped nto two reg ons based on Re y e ded: Lam nar reg on (Re < 2100): A = 1.308 × 10−3 a = 0.427 b = 0.930 c = −0.121 (6.89) Turbu ent reg on (Re > 3400): A = 25.64 a = 0.532 b = −0.327 c = 0.021 (6.90) Interpo at ons based on G and x are used to compute the pressure drop for the trans t on reg on. Th s mode pred cted 87% of the data w th n ±20%. These pred ct ons nc ude not on y the annu ar f ow reg on, but a so the m st and d sperse-wave f ow data, a though the mode deve opment uses a phys ca representat on where the qu d forms an annu ar f m around a gas core. Gar me a et a . (2005) exp a ned th s based on the fact that the annu ar f ow reg me ncreases n extent for sma er tubes, caus ng severa data c ass f ed nto the wavy and m st f ow reg ons us ng Dh = 1 mm trans t on cr ter a to be n fact n the annu ar f ow reg me. Representat ve pressure drop trends pred cted by th s mode are shown n F g. 6.25. 6.4.5. A r–water stud es (Dh < ~500 µm) Kawahara et a . (2002) measured pressure drops for a r–water f ow through a 100-µm c rcu ar tube (the correspond ng resu ts on f ow patterns and vo d fract ons were d scussed Chapter 6. Condensat on n m n channe s and m crochanne s 100 0.506 mm 0.761 mm 1.52 mm 3.05 mm 4.93 mm G 750 kg/m2-s 90 80 70 60 50 40 303 Interm ttent f ow mode D sperse/annu ar/m st f ow mode 30 20 Over ap zone 10 0 100 G 600 kg/m2-s 90 P (kPa) 80 70 60 Interm ttent f ow mode 50 40 Over ap zone 30 D sperse/annu ar/m st f ow mode 20 10 0 100 90 G 450 kg/m2-s 80 70 60 50 40 30 Interm ttent f ow mode D sperse/annu ar/m st f ow mode Over ap zone 20 10 0 0.0 0.2 0.4 0.6 Qua ty (x) 0.8 1.0 F g. 6.25. Pred ct ons of Gar me a et a . (2005) mu t p e-f ow-reg me mode for refr gerant condensat on pressure drops n m crochanne s. From Gar me a, S., Agarwa , A., and K on, J. D. Condensat on pressure drop n c rcu ar m crochanne s, Heat Transfer Eng neer ng, 26(3), pp. 1–8 (2005). n prev ous sect ons). Test sect on entrance (contract on) osses, and acce erat on osses due to the reduct on n dens ty across the test sect on were removed from the measured pressure drop n a manner s m ar to that d scussed n connect on w th the work of Gar me a et a . (2005) above. The vo d fract on requ red for the acce erat on osses was eva uated us ng 304 Heat transfer and f u d f ow n m n channe s and m crochanne s Eq. (6.35) deve oped from the r mage ana ys s work. The contr but ons of these contract on and acce erat on osses to the tota pressure drop were found to be 0.05–9% and 0–4.5%, respect ve y, of the measured pressure drop. A though the r f ow pattern and vo d fract on resu ts showed that the f ow was not we represented by a homogeneous mode , based on the success of others n us ng th s mode for sma channe s (Ungar and Cornwe , 1992; Tr p ett et a ., 1999a), they compared the r resu ts w th homogeneous mode us ng the homogeneous dens ty and homogeneous v scos ty mode s of Owens (1961), McAdams (1954), C cch tt et a . (1960), Duk er et a . (1964), Beatt e and Wha ey (1982), and L n et a . (1991). They found that none of these mode s were ab e to pred ct the r data adequate y (the Duk er et a . (1964) mode , µH = µG + (1 − )µL , prov ded somewhat reasonab e pred ct ons). When a Lockhart–Mart ne (1949) type two-phase mu t p er approach was used, the Ch sho m (1967) coeff c ent of C = 5 s gn f cant y overpred cted the data, wh e a C va ue of 0.66 from M sh ma and H b k s (1996) work resu ted n a 10% overpred ct on, and agreement w th n ± 10% was ach eved us ng Lee and Lee s (2001) corre at on for C. Therefore, they conducted a regress on ana ys s of the data to obta n a va ue of C = 0.24, wh ch a so y e ded a ± 10% agreement. In ater work on a 96 µm square channe (Chung et a ., 2004), they noted that channe shape d d not have any apprec ab e effect on twophase pressure drop, and prov ded rev sed va ues of C = 0.22 for c rcu ar channe s and C = 0.12 for square channe s for the C coeff c ent n Ch sho m s (1967) mode . Chung and Kawa (2004) a so nvest gated the effects of channe d ameter on pressure drop dur ng ad abat c f ow of n trogen–water through 530-, 250-, 100-, and 50 µm c rcu ar channe s. In th s study, they found that the Duk er et a . (1964) v scos ty mode (when used assum ng homogeneous f ow) d d not pred ct the pressure drops for the 530- and 250-µm channe s, a though the agreement was w th n ±20% for the 100- and 50-µm data. Beatt e and Wha ey s (1982) m xture v scos ty mode showed the oppos te trends, succeed ng somewhat n pred ct ng the 530- and 250-µm channe data, but s gn f cant y overpred ct ng the 100- and 50-µm channe data. They attr buted th s to the fact that Beatt e and Wha ey s (1982) mode y e ds a h gher m xture v scos ty than that of Duk er et a . (1964), resu t ng n better agreement w th the arger channe data because of the add t ona m x ng osses n the bubb y and churn f ows n these channe s. When predom nated by the more am nar s ug f ow w th weak momentum coup ng of the phases, however, as s the case n the sma er channe s, the ower v scos t es pred cted by Duk er et a . (1964) mode are more appropr ate. When attempt ng the use of a two-phase mu t p er approach as d scussed above, they noted a mass f ux effect on the fr ct on mu t p er for the 530- and 250-µm channe s. The data at d fferent mass f uxes for the 100- and 50-µm channe s, however, were pred cted we by a s ng e C va ue (C = 0.22 for 100 µm, C = 0.15 for 50 µm). Th s decrease n C va ue (based on the above d scuss on of Lee and Lee s (2001) work) w th a decrease n channe d ameter mp es a decrease n momentum coup ng of the two phases; that s, comp ete y separated am nar f ow of gas and qu d, as a so noted byA et a . (1993). Based on the somewhat nadequate pred ct ons of the homogeneous and two-phase mu t p er approaches, they adapted the nterm ttent f ow mode of Gar me a et a . (2002) to ca cu ate the pressure drops n the 50- and 100-µm channe s, wh ch have predom nant y nterm ttent f ows. They a so used the un t-ce concept cons st ng of gas bubb es surrounded by a qu d f m and f ow ng through s ower mov ng qu d s ugs. A though n accordance w th the mode of Gar me a et a . (2002), they represented the tota pressure Chapter 6. Condensat on n m n channe s and m crochanne s 305 drop as the summat on of the fr ct ona pressure drop n each of these reg ons, they gnored the m x ng osses at the trans t ons between these reg ons. The rat ona e used was that n these sma er channe s, the qu d f ow s typ ca y am nar, w th suppressed m x ng ead ng to neg g b y sma trans t on osses. For the s ng e-phase qu d reg on, they computed the average qu d ve oc ty UL = L /(1 − ), wh ch y e ded ReL = (L UL D)/µL . Lam nar or turbu ent s ng e-phase fr ct on factors were ca cu ated based on ReL . To compute the pressure drop n the f m/bubb e reg on, they assumed the average ve oc ty of the gas phase and the gas bubb e to be equa , UG = UB , where UG = G /. Th s assumpt on s d fferent from the assumpt on by Gar me a et a . (2002) that UB = 1.2US , wh ch, accord ng to Chung and Kawa (2004), s equ va ent to assum ng that the Armand (1946) corre at on for vo d fract on approx mate y ho ds. Instead, they used the non near express on for vo d fract on der ved by them and d scussed here n the prev ous sect on. Def n ng the bubb e ReB n terms of the re at ve ve oc ty between the bubb e and the nterface, the computat on of the pressure drop n the f m–bubb e reg on deve ops much ke that n Gar me a et a . (2002), w th an assumpt on that DB = 0.9D. The nterface ve oc ty, the f m–bubb e pressure grad ent, and the bubb e Reyno ds number are then obta ned terat ve y. To ca cu ate the re at ve engths of the bubb e and s ug reg ons, they re ed on the vo d fract on measured by them. Thus, treat ng the bubb e as a cy nder w th f at ends, the bubb e ength s g ven by: D 2 LB = LUC DB (6.91) F na y, the tota pressure drop n the two reg ons s g ven by: dPf dz

= TP dP dz F/B LB + LUC

dP dz L LL LUC (6.92) The mode was shown to agree better w th the 50- and 100-µm channe pressure drop data than the homogeneous and two-phase mu t p er approaches. A so, the pred ct ons of the 50-µm channe data were better than those of the 100-µm data, presumab y because the sma er channe exc us ve y exh b ts s ug f ow, wh e the arger channe may have some reg ons of other k nds of f ow. Th s a so po nts out that th s mode shou d not be app ed to arger (Dh > 100 µm) channe s where the f ow s not exc us ve y nterm ttent. 6.4.6. Summary observat ons and recommendat ons The ava ab e nformat on on pressure drop n two-phase f ows was d scussed above. One overa observat on s that most of the stud es cont nue to re y on the c ass ca works of Lockhart and Mart ne (1949), Ch sho m (1967), and Fr ede (1979) for the eva uat on of separated f ow pressure drops. As n the case of f ow reg me and vo d fract on eva uat on, most of the work n the sma channe s has been on ad abat c f ows of a r–water m xtures. It appears that the or g na two-phase mu t p ers proposed n these stud es have been mod f ed by the atter nvest gators (spec f ca y the C coeff c ent n the Ch sho m (1967) mode of 306 Heat transfer and f u d f ow n m n channe s and m crochanne s the Lockhart–Mart ne (1949) mu t p er) to match the data on the sma er channe s. Th s nd cates that of the qu d-phase, gas-phase, and gas– qu d nteract on parameters, the gas– qu d nteract on parameter has requ red the most mod f cat on to make t su tab e for sma channe s. Examp es nc ude the work of M sh ma and H b k (1996) and Lee and Lee (2001). As the channe approaches the sma er d mens ons, Kawahara et a . (2002), Chung and Kawa (2004), and others have found that the pressure drop pred ct ons n the 530- and 250-µm channe s and those for 100- and 50-µm channe s requ red d fferent m xture v scos ty mode s to make the homogeneous f ow mode s work. The standard twophase f ow mu t p er approach w th much sma er va ues of the C coeff c ent a so y e ded some success. However, t seems that for the p ug/s ug f ows n the sma channe s, an approach that recogn zes the nterm ttent nature of the f ow, and accounts for the pressure drops n the qu d s ug, the gas bubb e, and n some cases, the m x ng osses between the bubb es and s ugs, as proposed by Gar me a et a . (2002; 2003b) and ater adapted by Chung and Kawa (2004) s the best approach. Account ng for the range of f ow reg mes encountered n m crochanne s; that s, nterm ttent, annu ar, and to a certa n extent, m st f ow, has ust recent y been attempted. One mu t -reg on mode that accounts for the f ow mechan sms n each of these reg mes and resu ts n smooth trans t ons between these reg mes for condens ng f ows n the 0.4 < Dh < 5 mm range s the mode by Gar me a et a . (2005). The var ous techn ques for pred ct ng pressure drops n m crochanne s are ustrated n Examp e 6.3. 6.5. Heat transfer coeff c ents In the above sect ons, the ava ab e nformat on on f ow reg mes, vo d fract ons, and pressure drop n two-phase f ow was presented, nc ud ng ad abat c f ow of a r–water m xtures and condens ng f ows. Th s s because the arge body of work on ad abat c two-phase f ows can be used as a start ng po nt to understand the correspond ng phenomena dur ng phase change. However, for the eva uat on of heat transfer coeff c ents, on y the terature on condens ng f ows s re evant. The measurement and understand ng of heat transfer n the rea y sma channe s (Dh < 1 mm) has proved to be part cu ar y cha eng ng n v ew of the conf ned spaces for measurement probes, the sma f ow rates n m crochanne s that correspond to sma heat transfer rates that are therefore d ff cu t to measure accurate y, and the arge heat transfer coeff c ents that ead to very sma temperature d fferences, wh ch are a so d ff cu t to measure. There wou d be tt e work to report, ndeed, f the scope s restr cted to channe s w th Dh < 1 mm (not much nformat on s ava ab e even for channe s w th Dh < ~7 mm.) Therefore, n th s sect on, an overv ew of the c ass ca mode s of condensat on s presented f rst to serve as a background for the work on the re at ve y sma er channe s. Grav ty-dr ven, shear-dr ven (wh ch are the two dea zed modes that have been stud ed the most) and mu t -reg me mode s are addressed, fo owed by a presentat on of the m ted nformat on on channe s n the 400 µm to 3 mm range, that have oose y been referred to as m crochanne s by the a r-cond t on ng and refr gerat on ndustry. The focus here s on condensat on ns de hor zonta tubes (Tab e 6.4). Tab e 6.4 Summary of heat transfer stud es. Invest gator Hydrau c d ameter (mm) F u ds Or entat on/ cond t on Range/app cab ty Convent ona channe mode s and corre at ons: grav ty-dr ven condensat on Chato (1962) 27.94 mm A r–water Hor zonta , ReV < 35,000 14.53 mm R113 nc ned Techn ques, bas s, observat ons • Ana yzed strat f ed-f ow condensat on ns de hor zonta and nc ned tubes under ow vapor shear cond t ons • Heat transfer n the qu d poo neg ected Rosson and Myers (1965) 9.53 mm Methano , acetone Hor zonta Atmospher c pressure 2000 < ReSV < 40,000 60 < ReSL < 1500 • Measured c rcumferent a var at on of h dur ng strat f ed condensat on • h ~ constant at top of tube ( < 10 ), then decreased near y to po nt where t aga n became approx mate y constant Rufer and Kez os (1966) 9.75, 15.875 mm R-22, A r–water Hor zonta , nc ned T = 43.3 C for R-22 • Ana yzed strat f ed condensat on n hor zonta / nc ned tubes • Pred cted qu d eve s to f nd area for f m condensat on n the upper port on of tube, and therefore, overa effect ve h • L qu d poo depths and ax a var at on d fferent from open-channe hydrau cs resu ts of Chato (1962) • Numer ca mode for strat f ed condensat on n smooth hor zonta tube • Effects of nterfac a shear on condens ng f m and qu d poo , ax a P, Tsat , wa -tosaturat on T and var at on of qu d poo he ght taken nto account • Ana yt ca express on based on d mens ona ana ys s a so presented Chen and 10.8–16.6 mm Kocamustafaogu ar (1987) R-12, R-113, Hor zonta water, methano , acetone, propane S ngh et a . (1996) 11 mm R-134a Hor zonta T = 35 C 50 < G < 300 kg/m2 -s Guo and Anand (2000) Rectangu ar R-410A 12.7 × 25.4 mm Hor zonta ( ong s de vert ca ) • Ana yt ca mode for strat f ed condensat on T = 7.8–36.7 C 30 < G < 2200 kg/m2 -s • Area-we ghted average of the top, bottom, and vert ca wa h • Accounted for ax a vapor shear on f m condensat on • Strong mass f ux dependence for h n wavy–annu ar and annu ar f ow, ess n strat f ed wavy reg me • Trav ss et a . (1973), Shah (1979), Dobson et a . (1994), Chato (1962), and Akers et a . (1959) corre at ons underpred cted data • Accounted for forced convect on n qu d poo (Cont nued ) Tab e 6.4 (Cont nued ) Invest gator Hydrau c d ameter (mm) F u ds Or entat on/ cond t on Range/app cab ty Convent ona channe mode s and corre at ons: shear-dr ven condensat on Carpenter and 11.66 mm Steam, methano , Vert ca Co burn (1951) ethano , to uene, tr ch oroethy ene • Data for oca and average h • Th n condensate f m becomes turbu ent at Re ~ 240 • Cons dered vapor shear, grav ty, and momentum change due to condensat on act ng upon condensate ayer • Ma or therma res stance n am nar sub ayer • von Karman ve oc ty prof e for qu d f m to obta n oca h as funct on of f m th ckness and sum of forces on condensate • Condensat on d v ded nto: am nar condensate entrance reg on, turbu ent condensate, am nar vapor ex t (Nusse t ana ys s) Re > 5000 0.5 µv Rev > µ v Akers et a . (1959) 20,000 Hor zonta DGv 1000 < µ < 100,000 Techn ques, bas s, observat ons

v • Equ va ent a - qu d mass f ux to rep ace vapor core, and prov ded same h for condens ng annu ar f ow • Enab es s ng e-phase f ow treatment 0.5 • Three d fferent exper menta y va dated corre at ons for “sem strat f ed” f ow, am nar, and turbu ent annu ar f ow • Mu t p ers to s ng e-phase Nu based on data Akers and Rosson (1960) 15.88 mm Methano , R-12 So man et a . (1968) 7.44 < D < 11.66 mm Steam, R-113, Hor zonta ethano , methano , vert ca to uene, tr ch oro- downward ethy ene, R-22 1 < Pr < 10, 6 < Uv < 300 m/s, 0.03 < x < 0.99 Trav ss et a . (1973) 8-mm tube R-12 and R-22 • Heat–momentum ana ogy and von Karman ve oc ty prof e n qu d T = 25–58.3 C f m to corre ate Nu n annu ar f ow 161 < G < 1533 kg/m2 -s • Fr ct ona term from two-phase mu t p er • Three-reg on turbu ent f m mode Hor zonta • Mod f cat ons to Carpenter and Co burn (1951) shear stress • Wa shear stress comb nat on of fr ct on, momentum, and grav ty contr but ons • Wa shear stress used to deve op mode for h for annu ar f ow • Parametr c ana yses to h gh ght mportance of d fferent forces Shah (1979) 7 < D < 40 mm R-11, R-12, R-22, R-113, methano , ethano , benzene, to uene, tr ch oroethy ene Hor zonta , nc ned and vert ca tubes, annu us 11 < G < 211 kg/m2 -s 21 < T < 310 C 3 < Uv < 300 m/s 0.002 < Pr < 0.44 1 < Pr < 13 Re > 350 • W de y used genera purpose emp r ca corre at on • Large database from 21 sources • Bas s: condensat on h s m ar to evaporat ve h n absence of nuc eate bo ng • Two-phase mu t p er n terms of pr and x So man (1986) 7.4 < D < 12.7 mm Hor zonta vert ca 21 < T < 310 C 80 < G < 1610 kg/m2 -s 0.20 < x < 0.95 8 < We < 140 1 < Pr < 7.7 • Corre at on for annu ar–m st f ow • Heat–momentum ana ogy for s ng e-phase turbu ent f ow used as start ng po nt • Homogeneous f ow n thermodynam c equ br um • M xture v scos ty bas s Co-current annu ar f m n vert ca and hor zonta tube • F m condensat on corre at on based on ana yt ca and emp r ca approach to asymptot ca y comb ne va ues at m ts • Interfac a shear, nterface wav ness, and turbu ent transport • Inc uded effects of nterfac a shear, nterface wav ness, and turbu ent transport • Average and oca Nu from asymptot c m ts • At same Gv , h for R-22 h gher than for R-12 20 < T < 40 C 175 < G < 560 kg/m2 -s • h trends s ope change at Rev ~3 × 104 ; based on trans t on, proposed wavy, and annu ar/sem -annu ar f ow corre at ons Steam, R-113, and R-12 Chen et a . (1987) Tandon et a . (1995) 10 mm R-12, R-22 Hor zonta Ch tt and Anand (1995; 1996) 8 mm R-22, R-32/R-125 w th o Hor zonta Moser et a . (1998) 3.14 < D < 20 mm R-22, R-134a, R-410a, R-12, R-125, R-11 Hur burt and 3–10 mm Newe (1999) R-22, R-134a, R-11, R-12, R-22 • Ana yt ca mode for annu ar condensat on 32.2 < T < 51.7 C 148 < G < 450 kg/m2 -s • Used Prandt m x ng ength theory w th van Dr est s (1956) hypothes s ( aw of wa ) to obta n m , Prt = 0.9 • h for R-32/R-125 15% h gher than for R-22, effect of o m n ma • Akers et a . (1959) Geq mode cons stent y underpred cts data • Akers assumpt ons of constant and equa fv , f not correct, a so T of Akers et a . not correct 2 , boundary ayer ana ys s to obta n • New Reeq w th Fr ede (1979) o equ va ent Nu Hor zonta 0.2 < x < 0.9 200 < G < 650 kg/m2 s • Sca ng equat ons to trans ate between , P, h for d fferent refr gerants at d fferent cond t ons and geometr es (Cont nued ) Tab e 6.4 (Cont nued ) Invest gator Hydrau c d ameter (mm) F u ds Or entat on/ cond t on Range/app cab ty Convent ona channe mode s and corre at ons: mu t -reg me condensat on Jaster and 12.5 mm Steam Hor zonta 2 × 104 < P < 1.7 × 105 12.6 < G < 145 kg/m2 -s Kosky (1976) Breber et a . (1979; 1980) 4.8–50.8 mm R-11, R-12, R-113, steam, n-pentane N theanandan et a . (1990) 7.4–15.9 mm R-12, R-113, and steam Dobson et a . (1994) 4.57 mm R-12, R-134a Dobson and Chato (1998) 31.4, 4.6, and 7.04 mm R-12, R-134a, R-22, Near az otrop c b ends of R-32/ R-125 W son et a . (2003) D = 7.79, 6.37, R-134a and 4.40, and 1.84 R-410A Techn ques, bas s, observat ons • A gebra c express on for f m th ckness fac tates computat ons • Stress rat o to d st ngu sh between annu ar/strat f ed reg mes • Heat–momentum transfer ana ogy (annu ar f ow) w th two-phase mu t p er for shear stress • For strat f ed f ow, re ated qu d poo ang e to vo d fract on • L near nterpo at on for trans t on reg on 108.2 < P < 1250 kPa 17.6 < G < 990 kg/m2 -s • Forced-convect ve corre at on w th two-phase mu t p er for annu ar, nterm ttent and bubb y f ow • Nusse t-type equat on for strat f ed–wavy • Interpo at ons n trans t on zones Data from var ous sources • • • • Hor zonta 75 < G < 500 kg/m2 -s T = 35 C and 60 C • Annu ar f ow: h ncreased w th G, x due to ncreased shear and th nn ng of qu d f m – used two-phase mu t p er • Wavy f ow: h ndependent of G, s ght ncrease w th x – Chato (1962) w th ead ng constant changed to funct on of Xtt Hor zonta 25 < G < 800 kg/m2 -s T = 35–60 C • • • • Extended Dobson et a . (1994) – add t ona tubes, f u ds F u d propert es effect not very s gn f cant No s gn f cant effect of d ameter reduct on Wavy and annu ar corre at ons updated, forced convect on n qu d poo and f m condensat on at the top, vo d fract on for poo depth est mate • Frso and G used to separate reg mes Hor zonta T = 35 C 75 < G < 400 kg/m2 -s • Progress ve y f attened 8.91 mm round smooth/m crof n tubes • h ncreases w th f atten ng, h h ghest for he ca m crof ns • No pred ct ve mode s or corre at ons Hor zonta Eva uat on of corre at ons n terature M st f ow (We ≥ 40, Fr ≥ 7): So man (1986) Annu ar f ow (We < 40, Fr ≥ 7): Shah (1979) Wavy f ow (Fr < 7): Akers and Rosson (1960) Cava n et a . (2001; 2002a; 2002b) 8 mm R-22, R-134a, R-125, R-32, R-236ea, R-407C, and R-410A Hor zonta 30 < T < 50 C 100 < G < 750 kg/m2 -s • Breber et a . (1979; 1980) type approach for h recommendat ons • Own data and database from others used for corre at ons • Kosky and Staub (1971) for annu ar h w th mod f ed Fr ede (1979; 1980) for shear stress • Strat f ed: f m condensat on and forced convect on n poo • Interpo at ons for other reg mes, bubb y treated as annu ar • Mass f ux cr ter on for wavy–s ug trans t on Thome et a . (2003) 3.1 < D < 21.4 mm R-11, R-12, R-113, R-32/ R-125, propane, n-butane, so-butane, and propy ene Hor zonta 24 < G < 1022 kg/m2 -s 0.02 < pr < 0.8 0.03 < x < 0.97 • Curve-f tt ng from arge database • h data used to ref ne vo d fract on mode , wh ch s used to pred ct trans t ons, n turn used to pred ct tuned b end of strat f ed–convect ve heat transfer coeff c ents • Strat f ed poo spread out as part a annu ar f m • Inc udes effect of nterfac a waves on heat transfer • Smooth y vary ng h across reg mes through nterpo at ons • No new nterm ttent, bubb y, or m st mode s; treated as annu ar Goto et a . (2003) He ca , herr ngbone grooved tubes, 8.00 mm R-410A and R-22 Hor zonta 30 < T < 40 C 130 < G < 400 kg/m2 -s • h tests n comp ete vapor compress on system • h of herr ngbone grooved tube ~2 × he ca grooved tubes • Mod f ed Koyama and Yu s (1996) corre at on for strat f ed and annu ar condensat on, new express ons for annu ar f ow part Hor zonta T = 65 C 400 < G < 1400 kg/m2 -s • Shah (1979) s gn f cant y overpred cted data, Akers et a . (1959) better agreement, except at h gh G • Enhancement due to m crof ns decreased w th ncreas ng G • h showed heat f ux dependence • Surface tens on dra nage rat ona e for m crof n enhancement • Yang and Webb (1996b) conc uded surface tens on d d not p ay ro e n P n these same tubes Condensat on n sma channe s Yang and Rectangu ar R-12 Webb p a n (1996a) Dh = 2.64, m crof n tubes Dh = 1.56 mm (Cont nued ) Tab e 6.4 (Cont nued ) Invest gator Hydrau c d ameter (mm) F u ds Or entat on/ cond t on Range/app cab ty Techn ques, bas s, observat ons • G = 400 kg/m2 -s and x > 0.5, surface tens on contr but on equa s and exceeds vapor shear term • G = 1400 kg/m2 -s, surface tens on contr but on very sma • h mode based on shear and surface tens on dra nage (f ooded and unf ooded parts) • Sma f n t p rad us enhanced h, arge nter-f n dra nage area act vated surface tens on effect at ower x Yang and Webb (1997) Extruded m crochanne s (Dh = 1.41 and 1.56 mm) w th m crof ns R-12, R-134a K m et a . (2003) Dh = 1.41 smooth, 1.56 mm m crof n R-22 and R-410A Hor zonta T = 45 C 200 < G < 600 kg/m2 -s • In smooth tubes, R-410A h s ght y h gher than R-22, oppos te true for m crof n tubes • Many qua tat ve exp anat ons for vary ng enhancement trends • Recommend Moser et a . (1998) mode w th mod f ed two-phase mu t p er for smooth tubes, and Yang and Webb s (1997) mode w th m nor mod f cat ons for m crof n tubes Yan and L n (1999) 2 mm R-134a Hor zonta T = 40–50 C • Condensat on h h gher at ower Tsat , espec a y at h gher x • h decreased s gn f cant y as q ncreased, part cu ar y at h gh x; some unusua trends n data • Reeq based h corre at on Wang et a . (2002) Rectangu ar Dh = 1.46 mm (1.50 × 1.40 mm) R-134a Hor zonta T = 61.5–66 C 75 < G < 750 kg/m2 -s • Large var at ons n x across test sect on • Strat f cat on seen even at such sma Dh • Akers et a . (1959) Reeq agreed w th annu ar data, Jaster and Kosky (1976) agreed best w th strat f ed f ow data • Breber et a . (1980) and So man (1982; 1986) trans t on cr ter a • Curve-f ts for two-phase mu t p er, d mens on ess temperature • Strat f ed and annu ar f ow corre at ons w th nterpo at ons Koyama et a . (2003b) Mu t -port extruded a tubes R-134a Hor zonta T = 60 C 100 < G < 700 kg/m2 -s • Loca h measured every 75 mm of us ng heat f ux sensors • Comb nat on of convect ve and f m condensat on terms. Annu ar and strat f ed terms from Haraguch et a . (1994a, b), two-phase mu t p er rep aced by M sh ma and H b k (1996) Wang, Rose, and co-workers (Wang and Rose, 2004; Wang et a ., 2004) Tr angu ar (Wang and Rose, 2004) Dh = 0.577 mm Square (Wang et a ., 2004) R-134a Hor zonta T = 50 C 100 < G < 1300 kg/m2 -s • Wang and Rose (2004) numer ca ana yses to pred ct vary ng condensate f ow pattern across cross-sect on and ength • Wang et a . (2004) mode for f m condensat on of R-134a n square, hor zonta , 1-mm m crochanne s • Surface tens on, grav ty, and shear terms can be turned on or off to demonstrate nd v dua effects Cava n et a . (2005) Mu t p e para e 1.4-mm channe s R-134a R-410A T = 40 C 200 < G < 1000 kg/m2 -s • h from Twa measurements, ava ab e mode s (Akers et a ., 1959; Moser et a ., 1998; Zhang and Webb, 2001; Cava n et a ., 2002a; Wang et a ., 2002; Koyama et a ., 2003a) underest mate resu ts • M st f ow n exper ments m ght ead to h gh h Sh n and K m (2005) C rcu ar and square channe s 0.5 < Dh < 1 mm R-134a Hor zonta T = 40 C 100 < G < 600 kg/m2 -s • Techn que matched Tout of e ectr ca y heated a r stream w th s m ar a r stream heated by condens ng refr gerant R-134a • Measured sma , oca condensat on Q • Most ava ab e mode s and corre at ons (Akers et a ., 1959; So man et a ., 1968; Trav ss et a ., 1973; Cava n and Zecch n, 1974; Shah, 1979; Dobson, 1994; Moser et a ., 1998) underpred ct data at ow G • At ower G, square channe s had h gher h than c rcu ar channe s, reverse was true for h gh G Gar me a and Bandhauer (2001); Bandhauer et a . (2005) 0.4 < Dh < 4.9 mm R-134a Hor zonta 150 < G < 750 kg/m2 -s • Therma amp f cat on techn que for accurate h measurement • h mode for c rcu ar tubes based on Trav ss et a . (1973) boundary- ayer ana yses • Two-reg on d mens on ess f m temperature • Interfac a shear stress from mode s deve oped spec f ca y for m crochanne s • Addresses annu ar, m st, and d sperse wave reg mes 314 Heat transfer and f u d f ow n m n channe s and m crochanne s 6.5.1. Convent ona channe mode s and corre at ons 6.5.1.1. Grav ty-dr ven condensat on Grav ty-dr ven condensat on s expected to be ess mportant n m crochanne two-phase f ows than n convent ona channe s; however, var ous authors have assumed that condensat on occurs e ther n a grav ty-dr ven mode or a shear-dr ven mode, even n channe s as sma as ~3 mm. The re at ve preva ence of these modes s governed by f ow cond t ons and channe s ze, as d scussed at ength n the prev ous sect ons. A br ef overv ew of the treatment of grav ty-dr ven condensat on s prov ded here. Much of the ear y work on condensat on heat transfer s based upon the fa ng-f m condensat on ana ys s of Nusse t (1916), who assumed that a th n am nar condensate f m formed on an sotherma f at p ate w th a surface temperature ess than the saturat on temperature, surrounded by a stat onary vapor at the saturat on temperature. The condensate f ows under the nf uence of grav ty and the f m th ckens as t moves down the p ate. Nusse t further assumed that heat transfer s due to conduct on across the f m on y. In add t on, he assumed constant f m propert es, neg g b e f m subcoo ng, and a smooth qu d–vapor nterface. After conduct ng an energy ba ance around a f m e ement and ntegrat ng w th appropr ate re at onsh ps for the mass f ow rate, the re at onsh p for f m th ckness s found, wh ch s used to der ve the fo ow ng re at onsh p for the average Nusse t number a ong a p ate w th ength L: 1/4 h¯ L ( − v )g × h v × L3 NuL = = 0.943 k k × µ (Tsat − Twa ) (6.93) Th s resu t prov des the foundat on for much of the terature on grav ty-dr ven condensat on heat transfer. Rohsenow (1956) presented the so ut on for the temperature d str but on for the condensate f m n fa ng-f m condensat on, wh ch was emp oyed n Rohsenow et a . (1956) to account for the non near ty n f m temperature. The ana ys s was s m ar to Nusse t s fa ng-f m condensat on mode , except that the net entha py change n the f u d was accounted for n the energy ba ance. Therefore, a mod f ed entha py of vapor zat on, as shown be ow, was ntroduced to account for f m subcoo ng: h v = h v (1 + 0.68Ja ) (6.94) Chato (1962) ana yzed strat f ed-f ow condensat on ns de hor zonta and nc ned tubes under ow vapor shear cond t ons us ng s m ar ty so ut ons, and va dated the ana yses us ng data on a r/water m xtures f ow ng ns de a 27.94-mm tube, and R-113 condens ng ns de a 14.53-mm tube at var ous nc nat ons. He showed that for severa app cat ons, heat transfer n the qu d poo can be neg ected, and through measurements and ana ys s, found that the mean vapor ang e n the tube was re at ve y constant at about 120 . He a so found that heat transfer coeff c ents ncrease when the tube s nc ned; however, they decrease upon further ncreases n the nc nat on to y e d an opt mum nc nat on ang e of 10–20 . Due to the s gn f cant nf uence of vapor shear on qu d poo shape, h s work s on y app cab e for the entrance vapor Reyno ds number Rev < 35,000. The fo ow ng Chapter 6. Condensat on n m n channe s and m crochanne s 315 equat on y e ds sat sfactory agreement w th the resu ts for a but f u ds of ow qu d Prandt numbers: 1/4 ( − v ) g × h v × D3 hD Nu = = 0.468 × K (6.95) k k × µ (Tsat − Twa ) The constant K (a correct on for the qu d Prandt number) was presented graph ca y, and a nom na va ue of 0.555 = 0.468K may be used. Th s resu t for the Nusse t number for condensat on ns de tubes s very s m ar to Nusse t s (1916) resu t for fa ng-f m condensat on on a f at p ate w th Rohsenow s (1956) correct on for f m subcoo ng. However, the constant mu t p er was ower n Chato s ana ys s, pr mar y due to the ncreased therma res stance resu t ng from the th cker qu d poo of condensate at the bottom of the tube. Rosson and Myers (1965) measured the c rcumferent a var at on of heat transfer coeff c ents dur ng strat f ed condensat on of methano and acetone at atmospher c pressure n a 3/8 (9.53 mm) nom na sta n ess stee p pe w th a 1/8 (3.175 mm) wa th ckness. They found that the condensat on heat transfer coeff c ent was near y constant at the top of the tube ( < 10 ), then decreased near y to a po nt where t aga n became approx mate y constant. Based on th s resu t, for the top port on, they suggested a ead ng coeff c ent of 0.31Rev0.12 to the Nusse t condensat on equat on to account for vapor shear. For the bottom port on of the tube, the qu d poo heat transfer was not neg ected, and the von Karman ana ogy between heat transfer and momentum was used. By assum ng the wa shear was approx mate y equa to the buffer ayer shear and turbu ent Prt = 1, a re at onsh p between the Nusse t number and wa shear stress was obta ned. After assum ng that the average shear was equa to the oca shear and emp oy ng the Lockhart and Mart ne (1949) method for shear stress, the fo ow ng Nusse t number corre at on for the bottom port on of the tube was obta ned: √ 8Re (6.96) Nu = 5 + 5Pr −1 n(1 + 5Pr ) They a so def ned an nterpo at on scheme for reg ons between the top and the bottom. The near nterpo at on was based on the c rcumferent a ocat on at wh ch the heat transfer coeff c ent s the ar thmet c mean of the top and bottom heat transfer coeff c ents, wh ch was re ated to the vapor and qu d Reyno ds numbers and the Ga eo number, and the s ope of the heat transfer coeff c ent at th s ocat on. Rufer and Kez os (1966) a so ana yzed strat f ed condensat on n hor zonta and nc ned tubes us ng an ana yt ca approach; however, the r resu ts about the qu d poo depth were d fferent from those of Chato (1962), prompt ng an nterest ng d scuss on between these authors n the r paper about the va d ty of these two mode s. They mode am nar condensat on occurr ng n the upper port on of the tube ak n to the Nusse t ana ys s, comb ned w th a poo of qu d of vary ng depth at the bottom. They quest oned the openchanne hydrau cs bas s used n Chato s paper and n the work of Chaddock (1957), n wh ch t s assumed that the energy requ red to overcome fr ct on and acce erate the qu d s supp ed by the hydrau c grad ent. They a so stated that un ke Chato and Chaddock s treatments, the ang e subtended by the qu d poo shou d not be affected by the ex t 316 Heat transfer and f u d f ow n m n channe s and m crochanne s cond t on. Wh e they neg ected the effect of vapor shear on the am nar condens ng f m, for the qu d poo and the nterface, they accounted for the forces represented by the twophase pressure grad ent, and changes n momentum, hydrostat c, grav ty, and wa fr ct on forces. They compared the resu t ng var at on of the qu d eve a ong the condensat on path w th the vapor ho dup (vo d fract on) corre at ons of severa authors, and found that the Hughmark (1962) mode prov ded the best agreement. S m ar to the f nd ngs of Chato, they a so found that the qu d eve decreases steep y (thus ncreas ng heat transfer) w th tube nc nat on at sma ang es of nc nat on, w th th s effect be ng ess prom nent as the nc nat on s ncreased further. Contrary to the f nd ngs of Chato, they found that the f ow eve ang e (and therefore the qu d poo depth) ncreases as the condensat on proceeds n the tube. A so, they found that at ow mass f uxes, the qu d eve d d not change much as qua ty changed, wh e at h gh mass f uxes, the qu d eve ncreased cons derab y toward the ex t of the condenser. At a g ven qua ty, the qu d eve ncreased as the mass f ux was ncreased. Based on the manner n wh ch the condensat on mode s deve oped, the pred ct on of these qu d eve s d rect y determ nes the surface area ava ab e for the Nusse t f m condensat on n the upper port on of the tube, and therefore the overa effect ve heat transfer coeff c ent. A cons derab e amount of d scuss on on the contrad ctory trends pred cted by Rufer and Kez os (1966) and Chato (1962) appears at the end of the former paper. Th s d scuss on s pr mar y about whether the qu d eve shou d n fact r se or fa as the condensat on proceeds, about whether the tube shou d be f ed w th qu d at the end of the condensat on process, and about whether ex t cond t ons shou d govern the ho dup n the tube. The nterested reader s referred to the paper by Rufer and Kez os (1966) for further e aborat on. Chen and Kocamustafaogu ar (1987) presented a numer ca mode for strat f ed condensat on ns de a smooth hor zonta tube, n wh ch a condensate f m formed on the top port on of the tube and dra ned to a condensate poo at the bottom, where t moved ax a y. The effects of nterfac a shear on the th n condens ng f m and the condensate poo , ax a pressure grad ent, saturat on temperature, wa -to-saturat on temperature d fference, and the var at on of the qu d poo he ght were taken nto account. Cont nu ty, momentum, and energy equat ons were app ed to the vapor core and am nar fa ng f m, and the f m reg on was so ved assum ng a th n f m so that the effect of curvature cou d be neg ected. The nterfac a shear was computed based on the pressure grad ent n the vapor core. Heat transfer coeff c ents were ca cu ated for the am nar fa ng f m and the condensate poo , w th the tota heat transfer coeff c ent be ng a funct on of both of these and the subtended qu d poo eve ang e: h(z) = [m hc (z) + ( − m )h (z)]/ (6.97) where hc refers to the f m heat transfer coeff c ent, wh e h s the heat transfer coeff c ent n the qu d poo , and m s the ang e subtended by the poo to the vert ca . The f m heat transfer coeff c ent was an ntegrated average of the oca va ue, wh ch was ca cu ated from the numer ca y determ ned f m th ckness. The condensate poo heat transfer was found by us ng the express on deve oped by Rosson and Myers (1965) (d scussed above), wh ch s based on the von Karman ana ogy between momentum and heat transfer. Th s method y e ded reasonab e agreement w th an ex st ng condensat on heat transfer database, Chapter 6. Condensat on n m n channe s and m crochanne s 317 wh ch nc udes d ameters rang ng from 10.8 to 16.6 mm, and the f u ds R-12, R-113, water, methano , acetone, and propane. They a so conducted a d mens ona ana ys s to deve op an approx mate equat on for the heat transfer coeff c ent from the resu ts of the r numer ca ana ys s for des gn purposes. They noted that s nce the qu d poo heat transfer s a sma fract on of the tota heat transfer, the appropr ate d mens on ess parameters cou d be obta ned from the f m condensat on reg on a one, resu t ng n the fo ow ng equat on: 0.05 Nu = 0.492Ku−0.27 Pr 0.25 Ga0.73 ( /D)−0.03 Reg,s (1 + Re ,s )−0.01 (6.98) where Ku s the Kutate adze number Ku = Cp T /hfg , Ga s the Ga eo number Ga = D(g /2 )1/3 , and Reg,s and Re ,s are the superf c a gas and qu d phase ve oc t es at the n et, respect ve y. Th s corre at on was w th n 2.8% of the numer ca so ut on. S ngh et a . (1996) conducted exper ments on R-134a condens ng at 35 C ns de a 11-mm tube for the mass f ux range 50 < G < 300 kg/m2 -s. The data were found to be n the strat f ed–wavy, wavy-annu ar, and annu ar f ow reg mes. In the atter two reg mes, a strong mass f ux dependence was found for the heat transfer coeff c ent, wh e n the strat f ed–wavy reg me, the dependence was ess, a though not neg g b e. They found that the Trav ss et a . (1973), Shah (1979), Dobson et a . (1994), Chato (1962), and Akers et a . (1959) corre at ons underpred cted the data. Therefore, they proposed a corre at on that uses the Chato (1962) corre at on (w th a much ower ead ng constant of 0.0925) for the top port on of the tube ( .e. f m condensat on). For the forced convect ve heat transfer through the condensate at the bottom port on of the tube, they used a two-phase mu t p er correct on to the Gn e nsk (1976) s ng e-phase corre at on for Re > 2300: 0.2332 hfo = h 1 + 1.4020 (6.99) Xtt and a mod f cat on to the D ttus–Boe ter (1930) corre at on for Re < 2300: h = k 0.2339Re 0.5000 Pr 0.33 D (6.100) The top and bottom port ons of the heat transfer coeff c ent were comb ned n the usua manner, h = hfo + (2 − )hf , w th the subtended ang e g ven by » 2cos−1 (2 − 1), where the vo d fract on s ca cu ated us ng the Z v (1964) corre at on. The f ow reg mes were d st ngu shed us ng the mod f ed Froude number cr ter on proposed by Dobson et a . (1994): 1.5 Re 1.59 1 + 1.09Xtt0.039 Re ≤ 1250 Fr m = 0.025 Xtt Ga0.5 (6.101) 1.5 Re 1.04 1 + 1.09Xtt0.039 Re > 1250 Fr m = 1.26 Xtt Ga0.5 Thus, the f ow was cons dered strat f ed–wavy for Frm ≤ 7 and annu ar for Frm > 7. It shou d be noted that even though the r data showed a trans t on (wavy-annu ar) reg on, for the mode deve opment, an abrupt trans t on was chosen. 318 Heat transfer and f u d f ow n m n channe s and m crochanne s More recent y, Guo and Anand (2000) presented an ana yt ca mode for strat f ed condensat on of R-410A (50/50 m xture of R-32 and R-125) at 30 < G < 2200 kg/m2 -s n a 12.7 × 25.4-mm hor zonta rectangu ar channe , w th the ong s de vert ca . The heat transfer coeff c ent was the area-we ghted average of the top, bottom, and vert ca wa heat transfer coeff c ents. For the top wa , the von Karman ana ogy between heat transfer and momentum was app ed. The energy equat on was ntegrated across the f m am nar and buffer ayers, w th the shear n the buffer ayer assumed to equa the wa shear, and a so w th the assumpt on of Prt = 1. The shear stress was ca cu ated us ng the Lockhart and Mart ne (1949) two-phase mu t p er approach, resu t ng n the fo ow ng express on for the Nusse t number at the top: Nutop = # Re f s /2 5+ 5 Pr n(5Pr + 1) (6.102) where f s represents the superf c a qu d phase fr ct on factor. For the s de wa s, they pa d part cu ar attent on to the comb ned effects of Nusse t-type grav ty-dr ven condensat on and the vapor shear n the ax a d rect on. A mass, momentum, and energy ba ance was conducted across a twod mens ona e ement (vert ca and hor zonta axes), wh ch was used to obta n the f m th ckness dependent on the ax a and vert ca d rect on. The resu t ng quas - near, f rst-order part a d fferent a equat on was so ved numer ca y to get the f m th ckness and the average heat transfer coeff c ent for the vert ca wa s. Heat transfer through the bottom wa was assumed to be so e y due to conduct on across the qu d poo , wh ch n turn represented the accumu ated condensate from condensat on at the top and vert ca wa s. The r resu ts showed that the f m th ckness on the vert ca wa s was arger as the vapor ve oc t es decreased a ong the f ow d rect on (at decreas ng qua t es), w th ower nterfac a shear ead ng to a th cker f m. The f m th ckness decreased w th h gher mass f uxes due to the ncreased ve oc t es. The decrease n the heat transfer coeff c ent n the ax a f ow d rect on was ustrated through graphs of ncreas ng f m th ckness, wh ch const tuted an ncreas ng therma res stance. A so, the s ower growth of the f m th ckness a ong the ength represented the ower condensat on rates toward the end of the condensat on process due to the ower heat transfer coeff c ents. These ana yt ca and numer ca resu ts agreed we w th the data of Guo (1998) (mean dev at on of 6.75%), but not the corre at ons of Rosson and Myers (1965) and Dobson (1994). Th s cou d be because condensat on on the bottom poo was neg ected by Guo and Anand (2000), and a so due to d fferences between the c rcu ar and rectangu ar geometr es. 6.5.1.2. Shear-dr ven condensat on Carpenter and Co burn (1951) obta ned data for oca and average heat transfer coeff c ents for the condensat on of steam, methano , ethano , to uene, and tr ch oroethy ene ns de a vert ca 11.66-mm ID tube. The data were compared w th the Nusse t equat on, wh ch underpred cted the va ues due to the nf uence of vapor shear. Carpenter and Co burn stated that the th n condensate ayer for annu ar f ow m ght become turbu ent at very ow qu d Reyno ds numbers (~240). They cons dered vapor shear, grav ty, and the momentum change due to condensat on act ng upon th s condensate ayer. The ma or therma Chapter 6. Condensat on n m n channe s and m crochanne s 319 res stance was cons dered to be due to the am nar sub ayer. Us ng the von Karman un versa ve oc ty prof e for the qu d f m, a re at onsh p for the oca condensat on heat transfer coeff c ent was der ved, wh ch was a funct on of the f m th ckness ( .e. the am nar sub ayer th ckness), f u d propert es, and the sum of the forces act ng on the condensate. The grav ty force was deemed to be un mportant, wh ch s the case for annu ar f ow. The vapor fr ct on force was determ ned from an equ va ent vapor fr ct on factor, wh ch was h gher than for s ng e-phase f ow due to the presence of the qu d f m. They a so deve oped a separate corre at on for a am nar condensate f m w th a near ve oc ty prof e. Hence, the condensat on process was d v ded nto three d fferent reg ons: am nar condensate entrance reg on, turbu ent condensate, and am nar vapor ex t reg on (where a Nusse t ana ys s was more appropr ate). As d scussed n the prev ous sect on, Akers et a . (1959) def ned an a - qu d f ow rate that prov ded the same heat transfer coeff c ent for condens ng annu ar f ow. Th s a qu d f ow rate was expressed by an “equ va ent” mass f ux, wh ch was used to def ne an equ va ent Reyno ds number. Th s equ va ent Reyno ds number was subst tuted n a s ng ephase heat transfer equat on, wh ch was a funct on of the qu d Prandt and Reyno ds numbers, to pred ct the two-phase condensat on Nusse t number. Thus, the equ va ent qu d mass f ux Geq s composed of the actua qu d condensate f ux G , n add t on to a qu d f ux G that rep aces the vapor core mass f ux and produces the same nterfac a shear stress as the vapor core; Geq = G + G . By equat ng the shear stress due to the new qu d f ux G and the or g na vapor core f ux Gv , they obta ned: G = Gv

$ v fv f (6.103) At th s po nt, they assumed that f and fv wou d be constant due√to a presumed fu y rough nterface, and a so that f = fv , wh ch resu ted n Geq = G + Gv /v . Th s Geq was used to def ne Reeq , wh ch when subst tuted nto a typ ca s ng e-phase turbu ent heat transfer equat on y e ds: 0.8 1/3 Pr Nu = 0.0265Reeq (6.104) Th s equat on s typ ca y recommended for Re > 5000 and Rev (µv /µ )( /v )0.5 > 20,000. Akers and Rosson (1960) deve oped three d fferent exper menta y va dated heat transfer corre at ons for condensat on ns de hor zonta tubes for “sem -strat f ed” f ow (annu ar condensat on and run down super mposed on strat f ed f ow), am nar annu ar f ow, and turbu ent annu ar f ow. Loca va ues of the condensat on heat transfer coeff c ent were determ ned exper menta y for methano and R-12 f ow ng ns de a 15.88-mm hor zonta tube. For Re < 5000, the heat transfer coeff c ent was a funct on of the Prandt number, vapor Reyno ds number, and wa temperature d fference. For th s reg on, the constant mu t p er for the Nusse t number was determ ned from the exper menta data. Th s constant and the exponent for Rev were d fferent for Rev above and be ow 20,000, the resu t ng 320 Heat transfer and f u d f ow n m n channe s and m crochanne s express ons be ng: DGv 1000 < µ

v DGv 20, 000 < µ

0.5 hD = 13.8Pr 1/3 k < 20, 000 v 0.5 < 100, 000 hD = 0.1Pr 1/3 k

hfg Cp T hfg Cp T 1/6 1/6 DGv µ DGv µ

v v 0.5 0.2 0.5 2/3 (6.105) For turbu ent qu d f ms, the corre at on deve oped by Akers et a . (1959), wh ch neg ected the wa temperature d fference, was emp oyed. Anan ev et a . (1961) suggested that the two-phase heat transfer coeff c ent can be re ated to the correspond ng s ng e-phase heat transfer coeff c ent w th the ent re √ mass f ux f ow ng as a qu d, h = ho /m , where the m xture dens ty s g ven by 1/m = (1/ )(1 − x)+(1/v )x. Boyko and Kruzh n (1967) used th s approach to corre ate steam condensat on data for 10 < D < 17 mm from M ropo sky (1962), w th a D ttus– Boe ter-type s ng e-phase heat transfer coeff c ent proposed by M kheev (1956) of the form (ho D/k) = CRe0.8 Pr b0.43 (Prb /Prw )0.25 . D fferent ead ng constants C were requ red for the s ng e-phase heat transfer coeff c ent to f t the data from tubes of d fferent mater a s. So man et a . (1968) deve oped a mode for pred ct ng the condensat on heat transfer coeff c ent for annu ar f ow. In th s work, they note that Carpenter and Co burn (1951) d d not appropr ate y account for the momentum contr but on to the wa shear stress. They eva uated the wa shear stress as a comb nat on of fr ct on, momentum, and grav ty contr but ons, and used the resu t ng express on to eva uate the heat transfer coeff c ent, much ke the approach used by Carpenter and Co burn (1951). So man et a . (1968) performed momentum ba ances around the vapor core and the qu d f m. To obta n the shear stress at the vapor– qu d nterface, they used the Lockhart–Mart ne (1949) twophase mu t p er, represent ng t as v = 1 + 2.85Xtt0.523 , y e d ng = D dP − 4 dz F − dP dz F dP = g2 − dz v (6.106) n the usua manner. The momentum term accounts for the s ow ng down of the condens ng vapor as t converts nto the qu d phase. Depend ng on whether the qu d f m s am nar or turbu ent, the nterface ve oc ty s re ated to the mean f m ve oc ty as U = U , where = 2 for a am nar f m, and = 1.25 for a turbu ent f m. The shear due to momentum change s then represented as fo ows: m = D 4

G2 v

n/3 5 dx v an dz (6.107) n=1 The respect ve coeff c ents a1 . . . a5 n terms of the oca qua ty and the ve oc ty rat o are ava ab e n the r paper as we as n Carey (1992). For the genera case of an nc ned Chapter 6. Condensat on n m n channe s and m crochanne s 321 tube, the shear due to grav ty s g ven by: g = D (1 − )( − g )g s n 4 (6.108) where the vo d fract on s ca cu ated us ng the Z v (1964) corre at on. The three shear stress terms are summed as fo ows to y e d the wa shear stress: w = + m + g (6.109) Parametr c ana yses on the var ous components of the shear term were a so conducted to determ ne the r re at ve s gn f cance a ong the condensat on path. The fr ct on term dom nated at h gh and ntermed ate x, and progress ve y decreased toward the end of the condenser due to the decreas ng vapor ve oc t es. The grav ty term s neg g b e at h gh x, and ncreases w th ncreas ng f m th ckness as condensat on proceeds. The momentum term cons sts of the contr but on due to the momentum added to the qu d f m by the condens ng vapor (pos t ve term), and the reverse shear stress at the wa due to momentum recovery of the vapor (on y s gn f cant at ow vo d fract ons). Momentum effects are s gn f cant for h gh qu d-to-gas dens ty rat os and become mportant at ow x n the absence of a grav ty f e d. These cons derat ons were a so used to pred ct the onset of f ood ng n countercurrent gas– qu d f ow. They commented that wh e some nvest gators may have obta ned reasonab e agreement between the r heat transfer data and mode s even when gnor ng or ncorrect y account ng for momentum effects, th s may s mp y be due to the re at ve un mportance of these terms at the tested cond t ons. Due to the var at ons of the const tuent terms n the overa shear stress, however, these terms must be proper y accounted for, so that the mode can be va d over a w de range of cond t ons. W th the wa shear eva uated, the fo ow ng express on for the heat transfer coeff c ent was deve oped us ng the data from severa nvest gators: hµ k 1/2 = 0.036Pr 0.65 w1/2 (6.110) The above equat on s va d for 7.44 < D < 11.66 mm, the f u ds steam, R-113, ethano , methano , to uene, tr ch oroethy ene, and R-22, 1 < Pr < 10, 6 < Uv < 300 m/s, 0.03 < x < 0.99, for hor zonta as we as vert ca downward condensat on when annu ar f ow preva s. Trav ss et a . (1973) used the heat–momentum ana ogy and the von Karman un versa ve oc ty d str but on n the qu d f m to deve op a corre at on for the Nusse t number n annu ar f ow condensat on. The turbu ent vapor core prompted the assumpt on that the vapor core and vapor– qu d nterface temperatures were the same. The pressure grad ent was cons dered to be due to fr ct on, grav ty, and momentum change. In a manner ana ogous to that used by So man (1968), they computed the fr ct ona term us ng the Lockhart– Mart ne (1949) two-phase mu t p er approach w th the express on for g deve oped by So man (1968). They a so used Z v s (1964) vo d fract on mode to compute the grav tat ona term. For the momentum term, assum ng that the qu d f m was th n compared to 322 Heat transfer and f u d f ow n m n channe s and m crochanne s the tube ength, f at-p ate f ow was used. The rat o of the nterface ve oc ty to the average qu d f m ve oc ty, (def ned above n the d scuss on of the work of So man (1968)), was obta ned from the un versa ve oc ty prof e as a funct on of the non-d mens ona f m th ckness. The eddy d ffus v ty rat o ( .e. the turbu ent Prandt number) was assumed to be un ty. Us ng the assumed qu d ve oc ty prof e, a re at onsh p for the condensat on heat transfer coeff c ent was determ ned as a funct on of the turbu ent f m th ckness. They then der ved a re at onsh p for the qu d Reyno ds number as a funct on of th s f m th ckness. By argu ng that the nterfac a shear to wa shear rat o was approx mate y un ty, the f na re at onsh p for the condensat on heat transfer coeff c ent was deve oped as fo ows: 0.15Pr Re 0.9 hD = k FT

1 2.85 + 0.476 Xtt Xtt Re = G(1 − x)D µ (6.111) where FT s g ven as fo ows: FT = 5Pr + 5 n(1 + 5Pr ) + 2.5 n(0.0031Re 0.812 ) Re > 1125 = 5Pr + 5 n 1 + Pr (0.0964Re 0.585 − 1) 50 < Re < 1125 = 0.707Pr Re 0.5 (6.112) Re < 50 The resu ts were compared w th exper ments conducted on R-12 and R-22 condens ng n an 8-mm tube for 161 < G < 1533 kg/m2 -s. Agreement w th the exper menta data was good for qua t es as ow as 0.1, and the qua ty range from 0 to 0.1 (presumed by them to be s ug f ow) was pred cted we by conduct ng a near nterpo at on between th s mode and a s ng e-phase heat transfer corre at on. When the turbu ent Mart ne parameter was be ow 0.155, the corre at on produced good resu ts. However, for Mart ne parameters above 0.155, most probab y a m st f ow (h gh qua ty and mass f ux) cond t on, the exper menta data were underpred cted by the corre at on. A correct on factor was proposed to mprove pred ct ons n th s area. One of the most w de y used genera purpose condensat on corre at ons, due to the arge database from 21 nvest gators used for ts deve opment, and a so ts ease of use, s the Shah (1979) corre at on. Shah reasoned that n the absence of nuc eate bo ng, condensat on heat transfer shou d be s m ar to evaporat ve heat transfer when the tube s comp ete y wet, and extended the corre at on deve oped prev ous y (Shah, 1976) for evaporat on to condensat on as fo ows: h 3.8x0.76 (1 − x)0.04 = (1 − x)0.8 + h o (P/Pcr t )0.38

k GD 0.8Pr 0.4 h o = 0.023 D µ (6.113) He va dated the corre at on us ng data for water, R-11, R-12, R-22, R-113, methano , ethano , benzene, to uene, and tr ch oroethy ene condens ng ns de hor zonta , nc ned and vert ca tubes, as we as an annu us. The mean dev at on between the pred ct ons and the 474 data po nts used for corre at on deve opment was 17%. Some not ceab e d screpanc es Chapter 6. Condensat on n m n channe s and m crochanne s 323 nc uded the underpred ct on of the data at h gh qua t es (85–100%), wh ch cou d be due to entrance or entra nment effects. The operat ng cond t ons nc uded 11 < G < 211 kg/m2 -s, 21 C < Tsat < 310 C, 3 < Uv < 300 m/s, reduced pressure from 0.002 to 0.44, and 1 < Pr < 13, for tube d ameters between 7 and 40 mm. Shah recommended that the corre at on shou d be used on y for Re > 350 due to the ack of ower Re data for the deve opment of the corre at on. So man (1986) deve oped a heat transfer corre at on for condensat on n annu ar–m st f ow. The annu ar–m st trans t on cr ter on deve oped by h m was d scussed n a prev ous sect on of th s chapter. He stated that prev ous approaches for mode ng m st f ow heat transfer had pr mar y tr ed to mod fy annu ar f ow mode s, w th m ted success. Therefore, he deve oped a mode spec f ca y for th s reg me, us ng data from severa nvest gators, se ected because of the ex stence of m st f ow. Thus, the se ected data nc uded data for steam, R-113, and R-12, 7.4 < D < 12.7 mm, hor zonta and vert ca or entat ons, 21 C < Tsat < 310 C, 80 < G < 1610 kg/m2 -s, 0.20 < x < 0.95, 8 < We < 140, and 1 < Pr < 7.7. Of the f ve data sets chosen, one was n the annu ar f ow reg me, one was n annu ar and m st f ow, and three were predom nant y n m st f ow, fac tat ng the deve opment of th s corre at on. He found that a though the Akers et a . (1959) corre at on was successfu n pred ct ng the data for We > 20, the data were ser ous y underpred cted for We > 30. The Trav ss et a . (1973) corre at on was a so found to ser ous y underpred ct the data. Based on these observat ons, So man assumed that the f ow wou d occur as a homogeneous m xture n thermodynam c equ br um. Vapor shear wou d entra n the condensate nto the bu k f ow, eav ng a ternate dry spots and th n r dges of qu d at the wa . Heat transfer to th s a ternate y dry or th n-f m covered wa shou d therefore resu t n h gher heat transfer coeff c ents than those pred cted by annu ar f ow mode s. The ana ogy between heat and momentum transfer for s ng e-phase turbu ent f ow was used as a start ng po nt. Thereafter, the effect of wa temperature d fference was accounted for, wh ch resu ted n the fo ow ng corre at on (va d for We > ~30–35): Nu = 0.9 0.00345Rem µG hLG kG (Tsat − Tw ) 1/3 (6.114) where the m xture Reyno ds number s g ven by Rem = GD/µm , and the m xture v scos ty g ven by 1/µm = x/µG + (1 − x)/µL . Chen et a . (1987) deve oped a f m condensat on heat transfer corre at on based on ana yt ca and emp r ca approaches. They noted that at moderate f m Re, waves on the f m ncrease the surface area, wh ch enhances heat transfer above the Nusse t pred ct ons. At h gher f m Re (Re > 1800), the enhancement was attr buted to turbu ent transport n the condensate f m. At h gh vapor ve oc t es, as noted by others, they state that vapor shear mod f es the condensate f m, mak ng t th nner n co-current f ow, and th cker n countercurrent f ow. Thus, n th s work, they nc uded the effects of nterfac a shear, nterface wav ness, and turbu ent transport. They cons dered the m t ng cases of f m condensat on and nterpo ated between these m ts to obta n a more genera corre at on. For am nar f m condensat on n a qu escent vapor, they accounted for the effect of waves at the nterface by us ng Chun and Seban s (1971) express on Nu = 0.823Re−0.22 . For turbu ent x f ms, they used the data of B angett and Sch under (1978) for steam condensat on n 324 Heat transfer and f u d f ow n m n channe s and m crochanne s a vert ca tube, wh ch eads to ncreas ng Nu at ncreas ng Re due to h gher turbu ent ntens t es, to express the Nusse t number as NuT = 0.00402Rex0.4 Pr 0.65 . For condensat on w th h gh nterfac a shear stress, they neg ected the grav tat ona and momentum terms at h gh qua t es to deve op a s mp e re at onsh p assum ng that the wa shear stress was equa to the nterfac a shear stress. Thus, the equat on deve oped by So man et a . (1968) 1/2 was s mp f ed to Nu = 0.036Pr 0.65 * , where * = /L (gL )2/3 for vert ca as we as hor zonta f m condensat on. Hav ng deve oped express ons for nd v dua reg mes, they comb ned these us ng the techn que of Church and Usag (1972). They f rst deve oped an express on for grav ty-dom nated f m condensat on n a qu escent vapor by comb n ng the wavy– am nar and turbu ent express ons d scussed above as fo ows: 1/n Nuo = (NunL1 ) + (NunT1 ) 1 (6.115) The resu t ng Nuo for qu escent vapor condensat on was comb ned w th Nu for h gh nterfac a shear condensat on n ke manner to y e d: 1/n Nux = (Nuno2 ) + (Nun 2 ) 2 (6.116) Us ng the data of B angett and Sch under for downf ow condensat on of steam n a 30-mm vert ca tube over a w de range of f m Re and vapor ve oc t es, they determ ned that n1 = 6 and n2 = 2, wh ch y e ds the fo ow ng corre at on for oca f m condensat on heat transfer: hx Nux = kL

L g 1/3 = 0.31Re−1.32 x Rex2.4 Pr 3.9 + 2.37 × 1014 1/3 Pr 1.3 * + 771.6 1/2 (6.117) Based on the f nd ngs of Chen et a . (1984), they assumed that the nterfac a shear for a turbu ent vapor core wou d be the same for condens ng and ad abat c f ows, and used the shear stress re at onsh p of Duk er (1960) to compute the Nusse t number above: * = A(ReT − Rex )1.4 Rex0.4 A= 0.252µL1.177 µg0.156 D2 g 2/3 L0.553 g0.78 (6.118) where ReT s the f m Re f the ent re vapor f ow condenses, and the parameter A accounts for grav tat ona and v scous forces. The resu t ng Nusse t number s va d for co-current annu ar f m condensat on n vert ca tubes. For hor zonta annu ar f m condensat on, the grav ty terms are neg ected to y e d a s mp f ed vers on of the Nusse t number equat on as fo ows: Nux = 0.036Pr 0.65 A0.5 (ReT − Rex )0.7 Rex0.2 (6.119) Chen et a . caut oned that these equat ons wou d not app y at the very entrance of the tube when h gh vapor ve oc t es cou d cause s gn f cant entra nment and m st f ow, and toward the end of the condensat on process, where strat f ed and s ug f ows wou d preva . Chapter 6. Condensat on n m n channe s and m crochanne s 325 They a so deve oped express ons for the average Nusse t numbers for f m condensat on us ng the correspond ng m t ng express ons as the bas s (rather than ntegrat ng the oca Nux ), wh ch resu ted n the fo ow ng: Nu = Re−0.44 T ReT0.8 Pr 1.3 APr 1.3 ReT1.8 + + 1.718 × 105 2075.3 1/2 (6.120) for vert ca co-current condensat on, and Nu = 0.036Pr 0.65 − *1/2

2 ReL ReL −1.6 ReL −2 * 1.25 + 0.39 = A 0.2 ReT ReT ReT (6.121) for hor zonta condensat on. Us ng s m ar cons derat ons, Chen et a . a so deve oped express ons for the Nusse t number n countercurrent annu ar f ow (ref ux condensat on), deta s of wh ch are ava ab e n the r paper. Tandon et a . (1995) conducted condensat on exper ments for R-12 and R-22 f ow ng through a 10-mm hor zonta tube, pr mar y n the wavy f ow reg me. They comb ned these data w th prev ous y reported data n the annu ar and sem -annu ar reg mes for an overa range of 20 C < Tsat < 40 C and 175 < G < 560 kg/m2 -s to propose mod f cat ons to the Akers–Rosson (1960) corre at ons. They found that at the same vapor mass f ux, the heat transfer coeff c ent for R-22 was h gher than that for R-12, pr mar y due to the h gher therma conduct v ty and atent heat of R-22. They noted that the heat transfer coeff c ent trends showed a change n s ope at Rev ~3 × 104 , nd cat ng a trans t on from wavy to annu ar or sem -annu ar f ow. Th s prompted them to propose the fo ow ng corre at ons for shear-contro ed and grav ty-contro ed f ow reg mes: Rev > 3 × 104 Rev < 3 × 104 1/6 hfg Nu = Rev0.67 Cp T 1/6 hfg 1/3 Nu = 23.1Pr L Re1/8 v Cp T 0.084Pr 1/3 L (6.122) Ch tt and Anand (1995) deve oped an ana yt ca mode for annu ar f ow condensat on n smooth hor zonta round tubes. The resu ts were compared w th data for R-22 condens ng ns de an 8-mm tube. The r mode used the Prandt m x ng ength theory w th van Dr est s hypothes s (1956) (for the aw of the wa ) to obta n the momentum d ffus v ty, and Prt = 0.9. Thus, the computat on of the shear stress d str but on, pressure drop, and vo d fract on was avo ded, and the heat f ux was eva uated through cont nu ty n the rad a d rect on, w th the f m th ckness be ng ca cu ated terat ve y. For c osure, the fr ct on ve oc ty s ca cu ated through the use of the emp r ca Lockhart–Mart ne (1949) two-phase mu t 2 from Azer et a . (1972). The mode showed reasonab e agreement w th the r data, p er vv w th a mean dev at on of 15.3%, about the same as the agreement ach eved w th the Trav ss et a . (1973) mode and the Shah (1979) corre at on. Agreement w th the equ va ent mass 326 Heat transfer and f u d f ow n m n channe s and m crochanne s f ux-type mode s of Akers et a . (1959) and Boyko and Kruzh n (1967), however, was not part cu ar y good, and they attr buted th s to the ack of a strong phys ca bas s for these atter mode s. Ch tt and Anand (1996) ater extended the exper ments to R-32/R-125 m xtures (50/50 by we ght) w th o concentrat ons of 0%, 2.6%, and 5.35%, and pure R-22 condens ng n the 8-mm hor zonta tube, pr mar y n the annu ar reg me based on the Ta te – Duk er (1976) map. They found a m n ma effect (~10%) of condens ng temperature on the heat transfer coeff c ent because the operat ng cond t ons (24–38 C) were we be ow the cr t ca temperature. Heat transfer coeff c ents for R-32/R-125 m xtures were about 15% h gher than those for R-22, and the effect of po yo ester o was a so found to be m n ma , w th n the uncerta nty est mates. Moser et a . (1998) noted that the Akers et a . (1959) equ va ent mass ve oc ty mode d scussed above had been shown to cons stent y underpred ct the data of severa nvest gators. They stated that th s was pr mar y due to two assumpt ons n the r mode that cou d not be adequate y supported: (1) the assumpt on of constant and equa fr ct on factors for the vapor core and the qu d f m, because the atter wou d most certa n y be affected by the Re , (2) the dr v ng temperature d fferences for s ng ephase (Tb − Tw ) and condens ng f ows (T − Tw ) are comp ete y d fferent, someth ng that the Akers et a . mode does not take nto account. To correct the f rst prob em, they re ated the fr ct on n the vapor and qu d phases through the two-phase mu t p er concept, rather than assum ng the fr ct on factors to be equa . Th s ed to the fo ow ng def n t on of the equ va ent Reyno ds number: 8/7 Re . To eva uate 2 , they recommended the use of the Fr ede (1979) corre aReeq = o o o t on. For the dr v ng temperature d fference, they re ated the s ng e-phase and condens ng Ts as fo ows: F = (Tb − Tw )/(T − Tw ). Us ng a boundary- ayer ana ys s s m ar to that of Trav ss et a . (1973), a ong w th the s ng e-phase T be ng determ ned by the Petukhov (1970) equat on, the correct on factor F was expressed as fo ows: −0.185 2 F = 1.31(R+ )C1 ReC Pr C1 = 0.126Pr −0.448 L C2 = −0.113Pr −0.563 (6.123) The d mens on ess p pe rad us was obta ned through the fr ct on ve oc ty and the use of 7/8 the B as us fr ct on factor, to y e d R+ = 0.0994Reeq . The f na Nusse t number equat on was as fo ows: Nu = 1+0.875C1 Pr 0.815 2 0.0994C1 ReC hD Reeq = k (1.58 n Reeq − 3.28)(2.58 n Reeq + 13.7Pr 2/3 − 19.1) (6.124) Th s mode was shown to pred ct data from a var ety of sources for 3.14 < D < 20 mm w th a mean dev at on of 13.6%, about as we as the emp r ca Shah (1979) corre at on, and better than the Trav ss et a . (1973) corre at on. In genera , the trend was toward underpred ct on of data, wh ch they stated cou d be corrected by the app cat on of better pressure drop mode s for the eva uat on of the two-phase mu t p er requ red for th s mode . In add t on, account ng for entra nment and strat f ed annu ar f m wou d y e d better pred ct ons. It was ment oned n the prev ous sect on that Hur burt and Newe (1999) deve oped sca ng equat ons for condens ng f ows that wou d enab e the pred ct on of vo d fract on, pressure drop, and heat transfer for a refr gerant at a g ven cond t on and tube d ameter from Chapter 6. Condensat on n m n channe s and m crochanne s 327 the ava ab e resu ts for another s m ar f u d (R-11, R-12, R-22, and R-134a) operat ng at a d fferent cond t on n the d ameter range 3–10 mm. They deve oped an equ va ent average f m th ckness mode assum ng annu ar f ow, wh ch enab ed the pred ct on of shear stress. L ke severa other researchers, they assumed that the pr mary res stance to heat transfer occurs n the v scous and buffer ayers of the f m, regard ess of the actua th ckness. F m and core f ows were mode ed after the approach of Rohsenow et a . (Rohsenow et a ., 1957; Bae et a ., 1971; Trav ss et a ., 1973), assum ng turbu ent vapor and qu d f ms, and a aw-of-the-wa un versa ve oc ty and temperature prof es. They used data from Sacks (1975) for R11, R-12, and R-22, and the R-22 and R-134a data of Dobson (1994) to va date the r mode . Us ng these cons derat ons, they deve oped the fo ow ng sca ng equat on for heat transfer pred ct on: h2 /h1 = (g2 /g1 )−0.375 (L2 /L1 )0.375 (µg2 /µg1 )−0.75 (µL2 /µL1 )0.062 1 − x2 0.19 ×(CpL2 /CpL1 )(Pr L2 /Pr L1 )−0.25 (G2 /G1 )(x2 /x1 )0.82 1 − x1 (6.125) Upon compar son w th the data of Dobson (1994), they commented that d ameter effects may not be fu y accounted for w th the above approach, and perhaps the above equat on shou d a so have nc uded the factor (D2 /D1 )−0.15 . A so, the f u d property exponents n th s equat on were d fferent from those found n the terature; spec f ca y, the therma conduct v ty exponent var ed from 0.35 for Chen et a . (1987), 0.38 for Shah (1979), and 0.65 for Cava n and Zecch n (1974), wh e the above equat on shows an exponent of 0.25 through the Prandt number. They noted that severa propert es together determ ne the th cknesses of the var ous ayers; therefore, d fferences n the dependence on an nd v dua property may not be too s gn f cant. 6.5.1.3. Mu t -reg me condensat on For f ms under the s mu taneous nf uence of shear and grav ty, Kosky (1971) prov ded a s mp e a gebra c equat on for the conven ent eva uat on of the non-d mens ona f m th ckness to fac tate c osed-form computat ons. Thus, the work of Duk er (1960) (cons dered accurate c ose to the wa and for th ck f ms) and Kutate adze (1963) (cons dered accurate for th ck f ms, + > ~10) were comb ned to obta n the fo ow ng s mp f ed express ons: + = (ReL /2)1/2 for + < 25, and + = 0.0504ReL 7/8 for + > 25. Jaster and Kosky (1976) performed exper ments on condens ng steam ns de a 12.5-mm hor zonta tube. Coup ed w th the exper menta resu ts from Kosky and Staub (1971), the mass f ux ranged from 12.6 to 145 kg/m2 -s. A stress rat o F = w /(L g) (ax a shear-to-grav ty) was used to d v de the data nto annu ar (F > 29), trans t on (29 ≥ F ≥ 5), and strat f ed (F < 5) reg mes. The heat and momentum transfer ana ogy was app ed to determ ne the annu ar f ow heat transfer, w th the two-phase gas mu t p er g2 used to determ ne the shear stress and the non-d mens ona f m th ckness obta ned from Kosky (1971). For strat f ed f ow, they re ated the ha f-ang e subtended by the qu d poo to the vo d fract on us ng Z v s (1964) mode approx mate y as cos() = 2 − 1, wh ch resu ted n a mod f ed vers on of the Nusse t condensat on equat on w th the ead ng constant of 0.725 and a 328 Heat transfer and f u d f ow n m n channe s and m crochanne s mu t p er of 3/4 . In the trans t on reg me, a near nterpo at on between the two was suggested: Nutr = Nuan + F − 29 (Nuan − Nustr ) 24 (6.126) It was stated n a prev ous sect on that Breber et a . (1979; 1980) deve oped a four f ow zone map based on data for a var ety of f u ds for d ameters as sma as 4.8 mm. The correspond ng trans t on cr ter a were sted w th that d scuss on. For Zone I (annu ar f ow, g* > 1.5, X < 1.0), they recommend convect ve-type corre at ons: k 0.14 Re > 1500 h = 0.024 Re 0.8 Pr 1/3 (µ /µw ) D

Re Pr D 1/3 k (µ /µw )0.14 Re < 1500 h = 1.86 D L hannu ar = h ( 2 )0.45 (6.127) For Zone II (wavy or strat f ed f ow, g* < 0.5, X < 1.0), they recommend a Nusse t-type corre at on: hstrat f ed = Fg k 3 ( − v ) gh v 4µ (Tsat − Twa ) D 1/4 (6.128) where the factor Fg s a funct on of the qu d vo ume fract on and the vapor Reyno ds number, but may be approx mated by Fg = 0.79 as suggested by Pa en et a . (1977). For Zone III ( nterm ttent f ow, g* < 1.5, X > 1.5), n v ew of the ack of appropr ate mode s, they suggest the use of the annu ar f ow corre at on, because n the r v ew, the convect ve mechan sm governs, and a so because t approaches the s ng e-phase convect on m t as x Õ 0. For Zone IV (bubb e f ow, g* > 1.5, X > 1.5), once aga n they suggest the convect ve equat on presented above. For trans t ons between Zones I and II, they recommend nterpo at on between hannu ar and hstrat f ed based on g* , wh e for trans t ons between Zones II and III, they recommend nterpo at on based on X . N theanandan et a . (1990) eva uated the ava ab e condensat on corre at ons us ng condensat on data from severa sources for f ow ns de hor zonta tubes w th the ntent on of deve op ng a more accurate approach spann ng mu t p e f ow reg mes for the des gn of condensers. The heat transfer database nc uded R-12, R-113, and steam condens ng ns de tubes w th d ameters rang ng from 7.4 to 15.9 mm, d v ded nto three reg mes: wavy, annu ar, and m st f ow. Fo ow ng the work of So man (1982; 1983; 1986), they recommended the use of the fo ow ng heat transfer corre at ons for the respect ve reg mes: M st f ow (We ≥ 40, Fr ≥ 7) So man (1986) Annu ar f ow (We < 40, Fr ≥ 7) Shah (1979) Wavy f ow (Fr < 7) Akers and Rosson (1960) It shou d be noted that the We = 40 cr ter on to determ ne the m st–annu ar trans t on was d fferent from the va ue of 30 recommended ear er by So man (1986). S m ar Chapter 6. Condensat on n m n channe s and m crochanne s 329 recommendat ons were a so made for condensat on n vert ca tubes based on the Hew tt and Roberts (1969) f ow reg me map. Dobson et a . (1994) nvest gated condensat on of R-12 and R-134a n a 4.57-mm hor zonta tube at 75 < G < 500 kg/m2 -s at 35 C and 60 C saturat on temperature. They found that the Fr = 7 cr ter on proposed by So man (1982) was adequate for estab sh ng the wavy to wavy-annu ar f ow trans t on. In add t on, they ntroduced Fr = 18 as the trans t on cr ter on for the wavy-annu ar to annu ar trans t on, and a so stated that the We = 30 cr ter on of So man (1986) was appropr ate for the annu ar-to-m st f ow trans t on. Hav ng thus d v ded the r data nto grav ty- and shear-dr ven reg ons, they nterpreted severa trends n heat transfer coeff c ents. In annu ar f ow, the heat transfer coeff c ent ncreased w th mass f ux and qua ty, due to the ncreased shear and the th nn ng of the qu d f m. The heat transfer coeff c ents for R-134a were about 20% h gher than those for R-12 at constant mass f ux. In wavy f ow, the heat transfer coeff c ent was ndependent of mass f ux, and showed a s ght ncrease w th qua ty. As the pr mary heat transfer n th s reg on occurs n the upper port on due to Nusse t-type condensat on, w th very tt e heat transfer through the qu d poo , the s ght ncrease n heat transfer coeff c ents w th qua ty was attr buted to the th nn ng of the poo and a sma er poo depth at the h gher qua t es. In wavy f ow, the R-134a heat transfer coeff c ents were about 10% h gher than those for R-12. The heat transfer coeff c ents for R-134a were about 15% h gher at 35 C than at 60 C because of the decrease n the rat o of the dens t es of the vapor and qu d phases, wh ch eads to ower s p rat os, and a so because of a decrease n the therma conduct v ty at the h gher temperature. To corre ate the r data n the wavy f ow reg me, they emp oyed the Chato (1962) type corre at on, but changed the ead ng coeff c ent from 0.555 to a funct on of the Lockhart– Mart ne parameter to account for the change n vo d fract on w th qua ty, y e d ng: Nugrav ty 0.375 = 0.23 Xtt

g ( − v ) D3 h v µ (Tsat − Twa ) k 0.25 For annu ar f ow, they used the fam ar two-phase mu t p er approach: 2.61 0.80 0.3 Nuannu ar = 0.023ReL Pr L Xtt0.805 (6.129) (6.130) They a so found that the annu ar f ow heat transfer coeff c ents pred cted by th s equat on are qu te s m ar to those for convect ve evaporat on found by Watte et et a . (1994), mp y ng s m ar heat transfer mechan sms for the two convect ve phase-change processes. F na y, they stated that severa w de y used corre at ons n the terature (Trav ss et a ., 1973; Cava n and Zecch n, 1974; Shah, 1979) overpred cted the r data. Subsequent y, Dobson and Chato (1998) extended th s work to 3.14- and 7.04-mm tubes, and a so to the f u ds R-22 and m xtures of R-32/R-125. Data from these atter two tubes were used for the deve opment of updated heat transfer corre at ons. The on y d scern b e effect of tube d ameter n the r work was that trans t on to annu ar f ow happened at h gher mass f uxes and qua t es n the arger tubes, a though they were ab e to pred ct th s us ng the arged ameter corre at ons. Thus, surface tens on effects were not seen w th n th s d ameter range. Due to a var ety of compensat ng nf uences such as thermophys ca propert es, d fferent reduced 330 Heat transfer and f u d f ow n m n channe s and m crochanne s pressures and d fferent vo d fract ons, the effect of the part cu ar f u d used on the heat transfer coeff c ent was at most 10% n the wavy reg me, w th a s ght y better performance of R-134a n annu ar f ow. In deve op ng the rev sed corre at ons, they noted that the boundary- ayer ana yses used by severa nvest gators, nc ud ng pr mar y Trav ss et a . (1973), cou d be shown to be s m ar n bas s to the two-phase mu t p er approach used by others. Not ng a so that the pr mary therma res stance n annu ar f ow occurs n the am nar and buffer ayers (even the presence of waves at the nterface or the vary ng f m th ckness around the c rcumference wou d not s gn f cant y affect the near-wa behav or), they d d not f nd t necessary to nc ude a mu t -reg on mode of the qu d f m res stance. The ro e of entra nment was a so not found to be very s gn f cant n affect ng heat transfer coeff c ents, because they found that rare y d d true m st f ow w thout a th n annu ar f m coat ng the wa ex st. W th these cons derat ons, the fo ow ng annu ar f ow corre at on was proposed: 2.22 0.80 0.3 Nuannu ar = 0.023ReL Pr L 1 + 0.89 (6.131) Xtt For the strat f ed wavy f ow reg on, they accounted for the f m condensat on at the top of the tube and forced convect on n the qu d poo as fo ows: 0.12 GaPr 0.25 0.23Revo Nuwavy = + (1 − /) Nuforced (6.132) Ja 1 + 1.11Xtt0.58 where the forced convect on term s g ven by: c1 Nuforced = 0.0195ReL0.80 Pr L0.4 1.376 + c2 Xtt (6.133) The constants c1 and c2 n the above equat on are g ven as fo ows: For 0 < Fr ≤ 0.7: c1 = 4.172 + 5.48Fr − 1.564Fr 2 c2 = 1.773 − 0.169Fr (6.134) and for Fr > 0.7, c1 = 7.242, c2 = 1.655. The ang e from the top of the tube to the qu d poo eve , , was approx mated as fo ows: 1− arccos(2 − 1) (6.135) The vo d fract on requ red for the above equat on was ca cu ated us ng Z v s (1964) corre at on. They recommended that the above equat ons shou d be used as fo ows: G ≥ 500 kg/m2 -s G < 500 kg/m2 -s Nu = Nuannu ar Nu = Nuannu ar Nu = Nuwavy for Fr so > 20 for Fr so < 20 (6.136) It shou d be noted that the s ng e-va ue demarcat on on the bas s of G shown above cou d resu t n sharp d scont nu t es between the heat transfer coeff c ent pred ct ons n the ne ghborhood of G = 500 kg/m2 -s as the annu ar or wavy corre at ons are used. The more gradua Chapter 6. Condensat on n m n channe s and m crochanne s 331 trans t on between wavy, wavy-annu ar, annu ar, and m st f ows n the r f ow v sua zat on work may have warranted correspond ng y gradua trans t ons between heat transfer coeff c ent mode s. A so, they caut oned that for the sma d ameter (3.14 mm) tube, the data had arge uncerta nt es due to d ff cu t es n measur ng sma heat transfer rates accurate y, wh ch ed to re at ve y arger dev at ons from the above corre at ons. W son et a . (2003) reported heat transfer coeff c ents for 7.79-, 6.37-, 4.40-, and 1.84-mm tubes made by progress ve y f atten ng 8.91-mm round smooth tubes and tubes w th ax a and he ca m crof n tubes. Refr gerants R-134a and R-410A were condensed at 35 C n these tubes at 75 < G < 400 kg/m2 -s. Heat transfer coeff c ents were seen to ncrease over correspond ng smooth-tube va ues as the tube was f attened. (It shou d be noted that the r smooth-tube condensat on heat transfer coeff c ents were cons stent y ower by 20–30% than the Dobson and Chato (1998) pred ct ons, somewhat surpr s ng because these exper ments were conducted by researchers from the same group; the Dobson and Chato pred ct ons were therefore art f c a y ncreased by the factor requ red to match the data. S nce the enhancements observed were pr mar y n the 1.5–3.5 range, presumab y nc ud ng the 60% enhancement due to surface area ncreases n the m crof nned tubes, these d screpanc es cou d affect the conc us ons cons derab y.) The enhancement was the h ghest for the f attened tubes w th 18 he x ang e m crof ns. Tentat ve exp anat ons for these trends were prov ded nc ud ng: potent a ear y trans t on from strat f ed to annu ar f ow, “mod f cat ons to the f ow f e d w thout a ter ng f ow f e d conf gurat on,” format on of d fferent f ow f e d conf gurat ons, and others. No pred ct ve mode s or corre at ons were proposed, but t was noted that f atten ng to 5-mm he ght wou d ead to about a 10% reduct on n condenser s ze, 70% ncrease n pressure drop, and 40% reduct on n refr gerant charge for the examp e case stud ed. A s m ar study on condensat on of R-404A n 9.52-mm OD tubes at 40 C and 200 < G < 600 kg/m2 -s by Infante Ferre ra et a . (2003) a so showed enhancements of 1.8 to 2.4 n m crof n and cross-hatched tubes. In a set of re ated papers, Cava n et a . (2001; 2002a, b) obta ned heat transfer and pressure drop data for a var ety of refr gerants and b ends (R-22, R-134a, R-125, R-32, R-236ea, R-407C, and R-410A) condens ng n smooth 8-mm hor zonta tubes, and deve oped mu t p e-f ow reg me mode s to pred ct the r own data as we as data from other nvest gators. The exper ments covered the temperature range 30 < Tsat < 50 C, and 100 < G < 750 kg/m2 -s. When compar ng the ava ab e corre at ons aga nst these data, they noted that the data were often outs de the recommended range of app cab ty for many of these corre at ons, espec a y for the newer h gh-pressure refr gerants such as R-125, R-32, and R-410A. The pred ct ons of some of the emp r ca corre at ons were a so not sat sfactory w th n the stated range of app cab ty. They systemat ca y p otted the var ous corre at ons n the terature n d fferent graphs for ow and h gh-pressure refr gerants, adher ng to the recommended ranges of app cab ty to demonstrate the nadequacy of most of these corre at ons as genera purpose pred ct ve too s. Therefore, they tr ed to deve op new procedures patterned after the approach of Breber et a . (1979; 1980) that wou d span the pr mary f ow reg mes encountered n condensat on n hor zonta tubes; that s, annu ar, strat f ed, wavy, and s ug. For annu ar f ow, for wh ch they chose g* > 2.5; Xtt < 1.6 as the trans t on cr ter on, they recommended the use of the Kosky and Staub (1971) mode , wh ch re ates the heat transfer coeff c ent to the fr ct ona pressure grad ent through the nterfac a shear stress . However, to compute the necessary fr ct ona pressure grad ent 332 Heat transfer and f u d f ow n m n channe s and m crochanne s for th s mode , they mod f ed the Fr ede (1979) corre at on to app y on y to annu ar f ow, whereas t had or g na y been ntended for annu ar and strat f ed reg mes. Th s restr cted vers on was ntended to match the s ng e-reg me data better. The r mod f ed annu ar f ow mode s g ven by the fo ow ng equat ons: hannu ar = L CpL (/L )0.5 T+ (6.137) where the d mens on ess temperature s g ven by the fo ow ng: T + = + PrL + ≤ 5 7 8 T + = 5 PrL + n 1 + PrL + /5 − 1 5 < + < 30 T + = 5 PrL + n(1 + 5PrL ) + 0.495 n + /30 + ≥ 30 (6.138) w th: + = (ReL /2)0.5 for ReL ≤ 1145; + = 0.0504Re7/8 L for ReL > 1145 (6.139) The shear stress s eva uated from the two-phase pressure drop n the usua manner as fo ows: 2 (dP/dz)f G 2 2 2 (dP/dz)LO = LO = ; where (dP/dz)f = LO (6.140) fLO 4 DL The two-phase mu t p er conta ns severa mod f cat ons from the or g na Fr ede (1979) mode ; therefore, t s presented comp ete y be ow: 2 LO =E+ 1.262FH We0.1458 (6.141) The parameters E, F, and H are g ven by: E = (1 − x)2 + x2 L fGO G fLO F = x0.6978

0.3278 µG −1.181 µG 3.477 L 1− H= G µL µL (6.142) The Weber number s def ned based on the tota mass f ux and the gas phase dens ty nstead of the m xture dens ty, We = G 2 D/G . For g * < 2.5; Xtt < 1.6, strat f cat on beg ns, and they compute the heat transfer coeff c ent n th s reg on us ng a comb nat on of the annu ar mode eva uated at the g* = 2.5 boundary and a strat f ed heat transfer mode . It shou d be noted that throughout the reg on 0 < g* < 2.5; Xtt < 1.6, a progress ve y decreas ng contr but on of the annu ar term s nc uded; that s, the f ow s never treated as exc us ve y Chapter 6. Condensat on n m n channe s and m crochanne s 333 strat f ed. Therefore they refer to th s reg on as “annu ar–strat f ed trans t on and strat f ed reg on.” The nterpo at on formu a recommended s as fo ows: * /2.5) + hstrat f ed htrans t on = hannu ar, G* =2.5 − hstrat f ed ( G (6.143) where the strat f ed f ow contr but on s eva uated n the usua manner as a comb nat on of f m condensat on at the top and forced convect on n the qu d poo : hstrat f ed

1−x = 0.725 1 + 0.82 x + hL 1 − 0.268 −1 kL3 L (L − G ) ghLG µL D (Tsat − Twa ) 0.25 (6.144) where the forced convect on heat transfer coeff c ent n the qu d poo s g ven by: hL = 0.023 kL D

G(1 − x)D µL 0.8 PrL 0.4 (6.145) and the qu d poo ang e s def ned us ng the Z v (1964) vo d fract on mode as fo ows: 1− arccos(2 − 1) = (6.146) Cava n et a . commented that the qu d poo heat transfer term s s gn f cant at h gh va ues of reduced pressure. For cond t ons where g* < 2.5; Xtt > 1.6, trans t on from strat f ed to nterm ttent f ows starts to occur. Th s trans t on, however, occurs at a vary ng va ue of Xtt , and s def ned by a trans t on mass f ux, G > GW , def ned by Rabas and Arman (2000) as the mass f ux requ red to f the tube when be ng d scharged to a gas-f ed space. For s ug f ow, they used data from Dobson and Chato (1998) and Tang (1997) to curve-f t a s ng ephase heat transfer corre at on. Smooth trans t on between the heat transfer coeff c ents n strat f ed and s ug f ows s prov ded once aga n by an nterpo at on techn que. Thus, the strat f ed– s ug trans t on and s ug f ow reg on heat transfer coeff c ent s g ven by: hstrat f ed−s ug = hLO + x xXtt =1.6 (hXtt =1.6 − hLO ) (6.147) The trans t on qua ty can be s mp y ca cu ated from the def n t on of Xtt as fo ows: xXtt =1.6 = (µL /µG )1/9 (G /L )5/9 1.686 + (µL /µG )1/9 (G /L )5/9 (6.148) The qu d-on y (hLO ) and qu d-phase hXtt =1.6 heat transfer coeff c ents requ red above are ca cu ated us ng the D ttus–Boe ter (1930) equat on at ReLO and ReL (at the x for Xtt = 1.6), respect ve y. 334 Heat transfer and f u d f ow n m n channe s and m crochanne s 100 Bubb y f ow Annu ar f ow JG 10 1 Trans t on and Wavy–strat f ed f ow S ug-f ow 0.1 Gw Strat f ed f ow 0.01 0.01 0.1 1 10 100 X tt F g. 6.26. I ustrat on of Cava n et a . (2002a) mode mp ementat on. From Cava n , A., Cens , G., De Co , D., Dorett , L., Longo, G. A. and Rossetto, L. Condensat on of ha ogenated refr gerants ns de smooth tubes, HVAC and R Research, 8(4), pp. 429–451 (2002) w th perm ss on from the Amer can Soc ety of Heat ng, Refr gerat ng and A r-Cond t on ng Eng neers. There s a so a poss b ty for an annu ar-to-s ug trans t on across g* = 2.5 at h gh va ues of Xtt ; however, n common pract ce, th s rare y occurs because t requ res G >1000 kg/m2 -s and very ow qua t es for most refr gerants. However, n such nstances, they recommend a near nterpo at on between hannu ar and hLO based on the vapor qua ty. At h gh va ues of g* and Xtt , espec a y at h gh reduced pressures, bubb y f ow s encountered, and once aga n, n the absence of re ab e corre at ons n th s reg on, they recommend the use of the annu ar f ow corre at on. Th s mu t reg on map s recommended for the refr gerants sted above, and for the fo ow ng cond t ons: 3 < D < 21 mm pR < 0.75 L /G > 4 (6.149) Th s compos te mode was shown to pred ct the r own data and data from severa nvest gators w th an average abso ute dev at on of about 10%. A chart for the use of the above heat transfer coeff c ent corre at ons s prov ded n F g. 6.26. For the pred ct on of pressure drops, Cava n et a . recommend the use of the mod f ed Fr ede two-phase mu t p er 2 deve oped by them for annu ar f ow when * > 2.5; for * < 2.5, the or g na Fr ede LO g g (1979) mode s recommended. E Ha a et a . (2003) and Thome et a . (2003) conducted another deta ed exerc se (s m ar to that of Cava n et a . (2001; 2002a, b; 2003) descr bed above) of deve op ng f ow reg me-based heat transfer corre at ons over a w de range of cond t ons by f tt ng data from severa d fferent nvest gators for many d fferent f u ds. They patterned th s work after the s m ar work of Kattan et a . (1998a, b, c) on f ow bo ng. The r bas c prem se s that vo d fract on s the most mportant var ab e n determ n ng f ow reg mes, pressure drop, and heat transfer. However, re ab e vo d fract on data were not ava ab e, espec a y at h gh reduced pressures. Therefore, they f rst used the Ste ner (1993) hor zonta tube Chapter 6. Condensat on n m n channe s and m crochanne s 335 vers on of the Rouhan –Axe sson (1970) dr ft-f ux vo d fract on map because t nc udes mass f ux and surface tens on effects: −1 x x 1.18 (1 − x) [g (L − V )]0.25 1−x [1 + 0.12(1 − x)] = + + V V L GL0.5 (6.150) The vo d fract on pred cted by th s mode does not approach the homogeneous vo d fract on as the cr t ca pressure s approached. Therefore, they set about try ng to mprove upon th s mode by nd rect means. They stated that turbu ent annu ar f ow condensat on heat transfer data cou d n fact be used to deduce the app cab e vo d fract on. Th s s because the correspond ng qu d f m heat transfer coeff c ent may be represented as h = c RenL Pr L0.5 (kL /), where the f m th ckness s s mp y by def n t on = D(1 − )/4. Subst tut ng th s express on back nto the heat transfer coeff c ent equat on, t s c ear that h ∝ (1 − x)n /(1 − ). S nce vo d fract ons n annu ar f ow are very arge, the denom nator n th s express on, representat ve of the qu d f m th ckness, s a sma quant ty that affects the heat transfer coeff c ent s gn f cant y. It s th s sens t v ty of the heat transfer coeff c ent to vo d fract on that they sought to exp o t n deduc ng the vo d fract on from ava ab e heat transfer data. Thus, they used the annu ar f ow data from Cava n et a . (2001; 2002a, b) to stat st ca y curve-f t the ead ng constant and exponent for ReL n the heat transfer express on, and assumed the Pr exponent to be 0.5 nstead of 0.4. A though reasonab e pred ct on was ach eved, the pred ct on accuracy seemed to show a strong saturat on pressure dependency. Therefore, they dec ded, somewhat arb trar y, that the actua vo d fract on wou d n fact e somewhere between the Axe sson–Rouhan pred ct on and the homogeneous va ue. They then chose a ogar thm c mean of these two va ues to represent the actua vo d fract on: = homogeneous − Axe sson–Rouhan homogeneous n Axe sson–Rouhan (6.151) Th s formu at on was then used to curve-f t the heat transfer data aga n, wh ch demonstrated an mprovement n the pred ct on of the heat transfer data over a arger range of reduced pressures, wh ch they took to be a va dat on of the vo d fract on curve-f t. Once th s was estab shed, they p otted the trans t ons between strat f ed, wavy, nterm ttent, annu ar, m st, and bubb y f ow on G–x coord nates, adapt ng the trans t on express ons presented by Kattan et a . (1998a, b, c) (w th updates by Zurcher et a . (1999)) for bo ng to the condens ng s tuat on. For examp e, a owance s made for the fact that dryout does not occur n condens ng f ows; thus they remove that port on of the wavy trans t on ne and arb trar y extend t from the m n ma of that trans t on to the x = 1 ocat on. Var ous qu d–vapor cross-sect ona areas and per meters are der ved from the vo d fract on to enab e p ott ng of the trans t on nes. A representat ve map constructed n th s manner s shown n F g. 6.27. It shou d be noted that because the vo d fract on s dependent on the mass f ux, str ct y speak ng, these maps have to be redrawn for each mass f ux under cons derat on, even though G s n fact one of the coord nates. However, E Ha a et a . state that the trans t on nes are not very sens t ve to the mass f ux, and a f rst approx mat on may be ach eved by choos ng a representat ve mass f ux to p ot the trans t on nes. They demonstrate qua tat ve agreement 336 Heat transfer and f u d f ow n m n channe s and m crochanne s R-134a, G 300 kg/m2-s, T 40°C, d 8 mm 1200 Gm st MF Mass ve oc ty (kg/m2-s) 1000 Xm n/m st XLA 800 q = 15 kW/m2 (Evaporat on) 600 I A 400 Gwavy Xm n/wavy 200 Gstart SW 0 0 0.2 0.4 0.6 Vapor qua ty Condensat on 0.8 1 F g. 6.27. F ow reg me map of E Ha a et a . (2003). Repr nted from E Ha a , J., Thome, J. R. and Cava n , A. Condensat on n hor zonta tubes, Part 1: Two-phase f ow pattern map, Internat ona Journa of Heat and Mass Transfer, 46(18), pp. 3349–3363 (2003) w th perm ss on from E sev er. w th severa of the w de y used f ow reg me maps d scussed n one of the prev ous sect ons of th s chapter. Qu te a b t of the agreement or otherw se s, however, sub ect to nterpretat on and def n t on of the respect ve f ow reg mes used by the var ous nvest gators. It s surpr s ng, however, that F g. 6.27 shows such a arge nterm ttent reg me for an 8-mm tube, even at qua t es as h gh as about 45% and mass f uxes as h gh as 1000 kg/m2 -s and above. It s unusua to see nterm ttent s ugs and p ugs at such h gh G, x va ues n such arge tubes. They state that the maps are app cab e for 16 < G < 1532 kg/m2 -s, 3.14 < D < 21.4 mm, 0.02 < pr < 0.8, 76 < (We/Fr)L < 884, and for a var ety of refr gerants and b ends. It shou d be noted, however, that the app cab ty of the maps down to 3.14 mm may need add t ona va dat on. Th s s because very few of the data on wh ch the maps are based were for such sma tubes, the vast ma or ty of the data be ng for approx mate y 8-mm tubes. A so, for examp e, Dobson and Chato s (1998) work, from wh ch much of the 3.14-mm data came, exp c t y stated that these data were assoc ated w th arge uncerta nt es. For the deve opment of the heat transfer mode , Thome et a . (2003) start w th an annu ar f ow formu at on. The turbu ent annu ar f ow forced-convect ve heat transfer equat on d scussed n E Ha a et a . (2003) s app ed around the per meter for annu ar f ow. (But t shou d be noted that near the entrance of the condenser, the qu d fract on s very ow, and the f m may n fact be am nar.) For strat f ed f ows, nstead of treat ng the qu d phase as a poo w th a f at upper surface at the bottom of the tube, they red str bute t as an annu ar r ng of th ckness , occupy ng the port on of the tube c rcumference that wou d y e d a qu d phase cross-sect on equ va ent to that of the strat f ed poo . The qu d phase crosssect on that shou d be red str buted s n turn determ ned from the vo d fract on corre at on descr bed n the compan on paper. The qu d f m th ckness n the f m condensat on reg on at the top s deemed neg g b e n th s re-apport on ng of areas. Th s transformat on Chapter 6. Condensat on n m n channe s and m crochanne s 337 a ows them to use a grav ty-dr ven Nusse t (1916) f m condensat on express on (w th an assumpt on of zero vapor shear) n the upper part of the tube, coup ed w th the forcedconvect ve annu ar f ow express on for the ower port on of the c rcumference for what wou d otherw se be a qu d poo . The rest of the mode s s mp y a we ghted average of these d fferent contr but ons based on the fract on of the c rcumference occup ed by the “strat f ed” port on: h= hfa ng−f m r + hconvect ve (2 − )r 2r (6.152) L ke most other mode s d scussed n th s sect on, a of the mu t p e reg mes dent f ed n the f rst part of the study are hand ed e ther as (a) fu y annu ar forced-convect ve, or as (b) cons st ng of vary ng comb nat ons of upper grav ty-dr ven, and ower forced-convect ve terms. It s stated that nterm ttent f ow s very comp ex, and s therefore assumed to be pred cted by annu ar f ow equat ons. S m ar y, m st f ow s hand ed as annu ar f ow, assum ng that the qu d nventory entra ned n the vapor phase can be v ewed as an unsteady annu ar f m. Bubb y f ow s not mode ed n th s work, part y because t s not common y encountered. The nove manner of hand ng the strat f ed poo does, however, y e d smoother trans t ons between the heat transfer coeff c ent pred ct ons across trans t ons. Further, the degree of strat f cat on s tse f obta ned through quadrat c nterpo at on across the f ow reg me trans t on nes, ensur ng a smooth ntroduct on of the strat f cat on component nto the heat transfer coeff c ent: Gwavy − G 0.5 = strat (6.153) Gwavy − Gstrat Another nnovat on ntroduced s the account ng of the effects of nterfac a waves, us ng concepts of the “most dangerous” wave ength from stab ty cons derat ons, and deve op ng an enhancement factor due to nterfac a roughness n terms of the s p rat o and the roughness amp tude: f = 1 + uV uL (L − V ) g2 k G Gstrat (6.154) The exponents and k are p cked to be ½ and ¼ w thout d scuss on or substant at on from phys ca pr nc p es. The presence of the surface tens on n the denom nator correct y mp es a decrease n the ntens ty (damp ng) of the nterfac a waves at h gh surface tens on va ues. In add t on, the ast term n the equat on ensures that the waves are progress ve y damped out upon approach to fu y strat f ed cond t ons, a though why th s term shou d be a near decrease (ev denced by the exponent of 1 for th s term) and not ra sed to another exponent s not made c ear. The authors present numerous graphs to demonstrate that the pred ct ons are not b ased toward underpred ct on or overpred ct on w th respect to the ndependent var ab es. The r mode pred cts the data from the arge database over a w de range of f u ds, operat ng cond t ons, and d ameters (w th the caut on stated above about the app cab ty to the 3.14-mm d ameter) we , w th 85% of the refr gerant heat transfer data pred cted w th n 20%. It s nterest ng to note that th s s about the same 338 Heat transfer and f u d f ow n m n channe s and m crochanne s Heat transfer coeff c ent (W/m2 k) 14,000 R-410A, d 8 mm, Tsat 40°C, q 40 kW/m2 G 30 kg/m2s G 200 kg/m2s G 500 kg/m2s G 800 kg/m2s 12,000 10,000 MF 8000 A 6000 A 4000 A SW 2000 S 0 0.2 0.4 0.6 Vapor qua ty 0.8 1 F g. 6.28. Heat transfer coeff c ent trends pred cted by the Thome et a . (2003) mode . Repr nted from Thome, J. R., E Ha a , J. and Cava n , A., Condensat on n hor zonta tubes, Part 2: New heat transfer mode based on f ow reg mes, Internat ona Journa of Heat and Mass Transfer, 46(18), pp. 3365–3387 (2003) w th perm ss on from E sev er. eve of pred ct ve accuracy as the ear er mu t -f ow reg me corre at ons proposed by Cava n et a . (2002a). Th s mode does offer some new approaches – heat transfer data are used to ref ne a vo d fract on mode , wh ch s then used to pred ct f ow reg me trans t ons, wh ch are n turn used to pred ct a “tuned b end” of strat f ed-convect ve heat transfer coeff c ents. It ach eves smooth y vary ng heat transfer coeff c ents across mu t p e reg mes (F g. 6.28) w th good accuracy, us ng some f rst pr nc p es concepts and severa key assumpt ons and emp r ca constants or curve-f tted terms at appropr ate ocat ons for c osure. Add t ona substant at on of the use of an average of homogeneous and Ste ner (1993)/Rouhan –Axe sson (1970) vo d fract ons, and the phys ca bas s for extens ons of trans t on nes determ ned from bo ng cons derat ons wou d end further credence to the mode . Measurement, rather than deduct on from heat transfer data, of the fundamenta under y ng quant ty (vo d fract on), to wh ch the authors cred t much of the success of th s approach, wou d a so enhance ts pred ct ve ab t es. Th s wou d be part cu ar y mportant f the mode were to be extended n the future to m crochanne s. Goto et a . (2003) measured heat transfer coeff c ents for the condensat on of R-410A and R-22 ns de f ve d fferent he ca and herr ngbone nterna y grooved tubes of about 8.00-mm OD. Loca heat transfer coeff c ents were measured for 130 < G < 400 kg/m2 -s and Tsat = 30 C and 40 C by d v d ng a 1-m ong test sect on nto 10 sma sect ons. Un ke most other stud es d scussed n th s chapter, the heat transfer tests were conducted n a comp ete vapor compress on system. The reported uncerta nt es n the heat transfer coeff c ents were very h gh, about 40%. Heat transfer coeff c ents of the herr ngbone grooved tube were found to be about tw ce as arge as those for the he ca grooved tubes. Us ng Koyama and Yu s (1996) Nu = (Nu2annu ar + Nu2strat f ed )0.5 corre at on for strat f ed Chapter 6. Condensat on n m n channe s and m crochanne s 339 and annu ar condensat on as a start ng po nt, they deve oped new express ons for the annu ar f ow port on based on the r data. A turbu ent qu d f m bas s, Nu = Re*L PrL /T * , √ * was used, w th the qu d Reyno ds number ReL = (L w /L D)/µL computed from the fr ct on ve oc ty w th the twophase mu t p er of Goto et a . (2001) v = 1 + 1.64Xtt0.79 for the computat on of the wa shear stress. Th s approach resu ted n fo ow ng corre at on for he ca and cross m crof ns: 0.1

# µL 0.1 x v Nuannu ar = 0.743 f ReL0.7 (6.155) Xtt µV 1−x wh e herr ngbone m crof n tubes requ red a ead ng constant of 2.34, and a Re exponent of 0.62. These corre at ons pred cted the data w th n 20%. Ind v dua equat ons were reported for each tube for the fr ct on factor requ red n the above equat on. 6.5.2. Condensat on n sma channe s There has been a s ow progress on toward the measurement and mode ng of heat transfer coeff c ents n sma channe s, and un ke the stud es on f ow reg mes, vo d fract on and pressure drop c ted n prev ous sect ons, there are few stud es that address Dh ≤ 1 mm. Yang and Webb (1996a) measured heat transfer n s ng e- and two-phase f ow of refr gerant R-12 at 65 C n rectangu ar p a n and m crof n tubes w th Dh = 2.64 and 1.56 mm, respect ve y, us ng the mod f ed W son p ot techn que. The compan on pressure drop study (Yang and Webb, 1996b) was d scussed n the prev ous sect on. The measurements were conducted at somewhat h gher mass f uxes (400 < G < 1400 kg/m2 -s) than those of nterest for refr gerat on and a r-cond t on ng app cat ons. They found that the Shah (1979) corre at on s gn f cant y overpred cted the data, wh e the Akers et a . (1959) corre at on showed better agreement, except at h gh mass f uxes, where t a so overpred cted the data. The m crof n tube heat transfer coeff c ents showed steeper s opes than those for the p a n tubes when p otted aga nst qua ty, espec a y at the ower mass f uxes. The heat transfer coeff c ents a so showed a heat f ux dependence (h ∝ q0.2 ); such heat f ux dependence s typ ca y seen n strat f ed f ows (wh ch s certa n y not the case at such h gh mass f uxes) and n bo ng. They exp a ned th s dependence based on the work of So man et a . (1968), who argued that the momentum contr but on w cause an ncrease n heat transfer coeff c ent when q s ncreased. The r resu ts a so showed that the heat transfer enhancement due to the m crof ns decreased w th ncreas ng mass f ux. By p ott ng the heat transfer data aga nst the equ va ent mass ve oc ty Reyno ds number proposed by Akers et a . (1959), they showed that wh e data for d fferent mass f uxes and qua t es co apsed to a s ng e curve for p a n tubes, th s on y occurred at ow qua t es for the m crof n tube. Th s ed them to exp a n the d fferent trends on the bas s of the dra nage of the qu d f m by the m crof ns. They reasoned that at h gh qua t es, the m crof ns are not fu y submerged n qu d; therefore, they act ke Gregor g (1962) surfaces, he p ng to dra n the f m to the base of the f ns, thus ncreas ng heat transfer over the p a n tube va ues. At ow qua t es, however, the f ns are submerged; therefore, the dra nage mechan sm s not act ve. Accord ng to them, th s mechan sm s part cu ar y mportant at ow mass f uxes, because at h gh mass f uxes, vapor shear exerts the dom nant nf uence on heat transfer. It shou d be noted 340 Heat transfer and f u d f ow n m n channe s and m crochanne s that n the compan on work, Yang and Webb (1996b) had conc uded that surface tens on d d not p ay a ro e n determ n ng the pressure drop n these same tubes. Yang and Webb (1997) cont nued th s work to deve op a heat transfer mode for condensat on of R-12 and R-134a n extruded m crochanne s (Dh = 1.41 and 1.56 mm) w th m crof ns (0.2- and 0.3-mm deep). The data for these channe s were reported n the r ear er papers c ted above. They used the observat ons d scussed above about the contr but on of vapor shear and surface tens on to represent the heat transfer coeff c ent as fo ows: h = hu Au Af + hf A A (6.156) where the subscr pts u and f refer to unf ooded and f ooded port ons of the channe , respect ve y. They ca cu ated the surface and cross-sect ona areas of the qu d n the dra nage reg ons us ng the geometr c features of the m crof n, and the Z v (1964) vo d fract on mode to est mate the qu d vo ume occupy ng the dra nage reg on. For f ooded cond t ons (e.g. at ow vapor qua t es) they used a s ght y mod f ed vers on of the Akers et a . (1959) mode (wh ch corresponds to the r exper menta y der ved s ng e-phase Nusse t number 0.73 Pr 1/3 ) to compute the shear-dr ven heat transfer coeff c ent. When the NuDh = 0.10ReD h m crof ns were unf ooded, they assumed a th n am nar f m dra n ng on the wa s of the m crof n and computed the pressure grad ents n the ax a and m crof n wa d rect ons due to vapor shear and surface tens on, respect ve y. These perpend cu ar vapor shear (sh) and surface tens on (st) stresses were comb ned to y e d the wa shear stress, wh ch was then re ated to the unf ooded heat transfer coeff c ent, h2u = h2sh + h2st . Re at ng the surface2 D )/( ), tens on-dr v ng force to the m crof n prof e and the Weber number We = (Geq h they deve oped the fo ow ng express on for the surface tens on contr but on to the heat transfer coeff c ent: hst = C z − d (1/r) Reeq Pr 1/3 k dp/dz ds We (6.157) where z refers to the ax a d rect on, subscr pt refers to the nterface, and C was der ved through regress on of the r data to be 0.0703. The re at ve fract ons of the surface tens on and vapor shear contr but ons to the heat transfer coeff c ent were p otted to show that at ow mass f ux (G = 400 kg/m2 -s) and x > 0.5, the surface tens on contr but on cou d equa and exceed the vapor shear term. At h gh mass f uxes (G = 1400 kg/m2 -s), the surface tens on contr but on was very sma . A sma f n t p rad us enhanced the heat transfer, wh e a arge nter-f n dra nage area a owed the surface tens on effect to be act vated at ower qua t es. K m et a . (2003) conducted a study s m ar to that of Yang and Webb (1996a, b; 1997) above, on s m ar tubes (Dh = 1.41 smooth, 1.56-mm m crof n w th 39% greater surface area than the smooth tube), w th a s m ar exper menta set up us ng R22 and R-410A for 200 < G < 600 kg/m2 -s at 45 C. Representat ve tube cross-sect ons and the test sect on used for the exper ments s shown n F g. 6.29. To test the tubes, they formed a acket around the channe w th a 1-mm gap. S nce the W son p ot techn que was used, t was mportant to m n m ze the coo ant-s de res stance (an ssue not recogn zed or stated by nvest gators often enough.) In the r tests, the coo ant s de const tuted approx mate y 1/3 of tota res stance. Further decreases cou d not be ach eved because the requ red h gher coo ant f ow rates wou d resu t n an extreme y sma coo ant temperature r se, wh ch Chapter 6. Condensat on n m n channe s and m crochanne s P rb e ro 341 g (a) M crof n tube Dh 1.56 mm (a) Smooth tube Dh 1.41 mm Inserts Water acket cover Water acket Water channe Test tube t b w F g. 6.29. Representat ve p a n and m crof n extruded channe s and test sect on used by Yang and Webb (1996a) and K m et a . (2003). Repr nted from K m, N.-H., Cho, J.-P., K m, J.-O. and Youn, B., Condensat on heat transfer of R–22 and R–410A n f at a um num mu t -channe tubes w th or w thout m cro-f ns, Internat ona Journa of Refr gerat on, 26(27), pp. 830–839 (2003) w th perm ss on from E sev er and Internat ona Inst tute of Refr gerat on ( f r@ f r.org or www. f r.org). wou d s gn f cant y decrease the accuracy of heat duty measurement. At the ow mass f ux cond t ons, the heat dut es were measured us ng a 1.3 C coo ant temperature r se. For R-22, the m crof n tubes y e ded h gher heat transfer coeff c ents than the smooth tube, w th the enhancement decreas ng to 1 at a mass f ux of 600 kg/m2 -s. For R-410A, the m crof n tube heat transfer coeff c ents were s m ar y h gher than those of the smooth tubes at ow mass f uxes, but n fact decreased to va ues ower than those for the smooth tube at G = 600 kg/m2 -s. Us ng the rat ona e prov ded by Carnavos (1979; 1980) for s ng e-phase performance n f nned tubes, they exp a ned th s decrease n heat transfer by postu at ng that f ns reduce the nter-f n ve oc ty, thus decreas ng heat transfer. They further po nt out that n condens ng f ows, the surface tens on dra nage force that s prom nent at ow mass f uxes compensates for the decrease n ve oc ty, whereas at h gh mass f ux, the ve oc ty reduct on effect dom nates. No deta ed va dat on of th s hypothes s s prov ded. They found a so that n smooth tubes, R-410A heat transfer coeff c ents were s ght y h gher 342 Heat transfer and f u d f ow n m n channe s and m crochanne s than those of R-22, wh e the oppos te was true for m crof n tubes. In smooth tubes, the h gher R-410A heat transfer coeff c ents were attr buted to ts arger therma conduct v ty and sma er v scos ty. In m crof n tubes, however, they noted that the Weber number of R-22 was 2.7 t mes ower than that of R-410A at the test cond t ons, ead ng to greater surface tens on dra nage and heat transfer. The two t mes arger vapor-to- qu d vo ume rat o of R22 was a so be eved to have resu ted n a arger fract on of the f n surface be ng exposed to the vapor, thus enhanc ng heat transfer. Severa other p aus b e exp anat ons n terms of the f u d propert es are prov ded, a though add t ona va dat on wou d be necessary to conf rm these exp anat ons of the sma d fferences n heat transfer coeff c ents, espec a y n v ew of the uncerta nt es n the measurement of heat transfer coeff c ents at these sma heat transfer rates. Based on these trends, they recommended the use of Moser et a . s (1998) mode w th a mod f ed two-phase mu t p er for the smooth tubes, and Yang and Webb s (1997) mode w th m nor mod f cat ons for the m crof n tubes. Yan and L n (1999) measured heat transfer coeff c ents dur ng condensat on of R-134a through a bank of 2-mm tubes arranged n para e . The r pressure drop resu ts were d scussed n the prev ous sect on. Copper p ates, 5-mm th ck, were so dered on to the bank of refr gerant tubes, and coo ant tubes were n turn so dered on to the copper p ates to prov de a crossf ow coo ng or entat on. The authors state that the tubes and the p ate had good therma contact, e m nat ng the need for avo d ng gaps between the tubes and the p ates. However, they prov de no substant at on that the therma res stance from the refr gerant to the coo ant through the refr gerant and coo ant tubes, the gaps between the p ates and the tubes, and the two 5-mm th ck copper p ates was e ther accounted for proper y, or was n fact sma n magn tude. The ssue of ma -d str but on through the 28 para e channe s was a so not d scussed n deta , except to state that the n et and out et headers were appropr ate y des gned. The r s ng e-phase fr ct on factors were s gn f cant y h gher than the pred ct ons of the B as us equat on, and the s ng e-phase heat transfer coeff c ents were a so h gher than the pred ct ons of the Gn e nsk (1976) corre at ons. Entrance ength effects and tube roughness may have accounted for some of th s, accord ng to the authors. Condensat on heat transfer coeff c ents were h gher at the ower saturat on temperatures, espec a y at the h gher qua t es. They attr buted th s to the ower therma conduct v ty of R-134a at the h gher temperatures. The heat transfer coeff c ents a so decreased s gn f cant y as the heat f ux was ncreased, part cu ar y at the h gher qua t es. Th s s not typ ca of condensat on heat transfer, where any heat f ux (or T ) dependence s usua y seen at the ower qua t es n grav tydom nated f ow. The graphs they presented to demonstrate the effect of mass f ux show nf ect on po nts where the heat transfer coeff c ent f rst ncreased sharp y as qua ty s ncreased from around 10–20%, fo owed by a f atten ng w th further ncreases n qua ty at the ower mass f uxes, f na y aga n ncreas ng w th qua ty. Adequate exp anat ons for these trends are not prov ded. They proposed the fo ow ng corre at on for the r data: hD −0.33 0.3 1.04 Pr Bo Re = 6.48Reeq k Bo = q hfg G (6.158) It s not c ear why the Reyno ds number as we as the equ va ent Reyno ds number accord ng to the Akers et a . (1959) mode were requ red to corre ate the data. Chapter 6. Condensat on n m n channe s and m crochanne s 343 Rectangu ar channe s w th Dh = 1.46 mm (1.50 × 1.40 mm) were nvest gated by Wang et a . (2002) for the condensat on of R-134a at 61.5–66 C over the mass f ux range 75–750 kg/m2 -s. The exper ments were conducted n 610-mm ong tubes w th f nned 10 mu t -port channe s coo ed by a r n crossf ow. F ow v sua zat on exper ments were a so conducted by rep ac ng one a um num wa of the tube w th a g ass w ndow to he p determ ne the app cab e f ow reg mes. The authors report n et qua t es to the test sect on, wh ch were contro ed over a w de range; however, the qua ty change across the test sect on was substant a , nd cat ng that for many of the tests, the exper ments d d not y e d oca heat transfer coeff c ents over sma ncrements of qua ty. From the comb nat on of f ow v sua zat on and heat transfer exper ments, they note that even n a channe th s sma , strat f ed (wavy) f ows were seen at the ow mass f uxes. The heat transfer coeff c ents were nsens t ve to qua ty at the ow mass f uxes. At the h gh mass f uxes, the dependence on qua ty was strong, s gn fy ng annu ar f ow. Of the ava ab e heat transfer corre at ons, the Akers et a . (1959) equ va ent Re corre at on was found to agree most w th the r data n the annu ar f ow reg me, wh e the Jaster and Kosky (1976) corre at on agreed best w th the r strat f ed f ow data. However, they noted that the Reeq concept of Akers et a . (1959) does not exp c t y account for the app cab e f ow reg me, because Reeq can be the same for d fferent f ow reg mes. They attempted to mprove the pred ct ons by deve op ng the r own strat f ed and annu ar f ow corre at ons. They separated the data nto strat f ed and annu ar reg mes us ng the Breber et a . (1980) and So man (1982; 1986) trans t on cr ter a. The r strat f ed–annu ar trans t on occurred at a ower gas superf c a ve oc ty ( g* = 0.24 m/s) than that pred cted by the Breber et a . map, because the corners of the rectangu ar channe a ded the format on of annu ar f ow. Th s ear er trans t on to annu ar f ows was a so observed w th respect to the So man map, wh e the annu ar–m st trans t on was not observed. For annu ar f ow, they conducted a boundary- ayer ana ys s s m ar to Trav ss et a . (1973), and through the ana ys s of the r data, proposed the fo ow ng curve-f ts for the two-phase mu t p er, and un ke most other researchers, for the d mens on ess temperature as we : v = 1.376 + 8Xtt1.655 T+

Re = 5.4269 x 0.2208 (6.159) The resu t ng annu ar f ow corre at on was as fo ows: Nuannu ar = 0.0274 Pr Re 0.6792 0.2208 x 1.376 + 8Xtt1.655 Xtt2 0.5 (6.160) For strat f ed f ows, they used the Chato (1962) corre at on for the f m condensat on port on, and the D ttus–Boe ter (1930) corre at on for forced convect on n the qu d poo , comb n ng the two as fo ows: Nustrat f ed = Nuf m + (1 − ) Nuconvect on (6.161) Therefore, they essent a y used the Z v (1964) vo d fract on as a measure of the qu d poo he ght n th s rectangu ar channe . In add t on, for des gn purposes, espec a y 344 Heat transfer and f u d f ow n m n channe s and m crochanne s s nce the r condensat on tests were conducted over arge qua ty changes, they proposed a we ghted average between these two corre at ons us ng the qua ty at So man Froude number Frso = 8 as the trans t on between the reg mes. Koyama et a . (2003b) conducted a study on the condensat on of R-134a n two mu t -port extruded a um num tubes w th e ght 1.11-mm channe s, and n neteen 0.80-mm channe s, respect ve y. Loca heat transfer coeff c ents were measured at every 75 mm of the 600-mm ong coo ng sect on us ng heat f ux sensors. They acknow edged that t s very d ff cu t to accurate y measure oca heat transfer coeff c ents n such sma channe s us ng the temperature r se n the coo ant or a W son p ot method due to the naccurac es assoc ated w th the measurement of ow heat transfer rates and sma temperature d fferences. The tests were conducted over the range 100 < G < 700 kg/m2 -s at 60 C. For corre at ng the r data, they used a comb nat on of convect ve and f m condensat on terms to y e d Nu = (Nu2annu ar + Nu2strat f ed )0.5 . The nd v dua annu ar and strat f ed terms were obta ned from the work of Haraguch et a . (1994a, b), except that the r two phase mu t p er was rep aced by the mu t p er proposed by M sh ma and H b k (1996). They stated that th s was an mprovement over severa other corre at ons; however, the graphs of pred cted and exper menta Nusse t numbers show cons stent and arge (as h gh as 80%) overpred ct ons of a the r data at a most a cond t ons. In some recent stud es, Wang, Rose, and co-workers have deve oped ana yt ca approaches for address ng condensat on heat transfer n tr angu ar (Wang and Rose, 2004) and square (Wang et a ., 2004; Wang and Rose, 2005) m crochanne s. For f m condensat on of R-134a n hor zonta 1-mm tr angu ar channe s, Wang and Rose (2004) assumed that the condensate was n am nar f ow, and deve oped one of the f rst mode s that accounts for surface tens on, shear stress, and grav ty. They were ab e to pred ct and substant ate the vary ng condensate f ow pattern across the cross-sect on as we as a ong the ength of the channe . Thus, they were ab e to mode the correspond ng var at ons n heat transfer coeff c ent. Us ng a s m ar approach, Wang et a . (2004) deve oped a mode for f m condensat on of R-134a n square, hor zonta , 1-mm m crochanne s. As these papers account for the three pr mary govern ng nf uences n m crochanne condensat on, they prov de a good start for the mode ng of phenomena spec f c to m crochanne s. Furthermore, the respect ve forces can be “sw tched on or off ” to understand the r respect ve s gn f cances (F g. 6.30). In another recent study, Cava n et a . (2005) conducted measurements of heat transfer coeff c ents and pressure drops dur ng condensat on of R-134a and R-410A ns de mu t p e para e 1.4-mm Dh channe s. The test sect on was d v ded nto three separate segments to prov de quas - oca pressure drops and heat transfer coeff c ents. They deduced the fr ct ona pressure drop from the measured drop n saturat on temperature and found good agreement between the data for R-134a and the corre at ons of Fr ede (1979), Zhang and Webb (2001), M sh ma and H b k (1996), and Mue er-Ste nhagen and Heck (1986). A of these corre at ons were found to overpred ct the R-410A data, however. The r heat transfer coeff c ents were obta ned from wa temperature measurements, and they found that the ava ab e mode s n the terature (Akers et a ., 1959; Moser et a ., 1998; Zhang and Webb, 2001; Cava n et a ., 2002a; Wang et a ., 2002; Koyama et a ., 2003a) underest mated the r resu ts, part cu ar y at h gh va ues of mass f ux. They attr buted these d fferences to the much h gher gas ve oc t es n the r exper ments where m st f ow m ght preva , whereas the ava ab e corre at ons are pr mar y for annu ar f ow n arger d ameter tubes. Chapter 6. Condensat on n m n channe s and m crochanne s 345 15 b Ts T G az (kW/m2-K) R-134a 10 1.0 mm 50°C 6K 500 kg/m2-s s 0, g 0 g0 t, s, g a nc uded 5 0 0 100 200 300 z (mm) 400 500 600 F g. 6.30. Var at on of mean heat transfer coeff c ent a ong a square 1-mm channe . Repr nted from Wang, H. S. and Rose, J. W., A theory of condensat on n hor zonta non-c rcu ar m crochanne s, ASME Journa of Heat Transfer, n press (2005). Sh n and K m (2005) used a techn que that matched the out et temperature of an e ectr ca y heated a r stream w th that of a s m ar a r stream heated by condens ng refr gerant R-134a to measure sma , oca condensat on heat transfer rates. C rcu ar and square channe s w th 0.5 < Dh < 1 mm were tested for the mass f ux range 100 < G < 600 kg/m2 -s at 40 C. For c rcu ar and square channe s, they a so found that most of the ava ab e mode s and corre at ons d scussed above (Akers et a ., 1959; So man et a ., 1968; Trav ss et a ., 1973; Cava n and Zecch n, 1974; Shah, 1979; Dobson, 1994; Moser et a ., 1998) underpred ct the r data at the ow mass f uxes, wh ch they deemed to be the mportant range for eng neer ng app cat ons. The agreement w th these corre at ons mproved somewhat at the h gher mass f uxes. A though they found no s gn f cant effect of the heat f ux, they noted that at ower mass f uxes, square channe s had h gher heat transfer coeff c ents than those for c rcu ar channe s, whereas the reverse was true for h gh mass f uxes. No sat sfactory exp anat on was prov ded for these trends. Gar me a and Bandhauer (2001) conducted heat transfer exper ments us ng the tubes (0.4 < Dh < 4.9 mm) that were used for the pressure drop exper ments of Gar me a et a . (2002; 2003a, b; 2005) descr bed above. They spec f ca y addressed the prob ems n heat transfer coeff c ent determ nat on due to the h gh heat transfer coeff c ents and ow mass f ow rates n m crochanne s. For the sma x requ red for oca measurements, the heat dut es at the mass f uxes of nterest are re at ve y sma , and to ensure reasonab e accurac es n heat duty measurement, the coo ant n et-to-out et temperature d fference must be ncreased to m n m ze uncerta nt es. However, s nce th s requ res ow coo ant f ow rates, the coo ant-s de therma res stance becomes dom nant, mak ng t d ff cu t to deduce the refr gerant-s de res stance from the measured UA. These conf ct ng requ rements for the accurate measurement of heat duty and the refr gerant heat transfer coeff c ents were reso ved by deve op ng a therma amp f cat on techn que (F g. 6.31) that decoup ed these two ssues. Thus, the test sect on was coo ed us ng water f ow ng n a c osed (pr mary) oop at a h gh f ow rate to ensure that the condensat on s de presented the govern ng therma 346 Heat transfer and f u d f ow n m n channe s and m crochanne s Secondary Low f ow rate f u d T M T Pr mary H gh f ow rate f u d oop Heat exchanger T M T Test sect on P T P Des red out et x Des red nt et x T F g. 6.31. Therma amp f cat on techn que. Repr nted from Bandhauer, T. M., Agarwa , A. and Gar me a, S., Measurement and mode ng of condensat on heat transfer coeff c ents n c rcu ar m crochanne s, Journa of Heat Transfer, n press (2006). res stance. Heat exchange between th s pr mary oop and a secondary coo ng water stream at a much ower f ow rate was used to obta n a arge temperature d fference, wh ch was n turn used to measure the condensat on duty. The secondary coo ant f ow rate was ad usted as the test cond t ons change to ma nta n a reasonab e T and a so sma condensat on dut es n the test sect on. By ensur ng that the pump heat d ss pat on n the pr mary oop and the amb ent heat oss were sma fract ons of the condensat on oad, the sens t v ty to these osses and ga ns was m n m zed. Loca heat transfer coeff c ents were therefore measured accurate y n sma ncrements for the ent re saturated vapor– qu d reg on. Add t ona deta s of th s therma amp f cat on techn que are prov ded n Gar me a and Bandhauer (2001). The therma amp f cat on resu ted n uncerta nt es typ ca y as ow as ±2% n the measurement of the secondary oop heat duty even at the extreme y sma heat transfer rates under cons derat on. Comb n ng the errors n the secondary oop duty, the pump heat add t on, and the amb ent heat oss, the oca condensat on duty n these sma channe s was typ ca y known to w th n a max mum uncerta nty of ±10%. The arge coo ant f ow rate and the enhancement n surface area ( nd rect area of about 4.7 t mes the d rect area) prov ded by the coo ant port wa s on both s des of the m crochanne tube resu ted Chapter 6. Condensat on n m n channe s and m crochanne s 347 Heat transfer coeff c ent (W/m2-K) 14,000 G 150 kg/m2-s G 300 kg/m2-s G 450 kg/m2-s G 600 kg/m2-s G 750 kg/m2-s 12,000 D 0.506 mm D 0.761 mm D 1.520 mm 10,000 8000 6000 4000 2000 0 0.1 0.2 0.3 0.4 0.5 Qua ty (x) 0.6 0.7 0.8 0.9 F g. 6.32. M crochanne condensat on heat transfer coeff c ents. From Bandhauer, T. M., Agarwa , A. and Gar me a, S., Measurement and mode ng of condensat on heat transfer coeff c ents n c rcu ar m crochanne s, Journa of Heat Transfer, n press (2006). n h gh refr gerant-to-coo ant res stance rat os (between 5 and 30). W th th s h gh res stance rat o, even an uncerta nty of ±25% n the tube-s de heat transfer coeff c ent d d not apprec ab y affect the refr gerant-s de heat transfer coeff c ent, ensur ng ow uncerta nt es. Representat ve heat transfer coeff c ents reported by Bandhauer et a . (2005) are shown n F g. 6.32. In genera , the heat transfer coeff c ent ncreases w th decreas ng d ameter, w th the effect of d ameter becom ng more s gn f cant at the h gher qua t es (~x > 0.45) and mass f uxes. Thus, h ncreases between 10% and 40% for x > 0.45 as Dh s reduced from 1.524 to 0.506 mm, w th the effect of Dh be ng more s gn f cant for a decrease n Dh from 0.761 to 0.506 mm. Grav tydr ven corre at ons were found to be poor pred ctors of the data, w th the Chato (1962) corre at on underpred ct ng the data cons derab y, the Jaster and Kosky (1976) corre at on resu t ng n s gn f cant overpred ct on, and the Rosson and Myers (1965) and Dobson and Chato (1998) corre at ons be ng somewhat better – these atter corre at ons account for heat transfer through the bottom of the tube. Th s s not surpr s ng because these mode s are not appropr ate for Gar me a and Bandhauer s data, wh ch were demonstrated to be not n the strat f ed wavy reg mes (Co eman and Gar me a, 2000a; 2003). Among corre at ons that use a two-phase mu t p er approach, the Moser et a . corre at on pred cted the data reasonab y we (14% average dev at on), w th the Shah (1979) and Cava n and Zecch n (1974) corre at ons resu t ng n some overpred ct on. Homogeneous f ow corre at ons of Boyko and Kruzh n (1967) and the m st f ow corre at on of So man (1986) were better at pred ct ng the data, w th the So man corre at on ead ng to some overpred ct on. Shear-dom nated annu ar f ow corre at ons us ng boundary- ayer treatments a so showed reasonab e agreement: So man et a . (1968) (18% average dev at on), Chen and Kocamustafaogu ar (1987) (13% average dev at on.) However, n sp te of the theoret ca bas s of the Trav ss et a . (Trav ss and Rohsenow, 1973; Trav ss et a ., 1973) corre at on, as reported by severa nvest gators, th s mode showed the argest average dev at on (38%) among shear-dr ven treatments. 348 Heat transfer and f u d f ow n m n channe s and m crochanne s Bandhauer et a . (2005) noted that dur ng the condensat on process, the f ow changes from m st f ow (where app cab e) to annu ar f m f ow to nterm ttent, w th arge over aps n the types of f ow resu t ng n trans t on f ow ( nterm ttent/annu ar, nterm ttent/annu ar/ m st, and annu ar/m st). Based on the work of Co eman and Gar me a (2000a; 2003), wavy f ow was not expected n any of the channe s cons dered. Due to the arge over ap reg ons between these f ows, w th the bu k of the data be ng n trans t on between annu ar f ow and other reg mes, they deve oped a heat transfer mode based on annu ar f ow cons derat ons. They noted that many of the ava ab e shear-dr ven mode s, though sound n formu at on, ed to poor pred ct ons because of the nadequate ca cu at on of shear stresses us ng pressure drop mode s that were not app cab e to m crochanne s. Thus, the r mode s based on boundary- ayer ana yses, but w th the requ s te shear stress be ng ca cu ated from the pressure drop mode s of Gar me a et a . (2005) deve oped spec f ca y for m crochanne s. S nce n annu ar f ow, the condensate ayer m ght become turbu ent at very ow Re (~240) (Carpenter and Co burn, 1951) due to the turbu ent vapor core and the destab z ng effect of condensat on (Sch cht ng and Gersten, 2000), a mode based on turbu ent parameters w th appropr ate √ mod f cat ons for the cond t ons under study was deve oped. The fr ct on ve oc ty u* = / was expressed n terms of the nterfac a shear stress, rather than the common y used wa shear stress. The turbu ent d mens on ess temperature was def ned as fo ows: T+ = × Cp × u * (T − Tw ) q (6.162) The shear stress and heat f ux were expressed n the usua manner: = (µ + × m ) du dy (6.163) q = −(k + h × × Cp) dT dy (6.164) Assum ng that the a of the heat s transferred n the qu d f m resu ts n the fo ow ng express on for the heat transfer coeff c ent, where the nterface s at the saturat on temperature: h= q × Cp × u * = (Tsat − Tw ) T+ (6.165) The heat f ux above y e ds the d mens on ess temperature grad ent: dT + = dy+

1 × h + Pr µ −1 (6.166) √ To ntegrate th s, the turbu ent f m th ckness = (1 − )D/2 s determ ned us ng the Baroczy (1965) mode , wh ch s d fferent from the manner n wh ch Trav ss et a . (Trav ss and Rohsenow, 1973; Trav ss et a ., 1973) determ ned f m th ckness. The d mens on ess Chapter 6. Condensat on n m n channe s and m crochanne s 349 turbu ent f m th ckness s then + = ( u * )/µ . A so = (1 − y/R)w y e ds the fo ow ng equat on: 1 − y+ /R+ m = −1 µ du+ /dy+ (6.167) In a manner ana ogous to the deve opment by Trav ss et a . (1973), w th the assumpt ons of sma f m th ckness compared to the tube rad us, and m ~ = h , the fo ow ng s mp f ed two-reg on turbu ent d mens on ess temperature express ons were proposed: for Re < 2100: + −1 +1 T + = 5Pr + 5 n Pr 5 (6.168) for Re > 2100: T + = 5Pr + 5 n(5Pr + 1) + + 30

1 Pr

dy+ −1 + y+ 5

1− y+ R+

(6.169) The on y unknown quant ty, the nterfac a shear stress for determ n ng u* , s obta ned from the annu ar f ow port on of the mu t p e-f ow reg me pressure drop mode by Gar me a et a . (2005) d scussed n the prev ous sect on. Thus, the nterfac a fr ct on factor (and then the nterfac a shear stress) s computed from the correspond ng qu d-phase Re and fr ct on factor, the Mart ne parameter, and the surface tens on parameter: f /f = AX a Reb c . Add t ona deta s are ava ab e n the prev ous sect on. In summary, to obta n the heat transfer coeff c ent, the nterfac a shear stress s f rst ca cu ated us ng the pressure drop mode . The resu t ng shear stress s used to compute the fr ct on ve oc ty u* and the d mens on ess f m th ckness + . The d mens on ess temperature T + s then ca cu ated, wh ch y e ds the heat transfer coeff c ent. Th s mode pred cted 86% of the data w th n ±20%, w th an average abso ute dev at on of 10%. A more exp c t account ng of the heat transfer n the qu d s ugs, vapor bubb es, and the f m-bubb e nterface n nterm ttent f ow, and of the entra nment of qu d nto the vapor core n m st f ow wou d mprove the pred ct ons further. The data and the pred ct ons at representat ve mass f uxes for each tube are shown n F g. 6.33. The heat transfer coeff c ent ncreases w th an ncrease n mass f ux and qua ty, and w th a decrease n the tube d ameter. As the qu d f m becomes th nner w th ncreas ng vapor qua ty, the heat transfer coeff c ent ncreases. The steep s ope at the h gher qua t es represents an approach to a van sh ng y th n f m. However, t must be noted that the actua behav or n th s reg on w be dependent on the n et cond t ons of a condenser, nc ud ng the n et superheat, the coo ant temperature, and the consequent wa subcoo ng – these phenomena are not accounted for n th s mode . A though the mode was deve oped based on an annu ar f ow mechan sm, t appears to pred ct the heat transfer coeff c ents n the m st and m st–annu ar over ap reg ons n the h gh mass f ux and h gh-qua ty cases adequate y. At the extreme end of the graph (x > 80%), the f ow may be exc us ve y n the m st f ow reg on. For a g ven f ow rate, the qua ty at wh ch the m st f ow reg me occurs cou d 350 Heat transfer and f u d f ow n m n channe s and m crochanne s 14,000 Data mode G 450 kg/m2-s 0.506 mm 0.761 mm 1.524 mm 12,000 10,000 8000 6000 4000 2000 0 Pred cted and exper menta h (W/m2-K) 14,000 12,000 G 600 kg/m2-s 10,000 8000 6000 4000 2000 0 14,000 12,000 G 750 kg/m2-s 10,000 8000 6000 4000 2000 0 0.0 0.2 0.4 0.6 0.8 1.0 Qua ty (x) F g. 6.33. M crochanne condensat on heat transfer coeff c ent pred ct ons. From Bandhauer, T. M., Agarwa , A. and Gar me a, S., Measurement and mode ng of condensat on heat transfer coeff c ents n c rcu ar m crochanne s, Journa of Heat Transfer, n press (2006). decrease w th decreas ng d ameter due to ncreased nterfac a shedd ng. As qu d entra nment ncreases, the heat transfer coeff c ent w ncrease due to the th nn ng of the qu d f m (So man, 1986). A though qu d entra nment and m st format on are not exp c t y accounted for n the heat transfer part of th s mode , the trends n the data (the steeper s ope n the data at the h gher qua t es, part cu ar y for sma Dh ) can be nterpreted on th s bas s. Chapter 6. Condensat on n m n channe s and m crochanne s 351 6.5.3. Summary observat ons and recommendat ons The above d scuss on of the ava ab e terature on condensat on heat transfer shows that much of the ava ab e nformat on s on tubes arger than about 7 mm. In these tubes, heat transfer mode s have treated the mu t tude of f ow reg mes dent f ed by many nvest gators and d scussed n the prev ous sect ons as e ther strat f ed or annu ar f ow reg mes. Thus, a most a the ava ab e mode s n essence st use on y grav ty- or shear-dom nated approaches. What d ffers from nvest gator to nvest gator s the spec f c ana ys s that addresses these two modes. In mode s of grav ty-dom nated condensat on for examp e, the ear est mode s were patterned very c ose y after the Nusse t condensat on ana ys s. They then progressed from neg ect ng the heat transfer n the qu d poo to account ng for t n some manner, usua y through the app cat on of s ng e-phase qu d forced-convect ve heat transfer corre at ons. The apport on ng of f m condensat on and forced convect on reg ons w th n th s strat f ed f ow has a so d ffered n the m nor deta s. Most of these mode s use some vo d fract on mode , usua y the one by Z v (1964), to determ ne the qu d and vapor fract ons of the cross-sect on (wh ch s usua y quant f ed as the qu d poo ang e), wh ch then a ows them to pred ct the surface area over wh ch these two submode s are app ed. In many cases, however, the vo d fract on mode s were or g na y based on annu ar f ow data or ana yses. A few mode s have a so attempted to account for vapor shear on the fa ng-f m condensat on port on of the tube. In annu ar f ow mode s, the techn que of re at ng the nterfac a shear (through a boundary- ayer ana ys s that usua y uses the von Karman ve oc ty prof e) to the heat transfer across the qu d f m has been used w de y, start ng w th the work of Carpenter and Co burn (1951), w th ref nements and mprovements to the contr but on of var ous forces to the nterfac a shear, for examp e by So man (1968), Trav ss et a . (1973), and others. There have a so been some attempts such as the work of Chen et a . (1987) to deve op genera purpose annu ar f ow corre at ons start ng w th asymptot c m ts, and b end ng them through some s mp e comb nat on of the terms at the respect ve m ts. Another approach has been to d rect y app y a mu t p cat on factor to a s ng e-phase qu d or vapor phase heat transfer corre at on, w th the mu t p er determ ned us ng adaptat ons of the Lockhart– Mart ne (1949) and Fr ede (1979) pressure drop mu t p ers. Th s approach s n fact ana ogous to the boundary- ayer shear stress ana ys s used by many nvest gators. The on y essent a d fference s n where the mu t p er s app ed: the f rst type of mode s use the mu t p er f rst for shear stress determ nat on, and ca cu ate the heat transfer coeff c ent d rect y thereafter; the second type ca cu ate the s ng e-phase heat transfer coeff c ent d rect y, and app y the mu t p er thereafter. An exp anat on of the s m ar ty of these two approaches s prov ded n Dobson and Chato (1998). One corre at on that cont nues to be used frequent y s the pure y emp r ca Shah (1979) corre at on, because of ts s mp c ty, the w de range of data that were ut zed n ts deve opment, and ts comparat ve y good pred ct ons for annu ar f ows. Another mode that has been used w de y unt recent y s the Akers et a . (1959) techn que of determ n ng an equ va ent mass f ux that wou d prov de the same shear as the two-phase f ow; thus rep ac ng the vapor core w th an add t ona qu d f ow rate, and then treat ng the comb ned f ux as be ng n s ng e-phase f ow. At the t me of ts deve opment, th s approach was perhaps mot vated by the des re to fac tate computat on of heat transfer coeff c ents n a manner more fam ar 352 Heat transfer and f u d f ow n m n channe s and m crochanne s to most des gners ( .e. s ng e-phase ana ys s). However, recent papers have shown that the pred ct ons of these mode s are not very good, and a so that the appropr ate fr ct on factors and dr v ng temperature d fference are not app ed when transform ng the two-phase f ow to an equ va ent s ng ephase f ow. A though corrected vers ons (Moser et a ., 1998) of th s mode are now ava ab e, mp ement ng the corrected vers ons renders them as nvo ved as the boundary- ayer ana yses, and does not seem to offer any add t ona ease of use. In recent years, arge exper menta efforts or ana yses of data from mu t p e researchers have been undertaken to deve op mode s that address the ent re process of condensat on from n et to out et, spann ng a w de range of mass f uxes, d ameters, and f u ds. Representat ve mode s nc ude Dobson and Chato (1998), Cava n et a . (2002a), and Thome et a . (2003). Wh e these stud es have y e ded smoother, and n genera , more accurate pred ct ons over a w de range of cond t ons, they too st group the mu t tude of f ow reg mes nto on y strat f ed/wavy and annu ar f ows. Th s s n stark contrast to the f ow v sua zat on stud es d scussed n prev ous sect ons, where the tendency s more toward dent fy ng and categor z ng every s ng e nuance of the f ow structure. Important f ow reg mes (and ncreas ng y so w th the progress on toward sma er d ameter channe s) such as nterm ttent and m st f ow have, unt now, not been successfu y mode ed, except as unsubstant ated extens ons of annu ar f ow. A quas -homogeneous mode of So man (1986) s one of the few examp es of a mode ded cated to one such f ow reg me (m st). The ava ab ty of re ab e oca condensat on heat transfer measurements and mode s n m crochanne s of D < 3 mm s espec a y m ted. (The wea th of grav ty-dom nated condensat on mode s n arge tubes are of tt e re evance to condensat on n m crochanne s.) A group of researchers ed by Webb (Yang and Webb, 1996a, b; 1997; Webb and Erm s, 2001; Zhang and Webb, 2001) has made some progress n measur ng heat transfer coeff c ents n extruded a um num tubes w th mu t p e para e ports of Dh < 3 mm, a though the mass f uxes they have focused on are at the h gh end of the range of nterest for common refr gerat on and a r-cond t on ng app cat ons. In these papers, they have attempted severa d fferent approaches to mode ng the heat transfer coeff c ents nc ud ng typ ca shear stress mode s and equ va ent mass f ux mode s. A re ab e mode that pred cts and exp a ns the var ety of trends seen n these resu ts has however yet to be deve oped. Wh e surface tens on has been ment oned rout ne y as an mportant and govern ng parameter n m crochanne f ows, there s a most no mode that exp c t y tracks surface tens on forces on a f rst pr nc p es bas s from pressure grad ent to heat transfer eva uat on. At best, some nvest gators such as Bandhauer et a . (2005) have nd rect y accounted for surface tens on through a surface tens on parameter n the pressure drop mode used for the shear stress ca cu at on, wh ch has y e ded accurate m crochanne heat transfer pred ct ons over a w de range of cond t ons. A so, Yang and Webb (1997) are one of the few researchers who exp c t y account for surface tens on forces n m crochanne s (w th m crof ns) by comput ng the dra nage of the qu d f m from the m crof n t ps and the assoc ated heat transfer enhancement when the f n t ps are not f ooded. The ana yt ca treatments proposed recent y by Wang et a . (Wang and Rose, 2004; Wang et a ., 2004) for m crochanne s w th D ~ 1 mm that account for the comb ned nf uence of surface tens on, shear, and grav ty as condensat on proceeds a so ho d prom se. The use of representat ve heat transfer corre at ons and mode s d scussed n th s sect on s ustrated n Examp e 6.4. Chapter 6. Condensat on n m n channe s and m crochanne s 353 6.6. Conc us ons Th s chapter has prov ded an overv ew of the ava ab e terature that s re evant to the mode ng of condensat on n m crochanne s. M crochanne def n t ons and govern ng nf uences, f ow reg mes and trans t on cr ter a, vo d fract on corre at ons, pressure drop and heat transfer were addressed. Based on the nformat on presented, t can be sa d that the bu d ng b ocks for the mode ng of condensat on n m crochanne s are s ow y becom ng ava ab e, a though the bu k of the ava ab e nformat on s on arger channe s. These macrochanne stud es and mode s serve as a start ng po nt for correspond ng stud es on m crochanne s. By far the most progress on address ng m crochanne f ows has been n the area of f ow reg me determ nat on: ad abat c and condens ng f ows have been documented n deta n m n channe s, wh e n channe s w th Dh < ~3 mm, there are few stud es of condens ng f ows, a most a the research be ng on ad abat c two-phase f ow. It s not c ear whether these resu ts for ad abat c f ows can be re ab y extrapo ated to refr gerants w th ower surface tens on, arger vapor dens t es, and other property d fferences. Vo d fract ons requ red for c osure of most heat transfer and pressure drop mode s have pr mar y been for annu ar f ows, w th recent advances y e d ng some emp r ca corre at ons based on v deo ana ys s for ad abat c f ows n channe s of Dh » 1 mm and ower. Mode s for pressure drops dur ng condensat on are pr mar y ava ab e for channe s w th Dh > about 3 mm, and they cont nue to use two-phase mu t p er approaches, w th some mod f cat ons to the mu t p ers to match the data. A few f ow reg me-based pressure drop mode s for condens ng f ows n m crochanne s are beg nn ng to emerge. Heat transfer coeff c ents for condensat on present add t ona cha enges that must be addressed w th great care wh e des gn ng and runn ng exper ments. The w de y used W son p ot techn ques that were ub qu tous y used for arge channe s are m ted n the r ut ty for sma channe s. Contro ed exper ments that can measure heat transfer rates of the order of a few watts for oca condensat on ( x » 0.05) n m crochanne s are part cu ar y cha eng ng. Innovat ve techn ques, nc ud ng non- ntrus ve measurements, and attent on to the many sources of uncerta nty are essent a to ensure prov d ng an accurate database for wh ch mode s can be deve oped. As stated n th s chapter, th s cha enge of sma heat dut es s coup ed w th the arge condens ng heat transfer coeff c ents, wh ch mp y that heat s often be ng transferred across T < ~0.5 C. Any apprec ab e uncerta nt es n temperature measurement wou d render the resu t ng heat transfer coeff c ents unre ab e and naccurate. On the ana ys s front, mode s for heat transfer coeff c ents ref ect the cumu at ve effects of re ab e or unre ab e estab shment of f ow reg mes, mode ng of vo d fract ons, and computat on of shear stress through pressure drop, and of course uncerta nt es n the measurement of heat transfer coeff c ents. Thus, mode s that account for the under y ng phenomena, nc ud ng surface tens on cons derat ons un que to m crochanne s, are part cu ar y mportant. The research needs a so extend to pract ca ssues such as ma -d str but on of refr gerant f ows through mu t p e channe s – many nnovat ve des gns fa or are nfeas b e because adequate cons derat on s not g ven to estab sh ng un form d str but on of two-phase f u d through para e channe s. S m ar y, mode s for “m nor” osses through expans ons, bends, and contract ons shou d be deve oped to prov de the des gner a comp ete set of too s. Env ronmenta and other concerns w cont nue to change the refr gerants that are acceptab e and a owab e for use n pract ca a r-cond t on ng and refr gerat on systems. 354 Heat transfer and f u d f ow n m n channe s and m crochanne s Therefore, the mode s deve oped must be robust enough to be used for a var ety of f u ds, nc ud ng refr gerant b ends. If the chosen refr gerants are zeotrop c b ends, techn ques for hand ng the add t ona mass transfer res stance must e ther be va dated us ng arge channe methodo og es as a bas s, or new mode ng techn ques shou d be deve oped. The acceptab e synthet c refr gerant b ends and natura refr gerants such as CO2 are ncreas ng y h gher-pressure refr gerants; therefore, the effect of the approach to cr t ca pressure on condensat on phenomena requ res spec a attent on. A so, f u ds such as CO2 and steam have h gh therma conduct v t es, spec f c heats, and atent heat of condensat on, wh ch make measurement of the resu t ng h gher heat transfer coeff c ents part cu ar y cha eng ng. These are but a few of the foreseeab e research cha enges and needs; g ven the state of nfancy of understand ng and mode ng condensat on n m crochanne s, a cons derab e amount of carefu effort s requ red to obta n a comprehens ve understand ng. The ach evement of such a fundamenta understand ng of condensat on at the m crosca es w y e d far reach ng benef ts not on y for the a r-cond t on ng and refr gerat on ndustr es, but a so for other as-yet untapped app cat ons such as portab e persona coo ng dev ces, hazardous duty, and h gh amb ent a r-cond t on ng, sensors, and med ca /surg ca dev ces, to name a few. Examp e 6.1. F ow reg mes determ nat on Determ ne the app cab e f ow reg me for the f ow of refr gerant R-134a n a 1 mm d ameter tube at a pressure of 1500 kPa, at mass f uxes of 150, 400, and 750 kg/m2 -s and qua t es of 0.2, 0.5, and 0.8. Use the f ow reg me maps and trans t on cr ter a of Sardesa et a . (1981), Tandon et a . (1982), Dobson and Chato (1998), Breber et a . (1980), So man (1982; 1986), Co eman and Gar me a (2000a, b; 2003) and Gar me a (2004), and Cava n et a . (2002a), and compare and comment on the f ow reg mes pred cted by each mode . Refr gerant propert es Saturat on temperature Surface tens on Tsat = 55.21 C = 0.00427 N/m L qu d phase Vapor phase = 1077 kg/m3 µ = 1.321 × 10−4 kg/m-s v = 76.5 kg/m3 µv = 1.357 × 10−5 kg/m-s For G = 400 kg/m2 -s and x = 0.5: L qu d Reyno ds number, Re = GD(1 − x) 400 × 0.001 × (1 − 0.5) = = 1514 µ 1.321 × 10−4 Chapter 6. Condensat on n m n channe s and m crochanne s 355 Vapor Reyno ds number, GDx 400 × 0.001 × 0.5 = = 14743 µv 1.357 × 10−5 Rev = Vapor-on y Reyno ds number, GD 400 × 0.001 = = 29486 µv 1.357 × 10−5 Revo = Us ng the Church (1977) corre at on for the s ng e-phase fr ct on factors, assum ng a smooth tube, f = 0.0423, and fv = 0.0279. The correspond ng s ng e-phase pressure drops are:

dP dz dP dz = f G 2 (1 − x)2 0.0423 × 4002 (1 − 0.5)2 = = 784.6 Pa/m 2D 2 × 0.001 × 1077 = fv G 2 x2 0.0279 × 4002 × 0.52 = 7304 Pa/m = 2Dv 2 × 0.001 × 76.5

v The pred ct ons from each mode sted above are computed as fo ows. Sardesa et a . (1981) Mart ne parameter, X = (dP/dz) (dP/dz)v 1/2 = 784.6 7304 1/2 = 0.3278 Superf c a gas ve oc ty, v = xG 0.5 × 400 = = 2.615 m/s v 76.5 Mod f ed Froude number: F= v v ×√ = − v Dg

76.5 2.615 ×√ = 7.298 1077 − 76.5 0.001 × 9.81 Funct on = (0.7X 2 + 2X + 0.85)F = (0.7 × 0.32782 + 2 × 0.3278 + 0.85) × 7.298 = 11.54. The correspond ng va ues for the other cond t ons are a so ca cu ated n a s m ar manner, and the reg mes are ass gned as fo ows: < 1.75: Strat f ed and strat f ed/wavy f ow ≥ 1.75: Annu ar f ow 356 Heat transfer and f u d f ow n m n channe s and m crochanne s The resu t ng f ow reg mes are sted and p otted be ow. G x X

F ow reg me 150 150 150 400 400 400 750 750 750 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 1.631 0.4666 0.01974 1.024 0.3278 0.1375 1.212 0.3527 0.1082 6.538 5.297 5.571 10.61 11.54 13.29 23.55 22.47 23.53 Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar F ow reg me map by Sardesa et a . Froude number (F) 102 Annu ar 101 X 1.6 100 Strat f ed and strat f ed/wavy 101 102 103 b 1.75 102 101 Mart ne parameter (X) 100 101 Tandon et a . (1982) For the samp e cond t on w th G = 400 kg/m2 -s and x = 0.5, the requ red parameters are ca cu ated be ow. D mens on ess gas ve oc ty: g* = # Gx Dgv ( − v ) =# 400 × 0.5 0.001 × 9.81 × 76.5(1077 − 76.5) Vo d fract on, ì é ùü / 1−x ï−1 0 ï í ý 0 + 0.4 v 1 − x ê x ú 1 v = 1+ 1−x û ë0.4 + 0.6 ï ï x 1 + 0.4 x î þ = 7.298 Chapter 6. Condensat on n m n channe s and m crochanne s ì í 76.5 Þ= 1+ î 1077 Þ

357 / é −1 1−0.5 ùü 0 1077 ý 0 + 0.4 1 − 0.5 ë = 0.858 0.4 + 0.61 76.5 0.5 û þ 0.5 1 + 0.4 1−0.5 0.5 1− = 0.1654 In a s m ar manner, these parameters are computed for each of the cond t ons, and the f ow reg mes ass gned accord ng to the trans t on cr ter a sted n Sect on 6.2.2. G x g* 1− F ow reg me 150 150 150 400 400 400 750 750 750 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 1.095 2.737 4.379 2.919 7.298 11.68 5.473 13.68 21.89 0.5322 0.1654 0.04535 0.5322 0.1654 0.04535 0.5322 0.1654 0.04535 Undes gnated Annu ar and sem -annu ar Annu ar and sem -annu ar Undes gnated Spray Spray Undes gnated Spray Spray F ow reg me map by Tandon et a . 10 Spray Annu ar and sem -annu ar 6.0 1.0 1 0.5 g* 0.5 0.1 S ug Wavy 0.01 0.01 P ug 0.001 0.001 0.01 0.1 1a a 1 10 358 Heat transfer and f u d f ow n m n channe s and m crochanne s Dobson and Chato (1998) For the samp e cond t on G = 400 kg/m2 -s and x = 0.5, the requ red parameters are as fo ows. Xtt = 1−x x Þ Xtt = 0.9 1 − 0.5 0.5 v 0.5 0.9 µ µg 76.5 1077 0.1 0.5 1.321 × 10−4 1.357 × 10−5 0.1 = 0.3346 Vo d fract on: 1−x = 1+ x

2/3 −1 v

1 − 0.5 Þ= 1+ 0.5 76.5 1077 2/3 −1 = 0.8536 Ga eo number, √ Ga = g ( − v ) D 3 µ2 √ Þ Ga = 9.81 × 1077(1077 − 76.5) Fr so = 0.8536 × 0.001 3 (1.321 × 10−4 )2 ì 1.5 ï 1 1 + 1.09Xtt0.039 ï 1.59 ï 0.025 × Re ï ï í Xtt Ga0.5 ï ï 0.039 1.5 ï 1 ï 1.04 1 + 1.09Xtt ï 1.26 × Re î Xtt Ga0.5 = 478,182 for Re ≤ 1250 for Re > 1250 Therefore for Re = 1514, Fr so = 1.26 × Re 1.04 1 + 1.09Xtt0.039 Xtt Þ Fr so = 1.26 × 1514 1.04

1.5 1 Ga0.5 1 + 1.09 × 0.33460.039 0.3346 1.5 1 = 55.86 478,1820.5 Chapter 6. Condensat on n m n channe s and m crochanne s 359 In a s m ar manner, these parameters are computed for each of the cond t ons, and the f ow reg mes ass gned accord ng to the trans t on cr ter a based on Frso and G sted n Sect on 6.2.2. The respect ve f ow reg mes are fo ows: G x Frso F ow reg me 150 150 150 400 400 400 750 750 750 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 5.797 13.08 17.49 19.12 55.86 83.17 36.77 107.4 226 Wavy Wavy Wavy Wavy Annu ar Annu ar Annu ar Annu ar Annu ar F ow reg me map by Dobson and Chato Mass f ux, G (kg/m2-s) 800 600 Annu ar 500 kg/m2-s 400 Frso 20 200 Wavy Frso 10 0 0.0 0.2 0.4 0.6 Qua ty (x) 0.8 1.0 Breber et a . (1980) For the samp e cond t on G = 400 kg/m2 -s and x = 0.5, the d mens on ess gas ve oc ty, g* = 7.298, and the Mart ne parameter Xtt = 0.3346 from the ca cu at ons above. The respect ve f ow reg mes are ass gned based on the spec f c va ues of these parameters as shown be ow. 360 Heat transfer and f u d f ow n m n channe s and m crochanne s g* G x X F ow reg me 150 150 150 400 0.2 0.5 0.8 0.2 1.165 0.3346 0.0961 1.165 1.095 2.737 4.379 2.919 400 400 750 0.5 0.8 0.2 0.3346 0.0961 1.165 7.298 11.68 5.473 750 750 0.5 0.8 0.3346 0.0961 13.68 21.89 Trans t on between a four reg mes Annu ar and m st annu ar Annu ar and m st annu ar Trans t on from annu ar and m st annu ar f ow reg me to bubb e f ow reg me Annu ar and m st annu ar Annu ar and m st annu ar Trans t on from annu ar and m st annu ar f ow reg me to bubb e f ow reg me Annu ar and m st annu ar Annu ar and m st annu ar F ow reg me map by Breber et a . Trans t on 101 Annu ar and m st annu ar Bubb e 1.5 100 Trans t on Trans t on 0.5 101 102 102 1.0 Wavy and strat f ed 101 Trans t on D mens on ess gas ve oc ty, g* 102 1.5 S ug and p ug 101 100 Mart ne parameter, x 102 So man (1982; 1986) The annu ar to wavy and nterm ttent trans t on (So man, 1982) was estab shed based on the Froude number and the annu ar to m st trans t on (So man, 1986) was based on the mod f ed Weber number. The Froude number s ca cu ated us ng the fo ow ng equat ons: Re = 10.18 Fr 0.625 Ga0.313 (v /Xtt )−0.938 Re = 0.79 Fr 0.962 Ga0.481 (v /Xtt )−1.442 for Re ≤ 1250 for Re > 1250 From the above ca cu at ons, for G = 400 kg/m2 -s and x = 0.5, qu d Reyno ds number Re = 1514, and Mart ne parameter Xtt = 0.3346. The Ga eo number for th s mode s Chapter 6. Condensat on n m n channe s and m crochanne s 361 def ned as Ga = gD3 ( /µ )2 . Thus, Ga = 9.81 × 0.001 3 1077 1.321 × 10−4 2 = 652, 650 The curve-f t for the square root of the two-phase mu t p er, s g ven by: v = 1 + 1.09Xtt0.039 = 1 + 1.09 × 0.33460.039 = 2.044 The va ues are subst tuted nto the Reyno ds number equat on to y e d: 1514 = 0.79 Fr 0.962 × 652650 0.481 2.044 0.3346 −1.442 ; Fr = 48.22 The mod f ed Weber number s g ven by: We = ì ï ï ï ï ï ï í 2.45 Rev0.64 µ2v v D 0.3 v−0.4 for Re ≤ 1250 ï 2 0.3 2 0.084 ï ï 0.157 µv µv ï 0.79 ï Xtt /v2.55 ï î0.85 Rev v D µ v for Re > 1250 For the present cond t on, therefore: We = 0.85 × 14743 0.79

× 1.357 × 10−5 1.321 × 10−4 0.3 2 1.357 × 10−5 76.5 × 4.27 × 10−3 × 0.001 2 1077 76.5 0.084 0.3346 2.0442.55 0.157 = 12 Now, Fr < 7 Fr > 7 and We < 20 Fr > 7 and We > 30 Wavy and nterm ttent A ways annu ar A ways m st Thus, at G = 400 kg/m2 -s and x = 0.5, f ow reg me s annu ar. The f ow reg mes for the other po nts are shown be ow. 362 Heat transfer and f u d f ow n m n channe s and m crochanne s G x Fr We F ow reg me 150 150 150 400 400 400 750 750 750 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 3.908 11.54 16.7 12.58 48.22 80.20 24.17 92.68 219.3 3.36 6.10 8.324 7.008 12.00 15.59 11.52 19.72 23.32 Wavy and nterm ttent Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar/m st trans t on F ow reg me map by So man 2000 Mass f ux, G (kg/m2-s) Lam nar (Re1 < 1250) Turbu ent (Re1 > 1250) 1500 We 30 M st Fr 7 1000 We 20 500 Annu ar Wavy and nterm ttent 0 0.0 0.2 0.4 0.6 Qua ty (x) 0.8 Co eman and Gar me a (2000b; 2000a; 2003) and Gar me a (2004) S mp e a gebra c curve-f ts to the r G-x trans t on nes are g ven be ow. Interm ttent f ow to nterm ttent and annu ar f m f ow: G= 547.7 − 1227x 1 + 2.856x Interm ttent and annu ar f m f ow to annu ar f m f ow: G = −56.13 + 125.4/x Annu ar f m f ow to annu ar f m and m st f ow: G = 206.1 + 85.8/x 1.0 Chapter 6. Condensat on n m n channe s and m crochanne s 363 Annu ar f m and m st f ow to m st f ow: G= 207.2 − 527.5x 1 − 1.774x D spersed bubb e f ow: G = 1376 − 97.1/x The respect ve nes and the cond t ons of nterest are shown n the graph be ow. Thus, the f ow reg mes are ass gned as fo ows: G x F ow reg me 150 150 150 400 400 400 750 750 750 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 Interm ttent f ow Interm ttent and annu ar f m f ow Annu ar f m f ow Interm ttent and annu ar f m f ow Annu ar f m and m st f ow Annu ar f m and m st f ow Annu ar f m and m st f ow Annu ar f m and m st f ow M st f ow F ow reg me map by Co eman and Gar me a 800 D spersed bubb e f ow 700 Mass f ux, G (kg/m2-s) M st f ow 600 Annu ar f m and m st f ow 500 400 300 Annu ar f m f ow 200 P ug/s ug f ow 100 0.0 0.1 P ug/s ug and annu ar f m f ow 0.2 0.3 0.4 0.5 0.6 Qua ty (x) 0.7 0.8 0.9 1.0 364 Heat transfer and f u d f ow n m n channe s and m crochanne s Cava n et a . (2002a) The Cava n et a . map s s m ar to the Breber et a . map, be ng based on the Mart ne parameter and the d mens on ess gas ve oc ty, both of wh ch were ca cu ated above. The f ow reg me ass gnment for a the cond t ons of nterest s shown be ow, and a so p otted on the r map. g* G x Xtt 150 0.2 1.165 1.095 150 150 400 400 400 750 750 750 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.3346 0.0961 1.165 0.3346 0.0961 1.165 0.3346 0.0961 2.737 4.379 2.919 7.298 11.68 5.473 13.68 21.89 F ow reg me Annu ar-strat f ed f ow trans t on and strat f ed f ow Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar Annu ar F ow reg me map by Cava n et a . D mens on ess gas ve oc t,y g* 102 Bubb y f ow Annu ar f ow 102 100 Trans t on and wavy-strat f ed f ow S ug f ow G w 101 fo rF r Strat f ed f ow 102 2 10 100 101 101 Mart ne parameter, Xtt 0. 54 102 A compar son of the pred ct ons of each of these maps s shown be ow. Th s compar son shows that the d fferent nvest gators pred ct a w de range of reg mes for the cond t ons of nterest n th s examp e. The bu k of the po nts are n annu ar f ow accord ng to most mode s and trans t on cr ter a, a though the cr ter a of Dobson and Chato pred ct wavy f ow for a the G = 150 kg/m2 -s cond t ons, as we as for the ow qua ty, G = 400 kg/m2 -s case. It shou d be noted that n Dobson and Chato s paper, a d scuss on appears about the 10 < Fr < 20 reg on be ng some form of a trans t on reg on between wavy and annu ar f ow, a though n the r recommended cr ter a, Fr = 20 s stated as a un que trans t on ne. Tandon et a ., Breber et a ., and Co eman and Gar me a pred ct Chapter 6. Condensat on n m n channe s and m crochanne s 365 some form of m st f ow, e ther n comb nat on w th annu ar f ow, or by tse f, at the h gher mass f uxes and qua t es. The s m ar t es n the bases between the Breber et a . and Cava n et a . maps are a so ev dent n th s tab e. Of these maps, on y the Co eman and Gar me a cr ter a are reported to app y for condensat on n tubes as sma as 1 mm, wh e the Dobson and Chato cr ter a app y down to D = 3.14 mm, and many of the other cr ter a were for tubes w th D > 4.8 mm, w th most be ng pr mar y based on tubes w th D ≥ 8 mm. F ow reg me stud es for the channe s much sma er than 1 mm were not represented n the above ustrat on because they are a for ad abat c a r–water or n trogen–water f ow, and many of them do not propose exp c t trans t on equat ons. Sardesa et a . Tandon et a . Dobson and Chato G = 150 kg/m2 -s x = 0.2 Annu ar – Wavy Trans t on between a four reg mes Wavy and nterm ttent G = 150 kg/m2 -s x = 0.5 Annu ar Annu ar and sem -annu ar Wavy G = 150 kg/m2 -s x = 0.8 Annu ar Annu ar and sem -annu ar Wavy G = 400 kg/m2 -s x = 0.2 Annu ar – Wavy G = 400 kg/m2 -s x = 0.5 Annu ar Spray Annu ar G = 400 kg/m2 -s x = 0.8 Annu ar Spray Annu ar G = 750 kg/m2 -s x = 0.2 Annu ar – Annu ar G = 750 kg/m2 -s x = 0.5 Annu ar Spray Annu ar G = 750 kg/m2 -s x = 0.8 Annu ar Spray Annu ar Annu ar and m st annu ar Annu ar and m st annu ar Trans t on from annu ar and m st annu ar f ow reg me to bubb e f ow reg me Annu ar and m st annu ar Annu ar and m st annu ar Trans t on from annu ar and m st annu ar f ow reg me to bubb e f ow reg me Annu ar and m st annu ar Annu ar and m st annu ar Cond t on Breber et a . Cava n et a . Co eman and Gar me a P ug/s ug f ow Annu ar Annu arstrat f ed f ow trans t on and strat f ed f ow Annu ar Annu ar Annu ar Annu ar Annu ar P ug/s ug and annu ar f m f ow Annu ar Annu ar Annu ar f m and m st f ow Annu ar Annu ar Annu ar f m and m st f ow Annu ar Annu ar Annu ar f m and m st f ow Annu ar Annu ar Annu ar f m and m st f ow Trans t on Annu ar M st f ow So man P ug/s ug and annu ar f m f ow Annu ar f m f ow 366 Heat transfer and f u d f ow n m n channe s and m crochanne s Examp e 6.2. Vo d fract on ca cu at on Compute the vo d fract on for refr gerant R-134a f ow ng through a 1-mm tube at a pressure of 1500 kPa, at mass f uxes 150, 400, and 750 kg/m2 -s and qua t es of 0.2, 0.5, and 0.8. Use the homogeneous mode and corre at ons by Kawahara et a . (2002), Baroczy (1965), Z v (1964), Lockhart and Mart ne (1949), Thom (1964), Ste ner (1993), Tandon et a . (1985b), E Ha a et a . (2003), Sm th (1969), Premo et a . (1971), and Yashar et a . (2001). Compare the va ues pred cted by each mode and comment. Refr gerant propert es Saturat on temperature: Tsat = 55.21 C Surface tens on: = 4.27 × 10−3 N/m Representat ve ca cu at ons are shown here for the G = 400 kg/m2 -s, x = 0.5 case. L qu d phase Vapor phase = 1077 kg/m3 µ = 1.321 × 10−4 kg/m-s v = 76.5 kg/m3 µv = 1.357 × 10−5 kg/m-s Homogeneous mode The vo d fract on s g ven by:

1 − x v −1 1 − 0.5 76.5 −1 = 1+ = 1+ = 0.9337 x 0.5 1077 Some nvest gators, for examp e, Tr p ett et a . (1999a) recommend the homogeneous mode for ca cu at on of vo d fract on n channe s w th Dh ~ 1 mm. Kawahara et a . (2002) Kawahara et a . (2002) recommend the fo ow ng corre at on for vo d fract on n tubes of d ameters 100 and 50 µm. = C1 h0.5 1 − C2 h0.5 where, h s the homogeneous vo d fract on 0.03 For D = 100 µm C1 = and 0.02 For D = 50 µm C2 = 0.97 0.98 For D = 100 µm For D = 50 µm Based on exper menta data, they recommend that for D > 250 µm, the vo d fract on be computed us ng the homogeneous mode . Therefore, for th s case (D = 1 mm), the vo d fract on s 0.9337. Chapter 6. Condensat on n m n channe s and m crochanne s 367 Baroczy (1965) The Baroczy vo d fract on s ca cu ated as fo ows:

−1 1 − x 0.74 v 0.65 µ 0.13 = 1+ x µv

Þ= 1+ 1 − 0.5 0.5 0.74 76.5 1077 0.65 1.321 × 10−4 1.357 × 10−5 0.13 −1 = 0.8059 Koyama et a . (2004) recommend th s corre at on for smooth tubes, but ment on that t does not pred ct the vo d fract on n m crof n tubes we . Z v (1964) Th s w de y used corre at on s g ven by: −1 1 − x v 2/3 = 1+ x 1 − 0.5 Þ= 1+ 0.5

76.5 1077 2/3 −1 = 0.8536 Lockhart and Mart ne (1949) Th s corre at on s g ven by:

−1 1 − x 0.64 v 0.36 µ 0.07 = 1 + 0.28 x µv

1 − 0.5 Þ = 1 + 0.28 0.5 0.64 76.5 1077 0.36 1.321 × 10−4 1.357 × 10−5 0.07 −1 = 0.8875 Based on exper menta data for a r–water f ow n 6.6 and 33.2 mm d ameter tubes, Ekberg et a . (1999) recommend the Lockhart and Mart ne corre at on for vo d fract on ca cu at ons. Thom (1964) The ca cu at on s as fo ows: 0.89 0.18 −1 1−x µ v = 1+ x µv Þ= 1+

1 − 0.5 0.5

76.5 1077 0.89 1.321 × 10−4 1.357 × 10−5 0.18 −1 = 0.8748 368 Heat transfer and f u d f ow n m n channe s and m crochanne s Ste ner (1993) −1 x 1.18(1 − x) [g × ( − v )]0.25 x 1−x [1 + 0.12(1 − x)] + = + v v G × 0.5

Þ=

0.5 0.5 1 − 0.5 [1 + 0.12(1 − 0.5)] + 76.5 76.5 1077 0.25 −1 1.18(1 − 0.5) 9.81 × 4.27 × 10−3 × (1077 − 76.5) + 400 × 10770.5 The resu t ng vo d fract on s = 0.8675. Tandon et a . (1985b) The qu d Reyno ds number requ red for th s corre at on s g ven by: Re = GD(1 − x) 400 × 0.001 × (1 − 0.5) = = 1514 µ 1.321 × 10−4 The Mart ne parameter s g ven by: Xtt = 1−x x 0.9 v 0.5 µ µv 0.1 = 1 − 0.5 0.5 0.9 76.5 1077 0.1 0.5 1.321 × 10−4 1.357 × 10−5 = 0.3346 The parameter F s g ven by: F(Xtt ) = 0.15 [Xtt−1 + 2.85 Xtt−0.476 ] Þ F(Xtt ) = 0.15[0.3346−1 + 2.85 × 0.3346−0.476 ] = 1.168 The vo d fract on s g ven by: 1 − 1.928 Re−0.315 [F(Xtt )]−1 + 0.9293 Re−0.315 [F(Xtt )]−2 = [F(Xtt )]−1 + 0.0361 Re −0.176 [F(Xtt )]−2 1 − 0.38 Re −0.088 50 < Re < 1125 Re > 1125 In the present case: = 1 − 0.38 × 1514−0.088 [1.168]−1 + 0.0361 × 1514−0.176 [1.168]−2 = 0.8365 E Ha a et a . (2003) E Ha a et a . (2003) proposed the fo ow ng corre at on that uses the vo d fract on ca cu ated by the homogeneous mode and the corre at on by Ste ner (1993). = h − Ste ner h n Ste ner Chapter 6. Condensat on n m n channe s and m crochanne s 369 where h and Ste ner are the vo d fract ons ca cu ated by the homogeneous mode and the corre at on by Ste ner (1993). In the present case, h = 0.9337, and Ste ner = 0.8675. Therefore, = 0.9337 − 0.8675 = 0.9002 n 0.9337 0.8675 Sm th (1969) Th s corre at on s g ven by: / ì é −1 1−x ùü 0 ï ï

0 í ý + K x v 1−x 1 v ú ê = 1+ × ëK + (1 − K) 1−x û ï ï x 1 + K î þ x Sm th found good agreement w th exper menta data (a r–water and steam– water, 6–38mm d ameter tubes) for K = 0.4. In the present case: / ì é −1 1−0.5 ùü 0 1077

í ý 0 + 0.4 76.5 1 − 0.5 76.5 0.5 = 1+ × ë0.4 + (1 − 0.4) 1 = 0.858 û î þ 1077 0.5 1 + 0.4 1−0.5 0.5 Tabataba and Faghr (2001) found the best agreement w th the r data for 4.8–15.88-mm tubes (refr gerants) and 1–12.3-mm tubes (a r–water) w th the Sm th and Lockhart– Mart ne corre at ons. They preferred the Sm th corre at on for ts w de range of app cab ty. Premo et a . (1971) The qu d-on y Reyno ds number s g ven by: Re o = GD 400 × 1 × 10−3 = = 3028 µ 1.321 × 10−4 The qu d-on y Weber number s g ven by: We o = G2 D 4002 × 1 × 10−3 = = 34.78 0.00427 × 1077 The parameters E1 and E2 are ca cu ated as fo ows: 0.22 1077 0.22 −0.19 E1 = 1.578 Re−0.19 = 1.578 × 3028 × = 0.6157 o v 76.5 E2 = 0.0273 We o Re−0.51 o

v −0.08 = 0.0273 × 34.78 × 3028 −0.51 × 1077 76.5 −0.08 = 0.01289 370 Heat transfer and f u d f ow n m n channe s and m crochanne s The parameter y s a funct on of the homogeneous vo d fract on: y= 0.9337 h = = = 14.08 1− 1 − h 1 − 0.9337 The s p rat o s ca cu ated as fo ows: S = 1 + E1 y 1 + yE2

Þ S = 1 + 0.6157 × 1/2 − yE2 14.08 1 + 14.08 × 0.01289

1/2 = 3.11 − 14.08 × 0.01289 F na y, the vo d fract on s g ven by: = x 0.5 = 0.8191 = x + S(1 − x) v / 0.5 + 3.11 × (1 − 0.5) × 76.5/1077 Yashar et a . (2001) The Froude rate def ned them s eva uated as fo ows: 0.5 0.5 G 2 x3 4002 × 0.53 Ft = = = 26.4 (1 − x) v2 gD (1 − 0.5) × 76.52 × 9.81 × 1 × 10−3 The Mart ne parameter was ca cu ated above, Xtt = 0.3346. The vo d fract on s ca cu ated n terms of these two parameters as fo ows: 1 = 1+ + Xtt Ft −0.321 −0.321 1 = 1+ = 0.9034 + 0.3346 26.4 The vo d fract ons pred cted by the d fferent mode s and corre at ons for a the cond t ons of nterest are shown be ow. G kg/m2 -s x Homo- Baroczy Z v geneous (1965) (1964) 150 150 150 400 400 400 750 750 750 0.2 0.5 0.8 0.2 0.5 0.8 0.2 0.5 0.8 0.7788 0.9337 0.9826 0.7788 0.9337 0.9826 0.7788 0.9337 0.9826 0.5981 0.8059 0.9205 0.5981 0.8059 0.9205 0.5981 0.8059 0.9205 0.5932 0.8536 0.9589 0.5932 0.8536 0.9589 0.5932 0.8536 0.9589 Lockhart and Mart ne Thom (1949) (1964) Tandon Ste ner et a . (1993) (1985b) E Ha a et a . Sm th (2003) (1969) Premo Yashar et a . et a . (1971) (2001) 0.7647 0.8875 0.9504 0.7647 0.8875 0.9504 0.7647 0.8875 0.9504 0.6274 0.8461 0.9489 0.6769 0.8675 0.9555 0.6922 0.8737 0.9574 0.7004 0.8892 0.9656 0.7267 0.9002 0.969 0.7347 0.9034 0.9699 0.5964 0.7909 0.9017 0.6239 0.8191 0.9264 0.6433 0.8414 0.9445 0.636 0.8748 0.9655 0.636 0.8748 0.9655 0.636 0.8748 0.9655 0.9639* 0.8685 0.8987 0.6694 0.8365 0.9256 0.6855 0.845 0.9299 0.6527 0.858 0.9566 0.6527 0.858 0.9566 0.6527 0.858 0.9566 0.7296 0.8904 0.9622 0.7597 0.9034 0.9676 0.7691 0.9071 0.9692 * Note: Re = 908.5 and thus the equat on for 50 < Re < 1125 was used. However, f the equat on for Re > 1125 s used, then = 0.6428, wh ch s much c oser to the resu ts from the other mode s. Chapter 6. Condensat on n m n channe s and m crochanne s 371 The resu ts shown above nd cate that the vo d fract ons pred cted by severa of the mode s are qu te c ose to each other. A graph dep ct ng these va ues s shown n Sect on 6.3. Caut on must be exerc sed, however, because, as stated by E Ha a et a . (2003) and Thome et a . (2003), (1 − ) s an nd cator of the f m th ckness n annu ar f ow, and sma changes n vo d fract on mp y arge changes n qu d f m th ckness, wh ch therefore a ters the heat transfer coeff c ent. It shou d a so be noted that many of the corre at ons assume annu ar f ow, and a though some of the authors recommend that they can be used rrespect ve of the f ow reg me, n the f ow reg mes spec f c to m crochanne s, th s shou d be va dated further. Examp e 6.3. Pressure drop ca cu at on Compute the fr ct ona pressure grad ent dur ng condensat on of refr gerant R-134a f ow ng through a 1 mm d ameter tube, at a mass f ux of 300 kg/m2 -s, at a mean qua ty of 0.5 and a pressure of 1500 kPa. Use the corre at ons by Lockhart and Mart ne (1949), Fr ede (1979; 1980), Ch sho m (1973), M sh ma and H b k (1996), Lee and Lee (2001), Tran et a . (2000), Wang et a . (1997b), Chen et a . (2001), W son et a . (2003), Souza et a . (1993), Cava n et a . (2001; 2002a; 2002b), and Gar me a et a . (2005) and compare the va ues pred cted by each mode . Refr gerant propert es Saturat on temperature Surface tens on Tsat = 55.21 C = 0.00427 N/m L qu d phase Vapor phase = 1077 kg/m3 µ = 1.321 × 10−4 kg/m-s v = 76.5 kg/m3 µv = 1.357 × 10−5 kg/m-s Some common parameters that are used by severa corre at ons are ca cu ated be ow f rst. L qu d Reyno ds number, Re = GD(1 − x) 300 × 0.001 × (1 − 0.5) = = 1136 µ 1.321 × 10−4 Vapor Reyno ds number, Rev = GDx 300 × 0.001 × 0.5 = = 11054 µv 1.357 × 10−5 L qu d-on y Reyno ds number, Re o = GD 300 × 0.001 = = 2271 µ 1.321 × 10−4 372 Heat transfer and f u d f ow n m n channe s and m crochanne s Vapor-on y Reyno ds number, Revo = GD 300 × 0.001 = = 22108 µv 1.357 × 10−5 For th s ustrat on, the s ng e-phase fr ct on factors are ca cu ated us ng the Church (1977) corre at on: é ù1/12 ü ì 16

12 í 16 ý−1.5 1 37530 ê 8 ú f = 8ë + 2.457 × n 0.9 + û 7 þ î Re Re + 0.27/D Re Assum ng a smooth tube, that s, = 0, f = 0.05635, fv = 0.03016, f o = 0.03047, and fvo = 0.0252. The correspond ng s ng e-phase pressure grad ents are g ven by: dP f G 2 (1 − x)2 0.05635 × 3002 (1 − 0.5)2 = = = 588.5 Pa/m dz 2 D 2 × 0.001 × 1077 dP fv G 2 x2 0.03016 × 3002 × 0.52 = = = 4436 Pa/m dz v 2 Dv 2 × 0.001 × 76.5 dP f o G 2 0.03047 × 3002 = = = 1273 Pa/m dz o 2 D 2 × 0.001 × 1077 dP fvo G 2 0.0252 × 3002 = 14823 Pa/m = = dz vo 2 Dv 2 × 0.001 × 76.5 Lockhart and Mart ne (1949) The Mart ne parameter s g ven by: (dP/dz) 1/2 588.5 1/2 X = = = 0.3642 (dP/dz)v 4436 Two-phase mu t p er s g ven by: 2 = 1 + C 1 + 2; X X ì L qu d ï ï ï ï í 20 Turbu ent C = 12 Lam nar ï ï 10 Turbu ent ï ï î 5 Lam nar Vapor Turbu ent Turbu ent Lam nar Lam nar In th s case: 2 = 1 + 1 1 12 12 + 2 =1+ + = 41.4865 X X 0.3642 0.36422 The two-phase pressure grad ent s: P dP = 41.4865 × 588.5 = 24, 414 Pa/m = 2 L dz Chapter 6. Condensat on n m n channe s and m crochanne s 373 Fr ede (1979; 1980) The parameters E, F, and H are eva uated as fo ows: E = (1 − x)2 + x2 fvo v f o Þ E = (1 − 0.5)2 + 0.52 × 1077 × 0.0252 = 3.161 76.5 × 0.03047 F = x0.78 × (1 − x)0.24 = 0.50.78 × (1 − 0.5)0.24 = 0.4931 H= v 0.91

ÞH = 1077 76.5 µv µ 0.19 µv 1− µ 0.91 0.7 1.357 × 10−5 1.321 × 10−4 0.19 1.357 × 10−5 1− 1.321 × 10−4 0.7 = 6.677 The two-phase m xture dens ty s ca cu ated as fo ows: TP = x 1−x + v −1 = 1 − 0.5 0.5 + 76.5 1077 −1 = 142.8 kg/m3 The Froude and Weber numbers are g ven by: Fr = G2 3002 = = 449.6 2 9.81 × 0.001 × 142.82 gDTP We = G2 D 3002 × 0.001 = 147.5 = TP 142.8 × 4.27 × 10−3 The resu t ng two-phase mu t p er s now ca cu ated as fo ows: 2 o =E+ 3.24 × FH 3.24 × 0.4931 × 6.677 = 3.161 + = 9.967 0.045 0.035 0.045 × 147.50.035 449.6 Fr We F na y, the two-phase pressure grad ent s g ven by: P P 2 = o = 9.967 × 1273 = 12, 688 Pa/m L L o Ch sho m (1973) The rat o of the vapor- to qu d-on y pressure grad ents s g ven by: $ Y = (dP/dz)vo = (dP/dz) o

14,823 = 3.412 1273 374 Heat transfer and f u d f ow n m n channe s and m crochanne s The two-phase mu t p er s g ven by: 2 o = 1 + Y 2 − 1 [Bx(2−n)/2 (1 − x)(2−n)/2 + x2−n ] where the parameter B s g ven by: ì 55 ï √ ï ï ï G ï ï ï í 520 √ B= ï ï ïY G ï ï ï 15000 ï î √ Y2 G for 0 < Y < 9.5 for 9.5 < Y < 28 for 28 < Y and n s the power to wh ch Reyno ds number s ra sed n the fr ct on factor n= Re o ≤ 2100 Re o > 2100 1 0.25 Thus for the current examp e, 55 55 B= √ = √ = 3.175 and n = 0.25. G 300 The two-phase mu t p er can now be ca cu ated as fo ows: 2 = 1 + (Y 2 − 1) × [3.175x(2−0.25)/2 (1 − x)(2−0.25)/2 + x2−0.25 ] o = 1 + (3.4122 − 1) × [3.175 × 0.5(2−0.25)/2 (1 − 0.5)(2−0.25)/2 + 0.52−0.25 ] = 14.213 The pressure grad ent s eva uated as fo ows: P 2 = o L

P L = 14.213 × 1273 = 18, 093 Pa/m o M sh ma and H b k (1996) Mart ne parameter (ca cu ated above) s X = 0.3642. The two-phase mu t p er s g ven by: 2 = 1 + 1 C + 2 X X For sma d ameter tubes, M sh ma and H b k recommend the fo ow ng express on for C: C = 21(1 − e−0.319D ) Chapter 6. Condensat on n m n channe s and m crochanne s 375 Þ C = 21(1 − e−0.319×1 ) = 5.736 W th th s va ue of C, the two-phase mu t p er s g ven by: Þ 2 = 1 + 5.736 5.736 1 1 = 24.285 + 2 =1+ + X X 0.3642 0.36422 The two-phase pressure grad ent s then ca cu ated as fo ows: P = 2 L

dP dz = 24.285 × 588.5 = 14,292 Pa/m Lee and Lee (2001) The Mart ne parameter, ca cu ated above s X = 0.3642. For the C parameter n the twoµ2 phase mu t p er, Lee and Lee suggested C = A q r Res o , where, = D , = µ , and = qu d s ug ve oc ty, a so g ven by = V + L . The va ue of constants A, q, r, and s are determ ned based on the fo ow ng Tab e: F ow reg me L qu d Vapor A q r s X -Range Re o Range Lam nar Lam nar Turbu ent Turbu ent Lam nar Turbu ent Lam nar Turbu ent 6.833 × 10−8 6.185 × 10−2 3.627 0.408 −1.317 0 0 0 0.719 0 0 0 0.557 0.726 0.174 0.451 0.776–14.176 0.303–1.426 3.276–79.415 1.309–14.781 175–1480 293–1506 2606–17642 2675–17757 It shou d be noted that surface tens on effects are mportant on y n the am nar– am nar f ow reg mes. For a other f ow reg mes, the exponents of and are a most zero. Thus, for these f ow reg mes, C s mere y a funct on of Re o . The above tab e a so prov des the range of X and Re o for the data from wh ch these constants were determ ned. The qu d Reyno ds number for th s case, ca cu ated above, s Re o = 2271. For the current examp e ( am nar qu d f m and turbu ent vapor core), the coeff c ent C s eva uated as fo ows: 0.726 C = 6.185 × 10−2 × Re o = 6.185 × 10−2 × 22710.726 = 16.9 The two-phase mu t p er s therefore: 2 = 1 + 1 1 16.9 16.9 + 2 =1+ + = 54.953 X X 0.3642 0.36422 F na y, the two-phase pressure grad ent s ca cu ated as fo ows: P = 2 L

dP dz = 54.953 × 588.5 = 32,340 Pa/m 376 Heat transfer and f u d f ow n m n channe s and m crochanne s Tran et a . (2000) Tran et a . mod f ed the two-phase mu t p er corre at on of Ch sho m (1973) to the fo ow ng express on that nc udes the conf nement number: 2 o = 1 + 4.3Y 2 − 1 × Nconf x0.875 (1 − x)0.875 + x1.75 where, conf nement number s g ven by: Nconf = g( −v ) 1/2 = D 4.27×10−3 9.81(1077−76.5) 0.001 1/2 = 0.6595 for the current examp e. W th Y = 3.412 ca cu ated above, 2 o = 1 + (4.3 × 3.4122 − 1) × 0.6595 × 0.50.875 × (1 − 0.5)0.875 + 0.51.75 = 10.916 The two-phase pressure grad ent s eva uated n the usua manner: P 2 = o L

P L = 10.916 × 1273 = 13, 897 Pa/m o Wang et a . (1997b) Accord ng to the r corre at on, for G < 200 kg/m2 -s v2 = 1 + 9.4X 0.62 + 0.564X 2.45 . For G > 200 kg/m2 -s, they mod f ed the parameter C n the Lockhart–Mart ne (1949) corre at on as fo ows: −6 C = 4.566 × 10 X 0.128 0.938 × Re o v −2.15 µ µv 5.1 In th s examp e, G = 300 kg/m2 -s; therefore: C = 4.566 × 10−6 × 0.36420.128 × 22710.938

5.1 1077 −2.15 1.321 × 10−4 = 2.099 76.5 1.357 × 10−5 The two-phase mu t p er s then g ven by: 2 = 1 + 2.099 2.099 1 1 = 14.304 + 2 =1+ + X X 0.3642 0.36422 And the two-phase pressure grad ent s: P dP = 2 = 14.304 × 588.5 = 8416 Pa/m L dz Chapter 6. Condensat on n m n channe s and m crochanne s 377 Chen et a . (2001) Chen et a . mod f ed the homogenous mode and the Fr ede (1979; 1980) corre at on to determ ne the pressure drop n m crochanne s. The mod f cat on to the Fr ede corre at on s ustrated here. Thus, dP dP = ; dz dz Fr ede ì 0.45 ï 0.0333 × Re o ï ï ï í Re0.09 (1 + 0.4 exp (−Bo)) v = ï ï We0.2 ï ï î (2.5 + 0.06Bo) Bo < 2.5 Bo ≥ 2.5 2 where Weber number, We = GmD and Bond number, Bo = g( − v ) case: (D/2)2 Bo = g( − v )

(0.001/2)2 = 9.81(1077 − 76.5) 4.27 × 10−3

(D/2)2 . For th s = 0.5748 Because Bo < 2.5, the mod f er to the Fr ede corre at on s g ven by: Þ= 0.45 0.0333 × Re o 0.09 Rev (1 + 0.4 exp (−Bo)) = 0.0333 × 22710.45 110540.09 (1 + 0.4 exp (−0.5748)) = 0.3808 Therefore, the two-phase pressure grad ent s: dP dP = 12688 × 0.3808 = 4831 Pa/m = dz dz Fr ede W son et a . (2003) W son et a . recommended the qu d-on y two-phase mu t p er of Jung and Radermacher (1989): 2 o = 12.82Xtt−1.47 (1 − x)1.8 W th the Mart ne parameter (ca cu ated above), Xtt = 0.3346, the two-phase mu t p er s: 2 o = 12.82 × 0.3346−1.47 (1 − 0.5)1.8 = 18.413 wh ch y e ds a two-phase pressure grad ent of P 2 = o L

dP dz = 18.413 × 1273 = 23,434 Pa/m o 378 Heat transfer and f u d f ow n m n channe s and m crochanne s Souza et a . (1993) The two-phase qu d mu t p er g ven by them s as fo ows: 2 = 1.376 + C1 Xtt−C2 where, for Fr < 0.7 C1 = 4.172 + 5.48 Fr − 1.564 Fr 2 C2 = 1.773 − 0.169 Fr and for Fr > 0.7, C1 = 7.242; C2 = 1.655. In the present case, the Froude number s: Fr = 300 G = = 2.812 √ √ gD 1077 9.81 × 0.001 Therefore, C1 = 7.242; C2 = 1.655, wh ch y e ds a two-phase mu t p er of: 2 = 1.376 + 7.242 × 0.3346−1.655 = 45.725 For the correspond ng s ng ephase fr ct on factor, they recommend the Co ebrook (1939) fr ct on factor: 1 /D 2.51 + # = −2 og # 3.7 f Re f Thus for smooth tubes: 1 2.51 Þ f = 0.05983 # = −2 og 0 + # f 1136 f The s ng e-phase qu d pressure drop s ca cu ated as fo ows: dP dz = f G 2 (1 − x)2 0.05983 × 3002 (1 − 0.5)2 = = 624.8 Pa/m 2D 2 × 0.001 × 1077 F na y, the two-phase pressure drop, us ng the two-phase mu t p er s g ven by: P dP = 45.725 × 624.8 = 28,568 Pa/m = 2 L dz Cava n et a . (2001; 2002a, b) Cava n et a . recommended mod f cat ons to the Fr ede (1979; 1980) corre at on. The parameter E s the same as n h s corre at on, E = 3.161. The parameter F s mod f ed Chapter 6. Condensat on n m n channe s and m crochanne s 379 to F = x0.6978 , wh ch n th s case s F = 0.50.6978 = 0.6165. The mod f ed parameter H s g ven by: H= v 0.3278 µv µ −1.181 1− µv µ 3.477 For the present cond t on, H= 0.3278 1077 76.5 1.357 × 10−5 1.321 × 10−4 −1.181 1− 1.357 × 10−5 1.321 × 10−4 3.477 = 23.99 The Weber number s def ned n terms of the gas-phase dens ty: We = G2 D 3002 × 0.001 = = 275.5 v 76.5 × 4.27 × 10−3 The two-phase mu t p er s then ca cu ated as fo ows: 2 o =E+ 1.262 × FH 1.262 × 0.6165 × 23.99 = 3.161 + = 11.391 0.1458 275.50.1458 We The two-phase pressure drop s ca cu ated based on th s mu t p er as fo ows: P 2 = o L

P L = 11.391 × 1273 = 14,501 Pa/m o Gar me a et a . (2005) The case of G = 300 kg/m2 -s and x = 0.5 represents annu ar f ow, accord ng to the trans t on cr ter a of Co eman and Gar me a (2000a, b; 2003) and Gar me a (2004). Wh e Gar me a et a . deve oped a mu t -reg me ( nc ud ng nterm ttent/d screte wave and annu ar/m st/d sperse wave) mode to span the ent re qua ty range as descr bed n Sect on 6.4.4, for th s cond t on, the annu ar f ow mode app es. For the annu ar f ow mode , the nd v dua phase Reyno ds numbers are computed n terms of the cross-sect ona areas occup ed by the phases. Therefore, the vo d fract on s f rst ca cu ated us ng the Baroczy (1965) corre at on:

= 1+ 1−x x 0.74 v 0.65 µ µv 0.13 −1 For the present case, = 1+

1 − 0.5 0.5 0.74 76.5 1077 0.65 1.321 × 10−4 1.357 × 10−5 0.13 −1 = 0.8059 380 Heat transfer and f u d f ow n m n channe s and m crochanne s Therefore, the qu d and vapor Re va ues are g ven by: GD(1 − x) 300 × 0.001 × (1 − 0.5) Re = = = 598.5 √ √ 1 + µ (1 + 0.8059)1.321 × 10−4 GDx 300 × 0.001 × 0.5 = 12316 √ √ = µv 1.357 × 10−5 × 0.8059 Rev = The fr ct on factor for the am nar f m s f = 64/Re = 64/598.5 = 0.1069. The turbu ent vapor fr ct on factor s ca cu ated us ng the fv = 0.316 × Re −0.25 = 0.316×12316−0.25 = v 0.03. W th these fr ct on factors, the correspond ng s ng e-phase pressure grad ents are g ven by:

dP dz dP dz = f G 2 (1 − x)2 0.1069 × 3002 (1 − 0.5)2 = = 1117 Pa/m 2D 2 × 0.001 × 1077 = fv G 2 x2 0.03 × 3002 × 0.52 = 4412 Pa/m = 2Dv 2 × 0.001 × 76.5

v The Mart ne parameter s ca cu ated from the def n t on as fo ows: (dP/dz) X = (dP/dz)v 1/2

1117 = 4412 1/2 = 0.5031 The superf c a ve oc ty s g ven by: = 300 × (1 − 0.5) G(1 − x) = = 0.7172 m/s (1 − ) 1077 × (1 − 0.8059) Th s ve oc ty s used to eva uate the surface tens on parameter: = µ 0.7172 × 1.321 × 10−4 = 0.02218 = 4.27 × 10−3 The nterface fr ct on factor s then ca cu ated as fo ows: f = AX a Reb c f S nce Re < 2100, A = 1.308 × 10−3 ; a = 0.4273; b = 0.9295; c = −0.1211, wh ch y e ds: f = 1.308 × 10−3 × X 0.4273 × Re 0.9295 × −0.1211 f = 1.308 × 10−3 × 0.50310.4273 × 598.50.9295 × 0.02218−0.1211 = 0.5897 Chapter 6. Condensat on n m n channe s and m crochanne s 381 The resu t ng nterface fr ct on factor s f = 0.5897 × f = 0.5897 × 0.1069 = 0.06306. The nterfac a fr ct on factor s used to determ ne the pressure grad ent as fo ows: P 1 G 2 x2 1 = f L 2 v 2.5 D 1 3002 × 0.52 1 × 0.06306 × × 2 69.82 × 0.85092.5 0.001 = 15,907 Pa/m = A compar son of the pressure drops pred cted by each of these corre at ons s shown be ow. It can be seen that the pred cted pressure drops vary cons derab y, from 4.8 to 32.3 kPa. Th s arge var at on s attr buted to the cons derab y d fferent two-phase mu t p ers deve oped by the var ous nvest gators. The on y recommendat on that can be made s to choose a mode that s based on the geometry, f u d and operat ng cond t ons s m ar to those of nterest for a g ven app cat on. Th s nformat on s ava ab e n Sect on 6.4 as we as n Tab e 6.3. Pressure grad ent (kPa/m) 35 30 25 20 15 10 5 Lo ck ha r ta nd M ar t C ne h Fr sh ( 1 e de o m 94 9) ( M ( s Sou 197 197 h m za 9; 3) 1 a e an t a 98 0) . d ( W H 19 an b k 93 g ( ) Tr et a 199 an . 6 ) ( C et a 199 he 7 . C av Le n e (20 ) a e a t a 00 n e nd . (2 ) t a Le 00 W . (2 e (2 1) G so 00 00 ar n 1 m e ; 2 1) e t a 00 a . et (2 2) a 00 . ( 3) 20 05 ) 0 Examp e 6.4. Ca cu at on of heat transfer coeff c ents Compute the heat transfer coeff c ent for condensat on of refr gerant R-134a f ow ng through a 1mm tube, at a mass f ux of 300 kg/m2 -s, at a mean qua ty of 0.5 and at a pressure of 1500 kPa. The wa temperature of the tube s 52 C. Use the corre at ons by Shah (1979), So man et a . (1968), So man (1986), Trav ss et a . (1973), Dobson and 382 Heat transfer and f u d f ow n m n channe s and m crochanne s Chato (1998), Moser et a . (1998), Chato (1962), Boyko and Kruzh n (1967), Cava n et a . (2002a), and Bandhauer et a . (2005) and compare the va ues pred cted by each mode . Refr gerant propert es Saturat on temperature Cr t ca pressure Reduced pressure Surface tens on Heat of vapor zat on Shah (1979) App cab ty range: Tsat = 55.21 C Pcr t = 4059 kPa Pred = 0.37 = 0.00427 N/m h v = 145,394 J/kg L qu d phase Vapor phase = 1077 kg/m3 µ = 1.321 × 10−4 kg/m-s Cp = 1611 J/kg-K h = 131,657 J/kg k = 0.06749 W/m-K Pr = 3.152 v = 76.5 kg/m3 µv = 1.357 × 10−5 kg/m-s Cpv = 1312 J/kg-K hv = 277,051 J/kg kv = 0.0178 W/m-K Prv = 1 0.002 < Pred < 0.44 21 C < Tsat < 310 C 3 < Vv < 300 m/s 10.83 < G < 210.56 kg/m2 -s The ve oc ty of the vapor phase s g ven by: Vv = xG 0.5 × 300 = 1.961 m/s = v 76.5 In the present case, the vapor ve oc ty and mass f ux are beyond the range of app cab ty of the corre at on. The qu d-on y Reyno ds number Re o was ca cu ated n Examp e 6.3 to be Re o = 2271. The qu d-on y heat transfer coeff c ent s ca cu ated as fo ows: 0.8 h o = 0.023 Re o Pr 0.4 × k D = 0.023 × 22710.8 × 3.1520.4 × = 1190 W/m2 -K 0.06749 0.001 Chapter 6. Condensat on n m n channe s and m crochanne s 383 From th s s ng e-phase heat transfer coeff c ent, the condensat on heat transfer coeff c ent s obta ned by app y ng the mu t p er as fo ows: 3.8x0.76 (1 − x)0.04 0.8 h = h o (1 − x) + 0.38 Pred 3.8 × 0.50.76 (1 − 0.5)0.04 0.8 = 1190 (1 − 0.5) + 1500 0.38 4059 = 4474 W/m -K 2 So man et a . (1968) For th s corre at on, a qua ty change per un t ength s a so requ red, and for ustrat ve purposes, t s assumed that x = 0.10 over a ength of 0.3048 m. Th s corre at on was based on data for the fo ow ng range of cond t ons: 6.096 < Vv < 304.8 m/s 0.03 < x < 0.99 1 < Pr < 10 The vapor ve oc ty here s 1.961 m/s, wh ch s outs de the range of app cab ty of the corre at on. The vapor-on y Reyno ds number Revo was ca cu ated n Examp e 6.3 to be 22114, and the qu d Reyno ds number Re = 1136. Ff represents the effect of two-phase fr ct on: 0.261 0.0523 G 2 −0.2 1.8 µ 0.47 1.33 v (1 − x) Ff = 0.0225 × x + 5.7 Re x v vo µv µ + 8.11 µv 0.105 (1 − x) 0.94 0.86 x v 0.522 For the present case, 3002 Ff = 0.0225 × × 22114−0.2 76.5 0.0523 0.261 1.321 × 10−4 0.47 1.33 76.5 (1 × 0.51.8 + 5.7 × 0.5 − 0.5) 1.357 × 10−5 1077 1.321 × 10−4 + 8.11 1.357 × 10−5 = 6.999 0.105 (1 − 0.5)0.94 × 0.50.86 76.5 1077 0.522 384 Heat transfer and f u d f ow n m n channe s and m crochanne s Fm represents the effect of momentum change:

2/3 4/3 dx v v DG 2 1 − 2(1 − x) − 3 + 2x + + (2x − 1 − x) Fm = 4v L x 1/3 5/3 v v v × + 2 − − x + 2(1 − x − + x) x where, = 2 1.25 for Re ≤ 2000 for Re > 2000 For the present case, 0.1 0.001 × 3002 − Fm = 4 × 76.5 0.3048

76.5 2/3 1 76.5 4/3 × 2(1 − 0.5) + − 3 + 2 × 0.5 1077 0.5 1077 1/3

76.5 2 76.5 5/3 + (2 × 0.5 − 1 − 2 × 0.5) + 2×2− − 2 × 0.5 1077 0.5 1077 76.5 + 2(1 − 0.5 − 2 + 2 × 0.5) 1077 = 0.03144 F a represents the effect of the ax a grav tat ona f e d on the wa shear stress. In case of hor zonta condensat on n m crochanne s, th s term can be neg ected. The net shear stress s therefore: F 0 = Ff + F m ± F a = 6.999 + 0.03144 = 7.031 The heat transfer coeff c ent s g ven by: h = 0.036 × k × 0.5 × Pr 0.65 × F01/2 µ For the present case: h = 0.036 × 0.06749 × 10770.5 × 3.1520.65 × 7.0311/2 1.321 × 10−4 = 3377 W/m2 -K Chapter 6. Condensat on n m n channe s and m crochanne s 385 So man (1986) Th s corre at on shou d on y be used for m st f ow (We > 30), wh ch s not the case here, but the ca cu at ons are shown be ow to ustrate the procedure and nvest gate the va ue of the resu t ng heat transfer coeff c ent. W th Re = 1136, Rev = 11, 057, Xtt = 0.3346, v = 1 + 1.09 × Xtt0.039 = 2.044, a ustrated above or n the prev ous examp e, the Weber number can be ca cu ated as fo ows: ì 2 0.3 µv ï ï ï v D ï 0.64 í 2.45 × Rev for Re ≤ 1250 v0.4 We = 2 0.3 2 0.084 ï ï µv Xtt 0.157 µv ï ï for Re ≤ 1250 î 0.85 × Re0.79 v v D µ v v2.55 For Re = 1136: We = 2.45 × Rev0.64 0.3 µ2v v D v0.4 = 2.45 × 11,0570.64 (1.357×10−5 )2 76.5×0.00427×0.001 2.0440.4 0.3 = 9.509 C ear y, the f ow s not n the m st f ow reg me. The m xture v scos ty requ red for the heat transfer coeff c ent s g ven by: µm = x 1−x + µv µ −1 = 0.5 1 − 0.5 + 1.357 × 10−5 1.321 × 10−4 −1 = 2.46 × 10−5 kg/m-s W th th s m xture v scos ty, the correspond ng Reyno ds number s: Rem = GD 300 × 0.001 = = 12,193 µm 2.46 × 10−5 W th the Reyno ds number ca cu ated, the Nusse t number s g ven by: Nu = 0.9 0.00345 × Rem µv h v kv (Tsat − Twa ) 1/3 wh ch, for the present case y e ds: Nu = 0.00345 × 12193 0.9 1.357 × 10−5 × 145394 0.0178(55.21 − 52) = 53.46 The two-phase heat transfer coeff c ent s therefore: h = Nu 0.06749 k = 3608 W/m2 -K = 53.46 × 0.001 D 1/3 386 Heat transfer and f u d f ow n m n channe s and m crochanne s Trav ss et a . (1973) Th s corre at on was deve oped from data for the condensat on of R-22 n an 8-mm tube over the fo ow ng range of cond t ons: 25 C < Tsat < 58.3 C 161.4 < G < 1532 kg/m2 -s The requ red qu d Reyno ds number and Mart ne parameter are Re = 1136 and Xtt = 0.3346, respect ve y. W th these quant t es, the corre at on for the shear-re ated term F n the qu d f m s ca cu ated as fo ows: ì 0.5 for Re < 50 ï í 0.707 × Pr Re 0.585 F = 5 Pr + 5 n 1 + Pr 0.09636 Re −1 for 50 ≤ Re ≤ 1125 ï î 5 Pr +5 n(1 + 5 Pr ) + 2.5 n 0.00313 Re 0.812 for Re > 1125 For the present case: F = 5Pr + 5 n(1 + 5Pr ) + 2.5 n 0.00313 Re 0.812 = 5 × 3.152 + 5 n(1 + 5 × 3.152) + 2.5 n 0.00313 × 11360.812 = 29.72 F na y, the heat transfer coeff c ent s ca cu ated as fo ows: 1 0.15 × Pr × Re 0.9 2.85 + 0.476 Xtt F Xtt 0.06749 0.15 × 3.152 × 11360.9 2.85 1 = + 0.001 0.3346 0.33460.476 29.72 h= k D

= 4700 W/m2 -K Dobson and Chato (1998) The re evant bas c parameters (ca cu ated above) are as fo ows: Re = 1136, Revo = 22114, and Xtt = 0.3346. The vo d fract on s ca cu ated us ng Z v s (1964) mode : 1−x = 1+ x

v 2/3 −1

1 − 0.5 = 1+ 0.5

76.5 1077 2/3 −1 = 0.8536 Chapter 6. Condensat on n m n channe s and m crochanne s 387 The Ga eo number was ca cu ated n Examp e 6.1 for the present case, resu t ng n Ga = 478, 182. The mod f ed So man Froude number s then ca cu ated us ng the fo ow ng equat on: ì 1.5 1 1 + 1.09 × Xtt0.039 ï ï 1.59 ï 0.025 Re ï í Xtt Ga0.5 Frso = 1.5 ï ï ï 1 + 1.09 × Xtt0.039 1 ï î 1.26 Re 1.04 Xtt Ga0.5 for Re ≤ 1250 for Re > 1250 For the present case w th Re = 1136, Fr so 1.5 1 1 + 1.09 × Xtt0.039 = Xtt Ga0.5 0.039 1.5 1 1.59 1 + 1.09 × 0.3346 = 0.025 × 1136 0.3346 478,1820.5 0.025 Re 1.59 = 39.37 S nce Frso > 20, the annu ar f ow corre at on proposed by them s used: 2.22 Nuannu ar = 0.023 × Re 0.8 × Pr 0.4 1 + 0.89 Xtt Subst tut ng the re evant parameters, the Nusse t number for the present case s: Nu = 0.023 × 1136 0.8 × 3.152 0.4 2.22 1+ 0.33460.89

= 69.68 The resu t ng heat transfer coeff c ent s: h = Nu × 0.06749 k = 69.68 × = 4703 W/m2 -K D 0.001 Moser et a . (1998) Th s corre at on was based on data from tubes w th 3.14 < D < 20 mm. The var ous phase 300 × 0.001 Reyno ds numbers are, Re = 1136, Re o = 2271, and Revo = GD µv = 1.357 × 10−5 = 22,114. The two-phase homogeneous dens ty s g ven by: tp = x 1−x + v −1 = 1 − 0.5 0.5 + 76.5 1077 −1 = 142.8 kg/m3 388 Heat transfer and f u d f ow n m n channe s and m crochanne s The correspond ng Froude and Weber numbers are: Fr tp = G2 3002 = = 449.6 2 9.81 × 0.001 × 142.82 gDtp Wetp = G2 D 3002 × 0.001 = 147.5 = tp 0.00427 × 142.8 For the ca cu at on of the equ va ent Reyno ds number, the requ red fr ct on factors are obta ned as fo ows: f o = 0.079 × Re−0.25 = 0.079 × 2271−0.25 = 0.01144 o = 0.079 × 22114−0.25 = 0.006478 fvo = 0.079 × Re−0.25 vo Constants requ red for the two-phase mu t p er eva uat on are ca cu ated as fo ows:

fvo 1077 0.006478 = (1 − 0.5)2 + 0.52 = 2.243 A1 = (1 − x)2 + x2 v f o 76.5 0.01144

µv 0.70 0.91 µv 0.19 1− v µ µ 0.19 0.70 0.91 1.357 × 10−5 1.357 × 10−5 0.78 0.24 1077 (1 − 0.5) 1− = 0.5 76.5 1.321 × 10−4 1.321 × 10−4 A2 = x0.78 (1 − x)0.24 = 3.293 Two-phase mu t p er s obta ned from the parameters ca cu ated above: o2 = A1 + 3.24 × A2 3.24 × 3.293 = 2.243 + = 3.008 0.035 0.045 × 147.50.035 449.6 × Wetp 0.045 Fr tp The equ va ent Reyno ds Number s ca cu ated n terms of th s mu t p er as fo ows: 8/7 Reeq = o × Re o = 3.0088/7 × 2271 = 7996 The Nusse t number s ca cu ated n terms of th s equ va ent Reyno ds number as fo ows: −0.448 −0.113×Pr −0.563 1+0.11025×Pr −0.448 0.09940.126×Pr × Re × Reeq × Pr 0.815 Nu = 2/3 1.58 × n Reeq − 3.28 2.58 × n Reeq + 13.7 × Pr − 19.1 For the present case, −0.448 Nu = −0.563 −0.448 ×1136−0.113×3.152 ×79961+0.11025×3.152 ×3.1520.815 0.09940.126×3.152 (1.58 × n 7996 − 3.28) 2.58 × n 7996 + 13.7 × 3.1522/3 − 19.1 = 55.77 Chapter 6. Condensat on n m n channe s and m crochanne s 389 F na y, the heat transfer coeff c ent s g ven by: h = Nu × k 0.06749 = 55.77 × = 3764 W/m2 -K D 0.001 Chato (1962) Th s mode was proposed for strat f ed annu ar f ow cond t ons when Rev < 35,000. The mod f ed atent heat of vapor zat on, wh ch accounts for wa subcoo ng s g ven by: Cp × (Tsat − Twa ) h v = h v × 1 + 0.68 × h v 1611 × (55.21 − 52) = 145,394 × 1 + 0.68 × 145,394 = 148,910 J/kg The heat transfer coeff c ent s obta ned as fo ows: 1/4 g × × ( − v ) × k 3 × h v h = 0.728 × Kc × µ × (Tsat − Twa ) × D W th Kc = 0.76, 9.81 × 1077 × (1077 − 76.5) × 0.067493 × 148910 Þ h = 0.728 × 0.76 × 1.321 × 10−4 × (55.21 − 52) × 0.001 1/4 = 3216 W/m2 -K Boyko and Kruzh n (1967) Th s corre at on re ates the heat transfer coeff c ent n s ng e-phase f ow w th the ent re f u d f ow ng as a qu d to the two-phase heat transfer coeff c ent based on the m xture dens ty, m = 142.8 kg/m3 , ca cu ated above. The bu k and wa Prandt numbers are a so requ red. Here the Prandt number at the refr gerant pressure and wa temperature, Prw = 3.144. The s ng e-phase heat transfer coeff c ent s ca cu ated as fo ows: h0 = 0.024 ×

D 0.8 k Pr 0.25 × G× × Pr 0.43 × D µ Pr w For the present case, 0.8

0.001 0.06749 3.152 0.25 0.43 h0 = 0.024 × × 300 × × 3.152 × 0.001 1.321 × 10−4 3.144 = 1286 W/m2 -K 390 Heat transfer and f u d f ow n m n channe s and m crochanne s The two-phase heat transfer coeff c ent s obta ned as fo ows: 1077 = 1286 = 3531 W/m2 -K h = h0 m 142.8 Cava n et a . (2002a) The app cab e f ow reg me s f rst found from the Mart ne parameter Xtt = 0.3346, and the d mens on ess gas ve oc ty: g* = √ Gx 300 × 0.5 = 5.473 =√ Dgv ( − v ) 0.001 × 9.81 × 76.5(1077 − 76.5) Based on the trans t on cr ter a presented n examp e 6.1, the f ow s n the annu ar reg me. The two-phase mu t p er and the pressure grad ent requ red for the ca cu at on of the shear stress were presented n Examp e 6.3. Thus, P/L = 14,500 Pa/m. The shear stress, s g ven by: 0.001 dp D = 14500 × = 3.625 Pa = dz f 4 4 The d mens on ess f m th ckness s based on the qu d-phase Reyno ds number: ì 0.5 ï í Re for Re ≤ 1145 2 + = ï î 0.0504 × Re7/8 for Re > 1145 In the present case Re = 1136, wh ch y e ds:

Re 0.5 1136 0.5 = = 23.83 + = 2 2 The d mens on ess temperature s g ven by: ì + Pr ï ï ï ï ï + ï í T + = 5 Pr + n 1 + Pr 5 − 1 ï ï + ï ï ï ï î 5 Pr + n(1 + 5 Pr ) + 0.495 n 30 + ≤ 5 5 < + < 30 + ≥ 30 For the present case, + −1 T + = 5 Pr + n 1 + Pr 5 23.83 = 5 3.152 + n 1 + 3.152 −1 = 28.54 5 Chapter 6. Condensat on n m n channe s and m crochanne s 391 F na y, the heat transfer coeff c ent s ca cu ated as fo ows: 0.5 0.5 Cp 1077 × 1611 × 3.625 1077 = = 3527 W/m2 -K h= T+ 28.54 Bandhauer et a . (2005) Th s mode was based on heat transfer data for 0.5 < D < 1.5 mm and 150 < G < 750 kg/m2 -s, and the under y ng pressure drop mode was based on data for a much w der range of hydrau c d ameters (0.4 < D < 4.9 mm). The pressure grad ent ca cu at on was d scussed n Examp e 6.3, and resu ted n a va ue of P/L = 15, 907 Pa/m. The correspond ng nterfac a shear stress s:

√ P D P ×D = = L 4 L 4 √ 0.8059 × 0.001 = 15.907 × 103 × = 3.57 Pa 4 The fr ct on ve oc ty s ca cu ated from the nterfac a shear stress as fo ows: 3.57 * = 0.05756 m/s = u = 1077 The f m th ckness s d rect y ca cu ated from the def n t on of the vo d fract on as fo ows: 0.001 √ √ D = 1 − 0.8059 = 5.115 × 10−5 m = 1− 2 2 Th s th ckness s used to obta n the d mens on ess f m th ckness as fo ows: + = × × u * 5.115 × 10−5 × 1077 × 0.05756 = = 24.02 µ 1.321 × 10−4 The turbu ent d mens on ess temperature s g ven by: ì + ï ï −1 +1 5 Pr +5 n Pr ï ï í 5 + T = + dy+ ï ï 5 Pr +5 n(5 Pr +1) + ï 30 + ï 1 î −1 + y 1− Pr 5 f Re < 2100 y+ R+

Thus, for the current examp e, w th Re = 598.5: + −1 +1 T + = 5 Pr + 5 n Pr 5 24.02 = 5 × 3.152 + 5 n 3.152 − 1 + 1 = 28.58 5 f Re ≥ 2100 392 Heat transfer and f u d f ow n m n channe s and m crochanne s Therefore, the heat transfer coeff c ent s: h= × Cp × u * 1077 × 1611 × 0.05756 = = 3495 W/m2 -K + T 28.58 The heat transfer coeff c ents pred cted by the above mode s are dep cted n the fo ow ng graph. Heat transfer coeff c ent (W/m2-K) 5000 4000 3000 2000 1000 Bo yk o an Cha d to K (1 So ruz 96 m h 2) an n ( et 196 Tr av 7) a .( s s et 196 8) a .( 1 97 Sh D 3) a ob so So n (1 9 m n an 79 an ) d ( 1 C ha 98 M 6) os t er o (1 C 9 e av 98 ta a ) . Ba n (1 99 nd e 8) ha t a ue . ( 2 re 0 t a 00 ) . (2 00 5) 0 It can be seen from th s graph that there are cons derab e var at ons between the heat transfer coeff c ents pred cted by these d fferent mode s and corre at ons. One of the ma n reasons for these var at ons s that most of these corre at ons have been app ed n th s case outs de the r ranges of app cab ty, e ther n the mass f ux, the phase ve oc t es, or parameters such as the Weber and Froude numbers as app cab e. But the most mportant reason for the d fferences s that a except the Bandhauer et a . (2005) mode were based pr mar y on tube d ameters n the ~8 mm range, w th a few corre at ons be ng proposed based on m ted data on ~3 mm tubes. The channe under cons derat on n th s examp e s a 1-mm channe . 6.7. Exerc ses (1) P ot s mu ated a r–water data po nts at d fferent equa y spaced comb nat ons of gas and qu d superf c a ve oc t es n the range 0.1–100 m/s, and 0.01–10.0 m/s, respect ve y, as s typ ca y done n a r–water stud es. Now p ot on the same graph, refr gerant R-134a (at 40 C) po nts for the mass f ux range 100–800 kg/m2 -s n 100 kg/m2 -s ncrements and qua ty ncrements of 0.10; that s compute the vapor and qu d superf c a ve oc t es for these G, x comb nat ons and p ot on the a r–water Chapter 6. Condensat on n m n channe s and m crochanne s (2) (3) (4) (5) (6) (7) (8) 393 superf c a ve oc ty graph. Assume a 3 mm ID channe . Use th s graph to comment on extend ng a r–water f ow reg me maps to refr gerant condensat on. Propose appropr ate means to transform a r–water resu ts to refr gerant pred ct ons based on compar sons of the re evant propert es. F ow reg me maps are often p otted on the bas s of superf c a ve oc t es, n wh ch the phases are assumed to occupy the ent re channe cross sect on. Wh e th s s a common y used pract ce out of conven ence, what are the d sadvantages of choos ng superf c a ve oc t es nstead of actua phase ve oc t es? Quant fy for a range of f ow patterns. Conduct a d mens ona ana ys s to dent fy the d mens on ess parameters that wou d quant fy trans t ons between the four pr nc pa reg mes: nterm ttent, wavy, annu ar, and m st f ow, by cons der ng v scous, surface tens on, nert a, and grav ty forces. Conduct parametr c ana yses to choose the most s gn f cant parameters across the sca es. Est mate qu d f m th ckness as a funct on of three d fferent vo d fract on mode s n annu ar f ow, and ts effect on the heat transfer coeff c ent. Refer to E Ha a (2003) and Thome (2003) for gu dance. How wou d your conc us ons change for nterm ttent or strat f ed f ows? One of the uncerta nt es often over ooked n report ng pressure drop resu ts n m crochanne s s the effect to erances n tube d ameter. Conduct a parametr c ana ys s of the change n pressure drop n a 1-mm channe dur ng annu ar f ow condensat on of refr gerant R-22 across the qua ty range 0.3 < x < 0.8. Use a reasonab e range of to erances n the tube d ameter to ustrate these effects. Use a mass f ux of 400 kg/m2 -s, at a saturat on temperature of 45 C. Comment on how th s var at on wou d change at evaporat ng cond t ons (e.g. 5 C). Deve op a quant tat ve nd cator of momentum coup ng of qu d and vapor phases n condens ng f ows based on the papers of Lee and Lee (2001), the or g n of the C coeff c ent express on deve oped by Ch sho m (1967) for the Lockhart–Mart ne (1949) two-phase mu t p er approach, and the d scuss ons n the papers by Kawahara et a . (2002) and Chung and Kawa (2004). Est mate numer ca va ues of th s measure for 10 µm < Dh < 10 mm. Est mate the d fference n saturat on temperature decrease due to the f ow of refr gerant R-22 n a 0.5-mm channe at a mass f ux of 200 kg/m2 -s by comput ng the condensat on n 10% qua ty ncrements and account ng for fr ct ona and dece erat on contr but ons. Assume an appropr ate bu k-wa temperature d fference. P ot the pressure drop n each ncrement, and the correspond ng decrease n saturat on temperature. Use two d fferent pressure drop mode s, one based on a r–water stud es, another based on refr gerant condensat on. M crochanne s are espec a y attract ve due to the h gh heat transfer coeff c ents they offer. In measur ng these heat transfer coeff c ents, one has to measure the heat transfer rate us ng, typ ca y, a r se n coo ant temperature, and a temperature d fference between the refr gerant and the wa . Assume channe Dh = 0.5 mm. Assume further that the heat transfer coeff c ent needs to be measured n ncrements of x = 0.10. F rst est mate the heat transfer rate to ach eve th s condensat on at th s mass f ux. How accurate y can th s heat transfer rate be measured n a s ng e m crochanne ? 394 (9) (10) (11) (12) (13) (14) Heat transfer and f u d f ow n m n channe s and m crochanne s What changes n coo ant temperature wou d be requ red to effect th s heat transfer across the test sect on? Is th s rea st c – est mate the uncerta nt es n heat transfer rate measurement us ng reasonab e thermocoup e and mass f ow rate measurement uncerta nt es. In an attempt to mprove the uncerta nt es, mu t p e channe s n para e are proposed. How many channe s wou d be requ red to ach eve accurate y measurab e heat transfer rates? What are the drawbacks to us ng mu t p e channe s. Next, est mate the heat transfer coeff c ent n condens ng R-134a at 450 kg/m2 -s and 50 C us ng one of the ava ab e mode s. What refr gerant-wa temperature d fference wou d th s cause n a channe that accomp shes the 10% qua ty change? (you w have to choose a reasonab e test sect on ength). How accurate y can th s temperature d fference be measured? What wou d be the resu t ng heat transfer coeff c ent uncerta nty? How wou d the uncerta nty vary w th mass f ux – quant fy? How wou d the uncerta nty change f R-134a s subst tuted w th steam? It has been stated n th s chapter that boundary- ayer ana yses n the qu d f m and two-phase mu t p er approaches are essent a y equ va ent methods of ana yz ng annu ar f ow heat transfer. Refer to Dobson and Chato (1998) for background on th s matter. Conduct para e der vat ons of these two approaches to po nt out the essent a s m ar t es. A so comment on any s gn f cant d fferences. Most heat transfer mode s recommend s mp y extend ng annu ar f ow mode s to the nterm ttent f ow. Deve op a methodo ogy for conduct ng th s extens on on a rat ona bas s by account ng for the s ug/bubb e frequency. W nterm ttent f ow heat transfer coeff c ents be h gher or ower than those n annu ar f ow? Why? W they be h gher or ower than those n s ng e-phase gas or s ng e-phase qu d f ow? Why? Demonstrate through approx mate mode s, us ng R-134a condens ng at a 35 C through a 200 µm channe at 100 kg/m2 -s. Assum ng that condens ng f ows wou d go through a progress on of annu ar–m sthomogeneous f ow as the reduced pressure s ncreased, dent fy the most mportant f u d propert es and the correspond ng d mens on ess parameters, and nvest gate changes n these quant t es as the pressure s ncreased from 0.2 × pr to 0.9 × pr . How wou d the pressure drop and heat transfer be affected n th s progress on? Des gn an R-22 condenser that transfers 3 kW at a saturat on temperature of 55 C, w th the refr gerant enter ng as a saturated vapor and eav ng as a saturated qu d. Dry a r f ow ng n crossf ow at 4.8E-01 m3 /s, w th an n et temperature of 35 C and a heat transfer coeff c ent of 50 W/m2 -K s the coo ng med um. Use copper tub ng w th copper f ns on the a r s de that y e d a factor of 10 surface area enhancement. The eff c ency of the f ns can be assumed to be 90%. Choose the tub ng s ze and wa th ckness from standard tab es such that the drop n refr gerant saturat on temperature (due to the pressure drop) s ess than 1.5 C. Account for the f ow reg me var at on over the ength of the heat exchanger. Choose qua ty and/or tube ength ncrements of reasonab e s ze that demonstrate the f ow phenomena and the heat transfer coeff c ent and pressure drop var at ons. Assume a hor zonta or entat on. Repeat prob em 13 by chang ng the des gn to an opt ma para e -serpent ne arrangement of extruded A um num tubes w th twenty 0.7 mm m crochanne tubes w th the same amount of a r-s de A um num f n surface area enhancement as n prob em 13. Thus, decrease the number or tubes per pass as condensat on proceeds so that a Chapter 6. Condensat on n m n channe s and m crochanne s 395 arger f ow area s prov ded n the n t a passes to account for the h gher f ow ve oc t es, and decrease the number of para e tubes n subsequent passes. 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R., and Favrat, D., Evaporat on of ammon a n a smooth hor zonta tube: heat transfer measurements and pred ct ons, J. Heat Trans., Trans. ASME, 121(1), 89–101, 1999. Chapter 7 BIOMEDICAL APPLICATIONS OF MICROCHANNEL FLOWS M chae R. K ng Departments of B omed ca Eng neer ng and Chem ca Eng neer ng, Un vers ty of Rochester, Rochester, NY, USA 7.1. Introduct on As more s earned about the mo ecu ar and ce u ar b o ogy of human and mode organ sm ce s n the contexts of norma phys o og ca funct on and d sease patho ogy, t s the ro e of b oeng neers to app y th s know edge towards the mprovement of pub c hea th and a ev at on of human suffer ng. The contro and man pu at on of v ng ce s outs de of the body presents techn ca cha enges wh ch phys ca cond t ons such as temperature and pH must be ma nta ned c ose to the r phys o og ca va ues of 37 C and 7.4, respect ve y; f these quant t es dev ate by as much as 10% then ce s cannot surv ve for ong per ods of t me. W th greater dev at ons from phys o og ca base ne, prote ns and other macromo ecu es themse ves beg n to degrade. The comp ete sequenc ng of the human genome at the start of the 21st century has p aced an ncreased nterest n the deve opment of h gh-throughput m crof u d c techno og es to assay thousands of genes s mu taneous y. The nterest n transport ng, man pu at ng, and synthes z ng b omo ecu es n sma er and sma er condu ts s mot vated by a des re to automate b ochem ca protoco s and e m nate some of the most mundane techn ca tasks, as we as a need to operate on extreme y sma samp e s zes: for nstance to dent fy nd v dua s based on m nute samp es n forens c app cat ons cr me scenes or to detect b ochem ca factors that may be produced n the body s t ssues at nanomo ar concentrat ons. M crof u d cs and m crochanne transport processes have a so ed to the deve opment of new research too s. We now have the ab ty to peer ns de of m croscop c ce s and measure the nter or rheo ogy of the r cytop asm, and v sua ze the co- oca zat on of spec f c prote ns w th n var ous organe es. M cron-s zed p pettes are ab e to enter a ce , remove ts nuc ear mater a , and n ect th s DNA “soup” nto a d fferent ce to ach eve c on ng of an ma s and humans ( n the case of humans, therapeut c rather than reproduct ve). Another E-ma : m ke_k [email protected] 409 410 Heat transfer and f u d f ow n m n channe s and m crochanne s area of great research nterest s n the prec se measurement of the spec f c adhes on of ce s w th other contact ng surfaces. Many of the surface receptor prote ns presented by ce s are themse ves exqu s te y sens t ve detectors of so ub e chem ca concentrat on, and force transducers that can n t ate rap d cha n react ons w th n the ce compartments. In th s chapter we d scuss some of the d fferent techno og ca app cat ons of m crochanne s and m n channe s to measure the strength and frequency of nd v dua ce adhes on, grow ce s to rep ace v ta organs, pee ce s from prote n-coated surfaces us ng we -def ned forces, and character ze the deformab ty of d fferent ce types. In each case, we app y fundamenta s from transport phenomena to understand and mode these b o og ca app cat ons. 7.2. M crochanne s to probe trans ent ce adhes on under f ow M crochanne s w th funct ona zed b ochem ca surfaces have been qu te usefu n e uc dat ng the mo ecu ar mechan sms govern ng the trans ent adhes on between c rcu at ng wh te b ood ce s ( eukocytes) and the b ood vesse wa dur ng nf ammat on. The se ect ns are a fam y of adhes on mo ecu es that recogn ze a tetrasacchar de mo ety ca ed s a y Lew sx presented by many g ycoprote n gands expressed on ce surfaces, and are nvo ved n the traff ck ng of eukocytes throughout the system c and ymphat c c rcu at ons (Lasky, 1995; Ebnet and Vestweber, 1999). The se ect n-carbohydrate bond s of h gh spec f c ty, h gh aff n ty, and exh b ts fast rates of format on and d ssoc at on. The rate of d ssoc at on s a dom nant phys ca parameter, d st nct for each of the three dent f ed se ect ns P-, E-, and L-se ect n, and has been demonstrated to exh b t an exponent a dependence on the force oad ng on the bond (Sm th et a ., 1999). The same mo ecu ar mechan sms that med ate acute nf ammatory responses under phys o og ca cond t ons are a so respons b e for chron c nf ammat on, vesse occ us on, and other d sease states featur ng abnorma accumu at ons of adherent eukocytes or p ate ets (Ramos et a ., 1999). Many of the b o og ca comp ex t es of th s myr ad of surface nteract ons between eukocytes, p ate ets, and endothe a ce s have been revea ed us ng dea zed m crochanne assays and sem quant f ed us ng knockout m ce. There s current y a tremendous opportun ty to advance our understand ng of se ect n b o ogy n the context of nf ammatory and card ovascu ar d sease by app y ng eng neer ng too s to ntegrate th s know edge nto descr pt ve and pred ct ve mode s of b ood ce nteract ons n more rea st c sett ngs. In add t on to creat ng a we -def ned f ow f e d to study the ba ance between f u d forces and chem ca adhes on of b ood ce s, m crochanne s, m n channe s, and correspond ng numer ca s mu at ons have been successfu y deve oped to more c ose y m m c f ow rregu ar t es found n the d seased human c rcu atory system. The depos t on of p ate ets on surfaces n comp ex f ow geometr es such as stenos s, aneurysm, or f ow over a protrus on, has been character zed n v tro by severa groups (Kar no and Go dsm th, 1979; T ppe et a ., 1992; Schoephoerster et a ., 1993; S ack and Tur tto, 1994). These stud es support the dea that f ow non- dea t es are mportant n modu at ng thrombot c phenomena, yet ack the spat a and tempora reso ut on necessary to support deta ed ce u ar-sca e mode ng. In hypercho estero em c rabb ts and m ce, monocytes preferent a y depos t around aort c and ce ac or f ces, due to oca f ow character st cs n these reg ons (Back et a ., 1995; Wa po a et a ., 1995; Nakash ma et a ., 1998; I yama et a ., 1999; Chapter 7. B omed ca app cat ons of m crochanne f ows 411 Truskey et a ., 1999). Endothe a ce ayers respond to d sturbed shear f e ds (e.g. f ow over a rectangu ar obstac e) by upregu at ng express on of adhes on mo ecu es, resu t ng n a genera ncrease n the number of ce s recru ted downstream of such an obstac e (DePao a et a ., 1992; Pr tchard et a ., 1995; Barber et a ., 1998). Mu t p e groups have attempted to mode the part cu ate f ow of b ood through comp ex vesse geometr es w th and w thout part c e depos t on at the wa s, wh e neg ect ng hydrodynam c ce –ce nteract ons (Perkto d, 1987; Le et a ., 1997). Desp te theoret ca m tat ons, such stud es have been app ed towards prov d ng recommendat ons for surg ca reconstruct on procedures, and the des gn of commerc a grafts that m n m ze part c e depos t on based on stresses exerted by the f u d on the wa surfaces. Others have computat ona y stud ed shearact vat on and surface depos t on of p ate ets by treat ng the ce s as nf n tes ma po nts (B ueste n et a ., 1999; Kuharsk and Foge son, 2001). The MAD s mu at on (K ng and Hammer, 2001a, b) o ns a r gorous ca cu at on of mu t part c e f u d f ow w th a rea st c mode for spec f c ce –ce adhes on. Work w th MAD has revea ed b ophys ca mechan sms that contro dynam c ce adhes on under f ow, that have been ver f ed exper menta y both n m crochanne exper ments (K ng and Hammer, 2003) and n v vo (K ng et a ., 2003). Further understand ng of the rate of atherogenes s and r sk of acute thrombot c events w occur through carefu computer and n v tro mode s that are deve oped to cons der the near-wa ce –ce hydrodynam c nteract ons that occur dur ng ce depos t on n d seased vesse geometr es. 7.2.1. D fferent types of m crosca e f ow chambers The most common types of m crochanne s used n ce adhes on and f ow stud es are the rad a f ow chamber (Kuo and Lauffenberger, 1993; Go dste n and D M a, 1998), and the para e -p ate f ow chamber (Sung et a ., 1985; Pa ecek et a ., 1997). Such systems have found w despread use for the fo ow ng app cat ons: (a) measurement of trans ent adhes on of c rcu at ng ( .e. b ood) ce s to react ve surfaces under f ow, (b) use of we -def ned shear stresses to measure the detachment force of strong y adherent ce s, (c) measurement of the b o og ca response of a ce ayer to ong-term exposure to phys o og ca eve s of steady or pu sat e shear stress, (d) mon tor ng the rea -t me format on of b of ms on f ow-exposed surfaces. F gure 7.1 shows a d agrammat c representat on of the rad a and para e -p ate f ow chamber geometr es. The ma n advantage of the rad a f ow chamber comes from the fact that as the f u d ntroduced at the center of the chamber f ows rad a y outward, t passes through a arger and arger cross-sect on around the per meter and thus the near ve oc ty of the f u d decreases smooth y as t f ows from the nner to outer reg ons. As a resu t, the f u d samp es a w de range of wa shear stresses, from a h gh va ue near the center to a m n mum va ue at the outer edge of the chamber. If the ent re surface of nterest s maged from be ow at ow magn f cat on, then one can obta n nteract on data such as the number of adherent ce s or the ce trans at ona ve oc t es for a range of shear stress va ues at once rather than hav ng to perform repeated exper ments at many d fferent f ow rates. It 412 Heat transfer and f u d f ow n m n channe s and m crochanne s Vacuum Ce suspens on Bottom surface coated w th adhes ve mo ecu es (a) Syr nge pump Gasket (b) G ass or p ast c m cros de F g. 7.1. D agram of (a) rad a and (b) para e -p ate ce perfus on chambers. s stra ghtforward to show (see Prob em 7.1) that the wa shear stress n the rad a f ow assay w exh b t the fo ow ng dependence on rad a pos t on r: = 3Qµ rH 2 (7.1) where Q s the vo umetr c f owrate, µ s the f u d v scos ty, and H s the spac ng between the upper and ower surfaces. A more deta ed d agram of the para e -p ate f ow chamber assay s dep cted n F g. 7.1(b). Th s shows a common des gn, where a reusab e upper surface s brought nto contact w th a rubber gasket a d over a d sposab e g ass or p ast c m cros de. The m cros de can be ncubated w th re evant adhes on or extrace u ar matr x prote ns of nterest, or e se cu tured w th ce s unt a conf uent mono ayer s ach eved. The upper surface of the f ow chamber s f tted w th uer f tt ngs that attach to n et and out et streams to a ow for the perfus on of ce med a, or suspens ons of ce s or m crosphere part c es coated w th the r own appropr ate mo ecu es. A th rd f tt ng s common y attached to a vacuum ne, wh ch for typ ca chamber s zes and phys o og ca f u d f owrates s suff c ent to ho d together the top and bottom of the chamber w thout the use of bo ts or mechan ca c amps. The f ow doma n s def ned by a rectangu ar, cut-out reg on n the rubber gasket, wh ch necessar y over aps w th the n et and out et ports (but not w th the vacuum port). Most ce perfus on chambers are spec a y des gned so that the outer d mens ons are compat b e w th standard s zes of po ystyrene t ssue cu ture p ates or round g ass covers ps, wh ch fac tate the exchange of mater a s and protoco s between d fferent aborator es. Many such para e p ate ce perfus on assays are now commerc a y ava ab e, n sma er and sma er vo umes to m n m ze the amount of rare and va uab e mater a s needed to conduct each exper ment. Tab e 7. shows a samp ng of the para e -p ate chambers current y ava ab e commerc a y, a ong w th the reported ranges of channe d mens on. 7.2.2. Inverted systems: we -def ned f ow and ce v sua zat on The f at ower wa of the para e -p ate m crochanne prov des an exce ent surface to study ce u ar adhes ve nteract ons. As d scussed be ow, the s ght y greater dens ty of ce s re at ve to sa ne buffer or ce med a (»water) causes sed mentat on of ce s to the ower Chapter 7. B omed ca app cat ons of m crochanne f ows 413 Tab e 7.1 Some commerc a y ava ab e m crochanne s and m n channe s for ce perfus on. Mode Manufacturer Channe he ght H (µm) ECIS f ow system Stova f ow ce DH-40 m cro- ncubator F ow chamber system Ce adhes on f ow chamber Vacu-ce C rcu ar f ow chamber Focht chamber system 2 App ed B oPhys cs Stova L fe Sc ence, Inc. Warner Instruments, Inc. O gene Immunogen cs, Inc. C & L Instruments, Inc. G ycotech Corporat on Intrace Ltd. 1000 1000 800–980 300–500 250 250 125–250 100–1000 surface. Thus, re at ve y d ute systems of 106 ce s/m (vo ume fract on ~5 × 10−4 for 10-µm-d ameter ce s) can be used wh e st obta n ng a arge number of nd v dua nteract ons for study. A though the geometry of the m croc rcu at on more c ose y resemb es c rcu ar g ass cap ar es, and cap ar es can be commerc a y purchased w th very accurate nner d ameters rang ng from 3–1000 µm, round cap ar es have some d sadvantages compared to p anar channe s. The c rcu ar outer geometry of cap ar es creates ens ng effects wh ch nterfere w th accurate mag ng of the nner surface. Submers on of the ent re cap ary w th n ndex of refract on-matched ob ect ve o mproves th s to some degree. However, due to sed mentat on effects, the surface area ava ab e for ce u ar nteract ons s ess n the cap ary geometry. For these reasons as we as to s mp fy theoret ca mode ng, the para e -p ate geometry s the most popu ar method for study ng ce –surface nteract ons under f ow. H gh-qua ty mages of ce s adhes ve y attach ng to mo ecu ar surfaces can be obta ned as shown n F g. 7.2, and ce s grown under a var ety of cond t ons can be exposed to f u d f ow or brought nto contact w th suspended part c es of nterest. A cruc a e ement of b oeng neer ng exper ments n para e -p ate f ow chambers s that near the ower wa of the m crochanne , the parabo c f u d ve oc ty prof e can be approx mated as near shear f ow. In most cases th s assumpt on s va d, and s mp f es the ana ys s. F gure 7.3 shows a compar son of parabo c versus near f ow n a 200 µm channe , w th n 10 µm of the ower wa . Th s d stance s re evant to most ce adhes on stud es, where ce s above th s d stance are read y dent f ed and gnored s nce they are too far from the mo ecu ar ower surface to nteract adhes ve y. In F g. 7.3 the parabo c ve oc ty es w th n 5% of the near approx mat on for x ≤ 10 µm. D fferent opt ca techn ques are usefu for track ng the mot on of nd v dua ce s near the surface when phys o og ca concentrat ons of b ood ce s are used. For nstance, t s often d ff cu t to obta n c ear mages through who e b ood n e ther br ght f e d or phase contrast mag ng modes, s nce v sua zat on of the ower surface on an nverted m croscope us ng a trans- um nat on moda ty requ res ght transm ss on from the condenser through the ent re samp e of the dense suspens on of red ce s. Ep f uorescence mode, on the other hand, s um nated from be ow and thus there s m n ma ght oss due to the red ce s f ow ng above the surface. Add t ona y, for arge-sca e ce depos t on stud es t s not necessary to reso ve nd v dua ce s, but rather, an ntens ty average over the ent re v ew can be taken and re ated to the average concentrat on of f uorescent y abe ed ce s w th n the mage. 414 Heat transfer and f u d f ow n m n channe s and m crochanne s F g. 7.2. Bov ne aort c endothe a ce f uorescent y abe ed w th pha o d n, wh ch shows the nter or cytoske eton of the ce . The ce s adher ng to a 3 × 3 µm m cropattern of f bronect n prote n, a so f uorescent y abe ed. Such ce growth exper ments can be performed on the ower surface of a para e -p ate m crochanne , to ater expose ce s to f u d shear stress or to eas y exchange the f u d env ronment. Image courtesy of M. Manc n , K. Fu wara, G. Csucs, and M. K ng. Many f uorescent dyes are ava ab e for d fferent b ood ce s that do not affect the r surface adhes veness, such as f uorochrome carboxyf uoresce n d acetate succ n m dy ester for b ood p ate ets, or ca ce n-AM for eukocytes. Desp te the obv ous advantages to the nverted m crochanne geometry, one must be aware of anoma ous behav or that may be ess mportant w th n the body. Most prev ous stud es of eukocyte adhes on under f ow have been performed w th d ute eukocyte suspens ons n a para e -p ate f ow chamber, w th observat on of the ower surface on an nverted m croscope. There t s recogn zed that grav tat ona sed mentat on promotes n t a contact between the ce s and the react ve surface. Indeed, n s m ar d ute exper ments Lawrence et a . (1997) showed that nvert ng the f ow chamber comp ete y abo shes new nteract ons, but prev ous y ro ng ce s w cont nue to ro under f ow. In v vo, at Chapter 7. B omed ca app cat ons of m crochanne f ows 10 Po seu e f ow Shear f ow 8 X (m) 415 6 4 2 0 0 200 400 600 V (m/s) 800 1000 F g. 7.3. Compar son of parabo c Po seu e f ow and near shear f ow n the v c n ty of the ower wa n a 200 µm m crochanne . The wa shear rate has been set at 100 s−1 . phys o og ca concentrat ons of red b ood ce s, the s tuat on s qu te d fferent, w th red ce s m grat ng toward the center of the vesse and d sp ac ng the ess deformab e eukocytes to the near-wa reg on (a process ca ed “marg nat on”; see for nstance Go dsm th and Spa n, 1984). An erythrocytedep eted p asma ayer, conta n ng eukocytes and p ate ets, s formed ad acent to the vesse wa . Ce co s ons can promote n t a contact between eukocytes and the vesse wa ; th s may he p to exp a n how rough y equa numbers of ro ng eukocytes are v s b e on the upper and ower wa s of ntact mouse and hamster m crovesse s. There s a subt e effect of grav ty n rea vesse s however, as ev denced by a study by B shop et a . (2001) who observed f ow of who e b ood n hor zonta y and vert ca y or ented m crovesse s n rat sp notrapez us musc e. They observed a symmetr ca p asma ayer n vert ca y or ented venu es, whereas n hor zonta y or ented venu es the p asma ayer formed near the upper wa . S m ar y, grav tat ona effects of who e b ood f ow n rectangu ar channe s can be m n m zed by or ent ng the channe vert ca y (Abb tt and Nash, 2001), yet when opt ca y mag ng eukocytes through reg ons >40 µm the same comp cat ons of f uorescence abe ng occur as descr bed above. A theoret ca compar son of grav tat ona effects and norma forces at the t ps of eukocyte m crov suggest that surface nteract ons can dom nate over grav ty under the proper cond t ons (Zhao et a ., 2001). To summar ze, grav ty p ays a more mportant ro e n m crochanne f ow chamber exper ments compared to n v vo. However, contro exper ments can be performed by dens ty-match ng the so ut on to the suspended ce s, v a add t on of h ghmo ecu ar we ght dextran (shown to not nterfere w th se ect n adhes on; Chen and Spr nger, 2001) to separate out any art factua effects of grav tat ona sed mentat on. 7.2.3. Lubr cat on approx mat on for a gradua y converg ng (or d verg ng) channe One potent a app cat on of m crochanne s n the study and man pu at on of c rcu at ng ce s s the use of very sma -gap channe s (H ≤ 10 µm) to cont nuous y exert 416 Heat transfer and f u d f ow n m n channe s and m crochanne s h gh mechan ca stresses on 8–10 µm b ood ce s or b ood ce precursors as they pass through such a dev ce. Indeed, such mechan ca stresses, that approx mate the c osepack ng of ce s w th n the bone marrow, may n fact be necessary for proper maturat on of red b ood ce s and exp a n why product on of hea thy red ce s has proven to be so cha eng ng n v tro (Waugh et a ., 2001). In such sma m crochanne s created by the or entat on of two para e p ates n c ose prox m ty to each other, whether the gap s ma nta ned by rubber gaskets or part cu ate spacers, one must be concerned about the exact a gnment of the bound ng wa s. Spec f ca y, a s ght t t to e ther surface w th respect to the pr mary f ow d rect on w create e ther a converg ng or d verg ng f ow, wh ch cou d s gn f cant y nf uence bu k f ow character st cs and des gn ssues such as the tota pressure drop across the m crochanne . If po ystyrene s des are used as the bound ng surfaces, f ex ng of the wa at h gh pressures cou d a so resu t n a dev at on from para e a gnment. F gure 7.4 shows a d agram of the gradua y converg ng channe geometry, symmetr c about the center ne x = 0. The correspond ng p ane Po seu e f ow so ut on for the ve oc ty prof e s:

2 H2 dp 2x vz = (7.2) − 1− 8µ dz H where the d fferent a vers on of the pressure drop s used, to a ow genera zat on to a s ow y vary ng channe cross-sect on. S m ar y, for p ane Po seu e f ow the f ow rate per un t w dth, q s: H3 dp q= − (7.3) 12µ dz For sma var at on n the channe spac ng, (H1 − H2 )/H1 1, t can be assumed that the f ow w dev ate from the p ane Po seu e so ut on on y s ght y. Thus, we are ust f ed n assum ng that Eqs. (7.2) and (7.3) w rema n va d for the case dep cted n F g. 7.4, on y w th H as a s ow y vary ng funct on of z. S nce the f owrate through any cross-sect on must be a constant, we conc ude that the oca pressure grad ent must have the fo ow ng spat a dependence: 1 dp ~ 3 dz H (z) (7.4) X H1 H2 Z F g. 7.4. Pressure-dr ven f ow through a gradua y converg ng channe . In the genera case of a curved upper surface, the pos t on of the upper wa s def ned by x = H (z). Chapter 7. B omed ca app cat ons of m crochanne f ows 417 Thus, rearrang ng Eq. (7.3) for the oca pressure drop n the gradua y converg ng channe y e ds the express on: dp 12µq =− 3 dz H (z) (7.5) Equat on (7.5) can be ntegrated over the ength of the channe L = z2 − z1 , to g ve the overa pressure drop across the converg ng channe , z2 p = −12µq H −3 (z)dz (7.6) z1 For a s mp e near var at on n channe he ght of the form H1 − H2 (z − z1 ) H (z) = H1 + z1 − z 2 (7.7) the so ut on to Eq. (7.6) becomes L 1 1 p = −6µq − 2 H1 − H 2 H22 H1 (7.8) Th s can be compared to the pressure drop across a un form p anar channe , L 1 p = −12µq H H2 (7.9) F gure 7.5 shows the rat o of the tota pressure drop across the converg ng channe to the tota pressure drop n a tru y para e channe eva uated at e ther the n t a he ght (upper, p/p(H1) or p/p(Havg) 1.8 1.7 1.6 1.5 1.4 p/p(H1) 1.3 1.2 p/p(Havg) 1.1 1 0 0.05 0.1 0.15 0.2 0.25 (H1 H2)/H1 F g. 7.5. P ot of the rat o of the pressure drop over a gradua y converg ng channe to the pressure drop n a para e channe eva uated at e ther the n t a he ght (upper, so d ne) or the average he ght ( ower, dashed), as a funct on of the d mens on ess he ght d fference. Note from the upper curve that a d mens on ess he ght d fference of (H1 − H2 )/H1 = 0.25 resu ts n an ncrease n tota pressure drop of 55% re at ve to a para e channe w th a spac ng of H1 (upper curve), however, eva uat ng the pressure drop at the mean channe he ght ( ower curve) captures a but 5% of th s ncrease. 418 Heat transfer and f u d f ow n m n channe s and m crochanne s so d curve) or eva uated at the average he ght ( ower, dashed curve). Note that us ng the average he ght n Eq. (7.9) y e ds an est mate for the pressure drop n a converg ng channe that s accurate to w th n 90%. 7.3. M crochanne s and m n channe s as b oreactors for ong-term ce cu ture B oreactors are qu d vesse s n wh ch ve popu at ons of suspended or surface-attached ce s can be ma nta ned for extended per ods of t me, e ther for expans on and d fferent at on of a des red ce type, or for the mass product on of a b omo ecu e produced by the ce s. B oreactors usua y feature gas and/or qu d exchange, to rep en sh nutr ents to the ce s as they are consumed and to remove waste and other products made by the ce s. Wh e commerc a ce -based product on of sma mo ecu es (e.g., ethano ) or prote ns nvo ves the sca e up of b oreactors to >1000 ga on tanks, there s much nterest n m n atur z ng b oreactors for use as b oart f c a organ rep acement of the pancreas (Su van et a ., 1991), ver (Juaregu et a ., 1997), and k dney (C es nsk and Humes, 1994). These b oart f c a organs, n wh ch one or more ve ce popu at ons are comb ned w th art f c a eng neered mater a s, must rep ace the fo ow ng funct ons of the nat ve organs (Fourn er, 1999): ( ) Pancreas: The ma n purpose of the pancreas s to regu ate g ucose eve s n the body by host ng nsu n-produc ng ce s n sma c usters of ce s ca ed the “ s ets of Langerhans”. There s a great need for the restorat on of pancreat c funct on among those aff cted w th nsu n dependent d abetes me tus (IDDM). ( ) L ver: The human ver s a arge (~1500 g), h gh y vascu ar zed organ that rece ves 25% of the card ac output, and performs a var ety of cruc a fe funct ons. The ver stores and re eases excess g ucose, s the pr mary organ for fat metabo sm, and produces v rtua y a of the p asma prote ns other than ant bod es. The ma n eng neer ng cha enge n deve op ng a b oart f c a ver rep acement s that the organ s home to near y 250 b on hepatocyte ce s, each around 25 µm n d ameter! ( ) K dney: Among the v ta respons b t es of the k dneys are f trat on of, and waste remova from, the b ood, regu at on of red ce product on w th n the bone marrow, and rap d contro of b ood pressure through the product on of vasoact ve mo ecu es. M crochanne s and m n channe s prov de an exce ent opportun ty to max m ze surface area ava ab e for mono ayer ce attachment, wh e enab ng good temperature regu at on and rap d exchange of qu d med a. Successfu m n atur zat on of b oreactors for organ rep acement cou d resu t n a dev ce that may be comfortab y worn outs de the body, or even potent a y mp anted w th n the abdom na cav ty. Another potent a advantage of m crochanne s as b oreactors and the oca f ow env ronments that they are ab e to produce s that many ce types such as osteob asts (bone ce s) and endothe a ce s (b ood vesse n ng) have evo ved to prefer constant exposure to shear stresses >1 dyne/cm and exh b t norma phenotype on y n such an env ronment. 7.3.1. Rad a membrane m n channe s for hematopo et c b ood ce cu ture Hematopo et c stem and precursor ce s (HSPC) n the bone marrow are ab e to d fferent ate and produce a of the d fferent types of b ood ce s n the body. Bone marrow ce s grown Chapter 7. B omed ca app cat ons of m crochanne f ows 419 Regu ated gas f ow Gas phase H Gas-permeab e membrane L qu d phase Pumped- n ce med a 0 Ce s p ated on bottom wa Waste/ recyc e L X Z F g. 7.6. M n channe membrane b oreactor. ex v vo ma nta n th s p ur potent nature, and there s great nterest n expand ng b ood ce popu at ons n b oreactors for ater transp antat on to treat var ous b ood d sorders. The d ff cu ty n grow ng arge quant t es of mamma an ce s n cu ture s that oxygen s rap d y consumed and must be cont nuous y rep en shed somehow. Peng and Pa sson (1996) exam ned the potent a of a rad a membrane b oreactor to cu ture pr mary human mononuc ear ce s sub ected to severa growth factors ( nter euk n-3, granu ocyte-macrophage co onyst mu at ng factor, erythropo et n) known to nduce product on of wh te and red b ood ce s. F gure 7.6 shows a d agram of a rad a membrane b oreactor. Ce med a saturated w th oxygen enters the chamber from the eft-hand-s de, at a we -deve oped am nar ve oc ty u(x). Oxygen n the med a s consumed by the ce s near the ower surface. The f u d near the upper, gas-permeab e membrane s assumed to be n equ br um w th the gas, and at the saturat on oxygen concentrat on CS . In such a system, the f u d ve oc ty s chosen to be very ow – typ ca trans t t mes equa about 1.33 days. For compar son, the d ffus v ty of oxygen n prote n so ut ons at 37 C has been measured to be D = 2.69 × 10−5 cm2 /s (Go dst ck, 1966). For a channe he ght of 3 mm, the character st c t me for d ffus on from the upper membrane to the ce ayer at the bottom of the channe s 1 h. Thus, n th s case convect ve transport can be neg ected and the oxygen transport s mode ed as a one-d mens ona d ffus on equat on of the form, D ∂2 C =0 ∂x2 (7.10) The boundary cond t on at the upper membrane s that the f u d s saturated w th oxygen: C = CS @x = H (7.11) and at the ower wa the f ux s equa to the ce u ar uptake rate N0 : N0 = D ∂C ∂x @x = 0 (7.12) 420 Heat transfer and f u d f ow n m n channe s and m crochanne s Sub ect to these two cond t ons the so ut on to Eq. (7.10) s s mp y a near concentrat on prof e, C(x) = CS + N0 (x − H ) D (7.13) The uptake rate N0 obeys M chae s–Menten k net cs g ven by: N0 = qXC0 Km + C 0 (7.14) where C0 s the oxygen concentrat on at the ower wa , that s, the oxygen concentrat on exper enced by the ce s, q s the oxygen uptake rate on a per-ce bas s, and X s the ce dens ty. The M chae s constant Km , wh ch can be nterpreted as the concentrat on at wh ch the uptake rate s ha fmax mum, has been measured to be about 1–5% of CS (Ozturk, 1990). Th s mp es that the oxygen consumpt on operates at near zeroeth-order k net cs. Subst tut ng Eq. (7.14) nto Eq. (7.13) and eva uat ng the so ut on at x = 0 y e ds a quadrat c express on for C0 : C02 + (Km − CS + qXH /D)C0 − CS Km = 0 (7.15) In terms of the fract ona saturat on = C/CS , Eq. (7.15) becomes, 02 + ( + − 1)0 − = 0 (7.16) where we have ntroduced two d mens on ess parameters, a d mens on ess M chae s constant = Km /CS , and = qXH /DCS = H 2 /D HCS /qX (7.17) wh ch s the rat o of the character st c t me for d ffus on to the character st c t me for ce u ar oxygen uptake. The parameter depends on the ce dens ty, wh ch changes w th t me as the ce s on the surface d v de and doub e n number every 24–48 h. There s on y one phys ca y reasonab e root to Eq. (7.16), s nce the other w a ways be negat ve. Th s fract ona saturat on des gn equat on can be used to determ ne the range of parameter va ues for wh ch the ce s rece ve suff c ent oxygen to support pro ferat on. 7.3.2. The B oart f c a ver: membranes enhance mass transfer n p anar m crochanne s Many types of ce s can be grown n v tro more successfu y when co-cu tured n the presence of a second ce type such as 3T3-J2 f brob asts, wh ch are robust and can produce v ta nutr ents that support the growth of the ce s of nterest. T es et a . (2001) stud ed the co-cu ture of rat hepatocytes ( ver ce s) w th 3T3-J2 f brob asts n m crochanne Chapter 7. B omed ca app cat ons of m crochanne f ows 421 X Cg Z C0 So d wa U C0 U Po yurethane membrane H Ce ayer (a) (b) F g. 7.7. M crochanne b oreactors (a) w thout and (b) w th an nterna membrane oxygenator. b oreactors (H = 85–500 µm). The r ana ys s of the effects that an oxygenat ng membrane has on hepatocyte growth and prote n product on (see F g. 7.7) prov des a usefu set of des gn equat ons for the deve opment of m crochanne membrane-based b oreactors. For s mp c ty, we assume un form, or “p ug”, f ow w th n the m crochanne (see Prob em 7.4 for a mod f cat on of th s approx mat on). The f ow d rect on s def ned w th the z-coord nate, and oxygen d ffus on occurs n the x-d rect on norma to the wa and s consumed by hepatocytes at x = 0. Once the oxygen concentrat on C(x, z) s non-d mens ona zed w th the n et concentrat on C n , and the x and z var ab es nond mens ona zed w th the he ght (H ) and ength (L) of the b oreactor, respect ve y, then the d mens on ess steady-state convect on-d ffus on equat on becomes ∂2 C * ∂C * = ∂z Pe ∂x2 (7.18) Where we have ntroduced the d mens on ess ength rat o = L/H , and the Pec et number Pe = UH /D. Here U s the average ve oc ty and D s the oxygen d ffus v ty. The boundary cond t ons for Eq. (7.18) are C* = 1 ∂C * = Da ∂x * ∂C 0 = Sh[Cg* − C * ] ∂x @ z = 0, 0 ≤ x ≤ 1 @ x = 0, 0 ≤ z ≤ 1 (7.19) @ x = 1, 0 ≤ z ≤ 1 @ x = 1, 0 ≤ z ≤ 1 where there ex st two d fferent f ux cond t ons at the upper surface depend ng on whether there s no membrane oxygenat on, or w th membrane oxygenat on. We have ntroduced two add t ona d mens on ess groups, the Damkoh er number Da = Vm HX DC n (7.20) and the Sherwood number Sh = H D (7.21) 422 Heat transfer and f u d f ow n m n channe s and m crochanne s where the zeroeth-order uptake of oxygen by the hepatocytes s def ned as Vm , the surface dens ty of ce s as X , and s the membrane permeab ty to oxygen. We have nond mens ona zed the gas phase oxygen concentrat on w th C n as Cg* = Cg /C n . A s m ar d fferent a system has been so ved ana yt ca y be Cars aw and Jaeger (1959), wh ch can be app ed to y e d the ce surface oxygen concentrat on a ong the ength of the b oreactor e ther w thout nterna membrane oxygenat on: ∞ Da 2Da (−1)n n2 2 z C (0, z) = 1 − z Da − + 2 cos(n) exp − Pe 3 n2 Pe * (7.22) n=1 or w th nterna membrane oxygenat on: C * (0, z) = Cg* − Da − ∞ Da 2 z + Bn exp − n Sh Pe (7.23) n=1 In Eq. (7.23), the coeff c ents Bn are def ned by: Bn = 4Da(1 − cos n ) + 4n s n n (1 − Cg * + Da/Sh) [n s n 2n + 22n ] (7.24) and n are the roots of the transcendenta equat on: tan n = Sh n (7.25) Examp e 7.1 (see page 434) ustrates the use of these des gn equat ons (7.22–7.25). 7.3.3. Oxygen and actate transport n m cro-grooved m n channe s for ce cu ture As stated n Sect on 7.3.1, there s cons derab e nterest n cu tur ng and expand ng hematopo et c stem ce popu at ons ex v vo, for ater auto ogous (donor = rec p ent) or a ogene c (donor = rec p ent) transp antat on for the treatment of hemato og ca or mmuno og ca d sorders and to treat comp cat ons of cancer therap es. In the r nat ve env ronment n the bone marrow, hematopo et c ce s are not strong y adherent to the r surround ngs but are reta ned n the marrow n part due to adhes on to a second type of ce s ca ed stroma ce s. It has been recogn zed that ex v vo ce perfus on systems for ong-term cu ture that are ab e to reta n weak y adherent ce s wou d ho d many advantages over co-cu ture systems (Sandstrom et a ., 1996). Horner et a . (1998) ana yzed the so ute transport n a m cro-grooved m n channe des gned to reta n non-adherent ce s w th n m crosca e cav t es or ented perpend cu ar to the ma n f ow, as dep cted n F g. 7.8. F ow through the ma n m n channe generates a form of d-dr ven f ow w th n each cav ty known as nested Moffatt edd es (Moffatt, 1964). These edd es prov de convect on-enhanced mass transfer between the ce s rest ng at the bottom of each cav ty and the ma n chamber, wh e she ter ng the ce s from the h gh wa shear stresses that wou d be exper enced n a comparab e f at-bottomed channe . Chapter 7. B omed ca app cat ons of m crochanne f ows 423 5 mm channe X Z 200 m F g. 7.8. M cro-grooved, m n channe b oreactor for the cu ture of weak y adherent ce s. Note that the ce s are she tered w th n the grooves and do not become entra ned n the f ow. The exter or f ow nduces Moffett edd es that exchange so utes between the groove cav t es and the m n channe reg on. For a re at ve y w de channe of severa cent meters, the aspect rat o s such that the th rd d mens on (vort c ty d rect on) can be neg ected and the prob em treated as twod mens ona . As n our prev ous b oreactor examp es, we take the f ow d rect on as z and the ve oc ty grad ent d rect on as x. At steady-state, the Nav er–Stokes equat on n two-d mens ons s mp f es to:

2 ∂ 2 vx ∂p ∂ vx ∂vx ∂vx vx + + vz =− +µ ∂x ∂z ∂x ∂x2 ∂z 2

2 ∂p ∂ vz ∂vz ∂vz ∂ 2 vz vx + vz =− +µ + 2 ∂x ∂z ∂z ∂x2 ∂z (7.26) Note that un ke prev ous examp es where the f ow was assumed un d rect ona n the z-d rect on and d ffus on n the z-d rect on neg ected re at ve to convect on, ead ng to s mp f cat ons to the govern ng equat ons, the d-dr ven cav ty f ow s expected to be fu y two-d mens ona and these fu equat ons must be ntegrated. The so ute ba ance for spec es , at steady-state, takes the form: ∂C ∂C vx + vz = −D ∂x ∂z

∂ 2 C ∂ 2 C + ∂x2 ∂z 2 (7.27) The so utes of ma n b o og ca nterest (oxygen and actate) are d ute sma mo ecu es that do not apprec ab y affect the f u d dens ty and v scos ty. Thus, the f u d ve oc ty f e d may 424 Heat transfer and f u d f ow n m n channe s and m crochanne s be so ved ndependent y of the so ute ba ance equat on. It s reasonab e to mpose a we deve oped parabo c ve oc ty prof e at the channe n et, prov ded that the ce -conta n ng reg on occurs after the entrance ength g ven by: Lent = a Re H (7.28) where the Reyno ds number s def ned for channe f ow as: Re = vH µ (7.29) In Eq. (7.28), the coeff c ent a s equa to 0.034 to ensure that the center ne ve oc ty s w th n 98% of the theoret ca Po seu e f ow m t (Bodo a and Oster e, 1961), and the ang ed brackets denote an averaged quant ty. For typ ca b oreactor cond t ons, the entrance ength s reached w th n a fract on of one channe he ght (<0.05H ). No-s p cond t ons are enforced at the upper, ower, and m cro-groove wa s. The genera f ux cond t on at the ower wa of the m cro-groove cav t es, p ated w th hematopo et c ce s, s obta ned by sett ng the surface so ute f ux equa to a M chae s–Menten k net c rate of so ute consumpt on or product on (e.g., oxygen or actate, respect ve y). Th s boundary cond t on at the ce surface takes the form: N0 = −D ∂C qXC0 = ∂x Km + C 0 (7.30) where a s gn change from Eq. (7.12) s ntroduced for a oca coord nate system w th n the cav ty correspond ng to an upper cav ty open ng at x = 0 and the pos t ve x-coord nate d rected downward. The system of Eqs. (7.26), (7.27), and (7.30), sub ect to no-s p at the wa and g ven n et ve oc ty and so ute concentrat on cond t ons, can be read y so ved for the spat a concentrat on prof es us ng a commerc a computat ona f u d dynam cs (CFD) code such as FIDAP (F uent, Inc., Lebanon, NH). Horner et a . (1998) have used fu numer ca ana ys s of the m cro-grooved m n channe system to der ve const tut ve equat ons that descr be the so ute concentrat on w th n the groove cav t es as a funct on of var ous phys ca parameters such as f owrate, channe d mens ons, and zeroeth-order so ute consumpt on (product on) rate. 7.4. Generat on of norma forces n ce detachment assays Hydrodynam c f ow n para e -p ate m crochanne s s often used to detach f rm y adhered ce s from substrates, as a measure of the strength of ntegr n:matr x prote n bonds (F g. 7.9). However, the h gh f owrates necessary to detach spread ce s, when f ow ng over a f at ce w th a ra sed nuc ear reg on, may produce a negat ve pressure over the nuc eus (the so-ca ed “Bernou effect”) that can potent a y equa or exceed the shear forces that one s try ng to generate (F g. 7.10). Thus, here we mode th s prob em as nv sc d, rrotat ona f ow over a wavy wa as deve oped by M che son (1970), keep ng Chapter 7. B omed ca app cat ons of m crochanne f ows 425 H 250 m F ow F g. 7.9. M crochanne geometry for ce detachment. F ow F g. 7.10. The nuc e of adherent ce s pro ect nto the f ow f e d. n m nd that th s w on y be va d for Reyno ds numbers much greater than un ty (often ach eved n prev ous exper menta stud es as d scussed be ow). F gure 7.9 dep cts the d mens ons of a typ ca ce detachment f ow chamber. From the terature, the force requ red for ce detachment from an adhes ve substrate can be est mated as between 10−9 and 10−7 N. In sa ne buffer (µ = 1 cP), th s requ res f ow rates of Uavg = 1−100 cm/s. In such an exper ment, the h ghest Reyno ds number stud ed wou d be Re = Qw = 256 (7.31) Thus, not on y s nert a mportant n th s m crosca e system, but may be expected to dom nate over v scous effects. 7.4.1. Potent a f ow near an nf n te wa Our start ng po nt for mode ng the f ow over a ce mono ayer s the set of non- near Eu er s equat ons: ∂u ∂u 1 ∂p ∂U +v + =− ∂x ∂y ∂x ∂x ∂v ∂v 1 ∂p ∂U u +v + =− ∂x ∂y ∂y ∂y u (7.32) wh ch descr bes steady, two-d mens ona fr ct on ess f ows sub ect to conservat ve forces ( .e. der ved from a potent a U ) such as grav ty. When a r g d body rotates at angu ar speed , the ve oc ty components of a po nt ocated at (x, y) are u = −y, v = x (7.33) 426 Heat transfer and f u d f ow n m n channe s and m crochanne s E m nat ng x and y from Eq. (7.33) y e ds a def n t on for the vort c ty , ∂v ∂u 2 = − ∂x ∂y (7.34) a measure of the oca f u d rotat on. Ke v n s theorem states that the vort c ty s a constant n ncompress b e, fr ct on ess f ows, when the on y forces are der ved from potent a s such as grav ty. Therefore, Ke v n s theorem g ves us an add t ona re at on: ∂v ∂u − =0 ∂x ∂y (7.35) a ong w th the cont nu ty equat on for ncompress b e f ow, ∂u ∂v + =0 ∂x ∂y (7.36) We def ne the ve oc ty potent a (x, y) as u=− ∂ , ∂x v=− ∂ ∂y (7.37) wh ch dent ca y sat sf es the rrotat ona cond t on. Subst tut ng th s def n t on nto the cont nu ty equat on y e ds ∂2 ∂2 + 2 =0 ∂x2 ∂y (7.38) Thus, the ve oc ty potent a for p ane rrotat ona f ow of an ncompress b e f u d s a so ut on to Lap ace s equat on. Th s equat on s so ved for the un form f ow past a f at wa n Examp e 7.2 (see page 436). 7.4.2. L near zed ana ys s of un form f ow past a wavy wa To mode the d sturbance f ow generated by a un form f ow over a per od c ayer of ce s, one of the stream nes of un form f ow s mod f ed to be everywhere on y near y para e to the x-ax s. We expect that the ve oc t es e sewhere w d ffer s ght y from U, and the assoc ated pressure var at ons w be sma . F g. 7.11 shows a s ght y curved boundary wa of s nuso da form, y = h s n x (7.39) The ength of the who e cyc e of the s ne wave s L = 2/ and here h s the wave he ght. We requ re that h be sma as: h L wh ch corresponds to an assumpt on of “s ght wav ness”. The f ow ve oc ty components are assumed to d ffer on y s ght y from (U , 0) and can be wr tten as: u = U + u , v = v (7.40) Chapter 7. B omed ca app cat ons of m crochanne f ows 427 U y 0 x y h s n x 2p/ F g. 7.11. Geometry of the near zed ana ys s near a wavy wa . where the pr mes nd cate ve oc ty perturbat ons that van sh w th h, and u U , v U (7.41) Because of the rrotat ona ty cond t on, ∂v ∂u − =0 ∂x ∂y (7.42) the ve oc ty potent a (x, y) may be taken to refer on y to the perturbat ons u = − ∂ , ∂x v = − ∂ ∂y (7.43) Th s y e ds the fo ow ng express on for the tota f ow: u=U− ∂ , ∂x v=− ∂ ∂y (7.44) Bernou s theorem can now be app ed: u2 p + + potent a = constant 2 (7.45) wh ch s va d at d fferent po nts a ong a s ng e stream ne. Thus, the near zed vers on of the govern ng equat on becomes p(x, y) − p0 = U ∂ ∂x (7.46) 428 Heat transfer and f u d f ow n m n channe s and m crochanne s where we have subtracted off the quant ty p0 + U 2 = cst. eva uated far from the wa . In Eq. (7.46), p0 denotes the pressure n those parts of the f ow where the ve oc ty reaches ts un form va ue U. The ve oc ty potent a s a harmon c funct on g ven by the equat on ∂2 ∂2 + 2 =0 ∂x2 ∂y (7.47) We must now f nd the part cu ar so ut on to Eq. (7.47) out of the many genera so ut ons to Lap ace s equat on. F rst, we note the farf e d cond t on (x, ∞) = 0. At the boundary, we enforce that the s ope of the ve oc ty vector equa s the boundary s ope: v = h cos x u (7.48) From our ear er deve opment, the eft-hand-s de of Eq. (7.48) s g ven by LHS = − 1 ∂ ∂/∂y =− U − ∂/∂x U ∂y (7.49) where we have neg ected h gher order terms. We may wr te an approx mate boundary cond t on eva uated at y = 0: − ∂ (x, 0) = Uh cos x ∂y (7.50) Note that the d fferent a equat on, boundary cond t ons, and pressure re at onsh p are a near n -der vat ves. Th s prob em adm ts a separab e so ut on of the form: X (x) = A cos k x + B s n k x Y (y) = C exp(−k y) + D exp(k y) (7.51) wh ch comb ne to g ve the fo ow ng potent a funct on: (x, y) = {A cos k x + B s n k x}{C exp(−k y)+D exp(ky)} (7.52) where the term D exp(ky) van shes n order to sat sfy the farf e d cond t on. We are thus eft w th two rema n ng constants AC and BC, and can set C = 1 w thout oss of genera ty. The boundary cond t on at y = 0 takes the form Uh cos x = k{A cos k x + B s n k x} (7.53) wh ch, upon man pu at on, y e ds the resu t Uh = kA, B = 0, k= (7.54) These constants can be nserted nto our genera so ut on to g ve (x, y) = Uh exp(−y) cos x (7.55) Chapter 7. B omed ca app cat ons of m crochanne f ows 429 p p F g. 7.12. Pressure d str but on at the surface of the wa . for the ve oc ty potent a , and the fo ow ng express on for the pressure at any po nt n the f ow: p(x, y) = p0 − U 2 h s n x exp(−y) (7.56) Thus, the near approx mat on to the pressure on the boundary wa y = 0 s p(x, 0) = p0 − U 2 h s n x (7.57) Equat on (7.57) mp es that the pressure s a m n mum at the peaks of the wavy wa (x = /2) and reaches a max mum at the wave troughs, as dep cted n F g. 7.12. Typ ca parameter va ues for ce detachment exper ments y e d the fo ow ng est mate for the peak (negat ve) pressure above the ce nuc eus: U 2 h = (1000 kg/m3 )(1 m/s)2 (2 × 10−6 m)(2/20 × 10−6 m) (7.58) Thus, the pressure above the ce nuc eus s pred cted to have magn tude p = 628 N/m2 , or assum ng a character st c ce area of 100 µm2 , a negat ve pressure of magn tude F = |63 nN|. Th s va ue for the norma force nduced by the Bernou effect s of the same order as the shear forces generated by the mposed f ow! Reana ys s of many prev ous exper ments us ng ce detachment assays show the generat on of arge norma forces n both the m crochanne geometry (VanKooten et a ., 1992; Schn tt er et a ., 1993; Sank et a ., 1994; Thompson et a ., 1994; Wechazak et a ., 1994; Kapur and Rando ph, 1998; S ro s et a ., 1998; Chan et a ., 1999; Ma ek et a ., 1999; Grandas et a ., 2001) as we as the sp nn ng d sk assay (Garc a et a ., 1997; Boett ger et a ., 2001; De g ann et a ., 2001; M er and Boett ger, 2003). To reduce the Reyno ds number, and thus reduce the norma forces re at ve to shear forces, one shou d s mp y ncrease the f u d v scos ty w th a h ghmo ecu ar we ght mo ecu e such as dextran. For nstance, a 7.5 we ght percent dextran so ut on, of mo ecu ar we ght MW = 2 × 106 g/mo , wou d resu t n a re at ve v scos ty of µ = 20.5 µ0 7.5. Sma -bore m crocap ar es to measure ce mechan cs and adhes on The human c rcu atory system represents a cha eng ng system for eng neer ng mode ng, due to the w de var at on n vesse geometry, near f ow ve oc ty, and Reyno ds 430 Heat transfer and f u d f ow n m n channe s and m crochanne s Tab e 7.2 Typ ca f ow parameters n the human system c c rcu at on. Structure D ameter (cm) B ood ve oc ty (cm/s) Tube Reyno ds number Ascend ng aorta Descend ng aorta Large arter es Arter o es Cap ar es Venu es Large ve ns Vena cavae 2.0–3.2 1.6–2.0 0.2–0.6 0.001–0.015 0.0005–0.001 0.001–0.02 0.5–1.0 2.0 63 27 20–50 0.5–1.0 0.05–0.1 0.1–0.2 15–20 11–16 3600–5800 1200–1500 110–850 0.014–0.43 0.0007–0.003 0.0029–0.11 210–570 630–900 Data from Cooney, 1976; Wh tmore, 1968; Schneck, 2000. numbers, start ng at the ma or aort c vesse s down to the sma est m croscop c cap ar es. Tab e 7.2 shows the range of these va ues n a typ ca human, assum ng a constant b ood v scos ty of 0.035 P and mean peak ve oc t es reported for the pu sat e f ow of arter a f ow. It s ev dent that the human c rcu at on covers the ent re range of f ow behav or from creep ng (non- nert a ) f ow up to fu y deve oped turbu ence. We see n the arter o es, cap ar es, and venu es that the ength sca es span the range of f u d phys cs encountered n m crochanne and m n channe f ow. In the argest vesse s of the body, b ood behaves as a non-Newton an f u d and const tut ve equat ons are usua y successfu n descr b ng behav or (e.g. b unted ve oc ty prof es) at the cont nuum eve . At the sca e of the sma est vesse s however, the part cu ate nature of b ood and ts const tuent ce s must be taken nto account (K ng et a ., 2004). In the cap ar es of the ungs, for nstance, the mechan ca propert es of nd v dua wh te b ood ce s s gn f cant y nf uence the trans t t me through these vesse s (Bathe et a ., 2002). Suspens ons of erythrocytes (red b ood ce s, or RBCs) f ow ng through m crocap ar es (d ~ 40 µm) are we known to concentrate at the tube center, resu t ng n ve oc ty prof es b unted from parabo c. Tra ector es of nd v dua RBCs have been v sua zed both n v tro (Go dsm th and Mar ow, 1979) and n v vo (Lom nadze and Mched shv , 1999), show ng rad a dr ft ve oc t es and random wa ks that attenuate as the ce approaches the center ne. The s ng e-f e mot on of RBCs through sma cap ar es (d ~ 4–20 µm) has been mode ed theoret ca y sub ect to var ous s mp fy ng assumpt ons: x–y two-d mens ona ty (Sug hara-Sek et a ., 1990), symmetry about the center ax s (Secomb and Hsu, 1996), or approx mat on v a mu t po e expans on (O a, 1999). The rad a dr ft of RBCs s known to produce a ce free p asma ayer n m crovesse s (Yamaguch et a ., 1992), suggest ng that RBCs have on y a secondary nf uence on the dynam cs of eukocytes once they are d sp aced to the vesse wa (Go dsm th and Spa n, 1984). Furthermore, eukocytes have a dramat c nf uence on tota b ood v scos ty n the m croc rcu at on desp te the r re at ve y ow numbers n b ood as compared to RBCs (He mke et a ., 1998), h gh ght ng the need to better understand the mot on of eukocytes through m crocap ar es. Examp e 7.3 (see page 437) prov des an est mate of the b ood trans t t me through a cap ary, where the ma or ty of oxygen exchange takes p ace. Chapter 7. B omed ca app cat ons of m crochanne f ows 431 Ce samp e Sheath f u d Hydrodynam c focus ng reg on Forward scatter detector Laser S de scatter detector

Charged p ates to steer ce s to the eft or r ght F g. 7.13. Schemat c of a f ow cytometer, and f uorescence act vated ce sort ng (FACS). 7.5.1. F ow cytometry F ow cytometry s a powerfu ana ys s too n b omed ca research, that ut zes m crosca e f ow for automated s ng e ce character zat on and sort ng. The concentrat on of any f uorescent y abe ed mo ecu e on the surface of the ce can be measured re at ve to some standard, and these ce s can furthermore be sorted accord ng to the concentrat on of th s marker or accord ng to s ze. Ant bod es aga nst many of the known receptors on the surface of human or an ma ce s can be obta ned commerc a y. F gure 7.13 shows a schemat c of a typ ca system. The ce suspens on (poss b y conta n ng a m xture of d fferent ce types) s ntroduced through a c rcu ar condu t. Th s condu t opens nto a arger f ow doma n conta n ng a f ow ng annu us of “sheath f u d” wh ch acts to focus the stream of ce s nto a s ng e-f e conf gurat on w th n a narrow et of f u d (<100 µm d ameter). Th s stream of ce s then passes through a quartz chamber where they are character zed for the r scatter ng and f uorescent propert es. Most common y, a beam of ght from an argon aser s used to exc te the f uorophore of nterest. The em tted ght from th s 432 Heat transfer and f u d f ow n m n channe s and m crochanne s f uorophore s then co ected at one of two photomu t p er tubes. Downstream of these opt cs, the out et stream s ruptured nto a ser es of drop ets, each conta n ng a s ng e ce , v a p ezoe ectr c v brat on. An e ectr ca f e d s then used to steer pos t ve y or negat ve y charged nd v dua drop ets (a charge mposed depend ng on the prev ous f uorescence ntens ty of the correspond ng ce ) nto mu t p e co ect on vesse s. The who e process s automated so that commerc a y ava ab e f ow cytometers can sort ce s at a rate of over 104 ce s/s. In add t on to character z ng nd v dua ce s for c n ca app cat ons, f ow cytometry has been used to measure the s ze of sma b ood ce aggregates, where up to pentup ets of neutroph s can be successfu y dent f ed (S mon et a ., 1990; Nee amegham et a ., 2000). 7.5.2. M crop pette asp rat on Sma -bore g ass m crocap ar es are often used to e ther character ze the mechan cs and deformab ty of nd v dua ce s, or to test the surface adhes on of ce s contact ng other ce s or art f c a bead surfaces (Lomak na et a ., 2004; Sp mann et a ., 2004). Cap ar es w th nner d ameter ess than the ma or d ameter of ce s such as b ood ce s are used to part a y asp rate the ce , w th the pro ect on ength of the ce w th n the cap ary nter or be ng a measure of the cort ca tens on of the ce exter or (Herant et a ., 2003). Once the ce s e ected from the m crop pette, ts shape recovery can a so y e d nformat on on the v scoe ast c parameters of the ce (Dong et a ., 1988). Cap ar es w th nner d ameter arger than the ce d ameter can be used to trans ate the ce back and forth w th n the p pette to br ng a ce nto repeated contact w th another nearby surface. In th s manner, the probab ty for ce adhes on can be determ ned n the presence of var ous b o og ca st mu . Shao and Hochmuth (1996) ca cu ated the pressure drop caused by the gap f ow n the annu ar reg on between a spher ca ce and a arger cy ndr ca m crop pette. The geometry cons dered s dep cted n F g. 7.14. Assum ng ax symmetr c f ow, and neg ect ng nert a, y e ds the fo ow ng form of the Nav er–Stokes equat ons n cy ndr ca coord nates: 1 ∂ ∂w (ru) + =0 r ∂r ∂z 2 ∂p ∂2 u ∂ u 1 ∂u u + =µ − 2+ 2 ∂r ∂r 2 r ∂r r ∂z 2 ∂p ∂ w 1 ∂w ∂2 w + =µ + 2 ∂z ∂r 2 r ∂r ∂z (7.59) where p s the pressure, u and w are the oca f u d ve oc t es n the r- and z-d rect ons, respect ve y, and µ s the f u d v scos ty. No-s p cond t ons are enforced at a boundar es, and the tota vo umetr c f owrate through any c rcu ar cross-sect on of the p pette s a constant Q. After cons derab e man pu at ons, usefu ana yt ca express ons can be obta ned re at ng the tota pressure drop p, sphere ve oc ty U , and externa force act ng on the sphere F, for two s mp f ed cases: ( ) stat onary sphere, and ( ) force-free sphere. Chapter 7. B omed ca app cat ons of m crochanne f ows 433 r U F Ds z (a)

Rp (b) F g. 7.14. (a) Spher ca ce pos t oned at the center ne w th n a arger m crop pette. (b) C ose-up of the upper ha f of the cy ndr ca reg on. F rst, the force necessary to ho d a sphere stat onary w th n a pressure-dr ven f ow n a m crop pette s: F= 1 + 43 ¯ + R2p p √ 8(Leq −Ds ) Rp 2 2 5/2 ¯ 9 − 8 3 ¯ 92

(7.60) where Leq s the tota p pette ength, and the gap spac ng has been non-d mens ona zed w th the p pette rad us. For a neutra y buoyant or force-free sphere, on the other hand, the re at onsh p between the tota pressure drop and the sphere ve oc ty s g ven by: √ 8 Leq − Ds 4 2 71 4 µU 1 − ¯ − p = + (7.61) Rp 5 Rp 3 ¯ 1/2 7.5.3. Part c e transport n rectangu ar m crochanne s The transport of suspended ce s n rectangu ar m crochanne s s of nterest n the ana ys s of red ce d mens ons (G fford et a ., 2003), the study of b ood coagu at on and thrombot c vesse occ us on (Kamada et a ., 2004), measurement of ntrace u ar s gna ng n Jurkat ce s (L et a ., 2004), and v ra -based transfect on (Wa ker et a ., 2004). When the channe he ght approaches the same order of magn tude as the d ameter of the transported ce s, the part c es are nf uenced by the presence of the wa s and assum ng that the part c es trave w th the oca f u d ve oc ty no onger rema ns va d. Staben et a . (2003) numer ca y ca cu ated us ng a boundary- ntegra a gor thm the mot on of spher ca and sphero da part c es w th n a rectangu ar channe . Both the trans at ona and rotat ona ve oc t es of nd v dua part c es are affected by the presence of the wa s of the channe . As can be seen 434 Heat transfer and f u d f ow n m n channe s and m crochanne s 0.5 0.4 Y/H 0.3 0.2 0.1 0 0 0.2 0.4 0.6 U/Uc 0.8 1 1.2 F g. 7.15. The d mens on ess trans at ona ve oc ty of a sphere n a two-d mens ona Po seu e f ow n a rectangu ar channe . So d curve represents the f u d ve oc ty, and the symbo s represent part c es, wh ch when non-d mens ona zed w th the he ght of the channe , have the fo ow ng d ameters: 0.2 (c rc es), 0.4 (squares), 0.6 (d amonds), 0.8 (tr ang es), and 0.9 (stars). A ve oc ty curves are symmetr c about Y /H = 0.5. Data from Staben et a . (2003). n F g. 7.15 th s retardat on effect can be qu te pronounced when the ce d ameter s 20– 90% of the ent re gap w dth. Interest ng y, because the part c e centers are exc uded from a reg on one rad us th ck at the upper and ower wa s (where the f u d ve oc ty s sma est), the average part c e ve oc ty for a co ect on of spheres can actua y be greater than the average ve oc ty of the suspend ng f u d tse f. Spec f ca y, Staben et a . showed that random y-d str buted part c es w th d ameter ≤82% of the gap he ght w have an average ve oc ty greater than the average f u d ve oc ty, and that th s effect s most pronounced for part c es w th d ameter d = 0.42 H , wh ch w trave at a ve oc ty 18% greater than the average f u d ve oc ty. 7.6. So ved examp es Examp e 7.1 Compare graph ca y the two so ut ons descr b ng the spat a dependence of the ce surface oxygen concentrat ons w thout and w th an nterna membrane oxygenator, for parameter va ues of = 100, Da = 0.25, Pe = 25, Sh = Cg = 1. We w use the Mat ab programm ng anguage to ca cu ate and d sp ay our two so ut ons. F rst, we p ot the eft- and r ght-hand-s des of our transcendenta Eq. (7.25) to approx mate y ocate the va ue of the f rst few roots n , us ng the commands: ambda = nspace(1e-4,12,1000); sem ogy( ambda,tan( ambda), ambda,1./ ambda, − ) Chapter 7. B omed ca app cat ons of m crochanne f ows 435 102 tan(n) or 1/n 101 100 101 102 0 2 4 6 8 10 12 n F g. 7.16. Mat ab p ot to determ ne the approx mate ocat on of the f rst few roots to the transcendenta equat on for n . where m nor f gure formatt ng commands have been om tted. Note that the ambda vector was started at 10−4 to avo d the s ngu ar ty at = 0. F gure 7.16 shows th s p ot, where t s ev dent that the two s des of the transcendenta equat on cross each other at around 1, 3, 6, 9, and t s c ear that the h gher order roots w be separated by a d stance of about 3.14. Now that we know the approx mate ocat on of the f rst few roots, we can use a bu t- n non- near root-f nd ng rout ne to f nd these more accurate y. The fo ow ng commands accomp sh th s: guess = [1 3 6 9 12]; for =1: ength(guess) ambda_n( ) = fzero( tan(x)-1/x ,guess( )); end Execut ng the above commands produces the fo ow ng roots for the f rst f ve n s: 0.8603, 3.4256, 6.4373, 9.5293, 12.6453. Now we are ready to ca cu ate the two surface concentrat on d str but ons and compare them. We w store the f rst f ve e genfunct ons n the second through s xth rows of two matr ces, ca ed C_no_mem and C_mem, and sum them up before p ott ng: z= nspace(0,1,100); Pe=25; Da=0.25; C_no_mem(1,:)=1-z*100/Pe*Da-Da/3; C_mem(1,:)=(1-Da-Da)*ones(s ze(x)); for n=1:5 C_no_mem(n+1,:)=2*Da/p ˆ2*(-1)ˆn/nˆ2*cos(n*p )*... exp(-100*nˆ2*p ˆ2*z/Pe); 436 Heat transfer and f u d f ow n m n channe s and m crochanne s 1.2 1 C *(0, z) 0.8 W th membrane oxygenator 0.6 0.4 W thout membrane oxygenator 0.2 0 0.2 0 0.2 0.4 0.6 0.8 1 z /L F g. 7.17. Resu ts from the ana yt ca mode of oxygen transport n a m crochanne b oreactor. Note that w thout a membrane oxygenator, the ce surface w be comp ete y devo d of oxygen before the end of the channe s reached. Bn=(4*Da*(1-cos( ambda_n(n)))+... 4* ambda_n(n)*s n( ambda_n(n))*(1-1+Da))/... ( ambda_n(n)*s n(2* ambda_n(n))+2* ambda_n(n)ˆ2); C_mem(n+1,:)=Bn*exp(-100* ambda_n(n)ˆ2*z/Pe); end p ot(z,sum(C_no_mem),z,sum(C_mem)) F gure 7.17 shows these resu ts, where t s ev dent that the m crochanne w thout membrane oxygenat on w be fu y dep eted of oxygen at the surface at a d mens on ess pos t on of about z = 0.9. Examp e 7.2 As an examp e we cons der the case of un form potent a f ow. Th s serves as the base state for our ana ys s of the f ow over a wavy ce surface n Sect on 7.4.2. In un form f ow, the potent a s g ven by: 1 = −Ux (7.62) where U s a constant. The correspond ng ve oc ty components are then g ven by: ∂1 = U = constant ∂x ∂1 v1 = − =0 ∂y u1 = − as expected. (7.63) Chapter 7. B omed ca app cat ons of m crochanne f ows 437 Examp e 7.3 Cons der the demand for effect ve mass transport n the m croc rcu at on. Cooney (1976) showed through s mp e sca ng arguments that the rate of mass transport rates across b ood cap ar es must be qu te effect ve ow ng to the re at ve y short res dence t me of b ood n these vesse s. The typ ca cap ary d ameter s about 8 µm, wh ch a ows deformab e red and wh te b ood ce s (d ~ 8–10 µm) to squeeze through these sma est vesse s. The number of system c cap ar es n the body can be est mated to be about n = 109 , and the card ac output about Q = 5 /m n. Thus, the near ve oc ty of b ood f ow n the cap ar es can be read y ca cu ated as: v= 5000 cm3/m n Q = = 9.95 cm/m n = 0.166 cm/s nr 2 109 (3.142)(4 × 10−4 cm)2 (7.64) If we assume that each cap ary s 1 mm ong, the character st c b ood res dence t me n the cap ary s thus: t= 1 mm = 0.6 s 1.66 mm/s (7.65) Thus, mass transfer of metabo tes and waste products through the cap ary wa cannot encounter much barr er to mass transport s nce so tt e t me for exchange s prov ded before the b ood beg ns f ow ng back through the arter a tree back to the heart. 7.7. Pract ce prob ems 1. Ca cu ate the ve oc ty prof e for a rad a f ow assay as a funct on of r and z, n terms of the vo umetr c f owrate and the channe d mens ons. Neg ect the centra reg on and assume that the n et rad us s sma . Then, use your so ut on for the ve oc ty f e d to der ve the wa shear stress express on g ven by Eq. (7.1). 2. Cons der a s ow y converg ng channe . Assum ng a des red gap he ght of H = 8 µm, a channe w dth and ength of W = 25 mm and L = 75 mm, respect ve y, but an actua gap he ght ach eved of H1 = 8 and H2 = 7 µm, ca cu ate the requ red pressure head by neg ect ng the add t ona pressure drop across f tt ngs and tub ng. Based on your va ue, w grav ty-dr ven f ow or a commerc a y ava ab e syr nge pump be more appropr ate for th s app cat on? Not ng that the wa shear stress var es as the nverse of H squared, w the average wa shear stress w th n the channe s mp y be that for a stra ght channe us ng the average channe he ght? Ca cu ate the average wa shear stress n such a channe , for arb trary he ght change, by comput ng the area-averaged ntegra of the wa shear stress down the ength of the f ow doma n. 3. Ca cu ate the tota pressure drop n a m crochanne that exper ences a symmetr c buck ng of the upper and ower wa s n a parabo c shape, n terms of the f u d propert es, f ow rate, the n t a /f na he ght, and the he ght at the center of the channe . How much 438 Heat transfer and f u d f ow n m n channe s and m crochanne s Tab e 7.3 Parameter va ues for hepatocyte/3T3 f brob ast co-cu ture n a m crochanne b oreactor. Parameter Va ue H Q U Vm D Pe Da Sh 100 µm 105 hepatocytes/cm2 0.1 mL/m n 7 × 10−2 cm/s 0.4 nmo /s per 106 ce s 5 µm/s 2 × 10−5 cm2 /s 30 0.1 0.25 error s ntroduced by on y measur ng the gap he ght n the center of the channe and assum ng that the gap s constant? 4. Reformu ate the mode of Sect on 7.3.2 for a parabo c ve oc ty prof e, nstead of assum ng a un form ve oc ty. For the parameter va ues of Tab e 7.3, obta n a numer ca so ut on to th s mod f ed system of equat ons, and ca cu ate how much error s ntroduced by the s mp f ed ve oc ty prof e. Show that w thout a membrane oxygenator, a fract on of the hepatocyte cu ture w not rece ve suff c ent oxygen, and report th s fract on. 5. Ca cu ate the force exerted on a stat onary spher ca ce w th d ameter 9 µm, w th n a cy ndr ca m crop pette of nner d ameter 10 µm, n terms of the tota pressure drop. Compare th s to the so ut on f curvature s neg ected and the f ow w th n the gap s approx mated as p anar channe f ow. 6. If part c es enter a m crochanne at the center ne he ght, and s ow y sed ment towards s ower mov ng stream nes c ose to the ower wa as they f ow through the chamber, w th s resu t n a net accumu at on of part c es w th n the f ow chamber? Use the resu ts of Sect on 7.5.3 to est mate the extent of th s effect for 12 µm d ameter part c es w th dens ty 1.05 g/cm3 , suspended n water. Assume that the channe has he ght 30 µm and ength 1.0 cm, and neg ect the effect of the s de wa s ( .e., two-d mens ona channe ). 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SUBJECT INDEX Baroczy vo d fract on mode , 302 BGK–Burnett (BGKB) equat on, 33 b nary ntermo ecu ar co s on, 11–14, 36 b oart f c a ver, 420–422 b omed ca app cat ons, 409 b omo ecu es, 409 b oreactors, 418–424 B as us equat on, 295, 302, 342 b ood v scos ty, 430 bo ng number, 182 Bond number, 182, 184, 251, 289 boundary cond t ons, 18, 23, 25–28, 28–29, 49, 113, 159, 421, 424, 428 Bov ne aort c endothe a ce , 414 Boy e–Mar otte s aw, 21 br dges, 246, 247 bubb e, 177, 229, 230, 232, 239, 241, 246, 248, 249 bubb e-free e ectroosmot c f ow, 157 bubb e growth, 176, 177, 180, 190, 191–192, 193, 194–195, 195–197 bubb e ength, 305 bubb e nuc eat on, 176 bubb e spac ng, 249 bubb e ve oc ty, 299 bubb es coa escence, 244, 248, 293 bubb es co s on, 248 buoyancy effects, 239 Burnett equat ons, 30–36 AC e ectr c f e d, 157 AC e ectroosmot c f ow, 157–163 acce erat on pressure drop, 205, 218, 219 accommodat on coeff c ents, 29–30, 57, 62, 71, 72, 74 accommodat on pump ng, 69–70 accumu at on techn ques, 54 adhes on mo ecu es express on, 410–411 ad abat c a r–water f ow, 239, 244, 245, 248, 258, 263, 280 ad abat c f ow, 232, 264, 272, 280, 281, 286, 288, 292, 294, 304, 353 ad abat c qu d–vapor stud es, 288–290 ad abat c two-phase f ow, 206, 306 a r-s de pressure drops, 228 a r–water cr ter a, 239 a r–water f ow, 243, 244, 245, 251, 252, 273, 278, 292 a r–water pa r, 239 a r–water stud es, 291–294, 302–305 a r–water tests, 289 annu ar f m condensat on, 324 annu ar f ow, 192, 193, 197, 239, 240, 241, 244, 247, 248, 249, 250, 254, 258, 260, 262, 270, 271, 272, 274, 277, 280, 291, 321, 329, 332, 343 annu ar–wavy trans t on cr ter on, 255 annu us, 259, 299 apparent fr ct on factor, 93, 94, 120, 127 app ed e ectr ca f e d, 163, 168, 170 area rat o, 297 Armand corre at on, 272, 278, 279, 293 art f c a cav ty, 197 aspect rat o, 39, 52, 53, 73–74, 110, 145, 247–248, 263 augmented Burnett (AB) equat on, 33 average e ectroosmot c f ow ve oc ty, 148, 150 average f m ve oc ty, 300 average gas bubb e ve oc ty, 305 average gas phase ve oc ty, 305 average heat transfer coeff c ent, 318 average qu d f m ve oc ty, 322 average qu d ve oc ty, 305 average wa shear, 41 ax a coord nate, 39 C coeff c ent, 304 C parameter, 288, 292, 293 caged f uorescent dye, 146 cand date app cat ons, 227 capac tance sensors, 280 cap ary bubb e, 246 cap ary forces, 240 cap ary geometry, 413 cap ary number, 182, 251–252 ce adhes on, 410, 411, 432 ce co s ons, 415 ce dens ty, 420 ce detachment, 424–429 ce mechan cs and adhes on, 429–434 ce perfus on, 413 ce u ar uptake rate, 419, 420 443 444 Sub ect ndex channe cross-sect on, 1, 40, 91, 98, 191, 193, 416 channe d ameter, 2, 175, 192, 244, 250, 251, 279, 304 channe ob ect ves, 1 channe spac ng var at on, 416 channe wa , 1, 91, 101, 139, 184, 193, 197 channe s, 116, 139, 250–253, 263, 264, 287, 292 Chapman–Enskog method, 12, 36 Chato corre at on, 343, 347 Ch sho m corre at on, 283, 286, 293 Ch sho m parameter, 287, 288, 292 churn f ow, 244, 249, 274, 275 c rcu ar channe , 252 c rcu ar m crochanne s, 244 c rcu ar m crotube heat transfer, 73 c rcu ar m crotubes, 39, 44–46, 73 c rcu ar tube, 252 c ass c boundary cond t ons, 22, 23 c ass ca corre at ons, 286–287 C aus us–C apeyron equat on, 178 co ars, 246, 247 co s on mode s, 11–14, 14, 27 co s on rate, 11, 13, 14 compact dev ces, 227 compact geometr es, 227 compact ayer, 138 comp ex f ow geometr es, 410 comp mentary ana ys s techn que, 146 compress b ty effect, 20, 42 compress b e f ow, 73 concentrat on f e d, 166, 167, 168, 170 condensat on, 227, 230–231, 232, 259, 260, 261, 288, 294–302, 339 condens ng f ow, 254–263, 288, 306, 326 conf ned bubb e, 184, 189 conf nement number, 230, 293, 376 conservat on equat ons, 23, 24, 196 constant of proport ona ty, 291 constant-pressure techn que, 54 constant-vo ume techn que, 54 contact ang e, 179, 253–254 cont nu ty equat on, 24, 49, 153, 169, 426 cont nuum assumpt on, 2, 14–17, 90 cont nuum f ow reg me, 18, 22–23 cont nuum NS–QGD–QHD equat ons, 24–25 cont nuum theory, 3 contract on coeff c ent, 203, 297 contro vo ume ana ys s, 299 convect on-d ffus on equat on, 421 convect on number, 182, 210 convent ona channe mode s and corre at ons, 314 convent ona tubes, 232, 265 converg ng channe , 415–418 Couette–Po seu e f ow, 299 cr t ca heat f ux (CHF), 183, 194–195 cy ndr ca reg on, 433 Damkoh er number, 421 Darcy form, 301 Darcy fr ct on factor, 99, 103, 105, 302 Debye–Hucke parameter, 141 deep react ve on etch ng (DRIE), 39 De ss er boundary cond t on, 49, 53 dense gas, 11, 25 dens ty rat o, 239, 321 deve op ng am nar f ow, 92–95 deve op ng turbu ent f ow, 95–96 d ffuse ayer of EDL, 138, 139 d ffuse ref ect on, 26, 28, 51, 70 d ffus on coeff c ent, 24, 153, 163, 165 d ute gases, 11, 14, 25 d mens on ess f m th ckness, 349, 390, 391 d mens on ess ength rat o, 421 d mens on ess M chae s constant, 420 d mens on ess temperature, 72, 332, 343, 348, 349, 390, 391 d mens on ess trans at ona ve oc ty, 434 d rect s mu at on Monte Car o (DSMC) method, 12, 28, 36–37, 70 d spens ng process, 168, 170–171 d str but on coeff c ent, 276 d str but on parameter, 273 D ttus–Boe ter corre at on, 317, 343 d verg ng channe , 415–418 dr ft-f ux mode , 267, 268, 273, 274 dr ft ve oc ty, 273, 276 DSMC f ow chart, 38 dye-based m crof ow, 146, 171 dye n ect on techn que, 146 dynam c d ode effect, 66 dynam c v scos ty, 11, 14, 22, 75, 78 eddy d ffus ve rat o, 322 e ectrode doub e ayer (EDL) f e d, 137, 138–139, 140, 142, 153, 158, 171 e ectrok net c mean, 151, 164 e ectrok net c m crom xer, 166, 167 Sub ect ndex e ectrok net c m x ng, 163–167 e ectrok net c process, 137, 139 e ectrok net c samp e d spens ng, 168–171 e ectroosmos s, 137, 139, 147, 157 e ectroosmot c f ow, 139–145, 145–151, 151–156, 157–163 e ectroosmot c pump ng, 140 e ectroosmot c ve oc ty prof e, 160, 161 e ectrophores s, 137, 151, 164 energy accommodat on coeff c ent, 27 energy equat on, 24, 70, 71, 73, 316, 318 enhanced m crochanne s, 118–121 enhanced m crof u d c m x ng, 163 entrance effects, 41, 45, 63–64 entrance oss, 203–206 entrance reg on effects, 112, 116, 203 Eotvos number, 241 ep f uorescence mode, 413 equ atera tr angu ar channe s, 244, 293 equ va ent fr ct on factor, 294 equ va ent qu d mass f ux, 319 equ va ent mass f ux, 319, 351, 352 equ va ent mass ve oc ty, 294, 326, 339 evaporat on nvest gat ons, 185–188 evaporat on terature, 188–189 evaporat on process, 183, 185–188 ex t oss, 97, 98, 100, 101, 203–206 extended hydrodynam c equat on (EHE), 30–32 extrapo at on, 231, 232, 264 fa ng-f m condensat on, 314, 315, 351 Fann ng fr ct on factor, 90–91, 92, 95, 99, 103, 104 f m-bubb e nterface, 299, 349 f m condensat on, 317, 323, 324, 325, 330, 333, 336, 337, 343, 344, 351 f m heat transfer coeff c ent, 316 f n spac ng rat o, 116, 117 f rst-order s p boundary cond t ons, 25–28, 79 f rstorder so ut on, 40–42, 44–46, 48–49 f oor d stance to mean ne, 104 f ow bo ng, 175–176, 189, 195–199, 200–203, 206–207 f ow channe c ass f cat on, 2–3 f ow cytometry, 431–432 f ow d rect on, 144, 177, 190, 421, 423 f ow f e d, 140, 151, 157, 163, 168, 169, 170, 331 f ow eve ang e, 316 445 f ow mechan sms, 231, 250, 251, 252, 253, 259, 306 f ow passage d mens on, 88, 89 f ow patterns, 155, 184, 197, 231, 232, 240, 241, 243, 244, 250, 252, 254 f ow rate measurements, 54–56, 56–62, 57, 100, 394 f ow reg mes, 231, 232, 233–238, 239, 240, 241, 244, 245, 246, 247, 248, 249, 251, 252, 253, 257, 259, 261, 262, 276, 298, 338, 354, 364, 365 f ow-re ated parameters, 231 f ow stab zat on, 197–199 f ow v sua zat on, 62, 146, 229, 239, 258, 259, 273, 331, 343 f u d f ow, 3, 5, 6, 9, 20, 37, 39, 53, 54, 57, 66, 87 f u d propert es effects, 239, 252, 258, 279 f u d property exponents, 327 f u d v scos ty, 412, 429, 432 f uorescence act vated ce sort ng (FACS), 431 forced convect on, 330, 333, 343, 351 Four er aw, 22 Four er transform ana ys s, 191 free mo ecu ar f ow, 18, 30–37 fr ct on factor, 4, 41, 88, 90, 91, 93, 96–101, 104–108, 287, 294–295 fr ct on mu t p er, 304 fr ct on ve oc ty, 325, 326, 339, 348, 349 fr ct ona component, 293, 297 fr ct ona pressure drop, 4, 91, 94, 205, 220, 254, 270, 292, 294, 300, 305, 344 fr ct ona pressure grad ent, 239, 331–332 Fr ede corre at on, 292, 295, 377 Fr ede corre at on mod f cat on, 290, 295, 377 Froude number, 254, 271, 360, 378 Froude rate parameter, 271, 280 fu y deve oped am nar f ow, 4, 91–92, 105, 109–110, 127, 216 fu y deve oped turbu ent f ow, 95 future research needs, 74 Ga eo number, 254, 315, 317, 358, 360–361 gas at mo ecu ar eve , 10–14 gas f ow, 20, 39, 44–53, 57, 65, 232 gas superf c a ve oc ty, 244, 274, 317, 343 gas ve oc ty, 254, 256, 273, 355, 356 geometry-dependent constants, 297–298 geometry opt m zat on, 116–118 grav tat ona drag, 271, 280 grav tat ona pressure drop, 206 446 Sub ect ndex grav ty-dom nated f m condensat on, 324, 351, 352 grav ty-dr ven condensat on, 255, 314–318 Green s funct on approach, 159 greenhouse gas em ss ons reduct on, 227 Hagenbach s factor, 93, 94, 98, 125, 130 Hagen–Po seu e ve oc ty prof e, 45, 91, 92 hard sphere (HS) mode , 12 heat d ss pat on, 87, 89, 116, 346 heat exchanger advantages, 115 heat remova system, 87 heat transfer, 3–5, 70–74, 88–89, 109–116, 184, 199–203, 219, 307–313, 327, 352 heat transfer coeff c ent, 4, 175, 227, 228, 265, 306, 314, 316, 322, 330, 331, 339, 340, 342, 349, 381 heat transfer mechan sms, 184, 192, 193 hematopo et c b ood ce cu ture, 418–420 hematopo et c stem and precursor ce s (HSPC), 418 hepatocytes, 422 heterogeneous m crochanne s, 151–156, 167 h gh aspect rat o, 247–249 h gh heat f uxes, 175, 206–207, 227 h gh heat transfer rates, 227 h gh qua ty annu ar f ow, 184, 192 h gher-order s p boundary cond t ons, 28–29 homogeneous boundary cond t on, 159 homogeneous f ow mode mod f cat on, 289 homogeneous m xture, 323 homogeneous vo d fract on emp r ca f t, 278 human system c c rcu at on, 430 hydrau c d ameter, 2, 3, 4, 93, 142, 144, 177, 261–262 hydron c coup ng, 228–229 dea gas, 21–22 dea refr gerant character st cs, 207 nc dent neutron beam, 241 ncompress b e f ow, 38, 43, 71–73 ncrementa pressure defect, 93 nert a-contro ed reg on, 196 nert a forces, 182, 192, 276, 291 nf n te y ong waves, 240 n et m x ng effects, 250 nstab ty, 184, 190, 192, 195 nstantaneous 2-D vo d fract on prof es, 275 nstantaneous vo d fract ons, 274 nsu n dependent d abetes me tus (IDDM), 418 nterface ve oc ty, 299, 305, 320, 322, 348 nterfac a fr ct on factor, 285, 301, 302, 349, 381 nterfac a shear stress, 289, 301, 319, 324, 331, 348, 349, 391 nterfac a ve oc ty, 320 nterm ttent f ow, 241, 261, 263, 272, 298, 336 nterna f ows, 90 nterna membrane oxygenator, 421 nverse power aw (IPL) mode , 11–14, 16, 22, 44 nverted systems, 412–415 so ated bubb e, 184, 189 Jakob number, 183 Ke v n s theorem, 426 Ke v n–He mho tz nstab ty, 240 k dney, 1, 6, 418 Knudsen ana ogy, 17–18, 23 Kundsen number, 16, 20–21, 22, 27–33, 36, 42, 49, 72, 75–81 Kutate adze number, 317, 327 ab-on-a-ch p dev ce, 137–139, 151, 163, 168 actate transport, 422–424 Lame coeff c ent, 22 am nar condensate f m, 314–316, 319 am nar f m condensat on, 323–324 am nar f ow, 20, 88, 91–92, 92–95, 98, 103, 105, 108, 109–110, 112–114, 127, 133, 175, 200, 208, 304, 344, 375 am nar reg on, 4, 99, 105, 108, 302 am nar-to-turbu ent trans t on, 101–102, 108–109 am nar-trans t on-turbu ent corre at on, 293 ap ace constant, 229 att ce Bo tzmann method (LBM), 30, 37 aw of res stance, 103 eukocytes, 410, 430 near concentrat on prof e, 420 near doub e ayer ana ys s, 157 near nterpo at on, 94, 101, 109, 123, 124, 200, 255, 302, 310, 315, 322, 328, 334 near ve oc ty prof e, 319 near zed ana ys s geometry, 426–429 near zed ana ys s of un form f ow, 426–429 nked shutoff va ves, 272, 280 Sub ect ndex qu d br dge, 237, 250, 253 qu d entry channe , 176, 203 qu d f ow at m crosca e, 87 qu d f ux, 319 qu d umps, 241, 253 qu d phase, 241, 250, 256, 271, 286, 294, 302, 320 qu d poo heat transfer coeff c ent, 316 qu d poo eve ang e, 316 qu d r ng f ow, 250, 253 qu d r ng mot on, 253 qu d s ug, 176, 249, 253, 277, 278, 291, 293, 298, 299, 300, 304 qu d vo ume fract on, 247, 328 tt e raref ed reg me, 57 ver, 418 oad ng process, 168, 170 oca fr ct on pressure grad ent, 204 oca heat transfer coeff c ents, 124–125, 184, 338, 344–346 oca rarefact on number, 18 Lockhart–Mart ne corre at on, 232, 270, 286, 288, 291, 292, 293 Lockhart–Mart ne parameter, 232, 271, 280, 329 Lockhart–Mart ne two-phase mu t p er, 286, 294, 304, 318, 320, 321, 325 ong-term ce cu ture, 418 ubr cat on approx mat on, 415–418 yse, 6 Mach number, 16, 20, 27, 38 MAD s mu at on, 411 Mart ne parameter, 182, 204, 254, 255, 265, 271, 286, 302, 322, 349 mass f ow rate, 41, 42, 43, 51–53, 54, 56, 57 mass f ux, 24, 31, 195, 199, 231, 232, 244, 254, 257–258, 259, 260, 261, 271, 272, 289, 317, 320, 335, 345 mass transfer, 1, 6, 70, 88, 354, 420–422, 437 maturat on of red b ood ce s, 416 max mum prof e peak he ght, 103 Maxwe mo ecu es (MM) mode , 13 mean f m ve oc ty, 320 mean free path, 10, 11, 13, 14, 16, 62 mean heat transfer coeff c ent, 345 mean qu d ve oc ty, 272 mean space of prof e rregu ar t es, 104 mean vo umetr c f ow rate, 56 method of moments, 36 M chae s constant, 420 447 M chae s–Menten k net cs, 420, 424 m cro e ectro mechan ca systems (MEMS), 9, 139 m cro part c e mage ve oc metry (m cro PIV), 145–146, 203 m crocap ar es, 429–434 m crochanne s, 89, 96–102, 116–118, 146, 227–230, 277–279, 347, 350, 415, 421 m croc rcu at on, 430 m crofabr cat on techno ogy, 117, 119 m crof n tubes, 88, 272, 290, 294, 339–342 m crof u d cs, 5, 9, 10, 94, 95, 168 m crograv ty, 249–250, 276–277 m cro-grooved m n channe , 422–424 m cromach n ng techno og es, 139–140 m cron-s zed p pettes, 409, 432–433 m crosca e, 2, 5–6, 68, 87–88, 97, 411–412, 425, 431 m croscop c ength sca es, 10–11 m n channe geometry opt m zat on, 116–118 m n channe membrane b oreactor, 419 m st–annu ar trans t on, 256, 328 m xture dens ty, 287, 320 m xture v scos ty, 323 m xture vo umetr c f ux, 273 mode ng f u d f ow, 2 mod f ed entha py of vapor zat on, 314 mod f ed Fr ede corre at on, 332 mod f ed Froude number, 255, 317, 355 mod f ed M er equat on, 105–106 mo ecu ar mechan sms, 410 mo ecu ar tagg ng ve oc metry (MTV), 62 momentum coup ng decrease, 304, 393 momentum d ffus v ty, 325 momentum equat on, 24, 39, 43–44, 47, 48, 49, 50, 70, 71, 78 mu t -reg me condensat on, 327 mu t ouver f ns, 228 mu t p e-f ow-reg me mode , 303, 331 mu t p e streams short paths, 115 nanomo ar concentrat ons, 409 narrow rectangu ar channe s, 275 Nav er–Stokes equat on, 18, 22–23, 63, 423, 432 near-azeotrop c b end, 257 negat ve subcoo ng, 180, 181 Nernst–P anch conservat on equat on, 152, 153, 155 nested Moffatt edd es, 422 net entha py f u d change, 314 448 Sub ect ndex neutron rad ography, 241, 273, 274, 280 Newton an f u d, 22, 49, 90 n trogen–water f ow, 250, 251, 278, 304 non-d mens ona numbers, 181–184 nonwett ng system trans t ons, 253 norma forces generat on, 424–429 nove techn que, 157 nuc eate bo ng absence, 322 nuc eat on, 176–181, 194, 197– 199 Nusse t f m condensat on, 316 Nusse t number, 4, 72, 73, 91, 92, 109–110, 111, 112–113, 115, 118, 120, 123, 124, 129, 133, 208, 314, 315, 318, 319, 321, 324–326, 340, 344, 385, 387–388 offset str p-f ns, 117, 118 Ohnesorge number, 182 one-d mens ona d ffus on equat on, 419 onset of nuc eate bo ng (ONB), 180 opt ca d stort on effects, 250 or g na vapor core f ux, 319 orthogona Herm te po ynom a , 36 out et Knudsen number, 42, 43–44, 53, 58–62 over ap zones, 251, 255 oxygen, 419, 420, 421, 422–424 ozone dep et on, 227 pancreas, 418 parabo c ve oc ty prof e, 300, 413 para e channe nstab t es, 190 para e -p ate ce perfus on chambers, 412 para e -p ate f ow chamber, 411 parameters comb nat on, 293 part c e depos t on, 411 part c e transport n rectangu ar m crochanne s, 433–434 peak f ow ve oc ty, 157 Pec et number, 153, 165, 167, 421 photo- n ect on process, 146 phys ca property correct ons, 239 p pe d ameters, 232, 239 p anar m crochanne s, 420–422 p ane f ow between para e p ates, 39–44 p ane m crochanne , 71–73 p ug, 139, 232, 240, 244, 246, 248, 291, 293 Po seu e f ow, 41, 299, 415, 416, 424, 434 Po seu e number, 41, 43, 47, 92, 105 Po sson–Bo tzmann equat on, 138, 139, 140, 158 po yd methy s oxane (PDMS) p ate, 166 po ymerase cha n react on (PCR), 5 potent a f ow near nf n te wa , 425–426 power aw prof e, 299 pract ca condensat on process, 258 pract ca coo ng systems, 206–207 Prandt m x ng ength, 257, 288, 325 Prandt number, 28, 72, 315, 319, 322 pressure change, 296, 297 pressure data, 62 pressure d str but on, 42, 46, 53, 429 pressure-dr ven f ow, 416 pressure-dr ven steady s p f ows, 37 pressure drop, 3–5, 90–96, 195–196, 198, 203–206, 231, 239, 254, 281, 282–285, 286, 288, 289, 290, 291, 292, 293, 298, 299, 300, 301, 302, 334, 371, 377, 417, 418 pressure f uctuat on, 65–66, 184, 190, 191, 239 pressure grad ent, 249, 294, 299, 316, 321, 340 pressure oss, 203, 300 pr mary res stance heat transfer, 289, 327 probab ty dens ty funct ons (PDFs), 276, 277 property rat o method, 115–116 pu sed gas f ows, 65–67 quadrat c nterpo at on, 337 quas -gas dynam c (QGD) equat on, 23, 24–25 quas -hydrodynam c (QHD) equat on, 23, 24–25 quas -steady state t me per od c so ut on, 160 qu escent vapor, 323, 324 rad a ce perfus on chambers, 412 rad a f ow chamber, 411 rad a membrane m n channe s, 418–420 rad a pos t on, 412 rarefact on n m crof ows, 10, 17–18, 20 raref ed f ows, 37, 67 react ve on etch ng (RIE), 39 rectangu ar channe , 91, 184, 189, 247–249 rectangu ar m crochanne , 47–53, 73–74 red b ood ce s (RBCs), 415, 430 refract ve ndex match ng, 250 refr gerant charge pred ct on, 271 refr gerant condensat on, 239, 296, 303 reg mes f ow c ass f cat on, 18 re at ve roughness, 88, 102, 103, 113 rep ac ng process, 148–151 representat ve d str but on, 274 reversed d ode effect, 67 reversed f ow, 195 Sub ect ndex Reyno ds number, 4, 20, 27, 96, 250, 251, 254, 256, 287, 288, 293, 294, 299, 302, 314, 315, 318, 319, 322, 323, 326, 328, 339, 342, 355, 371, 388, 424 r ng-s ug f ow pattern, 252 rough tubes, 109 roughness, 102–109, 113–114 samp ng vo ume, 14, 15–16 sca ng equat on, 289 Schm dt number, 24 second-order so ut on, 42–44, 46, 49–53 se ect n–carbohydrate bond, 410 sem -tr angu ar m crochanne s, 244 separated f ow mu t p er, 204, 297 Shah corre at on, 322–323 shear ba ance, 299 shear-dr ven condensat on, 318–327 shear p ane, 138 Sherwood number, 421 s mp e gas, 10 s ng e phase e ectrok net c f ow, 137 s ng e-phase fr ct on factor, 287, 290, 291, 293, 294, 300, 342 s ng e-phase gas f ow, 9 s ng e-phase qu d f ow, 87, 90–96, 102–109, 177, 209, 279 s ng e-phase ve oc ty prof es, 248 s ng e stab ty cr ter a, 240 s p, 18, 23–30, 53, 54, 58, 61, 65, 72, 265, 270, 275, 276, 279 s ug, 176, 232, 239, 242, 244, 246, 247, 249, 250, 253, 291, 293, 299, 300 s ug–annu ar f ow, 244, 277 s ugg ng, 240 So man-mod f ed Froude number, 258 spec es transport, 151, 164 specu ar ref ect on, 26 stab z ng effects of surface tens on, 240–241 Stanton number, 118, 121 stat st ca f uctuat on eve , 15 steady-state t me per od, 160, 161 steam–water f ow, 252 strat f ed f ow, 239, 240, 241, 247, 254, 257, 271, 280, 288, 301, 314, 318, 327, 333, 336, 339, 343 stress tensor, 22 subcoo ed entry, 176 subson c f ow, 20 superf c a gas phase ve oc ty, 255, 274, 317 superf c a qu d phase fr ct on factor, 318 449 superf c a qu d ve oc t es, 244, 274, 302, 317 superf c a ve oc t es, 246, 249, 250 superson c f ow, 20 surface area-to-vo ume rat os, 227, 228 surface effects n sma channe s, 253 surface heterogene ty, 151 surface phenomena, 230 surface propert es, 253–254 surface tens on, 184, 240, 246, 247, 250, 253, 279, 293, 302, 335, 340 surface wettab ty effects, 254 system energy eff c enc es, 227 Ta te –Duk er mode , 240, 242, 248, 250, 254, 255 tangent a momentum accommodat on coeff c ent, 26 Tay or bubb e, 249–250, 276 temperature ump d stance, 27, 72 therma amp f cat on techn que, 345, 346 therma boundary cond t on, 113 therma conduct v ty exponent, 327 therma entry ength, 110 therma neutrons, 241 therma res stance, 315, 318–319 therma transp rat on, 68 therma y deve op ng f ow, 110–111 therma y dr ven gas m crof ows, 67–70 thermodynam c equ br um, 14–17, 323 three f ow pattern, 184, 189 t me-averaged 2-D vo d fract on, 275 t me per od c e ectroosmot c f ow, 157 tota p pette ength, 433 tota s ug vo ume, 247 traff ck ng, 410 trans ent ce adhes on, 410–418 trans ent stage ve oc ty, 162 trans t on cr ter on, 230, 250, 255 trans t on f ow, 18, 30–37 trans t on mass f ux, 333 trans t on reg me, 255, 328 trans at ona k net c temperature, 21 transp rat on pump ng, 68–69 transport process, 1 tr angu ar channe , 246 T-shaped m crof u d c m x ng system, 163 tube d ameter, 195, 199, 204, 232, 241–242, 273, 288, 292, 300, 323, 329, 349, 392, 393 tube nc nat on, 316 tube shape effects, 240–241, 262, 338 tumb ng reg on, 151 450 Sub ect ndex turbu ent d mens on ess temperature, 348, 349, 391 turbu ent f m, 322, 323–324, 348–349 turbu ent f ow, 20, 95–96, 103, 115–116, 199, 205, 265, 266, 375 turbu ent reg on, 104, 106, 204, 302 turbu ent vapor core, 289, 321, 324, 348, 375 two-phase entry, 176, 203 two-phase f ow, 3, 181, 183, 190, 194, 212–214, 230, 231, 232, 239, 240, 241, 246, 247, 249, 250, 263, 274, 293, 294, 305, 351 two-phase hor zonta f ow patterns, 239 two-phase m xture dens ty, 287, 373 two-phase mu t p er, 204, 214, 232, 286, 288, 290, 291, 293, 294, 304, 317, 318, 321, 326, 329, 330, 332, 343, 361, 372, 373–379, 388 two-phase pressure drop, 182, 206, 214, 216, 232, 286, 294, 295, 297, 332, 378, 379 two-phase pressure grad ent, 294, 372, 373, 375, 376, 377 typ ca f ow parameters, 430 uncaged dye mages, 147 un t ce for nterm ttent f ow, 298 unstab e wave ength, 247 vacuum generat on, 9, 18, 67–70 vapor bubb e growth, 194, 195 vapor-cutback phenomenon, 194 vapor fr ct on force, 319 vapor k net c energy, 271, 280 vapor– qu d qua ty, 232 vapor mass f ux, 325 vapor phase nert a, 256 vapor shear, 314, 315, 316, 318, 323, 337, 339, 340, 351 var ab e hard sphere (VHS) mode , 12 var ab e property effects, 115–116 var ab e soft sphere (VSS) mode , 12 ve oc ty d str but on, 40, 43, 45, 47, 72, 73, 77, 78, 79, 93, 270 ve oc ty grad ent d rect on, 423 ve oc ty rat o, 320 vert ca equ atera tr angu ar channe s, 244 v scos ty rat o, 239, 295 v scous d ss pat on, 24, 101 v scous effect, 20, 101 v scous stress tensor, 22, 30 v sua zat on methods, 146, 171 vo d fract on, 231, 244, 246, 247, 249, 265, 266–269, 272, 273, 274, 275, 276, 277, 280, 281, 291, 321, 334–335, 351, 353, 366–371 vo d fract ons for a r–water f ow, 273, 278, 280 vo ume rat o, 1, 193, 270, 342 vo umetr c f ow rate, 56, 140, 144, 145 vo umetr c f ux, 273, 276 vo umetr c qua ty, 248 vo umetr c vo d fract on, 273, 274, 276, 279, 280 von Karman ana ogy, 315, 316, 318 von Karman un versa ve oc ty prof e, 254, 270, 319 wa effects n m crof ows, 10, 18–20 wa shear, 90, 239, 250, 254, 279, 315, 318, 320, 321, 322, 324, 339, 411, 412, 415, 422, 437 wa temperature d fference, 208, 319, 320, 323 Wa s d mens on ess gas ve oc ty, 254 wavy–s ug trans t on, 255, 311 Weber number, 183, 249, 250, 251, 256, 289, 295, 332, 340, 342, 360, 361, 369, 377, 379 wett ng qu d, 240 w de range qua t es, 258 W son p ot techn que, 339, 340, 344, 353 Yan and L n corre at on, 201, 342 zeroeth order oxygen uptake, 422 zeta potent a , 138, 140, 142, 144, 165, 172 Z v corre at on, 266, 317, 321, 330, 367 Z v vo d fract on, 266, 280, 321, 327, 333, 340, 343, 351, 370, 386

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