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1995
Vapour jet refrigeration systems (VJRS) Hassan Sharifi-Bidgoli University of Wollongong
Recommended Citation Sharifi-Bidgoli, Hassan, Vapour jet refrigeration systems (VJRS), Doctor of Philosophy thesis, Department of Mechanical Engineering, University of Wollongong, 1995. http://ro.uow.edu.au/theses/1602
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I «•
VAPOUR JET REFRIGERATION SYSTEMS (VJRS) by
HASSAN SHARIFI-BIDGOLI B.Eng., M.Eng.
A thesis submitted in fulfilment of the requirements for the award of the degree of
DOCTOR OF PHILOSOPHY from
UNIVERSITY OF WOLLONGONG
Department of Mechanical Engineering OCTOBER 1995 UNIVERSITY OF WOLLONGONG LIBRARY
In the N a m e of God, the Compassionate and the Merciful
DEDICATION
To m y parents, m y father-in-law, m y wife, Mahin Nazifi Arani, m y daughter, Fateme, my country, Islamic Republic of Iran, and the people of my country.
DECLARATION
This is to certify that the work presented in this thesis is carried out by the author in the Department of Mechanical Engineering of the University of Wollongong, Australia, and has not been submitted for a degree at any other university or institution. Hassan Sharifi Bidgoli
ACKNOWLEDGMENTS
The author would like to express his sincere gratitude and appreciation to his supervisor D r Paul Cooper, senior lecturer in the Department of Mechanical Engineering at University of Wollongong for introducing the most interesting filed "Vapour Jet Refrigeration Systems (VJRS)" and for his excellent guidance, supervision and encouragement during the course of the present work. H e also would like to extend his appreciation to Dr Paul Cooper for his excellent and friendly support all time during this study. Many thanks are due to Mr Masanori Kanashige (Research and Development Department, Nippon Oil C o m p a n y Ltd., Tokyo, Japan) for his assistance. The author would like to extend his appreciation to M r Kanashige for his friendly help. M a n y thanks are also due to M r Ajit Godbole for his friendly help. The author also gratefully thanks the ministry of culture and higher education of the Islamic Republic of Iran for awarding him a scholarship and providing financial support to m a k e this thesis possible. H e would like to extend his appreciation to Dr. M . Gezelayagh and Dr. M . H. Sorouraddin and M r Hossein Hizomi Arani (Head Representative of Kashan University in Tehran, Iran) for their help, support and assistance.
The author wishes to express his gratitude to Prof. M. West the head of the Department of Mechanical Engineering, Dr G. J. Montagner, the sub-dean of the Faculty of Engineering, D r W . K. Soh, the postgraduate coordinator in the Department of Mechanical Engineering and Dr P. W . W y p y c h the senior lecturer
iii
of the Mechanical Engineering Department, University of Wollongong for their help, support and assistance.
The author would also like to recognise the great help provided by the Engineering Workshop and Laboratory staffs in the Department of Mechanical Engineering in building and modification the experimental facilities. Special thanks are due to M r R. Young, the supervisor of the thermodynamics laboratory and M r M . Morillas, the technical staff for their invaluable help during the experimental work. Thanks are also due to M r I. Kirby, M r T. Kent and M r K. Maywald for their help and assistance. M a n y thanks are also due to M r . Des Jamieson for his valuable assistance in the use of computer facilities, to department's administrative staffs Mrs R. Hamlet and M r s B. Butler and also to M s N . Eager for their help and assistance. Many thanks are due to Dr Mohammad T. Shervani Tabar, Dr Ali A. Karimi, Mr Hassan S. Jahanshahi and M r Seyed A. Moosavian for their help and assistance. The author wishes to express his gratitude to Mrs P. Tibbs, the teaching staff of the Wollongong English Language Centre for her help to edit some parts of this thesis.
The author acknowledges gratefully to his parents, his father-in-law, his brot his brothers-in-law and his sister-in-law for their support and encouragement.
Finally the author is extremely grateful and indebted to his wife, Mahin Nazif Arani and his daughter, Fateme, for their constant support, understanding and encouragement and for their unending patience.
iv
ABSTRACT
A Vapour Jet Refrigeration System (VJRS) is an alternative to the conventional mechanically driven vapour-compression refrigeration system. The V J R S utilises a supersonic ejector as a thermal compressor and has the potential significandy to reduce energy consumption in air conditioning systems. Experimental investigations and analysis of large supersonic jet ejectors such as those used for steam jet refrigeration, has been carried out in past literature. In the present study the performance characteristics of small V J R S ejectors using R 1 2 as the working fluid (ie ejectors with throat diameters of several millimetres only) have been investigated using experimental, analytical and C F D (computational fluid dynamics) techniques. In practice ejector performance may be affected by the choking phenomena in the secondary stream, superheating of vapour leaving the evaporator and generator, nozzle and diffuser efficiency. The present author developed a model for a small, single-fluid V J R S ejector using the theory of secondary vapour choking which was introduced by M u n d a y and Bagster (1977). This theory was employed by the present researcher to design and test a small R 1 2 ejector. Results of a computer simulation that models secondary choking in the converging part of the ejector and the effects of superheat conditions on the ejector performance are presented. In the present work further developed has been carried out to examine the optimum nozzle position w h e n the ejectors operated at fully developed choked conditions. It has been confirmed that for the small ejectors the choking v
phenomenon plays an important role in ejector performance. It was also recognised that the nozzle position is a very important factor of the ejector geometry for small V J R S ejectors. In the secondary choking theory, ejector performance may be taken to be a function of the "effective area" available to the secondary fluid within the mixing chamber of the ejector. The present study involved experimental investigation of h o w this effective area is influenced by operating conditions such as evaporator temperature and nozzle position. In the present investigation the geometry of the converging part of the ejector has also been studied and the shock pressure recovery process examined in the constant area mixing tube. The present author has developed a CFD simulation of the ejector using the P H O E N I C S - B F C code based on the finite volumetric technique. Comparisons have been m a d e between the predictions of the entrainment ratio using the onedimensional, constant area analysis, C F D results and experimental results.
In the present study analysis of two-fluid VJRS ejectors carried out based on th one-dimensional analysis, constant area method. It was found that using an appropriate fluid-pair, the entrainment ratio is likely to be substantially better than for the single fluid V J R S .
vi
TABLE OF CONTENTS
DECLARATION
ii
ACKNOWLEDGMENT iii ABSTRACT v TABLE OF CONTENTS vii LIST OF FIGURES xiii LIST OF TABLES xix LIST OF PUBLICATIONS xxi NOMENCLATURE xxii CHAPTER ONE - INTRODUCTION 1 1.1 General Considerations
1
1.2 Characteristics of the VJRS
5
1.3 VJRS Operation
8
1.4 Analysis and Performance of the Ejector Cycle
11
1.4.1 Power sub-cycle
11
1.4.2 Refrigerant sub-cycle
12
1.5 Historical Background of VJRS Development
14
1.6 Review of Previous Work by Kanashige
18
1.7 Refrigerant Selection
21 vii
1.7.1 General specification
21
1.7.2 Thermodynamic characteristics and properties of refrigerants
22
1.7.3 Chemical properties and environmental problems
22
1.7.4 Selection of the working fluid
25
1.8 Potential applications of ejectors
28
1.8.1 Refrigeration applications
28
1.8.2 Automobile V J R S
29
1.8.3 Steam jet ejectors
31
1.8.4 Power Plant application of ejectors
32
1.9 Scope of Present Study
33
CHAPTER 2 - THEORETICAL MODELLING 35 2.1 Introduction
35
2.2 Mixing Process
36
2.2.1 General specification
36
2.2.2 Turbulent jet mixing and supersonic jet development in a compressible flow
37
2.2.3 Constant area mixing tube
40
2.2.4 Shockwaves
41
2.2.5 Ejector choking
43
2.3 Governing Equations for One-Dimensional Analysis
45
2.4 Sensitivity Study on the Effect of the Nozzle Efficiency and Diffuser Efficiency on Ejector Performance in a Small Single Fluid V J R S
56
2.4.1 Sensitivity study on the effect of nozzle efficiency
56
2.4.2 Sensitivity study on the effect of diffuser efficiency
57
2.5 Effect of Superheat on Ejector Performance 2.5.1 Analysis of superheating effects viii
59 59
2.5.2 Theoretical results
60
2.6 Secondary Vapour Stream Choking
63
2.6.1 Analysis of the secondary choking theory
63
2.6.2 Application of secondary choking theory to small ejectors
66
2.7 Computational Fluid Dynamics (CFD)
70
2.7.1 General specification and objective
70
2.7.2 Introduction to P H O E N I C S as a C F D package
71
2.8 Validation of Computational Analysis by P H O E N I C S
74
2.9 Modelling of the Supersonic Ejector by P H O E N I C S
76
2.9.1 Computational procedures by P H O E N I C S
78
2.9.2 Running procedures
79
2.9.3 C F D results
80
2.10 Discussion
88
CHAPTER 3 - EXPERIMENTAL WORK 90 3.1 Scope and Objectives of the Experiments
90
3.1.1 Scope of experiments
90
3.1.2 Objectives
91
3.2 Experimental Design
93
3.2.1 Ejector geometry
93
3.2.2 Experimental apparatus
95
3.3 Initial Replication of Previous Experiments
97
3.4 Instrumentation and Modification of Test Rig to Extend the Range of the Test Conditions
101
3.5 Ejector Used in the Present Experimental W o r k
108
3.6 Experimental W o r k on Small Ejector using Constant Area Analysis Design in the Modified Test Rig
109
ix
3.6.1 Experimental results for the ejector with 20° converging half angle (Ejector La)
Ill
3.6.2 Experimental results for the ejector with 10° converging half angle (Ejector Lb)
120
3.6.3 Comparison of the results for Ejectors La and Lb
124
3.7 Experimental Tests on the Small Ejector Design, Based on the Secondary Choking Theory in the Modified Test Rig
127
3.7.1 Experimental results for the ejector with 20° converging half angle (Ejector 2.a)
127
3.7.2 Experimental results for the ejector with 10° converging half angle (Ejector 2.b)
132
3.7.3 Comparison of the results for Ejectors 2.a and 2.b
134
3.8 Experimental Investigation of Effective Area Ratio , Ab/At
136
3.9 Comparisons the Experimental Results of the Shock Pressure Recovery With One Dimensional Calculations
140
3.10 Discussion
142
CHAPTER 4 - THEORETICAL INVESTIGATION OF THE FEASIBILITY O F USING T W O W O R K I N G FLUIDS O F DIFFERENT M O L E C U L A R W E I G H T S IN A V J R S
147
4.1 Introduction
147
4.2 Objectives
150
4.3 Governing Equations for One-Dimensional Analysis
151
4.3.1 Equations of performance characteristics
151
4.3.2 Equations for condensation of mixtures
151
4.4 Condensation of Mixtures
154
4.5 Two-Fluid VJRS Simulation
160
4.6 Simulation Results
163 x
4.7 Comparison With Simulation Results for a Single Fluid V J R S
167
4.8 Feasibility of Separation of Fluids in the Condenser
168
4.9 Conclusion
170
CHAPTER 5 - IMPLICATIONS FOR THE PRACTICAL USE OF SMALL VJRS EJECTORS
171
5.1 General Considerations
171
5.2 Implications for Practical Use
173
5.2.1 Entrainment ratio
173
5.2.2 Condenser size
173
5.2.3 Heat input to drive the VJRS 5.2.4 Refrigerant pump
174 175
CHAPTER 6 - CONCLUSIONS AND FUTURE WORK 177 6.1 Analysis of Single Fluid Ejector Refrigeration Systems
177
6.1.1 One-dimensional analysis
177
6.1.2 C F D analysis
178
6.2 Performance of the Small Single Fluid V J R S
180
6.3 Analysis of Two-Fluid V J R S
183
6.4 Future W o r k
184
6.4.1 Practical work
184
6.4.2 Theoretical modelling
185
APPENDTXA 197 A.1 The Ql File
197
A.2 The Result File
203
xi
APPENDED B
238
B.l Additional Experimental Results
238
B.2 Pressure Transducers
242
APPENDIX C 243 C.l Simulation Analysis Program for Secondary Choking Theory
243
C.2 Subroutines for Calculation of R12 Thermo-Physical Properties (Kanashige, 1992)
262
C.3 Simulation Program for Two-Fluid VJRS Ejectors
283
C.4 Derivation of the Governing Equation for COPCarnot for a Vapour Jet Refrigeration System (VJRS)
300
xii
LIST OF FIGURES
Figure 1.1 Schematic Diagram of the Ejector Refrigeration System
7
Figure 1.2 Schematic of the Conventional Refrigeration System 7 Figure 1.3 Schematic Diagram of a Supersonic Nozzle 10 Figure 1.4 Schematic of the Supersonic Ejector 10 Figure 1.5 Thermodynamic Cycle of the VJRS 12 Figure 1.6 The Performance Map of Vapour Jet Refrigeration System 20 Figure 1.7 Schematic Illustration of an Automobile VJRS 30 Figure 2.1 Expansion of Jet Flow in a Cylindrical Mixing Chamber 39 Figure 2.2 A One Dimensional Normal Shock on an Enthalpy - Entropy Diagram (Van Wylen and Sonntag, 1985)
42
Figure 2.3 Nozzle Pressure Ratio As a Function of Backpressure for a Reversible Supersonic Nozzle (Van Wylen and Sonntag, 1985)
43
Figure 2.4 Ejector Choking 44 Figure 2.5 Schematic Diagram of an Ejector 45 Figure 2.6 Schematic Diagram of the Shock Process in the Constant Area Mixing Tube
53
xiii
Figure 2.7 Plot of Diffuser Efficiency Versus the Entrainment Ratio
58
Figure 2.8 Actual Thermodynamic Cycle of the Vapour Jet Refrigeration Systems 60 Figure 2.9 Superheat of the Generator and Evaporator Versus Entrainment Ratio 61 Figure 2.10 Superheat of the Generator and Evaporator Against the Critical Condenser Temperature 62 Figure 2.11 Superheat of the Generator and Evaporator Versus the Cooling Capacity 62 Figure 2.12 Schematic of Choking of the Secondary Vapour Stream in an Ejector 65 Figure 2.13 Entrainment Ratio Versus Condenser Temperature As Secondary Vapour Stream Choking Area is Varied 67 Figure 2.14 Velocity Fields at a Cross Section of the Cylindrical Mixing Chamber (6.7 Chamber Diameters from the Nozzle) 75 Figure 2.15 Velocity, Ul, Against the Number of Grid-Cells 82 Figure 2.16 PHOENICS Mesh for a 70 (axial) by 30 (radial) Axisymmetric Ejector {Ejector 2.a, Described in Section 3.7.1} 84 Figure 2.17 Velocity Field for an Axisymmetric Ejector (Based on CFD Analysis) (Ejector 2.a, Described in Section 3.7.1} 85 Figure 2.18 Pressure Distribution for an Axisymmetric Ejector (Based on CFD Analysis) {Ejector 2.a, Described in Section 3.7.1} 86
xiv
Figure 2.19 Contours of Mach Number for an Axisymmetric Ejector (Based on C F D Analysis) {Ejector 2.a, Described in Section 3.7.1}
87
Figure 3.1 Ejector Geometry and Schematic Diagram of Mixing Chamber 94 Figure 3.2 Experimental Apparatus 96 Figure 3.3 Relation Between Entrainment Ratio and Nozzle Position 98 Figure 3.4 Relationship between Entrainment Ratio and Condenser Temperature 99 Figure 3.5a General View of the Test Rig 105 Figure 3.5b External Appearance of the Supersonic Ejector 106 Figure 3.5c Refrigerant Pump 107 Figure 3.6 Entrainment Ratio Versus the Condenser Temperature {Ejector La, Evaporator Temperature, T e = 5.0°C}
Ill
Figure 3.7 Entrainment Ratio Versus the Condenser Temperature {Ejector La, Evaporator Temperature, T e = 7.5°C}
112
Figure 3.8 Entrainment Ratio Versus the Condenser Temperature {Ejector La, Evaporator Temperature, T e = 0.0°C}
112
Figure 3.9 Relation Between Entrainment Ratio and Nozzle Position {Ejector La, Evaporator Temperature, T e = 3.5°C, Generator Temperature, T g = 74.5°C}
113
Figure 3.10 Relation Between Entrainment Ratio and Nozzle Position {Ejector La, Evaporator Temperature, T e = 7.5°C, Generator Temperature, T g = 76.7°C}
114
Figure 3.11a Theoretical Performance Map {Ejector La, A^At = 3.0, Dt = 2.0 m m , D 2 = 3.5 m m }
116 xv
Figure 3.11b Experimental Performance M a p {Ejector La, A2/At = 3.0, D t = 2.0 m m , D 2 = 3.5 m m }
117
Figure 3.12 Pressure Distribution against Distance from Exit of the Nozzle {Ejector La}
119
Figure 3.13 Entrainment Ratio as a Function of Condenser Temperature {Ejector Lb, Evaporator Temperature, T e = 7.5°C, Generator Temperature, T g = 73.3°C}
121
Figure 3.14 Relation Between Entrainment Ratio and Nozzle Position {Ejector Lb, Evaporator Temperature, T e = 7.5°C, Generator Temperature, T g = 76.9°C}
122
Figure 3.15 Pressure Distribution Along the Supersonic Ejector {Ejector Lb} 123 Figure 3.16 Plot of Entrainment Ratio Versus the Condenser Temperature {Ejector L a and Lb}
125
Figure 3.17 Plot of Entrainment Ratio versus the Condenser Temperature {Ejector 2.a, Evaporator Temperature, T e = 12.5°C, Generator Temperature, T g = 73.0°C, 66.5°C}
128
Figure 3.18 Relation Between Entrainment Ratio and Nozzle Position {Ejector 2.a, Evaporator Temperature, T e = 12.5°C, Generator Temperature, T g = 73.0°C}
129
Figure 3.19 Ratio of the Practical Entrainment Ratio to the Simulation Value {Ejector 2.a}
130
Figure 3.20 Pressure Distribution Along the Supersonic Ejector {Ejector 2.a} 131 Figure 3.21 Plot of Entrainment Ratio Against the Condenser Temperature {Ejector 2.b, Evaporator Temperature, T e = 5.0°C & 12.5°C, T g = 66.5°C}
xvi
132
Figure 3.22 Relation Between Entrainment Ratio and Nozzle Position {Ejector 2.b, Evaporator Temperature, T e = 7.5°C,Tg = 76.7°C}
133
Figure 3.22a Entrainment Ratio Against the Condenser Temperature {Ejectors 2.a and 2.b, T g = 66.5±0.4°C, Te = 12.5±0.2°C}
135
Figure 3.23 Variation of Effective Area Ratio with Different Operation Conditions of Evaporator Temperature {Ejector La}
137
Figure 3.24 Variation of Effective Area Ratio with Different Operation Conditions of Evaporator Temperature {Ejector 2.a}
137
Figure 3.25 Variation of Effective Area Ratio Versus Nozzle Position {Ejector 2.a}
138
Figure 3.26 Variation of Effective Area Ratio Versus Nozzle Position {Ejector 2.b}
138
Figure 3.27 Variation of Optimum Nozzle Position Against Evaporator Temperature {Ejector 2.b}
139
Figure 3.28 Plot of Aminb/At Versus the Nozzle Position {Ejector La and Lb} 146 Figure 4.1 Equilibrium Diagram for Liquid Vapour Phase of the Rl 1-R12 at a Total Pressure of 350 kPa
157
Figure 4.2 Plot of the Saturation Temperature for Pure Rll and Pure R12 159 Figure 4.3 The Schematic Diagram of the Two Component Entrainment Ejector 161 Figure 4.4 Equilibrium Diagram for Liquid Vapour Phase of the Rl 1 - R12 at the Pressure of 437 kPa
165
xvii
Figure B.l Relation Between Entrainment Ratio and Nozzle Position {Ejector 2.a, Tg = 66.5±0.2°C, Te = 12.5±0.2°C} 239 Figure B.2 Relation Between Entrainment Ratio and Nozzle Position {Ejector 2.b, Tg = 66.5±*).3°C, Te = 5.0±0.1°C} 240 Figure B.3 Relation Between Entrainment Ratio and Nozzle Position {Ejector 2.b, Tg = 73.0±O.2°C, Te = 12.5±0.2°C} 240 Figure B.4 Pressure Distribution Along the Supersonic Ejector {Ejector 2.b, Tg = 66.5±0.3°C, Te = 5.0±0.1°C} 241 Figure B.5 Absolute Pressure Versus the Voltage for the Pressure Transducer 242
xviii
LIST OF TABLES
Table 1.1 The Simulation Design Conditions in One Dimensional Constant Area Modelling by Kanashige (1992)
19
Table 1.2 Thermodynamic Characteristics of Some Refrigerants (Stoecker and Jones, 1982)
22
Table 1.3 Environmental Characteristics of the Principal CFCs 23
Table 2.1 Results of the Theoretical Sensitivity Study on the Effect of Diffuser Efficiency on Ejectors
58
Table 2.2 Number of Cells for the Grids 80 Table 2.3 Comparisons of Entrainment Ratio for the Experimental Results, ID Analysis and C F D results
83
Table 3.1 The Experimental Data Compared with the Simulation Results 100 Table 3.2a Ejector Dimensions Used in The Present Experiments 108 Table 3.2b Ejectors Used for the Present Study 108 Table 3.3 Comparison of the Experimental Data and Simulation Results 115 Table 3.4 Comparison of the Experimental Data and Simulation Results for Ejectors L a and L b
xix
126
Table 3.5 Experimental Data Compared with the Simulation Results for Ejector 2.a
130
Table 3.6 Comparison of the Experimental Data and Simulation Results for Ejectors 2.a and 2.b
135
Table 3.7 Comparisons the one Dimensional Analysis of the Shock Pressure Recovery With the Practical Value
141
Table 4.1 Some Possible Refrigerant Mixtures 148 Table 4.2 Data for Calculation of Mole Fraction of Vapours in the Mixture 156 Table 4.3 Calculation Procedure to Find the Dew Point Temperature of the Mixture
156
Table 4.4 Temperature Variation During Condensation of Pure Refrigerants (Rl 1 andR12) and a Mixture of R11-R12 (44% and 5 6 % , respectively)
158
Table 4.5 The Simulation Conditions in One Dimensional Modelling 162 Table 4.6 The Entrainment Ratio and Condenser Backpressure for Two-Fluid V J R S Simulation (^ = 0.95, ^=0.80, and Q e = 1 k W )
164
Table 4.7 Calculated Dew Point Temperatures and Temperature Glide for Rl 1R12 ejector
166
Table 4.8 Comparisons Between Simulation Results for the Two-Fluid Ejector and the Single Fluid V J R S
167
Table 4.9 Thermodynamic and Physical Properties of Rll and R12 169 Table 5.1 Performance Characteristics and Dimensions of the VJRS Ejectors 172
xx
LIST OF PUBLICATIONS
H Sharifi Bidgoli and P. Cooper, "Analysis of Vapour Jet Refrigeration System (VJRS) Supersonic Ejector Performance", Proceedings of the Six Asian Congress of Fluid Mechanics, Singapore, M a y 22-26,1995, pp. 1308-1311.
xxi
NOMENCLATURE
ID
one-dimensional
2D two-dimensional A area [m2] C* critical speed of sound D diameter [m] L length [m] M Mach number M* dimensionless velocity [M* = V/C*] P pressure [kPa] Q heat transfer [W] R gas constant [J/kg.°K] S source rate of the conserved property T temperature [°K] U x-axis velocity [m/sec] V velocity [m/s], but in Chapter 2 denotes y-axis velocity [m/sec] W work [W], but in Chapter 2 denotes z-axis velocity [m/sec] X Position of the motive nozzle related to the entrance of the constant area mixing tube
xxii
c
specific heat [J/kg.°K]
carnot Camot cycle h enthalpy [J/kg] m mass flow rate [kg/s] s entropy [J/kg.°K] t time v specific volume [m3/kg] w molecular weight [kg/kgmol] x mole fraction of the liquid components in the mixture y mole fraction of the vapour components in the mixture
Greek Letters "
efficiency
0
angle [degree]
0
entrainment ratio
the conserved pro
P
density [kg/m 3 ]
Y
specific heat ratio
Subscripts 0 stagnation condition 2 cross section of the constant area mixing tube xxiii
a
primary stream
b
secondary stream
c
condenser
cr
critical condition
d
diffuser, but in Chapter 4 denotes "dew point temperature"
e
evaporator
ej
ejector
g
generator
i
phase i
m
mixing chamber
n
nozzle pump relative position of the motive nozzle exit to the entrance of the mixing tube nozzle throat the mixed flow before the shock the mixed flow after the shock
Superscripts *
critical condition
xxiv
CHAPTER ONE INTRODUCTION
1.1 General Considerations
With population growth and industrial development there is a continuing increas in energy consumption throughout the world. Environmental problems such as the greenhouse effect, air pollution, production of chemical and toxic gases and ozone depletion are the most important effects of intensive energy consumption. The greatest part of our energy resources is made up by fossil fuels which are non renewable. The rate of increase in extraction of such resources is very high. Increasing energy demand could further increase the consumption of fossil fuels rapidly. Therefore, if this trend continues the world will soon face a n e w and major energy crisis. The conventional mechanically driven vapour compression refrigeration system is a significant consumer of high grade energy. For example, the air conditioning systems for vehicles are great consumers of energy. S o m e alternative refrigeration systems have been proposed in the past literature (eg Chen, 1978) that would reduce or eliminate this fuel consumption. T w o of the most viable alternatives are the lithium bromide water and aqua ammonia vapour absorption
1
Introduction
CHAPTER
ONE
refrigeration systems. M a n y problems, such as providing high grade thermal energy to drive those alternative systems, have mitigated against their commercial development to date.
Nowadays more attention is being focussed on the decrease in the consumption energy. The vapour jet refrigeration system (VJRS) is also an alternative to the conventional refrigeration system. L o w grade thermal resources such as industrial waste heat and wast energy from the exhaust of automobiles m a y be used as the heat source to drive the V J R S as part of the air conditioning system. Solar energy m a y be another heat source used to drive a V J R S and residential cooling air conditioning systems. Several theoretical studies of the possibility of providing cooling and air conditioning through the use of solar energy have been carried out (eg Zeren et al, 1978, Grossman and Johannsen, 1981, Zeren and Holmes, 1981, Sokolov and Hershgal, 1993). Also the feasibility of utilising solar power absorption air conditioning systems in H o n g K o n g has been carried out by Yeung et al. (1992). The V J R S is an alternative to the conventional mechanically driven vapour compression refrigeration system. A vapour jet compression cycle offers mechanical simplification, because it has no moving parts, except those in a refrigerant p u m p . In the V J R S there is a supersonic ejector as a thermal compressor that has the potential to significantly reduce energy consumption in air conditioning systems. Also it is possible to provide low cost heat energy from wasted energy such as industrial waste heat resources. These are the major benefits of using a V J R S .
The VJRS has been studied by researchers such as Kenkare and Maitala (1984), Chen and H s u (1987), Kanashige (1992) and others. The present research work
2
Introduction
CHAPTER
ONE
has included further development of the Kanashige's (1992) study. The present author has further developed the computer programs to investigate the effect of the superheating vapour leaving the evaporator and generator on the entrainment ratio and the critical condenser temperature, T c , and also the effect of the diffuser efficiency on the ejector performance and optimal ejector geometry. These topics are presented in Chapters 2 and 3.
Kanashige's (1992) design was based on the "one dimensional constant area analysis method". Fully developed choked flow was not examined in his work, because there w a s not sufficient cooling capacity of the condenser to operate below the critical conditions. Therefore, the m a x i m u m ejector performance was not observed from his experimental results on the testrig.O n e of the objectives of the present study was to find out the best ejector performance. Thus, some improvements were necessary to the apparatus test rig. M o r e details are expressed in Chapter 3.
The following matters are discussed later in the next two chapter. Firstly, the theoretical study and experimental results of the effect of the choking phenomenon on the ejector performance and secondly, the behaviour of the pressure distribution along the length of the ejector.
In the present study based on the one dimensional-analysis, further development has been attempted to more accurately model the real situation of ejectors using the secondary choking theory which was introduced by M u n d a y and Bagster (1977). This theory has been applied to design and analysis of the performance characteristics of a small R 1 2 ejector (see Section 1.5 and Chapter 2 for more details).
3
Introduction
CHAPTER ONE
In addition, in the present work the results of a numerical simulation of the flow field within the ejector using the computational fluid dynamics code, P H O E N I C S , is also presented in Chapter 2 and comparisons made between the C F D prediction of the flow field and that assumed in the secondary choking theory.
4
CHAPTER ONE
Introduction
1.2 Characteristics of the V J R S The vapour jet refrigeration system, Figure 1.1, uses a supersonic ejector (see Figure 1.1). T h e ejector is the key component of the system which acts as a thermal compressor. T h e ejector is effectively a substitute for the mechanical compressor in a conventional vapour compression system. The system is driven by low grade thermal energy. That energy is used by the generator to produce the high pressure motive stream. That stream drives the ejector by passing through a supersonic nozzle.
Several advantages may be seen in the ejector refrigeration system compared with the conventional one. The conventional refrigeration system (Figure 1.2) utilises a mechanical compressor, evaporator and condenser. The ejector refrigeration system has no moving parts except a refrigerant pump. Its operation and design are fundamentally simple and reliable in service. It operates with readily available refrigerants such as Rll and R12. L o w grade thermal energy at temperatures of less than 80°C can drive the system. A s mentioned in the previous section, the solar energy and industrial waste heat m a y be used as the heat sources of the thermal energy to drive the system. Also compared with the conventional systems, it requires less maintenance. Finally due to the absence of moving parts in the system, there is no lubrication problem. In contrast, some problems may be seen in the application of VJRS for air conditioning systems. T h e major disadvantage of the ejector refrigeration system is its low system efficiency. The efficiency of the ejector m a y be defined as the entrainment ratio,
, which is the ratio of mass flow rate of the evaporator to mass flow rate of the generator, or the overall system performance which is called
5
CHAPTER ONE
Introduction
Coefficient O f Performance, C O P . For example, in the present work using R 1 2 as the working fluid the predicted entrainment ratio was 0.165 based on onedimensional analysis w h e n the generator and evaporator temperatures were 80.0°C and 0.0°C, respectively. This low efficiency can be introduced as one parameter against the feasible application of such system for air conditioning. The low efficiency is affected on the condenser size. More details are expressed in Chapter 5. Some ways such as maximising the VJRS performance may be found to overcome problems . The present research study is directed at maximising the efficiency of the ejectors by testing some ejectors/nozzles which have been designed by application of some existing theory of ejector analysis.
6
Introduction
CHAPTER ONE
Control Volume
MOTIVE NOZZLE GENERATOR
^
I
j SECTOR
1
EVAPORATOR «%
CONDENSER
r
PUMP
EXPANSION DEVICE RECEIVER
Figure 1.1 Schematic Diagram of the Ejector Refrigeration System
Compressor
Expansion Valve
Figure 1.2 Schematic of the Conventional Refrigeration System
7
Introduction
CHAPTER ONE
1.3 VJRS Operation An ejector heat pump cycle differs from the conventional refrigeration cycle in that the ejector works as a thermal compressor instead of a mechanical compressor. A ejector sucks the secondary vapour from the evaporator to compress and to deliver it to the condenser (Chen, 1978). This replaces the mechanical compressor with a thermal one that m a y be driven by heat. The VJRS utilises a supersonic nozzle which consists of a converging entry section and a diverging exhaust part. T h e m i n i m u m cross section of the converging diverging part of the nozzle is called the nozzle throat. Generally, supersonic or subsonic gas or vapour under high pressure, escapes through the nozzle. In the subsonic case, after expansion in the entry section, the flow is compressed in the exhaust section and remains as a subsonic flow. Under choked conditions the supersonic nozzle in the V J R S has a "critical pressure", Per, in the throat of the nozzle which causes the flow to attain supersonic velocity on passing the throat and continues to expand from this point on. Therefore, the velocity will continue to increase as the stream passes through the divergent part of the nozzle. A schematic diagram of a supersonic nozzle is given in Figure 1.3. In the diverging part of the nozzle, the motive flow obtains a low pressure and a considerable kinetic energy. That low pressure sucks up the flow from the evaporator to mix with it. The kinetic energy is utilised in the entrainment of the secondary stream (Work and Haedrich, 1939). Mixing occurs by eddy diffusion, m o m e n t u m is conserved and the jet spreads in the manner of a free turbulent jet (Hoggarth, 1971). A schematic diagram of a supersonic ejector is shown in Figure 1.4.
8
Introduction
CHAPTER
ONE
The mixing process of the motive and secondary vapours is, in reality, complex but has been considered to occur at constant pressure (Elrod, H. G., 1945, Keenan et al, 1950). Generally, the velocity of the mixed streams in the converging part of the ejector is supersonic and a shock will change the mixed, supersonic stream to the subsonic condition (Bagster and M u n d a y , 1976). The mixing of the primary and secondary flows m a y occur in the constant area mixing tube (DeFrate and Hoerl, 1959, and Dutton and Carroll, 1986). M o r e details are presented in Section 1.5.
The subsonic diffuser raises the pressure of the mixed stream. The exhaust strea in the diffuser m a y be compressed to raise the pressure by the kinetic energy of the mixture (Chen, 1978). In other words the diffuser converts some of the kinetic energy of the mixture to enthalpy to compress it to the condenser pressure (Stoecker, 1958). The ejector performance is affected by the choking phenomenon (Munday and Bagster, 1977, Huang, et al, 1985). W h e n the ejector back pressure decreases the entrainment ratio, , increases. Below the critical value of the back pressure, P cr , the ejector operates in choking conditions in which the entrainment ratio will remain constant. W h e n the ejector operates in the choked range of the condenser pressure, the ejector performance has the m a x i m u m value and it is independent of the condenser pressure. In experimental work of the present study a fully developed choking has been observed. M o r e details are given in Chapters 2 and 3. In conclusion, in a simple analysis of V J R S operation it m a y be seen that the motive or supersonic stream is mixed with secondary vapour from the evaporator to compress and to deliver it to the condenser.
9
Introduction
CHAPTER
Nozzle Throat with Critical Pressure
Velocity A Increases '
Velocity 4 Increases '
Subsonic
M<1 Region
M=l
Supersonic Region
Pressure ! Decreases 1
M > 1
Pressure Decreases
Figure 1.3 Schematic Diagram of a Supersonic Nozzle
Mixing Chamber
Constant Area Mixing Tube
Motive Nozzle Motive stream (a) °
• 1
£ 3 fr es
1 Figure 1.4 Schematic of the Supersonic Ejector
10
Subsonic Diffuser
ONE
Introduction
CHAPTER ONE
1.4 Analysis and Performance of the Ejector Cycle The ejector cycle may be thought of as being composed of two sub-cycles, the power and the refrigeration sub-cycles.
1.4.1 Power sub-cycle
The power sub cycle consists of a generator (or boiler), an ejector, a condenser and a refrigeration p u m p . The p u m p is the only component of the cycle that requires input of mechanical energy. The power sub-cycle 0-4-3-5-7-0, Figure 1.5 (and see also Figure 1.1), generates the primary flow as a motive stream.
State point 7 corresponds to the outlet from the liquid pump. In process 7-0 heat is absorbed in the generator and the fluid leaves as a saturated or superheated vapour. The total pressure at state point 0, corresponds to the generator pressure, P g . This high pressure motive fluid expands through the motive nozzle in process 0-1 to form a supersonic stream.
As a result of the mixing process the motive stream draws vapour from the evaporator, then some kinetic energy in the stream is converted to static pressure in process a-3 within the diffuser.
The combined working fluid streams leave the diffuser and enter the condenser at the state point 3, at the condenser pressure, P c . During the process 3-5 the fluid moves through the condenser at essentially constant pressure where the superheat and latent heat of vaporisation are removed. In process 5-7, a fraction of the condensed liquid refrigerant enters the generator with increased pressure from the
liquid pump.
11 3 0009 03140255 0
CHAPTER ONE
Introduction
u 3 ! ft-
Enthalpy
Figure 1.5 Thermodynamic Cycle of the VJRS
1.4.2 Refrigerant sub-cycle
The refrigeration sub-cycle 6-4-3-5-6 of Figure 1.5, includes an expansion v
and an evaporator in addition to the ejector and the condenser which are shar with the power cycle. The refrigerant flow passing through the evaporator in refrigeration cycle becomes the secondary vapour stream.
The condensed liquid refrigerant divides into two streams, one part as mentio above enters the generator. The other part passes through an expansion valve
reduce the pressure to Pe at point 6. It then enters the evaporator. The stat
corresponds to that of the evaporator pressure, Pe. In process 6-4, the fluid
absorbs heat from the environment and is vaporised in the evaporator. In proc
12
Introduction
CHAPTER
ONE
4-a, the secondary vapour is mixed with the supersonic motive vapour. The mixed vapour is compressed through the diffuser section to the condenser pressure, Pc, at state point 3 in the process a-3.
13
Introduction
CHAPTER
ONE
1.5 Historical Background of V J R S Development
Steam jet and air jet ejectors have existed since the early 1900s. The invest of Keenan and N e u m a n n (1942) used a simple air ejector without the diffuser. The object of their investigation was to study the performance of the simplest form of ejector. They assumed that the expansion of the primary and secondary flow from a chamber is reversible and adiabatic and all mixing of the two streams occurs adiabatically in the cylindrical mixing tube. Also it was assumed that the wall friction is negligible. According to their analysis the ratio of the secondary mass flow rate to the primary mass flow rate increases with the area ratio of the constant area mixing tube to the nozzle throat area (see Figure 1.4). Further development was carried out by Keenan et al (1950). They added a subsonic diffuser to the ejector system. A one-dimensional analysis of ejectors was employed. Their analysis was carried out by assuming that the mixing of the motive and secondary streams occurs in the constant area mixing tube. They applied the continuity, m o m e n t u m and energy equations and assumed the flow to be adiabatic and frictionless at the walls of the ejector. They showed the comparison of the analytical and experimental results for a greater range of variables, such as the pressure ratios and area ratios, than presented by Keenan and N e u m a n n (1942). They attempted to determine experimentally the length required for the mixing process and found that the length depends on both the mixing process and the shock phenomenon. The results and the performance diagrams were based on the case in which the molecular weight of the motive stream w a s the same as the secondary vapour.
14
Introduction
CHAPTER
ONE
D e Frate and Hoerl (1959) employed a one-dimensional, constant area analysis of ejectors based on compressible fluid flow theory to design the refrigeration ejector. Their research contains calculated performance curves for ejector performance. They used s o m e arbitrary values of the performance parameters, such as the ratio of the operating pressures of the secondary to primary and the area ratio of the constant area mixing tube to the throat area of the nozzle, to cover a range of the operating conditions. They found that if the m a x i m u m discharge pressure increases by a small amount this could cause a remarkable decrease in the ejector performance. Their research work extended the analysis of Keenan et al (1950) to design an optimum jet ejector for the primary and secondary streams of different molecular weights.
Chen (1978) applied a similar approach to that of Keenan et al (1950). He employed their theory of jet ejector to incorporate the ejector into a heat driven mobile refrigeration cycle. His research study considered the potential application of the ejector refrigeration system to vehicle air conditioning. His attention was focused on the possibility of using the fuel energy wasted as heat in the internal combustion engines to drive an ejector. H e only analysed the cycle performance of a proposed R 1 1 3 single fluid V J R S for the application of automobile air conditioning. Hsu (1984) applied a similar approach to the latter to determine the optimum mixing section area of an ejector. H e applied the principle of the ejector to a heat p u m p system that was powered by a heat source and both the secondary and motive fluids were in the vapour phase. His analysis did not consider a particular working fluid, but the halocarbon refrigerants were recommended because their
15
Introduction
CHAPTER
ONE
physical and thermodynamic properties are suitable to use in V J R S driven by low grade thermal energy.
Application of ejector compression systems for cooling was studied by Tyagi a Murty (1985). In a theoretical study they mentioned that the heat input to the system m a y be provided from low grade thermal energy such as waste steam and exhaust from automobiles in the temperature range of 70 to 90° C. Their study analysed the ejector performance and the condenser heat capacity for s o m e halocarbon refrigerants such as R l l and R113. They found that R l l is an attractive refrigerant due to the higher C O P compared with R 1 1 3 for identical values of the generator, evaporator and condenser temperatures.
Sokolov and Hershgal (1989) developed their research study to show the effect compression enhancement in C O P increase in ejector refrigeration systems. Their attention w a s given to modify the conventional system components like the evaporator and generator. Kanashige (1992) employed the one dimensional theory of compressible flow to analyse and design a refrigeration ejector and developed several computer simulation programs. The research included the calculation of predicted values of the entrainment ratio. Also experimental work has been carried out to compare the actual value of entrainment ratio with that predicted. M o r e details of his study are presented in Section 1.6.
The above one-dimensional models of supersonic ejector performance were based on simple mass, m o m e n t u m and energy balances on a control volume with inlets from the evaporator and generator and one outlet to the condenser. The pressure at the outlet of the primary vapour nozzle was varied to maximise the entrainment
16
CHAPTER
Introduction
ONE
ratio. In reality, the fluid dynamics of supersonic jet ejectors are extremely complex, involving multiple shocks in a non-trivial geometry. In an attempt to more accurately model the real situation the secondary stream choking theory was introduced by M u n d a y and Bagster (1977). They applied that theory to analyse the ejector performance for steam jet refrigeration systems. They postulated that the primary jet expands laterally during its flow d o w n the converging section of the ejector and that it effectively provides a converging duct for the initially subsonic secondary vapour. They assumed that in this converging duct the secondary stream must reach sonic velocity before mixing can take place and is therefore effectively choked at some cross section of the ejector. Using a numerical solution based on two-dimensional analysis can give more details of the flow field within the ejector. Two-dimensional analysis of flow in the ejectors w a s carried out by Hill (1973) and others. The theoretical and experimental investigation of ejectors using computational fluid dynamics, C F D , has been carried out by some researchers (eg. Hedges, 1972). Hedges (1972) studied the m e a n velocity and pressure fields of an axisymmetric mixing of compressible and parallel streams in the converging-diverging mixing sections. His work w a s both an experimental study and theoretical modelling based on finite difference method. The turbulence characteristics of a free jet such as the velocity profile and the averaged mixing length is considered by references such as Abramovich (1963). Hedges (1972) developed descriptions of the m e a n velocity and pressure fields in detail. His model does not accommodate strong shocks and it was restricted to single phase flows. Chapter 2 expresses details of C F D analysis based on the finite volumetric technique used for the present research.
17
Introduction
CHAPTER ONE
1.6 Review of Previous W o r k by Kanashige The present work follows on from that of Kanashige (1992). As mentioned in Section 1.5, based on the D e Frate and Hoerl (1959) research study, a "one dimensional compressible fluid flow theory" was applied by Kanashige (1992) to analyse and design
an ejector. H e developed several computer simulation
programs. Experimental results were below the performance predicted by the simulation program. The actual entrainment ratio, <2>, obtained lay between 5 0 % to 6 0 % of the predicted values. H e used R 1 2 as the working fluid.
In Kanashige's (1992) computer program the exit pressure of the nozzle, Pi, w varied up to the evaporator pressure, P e , to calculate the entrainment ratio iteratively to maximise it (see Figures 1.1 and 1.5). Kanashige (1992) designed the ejector refrigeration system based on the conditions given in Table 1.1.
The design simulation results of his study contain information such as the entrainment ratio, O , the throat diameter of the nozzle, Dt, diameter of the constant area mixing tube, D2, and C O P of the system for m a x i m u m entrainment ratio under the design conditions. His experimental measurements included the mass flow rates through the generator and evaporator to obtain the actual value of the ejector performance (entrainment ratio), and measurement of the actual pressures and temperatures of some important components such as the inlet and outlet points of the condenser, generator and evaporator.
18
Introduction
CHAPTER ONE
Simulation Design Modelling Motive and Secondary Streams
R12
Evaporator Temperature, T e
0.0°C
Generator Temperature, T g
80.0°C
Condenser Temperature, T c
40.0°C
Nozzle Efficiency, r\n
0.95
Diffuser Efficiency, r\d
0.80
Area Ratio, A^Jki
3.0
Cooling Capacity, Q e
lkW
j
Table 1.1 T h e Simulation Design Conditions in O n e Dimensional Constant Area Modelling by Kanashige (1992) Kanashige (1992) presented the simulation results for off-design performance the ejector in a performance m a p which is reproduced in Figure 1.6. O n this m a p the C O P value is plotted against the critical condenser temperature with generator and evaporator temperatures as parameters, T g and T e , respectively. Results of simulation only apply to conditions above the broken line in this figure. Under choked conditions the entrainment ratio will remain constant. In Kanashige's (1992) experimental results just a tendency towards the choking was observed because there was not sufficient cooling capacity of the condenser to operate below the critical conditions. Thus, experimentally he recognised that the condenser temperature has a great effect on the entrainment ratio, such that when the former increases one degree Centigrade, the latter decreases about 20 percent. The reason is simply that the condenser temperature was not sufficiently low to satisfy the choked condition. W h e n the condenser is at less than the critical
19
Introduction
CHAPTER
ONE
backpressure, the ejector operates in the fully developed choked m o d e (see Section 2.2.5).
The limitations of Kanashige's (1992) work and subsequent modifications to the test rig which was carried out by the present author are described in Chapter
0.6
0.5
Tg=60 (deg. C) Te=15 (deg. C)
0.4
a* 0.3 O
u 0.2
0.1
'
0.0
•
•
20 25 30 35 40 45 50 CRITICAL CONDENSER TEMP. (deg. C)
55
Figure 1.6 T h e Performance M a p of Vapour Jet Refrigeration System
(The area ratio of the nozzle exit to the throat area, An/At = 1.69) (Kanashige, 1992)
20
Introduction
CHAPTER
ONE
1.7 Refrigerant Selection
Refrigerant selection is a serious issue in regards to both the efficiency of t ejector and environmental problems. Recently there has been increasing concern about finding the most effective, safe and harmless refrigerants, especially in regard to h u m a n health and the environment. It is mentioned by Harris (1990) that the Montreal protocol leading to a programme to reduce the environmental chemical effects which m a y protect some environmental problems such as greenhouse warming potential in the future.
1.7.1 General specification The most important groups of refrigerants are as follows. Halocarbon compounds, such as R l l and R 1 2 , which contain one or more of the three halogens chlorine, fluorine and bromine; inorganic refrigerants like ammonia, air and carbon dioxide which were the early refrigerants; and the hydrocarbon refrigerants including methane and ethane which are suitable as working fluids, specially in the petroleum and petrochemical industry (Stoecker and Jones, 1982). Azeotropes are another group of refrigerants which are a mixture of two substances. Azeotropes can be separated but not by simple distillation. R 5 0 2 is the most well k n o w n azeotrope and is a mixture of R 2 2 and R l 15 by 48.8 percent and 51.2 percent, respectively.
21
Introduction
CHAPTER ONE
1.7.2 T h e r m o d y n a m i c characteristics and properties of refrigerants
Some of the general criteria used to select a refrigerant are as follows. One them is the proper operating pressure for a given application. The latent heat of vaporisation at the evaporating temperature should be high to increase the cooling capacity, Q e . The motive stream reduces the heat input to the generator by a low latent heat of vaporisation at the generator temperature. For example, R 1 2 is a suitable refrigerant for V J R S ejectors. M o r e information about thermodynamic characteristics of R 1 2 and Rll are shown in Table 1.2. This table shows the operation of the refrigerants on a standard vapour compression cycle with an evaporating temperature of -15° C and 30° C of condensing temperature (Stoecker and Jones, 1982).
Refrigerant Evaporator
Condensing
Refrigerating
COP
number
Pressure [kPaj Pressure [kPa] Effect [kJ/kg]
11
20.4
125.5
155.4
5.03
12
182.7
744.6
116.3
4.70
Table 1.2 Thermodynamic Characteristics of S o m e Refrigerants (Stoecker and Jones, 1982)
1.7.3 Chemical properties and environmental problems
a: Toxicity and flammability
These characteristics are two important factors for safety. For example, ammo is flammable with 16 to 25 percent of it by volume in air. R 1 2 is considered non
22
Introduction
CHAPTER ONE
toxic but Rll, R22 and R502 are slightly more toxic than R12 (Stoecker and Jones, 1982). The toxicity and flammability of some refrigerants are given Table 1.3 (Australian Environment Council, 1989).
Refrigeran Formula
Boiling Atoms
t
Point
Life
[°C]
[Years] Potential
Ozone
Flammable Toxicity Greenhouse
Depletion
Warming Potential
Rll
CFC1 3
24.0
60.0
1.0
No
Low
0.4
R12
CC1 2 F 2
-29.8
120.0
1.0
No
Low
1.0
R22
CHF 2 C1
-40.8
15.3
0.05
Yes
Low
0.07
R134a
CH2FCF3 -26.5
15.5
0.0
No
Testing
<0.1
R141b
CH 3 CC1 2 F 32.0
7.8
<0.05
Yes
Testing
<0.1
R142b
CH3CCIF2 -9.2
19.1
<0.05
Yes
Low
<0.2
R123
CHCI2CF3 28.7
1.6
0.02
No
Low
<0.1
R124
C H C 1 F C E -12.0
6.6
0.02
No
Low
<0.1
R143a
CH3CF3
-47.6
41
0.0
Yes
Testing
<0.3
Table 1.3 Environmental Characteristics of the Principal C F C s (Australian environment council, 1989, and Harris, 1990) Notes for Table 1.3: (1) The ODP values of CFCs are relative to Rl 1, which is given the value and are based on one-dimensional treatment averaged between models and normalised to a standard methyl chloroform reference lifetime.
(2) The GWP values of CFCs are relative to R12, which is given the value o
23
CHAPTER
Introduction
ONE
b: O z o n e Depletion Potential ( O D P )
Ozone depletion potential is another serious issue in the selection of a refri The harmful ultraviolet ( U V ) radiation is absorbed and most of it is stopped from reaching the Earth by the ozone layer. W h e n ozone concentration decreases it will lead to an increase in U V radiation, such that a one percent change in the former will cause a two percent change in the latter. The U V radiation can damage both proteins and genetic material. It could cause death of cells or skin cancer in which Australia has the highest incidence in the world. Generally, it had been increasing the ozone layer decreasing over Antarctica between 1979 to 1988. Also the hole in the ozone layer w a s 40 percent greater in 1988 compared with 1987 (Australian Environment Council, 1989). Some refrigerants may have environmental problems in regards to ODP. For example, the chlorine which released from halogenated hydrocarbons to the environment combined with ozone in the stratosphere (Australian Environment Council, 1989). It has been stated by Harris (1990), that "Even today there is certainly no but the weight of evidence has greatly increased in support of the view that C F C s can indeed deplete stratospheric ozone, that this depletion can affect ozone more severely than had been realised by the scientific community, and that there has already been slight, but statistically significant, ozone depletion even outside the polar regions". The ODP of some refrigerants is shown in Table 1.3. It may be seen on that Table, in regard to O D P , s o m e refrigerants such as R 1 2 5 and R 1 3 4 a are environmentaly safe.
24
Introduction
CHAPTER ONE
c: Greenhouse W a r m i n g Potential ( G W P )
The global warming of the planet takes the name of the Greenhouse Warming Potential, G W P . Greenhouse gases in the atmosphere allow sunlight through to heat the Earth, but traps the warmth that radiates back towards space. Thus global warming will result. Global temperature has increased over the last century and could cause widespread climate changes. Some refrigerants including CFCs and halons contribute to increase GWP (Australian Environment Council, 1989). The greenhouse warming effect of some refrigerants is shown in Table 1.3. Therefore reduction of emissions of the refrigerants with high G W P such as R 1 2 m a y assist in ameliorating the greenhouse effect. More details about chemical properties and chemical reactions of refrigerants presented in references such as A S H R A E (1986) and Spauschus (1988).
1.7.4 Selection of the working fluid
a: CFCs Selection CFCs and bromofluorocarbons (halons) are widely used in refrigeration systems because of their unusual thermal properties, inherent stability and low toxicity. Nowadays there is increasing concern about the release of certain chemicals such as C F C s and halons. The reason is that those refrigerants are depleting stratospheric ozone, specially over Antarctica (Australian Environment Council, 1989).
25
Introduction
CHAPTER
ONE
C F C s and halons contribute to the warming of the planet. Initially it is estimated that by 2030, the C F C s m a y contribute about 20 percent to the greenhouse effect (Australian Environment Council, 1989). A n y programme towards the reduction of C F C s and halons will assist in improving the greenhouse effect. Table 1.3 gives the G W P relation for some CFCs.
It can be seen that an urgent transition away from CFCs is necessary. That non easy task falls to industry to introduce the n e w processes, n e w technologies and n e w chemicals to satisfy the current social needs of C F C s and halons.
b: Ammonia Ammonia has a big heat capacity, but it is not suitable to use in a VJRS as the working fluid because of the high pressure and safety hazards (Tyagi and Murty, 1985). Generally, for large industrial refrigeration plants it is c o m m o n to use ammonia as a working fluid. The ammonia refrigeration system is a conventional one and it is larger and is less efficient than the halocarbon's system (Bedwell, 1990). Such a system contains a compressor, condenser and evaporator.
As mentioned in Section 1.7.2, some refrigerants like R12 are a suitable working fluid for a V J R S ejector. Also, the pressure of R 1 2 is less than ammonia at the same operating condition, eg at 50°C the saturated pressures of the former and latter are 1219 and 2036 kPa, respectively. Therefore, R 1 2 was selected as the working fluid for the present work.
26
Introduction
CHAPTER
ONE
c: R l 34a as an alternative refrigerant
Thermodynamic properties of R134a (tetrafluoroethane, CH2FCF3) are similar to those of R12. A s shown in Table 1.3 the molecule of R134a contains no chlorine and has a zero ozone depletion potential. Also the greenhouse warming potential of R134a is less than 0.1, i.e. more than ten times lower than the value of R12. Therefore, R134a is proposed as an alternative to R 1 2 in refrigeration systems including V J R S and the conventional air conditioning applications and it is a suitable refrigerant for V J R S ejectors in the future.
d: Choice of working fluid for the experimental rig In the present research R12 was selected as the working fluid. The reasons for choice can be summarised as follows. Firstly, the previously existing facility was already designed for R12. Secondly, there was a lack of thermodynamic and physical properties at the time of study. Also, for alternatives such as R134a, the pressure of this refrigerant (R134a) is higher than R 1 2 at the same operating condition. For example at 80°C the saturated pressure of R134a is 2621 kPa and for R 1 2 it is 2304 kPa. The refrigerant p u m p was not able to operate at the pressures higher than 2300 kPa (see Chapter 5 for more details) and other components such as flow meters had also limited pressure capability. Thus, R 1 2 was selected as the working fluid for the present investigation.
27
Introduction
CHAPTER
ONE
1.8 Potential applications of ejectors Depending on the working fluid there are various kinds of ejectors such as the steam jet, air jet and vapour jet ejectors. Ejectors have been used extensively in power plants and chemical industries. Their application has been greatly broadened by using them in refrigeration systems.
Some applications of ejectors are described in the following sections.
1.8.1 Refrigeration applications It is known that the mechanical drive to the conventional refrigeration system currently well developed but it is a great consumer of high grade, mechanical energy. The ejector could be applied in a refrigeration system as an alternative to the vapour compression system.
In regard to refrigeration applications, the ejector refrigeration systems coul used in air conditioning. Those systems m a y be driven by the thermal heat sources, such as industrial waste heat. Using waste heat is beneficial in two aspects, as follows. (a) The economical aspect, in which the waste heat may drive the VJRS. For example in food processing the waste heat which is normally produced by expensive energy sources m a y not be used. Such waste energy would be converted to a thermal energy to drive an ejector refrigeration system.
(b) The environmental aspect, in which the environmental problems, such as the greenhouse warming potential and ozone depletion problem could be avoided.
28
Introduction
CHAPTER
ONE
S o m e problems m a y be seen in the practical application for refrigeration systems, eg poor entrainment ratio, the condenser heat rejection and construction an appropriate refrigerant p u m p to operate safely and quietly at high pressure.
1.8.2 Automobile VJRS
The vehicle air conditioning systems are important in regard to comfort and safety for occupants of an automobile. Chen (1978) and others have suggested that a vapour jet refrigeration system m a y be applied to automobile air conditioning. H e has analysed the cycle performance of a single fluid system for that application.
Due to the thermal activation of vapour jet refrigeration systems, they may have the potential to reduce energy consumption in automobile air conditioning to virtually zero, since that the low grade thermal energy is able to drive the automobile V J R S . It m a y be seen that there are two thermal sources in an internal combustion engine to drive the V J R S . The exhaust gases m a y be used as one of the sources. Circulating hot water from the engine's jacket might be the other source. A schematic diagram of an automobile V J R S is shown in Figure 1.7 (Kanashige and Cooper, 1993). This figure shows a schematic illustration of an ejector refrigeration system that m a y possibly be used in an automobile V J R S .
In contrast, there are some problems in the application of this technology to automobile air conditioning systems. Generally, for the period of operation of car air conditioning systems there is not a situation of steady state conditions. Ejectors m a y operate at high entrainment ratio for fully developed choked flow under steady state conditions. It requires close control of the condenser and
29
Introduction
CHAPTER ONE
generator temperature, because usually the air temperature across the condenser varies due to variation of car speed and ambient temperature. Also when the ejector does not operate under the choking condition, the condenser temperature affects the entrainment ratio of the ejector, such that when the former increases the latter dramatically decreases. Working the condenser at a desired low temperature to improve the entrainment ratio requires the size of the condenser to be greater than that of conventional refrigeration systems. With a constant temperature of the evaporator, when the generator temperature increases, it is predicted to increase the critical condenser temperature and to decrease the COP of the system. Therefore, these problems should be investigated and examined for this application. More details of implications for practical applications are described in Chapter 5.
EXHAUST GAS ^
1 EJECTOR ENGINE EVAPORATOR GENERATOR•
PUMP water
-CD-
PUMP
LO
EXPANSION f DEVICE refrigerant
Figure 1.7 Schematic Illustration of an Automobile V J R S (Kanashige and Cooper, 1993)
30
JCONDENSER
o
Introduction
CHAPTER
ONE
1.8.3 Steam jet ejectors
Fundamental principal and applications of these systems have been studied by some researchers (eg Holton and Schuls, 1951, H o g e et al, 1959 and Vyas and Kar, 1975). T h e steam jet refrigeration cycle is similar to that of other refrigeration cycles and the working fluid of the system is water (Munday and Bagster, 1974). The advantages of the steam jet compression cycle are the relatively low capital cost, simplicity of its operation, reliability and very low maintenance costs. Also, the extremely high heat of vaporisation of water and inherent stability and safety m a k e this system attractive (Tyagi and Murty, 1985). Therefore the steam jet m a y be used in industrial plants when low cost steam is available. Some applications of the steam jet unit are as follows:
(a) Industrial applications such as chilling of water to moderate temperatures i industrial process (Stoecker, 1958). W h e n conventional refrigeration is not suitable or competitive the steam jet compression system m a y be an alternative of that. The concentration of fruit juices, dehydration pharmaceutical products such as antibiotics, and chilling of leafy vegetables are some typical examples of industrial applications. (b) V a c u u m p u m p , in which m a n y applications including noncontaminating gases and vapours can be performed economically by steam jets. That type of the steam jet system has no rotating parts, no lubrication or oil problems. It is mechanically one of the simplest types of the vacuum p u m p s and compressors (Harris and Fischer, 1964).
31
Introduction
CHAPTER
ONE
(c) This system m a y be used for air conditioning because it is able to produce chilled water at temperatures d o w n to 1.7°C.
In contrast, the system has some disadvantages. It requires all temperatures to above 0°C, because water is the working fluid. L o w efficiencies ordinarily achieved by these systems. Certainly, if higher efficiencies could be achieved, the number of practical uses m a y increase.
1.8.4 Power Plant application of ejectors For many years the most important of the ejector's applications have been in power generating plants and chemical industries in which they utilise the kinetic energy of the working fluid to affect the entrainment and m a y be operated with liquids, gases or vapours. In that application they are used in exhausting air from condensers (Work and Haedrich, 1939, and Doolittle and Hale, 1984). "Gas jet or steam jet ejectors are used in multistage ejectors of turbine plant series with respect to the induced flow of compression stage" (Belevich, 1984). usually the mixing chamber of the compression stages in the multistage steam jet ejectors has a conic shape. Certainly, there are some more applications such as ejector application in chemical industries and pneumatic conveying, which are not presented here.
32
Introduction
CHAPTER
ONE
1.9 Scope of Present Study The present thesis includes the following work:
1. Introduction. The literature of the ejector refrigeration system, the operation the V J R S , refrigerant selection and potential applications have been investigated in this chapter.
2. Theoretical modelling. One-dimensional analyses, both the basic constant area method and the secondary choking theory, are applied to model the performance characteristics and design of small ejectors. The governing equations are outlined, a sensitivity study on the effects of superheating vapours on ejector performance and C F D analysis using P H O E N I C S are presented in this chapter
3. Experimental results. Details of the experimental apparatus are given and modifications to the test rig are explained. Entrainment ratio, ejector choking, optimum nozzle position, shock pressure recovery and effective area ratio of the secondary vapour to the throat area of the nozzle, Ajj/At, for four different ejectors/mixing chambers are examined and experimental results presented.
4. Two-fluid ejectors. A theoretical investigation of the feasibility of using two working fluids of different molecular weights in a vapour jet refrigeration system is presented in this chapter. T h e d e w point temperature, temperature glide, performance characteristics and comparisons with a single fluid small ejector are studied.
33
CHAPTER
Introduction
ONE
5. Implications for practical applications. The practical difficulties such as poor entrainment ratio, problems for refrigerant pump and the condenser size are discussed in Chapter 5. 6. Conclusions of the present study is stated and future work is recommended in Chapter 6.
34
CHAPTER 2 THEORETICAL MODELLING
2.1 Introduction Vapour jet refrigeration systems have been in existence for many years. Experimental investigations and analysis of large supersonic jet ejectors such as those used for steam jet refrigeration, has been carried out by researchers such as Stoecker, 1958, Whitaker, 1975 and others. In the present study more attention has been paid to analysis and examination of the performance characteristics of small V J R S ejectors using halocarbon working fluids, ie ejectors with throat diameters of several millimetres only. Further development of this type of study for the small single fluid VJRS ejectors has been carried out by the present author using the theory of secondary vapour choking which was introduced by M u n d a y and Bagster (1977). T h e mixing process between the primary and secondary streams has been studied in this work to facilitate analysis of ejector performance including the entrainment ratio and shock pressure recovery. The present chapter includes theoretical modelling of performance characteristics using one-dimensional compressible fluid flow theory and C F D (computational fluid dynamics) analysis.
35
Theoretical Modelling
CHAPTER
TWO
2.2 Mixing Process 2.2.1 General specification
Turbulent jets and mixing process have been studied by Abramovich (1963), List (1982) and others. The V J R S ejectors are associated with this process. The V J R S operation is mentioned in Section 1.3. Figure 1.4 shows a supersonic ejector which has two fluid streams, the primary and secondary flows. The motive stream acts as a confined jet in the ejector and mixes with the secondary stream. Confined jets exhibit a number of basic turbulent flow phenomena. The details of these phenomena are not yet fully understood (Gibson, 1986). Mixing of the two streams m a y be considered as turbulent jet mixing of a compressible flow. This section will discuss the general specifications of the velocity profile and pressure distribution of a mixed stream in the ejector. In the most general case the velocity, temperature and molecular weights of the two streams could be different at the inlet of the ejector. Those streams m a y be mixed in the converging part of the ejector, Figure 1.4, which is called the mixing chamber. The non-uniformity of velocity, temperature and composition at the lateral cross section generally decreases with increase distance from the jet source (Abramovich, 1963). Thus, a uniform flow could be considered to form in a certain cross section of the ejector, eg. in the constant area mixing tube. In the phenomenon of coaxial turbulent jet mixing, momentum and mass transfer occur within the mixing region, causing energy dissipation. The mixture of the two streams causes an initial raise in static pressure. Kotwal et al (1968) in their experimental work observed an increase in pressure in mixing of two coaxial streams immediately downstream of the secondary flow due to mixing. That
36
Theoretical Modelling
CHAPTER
TWO
increase in pressure was followed by a subsequent pressure drop because of friction losses.
2.2.2 Turbulent jet mixing and supersonic jet development in a compressible flow For incompressible turbulent flows the effect of variations of density may be neglected. If the difference of the temperature between the two streams is very large, say several hundreds of degrees Centigrade, or the velocity is supersonic, both, the compressibility effect would not be expected to be negligible (Pai, 1954). In a supersonic ejector the motive stream is accelerated through a nozzle and its
pressure energy is converted into kinetic energy, forming a high velocity turbulen jet (Hoggarth, 1971). Therefore in a VJRS the effects of compressibility on the
turbulent jet mixing should be considered. These effects arise because of the high speeds and temperature differences involved. Some important characteristics may be noted in regard to the expansion of the cross section of the jet. For example, during development of a free jet from a nozzle into a stationary medium, it entrains the fluid flow from the surroundings (Wall et al, 1982). Some characteristics of the jet expansion are as follows. During the jet development the mass diffuses more rapidly than the momentum. The velocity of the motive stream decreases downstream of the exit nozzle in the mixing zone (Duggins, 1975). The relative rates of decay of velocity and density would depend on the relative densities of ambient and source fluids.
37
Theoretical Modelling
CHAPTER
TWO
The jets in ejectors are clearly confined and exhibit different characteristics
free-jets. Theoretical analysis of the velocity profile of a high velocity jet i constant diameter mixing tube has been carried out by several researchers (eg. Razinsky and Brighton, 1972, and Gibson, 1986). The jet development in a
cylindrical mixing chamber is depicted in Figure 2.1 that comprises three region as follows: (a) In region 1, the primary and secondary streams are separate. This region is
transition zone which includes two sub-regions. In sub-zone A, there is a core o
potential flow may be called as potential core. This potential core of the jet i eroded by an annular mixing layer. The length of the potential core depends on factors such as the velocity ratios of the secondary to primary and the nozzle geometry (Gibson, 1986). As shown in Figure 2.1, the sub-region B, starts from where the potential core ends and ends at the point where the profiles become approximately self-preserving.
(b) In the next region the potential core has been terminated by the spreading o
the mixing zone to the axis and there is a decay of centreline velocity (Duggins 1975). In region 2, the secondary flow may be considered as a uniform potential
flow. The velocity profile of the entrained (secondary) flow is a function of th
abscissa X and at cross sections it is practically constant (Barchilon and Curte
1964). As the motive jet spreads, it entrains the secondary flow. Then the veloci
of the free jet will be reduced to establish the positive axial pressure gradien (Gibson, 1986). (c) As shown in Figure 2.1, region 3 is that where the motive stream spreads to the wall and the secondary vapour is wholly entrained. Then a streamwise
38
Theoretical Modelling
CHAPTER
TWO
pressure gradient is established and the velocity profile changes shape (Gibson, 1986).
{ U s and U p denote the velocity for the secondary and primary flow, respectively} Figure 2.1 Expansion of Jet Flow in a Cylindrical Mixing Chamber
An analogy between fluid flow in a free jet and in a mixing chamber of an ejector m a y be seen. In a cylindrical mixing chamber the velocity profile at each cross section, appears as if the central part is bounded by the cylindrical walls of the chamber (Abramovich, 1963). In the VJRS the motive stream expands from the nozzle exit into the converging part of the ejector (mixing chamber) in a confined space, which is shown in Figure 1.4. The expanded vapour then acts as a confined jet to decrease the pressure and converts it to kinetic energy. Then, the low pressure of the primary stream entrains the secondary vapour from the evaporator. During mixing the m o m e n t u m from the high velocity motive vapour is transferred to evaporator secondary stream primarily through the turbulent mixing process (Gibson, 1986).
39
Theoretical Modelling
CHAPTER
TWO
2.2.3 Constant area mixing tube
The constant area mixing tube is shown in Figure 1.4. It is mentioned by Abramovich (1963) that a free jet can increase the pressure of mixed streams in the cylindrical mixing chamber. It was mentioned in Section 2.2.2 that the momentum is transferred from the primary vapour to the secondary stream
changes to pressure. In the radial direction the static pressure may stand const When the length of the constant area mixing tube theoretically is not infinite, uniform spanwise distribution of stream parameters does not exist (Abramovitch, 1963). Therefore, a short length mixing tube involves a higher inlet velocity
distortion factor to the subsonic diffuser. Subsequently the diffuser performanc is affected by highly distorted inlet flows which comes from the constant area mixing tube (Neve, 1993).
The wall skin friction is another factor that should be considered in the length
mixing tube. A long length mixing tube involves higher wall skin friction losses
compared with the short one. In a short length mixing tube the pressure drop may be calculated from the friction factor of the entry length of a turbulent flow.
40
Theoretical Modelling
CHAPTER
TWO
2.2.4 Shockwaves
Supersonic ejectors use the shock pressure recovery phenomenon to maintain the
pressure difference between the evaporator and condenser. The position of the sho
is affected by the back pressure of the ejector. For ejectors at choking conditio
shock pressure recovery process may appear in the constant cross sectional passage
When the back pressure increases beyond its critical value, it could cause the sho penetrate into the conical mixing section and the ejector performance starts to decrease rapidly (Huang et al, 1985).
Here, only some of the important aspects of the shock process are considered. For
example, a shock could be considered as an irreversible process. Thus, a supersoni ejector in practice cannot operate as an ideal device because of thermodynamic irreversibilities. If the velocity and temperature gradients are small, it may be assumed that the forces in the gas are because of variations in the pressure and due to friction, and the entropy remains constant. Otherwise the effect of the irreversible thermodynamical processes must be taken into account. Irreversible processes are generally significant in gases only in the narrow zones where the velocity and temperature gradients become very large. These processes may be described by sudden jump discontinuities (Courant and Friedrichs, 1967). A shock phenomenon involves an extremely rapid and steep change of state. When
this change occurs across a surface perpendicular to the direction of the flow, i called a normal shock. This shock may occur in an ideal gas flowing through a
nozzle. Normal shocks are special forms of pressure discontinuities within the flu When the discontinuities are inclined to the direction of oncoming flow they are called oblique shocks (Shapiro, 1953). The passage of a shock wave may influence mixing through modifications to both the fluctuating and steady flow properties 41
CHAPTER TWO
Theoretical Modelling
(Buttsworth and Morgan, 1992). Here only more details are expressed for a normal shock.
Flow through a shock is an example of a Rayleigh flow for which the heat gain loss is zero, and at the same time it is an example of Fanno flow for which the frictional effects are also zero (Fox, 1977). A s shown in Figure 2.2 (Van Wylen and Sonntag, 1985), in a normal shock the equations of energy and continuity combine to make the Fanno line in an h-s diagram. The Rayleigh line is the combination of the continuity and m o m e n t u m equations. Points a and b correspond to M = l at which the entropy is maximum. The regions of the supersonic and subsonic velocities are given in the figure. Points x and y satisfy all three equations. The normal shock, which is not isentropic, can proceed only from x (supersonic) before the shock to y (subsonic).
oi ™ * o » ,
M < 1 above points a and b M > 1 below points a and b M = 1 at a and b
Figure 2.2 A O n e Dimensional Normal Shock on an Enthalpy - Entropy Diagram (Van Wylen and Sonntag, 1985)
In a supersonic nozzle, the normal shock may be described as follows. Point d Figure 2.3) corresponds to the position that the exit pressure of the nozzle, P E , is just equal to the back pressure, P B , and the flow is maintained as an isentropic stream. W h e n P B increases, P E is not influenced up to point f, in which the increase from P E to P B occurs outside the nozzle. W h e n P B increases up to point g, the normal shock
42
Theoretical Modelling
CHAPTER
TWO
stands in the nozzle exit, in which downstream of the shock PE = PB, and the flow
leaving the nozzle is subsonic. If PB is raised from point g to h, the normal sho moves into the nozzle. In that case the divergence part of the nozzle acts as a diffuser. From h to c as PB is increased the shock moves towards upstream and disappears at the nozzle throat where the PB corresponds to c (Van Wylen and Sonntag, 1985)
i
v-o
«PM
i
Po To
^
1.0 p Po
Throal —
PB
i i
—
>
\ l
a b e n
f i
d t
Figure 2.3 Nozzle Pressure Ratio A s a Function of Backpressure for a Reversible Supersonic Nozzle (Van Wylen and Sonntag, 1985)
2.2.5 Ejector choking * Choking is a very important phenomenon in the supersonic ejector. As shown in
Figure 2.4, the entrainment ratio starts to build up and continues to increase q as the backpressure (ie condenser pressure) decreases below Pm, at which the
entrainment ratio is zero. At the critical backpressure, P*, the ejector perform
that the entrainment ratio is maximum. When the ejector operates below its critic backpressure, the entrainment ratio is virtually independent from the condenser pressure and the ejector is then "choked".
43
Theoretical Modelling
CHAPTER TWO
In the present work ejector choking has been studied in a small, single fluid VJRS. Also, in this study the hypothesis that choking may occur as a result of the
stream reaching sonic conditions in some cross section of the converging part ejector has been examined.
o •a at w
c o E c 2 c
Pc Ejector Backpressure
Figure 2.4 Ejector Choking
44
Pm
Theoretical Modelling
CHAPTER TWO
2.3 Governing Equations for One-Dimensional Analysis In the present study the following assumptions are made for one-dimensional analysis (see Figure 2.5). It is assumed that all processes are adiabatic, there is no wall friction, the potential energy of the stream is negligible, the working fluid is treated as an ideal gas. T h e outlet velocity from the subsonic diffuser is neglected. Also, it is assumed that the expansion and compression processes through the convergent-divergent sections are isentropic.
Mixing Chamber
Constant Area Mixing Tube
Diffuser Section
Motive Nozzle
MOTIVE 0 STREAM(a)
4 SECONDARY VAPOUR (b) Figure 2.5 Schematic Diagram of an Ejector The governing equations are given below and include the possibility of having different gases for the primary and secondary streams. In the present chapter the equations are used for single, fluid V J R S ejectors but in Chapter 4 these are used for two-fluid ejectors. Here, only the vapour phase is considered.
45
Theoretical Modelling
CHAPTER
TWO
(a) Equations of thermodynamics and performance analysis
The following governing equations are used for thermodynamic and performan
analysis of an ejector assuming all processes are reversible and adiabatic Figures 2.5 and 1.5 (or Figure 1.1)}. Heat absorbed by the evaporator, or cooling capacity, Qe, is given by: Qe = mb(h4-h6) (2.1) Heat input to the working fluid in the generator, Qg, is written as: Qg = ma(A,-A) (2.2) Heat rejected in the condenser, Qc, is defined as: Qc=(m0 + mb)(rh-h5) (2.3) Work done by the refrigeration pump, Wp, can be written as: W^m^hj-hs) (2.4)
The ratio of the mass flow rates of the secondary to primary streams is th entrainment ratio, O.
K The coefficient of performance, COP, is defined as: O
C0P =
^ h,-h,
j^K-hc
— ^ — =o 4 6 =0^—2_ Q,+WP h0a-h5 \a-h
(2.6)
•5
In the above equation it is assumed that the throttling device is isenthalpic, thus:
46
Theoretical Modelling
CHAPTER
TWO
1*6 = «5 (2.7) Similarly, COPcarnot, can he written as (see Appendix C, Section C.4 for derivation of COPcarnot for VJRS):
In the above equations, m , h and T denote mass flow rate, enthalpy and absolute temperature, respectively. (b) The perfect gas equations The perfect gas equations are as follows: P = pRT (2.9) h
= CpT = -l-RT (2.10) y-1
where p, R, P, cp and y denote density, gas constant, pressure, specific heat at constant pressure and specific heat ratio, respectively. (c) Equations for choked flow The following are the governing equations for choked flow (Cambel and Jennings, 1958).
The critical pressure, P, for choked flow, eg. for a supersonic nozzle, is given a (
r
2
> r-i
(2.11)
^0
where the subscript '0' denotes the stagnation condition.
47
CHAPTER TWO
Theoretical Modelling
The flow through the supersonic nozzle m a y be regarded as choked if the backpressure, or the pressure in the region into which the nozzle is discharging, is lower than the critical pressure, P*.
The m a x i m u m mass flow rate in a choked flow, can be obtained as: „, \ ° 5r
( m = APn RT
^ \2( -i) r
(2.12)
vT+ly
K oJ
The choked mass flow rate of the motive supersonic nozzle, ma, is given by: (2.13)
ma=P,AtVt
where the subscript 'a', denotes the primary vapour. hi above equation, pt, A t and Vt denote the density at the throat nozzle, the throat area of the nozzle and the sonic throat velocity, respectively. where p, is defined as:
r„-i
(2.14)
Pt=Poa Ya+l
The absolute temperature at the throat, Tt, and the sonic throat velocity, Vt, based on isentropic flow relations, are defined as: < Tt = T Oo(
2
(2.15a)
Ya+l.
r
vt =
^
2yaRT0i +l
y
0.5 0.5
= (YaRTt)
Ya
48
(2.15b)
CHAPTER TWO
Theoretical Modelling
The right hand side of Equation (2.15b), is also the velocity of sound, or the acoustic velocity in an ideal gas. The local acoustic velocity is independent of pressure. (d) Motive nozzle analysis
The governing equations for the motive nozzle are defined as follows (DeFrat and Hoerl, 1959). The critical speed of sound, C*, and the dimensionless velocity, M* are given by:
C=\-^-RT vr+i
(2.16a)
(2.16b) C where V denotes stream velocity. The dimensionless velocity of the motive stream at cross section 1, Figure 2.5, M*u, may be written as: ra-n 0.5 <
Ya
Ya+l I = n,
(2.17)
r.-i
It is assumed that the static pressures for the primary and secondary streams are equal. Similarly, the dimensionless velocity of the secondary flow at cross section 1, M*b, may be expressed as: r»-i'
Mlb =
n+i n-i
Pl
0.5
r.
(2.18)
p V
06.
The area ratio of the nozzle exit to the throat area, Aia/At, is given as:
49
Theoretical Modelling
CHAPTER TWO
f
2 ^
A*_ i A Mu KYa+h
i+V<
r„+i 2(r a -D
(2.19)
The Mach number at the exit of the nozzle, M j a , is written as: 0.5
M
rDQa^ Ya
= / fl -l
"I
(2.20)
^1
(e) Governing equations for mixing section The following are conservation equations for constant area analysis between sections 1 and 2. Continuity equation can be written as: (2.21)
(p™) l a +(P™)i6=M^ The momentum equation is:
= (ma + mb)V2 + P2A2
(2.22)
The energy equation is: /
( v2\(ma+mb)
f
/! + — I 'la
u V2\ m„ + z V Jib
(2.23)
h+— V 2 Ji
where A 2 denotes the area of the mixing tube.
Equations (2.17) to (2.23) are combined to give the following equations (DeF and Hoerl, 1959). The dimensionless velocity at Section 2, M*2, may be written as:
50
Theoretical Modelling
CHAPTER TWO
M*2=B-(B2-l)
, B>\
(2.24)
where
K+K^M±Ayi4if-' 2 Ca A V Ya M)a v+i r*
B =
:
(2-25>
^^(1 + OHr In above equation, the ratio of the critical speeds of sound for the secondary C* vapour to that of the primary stream, -\ , can be expressed as:
Q = r a +i Yb rob wa C'a V Ya Yb + lT0aWb where W
a
and W b are the molecular weights for the primary and secondary
streams, respectively. The specific heat ratio at point 2, y2, may be defined as: Ya
+
Yb Wa^
Y =1*
I Tt \J¥±—
(2 27)
l^^b~Yb~^ The ratio of the critical sound speeds for the mixed flow at point 2 to that of the
c; . . pnmary stream, —^, is given as:
£L=
C
r.+i r, IKYL
V Y.
Y2+lT0aW2
where C*, is the critical speed of sound.
51
(228)
Theoretical Modelling
CHAPTER TWO
The ratio of the molecular weights of the motive vapour to that of mixed stream can be expressed as:
W ^ = 5k- (2.29) W2 l+ O
^
)
The ratio of the stagnation temperature of the mixed flow to that of the primar stream, is: 7a • Yb Tpb *^a (ft
-*02 _ Ya~* Yb~l *0a "b .- - „. L
0a
,a
+
a Jh
(f) Equations for normal shock pressure recovery in a constant area tube
In a constant cross sectional area tube when the upstream flow is supersonic an downstream is subsonic a normal shock may occur. In a normal shock, the
frictional effects acting upon the fluid and heat gains or losses can be assume
negligible. The following relations can be used for the shock pressure recovery (John, 1984).
*±= l + r M > P2y l+y2M22x
(2.3i)
and p
u =Mix p + (y2-i)M22JC Pu M2yp + (y2-l)M22y The subscript '2x' and '2y' denote the mixed flow before and after the shock at point 2 (see Figure 2.6). Manipulating the two equations above gives M2X in terms of M2y, as follows:
52
Theoretical Modelling
CHAPTER
2 + (y2-l)M22x Ml
>
TWO
(2.33)
i2y2Mlx-{y2-l)
Motive Nozzle •
>
2y M>1 2x1 M<1
.
flow
Constant Area Tube
Figure 2.6 Schematic Diagram of the Shock Process in the Constant Area Mixing Tube (g) Entrainment ratio The entrainment ratio can be written as a function of the area ratio of the secondary vapour at cross section 1 to that of the motive stream, Aib/Aia as follows (Khoury et al 1967):
M„
\ Y A16
,0.5
To a
(2.34a)
K\P0a) Aa \i ob. Also, the entrainment ratio can be written as a function of the area ratio of the secondary vapour at cross section 1 to that of the motive nozzle throat, An/At follows: .0.5
^ _ Pob Ab | Too Poa A \Tob,
(2.34b)
For the case of constant area analysis for a supersonic ejector, the cross sectional
area of the constant area mixing tube, A2, is equal to (Aia + Aib). Therefore, t
entrainment ratio, may also be expressed as a function of area ratio of the cons area mixing tube to the throat area of the motive nozzle, A2/At as follows:
53
CHAPTER TWO
Theoretical Modelling
( pt Yr. * 0 6 - ^ b- ^Oa-
M;.C;
06
4> =
( P: YT M *- C *
Mlla *^
Pi W / a + l V 1 -1 V P o«y
"n». 06 P Oa W „ . Ini,.
V o«y (2.35)
(h) Governing equations for subsonic diffuser The pressure ratio of the diffuser outlet to that of the diffuser inlet, P3/P2, isentropic compression, may be defined as: Yi
1 x r W + iV-
M"
2
(2.36)
In above equation, the diffuser efficiency, r\d, and the M a c h number of the mixed stream, M2, are expressed respectively as:
*1J =
(2.37)
K~K 2
M2 =
K) 7
2 2
+l
(2.38)
1-K)27^-1 72+l
i
In Equation (2.37), h3 S is the enthalpy corresponding to point 3 of Figure 2.5, when the mixed flow is compressed isentropically through the subsonic diffuser.
54
Theoretical Modelling
CHAPTER TWO
(i) Critical condenser pressure When the ejector does not operate at choking conditions, the entrainment ratio,
, is affected by the backpressure, such that decreasing the latter could caus increase in the former. The critical condenser pressure, P*, can be obtained as follows (see Figure 2.4): f P
2 Pl ) *
P P -1-*- • 1
l
(2.39)
/max.*
in which P3/P2 can be calculated from Equation (2.36) and P2/P1, is given as:
M:+O.M;^-M;($+I4 %• = 1 + Pl
^
-7 ^-
A 1 Pi fr. + iV^ A7a^0aV 2 J
55
(2.40)
Theoretical Modelling
CHAPTER
TWO
2.4 Sensitivity Study on the Effect of the Nozzle Efficiency and 11
Diffuser Efficiency on Ejector Performance in a Small Single Fluid V J R S 2.4.1 Sensitivity study on the effect of nozzle efficiency
This sensitivity study is concerned with the effect of nozzle efficiency on ejector performance and the critical condenser temperature in the small, single fluid V J R S . T h e kinetic energy of the motive stream is affected by the nozzle efficiency. Also, the entrainment ratio, coefficient of performance, C O P , and the critical condenser temperature are functions of the nozzle efficiency. The effect of the nozzle efficiency on ejector performance (entrainment ratio, O) and the critical condenser temperature has been carried out using one-dimensional analysis of ejectors based on the constant area method. Theoretical results show that the entrainment ratio decreases, w h e n the nozzle efficiency used in the analyses decreases, such that one percent change in efficiency could cause a change in the range of 0.42% to 0.75%. Also, the critical condenser temperature is affected by the nozzle efficiency, such that w h e n the latter decreases one percent, the former decreases in the range of 0.5 to 1.5 percent.
56
Theoretical Modelling
CHAPTER
TWO
2.4.2 Sensitivity study on the effect of diffuser efficiency
Sensitivity studies of the effect of diffuser efficiency on ejector performance for large system of ejectors such as steam jet refrigeration systems have been made by some researchers (eg Hsu, 1984). The present author made a sensitivity study on these effects on entrainment ratio and the critical condenser temperature for small single fluid ejectors. Particular attention has been paid in this research to
observe the effect of the diffuser efficiency on optimal ejector dimensions such as the throat diameter of the nozzle. The present study has found that diffuser efficiency affects on entrainment ratio, critical condenser temperature and the ejector design dimensions. One set of data from the theoretical result are given in Table 2.1. These results show when the diffuser efficiency decreases the entrainment ratio also decreases, such that 1 percent change on the former could change the latter by approximately 0.15 percent. The resulting entrainment ratio values versus the diffuser efficiency are plotted in Figure 2.7. Also, it is seen that a one percent decrease in diffuser efficiency could cause a decrease in the critical condenser temperature of the order of 0.75°C. Table 2.1 shows that when the diffuser efficiency decreases, the design dimensions of the ejector (D2 and Dt) also decrease, such that every 10 percent change on the former, could change the latter in order of 0.84 percent. Therefore, the diffuser efficiency has a slight effect on the optimal ejector design dimensions.
57
Theoretical Modelling
.2
3 5 g E •| i fl
CHAPTER TWO
0.140 T 0.139:
1 ^
0138
"
^^^
:
0.137 ^ * ^ 0.136: ^ ^ 0.135: ^ ^ 0.134: >^ 0.133: 0.132 - ^ ^ 0.131 •; 0.130 | • | • | • |—i—|—i—I—•—T-1—T~~*— 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90
Diffuser Efficiency Figure 2.7 Plot of Diffuser Efficiency Versus the Entrainment Ratio {Simulation Conditions: Given in Table 2.1}
Nozzle Efficiency 0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760
Diffuser Efficiency 0.850 0.800 0.750 0.700 0.650 0.600 0.550 0.500
Entrainment Condenser Ratio Temp. [°C] 0.139 33.140 0.138 32.110 0.137 31.080 0.136 30.040 0.135 29.000 0.134 27.960 0.132 26.920 0.131 25.880
D2 [mm] 3.745 3.729 3.713 3.697 3.682 3.667 3.652 3.637
Dt [mm] 2.151 2.142 2.133 2.124 2.115 2:107 2.098 2.089
Table 2.1 Results of the Theoretical Sensitivity Study o n the Effect of Diffuser Efficiency on Ejectors {Tg = 80.0°C, Te = 0.0°C, Area Ratio, A2/At = 3.0, Qe = 1 kW}
58
Theoretical Modelling
CHAPTER
TWO
2.5 Effect of Superheat on Ejector Performance In practice the motive flow from the generator and the secondary vapour from the evaporator are often superheated. The effect of superheat on ejector performance has received little attention in the literature to date. In this study, the effects on a small R 1 2 V J R S have been investigated. The analysis is based on the onedimensional constant area method.
2.5.1 Analysis of superheating effects Compared with the standard vapour compression cycle, the actual cycle suffers from pressure drops and inefficiencies. B y taking into account the superheating effects, the standard cycle (Figure 1.5) is modified as Figure 2.8 to show the actual p-h cycle of the V J R S . The standard cycle, assumes that the vapour leaving the evaporator and generator is not superheated and no pressure drop occurs in the evaporator and condenser. In the actual cycle, however, there is a pressure drop of the refrigerant due to the friction. Those pressure drops could cause the compression process to require more work than in the standard cycle. Furthermore in the actual cycle the compression is not isentropic and there are inefficiencies because of friction and other losses (Stoecker and Jones, 1982).
The Carnot thermodynamic cycle gives the maximum Coefficient Of Performance, C O P , for a refrigerator. A non-ideal cycle of an ejector heat p u m p operates in a less than ideal cycle because of thermodynamic irreversibility including imperfect energy transfer and the friction losses (Chen and Hsu, 1987).
59
Theoretical Modelling
CHAPTER
Saturated Liquid
TWO
Saturated Vapour
g Superheating [Generator]
0
'Actual Compression
1
y
TCondenserl V
Standard Cycle
i__l? 6
[Evaporator Pressure Drop
Superheating
Enthalpy
Figure 2.8 Actual Thermodynamic Cycle of the Vapour Jet Refrigeration Systems When the temperature of the superheated vapour leaving the evaporator and
generator increases, the specific volume of the vapour also increases. In this w the effects of superheated vapour leaving the evaporator and generator on
entrainment ratio, critical condenser temperature and cooling capacity in a fixe ejector have been analysed.
2.5.2 Theoretical results
Here a small R12 ejector has been considered with the area ratio of the mixing tube to primary nozzle throat cross sectional area of A2/At = 3.0, a nominal cooling capacity of 1 kW and saturated generator and evaporator temperatures of
80°C and 0°C, respectively. In the present simulation it has been found that whe
60
Theoretical Modelling
CHAPTER TWO
the saturated temperatures of the evaporator and generator are maintained constant the entrainment ratio decreases with increasing superheat of either the motive or secondary vapour stream. A superheat of 1°C in the generator outiet results in a decrease of 1.2% in the entrainment ratio. It w a s also found that a superheat of 1°C in both the evaporator and generator outlets when the saturated temperatures were maintained constant, the
entrainment ratio decreased, such that 1 degree increase in the superheat caused a
decrease of about 1.3 percent in entrainment ratio. Thus, a slight effect on ejecto performance may be seen due to the superheating of vapour leaving the evaporator and generator. The resulting entrainment ratio, the critical condenser temperature and cooling capacity values versus the superheating conditions are plotted in Figures 2.9,2.10 and 2.11.
The limitation of this analysis is that the treatment of metastable equilibrium fo the superheated motive stream through the expansion in the convergent-divergent nozzle was not considered (Van Wylen and Sonntag, 1985). 0.166
1 2 3 Superheat of the Gen. and Evap. [°C]
Figure 2.9 Superheat of the Generator and Evaporator Versus Entrainment Ratio {T g ( s a t ) = 80°C and T e ( sa o = 0.0°C)
61
Theoretical Modelling
CHAPTER TWO
40.6
o. u
u
0 1 2 3 4 5 Superheat of the Gen. and Evap. [°C]
Figure 2.10 Superheat of the Generator and Evaporator Against the Critical Condenser Temperature {Tg(sat) = 80°C and Te(sat) = 0.0°C} 1000
Superheat of the Gen. and Evap. [°C]
Figure 2.11 Superheat of the Generator and Evaporator Versus the Cooling Capacity {Tg(sat) = 80°C and Te(sat) = 0.0°C}
62
Theoretical Modelling
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TWO
2.6 Secondary Vapour Stream Choking 2.6.1 Analysis of the secondary choking theory
A thorough literature survey on research into the performance characteristics and design analysis of the supersonic refrigeration ejectors has been carried out by the present author. T h e investigations of s o m e researchers were based on onedimensional compressible fluid flow theory. That analysis m a y employ either the constant area mixing (eg Khoury et al, 1967, Dutton et al, 1982, Dutton and Carroll, 1983, and Kanashige, 1992), or the constant pressure mixing assumptions (eg. Keenan et al, 1950, and Hsu, 1984). The analyses can be used to either determine the required dimensions for design of an ejector for m a x i m u m entrainment ratio at given operating pressures/temperatures (design), or to calculate actual operation of an ejector of given geometry (simulation). The one-dimensional analysis is based on simple mass, momentum and energy balances on a control volume with inlets from the generator and evaporator and one oudet to the condenser. In reality the fluid dynamics of the ejector are extremely complex involving supersonic flow with mixing of coaxial jets and multiple shocks in a non-trivial geometry. The flows involved are certainly not one-dimensional. Therefore, m u c h work remains to be done before ejector design can be optimised from a theoretical modelling approach. In an attempt to more accurately model the real situation, the theory of choking of the secondary stream was introduced by M u n d a y and Bagster (1977). They postulated that as the supersonic primary jet expands laterally during its flow d o w n the converging section of the ejector it effectively provides a converging duct for the initially subsonic secondary vapour. They assumed that the primary stream and the secondary vapour maintain their
63
Theoretical Modelling
CHAPTER
TWO
identity and in this converging duct the secondary vapour must reach sonic veloc
before mixing occurs and is therefore effectively choked at some cross section Y the ejector as shown in Figure 2.12a.
Analytically it may be assumed that both the secondary and primary fluids expand
from their respective stagnation pressures to the static pressure of the seconda choking at section Y-Y where their static pressures are equal.
A schematic diagram of the region in which the mixed stream is supersonic and th secondary choking may occur is shown in Figure 2.12b. Munday and Bagster (1977) developed the ejector modelling by using the secondary choking theory. They mentioned that for a given diffuser, as a limiting case, it is thermodynamically
impossible for a supersonic flow to decelerate below sonic conditions in a conve cone. Therefore, the supersonic mixed stream in the converging section must
decelerate (with pressure recovery), to sonic velocity (M = 1) just at the end of
cone. If secondary choking takes place in the constant area mixing tube, then the shock pressure recovery may happen there immediately after mixing (see Figure 2.12.b). Secondary choking is associated with a hypothesised "effective area" for the entrained (secondary) stream in the mixing zone (Huang and et al, 1985). The
effective area ratio may be introduced as the ratio of the area of the secondary
at choking to the throat area of the nozzle, AYYb/At. This "effective area ratio" then be used as a parameter to aid in calculation of the ejector performance.
64
CHAPTER
Theoretical Modelling
Mixing postulated to occur when secondary sream reaches sonic velocity Motive Nozzle
Motive stream (a) Constant Area Mixing Tube
Secondary stream (b)
(a) Secondary choking may occur in this region followed by deceleration of the supersonic mixed stream **>. v, M=l or M>1 ^^S^l If choking occurs in the constant area tube, /VxxPy^ I a shock may occur promptly after mixing Motive!
////^y'\ , rnnstnnt Area Tnhp. ^ //j£^ ' Mixing Section
(b) Figure 2.12 Schematic of Choking of the Secondary V a p o u r Stream in an Ejector
65
TWO
Theoretical Modelling
CHAPTER
2.6.2 Application of secondary choking theory to small ejectors
In the present study a computer program has been developed to model the secondary choking in the small R 1 2 ejector. In this analytical study, mixing is modelled as occurring when PYa=PYb and the local M a c h number for secondary stream at section Y - Y is unity, M y Y b = 1 (see Figure 2.2). The area of the secondary stream, A y Y b >
ma
Y be varied and the entrainment ratio is then
determined from the temperatures and pressures of the primary and secondary streams. The calculation of the entrainment ratio is repeated for various values of the effective area ratios of secondary vapour choking area to the throat area of the motive nozzle, AYYt/At, and of mixing tube to primary nozzle throat area, A2/At. A performance m a p similar to the one developed by M u n d a y and Bagster for a steam ejector is shown in Figure 2.13. program is listed in Appendix C, Section C. 1
Here, the simulation conditions were as follows. The generator temperature, evaporator temperature, nozzle efficiency, diffuser efficiency and cooling capacity were 80.0°C, 0.0°C, 0.95, 0.80 and 1 k W , respectively. The mass flow rate for the secondary stream was computed from Equation (2.1). The pressure for the secondary vapour, was determined from Equation (2.11) and Equation (2.12) was used to calculate the effective area for the secondary stream. The entrainment ratio and ejector dimensions such as the throat diameter of the nozzle were computed using Equations (2.13) to (2.30) and Equations (2.34a) to (2.35). The critical condenser pressure was obtained from Equations (2.36) to (2.40). Based on the saturated conditions for the condenser, the critical condenser temperature was determined. Then, the effective area ratio, Ab/At, was calculated. Finally, from Ab/A t , the entrainment ratio was computed from Equation (2.34b) iteratively.
66
TWO
Theoretical Modelling
CHAPTER TWO
50 A2/At = 2.5 1.92
\
A2/
^=3
„ AYYb/At ^
\ 3.To
o
Entrainment Ratio
Figure 2.13 Entrainment Ratio Versus Condenser Temperature As Secondary Vapour Stream Choking Area is Varied Simulation Conditions: generator at 80°C, evaporator at 0°C, nozzle efficiency 0.95, diffuser efficiency 0.8
Figure 2.13 shows the calculated entrainment ratio as a function of the conden
saturation temperature with area ratio of the constant area mixing tube to thr area of the nozzle, A2/At, as a parameter for a particular set of operating conditions as the area ratio, AYYb/At, is varied.
67
Theoretical Modelling
CHAPTER
TWO
There are three distinct regions of ejector operation shown in Figure 2.13. In Region 1 the equations of Section 2.3 do not model the flows in the ejector. In Region 2, secondary choking does occur and section Y - Y is situated within the converging section of the ejector. A s one moves from the boundary between Regions 1 and 2 towards Region 3, so
AYYT/AI
decreases equivalent in a given
ejector to section Y - Y approaching the entrance of the constant area mixing tube. At the boundary between Regions 2 and 3 the secondary vapour reaches sonic velocity just at the entrance to the constant area mixing tube. In region 3 the entrainment ratio remains constant as the condenser temperature decreases below the critical condenser temperature which reflects the "constant capacity" characteristic of supersonic ejectors that has been k n o w n empirically for many years. In practice other phenomena will influence the form of the performance m a p shown above. Boundary layer growth, for example, and the possibility of separation in an unfavourable pressure gradient will significantly affect performance. Some advantages are to be seen using the secondary choking theory to model the ejector performance compared with the constant area analysis. For example, the secondary choking modelling gives insights into the performance of the supersonic jet ejector that are not available w h e n using simpler theoretical models. Using this analysis to model small V J R S ejectors, some limitations can be seen. For example, the model proposed by M u n d a y and Bagster is idealised in that mixing is not considered to occur before the plane Y-Y, while in reality the primary jet entrains secondary fluid throughout its development.
In the next section details of the interaction of the primary and secondary stre are described based on computational fluid dynamics techniques with a view to
68
Theoretical Modelling
CHAPTER
gaining a greater understanding of the flow and mixing processes within the ejector.
69
TWO
Theoretical Modelling
CHAPTER
TWO
2.7 Computational Fluid Dynamics (CFD) 2.7.1 General specification and objective
Analysis of incompressible and compressible jet mixing in converging diverging axisymmetric ducts has been carried out in the past literature (eg Hill, 1967, Hickman et al, 1972). Numerical solution techniques may be used for CFD
analysis for the processes such as heat transfer and fluid flow when it is expres in terms of differential equations.
Some researchers used the finite difference method to solve the partial different conservation equations for compressible flow in ejectors to investigate the mean velocity and pressure fields in the converging diverging mixing section (eg Hedges and Hill, 1974). Davis et al (1986) used the PHOENICS-code to the prediction of Mach number distribution within the flow field of an underexpanded axisymmetric sonic nozzle. The analysis of diffuser performance in jet pumps using computational fluid dynamics has been carried out by Neve (1993). In the present work attention has been focused on numerical simulation of the
flow field within the small, single fluid ejector to compare the CFD prediction o the flow field to that assumed in the secondary choking theory and also compare the CFD prediction of entrainment ratio to that predicted using one-dimensional, constant area analysis and the experimental results. To achieve these objectives the PHOENICS-code based on the finite volume technique was used to model the flow field.
70
Theoretical Modelling
CHAPTER TWO
2.7.2 Introduction to PHOENICS as a CFD package In recent years numerical methods for the calculation of turbulent threedimensional flow have been developed. The PHOENICS-code simulates fluid flow, heat transfer and chemical reactions. The simulations are based on finite volumetric techniques which can be conducted on various scales. These simulations are mathematical deductions from established physical principles. The theoretical basis of this computational technique is set out in publications Patankar (1980) and Shaw (1992). PHOENICS provides solutions to the discretized versions of sets of differential equations having the general form (CHAM, 1992):
J^fU+di^r.p.v.cp.-r.r^.grad^r.S^ (2.41) dlr.p.tp.) In the differential equation presented above, the functions of
ll l
',
at
r.pvcp., rTtpgradtp. and r.Scp. are represented as the transient, convection, diffusion and source terms, respectively. Where, t, represents time; r., stands fo volume fraction of phase i; p-, denotes density of phase i; ©., stands for any
conserved property of phase i, such as enthalpy, momentum per unit mass and turbulence energy; v., denotes the velocity vector of phase i; Tcp., represents t exchange coefficient of the entity (p in phase i and Sip.t stands for the source rate of ©.. The continuity equation for phase i may be gained by setting the conserved mass property to unity in the above equation. This equation is given as: ^pJ- + divlr.p.v.) = r.S. (2.42)
71
Theoretical Modelling
CHAPTER TWO
where S. stands for the mass inflow rate into the phase, per unit volume of space.
In the case of a single-phase process, the volume fraction r. is omitted from the equations, therefore the general governing equation and the continuity equation become respectively: ii^
+
div{pv(p-pr(pgrad(p)j = S (2.43)
and:
M+
div{pv) = 0
(2.43)
dt
W h e n several phases are present, the volume fractions follows the relation:
5>i = 1 (2-43) The above equations are the instantaneously-valid ones that PHOENICS solves
for laminar flows. The solution for turbulent flows may be necessary for practic applications such as the VJRS ejector. It is the time-mean behaviour of these
flows that is usually of practical interest. Thus, the equations are converted i the time-averaged equations for turbulent flow. This conversion may be carried out by an averaging operation in which it is assumed that there are rapid and random fluctuations about the mean value (Patankar, 1980). However, for turbulent flows, PHOENICS can solve the equations that are time averaged.
The conservation equations for a two-dimensional, steady and time averaged for compressible flow ejectors may be written as follows (details are described in reference Hedges and Hill, 1974).
--dU -TrdU dP ^ 1 d pU-r-+pV—- = -—- + —— y
dx
dy
dx
ya dy
72
JLy^-U\pV)ya dy
(2.44)
Theoretical Modelling
dT
PUQ dx
CHAPTER
-TT^dT
T-TdP Id
TWO
kyajj--T'(pV) yaCf
•,-.N2
1
+V
dy
{
-u>(pv)fdU"
(2.45)
\dy j
where, a is a constant equal to unity for axisymmetric flow and zero for plane two-dimensional flow, U, V, p, CP,k, T, and \i, are time averaged velocity in x-direction, velocity in y-direction, fluid density, specific heat at constant pressure, thermal conductivity, temperature and absolute viscosity, respectively 7
and U' and U'(pV)
are instantaneous fluctuating x components of velocity and
turbulent shear stress, respectively.
73
Theoretical Modelling
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TWO
2.8 Validation of Computational Analysis by PHOENICS Validation of any computational analysis is very important, because either computational or programmer errors m a y occur during the modelling of a problem. The present author checked the C F D results at several increasing levels of complexity. For example, a laminar and incompressible flow in a constant area tube was modelled using P H O E N I C S . The computational results for the pressure and velocityfieldswere checked with analytical solution. G o o d agreement has been found with the C F D results. Here a typical example of internal flow of a jet in finite space is chosen (Abramovich, 1963) to check the C F D prediction of the velocity field for the V J R S simulation. This example concerns an air ejector in a cylindrical mixing chamber with two coaxial jets. In this ejector the area ratio of the nozzle exit to the area of the secondary flow at the inlet cross section of mixing chamber, Aa/Ab, was 0.1. The velocity at the nozzle exit is 355.0 m/s and the velocity of the secondary stream at the inlet of the mixing chamber is 101.0 m/s.
Abramovich (1963) presented results of the velocity field for this situation based on a one-dimensional analysis and on experimental data at a cross section of the cylindrical mixing chamber 6.7 chamber diameters from the nozzle. H e assumed the working fluid (air) to be incompressible. The axial velocity versus the nondimensional radius, r/R, based on the analytical one-dimensional model of Abramovich (1963) is shown in Figure 2.14 as a dashed line, in which data were taken from his results. Also, the experimental data in Figure 2.14 were taken from Abramovich (1963). The solid lines of Figure 2.14 represent the numerical results
74
Theoretical Modelling
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TWO
of the present author using P H O E N I C S with constant properties of air and isentropic flow.
According to this CFD calculation, the result of the velocity field agreed well with the experimental results and analytical solution.
1.0-
Abramovich (Theory)
0.8-
g
0.6-
&
Isentropic
'Ji
e c
0.4
Incompressible
B Abramovich (Experiment) ^ 0.2-
0.0
i
80
100
120
140
180
Velocity [m/s]
Figure 2.14 Velocity Fields at a Cross Section of the Cylindrical Mixing Chamber (6.7 Chamber Diameters from the Nozzle)
75
Theoretical Modelling
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TWO
2.9 Modelling of the Supersonic Ejector by P H O E N I C S PHOENICS has four main subprograms, Satellite, Earth, Ground and Photon. The procedures of using this computer-code are summarised as follows. First, a Q l file should be created based on P H O E N I C S input language, PJL. Thisfilem a y be written using an interactive m e n u or direct input by the user to the Q lfile.Then through the Satellite subprogram the Q l file is processed to create the data input file, Eardat. Calculations are carried out by Earth resulting a Resultfileand a Phi file. Using the Phifileby P H O T O N generates the graphs such as the contour of the pressure, velocityfieldand M a c h number distribution. Several steps such as validation of computational results and building an appropriate grid-cells should be carried out to simulate the supersonic jet ejector. Validation of the computational results was mentioned in Section 2.8. T o generate an appropriate grid, the exact dimensions of one of the experimental small R 1 2 V J R S ejectors were used (in which the throat diameter of the nozzle was 2.4 m m ) . This ejector w a s built and tested in the present research (Ejector 2.a) was described in Section 3.7.1. For this case one domain patch was used to specify the grid and geometry. T o create an appropriate geometry and grid-cells, the parametric number of cells in the lines and frames were used to m a k e it easy to change the number of cells (see Group 6 of Q l file in Appendix A ) . Here, x, y and z axes were the flow direction, radius and angle for an axisymmetric domain in a body-fitted-coordinate system, B F C , respectively. To solve the dependent variables such as the velocity and pressure, for a whole supersonic ejector is very complicated and involves subsonic conditions at the inlet of the nozzle, the transonic and supersonic conditions during the expansion
76
Theoretical Modelling
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TWO
downstream the throat of the nozzle and turbulent mixing in the converging part of the ejector. Therefore, the present domain was selected to be from the cross section at the throat of the nozzle to the end of the constant area mixing tube. The following boundary conditions were fixed for this modelling.
(a) Velocity, density and enthalpy at the throat of the nozzle were calculated from one-dimensional, compressible flow theory, and used as the boundary conditions for the primary vapour. The formulae used for these calculations are presented in Group 1 of the Ql file (see Appendix A). (b) The stagnation pressure of the secondary vapour and enthalpy were fixed as the secondary inlet boundary conditions.
(c) Pressure at the end of the constant area mixing tube was prescribed as the outlet boundary condition. (d) The wall skin friction was taken into account for the walls (see Group 13 in the Ql file for wall conditions). Group 13 in the Ql file in Appendix A shows the boundary conditions.
77
Theoretical Modelling
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TWO
2.9.1 Computational procedures by P H O E N I C S
To model and solve the present problem, the calculations had to be performed in Body Fitted Coordinate, B F C system and take account of turbulence effects. P H O E N I C S - B F C code offers such possibilities, thus, it was used for these calculations. In the present simulation the following dependent variables were calculated, U l , V I , Pl, H I , K E and E P as the velocity in x-direction (axial), velocity in y-direction (radial), the static pressure, the enthalpy, turbulence kinetic energy and rate of dissipation, respectively. The laminar kinematic viscosity for R12 was calculated from dynamic viscosity and density (data used from Perry and Chilton, 1973). The turbulence parameters, K E and E P were computed by P H O E N I C S (details are presented in references such as Chen, 1990, Shaw, 1992 and C H A M , 1992). The density of the working fluid was calculated assuming the ideal gas treatment for the fluid (see Group 9 in Q lfilein Appendix A ) . M o r e details are expressed in Section 2.9.3. If both partial differential equations for the turbulence parameters, KE and EP solved, then this is k n o w n as a two-equation turbulence model (Shaw, 1992). P H O E N I C S - B F C solves the dependent variables such as the velocity, pressure and kinetic energy in a slab by slab iterative procedure to obtain solution of the flowfieldin the ejector.
78
Theoretical Modelling
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TWO
2.9.2 Running procedures
In the present simulation the following procedures were carried out to run the Ql file for each mesh generated. (a) First, assuming the flow is laminar, the program was run for 500 sweeps until the solution was reasonably converged. Therefore, at this step KE and EP were
not computed. This Phi file was saved in the local directory. For the grid of 3600 cells (Grid 3 in Table 2.2) the running time for 500 sweeps was about 10 minutes.
(b) Then, to save time, these data (such as the velocity fields and enthalpy from
the Phi file of the laminar case) were selected to restart calculations for turbul flow. The initial values of KE and EP were given for this step for about 20 sweeps and the Phi file was saved again in the local directory. This Phi file included all the variables which should be calculated for the present case. Therefore, then the Ql file was running by restarting all variables including KE and EP for several hundreds of sweeps (about 1500) until the solution converged.
79
Theoretical Modelling
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TWO
2.9.3 C F D results
Here a 586 PENTIUM-S computer with 32 Mb RAM was employed and version 2.1 of PHOENICS was installed for the present CFD analysis. In a supersonic ejector the flow should be considered as a compressible one. Therefore, in the
present study the files of "compressible flow corrections" were compiled to this version of PHOENICS.
The present author generated three different grids in which the number of cells
different (see Table 2.2). The ejector is an axisymmetric device, therefore, thi
model is a two-dimensional simulation. Therefore, only one cell width (a circula sector) was needed in the tangential direction (z-axis).
Mesh
No. of Cells No. of Cells No. of Cells Total No. of in x-Axis
in y-Axis
in z-Axis
Cells
Gridl
70
30
1
2100
Grid 2
70
40
1
2800
Grid 3
120
30
1
3600
Table 2.2 N u m b e r of Cells for the Grids
In the present simulation, Ejector 2.a used in Section 3.7.1 has been modelled f the experimental data from Figure 3.17. In this experiment, Tg, Te, and optimum nozzle position, X/D2 were equal to 73.3°C, 12.5°C and 1.07, respectively. The downstream static pressure at the end of the constant area mixing tube was 8.1xl05Pa.
80
Theoretical Modelling
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TWO
The present author developed the Ql file for the present CFD modelling which is presented in Appendix A. The formulae used to calculate the boundary conditions such as the velocity at the throat of the nozzle are shown in Group 1 of Ql file.
The finer grids may give greater accuracy of prediction. Thus, the present author compared the axial velocity for three different grid cells. Using the result files from PHOENICS for Grid 1, Grid 2 and Grid 3, which have 2100, 2800 and 3600
grid-cells, respectively (see Table 2.2), the velocity in the x-direction, Ui, vers the number of grid-cells for some points are shown in Figure 2.15. These points A, B, and C were positioned in the start, middle and end of the constant area mixing tube, respectively. The value of y (r) coordinate of the points were 0.3 mm (distance from the centerline of the tube). As shown in Figure 2.15, the axial velocity, Ui of the flow increases when the total number of grid-cells increases.
In the present work, the mesh of grid-cells for Grid 1, the velocity field, pressur distribution, and Mach number from the throat of the nozzle until the start of the constant area mixing tube are presented in Figures 2.16, 2.17, 2.18 and 2.19 (data were taken from Phi file for Grid 2), respectively.
81
CHAPTER TWO
Theoretical Modelling
220
2000
2500
3000
3500
4000
N u m b e r of Grid-Cells
Figure 2.15 Velocity, Ui, Against the N u m b e r of Grid-Cells
The result file of Grid 2 was produced after 1500 sweeps (see Group 15 in Q l file) and about 40 minutes running time was required. This resultfileis presented in Appendix A. In the present study, the mass flow rates for the secondary and primary streams were taken from the net source of mass flow rate. The P H O E N I C S result file gives these values as "net source of R l " and it is the actual mass flow in kg/s through each source. Then the ratio of the mass flow rates of the secondary to primary streams, the entrainment ratio, O , was computed. Here the predicted values of the entrainment ratio using one-dimensional, constant area analysis, C F D analysis and the experimental result have been compared and shown in
Table 2.3.
82
Theoretical Modelling
CHAPTER
Case
Entrainment Ratio
Experimental Result
0.234
ID Analysis
0.321
C F D Result for Grid 1 (Table 2.2)
0.212
C F D Result for Grid 2 (Table 2.2)
0.221
C F D Result for Grid 3 (Table 2.2)
0.251
Table 2.3 Comparisons of Entrainment Ratio for the Experimental Results, ID Analysis and CFD results
83
TWO
« B O
84
a
^^f^^.
±
it
lo m
umitutit.
mmlm fpuniiiinn
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lo n
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,... m i i r i r I I I , . . . 1 1 1 1 1 1
I I
f
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t
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/ /
_ . . . n > f f tt! t t I t M ttl'.T i 7 I /
]. miittrt t t i tt,.,iutHt ttfi „...tti>MH t t t ,,.,ititllfrt t r tt,.ittl?ftti i 1
'ttrirtn >
85
i
H
ca o H SI
s CJ CU
X lO 10
10 10
+ + + + H H H W
X fit
in
ft
r-
m
•o CU
.J
u CO
nH
ca * U
©
6?
.2 >> « S
Q fe
U s o •o
cu CO
es
S u J
o CJ
cu
6? CJ
"C cu
o H SI
B SI
E E c u co
e
30) u S cn co cu
cu 00
ri
86
cu u 3 CD
e CJ
CU u 3 M
87
Theoretical Modelling
CHAPTER
TWO
2.10 Discussion Some advantages can be seen using a CFD analysis. For example, modelling V I R S ejectors based on this analysis gives more details of flowfieldwithin the ejector which cannot be seen in one-dimensional modelling. The present author developed ejector simulation using P H O E N I C S - B F C code based on finite volumetric technique. It was mentioned in Section 2.6 that the secondary choking theory assumes that the supersonic primary stream expands laterally during its flow d o w n the converging section of the ejector and effectively provides a converging duct for the initially subsonic secondary vapour which must reach sonic velocity before mixing occurs, but this was not confirmed by this C F D analysis. A s shown in Figure 2.19, the present C F D modelling did not indicate that the secondary vapour obtained the sonic velocity, M =1, in the converging section of the ejector or in the constant area mixing tube. Validation of computational results was carried out by the present researcher at several increasing levels of complexity and it was agreed well with analytical solution. In this modelling one of the experimental small single fluid V J R S ejectors, Ejector 2.a was used (see Section 3.7.1). Using Grid 1 (2100 grid-cells) and Grid 2 (2800 grid-cells) it was found that the predicted entrainment ratios were below the experimental result, but for Grid 3 with 3600 grid-cells it was higher (see Table 2.3). The results m a y be explained as follows. It was mentioned that the present domain was selected to be from the cross section at the throat of the nozzle to the end of the constant area mixing tube, because to solve the independent variables such as the velocity for a whole
88
Theoretical Modelling
CHAPTER
TWO
ejector is very complicated. The results for Grid 2 and Grid 3 which are finer than Grid 1, may be more accurate, but finer grids involve larger storage requirements for results and the analysis of such a large body of numbers can sometimes cause the accumulation of errors. A recirculation of the working fluid can be seen in Figure 2.17. This recirculation could cause a restriction on entrainment of the secondary flow cause to decrease the entrainment ratio.
The present author compared the prediction of the entrainment ratio using onedimensional, constant area analysis with CFD results (see Table 2.3). It has been found that the predicted entrainment ratios using CFD analysis were below the one-dimensional result, because in one-dimensional analysis the friction factor was not taken into account but in the CFD modelling it was. Also, in this study only three different grid-cells were used, but further work is required to ensure that the grid really is refined sufficiently.
89
CHAPTER 3 EXPERIMENTAL WORK 3.1 Scope and Objectives of the Experiments All the present experiments were carried out on small R12 ejectors in a single fluid vapour jet refrigeration system. In these ejectors the throat diameter of the motive nozzles was about 2.0 m m and the diameter of the constant area mixing tube was about 4 m m .
3.1.1 Scope of experiments (a) The constant area analysis design The following experimental work has been carried out on the ejector designed by Kanashige (1992) based on the constant area analysis design method: • Replication of the previous experiments carried out by Kanashige (1992) to validate the results of his experiments. • Modification of testrigto extend the range of test conditions.
• Test on a new geometry of mixing chamber.
90
Experimental Work
CHAPTER
THREE
(b) T h e secondary choking theory
Experiments on the ejector which was designed by the present author using the theory of secondary choking were as follows. • A complete series of tests on the ejector to compare the results with the ejector 3.1.1(a) above.
• Test on a new geometry of the mixing chamber/motive nozzle.
3.1.2 Objectives The aim of this research was to investigate:
• Ejector choking Ejector choking was examined to find out how entrainment ratio is affected by the backpressure. • Nozzle position Entrainment ratio as a function of nozzle position as a geometric factor has been tested in the present work.
• Pressure Distribution The purpose of measurement of the actual pressure in some specified points in the ejector w a s to investigate the shock pressure recovery process in supersonic ejectors.
91
Experimental Work
.
CHAPTER
THREE
• Comparison of Results
This research was undertaken to find out if there are any improvements in ejector performance when it is designed using the secondary choking theory as compared with the ejector performance when designed using the constant area analysis method. In the present work experimental results were compared when the geometry of mixing chamber, including the converging half angle of the ejector was changed.
• Effective area hypothesis This study was undertaken to find out how the "effective area ratio" of the secondary stream to the throat area of the nozzle, Ab/At, (see Figure 2.12a), is affected by the evaporator temperature and nozzle position.
92
Experimental Work
CHAPTER THREE
3.2 Experimental Design 3.2.1 Ejector geometry
The most important parts of the ejector geometry are given in Figure 3.1a. The principal dimensions such as the throat diameter, D tj diameter of the constant area mixing tube, D 2 , the length of the constant area mixing tube, L2, and the diverging half angle of the supersonic nozzle, Gn, m a y be calculated with the one dimensional analysis and the help of past literature (eg. Dutton and Carroll, 1986). For example, L 2 is not determined by the system simulation, however, from the literature suitable values range from 7 to 12 times the diameter of the mixing tube (Deleo et al, 1962, Engel, 1963). For a uniform and parallel supersonic flow at the nozzle exit, some researchers have discussed the optimum nozzle contour (eg. Puckett et al, 1946). In the present ejector the dimensions are small, therefore, the contour is replaced by a straight line and the diverging half angle of the nozzle, 0 n was chosen to be 5 degrees. The divergence angle of the diffuser, 0
93
Experimental Work
CHAPTER THREE
(a)
Motive Nozzle
ss s / z Z 7^X,
Head Angle
Mixing Chamber
(b) Figure 3.1 Ejector Geometry and Schematic Diagram of Mixing Chamber
94
Experimental Work
CHAPTER
THREE
3.2.2 Experimental apparatus The experimental facilities are shown in a schematic diagram, Figure 3.2. This test rig has been used in the present work to examine the ejectors/nozzles which have been designed using the "one dimensional, constant area analysis" and the "theory of the secondary choking stream". Pressure transducers and thermocouples with a digital temperature indicator were used in the test rig to measure the practical value of absolute pressure, P, and temperatures, T, of the refrigerant at the points shown in Figure 3.2. Also, the arrangement of the main components, such as the ejector, generator, condenser, evaporator and refrigerant pump are illustrated in the figure. More details about the instrumentation are presented in Section 3.4.
95
Experimental Work
CHAPTER
THREE
ea ci <
e
I •c 8.
96
Experimental Work
CHAPTER
THREE
3.3 Initial Replication of Previous Experiments In an initial experimental program at the start of this thesis, Kanashige's (1992) experimental results were confirmed by the present researcher by replication of his experiments. These experiments were carried out on the same testrigwith the same working fluid (R12) as Kanashige's.
The results are summarised as follows:
• Motive nozzle position The position of the exit of the motive nozzle in relation to the entrance of mixing tube is very important. This is one of the geometric factors which can be optimised to provide the best ejector performance in the V J R S . The relation between the entrainment ratio and relative position of the motive nozzle exit to the entrance of the mixing tube based on the experimental results of the present author and Kanashige (1992) are presented in Figure 3.3.
97
Experimental Work
CHAPTER
0.2
o oi
THREE
Kanashige's data
Present data
4> 0.1
•a
0.0 0
1 2 3 Nozzle Position [X/D2]
Operation Conditions
Generator Temp. [°C]
Evaporator Temp. [°C]
Present
66.1
14
Kanashige
65.5
13.1
{ X = Position of the motive nozzle related to the entrance of the constant area mixing tube} Figure 3.3 Relation Between Entrainment Ratio and Nozzle Position
• Ejector choking T h e relationship between the entrainment ratio and condenser saturation temperature for the present experiments was compared with Kanashige's (1992) results and these are depicted in Figure 3.4. Kanashige's (1992) experiments did not find conclusive evidence of the ejector choking phenomenon in his test rig and this w a s also observed by the present author. This was because of insufficient heat rejection capacity of the cooling plant leading to a high m i n i m u m condenser temperature achievable. In these experiments the condenser temperature ranged from 3 to 6 ° C below the simulation value of the critical temperature. T h e
98
Experimental Work
CHAPTER THREE
following sections discuss modifications m a d e to the test rig, including the condenser, to extend the range of the operating conditions to observe the choking phenomenon in the small VJRS.
0.22
0.14
33
34
35
36
37
38
39
Condenser Temperature [°C]
Operation Conditions
Generator Temp. [°C]
Evaporator Temp. [°C1
Present
73.6
14.5
Kanashige
74.1
15
Figure 3.4 Relationship between Entrainment Ratio and Condenser Temperature
• Entrainment ratio The entrainment ratios based on the experimental and simulation results in the present and Kanashige's (1992) work are given in Table 3.1. This table compares the experimentalresultswith the simulation values based on the one-dimensional, constant area analysis method. A s shown in the table, the practical entrainment ratio is very low compared with the simulation value, because in these tests the ejector did not operate under choked conditions amongst other reasons.
99
CHAPTER THREE
Experimental Work
Replication of the previous experiments by the present author confirmed the validity of the experimental work carried out by Kanashige (1992). Given many factors such as performance conditions and ambient temperature, a reasonable agreement between results is seen.
Case
Experimental Results
Simulation Value
O
TCT [°C]
4>
31.0
0.099
32.8
0.305
5.3
31.3
0.093
32.1
0.312
70.1
5.6
32.0
0.108
35.2
0.277
K
70.0
5.2
32.7
0.112 35.0
0.272
P
73.6
14.2
35.0
0.191
39.5
0.364
K
74
15.1
35.7
0.200 40.0
0.376
Tg [°C]
Te
Tc
[°C]
[°C]
P
66.2
5.4
K
65.2
P
Table 3.1 T h e Experimental Data Compared with the Simulation Results $ = Entrainment Ratio; T g = Generator Temperature; T e = Evaporator Temperature; T c = Condenser Temperature; P = Present study and K = Kanashige's results
100
Experimental Work
CHAPTER THREE
3.4 Instrumentation and Modification of Test Rig to Extend the Range of the Test Conditions Improvements to the previous vapour jet refrigeration system needed to be carried out. A s mentioned in Section 3.3, in Kanashige's work (1992) and in the present experiments, fully developed choking was not observed. Therefore, in those experiments the m a x i m u m entrainment ratio in practice was not tested. In Kanashige's (1992) ejector, the constant area mixing tube was m a d e of several hollow cylinders, but in the present rig these were replaced by a single cylinder and well machined to reduce friction and boundary layer effects. Three Alfa Laval plate heat exchangers in which the capacity of heat transfer for each is 5 k W , were installed for the condenser heat rejection and the generator heat absorption. The flow meter that was used for measuring the mass flow rate for the primary vapour (Fischer and Porter, Model 10A3567S/B10-STK) had the following specifications. Fittings were brass with buna O-Rings, tube was borosilicate glass and rating of temperature and absolute pressure were 120°C and 2200 kPa, respectively. The accuracy of the flow meter was ± 2 % of full scale reading. The flow meter of Fischer and Porter (Serial number, 1355-24-mm-SS-Y) was used for measuring the mass flow rate for the secondary stream. This is built for operation at temperatures to 120°C and absolute pressure to 1500 kPa. Based on manufacturer's information the accuracy of the flow meter was ± 5 % of full scale reading. In the present work the instrument error and uncertainty of entrainment ratio was calculated based on the accuracy of the flow meters for the primary and secondary flows. It has been found that this value for entrainment ratio was ± 7 % .
101
Experimental Work
CHAPTER
THREE
Practically, during the ejector operation, shocks always occur somewhere in the ejector. T h e shock process in the ejector has been investigated by some researchers in the past literature. For example Huang et al (1985), mentioned that shocks m a y appear in a constant area mixing tube of a supersonic ejector. Modifications were m a d e to the testrigso that the pressure distribution along the length of the ejector could also be measured to investigate the shock pressure recovery process for a small single fluid V J R S . A s shown in Figures 3.2 and 3.5.b, at three points along the constant area mixing tube, one point in the converging part of the ejector and one point at the end of the subsonic diffuser the actual pressure distribution in some experiments were measured by means of 3.0 m m diameter static pressure ports drilled through the body of the ejector. The series 2000 pressure transducers were used (part N o . 558140-0032), which provided 100 m V for full scale applied pressure, with 10 volts d.c. applied. These transducers were fitted with a hybrid amplifier providing various optional voltage outputs. The actual pressure ranges were compatible with the m a x i m u m pressure being measured by transducers. Based on the manufacturer's information, accuracy of the pressure transducers were ± 0 . 2 5 % of reading the scale. To examine the fully developed choked flow, the condenser temperature must be decreased below its practical critical temperature. Thus, an auxiliary air-cooled refrigeration system was constructed for the present test rig to provide chilled water for the condenser. Before installation, the heat transfer to the cooling coils immeresed in the circulation water tank from the water (the cooling water for the condenser) w a s tested to ensure the appropriate heat absorption from the condenser. T h e cooling capacity of the compressor w h e n the temperature of chilled water for the condenser was 24, 19 and 12°C were 9.2, 8.2 and 7.8 k W ,
102
Experimental Work
CHAPTER THREE
respectively. The operating temperature of the VJRS condenser could then be controlled by varying the mass flow rate and/or temperature of the chilled water.
The refrigerant pump used for the previous test rig was a vane pump and had some problems in practice, eg this p u m p experienced breakdown two times due to the rupture of a mechanical seal under the high pressure operating when the absolute pressure of the generator was more than 1800 kPa. The present ejectors were designed for a generator temperature of 80°C at which the absolute pressure of R 1 2 is 2304 kPa. Therefore, this p u m p was not safe for some experiments. T o avoid these problems a diaphragm cushion p u m p was installed in the new test rig. This p u m p was a Hydra-cell industrial p u m p (Model D 0 3 X R B T H H C ) . The p u m p body was m a d e of brass with buna O-Rings which R 1 2 does not have any effect on. According to the manufacturer's information this p u m p m a y be applied for pumping of hot fluids and chemicals. It has the characteristics such as quiet and smooth operation, running dry without damage and easy maintenance. This p u m p can operate at temperatures and pressures to 71°C and 6900 kPa, respectively. The old evaporator was the large thermal mass system. Therefore, there was a poor control of evaporator conditions. The vapour leaving the evaporator was usually superheated in practice. The effects of superheating vapour leaving the evaporator and generator on ejector performance have been discussed in Chapter 2. T o avoid these effects, a n e w evaporator (heat elements) was installed. Then, the evaporator temperature was controlled by a variable voltage auto transformer ( V A R I A C ) . Through variation of the current, the amount of heat absorbed by the evaporator was controlled very finely and the evaporator temperature could then be set to any desired value. 50 H z a.c. power was used and current was measured to find the heat input the evaporator. The refrigerant p u m p was able to vary the
103
Experimental Work
CHAPTER
THREE
pressure of the generator, Pg, which provides the high pressure motive vapour. Also the generator temperature, T g , w a s measured in the system. Thus, the saturation conditions for primary stream were obtained by checking the generator pressure and temperature. According to the manufacturer's design the accuracy of temperature indicator was ±0.2°C. The general view of the present testrig,external appearance of the ejector and the refrigerant p u m p are shown in Figures 3.5a, 3.5b and 3.5c, respectively. Also, one typical of the calibration graphs of the pressure transducers is presented in Figure B.5 in Appendix B. The thermocouples were not calibrated, because it was used srightly after Kanashige's (1992) experiments.
104
Experimental Work
CHAPTER THREE
Figure 3.5a General View of the Test Rig
105
Experimental Work
CHAPTER THREE
Figure 3.5b External Appearance of the Supersonic Ejector
106
CHAPTER THREE
Experimental Work
Figure 3.5c Refrigerant P u m p
107
Experimental Work
CHAPTER THREE
3.5 Ejector Used in the Present Experimental Work Two small R12 ejectors based on one-dimensional compressible fluid flow theory have been designed for the present study. Kanashige (1992) designed a supersonic ejector using constant area analysis method and the present author designed it based on secondary choking theory. The ejector dimensions are presented in Table 3.2a.
Number 1 2
Analysis
Dt
[mm] Constant 2.0 Area Secondary 2.4 Choking
D2 [mm]
D3
3.5
[mm] 11.0
L2 [mm] 42.0
4.2
13.0
42.0
Table 3.2a Ejector Dimensions Used in T h e Present Experiments
The present author, also designed and built two mixing chambers with converging half angles, 0 m , of 20° and 10° for both Ejectors 1 and 2 (Table 3.2a). These small R 1 2 ejectors were used in the present study are classified in Table 3.2b.
Number
Converging Half Angle of the Ejector, dm
Eiector La
20°
Ejector Lb
10°
Ejector 2.a
20°
Ejector 2.b
10°
Table 3.2b Ejectors Used for the Present Study
108
Experimental Work
CHAPTER
THREE
3.6 Experimental Work on Small Ejector using Constant Area Analysis Design in the Modified Test Rig This experimental investigation has been carried out on the modified VJRS test rig. A s mentioned in Section 3.4, in the modified testrig,the range of operation conditions was extended by installation of the n e w chilled water refrigeration unit and heat exchangers. T h e procedure for this practical work is summarised as follows.
(a) Before starting the test rig, the temperature of the auxiliary hot water tank wa fixed to provide the desired operating generator temperature. (b) Then, the refrigerant pump and the cooling water pump for the condenser were activated. T h e refrigerant p u m p was driven gently to avoid any possible damage. (c) A t the beginning of the V J R S operation, the m a x i m u m mass flow rate of the cooling water was maintained to reject sufficient heat from the condenser and therefore, satisfy the choking conditions for the ejector. The system was then operated at least for 10 minutes to obtain stable conditions. (d) Next, by adjusting the mass flow rate of the evaporator the operating evaporator temperature was fixed to be the intended temperature and waiting for several minutes to obtain steady state conditions. (e) At the next step the nozzle position was changed in increments of 0.5 mm to find the optimum nozzle position and it was then held at this position for at least 6 minutes. Then, the evaporator temperature was checked. If this temperature did not match the desired value for the performance conditions, as mentioned in
109
Experimental Work
CHAPTER
THREE
above (d), the mass flow rate of the secondary vapour was adjusted to obtain the desired evaporator temperature. Then the scales of the secondary and primary flow meters were recorded to measure the mass flow rate. The other performance conditions such as the condenser and generator temperatures were also recorded. This process was repeated for a number of nozzle positions. The optimum nozzle position was determined based on the m a x i m u m value of the entrainment ratio. (f) To observe ejector choking, the procedure was as follows. Firsdy, the optimum nozzle position was adjusted. Secondly, the cooling water for the condenser heat rejection was fixed at m a x i m u m and the test rig was operated without any changes for about 7 minutes to obtain system stability. Then the experimental data such as the scales of the flow meters, temperatures, dc. voltages of pressure transducers and the value of pressures from the pressure gauges were recorded. Next, the mass flow rate of the cooling water was decreased gradually by a control valve which was installed for this purpose. After at least 6 minutes the evaporator temperature was checked and the mass flow rate of the evaporator was fixed until the desired evaporator temperature was obtained. Then, all data including the condenser temperature was recorded. This process was replicated until the entrainment ratio became near zero. Between 2 to 3 hours were required for each experiment. This section presents the experimental results for the small ejector, which was designed by Kanashige (1992) based on the "constant area analysis method". Dimensions of this ejector are shown in Table 3.2a. R 1 2 was the working fluid for all experimental and analytical work in this thesis.
110
Experimental Work
CHAPTER THREE
3.6.1 Experimental results for the ejector with 20° converging half angle (Ejector L a ) In this particular ejector, Ejector La, the converging half angle of the ejector, 0
m
(see Figure 3.1), was 20 degrees. The results of the experiments, which have been carried out by the presentresearcherare as follows.
• Ejector choking The choking phenomenon of the ejector can be clearly observed in Figures 3.6, 3.7 and 3.8. These figures show that the m a x i m u m practical value of ejector performance has been achieved in the experiments. In the other words, fully developed choking was observed. In these figures entrainment ratio is depicted versus the inlet condenser temperature instead of the backpressure. The practical value of the critical condenser temperature as well as the m a x i m u m entrainment ratio in the experiments and C O P are given in Table 3.3. More details are discussed on pages 114-115 under system performance.
10
20
30
40
Condenser Temperature [°C]
Figure 3.6 Entrainment Ratio Versus the Condenser Temperature {Ejector 1 .a, Evaporator Temperature, T e = 5.0°C}
111
Experimental Work
CHAPTER THREE
0.3 O
Tg=73°C
••a
&
g=76.7°C
Tg=
« 0.2-
e
I c
•a £
0.1 H
u
0.0 20
1
1
1
30
1
1
40
50
Condenser Temperature [°C]
Figure 3.7 Entrainment Ratio Versus the Condenser Temperature {Ejector La, Evaporator Temperature, T e = 7.5°C}
0.15
&
Jg=73°C
0.10
e v
E a '3 § 0.05 W o.oo H — • — i — • — i — > — i — ' — i — • — 24 26 28 30 32 34
Condenser Temperature [°C]
Figure 3.8 Entrainment Ratio Versus the Condenser Temperature {Ejector La, Evaporator Temperature, Te = 0.0°C}
112
Experimental Work
CHAPTER THREE
• Motive nozzle position
The experimental apparatus was designed so that it was possible to change the position of the motive nozzle exit in relation to the entrance to the constant area mixing tube. The current experiments show that the axial location of the nozzle relative to the entrance to the constant area mixing tube is a very important parameter of ejector geometry. For example, in a fully choking ejector, the nozzle position must be optimised to test for the m a x i m u m ejector performance. The effect of the nozzle position on entrainment ratio can be seen in Figures 3.9 and 3.10. All data for thisfigurewas collected w h e n the ejector was operating with the condenser below the critical condenser temperature (see Chapter 6 for discussion). 0.14-j 0.12-
o "•3 ed
0.10-
asa £ a "3
0.08-
Uu
e
0.060.04•
0.02 0.00-
0
1
2
3
4
Nozzle Position [X/D2]
Figure 3.9 Relation Between Entrainment Ratio and Nozzle Position {Ejector La, T g = 74.5±0.3°C, T e = 3.5±0.1°C}
113
Experimental Work
CHAPTER THREE
0.18
3 ed
e B e '3
0.16-
0.14-
0.12
u C U
0.10 2
3
4
Nozzle Position [X/D2]
Figure 3.10 Relation Between Entrainment Ratio and Nozzle Position {Ejector La, T g = 76.7±0.2°C, T e = 7.5±0.2°C
• System performance The experimental and simulation results based on the one-dimensional, constant area analysis of the system are given in Table 3.3. This experimental investigation shows that the practical entrainment ratio lies between 65 to 75 percent of the simulation value. In these experiments the ejector operated under choked conditions. The actual critical condenser temperatures, ranged between 4.7 °C to 7.3°C below their simulated values. The experimental results are compared with the theoretical system performance m a p in Figures 3.11a and 3.11b. The theoretical values of entrainment ratio in the performance m a p (Figure 3.11a) are based on the simulation results of onedimensional constant area analysis method which was carried out by Kanashige
114
Experimental Work
CHAPTER THREE
(1992). As shown in these figures, ejector performance experimentally and theoretically is a function of the generator and evaporator temperature. For example, it is seen, that when the generator temperature increases while the evaporator temperature remains constant, the entrainment ratio decreases.
Case
Experimental Results Te
Tcr
COP
Simulation Value
O
COP
Qe
T CT
rwi
[°C]
0.112 0.092
323
39.2
0.170 0.143
0.134 0.111
331
38.1
0.177
76.7
5.0 33.5 0.153 0.128
371
39.1
0.227 0.193
4
76.7
7.5
33.8
0.176
0.147
442
39.6
0.255 0.219
5
73.0
0.0
28.9
0.148
0.122
381
35.8
0.194 0.161
6
73.0
5.0 29.5 0.181 0.152
430
36.8
0.249 6.214
7
73.0
7.5
30.5
0.193
0.164
470
37.6
0.280 6.243
8
73.0
12.5
32.0
0.223
0.183
554
38.6
0.328 0.278
9
66.5
5.0 27.7 0.192 0.159
465
32.9
0.295 0.246
10
66.5
7.5
28.4
0.235
0.197
578
33.5
0.334 0.287
11
66.5
12.5
29.8
0.266
0.220
680
34.8
0.404 0.337
Tg [°C]
[°C] [°C]
1
78.6
0.0
33.1
2
76.7
0.0
32.7
3
0.150
Table 3.3 Comparison of the Experimental Data and Simulation Results {Ejector La} fl> = Entrainment Ratio; T g = Generator Temperature; T e = Evaporator Temperature; T CT = Critical Condenser Temperature {Ejector La, A^At = 3.0, D t = 2.0 m m and D 2 = 3.5 m m }
115
Experimental Work
0.5
CHAPTER THREE
Tg=60°C
0.4Tg=70 °C
Tg=80°C •3
0.3
ed
. Te=10.°C
*J
C
E c '3 2 c
<
Te=5.°C
0.2 Te=0.0°C
w
- Te=-5°C 0.1-
0.0 20
-r40 40
30
50
Critical Condenser Temperature [°C]
Figure 3.11a Theoretical Performance M a p {Ejector La, A2/At = 3.0, Dt = 2.0 mm and D2 = 3.5 mm}
116
Experimental Work
CHAPTER THREE
Te(expt) = 12.5°C
e(expt) = 7.5°C Te(expt) = 5.0°C
a a
Te(expt)=0.0oC
B a
•a u
Critical Condenser Temperature [°C]
Figure 3.11b Experimental Performance M a p {Ejector La, A2/At = 3.0, Dt = 2.0 mm and D2 = 3.5 mm}
117
Experimental Work
CHAPTER
THREE
• Pressure distribution
The pressure recovery in a supersonic ejector can occur in a number of places by the presence of a shock, eg a shock may be located in the constant area mixing tube. The pressure distribution along the ejector has been measured in the present work to investigate the shock pressure recovery process in the ejector. The pressure distribution along the ejector is shown in Figure 3.12 for a number of experiments. As shown in the figure the shock moves upstream as the backpressure increases. Further details and comparisons of the experimental results of shock pressure recovery with one dimensional analysis are presented in Section 3.9.
118
V)
u ou oo
o oo (N
CM
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Ho CJ es ti cu
Experimental Work
CHAPTER
THREE
3.6.2 Experimental results for the ejector with 10° converging half angle (Ejector l.b)
A second mixing chamber was designed and tested by the present researcher. In this ejector, Ejector l.b, the converging half angle of the ejector, 0 m , was 10 degrees. The ejector choking as well as optimum nozzle position and pressure distribution have been tested with this mixing chamber (optimum nozzle position was not the same at different operation conditions). Comparisons of the practical results between the present mixing chamber and the one used in Section 3.6.1 are given in the next section.
• Ejector choking Fully developed choking has also been examined in this ejector. Entrainment ratio versus the condenser inlet temperature, instead of the backpressure is plotted in Figure 3.13. The ejector was operated with the nozzle adjusted to the optimum position at the operation conditions to observe the m a x i m u m performance of the ejector. A s shown in thisfigure,w h e n the condenser operates below the critical temperature, the entrainment ratio is not significantly affected by the backpresure and it remains approximately constant.
120
Experimental Work
0.20 - - ^ — ^
CHAPTER THREE
. -i
'•3 > Qtf
\
S \ cu g 0.10-
\ \
s '3
\ \
is e
*
ti
o.oo H—i—i—•—!—i—i—i—i—i— 28 30 Condenser Temperature [°C]
32
34
36
38
Figure 3.13 Entrainment Ratio as a Function of Condenser Temperature {Ejector l.b, Tg = 73.3±0.3°C, Te = 7.5±0.2°C and TCT = 31.5°C} • Motive nozzle position
The ejector performance is significantiy affected by the ejector geometry, such as
nozzle position and converging section of the ejector. This work tested the effect of the nozzle position on entrainment ratio for the ejector with t9m=10°. The
relationship between the entrainment ratio and relative position of the nozzle exi
to the entrance of the mixing tube, found experimentally, is given in Figure 3.14. Compared with the experimental results in Section 3.6.1 (Figure 3.10), under the same operation conditions, it is seen that the optimum nozzle position is changed and that the position is less critical.
121
Experimental Work
CHAPTER THREE
o ••a
ed
X fi
cu
E B ed fi ti •*-»
Nozzle Position [X/D2]
Figure 3.14 Relation Between Entrainment Ratio and Nozzle Position {Ejector l.b, Tg = 76.9±0.3°C, Te = 7.5±0.2°C}
Pressure distribution The pressure change along the ejector can be seen in Figure 3.15 for a number of experiments and the shock process may be seen to occur in the constant area
passage. In this case of operation, the critical condenser pressure, Pcr, was equ
to 1009 kPa, and the critical condenser temperature was 31.5°C. The ejector, als was working with the optimum position of nozzle at all times.
122
Experimental Work
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-Tc=35.8°C .Tcr=31.5°C Tc=28.1°C
-i
0
>
r
•
1
'
r
20 40 60 80 100 Distance F r o m Nozzle Exit [ m m ]
Figure 3.15 Pressure Distribution Along the Supersonic Ejector {Ejector l.b, Tc = Condenser Temperature[°C], TCT= Critical Condenser Temperature[°C], Tg = 73.3±0.3°C, Te = 7.7±*).20C}
123
Experimental Work
CHAPTER
THREE
3.6.3 Comparison of the results for Ejectors La and l.b
These experiments were conducted to see whether the ejector performance is a strong function of the ejector geometry such as the converging half angle of the ejector, 0m. Comparisons of the results for 0m=lO° and 20° are given in Figure 3.16 and Table 3.4. Only a slight effect on the ejector performance may be seen due to the geometry of the converging section. The present work shows that the ejector with the angle of 0m=lO° performs better than the one with 0m=2O°, by between 1 to 4 percent. As given in Table 3.4, under the same operation conditions the optimum nozzle position, in which the entrainment ratio is maximum, has been changed by changing the geometry of the converging section.
For example, the optimum position of the nozzle exit in relation to the entrance o the constant area mixing tube and the area available for the secondary flow around the nozzle exit, in Case 1 (0m=2O°), are 2.286 times of D2 and 17.3 times of the throat area of the nozzle, respectively. In Case 2 (0m=lO°), they are 3.286
times of D2 and 9.6 times of the throat area of the nozzle, respectively. These tw cases have the same operation conditions. Comparing Figures 3.10 and 3.14 (with the same operation conditions of Ejectors La and l.b), the smaller converging half angle of the ejector, gives a flat entrainment ratio versus the nozzle position. Therefore, the advantage of an ejector with a smaller 0m is that it may operate well over a wider range of operating conditions. In contrast, for the smaller 0m the possible blocking of the secondary vapour by the nozzle body should be considered.
124
Experimental Work
CHAPTER THREE
0.30
0.25- Tg=73.3°C, Conv. Angle=10°
£
0.20
ed
| 0,5
Tg= Conv. Angle=20
e '3 u E
0 10
ti
0.05-
0.00 H
26
1 — i
28
1
1 — • — I
30
' — I — ' — I — • "
32
34
36
38
Condenser Temperature [°C]
Figure 3.16 Plot of Entrainment Ratio Versus the Condenser Temperature {Ejectors La and l.b, Tg = (73.3±0.3, 73.0±0.3)°C, Te = 7.5±0.2°C}
125
Experimental Work
CHAPTER THREE
Case
Experimental Results Tg [°C]
Te Tcr [°C] [°C]
O
Simujajtion Value N.P.
[X/D2] 0.176 2.286
Tcr
i
[°C]
l(0m=2O°) 76.7
7.5
33.8
2(0m=lO°) 76.9
7.5
34.0 0.177
3.286
39.5 0.252
3(0m=2O°) 73.0
7.5
30.5
0.193
2.286
37.6 0.280
4(0m=lO°) 73.3
7.5
31.5
0.198
3.000
37.5" 0.277
5(0m=2O°) 66.5
9.0
29.3
0.229
2.000
34.0 0.355
6(0m=lO°) 66.2
9.2
29.5
0.240
3.143
33.9 0.360
39.6 0.255
Table 3.4 Comparison of the Experimental Data and Simulation Results for Ejectors La and l.b { = Entrainment Ratio, N.P. = Optimum Nozzle Position}
126
Experimental Work
CHAPTER
THREE
3.7 Experimental Tests on the Small Ejector Design, Based on the Secondary Choking Theory in the Modified Test Rig In these experiments, further development by the present writer has been carried out to examine another small ejector. This section presents results of the experimental investigation on this ejector which has been designed by the present author based on the secondary choking analysis when R 1 2 is the working fluid. Dimensions of this ejector such as the throat diameter of the nozzle are shown in Table 3.2a.
3.7.1 Experimental results for the ejector with 20° converging half angle (Ejector 2.a)
The same experimental work such as maximum of practical entrainment ratio which has been tested for Ejector L a was examined for this particular supersonic ejector (Ejector 2.a). For this purpose, the ejector operated under circumstances of the fully developed choking conditions and optimum nozzle position.
• Ejector choking
The results of examining the maximum entrainment ratio can be seen in Figure 3.17. Entrainment ratio is plotted in the figure versus the condenser temperature, instead of backpressure. In these experiments the critical condenser temperature ranges from 6 to 7°C below its simulation values.
127
CHAPTER THREE
Experimental Work
-Tg=66.5°C Tg=73.0°C ©
•a ed +•>
fi
I fi
'3 u 4->
fi ti
Condenser Temperature [°C]
Figure 3.17 Plot of Entrainment Ratio versus the Condenser Temperature {Ejector 2.a, Tg = 73±0.2°C, 66.5±0.3°C, Te = 12.5±0.2°C} • Motive nozzle position The entrainment ratio has been tested against nozzle position, when the ejector operated under fully developed choking conditions. The entrainment ratio is
plotted versus the nozzle position in Figures 3.18 and Figure B.l in Appendix B.
128
Experimental Work
CHAPTER THREE
0.24-
o ••c
ed fi cu
0.220.200.18-
E
0.16-
ed c
0.14-
«*•>
fi ti
0.120.10-
••
1
•
1
1
'
2
i
3
Nozzle Position [X/D21 Figure 3.18 Relation Between Entrainment Ratio and Nozzle Position {Ejector 2.a, T g = 73.0±0.3°C, T e = 12.5±0.2°C}
• System performance The entrainment ratio obtained from the experimental and the simulation results are given in Table 3.5. The experimental results show that the practical entrainment ratio lies between 66 to 78 percent of the simulation value. The ratio of the experimental entrainment ratio to the simulation values are given in Figure 3.19. In these experiments the ejector operated at choking conditions. A s shown in the table the practical value of the critical condenser temperatures, ranges between 4.9°C to 6.9°C below their simulated values based on the constant area analysis of the ejector.
129
Experimental Work
CHAPTER THREE
Case
Experimental Results
Simulation Value
Te
TCT
COP
Qe [W]
TCT [°C]
COP
[°(?1 [°C] [°C] 1
66.5
5.0
28.0
0.175
0.144
487
32.9
0.266
0.220
2
66.5
12.5
29.3
0.282
0.235
709
35.0
0.382
0.323
3
70
12.5
30.1
0.256
0.210
681
37.0
0.347
6.294
4
73
5.0
29.9
0.173
0.146
421
36.9
0.222
6.191
5
73
7.5
31.2
0.199
0.164
499
37.3
0.260 6.216
6
73
12.5
32.0
0.234
0.192
589
38.8
0.321
0.275
7
77.2
12.5
34.1
0.209
0.176
601
41.0
0.296
0.252
8
79.0
0.0
33.1
0.100
0.085
322
39.6
0.145
0.126
Table 3.5 Experimental Data Compared with the Simulation Results for Ejector 2.a
1.0-
o
Tg = 66.5°C
•o
0.9-
ed tf *-> fi cu
0.8-
Tg = 73°C 0.70.6-
E
0.5-
fi ed
0.4-
h* •M
fi ti CM
O O '•3
ed tf
0.30.20.1 -
0.0
-r -«—r -*—r 12 14 10 8 6 Evaporator Temperature [°C]
Figure 3.19 Ratio of the Practical Entrainment Ratio to the Simulation Value {Ejector 2.a}
130
Experimental Work
CHAPTER THREE
• Pressure distribution
Pressure distribution along this ejector is plotted in Figure 3.20. It can be seen the figure that the backpressure increases w h e n the condenser temperature increases and the shock pressure recovery m a y occur between points A and B. In this experiment the practical value of the condenser temperature, T c r , was 32.0°C and when the backpressure is 955 kPa, the entrainment ratio was zero.
Jc=36.1°C Tcr=32°C Tc=27.5°C
0
20
40
60
80
100
Distance From Start of Mixing Tube [ m m ]
Figure 3.20 Pressure Distribution Along the Supersonic Ejector {Ejector 2.a, T c = Condenser Temperature[°C]; T C T = Critical Condenser Temperature[°C], T g = 73.3±0.3°C, T e = 12.5±0.2°C}
131
Experimental Work
CHAPTER THREE
3.7.2 Experimental results for the ejector with 10° converging half angle (Ejector 2.b) Another geometry of the mixing chamber has been designed to test the ejector performance. In this ejector the converging half angle, 0m, is selected as 10 degrees. • Ejector choking Entrainment ratio versus the condenser temperature, Tc, instead of the backpressure is given in Figure 3.21. As shown in the figure, the maximum ejector performance has been tested in this ejector.
o c cu
E E
"3
a E
u
24
26
28
30
32
34
Condenser Temperature [°C]
Figure 3.21 Plot of Entrainment Ratio Against the Condenser Temperature {Ejector 2.b,Tg = 66.5±0.3°C}
132
Experimental Work
CHAPTER
THREE
• Nozzle Position
In the experiment, the ejector was operating at choking conditions. The entrainment ratio is plotted against the position of the nozzle related to the entrance of the mixing tube in Figure 3.22 and Figures B.2 and B.3 in Appendix B. 0.18-] 1
0.17- ^^^\. •°l 0.16- / ^s^ * 0.15- * § 0.14-
E C 'ed b
0.130.12-
fi
W 0.110.10-1—i—T 1 1 1 1 1 1 1 1 1—
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Nozzle Position [X/D2]
Figure 3.22 Relation Between Entrainment Ratio and Nozzle Position {Ejector 2.b, Tg = 76.7±0.3°C, Te = 7.5±0.1°C}
• Pressure Distribution
The shock pressure recovery has been observed in the constant area mixing tube of this ejector (see Figure B.4 in Appendix B). In this case of operation, the critical condenser pressure, Per, is about 919 kPa, and the practical value of critical condenser temperature is 28.1°C. This pressure distribution has been tested when the ejector was working in the optimum position of nozzle.
133
Experimental Work
CHAPTER THREE
3.7.3 Comparison of the results for Ejectors 2.a and 2.b
Comparisons of the results are shown in Table 3.6 and Figure 3.22a. Again in
these experiments only a slight effect on entrainment ratio may be seen because of the change in geometry of the converging section. The present work shows the small ejector with the angle of 0m=lO°, performs almost the same as one with 0m=2O°. Under the same operation conditions the optimum nozzle position, in which the entrainment ratio is maximum, is changed by changing the converging half angle of the ejector (see Table 3.6). By way of example, the optimum
position and the area available for the secondary stream around the nozzle exit, i Case 3 (0m=2O°), are 1.3 times of D2 and 12.1 times of the throat area of the nozzle, respectively, but in Case 4 (0m=lO°), they are 1.9 times of D2 and 7.4
times of the throat area of the nozzle, respectively. These two cases have the sam operation conditions.
Again, it may be seen that the smaller converging half angle of the ejector, gives
flat entrainment ratio versus the nozzle position (see Figures 3.18 and 3.22) and may operate well over a wider range of operating conditions.
134
Experimental Work
CHAPTER THREE
Experimental Results
Case Tg [°C]
Te TCT [°C] [°C]
O
Simulation Value N.P.
[xm2]
TCT [°C] ii
,,.,
l(0m=2O°) 66.5
5.0
28.0 0.175
1.071
32.9 0.266
2(0m=lO°) 66.5
5.0
28.2
0.181
1.667
32.9 0.266
3(0m=2O°) 66.5
12.5
29.3
0.282
1.310
35.0
6.382
4(0m=lO°) 66.6
12.5
29.0 0.283
1.905
35.0
6.38 i
5(0m=2O°) 73.0
12.5
32.0 0.234
1.071
38.8 0.321
11
ii.,
II.,,
6(0m=lO°) 73.0 12.5 32.3 0.236 2.024 38.8 0.321 $.6 Comparison of tlleExp erimenital Data and Simulationi Result Ejectors 2.a and 2.b { O = Entrainment Ratio, N.P. = Optimum Nozzle Position}
0.3
Conv. Angle=20°
Condenser Temperature [°C] Figure 3.22a Entrainment Ratio Against the Condenser Temperature Performance Conditions: Tg = 66.5±0.4°C, T e = 12.5±0.2°C {Ejectors 2.a and 2.b (see Sections 3.7.1 and 3.7.2)}
135
Experimental Work
CHAPTER
THREE
3.8 Experimental Investigation of Effective Area Ratio, Ab/At If the secondary stream choking model of Munday and Bagster (1977) is a realistic model of h o w ejectors operate, then the effective area ratio, Ah/At, has an important bearing on h o w the ejector operates in practice. T h e present experimental results have been used to show h o w the secondary fluid effective area is influenced by the evaporator temperature and nozzle position. In this work, the practical entrainment ratio used was that obtained at choking conditions of the ejector. T h e theoretical entrainment ratio w a s then computed by changing the value of the effective area ratio, Ab/At, until the calculated values of entrainment ratio and the experimental results of that were matched. Hence, the effective area ratios for the experiments were estimated (see Section 2.6 for more details). Here to estimate Ab/At, factors such as friction and recirculation that reduce the entrainment ratio were ignored. The values of the effective area ratios are plotted in Figures 3.23 to 3.26. A s shown in Figures 3.23 and 3.24, w h e n the ejector operates under different performance conditions but the generator temperature does not change, the effective area ratio increases w h e n the evaporator temperature increases. It may be seen in Sections 3.7.1 and 3.7.2 that entrainment ratio is affected by the nozzle position. A s mentioned in Section 2.6.2 the entrainment ratio was defined as a function of the effective area ratio, Ab/At. Therefore, effective area ratio, Ab/At, m a y be plotted against the nozzle position. Here variation of effective area ratio against nozzle position for Ejectors 2.a and 2.b are presented in Figures 3.25 and 3.26. Finally, variation of optimum nozzle position versus the evaporator temperature for Ejector 2.b is plotted in Figure 3.27.
136
Experimental Work
CHAPTER THREE
1.18
3 X!
6 8 10 12 Evaporator Temperature [°C]
14
Figure 3.23 Variation of Effective Area Ratio with Different Operation Conditions of Evaporator Temperature {Tg = 76.6±0.4°C, 0m = 20° (Ejector La)}
<
8 10 12 14 16 Evaporator Temperature [°C]
18
Figure 3.24 Variation of Effective Area Ratio with Different Operation Conditions of Evaporator Temperature {Tg = 73.0±0.3°C, 0m = 20° (Ejector 2.a)}
137
Experimental Work
CHAPTER THREE
1.21.1-
3 1.0< 0.9-
0.8-
0
1
2
3
Nozzle Position [X/D2]
Figure 3.25 Variation of Effective Area Ratio Versus Nozzle Position {Te = 12.5±0.2°C, Tg = 77.2+0.3, 0m = 20° (Ejector 2.a)} 1.1 -
1.0-
<
0.9-
0.8 | • 1 « 1 • 0
1
2
3
Nozzle Position [X/D2]
Figure 3.26 Variation of Effective Area Ratio Versus Nozzle Position {Te = 5.0±0.1°C, Tg = 66.5±0.3, 0m = 10° (Ejector 2.b)}
138
Experimental Work
CHAPTER THREE
fi
E 9
s a.
O
6 8 10 12 14 16 Evaporator Temperature [°C]
18
Figure 3.27 Variation of O p t i m u m Nozzle Position Against Evaporator Temperature {Tg = 73.0+D.3, 0m = 10° (Ejector 2.b)}
139
Experimental Work
CHAPTER
THREE
3.9 Comparisons the Experimental Results of the Shock Pressure Recovery With O n e Dimensional Calculations A sudden increase in pressure and reduction of velocity may occur in the flow through a constant area tube where a normal shock occurs, i.e. when upstream flow is supersonic and the downstream is subsonic (Fox, 1977). The pressure distribution along the small R 1 2 V J R S is plotted in Figures 3.12, 3.15 and 3.20. According to thesefiguresthe shock pressure recovery in small ejectors can be seen. Comparisons of the experimental results with the analytical values using onedimensional calculations is important to our understanding of the shock pressure recovery process in practice. A s shown in Figure 3.12, w h e n the condenser temperature is very low, say around 16°C, it appears that no shock was observed in the constant area mixing tube. W h e n the condenser temperature was between 19.7 to 22.5°C (backpressure between 630 to 690 kPa) the shock occurred in the second half part of the constant area mixing tube between points B and C. It is seen that, as the backpressure increases, the shock moves upstream. In this case of experiment (Figure 3.12), w h e n the backpressure is higher than 700 kPa, the shock pressure recovery occurs near the start of the constant cross section area tube, and w h e n the condenser temperature was 28.4°C (backpressure 786.2 kPa) the entrainment ratio was near zero. In the present work, one dimensional analysis of the shock pressure recovery has been carried out to compare it with the experimental result. Experimental data are selected from Figure 3.12 such that T g = 66.5°C, T e = -3.0°C and the condenser temperature, T c = 23.8°C (this is the practical value of the critical condenser
140
Experimental Work
CHAPTER THREE
temperature, T C T ). In this case, the actual pressures of the points at the end of the subsonic diffuser, P D , at the end of constant area mixing tube, Pc, and at the starting of the constant area mixing tube, P A , were 729, 540 and 370 kPa, respectively (see Figure 3.12). Based on the mass flow rate of the condenser and P c and P D , the M a c h number at the end of constant area mixing tube m a y be calculated. Then the pressure ratio [ P C / P A L is calculated according to the normal shock governing equations in constant cross section area tubes (Equations 2.32 and 2.33). In this case, the length of the constant area mixing tube is 12 times its diameter. Therefore, the friction factor for the constant area mixing tube is taken account for the case entrance length (not for fully developed flow) for turbulent flow (Rohsenow et al, 1985). The friction factor is not constant along the tube length. The comparisons are given in Table 3.7.
Assumptions
Friction Effects
Pressure Ratio
[PC/PA]
ID Analysis Experiment r]d = 0.75
TIA
1.78
1.46
r]d = 0.70
NTIA
1.72
1.46
77, = 0.70
TIA
1.62
1.46
Table 3.7 Comparisons the one Dimensional Analysis of the Shock Pressure Recovery With the Practical Value N T I A = Not Taken Into Account, TIA = Taken Into Account Operation Conditions: T g = 66.5°C, T e = -3°C, T c = 23.8°C Assumption: Nozzle Efficiency, r\n = 0.95
141
Experimental Work
CHAPTER
THREE
3.10 Discussion Ejector choking plays an important role in ejector performance and the maximum entrainment ratio cannot be observed when the ejector does not operate at fully developed choked conditions. A s mentioned in Section 3.4, the test rig was improved to observe the choking phenomenon. Therefore, the present researcher examined the m a x i m u m ejector performance for small single fluid ejectors (see Figures 3.6 3.7 3.8, 3.13 and 3.17). It was recognised that w h e n the ejector operated below the critical backpressure (ie condenser pressure), the entrainment ratio remains fairly constant.
It was found that the ratio of the entrainment ratio obtained experimentally to th simulated values ranged from 0.65 to 0.78 (see Tables 3.3, 3.4 and 3.5 and Figure 3.19). T h e results m a y be explained as follows. The simulation values of entrainment ratio was predicted using one-dimensional analysis of compressible fluid flow theory based on usual assumptions such as absence of wall friction and treating the working fluid as a perfect gas. The heat loss and pressure drop in the ejector were both estimated. It was found that the heat loss was negligible (less than 6.0 W ) , but the pressure drop for a turbulent flow in the constant area mixing tube for the Ejector 1 was 39.0 kPa. For this small R 1 2 ejector in which the condenser was operating between 30°C to 40°C from tables it can be seen that for this range of saturated temperatures the saturated pressure of R 1 2 decreases by about 20 kPa when the temperature decreases by 1.0°C. Therefore, by taking this pressure drop into account, the critical condenser temperature would be 2.0°C lower than the predicted value and this would also cause a decrease in the entrainment ratio. Furthermore, the ejector dimensions might not be optimal when the frictional pressure loss is taken into account. The length of the mixing tube,
142
Experimental Work
CHAPTER
THREE
for example, was 12 and 10 times D 2 for Ejectors 1 and 2,respectively.However, as was described in Chapter 2, a shorter length m a y be better to reduce the friction losses.
For given operating conditions, the optimum nozzle position was tested in the present experiments. Based on theresults,the ejector performance is a function of geometric parameters such as the nozzle position and converging section half angle, 0 m . Here it was found that this factor is a significant parameter to optimise the ejector performance. These results m a y be explained by the secondary choking theory. The velocity of the secondary vapour must be increased to reach the sonic conditions in a converging cone (Munday and Bagster, 1977). The flow area for the secondary stream at some cross section of M a c h number one is then called the effective secondary area. With an optimum nozzle position, the secondary stream m a y be accelerated to sonic conditions properly in the converging cone in which the secondary flow m a y be gained the effective area ratio of the secondary vapour to the throat area of the nozzle, Ab/At. The flow area for the secondary stream around the body of the nozzle would be changed by the nozzle position. The minimum area ratio of the flow area for the secondary vapour around the nozzle to the throat area of the nozzle, Amjnb/At, is plotted against the nozzle position in Figure 3.28. Comparing this area ratio (Amim/At) with the effective area ratio, Ai/At, deduced from the experiments, it m a y be concluded that the choking conditions for the secondary vapour occurred at some cross section of the converging part of the ejector after the nozzle exit and probably at the start of the constant area mixing tube. This concurs with the secondary choking theory which was introduced by M u n d a y and Bagster (1977). For example, for Case 1 from Table 3.4 in which the blocking of the converging
143
Experimental Work
CHAPTER
THREE
section by the body of the nozzle itself was not significant, the values of A ^ t / A t and Ab/A t were 17.30 and 1.17 times D t , respectively. In these experiments, it was also observed that the optimum nozzle position changes w h e n the evaporator temperature was changed while the generator temperature w a s constant (see Figures 3.23 and 3.24). This result m a y be expressed that the hypothesis effective area for the secondary vapour is a function of the evaporator temperature, such that when the latter increases the former also increases. For example, for Ejector 2.a, w h e n the evaporator temperature increased by 1.0°C, the optimum nozzle position, X/D2, increased between 0.048 to 0.110. Using the secondary choking theory, when the evaporator temperature changes the optimum nozzle position needs to be changed to form an appropriate converging cone for the secondary stream.
The shock pressure recovery may happen at several positions along the ejector depending on operating conditions, eg in the converging part of the ejector and at the constant area mixing tube. In the present study, the pressure distribution was measured for a number of experiments to examine the shock pressure recovery in the constant area mixing tube (see Figures 3.12, 3.15 and 3.20). It has been found that a shock pressure recovery occurs in the constant area tube for normal ejector operation. T h e shock m o v e d upstream as the backpressure (ie the condenser pressure) w a s increased. This process can be explained according to standard compressible flow theory (see Chapter 2). The geometry of the converging section of the ejector has been studied tofindout whether this is an important factor in the design of dimensions of a small single fluid V J R S . For 10° and 20° converging half angles, it was recognised that this geometric factor, 0 m , is not a strong factor for maximising entrainment ratio in
144
Experimental Work
CHAPTER
THREE
small ejector design. The results for 0m=2O° and 10° for Ejectors La and l.b, respectively, are given in Figure 3.16 and Table 3.4 and these results show that the ejector with the angle of 0 m = l O ° performs between 1 to 4 percent better than the other one, but for Ejectors 2.a and 2.b, that one with the angle of 0 m = l O ° , performs almost the same as one with 0 m = 2 O ° . However, some advantages and disadvantages may be seen for the small and large converging half angle of the ejector, 0 m . For example, for the smaller 0 m the nozzle position is less critical but possible blocking of the secondary vapour by the nozzle body should be considered. For larger 0 m , blocking of the secondary stream by the nozzle body is not a problem, but the nozzle position is critical. Comparing Figures 3.10 and 3.14, it is seen that for Ejector L a ( 0 m =20°), in practice the exact position of the nozzle is important, but for Ejector l.b ( 0 m =10°) the nozzle position is not so critical.
145
Experimental Work
CHAPTER THREE
Aminb Motive Nozzle
/ / /// a ZZZZ22Z Mixing Chamber
30 Conv. Half Angle = 20°
25-
20fi
E
15
<
10-
Conv.HalfAngle=10° • i • i • i • i • i
'~r
i ' l '
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Nozzle Position [X/D2]
Figure 3.28 Plot of Aminb/At Versus the Nozzle Position {Ejectors La and l.b]
146
CHAPTER 4 THEORETICAL INVESTIGATION OF THE FEASIBILITY OF USING TWO WORKING FLUIDS OF DIFFERENT MOLECULAR WEIGHTS IN A VJRS 4.1 Introduction The ejectors may be thought of as entrainment systems. These systems can be further classified as either self-entrainment or two component entrainment systems. The former using identical motive stream and secondary vapour, that m a y be called as a single fluid ejector. T h e self-entrainment in axisymmetric ducted jets was studied by researchers such as Curtet and Ricou (1964). In the two component system the primary and secondaryfluidshave different molecular weights and chemical formulae. The ejector performance in the case of the same working fluid (self entrainment) for both the motive vapour and entrained flow has been studied in the past literature (eg Schmitt et al, 1975, Wall et al, 1980, Hamner, 1980). A s mentioned in Chapters 1, 2 and 3, the behaviour of ejector performance in self entrainment systems has been investigated by the present author for small ejectors, in which R 1 2 was selected as the workingfluid.Such an ejector, m a y also operate with an
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FOUR
azeotrope refrigerant, like R502 say. This kind of refrigerant is a mixture of two substances that cannot be separated into its components by distillation. The refrigerant evaporates and condenses as a single working fluid like R22, but its properties are different from those of either constituent (Stoecker and Jones, 1982). Normally a VJRS operating with either an azeotrope or a pure refrigerant as working fluid will suffer from poor efficiency. By using a non-azeotropic mixture of refrigerants, it may be possible to improve the ejector performance. There are several parameters in the ejector such as the cooling capacity, as well as the COP that might be improved. Table 4.1 shows a list of some mixtures that may be used in conventional vapour-compression refrigeration cycles (Berntsson and Schnitzer, 1984, Tolouie et al, 1991).
Case
Mixture
1
R22 -R12
2
R22-R114
3
R22 -R142b
4
R12-R11
5
R12-R114
6
R12-R113
7
R12-R13
8
R152a-R13B1
Table 4.1 S o m e Possible Refrigerant Mixtures
Possible gains in performance by the use of mixed refrigerants, such as mixtures of R12-R22 and R22-R142b, for refrigeration systems has been studied by some
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Theoretical Investigation of...
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FOUR
researchers (eg. C h e n et al, 1991). Tolouie et al (1991) modelled operation of reciprocating compressorrefrigerationcycles using non-azeotropic mixtures and they found that w h e n the compressor operated with a mixture of R22-R142b ( 6 0 % R 2 2 ) , the cooling capacity was improved compared with that one using pure R 1 2 . They also found that this mixture m a y be considered as a possible R 1 2 replacement in this system.
This work detailed in this chapter was conducted to study the feasibility of two component entrainment systems. Performance of ejectors as a function of the molecular weights of the vapour has been studied by W o r k and Miller (1940), Holton, 1951 and D e Frate and Hoerl, (1959). Little attention has been paid to study those effects on small refrigeration ejectors, which m a y be applied to systems such as automobile air conditioning. In other words, the present study has been carried out to investigate the possibility of entrainment ratio in fluid pair V J R S , which are designed for relatively small cooling capacities. There are many possible fluid pairs having characteristics that may improve the performance of a small V J R S using refrigerant of two different molecular weights. The desirable characteristics for the fluid pairs can be listed as follows: • Low saturation pressure for one of the fluids of the pair. The reason for this is simply that driving the V J R S m a y be easier and more safe in practice compared with the motive fluids with a high saturated pressure at the generator temperature. The schematic diagram of a two-fluid V J R S with fluid pair R l 1 - R 1 2 is shown in Figure 4.3. In this two-fluid ejector R l 1 acts as the motive working fluid which m a y satisfy the requirement for low saturated pressure at a given generator temperature.
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FOUR
• The possibility of physical separation of the fluids after condensation is another necessary characteristic.
4.2 Objectives The aims of this study were as follows. (a) To investigate the effects of using two working fluids of different molecular weights on entrainment ratio in small ejectors. (b) The feasibility of using working fluids with different molecular weights in a vapour jet refrigeration system. (c) Comparisons of the simulated results of a two component entrainment VJRS with a single fluid one.
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Theoretical Investigation of...
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4.3 Governing Equations for One-Dimensional Analysis 4.3.1 Equations of performance characteristics
The governing equations for one-dimensional analysis of ejectors are the same for a two-fluid V J R S as those given in Chapter 2. Section 4.5 describes the equations used for this analysis.
4.3.2 Equations for condensation of mixtures Simply, the condensation process in a single fluid VJRS, may be carried out by heat rejection in the condenser. For a two-fluid V J R S this process is more complex and this is the major difference between this system and the single fluid V J R S . Details are expressed in Section 4.4. For a mixture of two refrigerants of different molecular weights the following equations m a y be used to find the mole fraction of vapour and liquid of each component, partial pressure and dew point temperature of the mixture (Goodger, 1984, V a n W y l e n and Sonntag, 1985). In the equations subscripts 'a' and 'b' denote the primary vapour and secondary stream, respectively. The governing equations for the mole fractions of two components in a mixture of vapours are:
y°-
Wa MJL+MJL
(4.1)
Wa Wb and
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Theoretical Investigation of...
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ML y =
>
w
M-M:b Wa W b
<4-2)
where y a +y fc = i
(4.3)
In the above equations, M, y and W, denote mass flow rate of the vapours, mole fraction of the vapour components in the mixture and molecular weights of working fluids, respectively. Partial pressures can be calculated according to Dalton's law (Cengel and Boles, 1994). Dalton's model holds both for the mixture and the separated components for ideal gas mixtures. Thus, it can be written: Pavm = yaRTm Pbvm=ybRTm p y = nt
fit
(4-4)
RT nt
In above equations, P a and Pb, are referred to as partial pressures, the subscript 'm' stands for mixture and R, is the gas constant. By substituting Equations (4.3) and (4.4), then: Pm = Pa + Pb ^ To simplify to analysis of the two-fluid VJRS the components of the mixture were assumed to be ideal gases. This assumption is reasonable given that under typical operating conditions the compressibility factor for R 1 2 is of the order
152
Theoretical Investigation of...
CHAPTER
0.95. On this basis Raoult's ideal gas model may be used to give (Cengel and Boles, 1994):
"a(sat)T
and
x.=J^
(4.7)
P *b(sat)T
In these equations, x, P m and P(sat)T> denote the mole fraction of the liquid components, the pressure of the mixture and the saturated pressure of each refrigerant at a given temperature. In a mixture of vapours containing just two components, such as R 2 2 and R142b, then Equation (4.3) must be satisfied. Also, at the d e w point temperature, the mole fractions in the liquid, should satisfy the following equation: *+*fc=l
(4-8)
153
FOUR
Theoretical Investigation of.
CHAPTER
FOUR
4.4 Condensation of Mixtures The present study employs the theory of condensation of mixtures to calculate the dew point temperature of the mixture, Td. In the two-fluid V J R S , condensation of the mixture of R l l and R 1 2 occurs in the condenser. Assuming the mixture is non-azeotropic one of the components will condense at a lower temperature compared to the other. In this case, the vapour-liquid equilibrium m a y be calculated using Raoult's law in an ideal mixture (Cengel and Boles, 1994). The calculated condenser pressure from the simulation program, P c , is used as the total pressure of the mixture, Pt, exiting the ejector. According to Raoult's L a w the composition of gas and liquid depends on the total pressure of the mixture and the vapour pressure of each component. The dew point temperature, Td, of the Rll-R12 mixture is calculated using the following procedure. For a given total pressure of the mixture and mole fractions of the vapour, one needs to determine the temperature at which the sum of the mole fractions for all components in the liquid is unity. The d e w point temperature is calculated by trial and error. Thus, the initial value for the dew point temperature, Td, should be guessed. Then the saturated pressure of each component at Td is determined from thermodynamic properties from tables. Using Equations (4.6) and (4.7) these pressures are then used to calculate the mole fraction of liquid for each workingfluid,x a and xb- If the total value of these mole fractions is equal to one, the guessed temperature can satisfy Equations (4.3) and (4.8), therefore it could be introduced as the d e w point temperature of the mixture. If these values do not equal one, then another temperature must be chosen. T h e process continues until the exact value of the d e w point temperature is found.
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O n e set of calculations is presented here to illustrate the procedure. In this example, R l l and R 1 2 are used as the primary and secondary working fluids of the ejector, respectively. The critical condenser pressure is taken as 350 kPa based on the one-dimensional simulation of constant area analysis under the conditions of T e , T g , 77,, r)d equal to 0°C, 80°C, 0.95 and 1.0, respectively. T h e mole percent of vapours are calculated from the mass flow rates of the working fluids as y a = 4 4 and yb = 56 percent for R l 1 and R 1 2 , respectively. Theresultsof the example calculations are shown in Tables 4.2 and 4.3. A s shown in Table 4.3, in Cases of Td = 20°C and Td = 40°C the sum of mole fractions (xa + xb ) is greater than one, but in Case of Td = 45°C, the sum of them is less than one. The d e w point temperature of the mixture is found to be 42.2°C. For a mixture of refrigerants, during condensation and evaporation, the condensation and evaporating temperatures are not constant. The temperature at which the mixture starts to condense is called the d e w point temperature, Td- The difference between the temperatures at which condensation starts and ends is k n o w n as "temperature glide". T h e d e w point curve and the liquid line for the mixture of this example is plotted in Figure 4.1. A s shown in this figure, for pure R 1 2 the saturation temperature is 3.9°C, while that for the pure R l 1 it is 64°C. It is seen also, in the mixture of R l l and R 1 2 with both phases present at a temperature of 42.2°C, the vapour will have a composition of 4 4 % of R l l and 5 6 % of R 1 2 (point A in Figure 4.1), but the liquid (point B of the figure) will have 8 1 % and 1 9 % of R l 1 and R 1 2 , respectively.
155
Theoretical Investigation of...
Refrigerant
CHAPTER
Molecular
Mass Flow
Weight
Rate[kg/s]
FOUR
Mole
Mole Fraction
Rll
137.5
Ma
(Ma/137.3)
• of Vapour From Eq. (4.1)
R12
120.9
Mb
(Mb/120.9)
From Eq. (4.2)
Table 4.2 Data for Calculation of Mole Fraction of Vapours in the Mixture
Td
xa
Xb
Xa+Xb
[°C]
Pm [kPa]
20
350
88.4
567.3
1.74
0.35
2.09
40
350
173.5
960.7
0.89
0.20
1.09
45
350
202.3
1084
0.76
0.18
0.94
42.2
350
188
1010
0.81
0.19
1.00
Pa(sat)T
Pb(sat)T
[kPa]
[kPa]
Table 4.3 Calculation Procedure to Find the D e w Point Temperature of the Mixture {Td = the dew point temperature of the mixture; P(sat)T = Saturated Pressure at T °C} Comparisons of the condensation temperatures of pure Rll and pure R12, with the mixture of Rll - R 1 2 for a condenser pressure of 350 kPa are given in Table 4.4. A s shown in this table for pure R l l and pure R 1 2 , the condensation temperature is constant, while for the vapour mixture of 4 4 % Rll and 5 6 % R 1 2 condensation occurs between 18.9°C and 42.2°C. Thus the temperature glide of this case is 23.3°C.
156
Theoretical Investigation of...
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Mole Percent of R12 100 80 60 40 20 0 80 | i I i I i I i I i I i I i I i I i I i | so
U o
70-
Dew Point Curve r (Vapour Line) -70
60-
-60
50-
-50
U o
UJ
V
IN
3
40-
-40
ea u v
£ u a s
30-
-30
E H
CM
s
20-
-20
10-
-10
H
• i • l ' l ' » ' i • I • i ' I ' I
0 10 20 30 40 50 60 70 80 90100 Mole Percent of R12
Figure 4.1 Equilibrium Diagram for Liquid Vapour Phase of the R11-R12 at a Total Pressure of 350 kPa
157
Theoretical Investigation of...
Case
CHAPTER
Pc
Condensation
Temp. Glide
[kPa]
Temp [°C]
[°C]
Pure Rll
350
64.0
0
PureR12
350
3.9
0
R11-R12
350
From 18.9 to 42.223.3
FOUR
Table 4.4 Temperature Variation During Condensation of Pure Refrigerants (Rll and R 1 2 ) and a Mixture of R 1 1 - R 1 2 ( 4 4 % and 5 6 % , respectively)
The condenser pressure versus the saturated temperature of the pure refrigerants of R l l and R 1 2 are plotted in Figure 4.2.
158
CHAPTER
Theoretical Investigation of...
FOUR
2500
i • i—•
0
i
•—r~>—r~"—i—•—i—'i'l
1
10 20 30 40 50 60 70 80 90 100
Saturation Temperature [°C]
Figure 4.2 Plot of the Saturation Temperature for Pure Rll and Pure R12
159
Theoretical Investigation of...
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FOUR
4.5 Two-Fluid V J R S Simulation Analysis of a vapour jet refrigeration system has been carried out by the present author to analyse the case of dissimilar molecular weights of workingfluidsfor the motive and secondary streams. T h e modelling w a s based on one-dimensional compressible fluid flow theory, using the constant area analysis method. The simulation conditions are shown in Table 4.5. The performance of the ejector is modelled by performing mass, momentum and energy balances on a control volume. S o m e of the performance characteristics such as mass flow rates of the primary and secondary streams, entrainment ratio and the condenser pressure have been calculated in a computer program using this one-dimensional theory (computer program is listed in Appendix C, Sction C.3). The schematic diagram of the system is shown in Figure 4.3. A s shown in this figure, the primary flow expands through a converging diverging motive nozzle to entrain the secondary vapour and mixing of the fluids is completed in the constant area passage of the ejector. In this study of an ejector as a two-component entrainment system, R l l is selected as the working fluid for the motive vapour and R 1 2 for that of the secondary stream. This one-dimensional simulation program calculates the entrainment ratio from Equation (2.35) and Equations (2.36) to (2.40) were used to calculate the ejector backpressure (ie the critical condenser pressure) for the given simulation conditions of 77„, f]d, T g , T e , A2/At and Q e . This is then used with the theory of condensation of mixtures mentioned in Section 4.4 to calculate the mole fractions of either the vapour or the liquid mixture and the d e w point temperature of the mixture, Td.
160
Theoretical Investigation of...
CHAPTER
Mixing Chamber
Constant Area Mixing Tube
FOUR
Subsonic Diffuser
Motive N o z z l e ^ s , Motive stream (Rll)
- ^
Mixture of (Rll - R 1 2 )
Condenser Condensation and separation of Rll andR12 Refrigerant
Pump
Figure 4.3 T h e Schematic Diagram of the T w o C o m p o n e n t Entrainment Ejector
161
Theoretical Investigation of...
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FOUR
Simulation Modelling Motive Stream
Rll
Secondary Vapour
R12
Nozzle Efficiency, r\n
0.95
Diffuser Efficiency, r\d
0.8
Evaporator Temperature, T e
*
Generator Temperature, Tg
*
Area Ratio, Aj)k\
*
Cooling Capacity, Q e
lkW
Table 4.5 T h e Simulation Conditions in O n e Dimensional Modelling The symbol [*] denotes these values are chosen for simulation conditions. For example to design the system for the operating conditions T g = 80 °C and T e = 0 °C, then those values must be given for simulation modelling.
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Theoretical Investigation of...
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4.6 Simulation Results In this program the performance characteristics of the ejector are analysed for the case of dissimilar molecular weights of refrigerants. A s mentioned previously, the analysis is based on the one-dimensional constant area method. The simulation results for several operating conditions are shown in Table 4.6. The entrainment ratio, 3>, and the condenser pressure, P c , are given in the table. The vapour line (dew point curve) and the liquid line of the mixture of Case 12 (T g = 85°C, T e = 10°C, A2/A t = 3.0) from Table 4.6, are depicted in Figure 4.4. The temperature glide in this case is 23°C (difference between 48°C and 25°C). A s shown in thisfigure,for the pure R 1 2 the saturated temperature is 11.1°C, while for the pure R l 1 that is 72.8°C. It is seen also, in the mixture of R l 1-R12 with both phases present, at a temperature of 48°C, the vapour will have a composition of 3 9 . 4 % of R l l and 6 0 . 6 % of R 1 2 (point A in Figure 4.4), but the liquid (point B of thefigure)will have mole fraction of 7 6 % and 2 4 % for Rll and R 1 2 , respectively. Only some of the cases presented in Table 4.6 represent ejectors that are useful for refrigeration purposes in practice as the condenser critical temperature needs to be below ambient temperature. M o r e details, including the d e w point temperature, temperature glide, mole fractions of vapour and liquid of the fluid pair (mixture of R11-R12), for various operating conditions are presented in Table 4.7 for s o m e of the more practically realistic cases. Condensation of the mixture in a practical system is very important. For example, it is seen from Table 4.7, that if Case 1 is selected the condenser should operate between 42.2°C to 18.9°C to liquefy the mixture, but if Case 4 is chosen, the range of temperature is
163
Theoretical Investigation of...
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54.4°C to 31.8°C. Thus, in regard the condenser operation, Case 4 (Tg = 90, T e = 15, A2/At = 3.0) may be taken as a feasible case for design of a practical VJRS using two working fluids of different molecular weights, providing ambient wetbulb temperatures remain below 30°C, say.
Case
Tg [°C]
Te
[°C]
A2/At Entrainment Pc Ratio, 4> [kPal
1
80
0
2.5
0.821
361
2
80
0
3.0
1.117
350
3
80
5
2.5
0.997
388
4
80
5
3.0
1.311
379
5
80
10
2.5
1.145
419
6
80
10
3.0
1.534
413
7
85
0
2.5
0.711
390
8
85
0
3.0
0.964
373
9
85
5
2.5
0.853
416
10
85
5
3.0
1.149
403
11
85
10
2.5
1.008
446
12
85
10
3.0
1.352
437
13
90
10
2.5
0.894
474
14
90
10
3.0
1.203
461
15
90
15
2.5
1.048
508
16
90
15
' 3.0
1.405
499
Table 4.6 The Entrainment Ratio and Condenser Backpressure for TwoFluid V J R S Simulation (r/„ = 0.95, 7^=0.80, and Q e = 1 k W )
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Theoretical Investigation of...
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Mole Percent of R 1 2 100 80 60 40 20 80 I • • • ' • • • • • • • • • ' •
70-
-70 -60
60o
80
D e w Point Curve (Vapour Line
50
50
V t-
9 e a u 9>
40
E
30
-40
CM
-30
v H
U o V -u 5 9ea w. a> CU
B -20
20 10 -
10
Liquid Line • •••
•••••
H
•••
•••••
0 10 20 30 40 50 60 70 80 90100 Mole Percent of Rll
Figure 4.4 Equilibrium Diagram for Liquid Vapour Phase of the Rll - R 1 2 at the Pressure of 437 kPa Tg = 85°C, Te = 10°C, A^At = 3.0, Nozzle efficiency, r)n = 0.95, Diffuser efficiency, rid=O.S0, and Qe = 1 kW
165
Theoretical Investigation of...
Case
Te Tg [°C] [°C]
O
CHAPTER
Pc Ya [kPa] (%)
Yb (%)
xa
Xb
(%) (%)
FOUR
Temp Td [°C] Glide[°C]
1
80
0
1.117 350
44
56
81
19
42.2 23.3
2
80
10
1.534 413
36
64
76
24
45.5 22.5
3
85
10
1.352 437
39.4 60.6 76
24
48.0 23.0
4
90
15
1.405 499
42.2 57.8 78
22
54.5 22.6 (1 •
_^
..
Table 4.7 Calculated D e w Point Temperatures and Temperature Glide for R11-R12 ejector Nozzle efficiency, TT, = 0.95, Diffuser efficiency, 7^=0.80, A^At = 3.00, 4> = Entrainment Ratio
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Theoretical Investigation of...
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FOUR
4.7 Comparison With Simulation Results for a Single Fluid V J R S Comparisons between the simulation results for the single fluid VJRS (using R12) and the case of the fluid pair R l l and R 1 2 are given in Table 4.8. S o m e very significant advantages m a y be seen for a V J R S using two working fluids RllR 1 2 , compared with the single fluid system of R 1 2 . These advantages can be summarised as follows:
(a) According to this theoretical investigation the two-fluid ejector performance (entrainment ratio) is up to seven times better than the single fluid V J R S .
(b) The two-fluid ejector system operates under a much lower generator pressure compared with the ejector system with pure R12. For example, in the single fluid V J R S w h e n the generator temperature is 80°C the saturated pressure of the generator is 2304.6 kPa, while in the two-fluid pair system R11-R12, this is reduced to 519.2 kPa.
Case
T e A2/At [°C] [°C]
0>
Tg
Temp Pc Td or T c [kPa] [°C] Glide[°C]
80
0
3.00
0.165 961
40
2(R11-R12) 80
0
3.00
1.117 350
42.2 (Td) 23.3
3(Pure R12) 80
10
3.00
0.264 1011 42.1
4(R11-R12) 80
10
3.00
1.534 413
l(PureR12)
0.0
0.0
45.5 (Td) 22.5
Table 4.8 Comparisons Between Simulation Results for the Two-Fluid Ejector and the Single Fluid V J R S Nozzle efficiency, T]„ = 0.95, Diffuser efficiency, 7?rf=0.80, Area Ratio, A2/A t = 3.00, O = Entrainment Ratio
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Theoretical Investigation of...
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FOUR
4.8 Feasibility of Separation of Fluids in the Condenser In a two-fluid VJRS separation of fluids would occur in the condenser or in a dedicated liquid separator. There are various methods that could be used in practice to separate a mixture of refrigerants. This section describes some possible methods. (a) Fractional distillation
This technique of separation is based on the difference between the boiling points of the two refrigerants. If there is a big difference between the boiling temperatures of the components then in the condenser one can be liquefied earlier than the other. In this case the fractional distillation of the refrigerants m a y help to separate the components in practice. (b) Diffusion method The molecule size of a substance depends on the molecular weight of it, such that w h e n the latter is bigger the former will also be bigger. B y this method, in the liquid mixture one refrigerant with smaller molecular weight can be separated in the condenser. Only if the molecular weights of the components in a mixture are significantly different then the diffusion method of separation m a y be used.
(c) Gravity If the refrigerant liquids are not miscible it would be possible to separate them simply by gravity. T h e physical and thermodynamic properties
of the fluid pair R 1 1 - R 1 2 are
presented in Table 4.9 (Perry and Chilton, 1973). Only a 13 percent difference
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Theoretical Investigation of...
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FOUR
between the molecular weights of R l 1 and R 1 2 can be seen, but the different of their boiling temperatures at atmospheric pressure is 54.1°C. Therefore, in this two-fluid V J R S (R11-R12) the diffusion method m a y not be useful in practice to separate the liquid mixture in the condenser, but the fractional distillation m a y be practically applicable for this separation.
Refrigerant
Molecular
Saturated Pressure
Boiling Point at
Weight
at 80°C [kPa]
Atmospheric Pressure [°C]
Rll
137.38
519.21
24.9
R12
120.92
2304.60
-29.2
Table 4.9 T h e r m o d y n a m i c and Physical Properties of R l l and R 1 2
169
Theoretical Investigation of...
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FOUR
4.9 Conclusion The work described in this chapter represents a preliminary study of a vapour jet refrigeration system using a fluid pair. In this study only the fluid pair of R l 1 and R 1 2 has been analysed for a small V J R S , w h e n R l 1 is the working fluid of the motive vapour and R 1 2 is that of the secondary stream. The reasons for selecting these fluids were,firstlythat data on thermodynamic and physical properties were readily available. Secondly, R l l has a low saturation pressure compared with other refrigerants such as R 1 2 . Thus, the motive vapour in the generator m a y operate at low pressure of about 500 kPa say, which is about four times less than the motive pressure for a small R 1 2 V J R S . Thirdly, this analysis has demonstrated (see Table 4.8) that the entrainment ratio is likely to be substantially better than for the singlefluidV J R S . Unfortunately, in regard to the environmental problems such as ozone depletion potential and chemical toxicity, Rll and R 1 2 cannot be selected as thefluidsfor future refrigeration systems in Australia ( A I R A H . Environment Protection Agency, A P A , September 1994). Therefore, other replacementfluidspairs need to be investigated in a feasibility study for a two-fluid V J R S . Practical alternatives to R l l and R 1 2 which are currently available include R 1 4 2 and R134a, respectively. The limitation of this work is that the one-dimensional analysis assumes the ideal gas treatment for the fluids and that the wall friction is negligible. Also, all equations were used to calculate the d e w point temperature and mole fraction of the components using Dalton's model assumes both the mixture and the separated components to be ideal gases.
170
CHAPTER 5 IMPLICATIONS FOR THE PRACTICAL USE OF SMALL VJRS EJECTORS
5.1 General Considerations Implications for the practical use of the small ejector is very important. Application of ejectors have been studied in the past literature. In the present investigation more attention has been paid to study implications for the practical use of a small V J R S . Applications of ejectors in power plant, chemical industries and refrigeration systems have been studied by several researchers (eg Whitaker, 1975, Doolittle and Hale, 1984). Chen (1978) and Kanashige and Cooper (1993) studied the potential applications of vapour jet refrigeration systems in vehicle air conditioning, and the present author considers this to be one of the most promising areas of application for small V J R S . The small V J R S m a y also be used for refrigeration systems. In this study the performance of small R 1 2 refrigeration ejectors has been tested and experimental results are presented in Chapter 3.
171
Implications for the Practical Use of Small VJRS Ejectors
CHAPTER FIV
In regard to ejector dimensions and performance characteristics of a small VJRS that m a y be applied to air conditioning (in vehicles say) one example is illustrated here. In this example it is considered that a small refrigeration system demands about 8 k W cooling capacity. For this purpose two R 1 2 ejectors are modelled using a one-dimensional simulation of constant area analysis. T h e simulation results of the ejector dimensions such as the throat diameter of the nozzle as well as the performance characteristics are shown in Table 5.1.
VJRS No.
Dt [mm]
D2 [mm]
O
COPcarnot
Qe [kW]
Qg [kW]
Qc [kW]
1
4.6
8.0
0.165
0.773
5.0
37.8
44.2
2
5.0
8.7
0.165
0.773
6.0
45.4
53.0
Table 5.1 Performance Characteristics and Dimensions of the V J R S Ejectors { T g = 80°C, T e = 0°C, Condenser Critical Temperature, T C T = 40°C, r\n = 0.95, r)d = 0.8,
Using VJRS ejectors for this purpose has several advantages compared with conventional systems. Operation is simple and less maintenance is required. Also, this system can operate with low grade thermal energy and it has no moving parts except those of the refrigerant pump. In the present study the small ejectors with a 1 k W nominal cooling capacity were used. Implications for the present practical system are discussed in the next section.
172
Implications for the Practical Use of Small VJRS Ejectors
CHAPTER FIVE
5.2 Implications for Practical Use In a practical application use of the small ejector should be considered in the light of some factors such as system efficiency. These factors m a y be summarised as follows. 5.2.1 Entrainment ratio
As mentioned in Chapters 2 and 3, the present theoretical and experimental study shows that the entrainment ratio for a single fluid V J R S is low. T h e low entrainment ratio would be not a real problem if waste heat energy was used to drive the system, since these systems can be driven by low grade thermal resources. Also, the theoretical study in Chapter 4, shows that using a two-fluid V J R S m a y substantially increase the entrainment ratio. Still there are some problems compared with the conventional refrigeration systems due to the poor entrainment ratio, leading to problems including the condenser size must be bigger than in a conventional vapour compression system and it m a y not be currently economic in some applications like vehicle air conditioning.
5.2.2 Condenser size Heatrejectionfrom the condenser is very important because both large and small ejectors are associated with the choking phenomenon in which the m a x i m u m entrainment ratio can be obtained only w h e n the system operates at or below the critical backpressure (ie condenser pressure). For example in the present experiments, as mentioned in Chapter 3, an auxiliary air cooled refrigeration
173
Implications for the Practical Use of Small VJRS Ejectors
CHAPTER FI
system was constructed in which when the condenser cooling water was 24°C its cooling capacity was 9.0 k W . Therefore, the size of the condenser since this study was undertaken has n o w been approximately halved (as seen in Table 5.1, i.e. 44.2 k W to 53.0 k W ) in a practical V J R S , while a normal conventional condenser for the same cooling capacity (5 k W to 6 k W ) needs 9.0 k W heat rejection. If the radiator is replaced, the effective "under the bonnet" space is the same as a car using a current conventionalrefrigerationsystem. These problems must be tested for vehicle air conditioning. However, the V J R S m a y not be feasible for large automobile air conditioning such as buses.
5.2.3 Heat input to drive the VJRS
In the present experiments an auxiliary electrical heater was used in the test rig provide the hot water (up to about 85°C) for the generator heat exchanger.
In practice, it may be possible to use the low grade heat energy to drive these systems. Circulating hot water from the engine's jacket and the waste energy from the exhaust gas in cars m a y be used to drive an automobile air conditioning (see Figure 1.7). For example, for a 2000 cc. automobile, under the normal operation the available heat energy from the engine's jacket would be about 35 k W (Taylor, 1971), but several important parameters such as full loading and passengers comfort should be considered to determine whether the available heat energy is enough to drive the vehicle V J R S . Also, using solar energy and industrial waste energy are s o m e other thermal resources which m a y be used to drive a small V J R S ejector.
174
Implications for the Practical Use of Small VJRS Ejectors
CHAPTER FIVE
5.2.4 Refrigerant p u m p
The refrigerant pump is the only moving part in the vapour jet refrigeration system. The high pressure motive vapour is an important issue in this regard. Development of a suitable practical refrigeration p u m p is very important. The desirable characteristics for the refrigerant p u m p can be listed as follows. (a) Trouble free operation at high pressure
In practice the refrigerant pump must operate at high pressure, eg using R12 as the working fluid, the desirable absolute pressure at 80°C is 2304 kPa. In the present work, it was difficult to find a commercial refrigerant p u m p which met the required high discharge pressure. First, a commercial vane p u m p was used, but this p u m p experienced breakdown several times due to the rupture of a mechanical seal during the experiment under the high pressure operating condition. Then, a diaphragm cushion p u m p was used in the testrigto avoid the high pressure problem (see Chapter 3 for more details). (b) Noise pollution
This factor is also very important in the practice. The refrigerant pump should operate quietly and smoothly at high pressure. The p u m p s used in the present work had this problem specially at high operating pressures.
(c) Operating temperature range
A suitable practical pump for VJRS application must operate for the desired temperature range. For example, in a R 1 2 V J R S , the refrigerant p u m p should work at 60 to 80°C without any problem. The diaphragm cushion p u m p and the
175
Implications for the Practical Use of Small VJRS Ejectors
CHAPTER FIVE
vane p u m p were used in the present study did not have this problem during the experiments.
(d) Chemical reaction
The refrigerant used does frequently dictate the material employed in the system (eg R 1 2 does notreactwith brass but ammonia does).
(e) pump size The particular application of the VJRS ejector dictates the pump size. The size of p u m p s used in the present test rig is not suitable for applications such as automobile air conditioning which needs a small size of the refrigerant pump. Therefore, an appropriate size of p u m p body should be designed and built for such applications.
176
CHAPTER 6 CONCLUSIONS AND FUTURE WORK
6.1 Analysis of Single Fluid Ejector Refrigeration Systems In the present research theoretical modelling and experimental investigations have been carried out to analyse and examine the performance characteristics of small V J R S ejectors. The significant conclusions m a y be summarised as follows.
6.1.1 One-dimensional analysis Theoretical analysis of the fluid flow within a supersonic jet ejector in a vapour jet refrigeration system is extremely complex. Here, further development has been carried out by the present author based on one-dimensional compressible flow analysis to model a single fluid V J R S using the secondary choking theory. S o m e advantages are to be seen using this theory to model the ejector performance compared with the constant area analysis. For example, the secondary choking modelling gives insights into the performance of the supersonic jet ejector that are not available w h e n using simpler theoretical models. The present researcher found that both the constant area analysis method and secondary choking theory are useful for prediction of the ejector performance.
177
Conclusions and Future Work
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In the present study, the conventional one-dimensional compressible flow analysis has been extended to determine the effects of superheating of the inlet vapour streams. It has been found that the entrainment ratio is only slightly affected by superheating of vapour leaving the evaporator and generator, such that in a R 1 2 small ejector a superheat of 1°C in the generator outlet results in a decrease of 1.2% in the entrainment ratio. A sensitivity study on the effect of the diffuser efficiency on ejector performance has been carried out in this work. The present author found that the diffuser efficiency has a little effect on the ejector performance and therefore, it affects on the optimal ejector design dimensions.
6.1.2 CFD analysis The present author developed a computational fluid dynamics simulation of a small V J R S ejector using P H O E N I C S - B F C code based on the finite volume technique. Validation of computational results was successfully carried out with analytical solutions at several increasing levels of complexity. The present domain was selected to be from the throat of the nozzle to the end of the constant area mixing tube. The velocity, density and enthalpy of vapour at the throat of the nozzle w a s calculated from one-dimensional, compressible flow theory and used as the boundary conditions for the primary vapour. The stagnation pressure of the secondary vapour w a s fixed as the secondary inlet boundary condition. Then, the experimental value of the pressure at the end of the constant area mixing tube was used as the outlet boundary condition. In the present simulation, Ejector 2.a used in Section 3.7.1 was modelled. In this simulation, the working fluid w a s R 1 2 and T g , T e , and X / D 2 were equal to
178
Conclusions and Future Work
CHAPTER SIX
73.0°C, 12.5°C and 1.07, respectively. T h e results of the entrainment ratio were compared with that from experiment. T h e C F D simulation matched the experimentalresultwithin 7 % . In this CFD modelling, it was found that the secondary vapour did not reach sonic velocity, M = 1 , in the converging section of the ejector or in the constant area mixing tube (see Figure 2.17). This is contrary to the assumption in the secondary choking theory that assumes the supersonic primary stream expands laterally during its flow d o w n the converging section of the ejector it effectively provides a converging duct for the initially subsonic secondary vapour which must reach sonic velocity before mixing occurs. This is possibly due to the effects of recirculation and in reality the primary jet entrains secondary stream throughout its development. In the present C F D analysis a recirculation of the working fluid was predicted (Figure 2.17) and this m a y restrict entrainment of the secondary flow and cause thus a decrease in the entrainment ratio. Also, it has been found that the predicted entrainment ratios using C F D analysis were below the one-dimensional result, because in the one-dimensional analysis the effects of friction were not taken into account
179
Conclusions and Future Work
CHAPTER SIX
6.2 Performance of the Small Single Fluid V J R S In the present experimental investigation of small VJRS using Rl2 as the working fluid, the following m a y be concluded as significant results. (a) Choking phenomenon Experimentally it has been confirmed that for the small ejectors the choking phenomenon plays an important role in ejector performance. It has been found that the entrainment ratio remains fairly constant for backpressure (ie condenser pressure) below a critical value and this means that the entrainment ratio is essentially independent of the condenser pressure w h e n the ejector operates at fully developed choked flow. It was observed when the ejector was not operated at choking conditions, the entrainment ratio was significantly affected by the condenser temperature, such that one degree increase on the latter, decreased entrainment ratio by 20 %. In the small ejectors studied there was a very clear indication of the critical condenser temperature in all cases.
(b) System performance In the present work the m a x i m u m entrainment ratio has been examined when the ejectors operated at fully developed choked conditions. It was recognised that the m a x i m u m entrainment ratio in practice was below the predicted value using onedimensional analysis of the ejector performance by between 22 to 35 percent. The experimental values of the critical condenser temperatures were also lower than those predicted by the analytical value by between 4.3°C to 7.3°C. Reasons for this are discussed as follows.
180
Conclusions and Future Work
CHAPTER SIX
Here the limitations of the one-dimensional analysis are summarised. The performance characteristics were predicted based on assumptions such as neglect of the pressure drop due to friction and assuming the flow is treated as an ideal gas. Heat loss and pressure drop in the ejector were calculated and it was found that the former is negligible, but the latter is not. Thus, the ejector dimensions might not be optimal. In this ejector (Ejector 1) length of the constant area mixing tube w a s the m a x i m u m found in the literature, 12 times D 2 , while the shorter length m a y reduce the friction effects. (c) Shock pressure recovery
From the pressure distribution along the ejector that was measured for a number of experiments, it has been found that a shock pressure recovery occurs in the constant area mixing tube. Experimentally, it was observed that the shock moves upstream w h e n the backpressure (ie condenser pressure) increases as expected from compressible flow theory. Considering that the ejector acts as a thermal compressor, then the importance of the shock pressure recovery which enhances the compression ratio and therefore, increases the entrainment ratio, can be seen. For example, in the present experimental work for Ejector l.b used in Section 3.6.2, w h e n the ratio of the pressure at start of the mixing tube to the condenser pressure was 1.96 and 1.38, the entrainment ratio was 0.209 and 0.081, respectively. In a practical application of small ejectors in which maximisation of the entrainment ratio is very important, then the importance of the shock process can be recognised.
181
Conclusions and Future Work
CHAPTER SIX
(d) O p t i m u m nozzle position
It was recognised in these experiments that the nozzle position is a very important factor of the ejector geometry for small V J R S ejectors. The importance of the prediction of this optimum nozzle position should be considered particularly if small V J R S ejectors were to be mass produced. The entrainment ratio was significantly affected by the nozzle position (eg see Figures 3.18 and 3.22). The m a x i m u m entrainment ratio was tested with an optimum nozzle position. Also, it has been found that the optimal nozzle position is sensitive to operating conditions. For example, w h e n the generator temperature was held constant and the evaporator temperature increased 1.0°C, the optimum nozzle position, X/D2, increased from 0.048 to 0.110. Expressed in terms of the secondary choking theory, this means w h e n the evaporator temperature is changed, nozzle position must be changed to form an appropriate converging cone for the secondary vapour. (e) Converging section of the ejector In the present investigation the geometry of the converging part of the ejector has been studied. The present author found some advantages and disadvantages for small and large converging half angles of the ejector, 0 m . For example, in regard to the entrainment ratio, 0
m
is not a significant factor for optimal design
dimensions of a small V J R S ejector. However, for smaller values of 0 m , the nozzle position is not so critical and the ejector m a y operate well over a wider range of operating conditions. Possible blocking of the secondary vapour by the nozzle body for smaller 6m should be considered. For larger 0 m , blocking of the
182
Conclusions and Future Work
CHAPTER SIX
secondary stream by the nozzle body is not a problem, but the nozzle position is critical (see Figures 3.10 and 3.14 and Section 3.10).
6.3 Analysis of Two-Fluid VJRS The work carried out in the present study used the one-dimensional analysis, constant area method, for a two-fluid V J R S ejector. In this analysis it was found that for an appropriatefluid-pair,the entrainment ratio is likely to be substantially better than for the single-fluid V J R S and the motive vapour in the generator m a y operate at lower pressure compared with the single fluid ejector. For example, it was found using a R11-R12 ejector in which R l 1 was the primary flow and R 1 2 the secondary one, the entrainment ratio m a y increase compared with a R 1 2 ejector and the generator pressure m a y be m u c h lower. In practical applications of small VJRS, like in automobile air conditioning, maximising the entrainment ratio as well as avoiding problems such as the condenser size which a single-fluid ejector involves, a two-fluid ejector m a y be a promising candidate. The present author was not able to investigate fluid-pairs other than R 1 1 - R 1 2 . Also this proposed two-fluid ejector was not tested experimentally due to lack of time.
183
Conclusions and Future Work
CHAPTER SIX
6.4 Future Work Experimental investigation and theoretical modelling of small ejectors have been carried out by the present author. There are still some problems to address such as the low entrainment ratio and implications for practical applications. Therefore, future work can be listed as follows.
6.4.1 Practical work Since there are optimum dimensions for a given singlefluidejector, optimising some dimensions m a y improve the system performance. For example for the mixing tubes length used in these experiments were 10 and 12 times the diameter of the tube in which according to the literature, 12 times is the largest length recommended.
Therefore, in future work shorter lengths need to be examined,
because this m a y reduce the friction effects. Only the converging half angles of 10° and 20° were tested in the present work. In the C F D analysis a recirculation of the flow was observed around the walls of the converging section of the ejector. Thus, several angles can be tested in future study to examine the possible effects of recirculation on system performance of the small ejectors. For two-fluid small V J R S ejectors, a preliminary study on the effect of dissimilar molecular weights on ejector performance has been carried out in the present research. Considering the importance of the ejector performance in a practical application, it was mentioned that using an appropriate fluid-pair, the entrainment ratio is likely to be better than for the single fluid ejector. T o investigate the
184
Conclusions and Future Work
CHAPTER SIX
practical problems and implications, in the future study a test rig needs to be designed to examine the ejector performance of a two-fluid V J R S .
6.4.2 Theoretical modelling The behaviour of a confined, turbulent supersonic jet is very complicated. In the present C F D analysis the flow w a s treated as an ideal gas and the domain was selected to be from the cross section at the throat of the nozzle to the end of the constant area mixing tube. M o r e detailed geometry could be looked at, eg diffuser included. The present author generated three different grids in which the number of cells were different, but further work is required to ensure that the grid really is refined sufficiendy. There are s o m e other limitations in the present work. For example, expanding a superheated vapour in a convergent-divergent nozzle m a y in reality be a vapour in metastable equilibrium or it m a y be a mixture of vapour and liquid. This was not considered in the present thesis. For two-fluid ejectors with regard to the environmental problems such as ozone depletion potential and chemical toxicity, other replacementfluid-pairsinstead of R 1 1 - R 1 2 could be investigated in a feasibility study for a two-fluid V J R S . Practical alternatives to R l l and R 1 2 which currently available would include R 1 4 2 and R134a, respectively. Also, the feasibility of the separation of the components in the condenser should be considered.
185
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Patankar, S. V., "Numerical Heat Transfer and Fluid Flow". McGraw Hill, 1980.
Perry, R. H. and Chilton, C. H. (editors), "Chemical Engineers' Handbook". Fif edition, McGraw Hill, 1985.
193
References
Puckett, A. E. and Pasadena, C, "Supersonic Nozzle Design", Journal of Applie Mechanics. December, 1946, pp. A-265 to A-270.
Razinsky, E. and Brighton, J. A., "A Theoretical Model for Nonseparated Mixin of a Confined Jet", Journal of Basic Engineering, Transaction of the A S M E , Vol. 94, Series D, September 1972, pp. 551-558.
Rohsenow, W. M., Hartnett, J. P. and Ganic, E. N. (editors), "Handbook of Heat Transfer Fundamentals". Second Edition, M c G r a w Hill, 1985.
Schmitt, H., Bertin and Cie, "Diversity of Jet Pumps and Ejector Techniques", Proceedings of the 2nd Symposium on Jet Pumps and Ejectors and Gas Lift Techniques, Published by B H R A Fluid Engineering, England, Paper A4, pp. A435 to A4-49, March, 24th- 26th, 1975. Shapiro, A. H., "The Dynamics and Thermodynamics of Compressible Fluid Flow", Vol. 1, The Ronald Press Company, N e w York, 1953. Shaw, C. T., "Using Computational Fluid Dynamics". Prentice Hall, N e w York, 1992. Sokolov, M . and Hershgal, D., "Compression Enhanced Ejector Refrigeration Cycle for L o w Grade Heat Utilisation", Proceedings of the 24th Inter Society Energy Conversion Engineering Conference, IECEC-89, Vol. 5, Washington, D.C., pp. 2543-2548, August 6-11,1989. Sokolov, M . and Hershgal, D., "Optimal Coupling and Feasibility of a Solar Powered Year-Round Ejector Air Conditioner", Tournal of Solar Energy, Vol. 50, No. 6, pp. 507-5016,1993.
194
References
Spauschus, H. O., " H F C 134a as a Substitute Refrigerant for C F C 12", Georgia Technical Research Institute, Atlanta, G A 30332, U.S.A., Nov., 1988.
Stoecker, W. F. and Jones, J. W., "Refrigeration and Air Conditioning". McGraw Hill, Singapore, 1982.
Stoecker, W. F., "Refrigeration and Air Conditioning". McGraw Hill, New York, N.Y., Chapter 13,1958. Tolouie, C. M., Behnia, M. and Leonardi, E., "Computer Simulation of Mixed Refrigerant Cycles", Journal of Australian Refrigeration. Air Conditioning and Heating (AJRAH). Vol. 45, No. 5, pp. 18-23, May, 1991. Tyagi, K. P. and Murty, K. N., "Ejector Compression Systems for Cooling; Utilising L o w Grade Waste Heat", Journal of Heat Recovery Systems. Vol. 5, No. 6, Printed in Great Britain, pp. 545-550, 1985. V a n Wylen, G. J. and Sonntag, R. E., "Fundamentals of Classical Thermodynamics". Third Edition, John Wiley and Sons, Inc., Singapore, 1985.
Vyas, B. D. and Kar, S., "Study of Entrainment and Mixing Process for an Air to Air Jet Ejector", Proceedings of the 2nd Symposium on Jet Pumps and Ejectors and Gas Lift Techniques, Published by B H R A Fluid Engineering, England, Paper C2, pp. C2-15 to C2-25, March, 24th- 26th, 1975. Wall, T. F., Nguyen, H., Subramanian, V., Mai-Viet, T. and Howley, P., "Direct Measurements of the Entrainment by Using and Double Concentric Jets in the Regions of Transition and Flow Establishment", Institution of Chemical Engineers, Trans IChem E, Vol. 58, pp. 237-241,1980.
195
References
Wall, T. F., Subramanian, V. and Howley, P., "An Experimental Study of Geometry, Mixing and Entrainment of Particle-Laden Jets up to Ten Diameters from the Nozzle", Institution of Chemical Engineers, Trans IChem E, Vol. 60, 0046-9858/82/040231-09,1982, pp. 231-239.
Whitaker, R, "An Experimental Study Into Cold Air Ejector Mixing Duct for Computerised Design Purposes", Proceedings of the 2nd Symposium on Jet Pumps and Ejectors and Gas Lift Techniques, Published by B H R A Fluid Engineering, England, Paper D2, pp. D2-25 to D2-44, March, 24th- 26th, 1975.
Work, L. T. and Haedrich, V. W., "Performance of Ejectors as a Functi Molecular Weghts of Vapours", Journal of Industrial and Engineering Chemistry. Vol. 31, No. 4, pp. 464-477, April, 1939. Work, L. T. and Miller, A., "Factor C in the Performance of Ejectors as a Function of Molecular Weights of Vapours", Journal of Industrial and Engineering Chemistry. Vol. 32, No. 9, September, 1940, pp. 1241-1245.
Yeung, M. R, Yuen, P. K., Dunn, A. and Cornish, L. S., "Performance of Powered Air Conditioning System in Hong Kong", Tournal of Solar Energy, Vol. 48, No. 5, Printed in USA, 1992, pp. 309-319. Zeren, F. and Holmes, R. E., "Performance Evaluation for a Jet Pump Solar Cooling System",
ARVIR
*1-WA/Sol-30.1981, pp. 1-8.
Zeren, F„ Holmes, R. E. and Jenkins, P. E , "Design of a Freon Jet Pump for Use in a Solar Cooling System", ASME, 78-WA/Sol-15, Aug., 1978, pp. 1-9
196
APPENDIX A
A.1 The Ql File The Q lfilefor the present C F D analysis using PHOENICS-code is presented on the next pages.
197
Appendix A
TALK=T;RUN( 1, 1);VDU=VGAM0USE IRUNN = 1 ;LIBREF = 0 ****************************************************
TEXT(R12 SUPERSONIC EJECTOR MODELLING FOR TURBULENT FLOW) real(MACH) REAL(GASCON,GAMMA,CP);GASCON=68.7;GAMMA=1.4 CP=GAMMA*GASCON/(GAMMA-1.)
•• Primary inlet parameters - calculations for nozzle throat REAL(PPRM,TPRm,DENPRIM,PINl,TINl,UINl,HINl,DENINl) PPRIM=2.0068e+06;TPRIM=73.+273.15;DENPRIM=123.10 ••• For choked flow at the throat of the nozzle, the pressure, temperat *** velocity may be calculated from Equations (2.11), (2.15a) and (2.15b), •••respectively as follows: PESfl=PPRIM+((27(GAMMA+l .))**(GAMM A/(GAMMA-1.))) TINl=TPRIM+(2./(l.+GAMMA)) UIN1=(GAMMA + GASCON + TIN1) + *0.5 HIN1=CP + TIN1 DENINl=DENPRIM+((2./(GAMMA+l.))**(l./(GAMMA-l.))) •• Secondary stream parameters REAL(PSEC,TSEC,DENSEC,PIN2,TIN2,HIN2) PSEC=4.574E+5;TSEC=12.5+273.15;DENSEC=26.203 PIN2=PSEC;TIN2=TSEC HIN2=TIN2 + CP •• Outiet Conditions REAL(POUT) POUT=8.1E+05 ************************************************************
Group 2. Transience STEADY = T ************************************************************
Groups 3,4,5 Grid Information •• Overall number of cells, RSET(M,NX,NY,NZ,tolerance) RSET(M,NX,NY,NZ) _ mrr AO _ •• Overall domain extent, RSET(D,name,XULAST,YVLAST,ZWLAST) RSETP,EJECTOR,1.080E-01,2.000E-02,1.00E-01) ************************************************************
Group 6. Body-Fitted coordinates INTEGER(NX1,NX1F,NX2,NX2F,NX3,NX) INTEGER(NYl,NYlF^rY2,NY2F,NY3,NY) NX1=30 ;NX1F=NX1+1 ;NX2=30 ;NX2F=NX1+NX2+1 ;NX3=60 ;NX=NX1+NX2+NX3 rfs t ^„ 1C NY1=14 ;NYlF=NYl+l ;NY2=l ;NY2F=NYl+NY2+l ;NY3=l5 ;NY=NY1+NY2+NY3 BFC=T •• Set points 198 GSET(P,P9, lO.0O0E-03,O.0O0OE+O0,0.0O0OE+0O) GSET(P,PlO,lO.OOOE-03,l.2lOOE-03,O.OOOOE+00) GSET(P,P11, lO.OOOOE-03,3.2 l00E-03,0.0000E+00)
Appendix A
GSET(P,P12,10.000E-03,5.376E-03,0.0000E+00) GSET(P,P13,1.45000E-02,3.738E-03,0.0000E+00) GSET(P,P14,1.45000E-02,1.5000E-03,O.OOOOE+00) GSET(P,P28,1.45000E-02,1.7000E-03,0.0000E+00) GSET(P,P29,1.9000E-02,1.7000E-03,0.0000E+00) GSET(P,P30,6.1000E-02,1.7000E-03,0.0000E+00) GSET(P,P15,1.45000E-02,0.0000E+00,0.0000E+00) GSET(P,P16,1.9000E-02,O.OOOOE+00,O.OOOOE+00) GSET(P,P17,1.900OE-02,1.5O00E-03,0.0000E+O0) GSET(P,P18,1.9000E-02,2.1000E-03,0.0000E+00) GSET(P,P19,6.1000E-02,2.1000E-03,0.0000E+O0) GSET(P,P20,6.1000E-02,1.5000E-03,0.0000E+00) GSET(P,P21,6.10O0E-O2,O.0000E+O0,0.0O00E+O0) •• Set lines/arcs GSET(L,L10,P11,P12,NY3,0.6) GSET(L,L42,P11,P10,NY2,1) GSET(L,L43,P28,P29,NX2,1) GSET(LX44,P29,P17,NY2,1) GSET(L,L45,P29,P30,NX3,1) GSET(L,L46,P20,P30,NY2,1) GSET(L,L13,P9,P10,NY1,0.6) GSET(L,L15,P12,P13,NX1,1.0) GSET(L,L16,P28,P13,NY3,0.6) GSET(L,L41,P14,P28,NY2,1) GSET(L,L17,P28,P11,NX1,1.0) GSET(L,L18,P14,P10,NX1,1.0) GSET(L,L19,P15,P14,NY1,0.6) GSET(L,L20,P15,P9,NX1,1.0) GSET(LX21,P13,P18,NX2,1.0) GSET(L,L22,P18,P29,NY3,1.0) GSET(L,L23,P17,P14,NX2,1.0) GSET(L,L24,P17,P16,NY1,1.0) GSET(L,L25,P16,P15,NX2,1.0) GSET(L,L26,P18,P19,NX3,1.0) GSET(L,L27,P30,P19,NY3,0.6) GSET(L,L28,P20,P17,NX3,1.0) GSET(L,L29,P21,P20,NY1,0.6) GSET(L,L30,P21,P16,NX3,1.0) •• Set frames GSET(F,F5,P9,-,P10,-,P14,-,P15,-) GSET(F,F6,P11,-,P12,-,P13,- J*28,-) GSET(F,F7,P28,-,P13,-,P18,-,P29,-) GSET(F,F8,P15,-,P14,-,P17,-,P16,-) GSET(F,F15,P10,-,P11,-,P28,-,P14,-) GSET(F,F16,P14,-,P28,-,P29,-,P17,-) GSET(F,F17,P17,-,P29,-,P30,-,P20,-) GSET(F,F9,P16,-,P17,-,P20,-,P21,-) GSET(F,F10,P29,-,P18,-,P19,-,P30,-) •+ Match a grid mesh GSET(M,F5,+J+I,1,1,1,TRANS) 199
Appendix A
GSET(M,F6,+J+I,1,NY2F,1,TRANS) GSET(M,F7,+J+I,NX1F,NY2F,1,TRANS) GSET(M,F8,+J+I,NX1F,1,1,TRANS) GSET(M,F15,+J+I,1,NY1F,1,TRANS) GSET(M,F16,+J+I,NX1F,NY1F,1,TRANS) GSET(M,F17,+J+I,NX2F,NY1F,1,TRANS) GSET(M,F9,+J+I,NX2F, 1,1 ,TRANS) GSET(M,F10,+J+I,NX2F,NY2F,1,TRANS) •• Copy/Transfer/Block grid planes GSET(C,K2,F,K1,1 ,NX, 1 ,NY,RX, 1 .OOOE-01,0.0,0. ,INC, 1) **********
NONORT = T •• X-cyclic boundaries switched VIEW(K,1) STOP ************************************************************
Group 7. Variables: STOREd,SOLVEd,NAMEd ONEPHS = T •• Non-default variable names NAME(47) = E N U T ; NAME(48) =VCRT NAME(49) =RH01 ; NAME(50) =UCRT; NAME(45)=MACH NAME(44)=UCMP ; NAME(43) =VCMP; NAME(42)=WCMP NAME(46)=TMP1 •• Solved variables list S0LVE(P1,U1,V1) SOLVE(Hl) SOLVE(KE,EP) •• Stored variables list STORE(RH01,ENUT,PTOT,VABS) STORE(MACH) STORE(UCMP,VCMP,WCMP) STORE(TMPl) •• Additional solver options SOLUTN(Pl ,Y,Y,Y,N,N,N) SOLUTN(Hl ,Y,Y,Y,N,N,N) ************************************************************
Group 8. Terms & Devices TERMS (KE ,N,Y,Y,Y,Y,N) TERMS (EP ,N,Y,Y,Y,Y,N) TERMS (U1,Y,Y,N,N,Y,N) TERMS (V1,Y,Y,N,N,Y,N) TERMS (H1,P,P,N,P,P,P) NEWRH1 = T NEWENT = T ************************************************************
Group 9. Properties RHOl =GRND5 200 RHOIB =l/GASCON ;RH01C = 1/GAMMA TMP1=GRND2 ELI PRESSO TMP1B=UCP = GRND4 = 1.000E+0
Appendix A
ENUL =9.856E-08 TURMOD(KEMODEL) ENUT =GRND3 DRH1DP =GRND5 ************************************************************
Group lO.Inter-Phase Transfer Processes ************************************************************
Group 11.Initialise Var/Porosity Fields RESTRT(ALL) FIINIT(Pl) = L000E+04 ;FITNIT(U1) = 1.000E+01;FnNIT(Vl) = 0.000E+00 FIINIT(Hl) = HIN1 No PATCHes used for this Group RSTGRD = F INIADD = F FnNTT(KE)=l.E-6 FIINIT(EP)=LE-6 CONPOR(BLOK1,0.,CELL,1,NX1,NY1F,NY1F,1,NZ) ************************************************************
Group 12. Convection and diffusion adjustments ************************************************************
Group 13. Boundary & Special Sources PATCH (KESOURCE,PHASEM,l,NX,l,NY,l,NZ,l,l) COVAL(KESOURCE,KE ,GRND4 ,GRND4 ) C O V A L (KESOURCE,EP , GRND4 , GRND4 ) ••• Primary Inlet ••• INLET(INL1,WEST,1,1,1,NY1,1,NZ,1,1) VALUE(INL1,P1,DENIN1+UIN1) VALUE(DSTL1,H1,HIN1) ••• Secondary Inlet ••• PATCH(INL2,WEST,1,1,NY2F,NY,1,NZ,1,1) COVAL(INL2,Pl,-2.+DENSEC,PIN2) COVAL(INL2,Ul,ONLYMS,SAME) COVALaNL2,Vl,ONLYMS,SAME) COVALaNL2,Hl,FTXVAL,HIN2) ••• Outlet Conditions ••• OUTLET(OUTl,EAST,NX,NX,l,NY,l,NZ,l,l) VALUE(OUTl,Pl,POUT) COVAL(OUTl,Ul,ONLYMS,SAME) COVAL(OUTl ,H1 ,ONLYMS,S AME) ••• Wall Conditions •• WALL(WALl,NORTH,l,NX,NY,NY,l,NZ,l,l) COVAL(WAL1,U1,GRND2,0.) C0VAL(WAL1,KE,GRND2,GRND2)
201
Appendix A
Group 15. Terminate Sweeps LSWEEP = 1000 SELREF = T RESFAC = 1.000E-02 ************************************************************
Group 16. Terminate Iterations ************************************************************
Group 17. Relaxation RELAX(P1 .LENTRLX, 1.000E-01) RELAX(U1 ,FALSDT, 2.857E-05) RELAX(V1 ,FALSDT, 2.857E-05) RELAX(hl ,FALSDT, 2.857E-05) RELAX(KE ,FALSDT, 2.857E-10) RELAX(EP ,FALSDT, 2.857E-10) KELIN =1 Group 18. Limits Group 19. EARTH Calls To GROUND Station GEND=T UCONV=T NAMGRD=CONV ************************************************************
Group 20. Preliminary Printout
ECHO =T ************************************************************
Group 21. Print-out of Variables ************************************************************
Group 22. Monitor Print-Out IXMON = 55 ;IYMON = TSTSWP = -2
15 ;IZMON =
1
************************************************************
Group 23.Field Print-Out & Plot Control No PATCHes used for this Group NXPRLN=2 ;NYPRIN=2 ************************************************************
Group 24. Dumps For Restarts ************************************************************
STOP
202
Appendix A
A.2 The Result File The result file for the present CFD analysis using PHOENICS-code is presented on the next pages. This resultfileis for Grid 2 (see Table 2.2).
203
Appendix A
************************************************************
CCCC H H H PHOENICS Version 2.1.1 - EARTH CCCCCCCC H H H H H (C) Copyright 1995 18.04.1995 CCCCCCC H H H H H H H H H H Concentration Heat and Momentum Ltd CCCCCCC H H H H H H H H H H H H All rights reserved. C C C C C C H H H H H H H H H H H H H Address: Bakery House, 40 High St CCCCCCC H H H H H H H H H H H H Wimbledon, London, SW19 5AU CCCCCCC H H H H H H H H H H Tel: 0181-947-7651 CCCCCCCC H H H H H Facsimile: 0181-879-3497 E-mail: [email protected] CCCC H H H This program forms part of the PHOENICS installation for:
CHAM The code expiry date is the end of: Aug 1995 ************************************************************
Number of F-array location available is 6214656 Number used before BFC allowance (if any) is 546321 Number of BFC-geometry F-array locations is 190400 754187 Total number of F-array locations used is 65 Number of BFC-geometry F-array variables is Number of BFC-geometry variables on disk is 0 ************************************************************
Group 1. Run Title and Number ************************************************************ ************************************************************
TEXT(R12 SUPERSONIC EJECTOR MODELLING ) ************************************************************ ************************************************************
IRUNN = 1;LIBREF= 0 ************************************************************
Group 2. Transience STEADY = T ************************************************************
Group 3. X-Direction Grid Spacing CARTES = T NX = 70 XULAST = 1.080E-01 ************************************************************
Group 4. Y-Direction Grid Spacing NY = 40 *********;**"************************************************
Group 5. Z-Direction Grid Spacing PARAB = F NZ = 1
*^£*T***V^^ Group 6. Body-Fitted Coordinates
204
Appendix A
BFC = T 0SfONORT= T NCRT = 1 RSTGEO = F ;SAVGEO= F UUP = F ;VUP = F ;WUP = F NGEOM =CHAM NAMXYZ =CHAM ************************************************************
Group 7. Variables: STOREd,SOLVEd,NAMEd ONEPHS = T NAME(1)=P1 ;NAME(3)=U1 NAME(5)=V1 ;NAME(12)=KE NAME(13)=EP ;NAME(14)=H1 NAME(39) =VPOR ;NAME(40) =TMP1 NAME(41) = W C M P ;NAME(42) =VCMP NAME(43) =UCMP ;NAME(44) = M A C H NAME(45) =VABS ;NAME(46) =PTOT NAME(47) =ENUT iNAME(48) =VCRT NAME(49) =RH01 ;NAME(50) =UCRT * Y in SOLUTN argument list denotes: * 1-stored 2-solved 3-whole-field * 4-point-by-point 5-explicit 6-hannonic averaging SOLUTN(Pl ,Y,Y,N,N,N,N) SOLUTN(Ul ,Y,Y,N,Y,N,Y) SOLUTNCV1 ,Y,Y,N,Y,N,Y) SOLUTN(KE ,Y,Y,N,N,N,Y) SOLUTN(EP ,Y,Y,N,N,N,Y) SOLUTN(Hl ,Y,Y,N,N,N,N) SOLUTN(VPOR,Y,N,N,N,N,N) SOLUTN(TMPl,Y,N,N,N,N,Y) SOLUTN(WCMP,Y,N,N,N,N,Y) SOLUTN(VCMP,Y,N,N,N,N,Y) SOLUTN(UCMP,Y,N,N,N,N,Y) SOLUTN(MACH,Y,N,N,N,N,Y) SOLUTN(VABS,Y,N,N,N,N,Y) SOLUTN(PTOT,Y,N,N,N,N,Y) SOLUTN(ENUT,Y,N,N,N,N,Y) SOLUTN(VCRT,Y,N,N,N)N,N) SOLUTN(RH01,YJSf,N,N,N,Y) SOLUTN(UCRT,Y,N,N,N,N,N) DEN1 = 49 VIST = 47 EPOR = 0;HPOR = 0;NPOR = 0;VPOR = 39 TEMPI = 40 ************************************************************
Group 8. Terms & Devices * Y in T E R M S argument list denotes: * 1-built-in source 2-convecdon 3-diffusion 4-transient * 5-first phase variable 6-interphase transport T E R M S (Pl ,Y,Y,Y,N,Y,N) T E R M S (Ul ,Y,Y,Y,N,Y,N) T E R M S (VI ,Y,Y,Y,N,Y,N) T E R M S (KE ,N,Y,Y,N,Y,N) T E R M S CEP ,N,Y,Y,N,Y,N) T E R M S (HI ,Y,Y,N,N,Y,N) DIFCUT = 5.000E-01 ;ZDIFAC = l.OOOE+00
GALA = F ;ADDDIF= F ;BLOCKZ= T NEWRH1 = T 205
Appendix A
NEWENT = T UCONV = T ISOLX = -1 ;ISOLY =
-1 ;ISOLZ =
0
************************************************************
Group 9. Properties RHOl = G R N D 5 ;TMP1 = G R N D 2 ;EL1 =GRND4 TSURR =0.000E+00 RHOl A = 0.000E+O0 -.RHOIB = 1.456E-02 ;RH01C = 7.692E-01 PRESSO = 1.000E+00 DRH1DP = GRND5 TMP1A = 0.000E+00 ;TMP1B = 3.359E-03 ;TMP1C = 0.000E+00 TEMPO = 0.000E+O0 EL1A = 0.000E+00 ,EL1B = O.OOOE+00 ;EL1C =0.OO0E+O0 ENUL =9.856E-08;ENUT =GRND3 ENUTA = 0.000E+00 ;ENUTB = 0.000E+00 ;ENUTC = O.OOOE+00 TENUTA = 0 PRNDTL(U1 )= 1.000E+00 ;PRNDTL(V1 )= 1.000E+00 PRNDTLCKE )= 1.000E+00 ;PRNDTLC3P )= 1.000E+00 PRT (Ul )= 1.000E+00;PRT (VI )= 1.000E+00 PRT (KE ) = 1.000E+00 ;PRT (EP )= 1.314E+00 ************************************************************
Group lO.Inter-Phase Transfer Processes ************************************************************
Group 1 Unitialise Var/Porosity Fields FUNTTCPl ) = READFI ;HINIT(U1 ) = READFI FITMT(V1 ) = READFI ;FIINIT(KE ) = R E A D H FIINIT(EP ) = R E A D H ;FnNIT(Hl ) = READFI FDNTTCVPOR) = 1.000E+00 ;HINIT(TMP1) = R E A D H FnNTT(WCMP) = R E A D H -^IINIT(VCMP) = READFI FIINIT(UCMP) = R E A D H ;HINIT(MACH) = R E A D H FnNTT(VABS)= R E A D H ;HINIT(PTOT) = READFI FBhrnXENUT) = R E A D H ;HINIT(VCRT) = READFI FIINITCRHOl) = READFI ;HINIT(UCRT) = R E A D H PATCH(BLOKl ,INIVAL, 1, 20, 20, 20, 1, 1, 1, 1) INJT(BLOKl ,VPOR, 0.000E+00, O.OOOE+00) INIADD = F RSTGRD = F FS W E E P = 1
NAMH =CHAM ************************************************************
Group 12. Patchwise adjustment of terms Patches for this group are printed with those for Group 13. Their names begin either with GP12 or & *************!********************************************** Group 13. Boundary & Special Sources PATCH(KESOURCE,PHASEM, 1, 70, 1, 40, 1, 1, 1, 1) COVAL(KESOURCE,KE , G R N D 4 , G R N D 4 ) COVAL(KESOURCE,EP , G R N D 4 , G R N D 4 ) PATCH(INL1 .WEST , 1, 1, 1, 19, 1, 1, 1, D COVAL(INLl ,P1 , H X F L U , 1.267E+04) C O V A L 0 N L 1 ,U1 , 0.000E+00,0.000E+00) COVAL(INLl ,V1 .O.OOOE+OO, 0.000E+00) COVAL(INLl ,KE , O.OOOE+00,0.000E+00)
206
Appendix A
COVAL0NL1 ,EP .0.000E+00, O.OOOE+00) COVALflNLl ,H1 ,0.000E+00, 8.961E+04) PATCH(LNL2 ,WEST , 1, 1, 21, 40, 1, 1, 1, 1) COVAL(INL2 J>1 .-5.810E+01.4.554E+05) COVAL(INL2 ,U1 , O.OOOE+00, SAME ) COVAL(INL2 ,V1 , O.OOOE+00, SAME ) COVAL(INL2 ,KE , 0.000E+00, O.OOOE+00) COVAL0NL2 ,EP , 0.000E+00, O.OOOE+00) COVAL(INL2 ,H1 ,HXVAL , 8.600E+04) PATCH(0UT1 .EAST , 70, 70, 1, 40, 1, 1, 1, 1) COVAL(OUTl ,P1 ,1.000E+03, 8.100E+05) COVAL(OUTl ,U1 ,0.000E+00, SAME ) COVAL(OUTl ,V1 ,0.000E+00, SAME ) COVAL(OUTl ,KE , O.OOOE+00, SAME ) COVAL(OUTl ,EP , O.OOOE+00, SAME ) COVAL(OUTl ,H1 ,0.O00E+00, SAME ) PATCH(WAL1 ,NWALL, 1, 70, 40, 40, 1, 1, 1, 1) COVALfWALl ,U1 ,GRND2 , O.OOOE+00) COVALfWALl ,KE , GRND2 , GRND2 ) COVALfWALl ,EP , GRND2 , GRND2 ) XCYCLE = F EGWF = F ************************************************************
Group 14. Downstream Pressure For PARAB ************************************************************
Group 15. Terminate Sweeps 1 LSWEEP = 1000 ;ISWC1 = LITHYD = 1 ;LITFLX = 1 ;LITC = 1 ;ITHC1 = SELREF = T RESFAC = 1.000E-02 RESREF(P1 )= 1.000E-08 ;RESREF(U1 )= 1.000E-08 RESREF(V1 )= 1.000E-08 ;RESREF(KE )= 1.000E-08 RESREF(EP )= 1.000E-08 ;RESREF(H1 )= 1.000E-08
1
************************************************************
Group 16. Terminate Iterations LITERCPl )= 20;LITER(U1 )= 1 1 ;LJTER (KE ) = 20 LITER (VI ) = LITER CEP )= 20 LITER (HI )= 20 ENDIT(P1 )= 1.000E-03 ;ENDIT (Ul )= 1.000E-03 ENDIT(V1 )= 1.000E-03;ENDIT(KE )= 1.000E-03 ENDIT(EP )= 1.000E-03;ENDIT(H1 )= 1.000E-03 ************************************************************
Group 17. Relaxation RELAX(P1 LINRLX, 1.000E-01) RELAXtUl ,FALSDT, 2.857E-05) RELAX(V1 J'ALSDT, 2.857E-05) RELAX(KE .FALSDT, 2.857E-10) RELAXCEP ,FALSDT, 2.857E-10) RELAXCH1 ^ALSDT, 2.857E-05) RELAX(TMP1LINRLX, 1.000E+00) RELAX(WCMP,LINRLX, 1.000E+00) RELAX(VCMP,LINRLX, 1.000E+00) RELAX(UCMP,LINRLX, 1.000E+00) RELAX(MACH,LINRLX, 1.000E+00)
207
Appendix A
RELAX(VABS,LINRLX, l.OOOE+00) RELAX(PTOT,LINRLX, l.OOOE+00) RELAX(ENUT,LINRLX, l.OOOE+00) RELAX(VCRT,LINRLX, 1.000E+09) RELAXCRHOILINRLX, l.OOOE+00) RELAX(UCRT,LINRLX, 1.000E+09) KELIN = 1 OVRRLX =0.000E+00 EXPERT = F ;NNORSL= F ************************************************************
Group 18. Limits V A R M A X ( P 1 ) = 1.000E+10 ;VARMIN(P1 )=-1.000E+10 V A R M A X ( U 1 )=1.000E+10;VARMINCU1 )=-1.000E+10 V A R M A X ( V 1 ) = 1.000E+10 ;VARMIN(V1 ) =-1.000E+10 V A R M A X ( K E ) = 1.000E+10 ;VARMIN(KE ) = 1.000E-10 V A R M A X ( E P ) = 1.000E+10 ;VARMIN(EP ) = 1.000E-10 V A R M A X ( H 1 ) = 1.000E+10 ;VARMIN(H1 ) =-1.000E+10 V A R M A X ( V P O R ) = 1.000E+10 ;VARMIN(VPOR) =-1.000E+10 V A R M A X f T M P l ) = 1.000E+10 ;VARMIN(TMP1) =-1.000E+10 V A R M A X ( W C M P ) = 1.000E+10 ;VARMIN(WCMP) =-1.000E+10 V A R M A X ( V C M P ) = 1.000E+10 ;VARMIN(VCMP) =-1.000E+10 V A R M A X ( U C M P ) = 1.000E+10 ;VARMIN(UCMP) =-1.000E+10 V A R M A X ( M A C H ) = l.OOOE+10 ;VARMIN(MACH) =-1.000E+10 V A R M A X ( V A B S ) = l.OOOE+10 ;VARMIN(VABS) =-1.000E+10 V A R M A X ( P T O T ) = l.OOOE+10 ;VARMIN(PTOT) =-1.000E+10 V A R M A X ( E N U T ) = l.OOOE+10 ;VARMIN(ENUT) =-1.000E+10 V A R M A X C V C R T ) = l.OOOE+10 ;VARMIN(VCRT) =-1.000E+10 V A R M A X C R H O l ) = l.OOOE+10 ;VARMIN(RH01) =-1.000E+10 V A R M A X ( U C R T ) = l.OOOE+10 ;VARMIN(UCRT) =-1.000E+10 ************************************************************
Group 19. EARTH Calls To GROUND Station USEGRD = T ;USEGRX= T NAMGRD =CONV GENK = T ************************************************************
Group 20. Preliminary Printout
ECHO = T ************************************************************
Group 21. Print-out of Variables INIFLD = F ; S U B W G R = F * Y in O U T P U T argument list denotes: * 1-field 2-correction-eq. monitor 3-selective dumping * 4-whoIe-field residual 5-spot-value table 6-residual table OUTPUT(Pl ,Y,N,N,Y,Y,Y) OUTPUT(Ul ,Y,N,N,Y,Y,Y) OUTPUT(Vl ,Y,N,N,Y,Y,Y) OUTPUT(KE ,Y,N,N,Y,Y,Y) OUTPUT(EP ,Y,N,N,Y,Y,Y) OUTPUT(Hl ,Y,N,N,Y,Y,Y) OUTPUT(VPOR,Y,N,N,N,N,N) OUTPUT(TMPl,Y,N,N,N,N,N) OUTPUT(WCMP,Y,N,N,N,N,N) OUTPUT(VCMP,Y,N,N,N,N,N) OUTPUT(UCMP,Y,N,N,N,N,N) OUTPUT(MACH,Y,N,N,N,N,N) OUTPUT(VABS,Y,N,N,N,N,N) OUTPUT(PTOT,Y,N,N,N,N,N)
208
Appendix A
OUTPUT(ENUT,Y,N,N,N,N,N) OUTPUT(VCRT,N,N,N,N,N,N) 0UTPUT(RH01,Y,N,N,N,N,N) OUTPUT(UCRT,N,N,N,N,N>N) *****************************************lk#*!(!!|!!|1<„k++lk+1)(!)t!|(+1|(i|ii|i
Group 22. Monitor Print-Out 30 ;IYMON = 20 ;IZMON = 1 IXMON = N P R M O N = 10000 ;NPRMNT= 10000 ;TSTSWP = 10002 U W A T C H = F ;USTEER = F HIGHLO m F **************************#+#^##^#^###^^:(!!t:!)c)(!!)c!|[!(!!)(!)(!|(!|[!)[!)c;<[#]|[!|[
Group 23.Field Print-Out & Plot Control 5 NPRJNT = 1000 jNUMCLS = NXPRJN = 2 ;KPRF = 1 ;DCPRL = 70 NYPRIN = 2 ;IYPRF = 1 ;IYPRL = 40 IPLTF = 1 ;IPLTL = 1000 ;NPLT = 50 ISWPRF = 1 ;ISWPRL = 10000 ITABL = 1 ;IPROF = 1 ABSIZ = 5.000E-01 ;ORSIZ = 4.000E-01 NTZPRF = l;NCOLPF= 50 ICHR = 2;NCOLCO= 45;NROWCO= 20 No PATCHes yet used for this Group •t^*************************************************,,.*.,,,,^,,.
Group 24. Dumps For Restarts SAVE = T ;AUTOPS = F ;NOWIPE = F NSAVE =CHAM — INTEGRATION OF EQUATIONS BEGINS —
*********************************************************,„** TTMESTP=
1 SWEEP NO= 1000ZSLABNO=
HTERNNO=
F L O W FIELD A T ITHYD= 1, IZ= 1, ISWEEP=1000, ISTEP= 1 FIELD VALUES OF Pl IY= 40 5.229E+05 4.929E+05 4.823E+05 4.868E+05 4.973E+05 IY= 38 5.174E+05 4.910E+05 4.819E+05 4.872E+05 4.979E+05 IY= 36 5.112E+05 4.888E+05 4.811E+05 4.875E+05 4.984E+05 IY= 34 5.069E+05 4.870E+05 4.802E+05 4.880E+05 4.991E+05 IY= 32 5.039E+05 4.855E+05 4.805E+05 4.892E+05 5.000E+05 IY= 30 4.986E+05 4.854E+05 4.830E+05 4.914E+05 5.OO9E+05 IY= 28 4.945E+05 4.878E+05 4.872E+05 4.941E+05 5.020E+05 IY= 26 4.933E+05 4.922E+05 4.920E+05 4.970E+05 5.034E+05 IY= 24 4.951E+05 4.973E+05 4.969E+05 5.001E+05 5.050E+05 IY= 22 4.995E+05 5.027E+05 5.013E+05 5.030E+05 5.070E+05 IY= 20 O.OOOE+00 0.000E+00 0.0O0E+0O O.OOOE+00 0.000E+00 IY= 18 2.888E+06 1.482E+06 1.175E+06 1.002E+06 8.794E+05 IY= 16 2.906E+06 1.489E+06 1.177E+06 1.003E+06 8.805E+05 IY= 14 2.922E+06 1.496E+06 1.180E+06 1.005E+06 8.814E+05 IY= 12 2.935E+06 1.505E+06 1.183E+06 1.006E+06 8.822E+05 IY= 10 2.946E+06 1.514E+06 1.187E+06 1.007E+06 8.828E+05 IY= 8 2.956E+06 1.523E+06 1.191E+06 1.008E+06 8.832E+05 IY= 6 2.965E+06 1.532E+06 1.195E+06 1.010E+06 8.835E+05 IY= 4 2.973E+06 1.541E+06 1.200E+06 1.011E+06 8.836E+05 IY= 2 2.979E+06 1.548E+06 1.204E+06 1.012E+06 8.836E+05
K=
1
3
5
7
9
IY= 40 5.051E+05 5.112E+05 5.177E+05 5.240E+05 5.292E+05
209
1
Appendix A
IY= 38 5.056E+05 5.116E+05 IY= 36 5.060E+05 5.121E+05 IY= 34 5.065E+05 5.126E+05 IY= 32 5.070E+05 5.131E+05 IY= 30 5.076E+05 5.136E+05 IY= 28 5.083E+05 5.142E+05 IY= 26 5.092E+05 5.149E+05 IY= 24 5.104E+05 5.160E+05 IY= 22 5.120E+05 5.174E+05 IY= 20 O.OOOE+00 0.000E+00 IY= 18 7.848E+05 7.083E+05 IY= 16 7.857E+05 7.091E+05 IY= 14 7.865E+05 7.098E+05 IY= 12 7.872E+05 7.105E+05 IY= 10 7.878E+05 7.112E+05 IY= 8 7.882E+05 7.118E+05 IY= 6 7.884E+05 7.123E+05 IY= 4 7.886E+05 7.127E+05 IY= 2 7.885E+05 7.130E+05 13 K= 11 15 IY= 40 5.334E+05 5.368E+05 IY= 38 5.334E+05 5.368E+05 IY= 36 5.337E+05 5.370E+05 IY= 34 5.340E+05 5.372E+05 IY= 32 5.343E+05 5.374E+05 IY= 30 5.344E+05 5.375E+05 IY= 28 5.345E+05 5.375E+05 IY= 26 5.345E+05 5.375E+05 IY= 24 5.346E+05 5.377E+05 IY= 22 5.348E+05 5.384E+05 IY= 20 5.234E+05 5.431E+05 IY= 18 5.187E+05 5.289E+05 IY= 16 5.123E+05 5.098E+05 IY= 14 5.090E+O5 4.946E+05 IY= 12 5.076E+05 4.845E+05 IY= 10 5.072E+05 4.788E+05 IY= 8 5.073E+05 4.759E+05 IY= 6 5.076E+05 4.747E+05 IY= 4 5.081E+05 4.743E+05 IY= 2 5.088E+05 4.744E+05 23 DG= 21 25 IY= 40 5.512E+05 5.529E+05 IY= 38 5.511E+05 5.528E+05 IY= 36 5.512E+05 5.528E+05 IY= 34 5.512E+05 5.528E+05 IY= 32 5.512E+05 5.528E+05 IY= 30 5.513E+05 5.528E+05 IY= 28 5.513E+05 5.527E+05 1Y= 26 5.513E+05 5.527E+05 IY= 24 5.512E+05 5.527E+05 IY= 22 5.509E+05 5.526E+05 IY= 20 5.503E+05 5.526E+05 IY= 18 5.476E+05 5.519E+05 IY= 16 5.394E+05 5.490E+05 IY= 14 5.293E+05 5.440E+05 IY= 12 5.191E+05 5.378E+05 IY= 10 5.089E+05 5.315E+05
5.180E+05 5.186E+05 5.189E+05 5.193E+05 5.197E+05 5.202E+05 5.208E+05 5.216E+05 5.228E+05 0.000E+00 6.448E+05 6.455E+05 6.462E+05 6.469E+05 6.476E+05 6.482E+05 6.489E+05 6.495E+05 6.500E+05 17 19 5.394E+05 5.394E+05 5.395E+05 5.397E+05 5.399E+05 5.403E+05 5.406E+05 5.409E+05 5.411E+05 5.413E+05 5.416E+05 5.323E+05 5.175E+05 5.003E+05 4.833E+05 4.692E+05 4.593E+05 4.530E+05 4.494E+05 4.473E+05 27 29 5.552E+05 5.552E+05 5.553E+05 5.554E+05 5.555E+05 5.556E+05 5.557E+05 5.557E+05 5.557E+05 5.558E+05 5.560E+05 5.564E+05 5.581E+05 5.591E+05 5.581E+05 5.557E+05
5.243E+05 5.247E+05 5.250E+05 5.253E+05 5.256E+05 5.259E+05 5.262E+05 5.268E+05 5.277E+05 O.OOOE+00 5.910E+05 5.916E+05 5.923E+05 5.929E+05 5.935E+05 5.942E+05 5.948E+05 5.954E+05 5.960E+05
5.294E+05 5.297E+05 5.300E+05 5.303E+05 5.305E+05 5.307E+05 5.308E+05 5.311E+05 5.316E+05 0.000E+00 5.451E+05 5.456E+05 5.460E+05 5.464E+05 5.469E+05 5.474E+05 5.480E+05 5.485E+05 5.490E+05
5.428E+05 5.428E+05 5.430E+05 5.433E+05 5.436E+05 5.441E+05 5.447E+05 5.451E+05 5.452E+05 5.451E+05 5.443E+05 5.361E+05 5.226E+05 5.085E+05 4.928E+05 4.763E+05 4.611E+05 4.488E+05 4.399E+05 4.340E+05
5.475E+05 5.475E+05 5.477E+05 5.479E+05 5.482E+05 5.485E+05 5.489E+05 5.493E+05 5.491E+05 5.486E+05 5.476E+05 5.420E+05 5.300E+05 5.172E+05 5.044E+05 4.907E+05 4.762E+05 4.624E+05 4.506E+05 4.418E+05
5.593E+05 5.593E+05 5.595E+05 5.597E+05 5.601E+05 5.604E+05 5.607E+05 5.610E+05 5.613E+05 5.615E+05 5.619E+05 5.626E+05 5.668E+05 5.730E+05 5.767E+05 5.773E+05
5.578E+05 5.591E+05 5.611E+05 5.631E+05 5.649E+05 5.665E+05 5.675E+05 5.684E+05 5.688E+05 5.689E+05 5.680E+05 5.679E+05 5.760E+05 5.870E+05 5.942E+05 5.969E+05
210
Appendix A
IY= 8 4.986E+05 5.253E+05 5.527E+05 5.760E+05 5.964E+05 1Y= 6 4.882E+05 5.195E+05 5.496E+05 5.734E+05 5.941E+05 IY= 4 4.788E+05 5.145E+05 5.468E+05 5.706E+05 5.913E+05 IY= 2 4.714E+05 5.110E+05 5.451E+05 5.685E+05 5.892E+05 33 K= 31 35 37 39 IY= 40 5.864E+05 6.203E+05 6.724E+05 6.616E+05 6.250E+05 IY= 38 5.865E+05 6.203E+05 6.724E+05 6.616E+05 6.249E+05 IY= 36 5.865E+05 6.202E+05 6.724E+05 6.616E+05 6.248E+05 IY= 34 5.865E+05 6.201E+05 6.724E+05 6.616E+05 6.248E+05 IY= 32 5.866E+05 6.20OE+05 6.724E+05 6.616E+05 6.247E+05 IY= 30 5.867E+05 6.199E+05 6.725E+05 6.616E+05 6.248E+05 IY= 28 5.867E+05 6.198E+05 6.725E+05 6.616E+05 6.248E+05 IY= 26 5.867E+05 6.197E+05 6.725E+05 6.616E+05 6.248E+05 IY= 24 5.868E+05 6.196E+05 6.726E+05 6.616E+05 6.249E+05 IY= 22 5.869E+05 6.195E+05 6.725E+05 6.616E+05 6.250E+05 IY= 20 5.873E+05 6.193E+05 IY= 18 5.880E+05 6.197E+05 6.723E+05 6.616E+05 6.249E+05 IY= 16 5.897E+05 6.204E+05 6.714E+05 6.614E+05 6.253E+05 IY= 14 5.960E+05 6.216E+05 6.702E+05 6.610E+05 6.259E+05 IY= 12 6.046E+05 6.229E+05 6.688E+05 6.605E+05 6.266E+05 IY= 10 6.124E+05 6.240E+05 6.675E+05 6.600E+05 6.271E+05 IY= 8 6.186E+05 6.250E+05 6.664E+05 6.596E+05 6.276E+05 IY= 6 6.231E+05 6.257E+05 6.656E+05 6.593E+05 6.280E+05 IY= 4 6.260E+05 6.262E+05 6.649E+05 6.590E+05 6.283E+05 IY= 2 6.276E+05 6.265E+05 6.645E+05 6.588E+05 6.285E+05 6.642E+05 6.587E+05 6.286E+05 43 IX= 41 45 49 IY= 40 6.021E+05 6.274E+05 47 IY= 38 6.021E+05 6.274E+05 6.250E+05 6.452E+05 6.560E+05 IY= 36 6.020E+05 6.274E+05 6.250E+05 6.452E+05 6.560E+05 IY= 34 6.020E+05 6.274E+05 6.250E+05 6.452E+05 6.560E+05 IY= 32 6.019E+05 6.275E+05 6.250E+05 6.452E+05 6.560E+05 IY= 30 6.018E+05 6.275E+05 6.250E+05 6.452E+05 6.560E+05 IY= 28 6.017E+05 6.275E+05 6.250E+05 6.452E+05 6.560E+05 IY= 26 6.016E+05 6.275E+05 6.250E+05 6.451E+05 6.560E+05 IY= 24 6.016E+05 6.275E+05 6.250E+05 6.451E+05 6.560E+05 IY= 22 6.016E+05 6.275E+05 6.250E+05 6.450E+05 6.560E+05 IY= 20 6.015E+05 6.278E+05 6.250E+05 6.448E+05 6.560E+05 IY= 18 6.013E+05 6.284E+05 6.253E+05 6.441E+05 6.563E+05 IY= 16 6.011E+05 6.289E+05 6.257E+05 6.429E+05 6.567E+05 IY= 14 6.010E+05 6.293E+05 6.262E+05 6.417E+05 6.572E+05 IY= 12 6.010E+05 6.296E+05 6.267E+05 6.404E+05 6.576E+05 IY= 10 6.011E+O5 6.298E+05 6.273E+05 6.393E+05 6.580E+05 IY= 8 6.012E+05 6.299E+05 6.278E+05 6.382E+05 6.583E+05 IY= 6 6.014E+05 6.299E+05 6.374E+05 6.585E+05 IY= 4 6.015E+05 6.300E+05 6.283E+05 6.287E+05 6.366E+05 6.586E+05 IY= 2 6.017E+05 6.300E+05 6.290E+05 6.361E+05 6.587E+05 53 55 IX= 51 IY= 40 6.544E+05 6.902E+05 6.293E+05 6.357E+05 6.587E+05 59 IY= 38 6.544E+05 6.902E+05 57 7.474E+05 8.257E+05 8.185E+05 IY= 36 6.544E+05 6.902E+05 IY= 34 6.544E+05 6.901E+05 7.473E+05 8.256E+05 8.185E+05 IY= 32 6.544E+05 6.901E+05 7.473E+05 8.256E+05 8.185E+05 IY= 30 6.544E+05 6.901E+05 7.473E+05 8.255E+05 8.185E+05 IY= 28 6.544E+05 6.900E+05 7.472E+05 8.254E+05 8.184E+05 IY= 26 6.544E+05 6.900E+05 7.472E+05 8.253E+05 8.183E+05 IY= 24 6.544E+05 6.899E+05 7.471E+05 8.252E+05 8.182E+05 IY= 22 6.545E+05 6.897E+05 7.471E+05 8.249E+05 8.181E+05 7.471E+05 8.246E+05 8.179E+05 7.471E+05 8.242E+05 8.177E+05
211
Appendix A
IY= 20 6.550E+05 6.892E+05 7.474E+05 8.240E+05 8.175E+05 IY= 18 6.558E+05 6.885E+05 7.477E+05 8.235E+05 8.170E+05 IY= 16 6.565E+05 6.878E+05 7.478E+05 8.236E+05 8.168E+05 IY= 14 6.573E+05 6.870E+05 7.480E+05 8.242E+05 8.169E+05 IY= 12 6.581E+05 6.863E+05 7.481E+05 8.252E+05 8.171E+05 IY= 10 6.590E+05 6.856E+05 7.482E+05 8.260E+05 8.175E+05 IY= 8 6.598E+05 6.850E+05 7.482E+05 8.265E+05 8.182E+05 IY= 6 6.605E+05 6.845E+05 7.482E+05 8.269E+05 8.187E+05 IY= 4 6.611E+05 6.840E+05 7.482E+05 8.273E+05 8.193E+05 IY= 2 6.617E+05 6.837E+05 7.481E+05 8.276E+05 8.197E+05 IX= 61 63 65 67 69 FIELD V A L U E S O F Ul IY= 40 -6.374E+01 -7.192E+01 -7.303E+01 -7.236E+01 -7.583E+01 IY= 38 -3.123E+01 -3.805E+01 -4.860E+01 -5.233E+01 -5.295E+01 IY= 36 -1.075E+01 -2.620E+01 -3.561E+01 -3.335E+01 -2.125E+01 IY= 34 1.296E+01 -1.080E+01 -1.841E+01 -1.330E+01 -4.903E+00 IY= 32 3.488E+01 1.932E+01 1.084E+01 1.312E+01 1.326E+01 IY= 30 4.637E+01 3.851E+01 3.793E+01 3.513E+01 2.921E+01 IY= 28 5.231E+01 5.319E+01 5.468E+01 4.557E+01 3.800E+01 IY= 26 5.495E+01 5.855E+01 5.495E+01 4.755E+01 4.003E+01 IY= 24 4.658E+01 5.054E+01 4.896E+01 4.392E+01 4.029E+01 IY= 22 2.997E+01 3.708E+01 3.908E+01 3.839E+01 3.734E+01 IY= 20 0.000E+00 0.000E+00 0.000E+00 O.OOOE+00 O.OOOE+00 IY= 18 8.664E+01 1.402E+02 1.614E+02 1.755E+02 1.863E+02 IY= 16 8.715E+01 1.410E+02 1.622E+02 1.762E+02 1.870E+02 IY= 14 8.738E+01 1.416E+02 1.629E+02 1.769E+02 1.876E+02 IY= 12 8.745E+01 1.420E+02 1.635E+02 1.775E+02 1.882E+02 IY= 10 8.744E+01 1.423E+02 1.640E+02 1.780E+02 1.887E+02 IY= 8 8.738E+01 1.423E+02 1.644E+02 1.785E+02 1.892E+02 IY= 6 8.729E+01 1.423E+02 1.646E+02 1.789E+02 1.896E+02 IY= 4 8.720E+01 1.423E+02 1.648E+02 1.793E+02 1.899E+02 IY= 2 8.711E+01 1.421E+02 1.649E+02 1.796E+02 1.903E+02 -6.448E+01-6.278E+01 IX= 1 3 5 7 9 IY= 40 -8.358E+01 -8.483E+01 -7.090E+01 -2.650E+01 -2.412E+01 IY= 38 -3.668E+01 -2.770E+01 -2.819E+01 -1.258E+01 -9.739E+00 IY= 36 -1.421E+01 -1.563E+01 -1.578E+01 -7.768E-01 1.959E+00 IY= 34 -2.309E+00 -4.214E+00 -3.865E+00 7.870E+00 9.397E+00 IY= 32 1.102E+01 8.110E+00 7.115E+O0 1.391E+01 1.310E+01 IY= 30 2.381E+01 1.886E+01 1.578E+01 1.833E+01 1.468E+01 IY= 28 3.164E+01 2.702E+01 2.255E+01 2.146E+01 1.573E+01 IY= 26 3.592E+01 3.226E+01 2.722E+01 2.312E+01 1.654E+01 IY= 24 3.790E+01 3.451E+01 2.945E+01 2.290E+01 1.667E+01 IY= 22 3.593E+01 3.320E+01 2.875E+01 O.OOOE+00 O.OOOE+00 IY= 20 O.OOOE+00 O.OOOE+00 O.OOOE+00 2.151E+02 2.197E+02 IY= 18 1.952E+02 2.027E+02 2.093E+02 2.157E+02 2.205E+02 IY= 16 1.958E+02 2.034E+02 2.099E+02 2.163E+02 2.212E+02 IY= 14 1.964E+02 2.039E+02 2.105E+02 2.167E+02 2.218E+02 IY= 12 1.970E+02 2.044E+02 2.109E+02 2.171E+02 2.222E+02 IY= 10 1.974E+02 2.049E+02 2.113E+02 2.175E+02 2.226E+02 IY= 8 1.978E+02 2.052E+02 2.117E+02 2.177E+02 2.229E+02 IY= 6 1.982E+02 2.055E+02 2.120E+02 2.180E+02 2.232E+02 IY= 4 1.985E+02 2.058E+02 2.122E+02 2.182E+02 2.234E+02 IY= 2 1.988E+02 2.061E+02 2.124E+02 -5.244E+01 -4.664E+01 DC= 11 13 15 17 19 -3.214E+01 -3.524E+01 IY= 40 -5.899E+01 -5.662E+01 -5.631E+01 -2.408E+01 -2.871E+01 IY= 38 -2.436E+01 -2.752E+01 -3.114E+01 IY= 36 -1.211E+01 -1.709E+01 -2.190E+01
212
Appendix A
7.968E+00 -1.604E+01 -1.960E+01 -2.671E+01 IY= 34 7.589E-01 • IY= 32 9.847E+00 7.158E+00 -6.328E+00 -1.474E+01 -2.456E+01 IY= 30 1.419E+01 1.479E+01 9.879E+00 -9.608E+00 -2.042E+01 IY= 28 1.438E+01 1.698E+01 1.415E+01 -4.037E+00 -1.573E+01 1Y= 26 1.286E+01 1.511E+01 1.569E+01 6.999E+00 -9.864E+00 IY= 24 1.170E+01 1.126E+01 1.389E+01 1.062E+01 3.590E+00 IY= 22 1.023E+01 6.141E+00 7.663E+00 9.861E+00 1.132E+01 IY= 20 2.134E+01 2.327E+01 1.502E+01 1.160E+01 1.281E+01 IY= 18 2.187E+02 2.180E+02 2.120E+02 1.950E+02 1.672E+02 IY= 16 2.216E+02 2.215E+02 2.211E+02 2.194E+02 2.141E+02 IY= 14 2.235E+02 2.242E+02 2.236E+02 2.227E+02 2.212E+02 IY= 12 2.248E+02 2.264E+02 2.261E+02 2.250E+02 2.235E+02 IY= 10 2.256E+02 2.282E+02 2.285E+02 2.272E+02 2.253E+02 IY= 8 2.261E+02 2.294E+02 2.305E+02 2.294E+02 2.271E+02 IY= 6 2.266E+02 2.302E+02 2.321E+02 2.315E+02 2.289E+02 IY= 4 2.270E+02 2.309E+02 2.333E+02 2.334E+02 2.307E+02 IY= 2 2.276E+02 2.316E+02 2.344E+02 2.349E+02 2.322E+02 K= 21 23 25 27 29 -4.963E+01 -5.138E+01 IY= 40 -4.872E+01 -5.296E+01 -5.285E+01 -6.600E+01 -7.720E+01 IY= 38 -4.797E+01 -5.624E+01 -6.064E+01 -6.661E+01 -7.873E+01 IY= 36 -4.260E+01 -5.064E+01 -5.746E+01 -6.497E+01 -7.712E+01 IY= 34 -3.742E+01 -4.448E+01 -5.3O0E+01 -6.110E+01 -7.251E+01 IY= 32 -3.129E+01 -3.833E+01 -4.754E+01 -5.579E+01 -6.589E+01 IY= 30 -2.557E+01 -3.224E+01 -4.153E+01 -4.962E+01 -5.833E+01 IY= 28 -1.991E+01 -2.632E+01 -3.575E+01 -4.306E+01 -5.007E+01 IY= 26 -1.414E+01 -2.186E+01 -3.079E+01 -3.711E+01 -4.051E+01 IY= 24 -8.805E+00 -1.867E+01 -2.612E+01 -3.085E+01 -2.640E+01 IY= 22 6.095E+00 -1.243E+01 -2.103E+01 2.495E+00 -1.276E+01 IY= 20 1.478E+01 1.290E+01 9.164E+00 5.886E+01 4.436E+01 IY= 18 1.320E+02 9.925E+01 7.452E+01 1.623E+02 1.659E+02 IY= 16 2.037E+02 1.900E+02 1.751E+02 2.083E+02 2.093E+02 IY= 14 2.190E+02 2.156E+02 2.116E+02 2.167E+02 2.159E+02 IY= 12 2.216E+02 2.197E+02 2.178E+02 2.169E+02 2.161E+02 IY= 10 2.230E+02 2.205E+02 2.183E+02 2.166E+02 2.156E+02 IY= 8 2.242E+02 2.211E+02 2.184E+02 2.165E+02 2.153E+02 IY= 6 2.253E+02 2.215E+02 2.185E+02 2.165E+02 2.150E+02 IY= 4 2.263E+02 2.219E+02 2.186E+02 2.165E+02 2.149E+02 IY= 2 2.272E+02 2.222E+02 2.186E+02 4.902E+01 «= 31 33 35 37 39 1.904E+01 -6.572E-01 -1.967E+01 IY= 40 -3.874E+01 -5.403E+01 -2.414E+01 4.767E+01 -3.909E+01 IY= 38 -6.712E+01 -7.165E+01 -3.915E+01 -6.434E+01 -5.182E+01 IY= 36 -7.082E+01 -7.674E+01 -5.074E+01 -6.647E+01 -5.244E+01 IY= 34 -7.078E+01 -7.625E+01 -5.338E+01 -6.122E+01 -4.478E+01 IY= 32 -6.650E+01 -6.955E+01 -4.707E+01 -5.150E+01 IY= 30 -5.877E+01 -5.766E+01 -3.327E+01 -3.152E+01 -3.930E+01 IY= 28 -5.023E+01 -4.403E+01 -1.047E+01 -1.211E+01 IY= 26 -4.351E+01 -2.733E+01 1.168E+01 2.367E+01 -2.579E+01 IY= 24 -3.947E+01 7.916E-01 2.564E+01 4.279E+01 -8.705E+00 7.408E+01 IY= 22 -3.660E+01 2.422E+01 4.285E+01 7.330E+01 1.672E+02 1.579E+02 IY= 20 -1.541E+01 4.161E+01 6.492E+01 2.047E+02 IY= 18 7.621E+01 1.256E+02 1.495E+02 1.938E+02 2.117E+02 2.046E+02 IY= 16 1.665E+02 1.791E+02 1.882E+02 2.120E+02 IY= 14 2.016E+02 2.082E+02 2.046E+02 2.060E+02 2.120E+02 IY= 12 2.156E+02 2.161E+02 2.073E+02 2.064E+02 2.118E+02 IY= 10 2.178E+02 2.170E+02 2.080E+02 2.067E+02 2.116E+02 1Y= 8 2.181E+02 2.174E+02 2.085E+02 2.067E+02 IY= 6 2.182E+02 2.176E+02 2.087E+02
213
Appendix A
IY= 4 2.182E+02 2.178E+02 :2.088E+02 2.066E+02 2.113E+02 IY= 2 2.181E+02 2.177E+02 2.087E+02 2.064E+02 2.110E+02 43 IX= 41 45 17 49 IY= 40 -3.375E+01 -3.796E+01 -4.680E+01 -2.459E+01 -3.554E+01 IY= 38 -8.972E+01 -9.446E+01 -8.873E+01 -7.181E+01 -7.678E+01 IY= 36 -9.449E+01 -9.157E+01 -7.446E+01 -6.034E+01 -6.505E+01 IY= 34 -8.812E+01 -7.591E+01 -4.814E+01 -4.449E+01 4.902E+01 IY= 32 -7.562E+01 -5.180E+01 -1.365E+01 -2.929E+01 -3.201E+01 IY= 30 -5.937E+01 -2.811E+01 1.415E+01 -1.504E+01 -1.622E+01 IY= 28 -4.333E+01 -6.766E+00 3.105E+01 8.244E+00 2.892E+00 IY= 26 -2.741E+01 2.765E+01 5.166E+01 3.201E+01 3.019E+01 IY= 24 3.222E+00 5.662E+01 7.581E+01 5.514E+01 5.670E+01 IY= 22 4.776E+01 8.927E+01 1.007E+02 8.233E+01 8.622E+01 IY= 20 1.066E+02 1.237E+02 1.267E+02 1.191E+02 1.203E+02 IY= 18 1.928E+02 1.962E+02 1.916E+02 1.770E+02 1.718E+02 IY= 16 2.111E+02 2.058E+02 2.008E+02 1.943E+02 1.928E+02 IY= 14 2.131E+02 2.069E+02 2.033E+02 2.010E+02 2.013E+02 1Y= 12 2.132E+02 2.071E+02 2.041E+02 2.033E+02 2.046E+02 IY= 10 2.131E+02 2.071E+02 2.044E+02 2.042E+02 2.058E+02 IY= 8 2.130E+02 2.069E+02 2.046E+02 2.046E+02 2.064E+02 IY= 6 2.127E+02 2.066E+02 2.047E+02 2.049E+02 2.066E+02 IY= 4 2.123E+02 2.064E+02 2.049E+02 2.050E+02 2.068E+02 IY= 2 2.121E+02 2.063E+02 2.050E+02 2.051E+02 2.069E+02 57 59 55 53 K= : 51 -6.568E+00 2.043E+01 -3.121E+01 -3.476E+01 40 -4.793E+01 IY= 9.584E+00 4.373E+01 -2.930E+01 -6.556E+01 -4.569E+01 38 IY= 1.175E+01 4.863E+01 -1.821E+01 -3.173E+01 36 -4.343E+01 IY= 1.335E+01 4.443E+01 -6.946E+00 -1.833E+01 34 -1.908E+01 IY= 3.778E+01 1.992E+01 4.613E+00 -7.275E+00 IY= 32 4.898E+00 3.889E+01 2.804E+01 1.627E+01 7.711E+O0 IY= 30 1.841E+01 IY= 28 3.447E+01 2.473E+01 2.933E+01 3.708E+01 5.267E+01 IY= 26 5.263E+01 4.404E+01 4.407E+01 4.752E+01 6.298E+01 IY= 24 7.345E+01 6.465E+01 6.020E+01 5.934E+01 7.213E+01 IY= 22 9.438E+01 8.547E+01 7.647E+01 7.260E+01 8.413E+01 IY= 20 1.172E+02 1.084E+02 9.479E+01 9.307E+01 1.061E+02 IY= 18 1.685E+02 1.587E+02 1.473E+02 1.471E+02 1.544E+02 IY= 16 1.884E+02 1.798E+02 1.699E+02 1.670E+02 1.693E+02 IY= 14 1.976E+02 1.908E+02 1.828E+02 1.789E+02 1.788E+02 IY= 12 2.017E+02 1.963E+02 1.898E+02 1.856E+02 1.842E+02 IY= 10 2.036E+02 1.990E+02 1.933E+02 1.889E+02 1.870E+02 IY= 8 2.045E+02 2.003E+02 1.949E+02 1.904E+02 1.884E+02 IY= 6 2.050E+02 2.009E+02 1.955E+02 1.910E+02 1.891E+02 IY= 4 2.053E+02 2.012E+02 1.958E+02 1.914E+02 1.895E+02 IY= 2 2.056E+02 2.015E+02 1.960E+02 1.916E+02 1.898E+02 67 69 65 63 IX= 61 FIELD V A L U E S O F IY= 39 8.863E+O0 IY= 37 -3.993E+00 IY= 35 -1.329E+01 IY= 33 -2.098E+01 IY= 31 -3.307E+01 IY= 29 -3.927E+01 IY= 27 -4.301E+01 IY= 25 -4.155E+01 IY= 23 -3.395E+01 IY= 21 -2.176E+01 IY= 19 0.0O0E+O0
VI 1.656E+01 9.844E+00 3.415E+00 -8.731E+00 -2.210E+01 -3.017E+01 -3.421E+01 -3.156E+01 -2.610E+01 -1.608E+01 O.OOOE+00
2.054E+01 1.357E+01 7.301E+00 7.762E-01 -1.261E+01 -1.923E+01 -2.051E+01 -2.011E+01 -1.793E+01 -1.249E+01 O.OOOE+00
2.286E+01 2.420E+01 1.344E+01 1.061E+01 9.645E+O0 8.1O4E+00 4.259E+00 5.360E+00 -2.025E+00 4.096E-01 -7.340E+00 -4.579E+00 -1.102E+01 -8.141E+O0 -1.273E+01 -1.075E+01 -1.337E+01 -1.211E+01 -1.085E+01 -1.055E+01 O.OOOE+00 O.OOOE+00
214
Appendix A
IY= 17 4.330E+00 8.518E+00 9.915E+00 1.071E+01 1.130E+01 IY= 15 3.445E+00 7.887E+00 9.408E+00 1.012E+01 1.060E+01 IY= 13 2.773E+00 7.125E+00 8.803E+O0 9.455E+00 9.837E+O0 IY= 11 2.242E+00 6.284E+00 8.087E+00 8.711E+O0 8.999E+00 IY= 9 1.804E+00 5.399E+00 7.249E+00 7.865E+00 8.070E+00 IY= 7 1.427E+00 4.485E+00 6.274E+00 6.885E+00 7.024E+00 1Y= 5 1.084E+00 3.537E+00 5.139E+00 5.721E+00 5.811E+00 IY= 3 7.502E-01 2.516E+00 3.777E+00 4.274E+00 4.331E+00 IY= 1 3.690E-01 11.263E+00 1.947E+00 2.239E+00 :2.267E+00 5 7 9 K= 1 3 IY= 39 2.449E+01 1.701E+01 1.609E+01 1.483E+01 1.439E+01 IY= 37 1.027E+01 1.217E+01 1.064E+01 8.447E+00 7.701E+00 IY= 35 8.188E+00 8.672E+00 7.270E+00 5.562E+00 4.57OE+O0 IY= 33 5.079E+00 4.813E+00 4.541E+00 4.000E+00 2.901E+00 IY= 31 5.244E-01 1.173E+00 2.291E+00 3.085E+00 2.5O8E+O0 IY= 29 -4.078E+00 -2.530E+00 1.049E-01 2.201E+00 2.777E+00 IY= 27 -7.608E+00 -5.546E+00 -2.040E+00 9.994E-01 2.686E+00 IY= 25 -1.004E+01 -7.565E+00 -3.972E+00 -5.454E-01 1.731E+00 IY= 23 -1.100E+01 -8.786E+00 -5.553E+00 -2.310E+00 3.934E-02 IY= 21 -1.024E+01 -9.054E+00 -6.831E+00 -4.346E+00 -2.372E+00 IY= 19 O.OOOE+00 O.OOOE+00 O.OOOE+00 O.OOOE+00 O.OOOE+00 IY= 17 1.179E+01 1.223E+01 1.262E+01 1.297E+01 1.320E+01 IY= 15 1.101E+01 1.140E+01 1.175E+01 1.208E+01 1.228E+01 IY= 13 1.017E+01 1.050E+01 1.082E+01 1.113E+01 1.132E+01 IY= 11 9.246E+00 9.521E+00 9.816E+00 1.010E+01 1.029E+01 IY= 9 8.236E+00 8.457E+00 8.718E+00 8.978E+O0 9.172E+00 IY= 7 7.115E+00 7.281E+00 7.5O5E+O0 7.738E+O0 7.926E+00 IY= 5 5.840E+00 5.952E+00 6.135E+00 6.334E+00 6.504E+00 IY= 3 4.316E+00 4.379E+00 4.514E+00 4.668E+00 4.805E+00 IY= 1 2.241E+00 2.263E+00 2.333E+O0 2.417E+00 2.493E+00
rx=
11
13
15
17
19
IY= 39 1.435E+01 1.501E+01 1.568E+01 1.548E+01 1.571E+01 IY= 37 8.608E+00 8.954E+00 1.029E+01 1.178E+01 1.237E+01 IY= 35 5.033E+00 6.356E+00 8.401E+00 1.101E+01 1.049E+01 IY= 33 1.509E+00 1.535E+00 5.846E+00 1.042E+01 8.552E+00 IY= 31 1.340E-01 -2.829E+00 -7.309E-01 8.970E+00 5.631E+O0 IY= 29 5.796E-01 -3.830E+00 -4.691E+00 4.600E+00 2.607E+00 IY= 27 1.873E+00 -2.698E+00 -7.000E+00 -3.417E+00 5.021E-01 IY= 25 2.496E+00 -1.090E-02 -7.994E+00 -8.5O5E+O0 -2.942E+00 IY= 23 1.696E+00 2.638E+00 -7.512E+00 -1.098E+01 -1.020E+01 IY= 21 -1.675E+00 5.070E+00 -5.932E+00 -1.054E+01 -1.130E+01 IY= 19 6.122E+00 -2.550E+00 -6.663E+00 -8.832E+00 -9.780E+O0 IY= 17 8.979E+00 -1.590E+00 -8.886E+00 -1.453E+01 -1.803E+01 IY= 15 1.002E+01 1.947E+00 -6.269E+00 -1.283E+01 -1.839E+01 IY= 13 1.012E+01 4.799E+00 -3.116E+00 -1.013E+01 -1.582E+01 IY= 11 9.669E+00 6.493E+00 9.692E-02 -7.183E+00 -1.305E+01 IY= 9 8.857E+00 7.072E+00 2.604E+00 -3.996E+00 -1.019E+01 IY= 7 7.769E+00 6.785E+O0 3.985E+O0 -1.185E+O0 -7.092E+00 • IY= 5 6.423E+00 5.869E+00 4.237E+00 6.836E-01 4.203E+00 1.395E+00 -1.994E+00 4.439E+00 3.534E+O0 IY= 3 4.763E+00 9.602E-01 -6.281E-01 2.326E+00 1.902E+00 IY= 1 2.516E+00 25 27 29 23 IX= 21 2.320E+01 2.601E+01 2.046E+01 2.170E+01 IY= 39 1.821E+01 2.366E+01 2.625E+01 IY= 37 1.506E+01 1.846E+01 2.U0E+01 2.279E+01 2.432E+01 IY= 35 1.215E+01 1.633E+01 1.969E+01 2.081E+01 2.155E+01 IY= 33 9.329E+00 1.417E+01 1.764E+01 1.807E+01 1.827E+01 IY= 31 6.897E+00 1.202E+01 1.521E+01
215
Appendix A
IY= 29 4.933E+00 9.919E+00 1.266E+01 1.491E+01 1.501E+01 IY= 27 3.143E+00 7.982E+00 1.021E+01 1.157E+01 1.200E+01 IY= 25 1.483E+00 6.108E+00 7.946E+00 8.357E+00 9.356E+00 IY= 23 -1.989E+00 3.390E+00 5.747E+00 5.613E+00 6.950E+00 IY= 21 -7.826E+00 -2.879E+00 1.598E+00 2.556E+00 5.118E+00 IY= 19 -8.631E+00 -7.524E+00 -4.838E+00 -1.286E+00 -5.138E+00 IY= 17 -1.881E+01 -1.681E+01 -1.260E+01 -7.260E+00 -8.544E+00 IY= 15 -2.216E+01 -2.245E+01 -1.987E+01 -1.514E+01 -9.626E+00 IY= 13 -2.018E4-01 -2.209E+01 -2.111E+01 -1.785E+01 -1.172E+01 IY= 11 -1.729E+01 -1.949E+01 -1.936E+01 -1.722E+01 -1.322E+01 IY= 9 -1.440E+01 -1.651E+01 -1.667E+01 -1.534E+01 -1.290E+01 IY= 7 -1.146E+01 -1.345E+01 -1.366E+01 -1.284E+01 -1.138E+01 IY= 5 -8.298E+00 -1.023E+01 -1.044E+01 -9.921E+00 -9.011E+00 IY= 3 -5.179E+O0 -6.698E+00 -6.810E+00 -6.444E+00 -5.861E+00 IY= 1 -2.193E+00 -2.872E+00 -2.799E+00 -2.518E+00 -2.160E+00 IX= 31 33 35 37 39 IY= 39 3.134E-01 -2.040E-01 2.507E-01 2.110E-01 1.612E-01 IY= 37 6.768E-01 -6.035E-01 7.452E-01 5.787E-01 -2.928E-01 IY= 35 9.091E-01 -1.047E+00 1.054E+00 7.699E-01 -6.958E-01 IY= 33 1.042E+00 -1.519E+00 1.213E+00 8.493E-01 -7.580E-01 IY= 31 1.123E+00 -1.961E+00 1.241E+00 8.475E-01 -7.459E-01 IY= 29 1.207E+00 -2.340E+00 1.125E+00 7.805E-01 -7.263E-01 IY= 27 1.324E+00 -2.677E+00 1.O43E+O0 6.413E-01 -7.345E-01 IY= 25 1.445E+00 -2.906E+00 1.135E+00 3.345E-01 -7.915E-01 IY= 23 1.5O3E+O0 -2.564E+00 1.217E+00 1.209E-01 -9.329E-01 IY= 21 1.375E+O0 -1.435E+00 1.316E+00 1.048E-01 -1.422E+00 IY= 19 2.278E+00 3.922E+00 2.747E+00 3.099E-01 -2.191E+00 IY= 17 2.767E+00 6.186E+00 3.602E+00 4.348E-01 -1.934E+00 IY= 15 2.240E+00 6.637E+00 3.943E+00 6.523E-01 -1.358E+00 IY= 13 7.588E-01 6.230E+00 3.679E+00 6.671E-01 -1.098E+00 IY= 11 -8.888E-02 5.216E+00 3.122E+00 5.963E-01 -9.084E-01 IY= 9 -7.083E-01 4.140E+00 2.558E+00 5.181E-01 -7.298E-01 IY= 7 -1.040E+00 3.157E+00 2.012E+00 4.303E-01 -5.645E-01 IY= 5 -1.040E+00 2.256E+00 1.479E+00 3.296E-01 -4.104E-01 IY= 3 -7.494E-01 1.426E+00 9.589E-01 2.143E-01 -2.639E-01 IY= 1 -2.682E-01 5.535E-01 3.865E-01 8.441E-02 -1.124E-01 IX= 41 43 45 47 49 IY= 39 -4.157E-01 -1.800E-02 -2.349E-02 1.500E-01 -2.728E-01 IY= 37 -1.243E+00 -8.297E-02 5.325E-02 3.388E-01 -4.354E-01 IY= 35 -1.895E+00 -5.855E-02 2.976E-01 4.623E-01 -5.162E-01 IY= 33 -2.366E+00 9.774E-02 6.897E-01 4.586E-01 -5.502E-01 IY= 31 -2.679E+00 4.237E-01 1.215E+00 2.439E-01 -5.422E-01 IY= 29 -2.841E+00 8.330E-01 1.841E+00 -1.612E-01 -5.174E-01 IY= 27 -2.900E+00 1.310E+00 2.112E+00 -5.691E-01 -3.961E-01 IY= 25 -2.908E+00 2.069E+00 2.377E+00 -8.221E-01 -3.949E-02 IY= 23 -2.567E+00 2.624E+00 2.602E+00 -1.110E+00 3.424E-01 IY= 21 -1.257E+00 3.081E+00 2.744E+00 -1.430E+00 6.793E-01 IY= 19 1.993E+00 4.220E+00 2.686E+00 -1.727E+00 6.983E-01 IY= 17 3.182E+00 4.051E+00 2.359E+00 -2.290E+00 4.836E-01 IY= 15 3.185E+00 3.488E+00 2.018E+00 -2.339E+00 2.726E-01 IY= 13 2.828E+00 2.929E+00 1.749E+00 -2.119E+00 9.860E-02 IY= 11 2.446E+00 2.446E+00 1.508E+00 -1.815E+O0 -3.555E-02 IY= 9 2.070E+00 2.012E+00 1.270E+00 -1.497E+00 -1.317E-01 IY= 7 1.680E+00 1.606E+00 1.035E+O0 -1.186E+00 -1.875E-01 IY= 5 1.273E+00 1.224E+00 8.030E-01 -8.852E-01 -2.023E-01 IY= 3 8.389E-01 8.304E-01 5.583E-01 -5.870E-01 -1.749E-01 IY= 1 3.562E-01 3.729E-01 2.599E-01 -2.634E-01 -9.578E-02
216
Appendix A
IX= 51 53 55 57 59 IY= 39 -2.186E-02 4.547E-02 9.636E-03 1.341E-01 9.770E-02 IY= 37 1.390E-01 2.016E-01 1.890E-01 5.226E-01 3.802E-01 IY= 35 4.346E-01 2.502E-01 4.068E-01 8.023E-01 7.019E-01 IY= 33 8.549E-01 1.722E-01 6.548E-01 9.626E-01 1.028E+00 IY= 31 1.441E+00 9.617E-03 9.204E-01 1.111E+00 1.269E+00 IY= 29 1.860E+00 -1.306E-01 1.101E+00 1.256E+00 1.350E+00 IY= 27 2.106E+00 -2.247E-01 1.219E+00 1.374E+00 1.476E+00 IY= 25 2.346E+00 -2.858E-01 1.277E+00 1.473E+00 1.677E+00 IY= 23 2.533E+00 -3.365E-01 1.264E+00 1.544E+00 1.877E+O0 IY= 21 2.638E+00 -4.046E-01 1.148E+00 1.566E+00 2.093E+00 IY= 19 2.348E+00 -7.090E-01 1.658E-01 1.827E+00 3.097E+00 IY= 17 2.224E+00 -9.050E-01 -4.279E-02 2.049E+00 3.257E+00 IY= 15 1.995E+O0 -9.585E-01 -1.892E-01 2.037E+00 3.222E+00 IY= 13 1.763E+00 -9.218E-01 -2.719E-01 1.887E+00 3.057E+00 IY= 11 1.545E+00 -8.175E-01 -3.048E-01 1.666E+00 2.807E+00 IY= 9 1.333E+00 -6.792E-01 -3.083E-01 1.449E+00 2.489E+00 IY= 7 1.117E+00 -5.329E-01 -2.941E-01 1.251E+00 2.122E+00 IY= 5 8.893E-01 -3.919E-01 -2.630E-01 1.013E+00 1.717E+00 IY= 3 6.356E-01 -2.575E-01 -2.097E-01 7.293E-01 1.252E+00 IY= 1 3.093E-01 -1.175E-01 -1.147E-01 3.636E-01 6.431E-01 63 DC= 61 65 67 69 FIELD V A L U E S O F K E IY= 40 3.028E+01 3.732E+01 3.796E+01 3.603E+01 3.429E+01 IY= 38 3.008E-02 4.595E-03 2.032E-03 1.883E-03 1.767E-03 IY= 36 7.261E-04 4.377E-03 4.756E-03 4.846E-03 4.997E-03 IY= 34 3.395E-03 5.136E-03 4.134E-03 4.385E-03 4.366E-03 IY= 32 2.539E-03 4.798E-03 4.222E-03 4.137E-03 3.466E-03 IY= 30 1.187E-03 1.962E-03 2.259E-03 2.410E-03 1.671E-03 IY= 28 6.985E-04 5.398E-04 6.113E-04 6.710E-04 4.847E-04 IY= 26 6.529E-04 3.191E-04 1.972E-04 1.574E-04 1.171E-04 IY= 24 7.193E-04 3.756E-04 1.763E-04 1.079E-04 7.841E-05 IY= 22 2.314E-03 9.453E-04 3.214E-04 1.641E-04 1.126E-04 IY= 20 O.OOOE+00 O.OOOE+00• O.OOOE+00 0.000E+00 O.OOOE+00 IY= 18 3.687E-03 1.832E-03 1.525E-04 6.201E-05 3.083E-05 IY= 16 1.632E-02 3.126E-03 1.564E-04 6.016E-05 2.987E-05 IY= 14 1.657E-02 3.196E-03 1.560E-04 5.889E-05 2.908E-05 IY= 12 1.679E-02 3.265E-03 1.591E-04 5.837E-05 2.851E-05 IY= 10 1.693E-02 3.321E-03 1.654E-04 5.862E-05 2.812E-05 IY= 8 1.699E-02 3.361E-03 1.740E-04 5.969E-05 2.791E-05 IY= 6 1.700E-02 3.385E-03 1.839E-04 6.156E-05 2.788E-05 IY= 4 1.696E-02 3.395E-03 1.943E-04 6.411E-05 2.804E-05 IY= 2 1.689E-02 3.396E-03 2.039E-04 6.704E-05 2.838E-05 7 9 DC= 1 3 5 IY= 40 3.997E+01 4.709E+01 4.350E+01 3.298E+01 2.903E+01 IY= 38 9.018E-O4 7.008E-04 7.265E-04 7.923E-04 8.344E-04 IY= 36 5.900E-03 6.259E-03 7.573E-03 9.908E-03 1.171E-02 IY= 34 3.482E-03 2.754E-03 2.382E-03 2.411E-03 3.146E-03 IY= 32 2.241E-03 1.673E-03 1.324E-03 1.171E-03 1.169E-03 IY= 30 1.158E-03 7.890E-04 5.669E-04 4.609E-04 4.277E-04 IY= 28 3.370E-04 2.272E-04 1.692E-04 1.446E-04 1.404E-04 IY= 26 7.902E-05 5.547E-05 4.818E-05 5.293E-05 6.288E-05 IY= 24 6.003E-05 4.760E-05 4.254E-05 4.589E-05 5.794E-05 IY= 22 8.896E-05 7.136E-05 6.256E-05 6.424E-05 8.698E-05 IY= 20 O.OOOE+00 O.OOOE+00 0.000E+00 O.OOOE+00 0.000E+00 IY= 18 1.908E-05 1.080E-05 7.528E-06 8.450E-06 1.658E-05 IY= 16 1.852E-05 1.314E-05 1.022E-05 8.873E-06 1.294E-05
217
Appendix A
IY= 14 1.798E-05 1.276E-05 9.926E-06 8.568E-06 1.075E-05 IY= 12 1.756E-05 1.246E-05 9.702E-06 8.336E-06 9.419E-06 IY= 10 1.720E-05 1.220E-05 9.526E-06 8.158E-06 8.575E-06 IY= 8 1.689E-05 1.197E-05 9.377E-06 8.019E-O6 8.019E-06 IY= 6 1.659E-05 1.174E-05 9.242E-06 7.909E-06 7.641E-06 IY= 4 1.632E-05 1.151E-05 9.111E-06 7.820E-06 7.379E-06 IY= 2 1.608E-05 1.128E-05 8.985E-06 7.749E-06 7.199E-06 11 13 15 17 19 K= IY= 40 2.746E+01 2.554E+01 2.506E+01 2.419E+01 2.043E+01 IY= 38 3.925E-04 3.877E-04 3.744E-04 3.165E-04 2.594E-04 IY= 36 1.385E-02 1.568E-02 1.565E-02 1.504E-02 1.494E-02 IY= 34 4.759E-03 6.716E-03 8.679E-03 9.333E-03 1.112E-02 IY= 32 1.527E-03 2.767E-03 4.187E-03 5.902E-03 9.470E-03 IY= 30 4.758E-04 6.854E-04 1.094E-03 2.126E-03 4.284E-03 IY= 28 1.494E-04 2.055E-O4 3.268E-04 1.029E-03 3.186E-03 IY= 26 7.347E-05 8.214E-05 1.513E-04 9.541E-04 3.711E-03 IY= 24 6.853E-05 7.863E-05 3.961E-04 2.482E-03 6.630E-03 IY= 22 1.478E-04 6.821E-04 8.415E-03 1.511E-02 1.693E-02 IY= 20 2.6O2E-01 7.700E-02 5.459E-02 4.896E-02 4.766E-02 IY= 18 8.156E-03 1.502E-03 2.881E-04 9.264E-04 3.859E-03 IY= 16 5.840E-04 8.484E-04 2.821E-04 4.168E-04 8.659E-04 IY= 14 1.029E-04 3.924E-04 1.837E-04 1.929E-04 2.830E-04 1Y= 12 3.487E-05 1.606E-04 1.431E-04 1.025E-04 1.884E-04 IY= 10 1.818E-05 6.851E-05 1.018E-04 6.837E-05 1.042E-04 IY= 8 1.231E-05 3.343E-05 6.430E-05 5.812E-05 5.456E-05 IY= 6 9.682E-06 1.935E-05 3.913E-05 4.888E-05 4.033E-05 IY= 4 8.295E-06 1.318E-05 2.494E-05 3.797E-05 3.866E-05 IY= 2 7.349E-06 1.012E-05 1.718E-05 2.797E-05 3.603E-05 27 29 2.341E+01 2.303E+01 21 23 25 K= 2.442E+01 1.069E-03 1.474E-03 IY= 40 38 1.897E+01 2.224E+01 IY= 36 3.226E-04 6.583E-04 8.671E-04 2.835E-02 3.648E-02 IY= 34 1.470E-02 1.744E-02 2.215E-02 2.075E-02 2.451E-02 IY= 32 1.310E-02 1.424E-02 1.624E-02 1.980E-02 2.102E-02 IY= 30 1.213E-02 1.394E-02 1.589E-02 1.503E-02 2.031E-02 IY= 28 8.053E-03 1.702E-02 1.880E-02 2.189E-02 1.821E-02 IY= 26 6.705E-03 1.345E-02 2.036E-02 2.686E-02 1.826E-02 IY= 24 6.743E-03 1.137E-02 1.572E-02 2.395E-02 3.206E-02 IY= 22 1.035E-02 1.344E-02 1.627E-02 2.115E-02 3.248E-02 IY= 20 1.871E-02 1.879E-02 1.803E-02 1.348E-02 4.226E-03 IY= 18 3.777E-02 2.557E-02 2.027E-02 1.244E-02 1.164E-02 IY= 16 7.183E-03 9.361E-03 1.170E-02 6.773E-03 4.888E-03 IY= 14 2.389E-03 5.658E-03 7.983E-03 3.528E-03 3.053E-03 IY= 12 4.386E-04 1.026E-03 2.508E-03 9.753E-04 1.007E-03 IY= 10 2.463E-04 3.259E-04 5.853E-04 3.999E-04 4.271E-04 IY= 8 1.939E-04 2.242E-04 2.737E-04 2.315E-04 2.118E-04 IY= 6 1.273E-04 1.907E-04 2.044E-04 2.358E-04 2.087E-04 IY= 4 6.619E-05 1.486E-04 1.932E-04 3.162E-04 3.301E-04 IY= 2 3.904E-05 9.261E-05 1.894E-04 3.859E-04 3.946E-04 3.355E-05 4.915E-05 1.726E-04 3.605E+00 1.829E+01 IY= 40 37 39 2.690E-04 1.876E-04 31 33 35 K= 38 2.322E+01 2.535E+01 9.582E+00 6.089E-02 4.058E-02 IY= 36 5.433E-04 2.170E-04 2.065E-04 7.306E-02 8.649E-02 IY= 34 1.879E-02 2.583E-02 4.955E-02 7.632E-02 6.949E-02 IY= 32 2.166E-02 3.094E-02 5.265E-02 6.688E-02 5.918E-02 IY= 30 2.118E-02 4.223E-02 5.993E-02 6.706E-02 6.020E-02 IY= 28 1.998E-02 4.477E-02 8.072E-02 IY= 2.163E-02 4.915E-02 8.171E-02 IY=
218
Appendix A
IY= 26 2.421E-02 5.328E-02 9.173E-02 8.535E-02 7.897E-02 IY= 24 2.820E-02 6.467E-02 1.257E-01 1.029E-01 9.805E-02 IY= 22 3.714E-02 7.118E-02 1.898E-01 1.405E-01 1.500E-01 IY= 20 1.606E-03 1.168E-03 1.118E-03 9.898E-04 1.162E-03 IY= 18 9.229E-03 8.148E-03 9.784E-03 9.547E-03 1.170E-02 IY= 16 3.058E-03 1.486E-03 1.428E-03 1.519E-03 1.935E-03 IY= 14 1.981E-03 8.329E-04 5.355E-04 4.370E-04 4.888E-04 IY= 12 6.183E-04 3.665E-04 1.988E-04 1.791E-04 2.040E-04 IY= 10 2.742E-04 2.256E-04 1.641E-04 1.252E-04 1.096E-04 IY= 8 1.010E-04 1.106E-04 1.078E-04 1.021E-04 8.980E-05 IY= 6 2.709E-05 3.401E-05 4.640E-05 5.875E-05 6.291E-05 IY= 4 2.276E-05 5.257E-06 8.784E-06 1.724E-05 2.676E-05 IY= 2 2.571E-05 1.574E-06 1.575E-06 2.159E-06 3.506E-06 45 41 43 47 49 IX= 40 8.579E+O0 1.224E+01 2.124E+01 1.219E+01 6.098E+00 IY= 1Y= 38 2.316E-04 3.203E-04 2.065E-04 2.250E-04 1.826E-04 7.130E-02 8.391E-02 9.577E-02 IY= 36 3.793E-02 3.493E-03 8.971E-02 1.123E-01 1.200E-01 IY= 34 8.745E-02 5.396E-02 7.397E-02 1.339E-01 1.023E-01 IY= 32 7.012E-02 6.860E-02 5.531E-02 9.142E-02 30 5.980E-02 6.578E-02 6.509E-02 IY= 8.000E-02 1.058E-01 28 5.652E-02 7.756E-02 6.032E-02 IY= 8.869E-02 1.267E-01 26 7.569E-02 8.255E-02 6.153E-02 IY= 1.128E-01 1.534E-01 8.848E-02 IY= 24 8.842E-02 8.319E-02 1.517E-01 1.283E-01 IY= 22 20 1.567E-01 6.984E-02 2.704E-03 1.626E-01 1.837E-03 1.855E-03 IY= 18 1.209E-03 2.381E-03 1.567E-02 1.213E-02 1.357E-02 IY= 16 1.420E-02 1.513E-02 3.735E-03 2.938E-03 3.256E-03 IY= 14 2.494E-03 2.949E-03 9.859E-04 8.054E-04 8.623E-04 IY= 12 5.974E-04 7.221E-04 3.299E-04 2.632E-04 2.844E-04 IY= 10 2.209E-04 2.434E-04 1.544E-04 1.263E-04 1.320E-04 IY= 8 1.123E-04 1.211E-04 9.441E-05 7.976E-05 8.451E-05 IY= 6 7.892E-05 7.593E-05 6.490E-05 5.829E-05 5.990E-05 IY= 4 6.231E-05 5.946E-05 4.280E-05 3.822E-05 4.020E-05 IY= 2 3.238E-05 3.616E-05 1.624E-05 1.209E-05 6.030E-06 9.227E-06 1.431E-05 IY= 57 59 2.075E+00 3.912E+00 55 53 DC= 40 51 1.257E+01 1.184E-04 1.438E-04 IY= 38 2.356E+01 1.656E+01 1.237E-04 1.337E-02 1.665E-02 IY= 36 1.460E-04 2.021E-04 2.120E-02 9.194E-02 3.321E-02 IY= 34 1.098E-01 1.209E-01 2.302E-01 9.929E-02 4.833E-02 1Y= 32 1.537E-01 2.360E-01 1.846E-01 1.146E-01 3.621E-02 IY= 30 1.493E-01 1.591E-01 1.627E-01 1.176E-01 3.028E-02 IY= 28 1.218E-01 1.104E-01 1.491E-01 1.230E-01 4.360E-02 9.148E-02 9.386E-02 IY= 7.204E-02 3.447E-02 4.293E-02 IY= 26 7.207E-02 1.001E-01 7.992E-02 2.340E-02 3.863E-02 IY= 24 1.041E-01 1.095E-01 7.298E-02 1.319E-03 8.985E-04 IY= 22 1.472E-01 1.402E-01 1.626E-03 2.554E-02 2.448E-02 18 IY= 20 1.925E-03 1.982E-03 2.532E-02 IY= 9.016E-03 1.362E-02 IY= 16 1.984E-02 1.995E-02 IY* 14 4.354E-03 4.321E-03 5.459E-03 3.760E-03 6.245E-03 IY= 12 1.079E-03 1.304E-03 1.876E-03 2.279E-03 3.328E-03 IY= 10 3.900E-04 5.478E-04 1.018E-03 1.405E-03 1.748E-03 IY= 8 1.921E-04 2.644E-04 4.797E-04 7.570E-04 1.242E-03 IY= 6 1.145E-04 1.628E-04 2.586E-04 3.065E-O4 6.908E-04 IY= 4 7.544E-05 1.043E-04 1.656E-04 1.363E-04 2.111E-04 IY= 2 4.789E-05 5.868E-05 8.306E-05 3.426E-05 4.544E-05 2.730E-05 IX= FIELD V 1.881E-05 A L U E S O F 2.244E-05 EP 67 69 65 61 63
219
Appendix A
IY= 40 2.170E+06 2.987E+06 3.083E+06 2.869E+06 2.679E+06 IY= 38 2.227E+03 4.207E+02 2.106E+02 2.006E+02 1.888E+02 IY= 36 2.287E+00 2.985E+00 2.973E+00 3.018E+00 3.208E+00 IY= 34 2.859E+00 3.638E+00 2.602E+00 2.690E+00 2.582E+00 IY= 32 1.769E+00 2.856E+00 2.578E+00 2.426E+00 1.672E+00 IY= 30 5.302E-01 7.672E-01 9.507E-01 9.830E-01 5.366E-01 IY= 28 2.212E-01 1.236E-01 1.374E-01 1.489E-01 8 793E-02 IY= 26 1.662E-01 5.006E-02 2.305E-02 1.633E-02 1045E-02 1Y= 24 1.729E-01 5.940E-02 1.889E-02 9.133E-03 5.637E-03 IY= 22 8.749E-01 2.296E-01 4.599E-02 1.712E-02 9 817E-03 IY= 20 0.000E+00 0.0O0E+O0 O.OOOE+00 O.OOOE+00 0.000E+00 IY= 18 2.017E+01 2.368E+00 3.669E-02 4.302E-03 1439E-03 IY= 16 2.056E+01 3.004E+00 3.968E-02 4.143E-03 1.376E-03 IY= 14 2.107E+01 3.102E+00 4.052E-02 4.040E-03 1.324E-03 IY= 12 2.150E+01 3.189E+00 4.199E-02 4.012E-03 1.287E-03 IY= 10 2.178E+01 3.253E+00 4.389E-02 4.062E-03 1.264E-03 IY= 8 2.188E+01 3.290E+00 4.604E-02 4.192E-03 1.252E-03 IY« 6 2.186E+01 3.304E+00 4.826E-02 4.397E-03 1.253E-03 IY= 4 2.175E+01 3.301E+O0 5.040E-02 4.669E-03 1.267E-03 IY= 2 2.159E+01 3.286E+00 5.227E-02 4.978E-03 1.292E-03 IX= 1 3 5 7 9 IY= 40 3.392E+06 4.364E+06 3.899E+06 2.590E+06 2.152E+06 1Y= 38 8.678E+01 6.274E+01 6.639E+01 7.534E+01 8.124E+01 IY= 36 3.969E+00 4.270E+00 5.891E+00 9.044E+00 1.175E+01 IY= 34 1.725E+00 1.174E+00 9.285E-01 9.648E-01 1.522E+00 IY= 32 8.583E-01 5.425E-01 3.784E-01 3.119E-01 3.099E-01 IY= 30 3.105E-01 1.758E-01 1.081E-01 7.940E-02 7.028E-02 IY= 28 5.091E-02 2.845E-02 1.828E-02 1.439E-02 1.367E-02 IY* 26 5.798E-03 3.447E-03 2.794E-03 3.190E-03 4.106E-03 IY= 24 3.778E-03 2.687E-03 2.284E-03 2.568E-03 3.677E-03 IY= 22 6.889E-03 4.976E-03 4.116E-03 4.317E-03 6.846E-03 IY= 20 0.000E+00 0.000E+00 0.0O0E+00 O.OOOE+00 0.000E+00 IY= 18 7.124E-04 4.669E-04 3.519E-04 2.551E-04 4.806E-01 IY= 16 6.802E-04 4.128E-04 2.865E-04 2.331E-04 4.101E-04 IY= 14 6.519E-04 3.953E-04 2.745E-04 2.217E-04 3.105E-04 IY= 12 6.298E-04 3.818E-04 2.656E-04 2.131E-04 2.550E-04 IY= 10 6.116E-04 3.706E-04 2.587E-04 2.065E-04 2.222E-04 IY= 8 5.956E-04 3.605E-04 2.528E-04 2.015E-04 2.015E-O4 IY= 6 5.810E-04 3.506E-04 2.476E-04 1.975E-04 1.878E-04 IY= 4 5.677E-04 3.406E-04 2.425E-04 1.943E-04 1.786E-04 IY= 2 5.562E-04 3.307E-04 2.376E-04 1.918E-04 1.723E-04 DC= 11 13 15 17 19 IY= 40 2.024E+06 1.949E+06 2.045E+06 2.108E+06 1.792E+06 IY= 38 2.790E+01 2.911E+01 3.024E+01 2.346E+01 1.039E+01 IY= 36 1.514E+01 1.919E+01 2.093E+01 2.140E+01 2.200E+01 IY= 34 2.905E+00 4.878E+00 7.383E+00 9.135E+00 1.271E+01 IY= 32 4.651E-01 1.141E+00 2.165E+00 3.685E+00 8.0Q6E+00 IY= 30 8.236E-02 1.421E-01 2.862E-01 7.641E-01 2.418E+00 IY= 28 1.497E-02 2.444E-02 4.930E-02 2.633E-01 1.601E+00 IY= 26 5.140E-03 6.171E-03 1.591E-02 2.495E-01 1.935E+00 1Y= 24 4.648E-03 5.691E-03 6.551E-02 1.132E+00 4.738E+00 IY= 22 1.447E-02 1.590E-01 6.844E+00 1.635E+01 1.999E+01 IY= 20 5.277E+03 6.420E+02 2.976E+02 1.965E+02 1.467E+02 IY= 18 1.008E+01 3.102E+00 3.403E+00 5.369E+00 1.634E+01 IY= 16 1.370E-01 2.159E-01 4.228E-02 7.073E-02 2.876E-01 IY= 14 9.551E-03 6.120E-02 2.137E-02 2.211E-02 3.886E-02 IY= 12 1.834E-03 1.654E-02 1.384E-02 8.475E-03 2.153E-02
220
Appendix A
IY= 10 6.828E-04 4.752E-03 8.170E-03 <4.643E-03 8.951E-03 IY= 8 3.800E-04 1.653E-03 4.181E-03 3.618E-03 3.327E-03 IY= 6 2.661E-04 7.378E-04 2.037E-03 2.783E-03 2.118E-03 1Y= 4 2.120E-04 4.187E-04 1.058E-03 1.931E-03 1.981E-03 IY= 2 1.778E-04 2.843E-04 6.162E-04 1.247E-03 1.793E-03 23 25 27 29 K= 21 1.771E+06 2.512E+06 3.272E+06 3.542E+06 4.080E+06 IY= 40 IY= 38 2.624E+00 1.969E+00 3.482E+O0 1.308E+01 6.611E+01 IY= 36 2.342E+01 3.161E+01 4.639E+01 6.667E+01 8.849E+01 IY= 34 1.727E+01 2.123E+01 2.707E+01 4.041E+01 5.623E+01 IY= 32 1.388E+01 1.938E+01 2.535E+01 3.660E+01 4.381E+01 IY= 30 7.126E+00 2.145E+01 2.920E+01 2.658E+01 3.897E+01 IY= 28 5.553E+O0 1.572E+01 3.150E+01 3.754E+01 3.457E+01 IY= 26 5.268E+00 1.190E+01 2.308E+01 4.774E+01 3.741E+01 IY= 24 9.644E+00 1.472E+01 1.998E+01 3.760E+01 6.926E+01 IY= 22 2.422E+01 2.516E+01 2.540E+01 3.829E+01 1.448E+02 IY= 20 9.431E+01 5.429E+01 3.635E+01 2.206E+01 1.907E+01 IY= 18 3.229E+01 3.952E+01 4.033E+01 3.510E+01 2.767E+01 IY= 16 1.859E+O0 7.701E+00 1.366E+01 1.153E+01 7.650E+00 IY= 14 8.361E-02 4.016E-01 1.773E+00 3.147E+O0 2.703E+00 IY= 12 3.154E-02 5.042E-02 1.515E-01 .3.896E-01 4.439E-01 IY= 10 2.231E-02 2.730E-02 3.749E-02 '7.295E-02 8.949E-02 IY= 8 1.211E-02 2.152E-02 2.352E-02 2.797E-02 2.496E-02 IY= 6 4.540E-03 1.493E-02 2.158E-02 2.857E-02 2.376E-02 IY= 4 2.021E-03 7.465E-03 2.110E-02 4.499E-02 4.782E-02 IY= 2 1.627E-03 2.919E-03 1.916E-02 6.213E-02 6.366E-02 33 35 37 39 K= 31 IY= 40 4.518E+06 5.296E+06 1.265E+06 3.003E+05 3.532E+06 IY= 38 3.765E+01 1.090E+01 1.399E+01 1.863E+01 1.701E+01 IY= 36 2.628E+01 5.315E+01 1.590E+02 2.361E+02 1.533E+02 IY= 34 3.847E+01 5.978E+01 1.221E+02 2.027E+02 2.494E+02 IY= 32 4.545E+01 9.618E+01 1.567E+02 2.229E+02 2.344E+02 IY= 30 4.465E+01 1.158E+02 2.688E+02 2.578E+02 2.353E+02 IY= 28 5.079E+01 1.420E+02 3.093E+02 2.806E+02 2.721E+02 IY= 26 5.910E+01 1.827E+02 3.745E+02 4.151E+02 3.958E+02 IY= 24 8.939E+01 2.856E+02 6.053E+02 5.783E+02 5.708E+02 IY= 22 3.297E+02 5.758E+02 1.358E+03 1.019E+03 1.096E+03 IY= 20 3.537E+01 1.505E+01 1.426E+01 7.403E+00 4.061E+00 IY= 18 1.421E+01 1.057E+01 1.202E+01 1.190E+01 1.520E+01 IY= 16 2.983E+O0 8.459E-01 6.649E-01 6.821E-01 8.893E-01 IY= 14 1.321E+O0 3.221E-01 1.456E-01 1.026E-01 1.071E-01 IY= 12 2.476E-01 1.144E-01 4.657E-02 3.374E-02 3.165E-02 IY= 10 5.998E-02 5.001E-02 3.279E-02 2.291E-02 1.692E-02 IY= 8 1.064E-02 1.396E-02 1.496E-02 1.546E-02 1.337E-02 IY= 6 2.045E-03 2.007E-03 3.438E-03 5.707E-03 7.207E-03 IY= 4 2.794E-03 1.208E-04 2.616E-04 7.390E-O4 1.592E-03 IY= 2 3.448E-03 2.072E-05 1.946E-05 3.136E-05 6.510E-05 45 47 49 43 IX= *1 IY= 40 1.169E+06 2.056E+O6 4.852E+06 2.181E+06 7.980E+05 IY= 38 1.825E+01 1.722E+01 1.992E+01 2.896E+01 1.680E+01 IY= 36 1.217E+02 1.374E+01 3.562E+02 3.534E+02 2.402E+02 IY= 34 2.997E+02 2.214E+02 4.560E+02 4.562E+02 3.599E+02 IY= 32 2.656E+02 2.951E+02 5.859E+02 4.656E+02 3.546E+02 IY= 30 2.605E+02 3.215E+02 4.514E+02 3.5O4E+02 2.975E+02 IY= 28 2.860E+02 4.021E+02 5.750E+02 3.638E+02 4.786E+02 IY= 26 4.213E+02 4.514E+02 7.250E+02 3.842E+02 5.530E+02 IY= 24 5.516E+02 5.012E+02 9.225E+02 5.653E+02 6.515E+02 t
221
Appendix A
IY= 22 1.144E+03 4.734E+02 8.186E+02 9.925E+02 8.333E+02 IY= 20 2.802E+00 2.186E+00 2.016E+00 1.616E+00 1.519E+O0 IY= 18 1.901E+01 2.122E+01 1.765E+01 1.959E+01 2.283E+01 IY= 16 1.212E+00 1.584E+00 1.705E+00 1.959E+00 2.394E+00 IY= 14 1.321E-01 1.741E-01 2.124E-01 2.482E-01 3.229E-01 IY= 12 3.138E-02 3.461E-02 3.912E-02 4.639E-02 6.202E-02 IY= 10 1.440E-02 1.385E-02 1.379E-02 1.499E-02 1.900E-02 IY= 8 1.051E-02 8.691E-03 7.985E-03 8.069E-03 9.063E-03 IY= 6 7.312E-03 6.544E-03 5.807E-03 5.526E-03 5.690E-03 IY= 4 2.438E-03 2.966E-03 3.139E-03 3.238E-03 3.352E-03 IY= 2 1.493E-04 2.964E-04 4.643E-04 6.062E-04 7.257E-04 51 53 55 57 59 DC= IY= 40 6.278E+06 3.838E+06 2.636E+06 1.839E+05 4.960E+05 IY= 38 1.762E+01 2.006E+01 1.858E+01 1.866E+01 9.317E+00 IY= 36 3.877E+02 5.407E+02 1.171E+02 5.194E+01 2.485E+01 IY= 34 6.563E+02 1.155E+03 1.109E+03 3.160E+02 5.789E+01 IY= 32 7.083E+02 7.746E+02 8.445E+02 3.581E+02 1.093E+02 IY= 30 6.498E+02 5.679E+02 7.891E+02 4.804E+02 1.050E+02 IY= 28 5.241E+02 5.394E+02 7.745E+02 5.2O6E+02 1.033E+02 IY= 26 4.262E+02 5.834E+02 3.643E+02 5.425E+02 1.544E+02 IY= 24 6.154E+02 5.908E+02 3.673E+02 1.402E+02 1.422E+02 IY= 22 7.790E+02 6.356E+02 2.786E+02 7.709E+01 9.692E+01 IY= 20 1.544E+00 1.388E+00 1.121E+00 9.581E-01 6.694E-01 IY= 18 2.970E+01 3.080E+01 4.202E+01 5.070E+01 5.506E+01 IY= 16 2.965E+O0 3.355E+00 5.415E+O0 1.156E+01 2.344E+01 IY= 14 3.974E-01 5.793E-01 1.180E+00 3.449E+00 7.970E+00 2.895E+00 IY= 12 8.640E-02 1.536E-01 4.060E-01 1.526E+00 1.O01E+O0 IY= 10 2.711E-02 4.862E-02 1.285E-01 6.216E-01 5.562E-01 IY= 8 1.200E-02 2.142E-02 4.905E-02 2.497E-01 2.214E-01 IY= 6 6.769E-03 1.066E-02 2.212E-02 6.011E-02 3.305E-02 IY= 4 3.757E-03 4.870E-03 7.843E-03 1.636E-02 3.117E-03 IY= 2 8.867E-04 1.129E-03 1.489E-03 2.059E-03 61 63 65 67 69 IX= 7.676E+04 FIELD V A L U E S O F HI 38 8.600E+04 7.793E+04 7.724E+04 7.694E+04 8.517E+04 IY= 40 IY= 36 8 600E+04 8.496E+04 8.456E+04 8.479E+04 8.555E+04 1Y= 34 8.600E+04 8.512E+04 8.481E+04 8.510E+04 8.559E+04 IY= 32 8.600E+04 8.501E+04 8.475E+04 8.511E+04 8.567E+04 IY= 30 8.600E+04 8.501E+04 8.473E+04 8.517E+04 8.581E+04 IY= 28 8 600E+04 8.519E+04 8.499E+04 8.539E+04 8.601E+04 IY= 26 8 600E+04 8.550E+04 8.540E+04 8.569E+04 8.619E+04 IY= 24 8.600E+04 8.583E+04 8.577E+04 8.595E+04 8.630E+04 IY= 22 8.600E+04 8.607E+04 8.602E+O4 8.613E+04 8.632E+04 IY= 20 8 600E+04 8.619E+04 8.613E+04 8.618E+04 O.OOOE+00 IY= 18 0000E+00 0.000E+00 0.000E+00 O.OOOE+00 6.985E+04 IY= 16 8 961E+04 7.846E+04 7.454E+04 7.193E+04 6.978E+04 IY= 14 8 961E+04 7.843E+04 7.448E+04 7.186E+04 6.971E+04 IY= 12 8 961E+04 7.843E+04 7.443E+04 7.180E+04 6.965E+04 IY= 10 8 961E+04 7.844E+04 7.439E+04 7.174E+04 6.960E+04 IY= 8 8 961E+04 7.847E+04 7.437E+04 7.169E+04 6.954E+04 IY= 6 8 961E+04 7.850E+04 7.436E+04 7.164E+04 6.949E+04 IY= 4 8 961E+04 7.853E+04 7.436E+04 7.161E+04 6.945E+04 IY= 2 8 961E+04 7.857E+04 7.437E+04 7.158E+04 6.941E+04 8 961E+04 7.861E+04 7.439E+04 7.156E+04 7.490E+04 DC= 8.401E+04 5 7 9 IY= 40 1 3 IY= 38 7 681E+04 7.690E+04 7.683E+04 7.571E+04 8 561E+04 8.598E+04 8.586E+04 8.520E+04
222
Appendix A
IY= 36 8.589E+04 8.617E+04 8.644E+04 8.664E+04 8.670E+04 IY= 34 8.592E+04 8.620E+04 8.649E+04 8.674E+04 8.690E+04 IY= 32 8.599E+04 8.627E+04 8.655E+04 8.680E+04 8.697E+04 IY= 30 8.611E+04 8.637E+04 8.664E+04 8.687E+04 8.703E+04 IY= 28 8.626E+04 8.649E+04 8.673E+04 8.693E+04 8.705E+04 IY= 26 8.640E+04 8.660E+04 8.680E+O4 8.696E+04 8.706E+04 IY= 24 8.648E+04 8.667E+04 8.684E+04 8.697E+04 8.704E+04 IY= 22 8.649E+04 8.666E+04 8.681E+04 8.692E+04 8.698E+04 IY= 20 0.O00E+O0 O.OOOE+00 0.000E+00 0.000E+00 O.OOOE+00 IY= 18 6.808E+O4 6.652E+04 6.512E+04 6.384E+04 6.268E+04 IY= 16 6.801E+04 6.645E+04 6.505E+04 6.377E+04 6.261E+04 IY= 14 6.794E+04 6.638E+04 6.499E+04 6.371E+04 6.254E+04 IY= 12 6.788E+04 6.633E+04 6.493E+04 6.366E+04 6.249E+04 IY= 10 6.783E+04 6.628E+04 6.488E+04 6.361E+04 6.244E+04 IY= 8 6.778E+04 6.623E+04 6.484E+04 6.357E+04 6.240E+04 IY= 6 6.773E+04 6.619E+04 6.481E+04 6.354E+04 6.237E+04 IY= 4 6.769E+04 6.615E+04 6.477E+04 6.351E+04 6.234E+04 IY= 2 6.764E+04 6.612E+04 6.475E+04 6.349E+04 6.231E+04 11 13 K= 15 17 19 IY= 40 7.439E+04 7.366E+04 7.316E+04 7.292E+04 7.283E+04 IY= 38 8.204E+04 7.883E+04 7.651E+04 7.493E+04 7.363E+04 IY= 36 8.599E+04 8.308E+04 8.005E+O4 7.734E+04 7.482E+04 IY= 34 8.699E+04 8.595E+04 8.318E+04 7.961E+04 7.590E+04 IY= 32 8.707E+04 8.709E+04 8.617E+04 8.209E+04 7.679E+04 IY= 30 8.711E+04 8.715E+04 8.717E+04 8.503E+04 7.771E+04 IY= 28 8.712E+04 8.716E+04 8.722E+04 8.728E+04 7.878E+04 IY= 26 8.710E+04 8.712E+04 8.722E+04 8.731E+04 8.055E+O4 IY= 24 8.706E+04 8.705E+04 8.718E+04 8.731E+04 8.420E+04 IY= 22 8.701E+04 8.696E+04 8.710E+04 8.726E+04 8.609E+04 IY= 20 7.753E+04 8.025E+04 8.177E+04 8.440E+04 8.587E+04 IY= 18 6.192E+04 6.220E+04 6.287E+04 6.487E+04 6.840E+04 IY= 16 6.168E+04 6.157E+04 6.178E+04 6.203E+04 6.281E+04 IY= 14 6.153E+04 6.108E+04 6.123E+04 6.145E+04 6.173E+04 IY= 12 6.143E+04 6.075E+04 6.069E+O4 6.095E+04 6.127E+04 IY= 10 6.137E+04 6.054E+04 6.024E+O4 6.043E+04 6.083E+O4 IY= 8 6.132E+04 6.042E+04 5.991E+04 5.994E+04 6.038E+04 IY= 6 6.128E+04 6.034E+O4 5.969E+04 5.955E+04 5.994E+04 IY= 4 6.126E+04 6.030E+O4 5.955E+04 5.925E+04 5.957E+04 IY= 2 6.124E+04 6.027E+O4 5.947E+04 5.906E+04 5.929E+04 IX= /21 23 27 29 25 IY= 40 7.288E+04 7.296E+04 7.302E+O4 7.317E+04 7.331E+04 IY= 38 7.319E+04 7.318E+04 7.315E+04 7.312E+04 7.303E+04 IY= 36 7.408E+04 7.414E+04 7.395E+04 7.373E+04 7.356E+04 IY= 34 7.519E+04 7.528E+04 7.492E+04 7.454E+04 7.435E+04 IY= 32 7.633E+04 7.644E+04 7.596E+04 7.549E+04 7.532E+04 IY= 30 7.746E+04 7.752E+04 7.693E+04 7.645E+04 7.635E+04 IY= 28 7.860E+04 7.853E+04 7.777E+04 7.731E+04 7.729E+04 IY= 26 7.985E+04 7.956E+04 7.856E+04 7.803E+04 7.808E+04 IY= 24 8.128E+04 8.072E+04 7.949E+04 7.879E+04 7.887E+04 IY= 22 8.368E+04 8.246E+04 8.056E+04 7.977E+04 7.988E+04 IY= 20 8.562E+04 8.508E+04 8.508E+O4 8.129E+04 8.032E+04 IY= 18 7.271E+04 7.684E+04 7.983E+04 8.138E+04 8.136E+04 IY= 16 6.447E+04 6.683E+04 6.933E+04 7.140E+04 7.287E+04 IY= 14 6.224E+04 6.314E+04 6.431E+04 6.548E+04 6.630E+04 IY= 12 6.169E+04 6.225E+04 6.291E+04 6.356E+04 6.414E+04 IY= 10 6.134E+04 6.196E+04 6.262E+04 6.320E+04 6.372E+04 IY= 8 (5.101E+04 <5.175E+04 (5.248E+04 i5.308E+04 i6.361E+04
223
Appendix A
IY= 6 6.069E+04 i5.157E+04 t5.237E+04 i5.298E+04 <6.350E+04 IY= 4 6.041E+04 i5.142E+04 i5.228E+04 i5.288E+04 .6.341E+04 IY= 2 6.019E+04 i6.132E+04 i6.223E+04 i5.282E+04 6.334E+04 :YI 39 35 K= 33 31 IY= 40 7.413E+04 7.487E+04 7.528E+04 7.456E+04 7.389E+04 IY= 38 7.378E+04 7.483E+04 7.609E+04 7.578E+04 7.480E+04 IY= 36 7.449E+04 7.547E+04 7.659E+04 7.643E+04 7.568E+04 IY= 34 7.538E+04 7.627E+04 7.704E+O4 7.666E+04 7.613E+04 IY= 32 7.640E+04 7.736E+04 7.780E+O4 7.673E+04 7.634E+04 IY= 30 7.738E+04 7.849E+04 7.918E+04 7.675E+04 7.641E+04 IY= 28 7.816E+04 7.926E+04 8.103E+O4 7.691E+04 7.638E+04 IY= 26 7.874E+04 7.975E+04 8.122E+04 7.772E+04 7.630E+04 IY= 24 7.926E+04 8.015E+04 8.053E+O4 7.978E+04 7.619E+04 IY= 22 7.985E+04 7.993E+04 7.988E+04 8.007E+04 7.607E+04 IY= 20 8.061E+04 7.896E+04 7.905E+04 7.903E+04 7.799E+04 IY= 18 8.017E+04 7.675E+04 7.501E+O4 7.449E+04 7.444E+04 IY= 16 7.306E+04 7.113E+04 7.046E+04 7.006E+04 6.988E+04 IY= 14 6.683E+04 6.680E+04 6.731E+04 6.7O4E+04 6.660E+04 IY= 12 6.464E+04 6.508E+04 6.598E+04 6.575E+04 6.510E+O4 IY= 10 6.422E+04 6.464E+04 6.559E+04 6.537E+04 6.463E+04 IY= 8 6.424E+04 6.455E+04 6.548E+04 6.527E+04 6.452E+04 IY= 6 6.429E+04 6.453E+04 6.543E+04 6.523E+04 6.448E+04 IY= 4 6.433E+04 6.452E+04 6.541E+04 6.520E+04 6.446E+04 IY= 2 6.435E+04 6.451E+04 6.539E+04 6.519E+04 6.446E+04 47 49 45 43 DC=: 41 7.982E+04 8.224E+04 7.965E+04 7.907E+04 IY= 40 7.739E+04 7.867E+04 7.957E+04 7.782E+04 7.794E+04 IY= 38 7.711E+04 7.824E+04 7.900E+04 7.741E+04 7.747E+04 IY= 36 7.673E+04 7.802E+04 7.885E+04 7.742E+04 7.739E+04 IY= 34 7.655E+04 7.779E+04 7.874E+04 7.756E+04 IY= 32 7.653E+04 7.750E+04 7.765E+04 7.857E+04 7.756E+04 7.728E+04 IY= 30 7.642E+04 7.799E+04 7.835E+04 7.756E+04 7.699E+04 IY= 28 7.622E+04 7.819E+04 7.804E+04 7.756E+04 7.724E+04 IY= 26 7.624E+04 7.821E+04 7.844E+04 7.746E+04 7.771E+04 IY= 24 7.639E+04 7.800E+04 7.872E+04 7.714E+04 7.763E+04 IY= 22 7.656E+04 7.718E+04 7.800E+04 7.642E+04 7.723E+04 IY= 20 7.707E+04 7.281E+04 7.396E+04 7.176E+04 7.274E+04 IY= 18 7.314E+04 7.050E+04 6.941E+04 6.845E+04 6.899E+04 IY= 16 6.888E+04 6.781E+04 6.686E+04 6.617E+04 IY= 14 6.591E+04 6.642E+04 IY= 12 6.454E+04 6.522E+04 6.509E+04 6.551E+04 6.626E+04 IY= 10 6.409E+04 6.483E+04 6.474E+04 6.499E+04 6.561E+04 IY= 8 6.397E+04 6.472E+04 6.465E+04 6.483E+04 6.540E+04 IY= 6 6.393E+04 6.468E+04 6.462E+04 6.476E+04 6.533E+04 IY= 4 6.391E+04 6.466E+04 6.461E+04 6.473E+04 6.530E+04 IY= 2 6.390E+04 6.465E+04 6.461E+04 6.470E+04 6.528E+04 57 59 55 53 K== 51 IY= 40 8.205E+04 8.193E+04 8.302E+04 8.465E+04 8.405E+04 IY= 38 7.905E+04 8.032E+04 8.231E+04 8.468E+04 8.432E+04 IY= 36 7.883E+04 8.000E+04 8.207E+O4 8.458E+04 8.427E+04 IY= 34 7.870E+04 7.980E+04 8.191E+04 8.446E+04 8.416E+04 1Y= 32 7.860E+04 7.970E+04 8.181E+04 8.434E+04 8.403E+04 IY= 30 7.856E+04 7.970E+04 8.172E+04 8.421E+04 8.389E+04 IY= 28 7.854E+04 7.972E+04 8.167E+04 8.407E+04 8.370E+04 IY= 26 7.856E+04 7.973E+04 8.161E+04 8.389E+04 8.341E+04 IY= 24 7.855E+04 7.968E+04 8.148E+04 8.364E+04 8.299E+04 IY= 22 7.831E+04 7.942E+04 8.119E+04 8.325E+04 8.232E+04 IY= 20 7.751E+04 7.855E+04 8.047E+O4 8.225E+04 8.070E+04
224
Appendix A
IY= 18 7.309E+04 7.410E+04 7.602E+O4 7.735E+04 7.571E+04 16 7.001E+04 7.101E+04 7.292E+04 7.437E+04 7.321E+04 6.850E+04i 7.032E+O4 7.189E+04 7.120E+04 6.696E+04t 6.863E+04 7.030E+04 6.990E+04 6.626E+04\ 6.780E+O4 6.950E+04 6.925E+04 6.600E+04 6.748E+04 6.919E+04 6.899E+04 6.591E+04 6.737E+04 6.908E+04 6.890E+04 6.587E+04 6.733E+04 6.904E+04 6.887E+04 6.585E+04 6.731E+04 6.903E+04 6.886E+04 67 69 65 FIELD V A L U E S O F M A C H IY= 40 4.656E-01 4.725E-01 4.971E-01 4.628E-01 4.681E-01 IY= 38 3.859E-01 2.083E-01 2.872E-01 2.900E-01 2.987E-01 IY= 36 3.686E-01 1.335E-01 2.153E-01 1.860E-01 1.374E-01 IY= 34 3.550E-01 6.180E-02 1.217E-01 5.750E-02 3.881E-02 IY= 32 3.705E-01 1.907E-01 6.744E-02 1.069E-01 9.759E-02 IY= 30 4.123E-01 2.814E-01 2.369E-01 2.494E-01 2.052E-01 IY= 28 4.906E-01 3.556E-01 3.496E-01 3.110E-01 2.598E-01 IY= 26 5.631E-01 3.708E-01 3.502E-01 3.162E-01 2.672E-01 IY= 24 5.694E-01 3.172E-01 3.096E-01 2.872E-01 2.650E-01 IY= 22 5.833E-01 2.330E-01 2.471E-01 2.496E-01 2.459E-01 IY= 20 9.492E-09 7.682E-10 8.880E-10 9.622E-10 1.018E-09 IY= 18 1.084E+00 7.979E-01 1.018E+00 1.150E+00 1.251E+00 IY= 16 1.166E+00 8.027E-01 1.024E+00 1.156E+00 1.257E+00 1Y= 14 1.218E+00 8.056E-01 1.029E+00 1.161E+00 1.262E+00 IY= 12 1.248E+00 8.069E-01 1.033E+00 1.166E+00 1.267E+00 IY= 10 1.264E+00 8.070E-01 1.035E+00 1.170E+00 1.271E+00 IY= 8 1.273E+00 8.065E-01 1.037E+00 1.173E+00 1.275E+00 IY= 6 1.277E+00 8.054E-01 1.038E+00 1.176E+00 1.278E+00 1Y= 4 1.279E+00 8.041E-01 1.038E+00 1.178E+00 1.281E+00 IY= 2 1.281E+00 8.028E-01 1.O38E+0O 1.180E+00 1.284E+00 1 3 7 9 K= 5 IY= 40 5.224E-01 5.600E-01 5.078E-01 4.397E-01 4.246E-01 IY= 38 2.745E-01 1.599E-01 1.483E-01 1.443E-01 1.368E-01 IY= 36 8.396E-02 7.005E-02 6.949E-02 5.749E-02 4.384E-02 IY= 34 1.829E-02 2.191E-02 2.793E-02 3.108E-02 3.494E-02 IY= 32 7.877E-02 7.453E-02 7.522E-02 7.818E-02 8.181E-02 IY= 30 1.637E-01 1.389E-01 1.270E-01 1.184E-01 1.106E-01 IY= 28 2.124E-01 1.879E-01 1.689E-01 1.485E-01 1.261E-01 IY= 26 2.371E-01 2.205E-01 1.978E-01 1.689E-01 1.346E-01 IY= 24 2.505E-01 2.355E-01 2.121E-01 1.787E-01 1.384E-01 IY= 22 2.402E-01 2.296E-01 2.085E-01 1.756E-01 1.355E-01 IY= 20 1.065E-09 1.105E-09 1.141E-09 1.173E-09 1.200E-09 IY= 18 1.336E+00 1.409E+00 1.475E+00 1.534E+00 1.588E+00 IY= 16 1.341E+00 1.415E+00 1.480E+00 1.539E+00 1.593E+00 IY= 14 1.346E+00 1.419E+00 1.485E+O0 1.544E+00 1.598E+O0 IY= 12 1.351E+O0 1.424E+00 1.489E+00 1.548E+00 1.602E+00 IY= 10 1.355E+00 1.427E+00 1.492E+00 1.551E+00 1.606E+00 IY= 8 1.358E+00 1.431E+00 1.495E+00 1.554E+00 1.609E+00 IY= 6 1.361E+00 1.433E+O0 1.498E+00 1.557E+00 1.611E+00 IY= 4 1.364E+00 1.436E+00 1.500E+00 1.559E+00 1.613E+00 IY= 2 1.367E+00 1.438E+00 1.502E+00 1.561E+00 1.615E+00 17 19 IX= 11 13 15 IY= 40 4.103E-01 3.940E-01 3.928E-01 3.734E-01 3.360E-01 IY= 38 1.402E-01 1.694E-01 2.019E-01 2.115E-01 2.275E-01 IY= 36 5.857E-02 1.008E-01 1.351E-01 1.514E-01 1.752E-01 IY= 34 3.656E-02 4.206E-02 9.501E-02 1.219E-01 1.589E-01
rv=
IY= 14 6.761E+04 IY= 12 6.621E+04 IY= 10 6.563E+04 IY= 8 6.544E+04 IY= 6 6.539E+04 IY= 4 6.538E+04 IY= 2 6.538E+04 63 K== 61
225
Appendix A
IY= 32 8.453E-02 6.366E-02 3.352E-02 8.689E-02 1.577E-01 IY= 30 1.080E-01 1.035E-01 8.451E-02 4.522E-02 1.459E-01 IY= 28 1.092E-01 1.113E-01 1.142E-01 6.555E-02 1.283E-01 IY= 26 1.021E-01 9.329E-02 1.207E-01 8.864E-02 9.754E-02 1Y= 24 9.564E-02 6.970E-02 1.038E-01 9.849E-02 5.767E-02 IY= 22 9.212E-02 5.297E-02 6.533E-02 8.478E-02 8.366E-02 IY= 20 1.504E-01 1.547E-01 1.475E-01 1.146E-01 1.005E-01 IY= 18 1.619E+00 1.600E+00 1.582E+00 1.485E+00 1.292E+00 IY= 16 1.634E+00 1.633E+00 1.627E+00 1.619E+00 1.589E+00 IY= 14 1.644E+00 1.661E+00 1.654E+00 1.645E+00 1.634E+00 IY= 12 1.651E+O0 1.681E+O0 1.682E+00 1.670E+00 1.655E+O0 IY= 10 1.655E+00 1.693E+00 1.706E+00 1.696E+00 1.677E+O0 IY= 8 1.659E+00 1.701E+00 1.724E+00 1.722E+00 1.700E+00 IY= 6 1.662E+00 1.706E+00 1.737E+O0 1.743E+O0 1.723E+O0 IY= 4 1.664E+00 1.7O9E+00 1.744E+00 1.758E+00 1.742E+00 IY= 2 1.665E+00 1.711E+00 1.749E+O0 1.768E+00 1.757E+00 IX= 21 23 25 27 29 1Y= 40 3.397E-01 3.661E-01 3.727E-01 3.706E-01 4.032E-01 IY= 38 3.129E-01 3.762E-01 4.109E-01 4.593E-01 5.477E-01 IY= 36 2.790E-01 3.359E-01 3.817E-01 4.546E-01 5.450E-01 IY= 34 2.509E-01 2.933E-01 3.469E-01 4.372E-01 5.273E-01 IY= 32 2.141E-01 2.508E-01 3.084E-01 4.083E-01 4.936E-01 IY= 30 1.790E-01 2.119E-01 2.685E-01 3.739E-01 4.491E-01 IY= 28 1.455E-01 1.738E-01 2.304E-01 3.359E-01 4.044E-01 IY= 26 1.102E-01 1.413E-01 2.020E-01 2.998E-01 3.563E-01 IY= 24 7.399E-02 1.231E-01 1.701E-01 2.762E-01 3.025E-01 IY= 22 6.176E-02 9.564E-02 1.364E-01 2.428E-01 2.367E-01 IY= 20 9.863E-02 8.659E-02 4.077E-02 9.799E-02 2.102E-01 IY= 18 1.031E+00 7.727E-01 5.693E-01 4.229E-01 2.356E-01 IY= 16 1.517E+00 1.407E+00 1.275E+O0 1.157E+00 1.103E+O0 IY= 14 1.615E+00 1.585E+00 1.543E+O0 1.498E+00 1.480E+00 IY= 12 1.638E+O0 1.616E+00 1.592E+00 1.571E+00 1.554E+O0 IY= 10 1.654E+00 1.627E+00 1.600E+00 1.578E+00 1.560E+O0 IY= 8 1.670E+00 1.636E+00 1.604E+00 1.579E+00 1.559E+O0 IY= 6 1.686E+00 1.645E+O0 1.608E+00 1.582E+00 1.561E+00 IY= 4 1.702E+00 1.653E+00 1.612E+00 1.584E+00 1.563E+00 IY= 2 1.714E+00 1.659E+O0 1.615E+O0 1.587E+00 1.564E+O0 K= 35 37 39 31 33 IY= 40 2.764E-01 3.492E-01 9.238E-02 9.494E-02 2.641E-01 IY= 38 4.625E-01 4.602E-01 2.153E-01 6.628E-02 2.687E-01 IY= 36 4.818E-01 4.914E-01 3.044E-01 2.359E-01 3.606E-01 IY= 34 4.763E-01 4.848E-01 3.300E-01 3.308E-01 4.471E-01 IY= 32 4.423E-01 4.340E-01 2.916E-0I 3.440E-01 4.521E-01 IY= 30 3.918E-01 3.385E-01 2.030E-01 3.O01E-01 4.135E-01 IY= 28 3.401E-01 2.144E-01 5.559E-02 2.167E-01 3.480E-01 IY= 26 3.001E-01 7.562E-02 7.681E-02 1.023E-01 2.681E-01 IY= 24 2.708E-01 1.022E-02 1.708E-01 2.036E-01 1.750E-01 IY= 22 2.354E-01 1.246E-02 2.757E-01 3.014E-01 7.050E-02 IY= 20 1.149E-01 3.239E-02 4.188E-01 4.831E-01 4.990E-01 IY= 18 4.676E-01 6.182E-01 9.810E-01 1.066E+00 1.094E+00 IY= 16 1.142E+00 1.145E+00 1.304E+00 1.341E+00 1.382E+00 IY= 14 1.420E+00 1.465E+00 1.468E+00 1.447E+00 1.469E+00 IY= 12 1.534E+00 1.564E+00 1.506E+00 1.471E+00 1.490E+00 IY= 10 1.547E+O0 1.573E+00 1.516E+00 1.479E+O0 1.497E+00 IY= 8 1.542E+00 1.573E+O0 1.522E+00 1.482E+00 1.498E+00 IY= 6 1.537E+00 1.573E+O0 1.525E+00 1.482E+00 1.497E+00 IY= 4 1.532E+O0 1.571E+00 1.527E+00 1.482E+00 1.496E+00
226
Appendix A
IY= 2 1.529E+O0 1.57OE+O0 1.525E+O0 1.481E+00 1.494E+00 IX= 41 43 45 47 49 IY= 40 2.608E-01 1.948E-01 2.696E-01 2.130E-01 2.093E-01 IY= 38 6.043E-01 5.898E-01 5.727E-01 5.002E-01 5.046E-01 IY= 36 6.325E-01 5.904E-01 4.927E-01 4.293E-01 4.371E-01 IY= 34 5.896E-01 5.046E-01 3.298E-01 3.273E-01 3.362E-01 IY= 32 5.071E-01 3.639E-01 1.120E-01 2.349E-01 2.261E-01 IY= 30 4.008E-01 2.023E-01 1.304E-02 1.640E-01 1.253E-01 IY= 28 2.934E-01 7.687E-02 1.476E-01 1.455E-01 2.195E-02 IY= 26 1.770E-01 6.751E-03 2.806E-01 2.796E-01 6.288E-02 IY= 24 8.331E-03 2.169E-01 4.548E-01 4.380E-01 2.516E-01 IY= 22 9.305E-03 4.686E-01 6.451E-01 6.079E-01 4.673E-01 IY= 20 5.798E-01 7.649E-01 8.511E-01 7.915E-01 7.515E-01 IY= 18 1.254E+00 1.327E+00 1.328E+00 1.236E+00 1.130E+00 IY= 16 1.457E+00 1.445E+00 1.423E+00 1.363E+O0 1.312E+00 IY= 14 1.508E+00 1.481E+00 1.464E+00 1.427E+00 1.401E+00 IY= 12 1.524E+00 1.496E+00 1.480E+00 1.456E+00 1.441E+00 IY= 10 1.529E+00 1.500E+00 1.485E+00 1.468E+00 1.458E+00 IY= 8 1.528E+00 1.499E+00 1.486E+O0 1.473E+00 1.464E+00 IY= 6 1.527E+O0 1.497E+00 1.486E+00 1.477E+00 1.466E+00 IY= 4 1.525E+00 1.495E+00 1.486E+00 1.479E+00 1.468E+00 IY= 2 1.523E+00 1.494E+00 1.486E+00 1.481E+00 1.468E+00 57 59 55 DC= 51 53 IY= 40 2.857E-01 2.541E-01 2.373E-01 2.781E-03 1.177E-01 IY= 38 4.255E-01 3.162E-01 2.106E-01 1.033E-03 2.220E-01 IY= 36 2.803E-01 2.208E-01 1.310E-01 1.759E-03 2.435E-01 IY= 34 1.326E-01 1.362E-01 5.639E-02 3.465E-02 2.170E-01 IY= 32 1.011E-02 7.090E-02 3.927E-03 7.319E-02 1.901E-01 IY= 30 2.345E-02 1.019E-01 5.222E-02 1.259E-01 2.299E-01 IY= 28 1.722E-01 2.011E-01 1.558E-01 1.932E-01 3.036E-01 IY= 26 2.952E-01 3.187E-01 2.674E-01 2.688E-01 3.628E-01 IY= 24 4.511E-01 4.477E-01 3.869E-01 3.525E-01 4.260E-01 IY= 22 6.123E-01 5.812E-01 5.059E-01 4.421E-01 5.070E-01 IY= 20 7.938E-01 7.292E-01 6.358E-01 5.602E-01 6.572E-01 IY= 18 1.153E+00 1.096E+00 9.929E-01 9.433E-01 1.012E+00 IY= 16 1.321E+00 1.260E+00 1.165E+00 1.113E+00 1.144E+00 IY= 14 1.409E+00 1.355E+00 1.276E+00 1.222E+00 1.233E+00 IY= 12 1.451E+00 1.406E+00 1.339E+00 1.286E+00 1.286E+00 IY= 10 1.469E+00 1.431E+O0 1.370E+00 1.317E+O0 1.313E+00 IY= 8 1.476E+O0 1.442E+00 1.384E+00 1.331E+00 1.326E+00 IY= 6 1.479E+00 1.448E+00 1.389E+00 1.336E+00 1.332E+00 IY= 4 1.480E+00 1.451E+00 1.392E+00 1.338E+00 1.335E+00 IY= 2 1.481E+00 1.454E+00 1.394E+00 1.340E+00 1.337E+00 67 69 65 1X= 61 63 FIELD V A L U E S O F V A B S IY= 40 7.479E+01 7.251E+01 7.592E+01 7.061E+01 7.128E+01 IY= 38 6.197E+01 3.327E+01 4.577E+01 4.627E+01 4.777E+01 IY= 36 5.918E+01 2.134E+01 3.436E+01 2.973E+01 2.203E+01 IY= 34 5.701E+01 9.869E+00 1.941E+01 9.191E+O0 6.224E+00 IY= 32 5.950E+01 3.045E+01 1.075E+01 1.710E+01 1.566E+01 IY= 30 6.620E+01 4.499E+01 3.785E+01 3.994E+01 3.295E+01 IY= 28 7.878E+01 5.697E+01 5.598E+01 4.990E+01 4.176E+01 IY= 26 9.043E+01 5.953E+01 5.622E+01 5.081E+01 4.300E+01 IY= 24 9.146E+01 5.099E+01 4.978E+01 4.621E+01 4.268E+01 IY= 22 9.371E+01 3.748E+01 3.975E+01 4.016E+01 3.960E+01 IY= 20 1.082E+02 8.759E+00 1.012E+01 1.097E+01 1.161E+01 1.812E+02 IY= 18 1.778E+02 1.224E+02 1.522E+02 1.690E+02
227
Appendix A
IY= 16 1.911E+02 1.231E+02 1.531E+02 1.698E+02 1.819E+02 IY= 14 1.997E+02 1.236E+02 1.537E+02 1.705E+02 1.826E+02 IY= 12 2.047E+02 1.238E+02 1.543E+02 1.711E+02 1.831E+02 IY= 10 2.073E+02 1.238E+02 1.546E+02 1.716E+02 1.837E+02 IY= 8 2.086E+02 1.238E+02 1.549E+02 1.720E+02 1.841E+02 IY= 6 2.093E+O2 1.236E+02 1.550E+02 1.724E+02 1.845E+02 IY= 4 2.097E+02 1.235E+02 1.551E+02 1.727E+02 1.849E+02 IY= 2 2.100E+02 1.233E+02 1.551E+02 1.729E+02 1.852E+02 5 7 9 IX= 1 3 6.666E+01 6.396E+01 8.517E+01 7.728E+01 1Y= 40 7.941E+01 IY= 38 4.399E+01 2.571E+01 2.385E+01 2.318E+01 2.184E+01 IY= 36 1.348E+01 1.127E+01 1.120E+01 9.280E+O0 7.083E+00 IY= 34 2.938E+00 3.525E+00 4.503E+00 5.019E+00 5.650E+00 IY= 32 1.266E+01 1.200E+01 1.213E+01 1.263E+01 1.323E+01 IY= 30 2.634E+01 2.238E+01 2.049E+01 1.914E+01 1.789E+01 IY= 28 3.419E+01 3.030E+01 2.727E+01 2.401E+01 2.040E+01 IY= 26 3.820E+01 3.557E+01 3.196E+01 2.732E+01 2.179E+01 IY= 24 4.039E+01 3.801E+01 3.426E+01 2.890E+01 2.240E+01 IY= 22 3.873E+01 3.706E+01 3.368E+01 2.839E+01 2.192E+01 IY= 20 1.214E+01 1.260E+01 1.301E+01 1.338E+01 1.368E+01 IY= 18 1.909E+02 1.991E+02 2.061E+02 2.123E+02 2.178E+02 IY= 16 1.916E+02 1.997E+02 2.067E+02 2.129E+02 2.184E+02 1Y= 14 1.922E+02 2.003E+02 2.073E+02 2.134E+02 2.189E+02 IY= 12 1.928E+02 2.008E+02 2.078E+02 2.139E+02 2.194E+02 IY= 10 1.932E+02 2.013E+02 2.082E+02 2.143E+02 2.198E+02 IY= 8 1.937E+02 2.016E+02 2.085E+02 2.146E+02 2.201E+02 IY= 6 1.941E+02 2.020E+O2 2.088E+02 2.149E+02 2.204E+02 IY= 4 1.944E+02 2.023E+02 2.091E+02 2.152E+02 2.206E+02 IY= 2 1.947E+02 2.025E+02 2.093E+02 2.154E+02 2.208E+02 15 17 19 5.532E+01 4.973E+01 IX= 11 13 IY= 40 6.153E+01 5.879E+01 5.834E+01 3.189E+01 3.394E+01 IY= 38 2.217E+01 2.630E+01 3.082E+01 2.328E+01 2.642E+01 IY= 36 9.442E+00 1.606E+01 2.113E+01 1.902E+01 2.416E+01 1Y= 34 5.916E+00 6.783E+O0 1.513E+01 1.378E+01 2.413E+01 IY= 32 1.368E+01 1.031E+01 5.412E+00 7.287E+00 2.245E+01 IY= 30 1.748E+01 1.677E+01 1.369E+01 1.062E+01 1.988E+01 IY= 28 1.768E+01 1.803E+01 1.851E+01 1.437E+01 1.534E+01 IY= 26 1.653E+01 1.5UE+01 1.955E+01 1.597E+01 9.253E+O0 IY= 24 1.548E+01 1.128E+01 1.681E+01 1.374E+01 1.352E+01 IY= 22 1.491E+01 8.568E+00 1.058E+01 1.822E+01 1.612E+01 IY= 20 2.299E+01 2.404E+01 2.313E+01 2.071E+02 1.849E+02 IY= 18 2.209E+02 2.189E+02 2.174E+02 2.211E+02 2.184E+02 IY= 16 2.224E+02 2.223E+02 2.218E+02 2.237E+02 2.226E+02 IY= 14 2.234E+02 2.251E+02 2.245E+02 2.261E+02 2.247E+02 IY= 12 2.241E+02 2.271E+02 2.272E+02 2.287E+02 2.269E+02 IY= 10 2.246E+02 2.283E+02 2.296E+02 2.310E+02 2.290E+02 IY= 8 2.250E+02 2.291E+02 2.313E+02 2.330E+02 2.312E+02 IY= 6 2.253E+02 2.296E+02 2.324E+02 2.344E+02 2.330E+02 IY= 4 2.256E+02 2.299E+02 2.332E+02 2.354E+02 2.344E+02 IY= 2 2.257E+02 2.301E+02 2.336E+02 5.482E+01 5.959E+01 25 27 29 6.791E+01 8.080E+01 TX= 21 23 IY= 40 5.024E+01 5.414E+01 5.511E+01 6.752E+01 8.071E+01 IY= 38 4.640E+01 5.572E+01 6.082E+01 6.533E+01 7.854E+01 IY= 36 4.164E+01 5.009E+01 5.683E+01 6.143E+01 7.406E+01 IY= 34 3.773E+01 4.410E+01 5.203E+01 5.666E+01 6.790E+01 IY= 32 3.244E+01 3.803E+01 4.661E+01 IY= 30 2.734E+01 3.239E+01 4.087E+01
228
Appendix A
IY= 28 2.240E+01 IY= 26 1.711E+01 IY= 24 1.161E+01 IY= 22 9.845E+00 IY= 20 1.585E+01 IY= 18 1.522E+02 IY= 16 2.111E+02 IY= 14 2.210E+02 IY= 12 2.231E+02 IY= 10 2.247E+02 1Y= 8 2.262E+02 IY= 6 2.278E+02 IY= 4 2.293E+02 IY= 2 2.304E+02 IX= 31 33 IY= 40 4.112E+01 IY= 38 6.861E+01 IY= 36 7.184E+01 IY= 34 7.146E+01 IY= 32 6.686E+01 IY= 30 5.966E+01 IY= 28 5.208E+01 IY= 26 4.612E+01 IY= 24 4.172E+01 IY= 22 3.640E+01 IY= 20 1.787E+01 IY= 18 7.274E+01 IY= 16 1.693E+02 IY= 14 2.013E+02 IY= 12 2.135E+02 IY= 10 2.146E+02 IY= 8 2.140E+02 IY= 6 2.134E+02 IY= 4 2.128E+02 IY= 2 2.124E+02
K = 41 43 IY= 40 3.952E+01 IY= 38 9.151E+01 IY= 36 9.557E+01 1Y= 34 8.899E+01 IY= 32 7.653E+01 IY= 30 6.044E+01 IY= 28 4.418E+01 IY= 26 2.668E+01 IY= 24 1.260E+00 IY= 22 1.4O8E+00 1Y= 20 8.812E+01 IY= 18 1.856E+02 IY= 16 2.092E+02 IY= 14 2.117E+02 IY= 12 2.118E+02 IY= 10 2.116E+02 IY= 8 2.115E+02 IY= 6 2.112E+02 IY= 4 2.109E+02 IY= 2 2.107E+02 IX= 51 53
2.676E+01 3.529E+01 5.121E+01 6.156E+01 2.193E+01 3.111E+01 4.591E+01 5.451E+01 1.925E+01 2.638E+01 4.249E+01 4.652E+01 1.513E+01 2.133E+01 3.756E+01 3.665E+01 1.387E+01 6.532E+00 1.529E+01 3.264E+01 1.173E+02 8.819E+01 6.611E+01 3.677E+01 1.992E+02 1.838E+02 1.695E+02 1.626E+02 2.183E+02 2.144E+02 2.099E+02 2.084E+02 2.211E+02 2.190E+02 2.171E+02 2.154E+02 2.222E+02 2.196E+02 2.175E+02 2.158E+02 2.230E+02 2.200E+02 2.175E+02 2.156E+02 2.239E+02 2.203E+02 2.177E+02 2.157E+02 2.247E+02 2.207E+02 2.180E+02 2.159E+02 2.253E+02 2.210E+02 2.182E+02 2.160E+02 35 37 39 5.201E+01 1.384E+01 1.418E+01 3.932E+01 6.852E+01 3.240E+01 9.964E+00 4.019E+01 7.350E+01 4.595E+01 3.563E+01 5.427E+01 7.296E+01 4.997E+01 5.006E+01 6.747E+01 6.585E+01 4.437E+01 5.208E+01 6.833E+01 5.183E+01 3.116E+01 4.543E+01 6.253E+01 3.307E+01 8.650E+00 3.282E+01 5.262E+01 1.170E+01 1.201E+01 1.555E+01 4.052E+01 1.581E+00 2.659E+01 3.142E+01 2.644E+01 1.916E+00 4.273E+01 4.679E+01 1.064E+01 4.968E+00 6.454E+01 7.464E+01 7.653E+01 9.331E+01 1.471E+02 1.596E+02 1.639E+02 1.665E+02 1.894E+02 1.948E+02 2.004E+02 2.066E+02 2.084E+02 2.054E+02 2.079E+02 2.176E+02 2.116E+02 2.067E+02 2.085E+02 2.182E+02 2.123E+02 2.071E+02 2.086E+02 2.181E+02 2.129E+02 2.074E+02 2.086E+02 2.180E+02 2.133E+02 2.074E+02 2.084E+02 2.178E+02 2.135E+02 2.073E+02 2.082E+02 2.177E+02 2.133E+02 2.071E+02 2.079E+02 45 47 49 2.981E+01 4.160E+01 3.294E+01 3.277E+01 8.991E+01 8.739E+01 7.682E+01 7.778E+01 8.973E+01 7.498E+01 6.576E+01 6.717E+01 7.665E+01 5.019E+01 5.006E+01 5.162E+01 5.533E+01 1.707E+01 3.588E+01 3.469E+01 3.074E+01 1.988E+00 2.5O4E+01 1.920E+01 1.164E+01 2.251E+01 2.229E+01 3.358E+00 1.021E+O0 4.283E+01 4.289E+01 9.597E+00 3.306E+01 6.945E+01 6.724E+01 3.852E+01 7.150E+01 9.837E+01 9.327E+01 7.182E+01 1.166E+02 1.292E+02 1.209E+02 1.151E+02 1.963E+02 1.953E+02 1.832E+02 1.685E+02 2.078E+02 2.042E+02 1.971E+02 1.910E+02 2.088E+02 2.062E+02 2.023E+02 1.998E+02 2.089E+02 2.067E+02 2.042E+02 2.030E+02 2.088E+02 2.068E+02 2.050E+02 2.043E+02 2.086E+02 2.068E+02 2.056E+02 2.049E+02 2.082E+02 2.067E+02 2.059E+02 2.051E+02 2.079E+02 2.067E+02 2.062E+02 2.053E+02 2.078E+02 2.067E+02 2.064E+02 2.054E+02 55
57
59
229
Appendix A
IY= 40 4.482E+01 3.983E+01 3.737E+01 4.417E-01 1.866E+01 IY= 38 6.545E+01 4.905E+01 3.303E+01 1.642E-01 3.526E+01 IY= 36 4.308E+01 3.419E+01 2.052E+01 2.795E-01 3.868E+01 IY= 34 2.036E+01 2.107E+01 8.829E+00 5.503E+00 3.445E+01 IY= 32 1.551E+00 1.097E+01 6.145E-01 1.162E+01 3.016E+01 IY= 30 3.599E+O0 1.576E+01 8.171E+00 1.997E+01 3.645E+01 IY= 28 2.643E+01 3.112E+01 2.438E+01 3.063E+01 4.811E+01 IY= 26 4.534E+01 4.934E+01 4.183E+01 4.258E+01 5.740E+01 IY= 24 6.932E+01 6.933E+01 6.051E+01 5.578E+01 6.725E+01 IY= 22 9.400E+01 8.989E+01 7.900E+01 6.981E+01 7.972E+01 IY= 20 1.213E+02 1.122E+02 9.888E+01 8.795E+01 1.023E+02 IY= 18 1.710E+02 1.636E+02 1.500E+02 1.435E+02 1.525E+02 IY= 16 1.915E+02 1.840E+02 1.723E+02 1.659E+02 1.695E+02 IY= 14 2.006E+02 1.942E+02 1.851E+02 1.791E+02 1.801E+02 IY= 12 2.044E+02 1.993E+02 1.919E+02 1.863E+02 1.860E+02 IY= 10 2.060E+02 2.017E+02 1.952E+02 1.898E+02 1.891E+02 IY= 8 2.067E+02 2.029E+O2 1.967E+02 1.913E+02 1.906E+02 IY= 6 2.070E+02 2.036E+02 1.974E+02 1.919E+02 1.913E+02 IY= 4 2.071E+02 2.040E+02 1.977E+02 1.922E+02 1.917E+02 IY= 2 2.073E+02 2.044E+02 1.979E+02 1.924E+02 1.920E+02 67 69 65 K= 61 63 E N U T FIELD V A L U E S O F IY= 40 3.802E-05 4.196E-05 4.207E-05 4.074E-05 3.950E-05 IY= 38 3.656E-08 4.516E-09 1.765E-09 1.590E-09 1.490E-09 IY= 36 2.075E-08 5.775E-07 6.850E-07 7.004E-07 7.004E-07 IY= 34 3.627E-07 6.527E-07 5.909E-07 6.432E-07 6.646E-07 IY= 32 3.280E-07 7.254E-07 6.223E-07 6.351E-07 6.469E-07 IY= 30 2.390E-07 4.514E-07 4.830E-07 5.318E-07 4.682E-07 IY= 28 1.985E-07 2.122E-07 2.448E-07 2.720E-07 2.405E-07 IY= 26 2.309E-07 1.831E-07 1.519E-07 1.366E-07 1.182E-07 IY= 24 2.693E-07 2.137E-07 1.481E-07 1.147E-07 9.817E-08 IY= 22 5.509E-07 3.502E-07 2.021E-07 1.415E-07 1.163E-07 IY= 20 O.OOOE+00 0.000E+00 O.OOOE+00 O.OOOE+00 0.000E+00 IY= 18 6.066E-08 1.275E-07 5.707E-08 8.046E-08 5.942E-08 IY= 16 1.166E-06 2.927E-07 5.551E-08 7.862E-08 5.838E-08 IY= 14 1.173E-06 2.964E-07 5.405E-08 7.727E-08 5.749E-08 IY= 12 1.180E-06 3.008E-07 5.428E-08 7.642E-08 5.682E-08 IY= 10 1.185E-06 3.052E-07 5.609E-08 7.614E-08 5.633E-08 5.600E-08 IY= 8 1.188E-06 3.090E-07 5.917E-08 7.651E-08 5.584E-08 7.757E-08 6.309E-08 IY= 6 1.189E-06 3.121E-07 5.588E-08 7.922E-08 6.740E-08 IY= 4 1.190E-06 3.144E-07 5.609E-08 8.126E-08 7.155E-08 IY= 2 1.189E-06 3.159E-07 7 9 5 1 3 3.524E-05 40 4.239E-05 4.573E-05 4.368E-05 3.780E-05 7.714E-10 IY= 7.499E-10 7.155E-10 38 8.435E-10 7.044E-10 1.051E-06 IY= 9.770E-07 8.761E-07 36 7.892E-07 8.256E-07 IY= 5.854E-07 5.421E-07 5.502E-07 34 6.325E-07 5.816E-07 IY= 3.968E-07 5.265E-07 4.641E-07 4.168E-07 3.955E-07 IY= 32 2.343E-07 2.408E-07 2.676E-07 3.187E-07 IY= 30 3.889E-07 1.298E-07 1.309E-07 1.634E-07 1.409E-07 IY= 28 2.008E-07 8.665E-08 7.903E-08 7.479E-08 IY= 26 9.693E-08 8.035E-08 7.132E-08 7.378E-08 8.216E-08 7.591E-08 8.585E-08 IY= 24 8.603E-08 9.947E-08 IY= 22 1.034E-07 9.212E-08 8.559E-08 O.OOOE+00 O.OOOE+00 IY= 20 O.OOOE+00 0.0O0E+00 O.OOOE+00 5.145E-11 2.519E-08 IY= 18 4.597E-08 2.248E-08 1.449E-08 3.677E-08 3.039E-08 IY= 16 4.538E-08 3.768E-08 3.283E-08 3.353E-08 2.981E-08 IY= 14 4.464E-08 3.706E-08 3.230E-08
rx=
230
Appendix A
IY= 12 4.405E-08 IY= 10 4.355E-08 IY= 8 4.309E-08 IY= 6 4.266E-08 IY= 4 4.224E-08 IY= 2 4.186E-08 13 K= 11 IY= 40 3.354E-05 IY= 38 4.970E-10 IY= 36 1.141E-06 IY= 34 7.017E-07 IY= 32 4.513E-07 IY= 30 2.474E-07 IY= 28 1.342E-07 IY= 26 9.452E-08 IY= 24 9.094E-08 IY= 22 1.360E-07 IY= 20 1.154E-06 IY= 18 5.942E-07 IY= 16 2.241E-07 IY= 14 9.974E-08 IY= 12 5.968E-08 IY= 10 4.359E-08 IY= 8 3.589E-08 IY= 6 3.171E-08 IY= 4 2.921E-08 IY= 2 2.734E-08
rx=:
21
23
IY= 40 1.829E-05 IY= 38 3.569E-09 IY= 36 8.299E-07 IY= 34 8.942E-07 IY= 32 9.540E-07 IY= 30 8.191E-07 IY= 28 7.286E-07 IY= 26 7.767E-07 IY= 24 9.994E-07 IY= 22 1.301E-06 IY= 20 1.361E-06 IY= 18 1.438E-07 IY= 16 2.764E-07 IY= 14 2.070E-07 IY= 12 1.731E-07 IY= 10 1.516E-07 IY= 8 1.205E-07 IY= 6 8.685E-08 IY= 4 6.787E-08 IY= 2 6.228E-08 33 IX= 31 IY= 40 1.074E-05 IY= 38 7.055E-10 IY= 36 1.209E-06 IY= 34 1.097E-06 IY= 32 8.886E-07 IY= 30 8.052E-07 IY= 28 8.289E-07 IY= 26 8.927E-07
3.657E-08 3.615E-08 3.577E-08 3.539E-08 3.500E-08 3.460E-08
3.189E-08 2.936E-08 3.131E-08 3.157E-08 2.900E-08 2.979E-08 3.130E-08 2.873E-08 2.873E-08 3.105E-08 2.851E-08 2.798E-08 3.081E-08 2.833E-08 2.744E-08 3.057E-08 2.819E-08 2.707E-08 15 17 19 3.012E-05 2.763E-05 2.499E-05 2.097E-05 4.647E-10 4.172E-10 3.844E-10 5.826E-10 1.153E-06 1.053E-06 9.518E-07 9.128E-07 8.323E-07 9.182E-07 8.582E-07 8.757E-07 6.041E-07 7.289E-07 8.506E-07 1.008E-06 2.975E-07 3.761E-07 5.325E-07 6.831E-07 1.555E-07 1.950E-07 3.618E-07 5.707E-07 9.840E-08 9.778E-08 1.295E-07 3.284E-07 6.404E-07 2.633E-07 2.156E-07 4.897E-07 8.352E-07 8.312E-07 9.313E-07 1.258E-06 1.290E-06 6.544E-08 9.012E-07 1.098E-06 1.394E-06 3.001E-07 2.195E-09 1.439E-08 8.200E-08 2.264E-07 1.694E-07 2.211E-07 2.347E-07 1.403E-07 1.421E-07 1.514E-07 1.854E-07 8.888E-08 1.331E-07 1.117E-07 1.485E-07 6.085E-08 1.142E-07 9.059E-08 1.091E-07 4.569E-08 8.902E-08 8.403E-08 8.054E-08 3.734E-08 6.767E-08 7.728E-08 6.911E-08 3.239E-08 5.288E-08 6.723E-08 6.791E-08 25 4.310E-08 5.646E-08 6.516E-08 1.773E-05 27 29 1.981E-08 1.640E-05 1.393E-05 1.170E-05 8.657E-07 1.943E-08 7.862E-09 2.956E-09 8.592E-07 9.520E-07 1.085E-06 1.354E-06 9.027E-07 8.772E-07 9.588E-07 9.615E-07 1.215E-06 8.968E-07 9.640E-07 9.074E-07 1.035E-06 1.089E-06 7.645E-07 9.530E-07 9.785E-07 1.184E-06 1.149E-06 8.635E-07 1.104E-06 9.638E-07 1.360E-06 8.023E-07 1.263E-06 1.193E-06 1.373E-06 1.336E-06 1.084E-06 1.152E-06 1.051E-06 6.557E-07 1.996E-07 1.017E-06 7.412E-07 8.428E-08 3.741E-07 2.358E-07 3.054E-07 3.967E-07 4.410E-07 1.896E-07 4.197E-07 3.580E-07 2.811E-07 1.657E-07 3.193E-07 3.559E-07 3.103E-07 1.522E-07 2.035E-07 2.197E-07 2.054E-07 1.332E-07 1.798E-07 1.973E-07 1.835E-07 1.034E-07 1.598E-07 1.725E-07 1.618E-07 7.448E-08 1.557E-07 1.752E-07 1.650E-07 1.531E-07 2.000E-07 2.051E-07 35 1.092E-05 1.400E-07 2.158E-07 2.202E-07 39 3.890E-10 37 6.534E-06 3.896E-06 8.522E-06 1.130E-06 2.744E-10 3.496E-10 1.861E-10 1.441E-06 1.669E-06 1.390E-06 1.413E-06 9.670E-07 1.558E-06 2.043E-06 2.370E-06 2.699E-06 1.532E-06 2.063E-06 2.352E-06 1.854E-06 1.399E-06 2.181E-06 1.562E-06 1.340E-06 1.943E-06 1.443E-06 1.199E-06 2.022E-06 1.579E-06 1.418E-06
231
Appendix A
IY= 24 8.007E-07 1.318E-06 2.351E-06 1.649E-06 1.516E-06 IY= 22 3.766E-07 7.919E-07 2.388E-06 1.745E-06 1.847E-06 IY= 20 6.562E-09 8.155E-09 7.884E-09 1.191E-08 2.995E-08 IY= 18 5.395E-07 5.654E-07 7.167E-07 6.896E-07 8.111E-07 IY= 16 2.821E-07 2.351E-07 2.761E-07 3.043E-07 3.789E-07 IY= 14 2.672E-07 1.938E-07 1.772E-07 1.676E-07 2.007E-07 1Y= 12 1.390E-07 1.057E-07 7.639E-08 8.557E-08 1.184E-07 IY= 10 1.128E-07 9.163E-08 7.387E-08 6.161E-08 6.385E-08 IY= 8 8.633E-08 7.891E-08 6.993E-08 6.072E-08 5.431E-08 IY= 6 3.230E-08 5.189E-08 5.637E-08 5.443E-08 4.942E-08 IY= 4 1.668E-08 2.059E-08 2.655E-08 3.620E-08 4.047E-08 IY= 2 1.726E-08 1.076E-08 1.147E-08 1.338E-08 1.699E-08 1X= 41 43 45 47 49 IY= 40 5.665E-06 6.561E-06 8.371E-06 6.137E-06 4.194E-06 IY= 38 2.645E-10 5.362E-10 1.926E-10 1.574E-10 1.786E-10 IY= 36 1.064E-06 7.990E-08 1.779E-06 2.336E-06 1.905E-06 IY= 34 2.297E-06 1.183E-06 2.491E-06 2.841E-06 2.012E-06 IY= 32 1.666E-06 1.435E-06 2.753E-06 2.025E-06 1.388E-06 IY= 30 1.236E-06 1.211E-06 1.666E-06 1.088E-06 9.254E-07 IY= 28 1.005E-06 1.347E-06 1.752E-06 9.002E-07 1.204E-06 IY= 26 1.224E-06 1.359E-06 1.994E-06 8.870E-07 1.280E-06 IY= 24 1.276E-06 1.243E-06 2.295E-06 1.247E-06 1.757E-06 IY= 22 1.932E-06 9.275E-07 1.810E-06 2.396E-06 2.485E-06 IY= 20 4.695E-08 2.335E-07 3.265E-07 1.917E-07 1.999E-07 IY= 18 9.547E-07 9.715E-07 7.496E-07 8.467E-07 9.685E-07 IY= 16 4.619E-07 4.942E-07 4.556E-07 4.871E-07 5.245E-07 IY= 14 2.432E-07 2.695E-07 2.748E-07 2.696E-07 2.709E-07 IY= 12 1.400E-07 1.541E-07 1.593E-07 1.569E-07 1.580E-07 IY= 10 7.887E-08 9.537E-08 1.042E-07 1.046E-07 1.130E-07 IY= 8 5.334E-08 5.970E-08 7.170E-08 7.967E-08 8.851E-08 IY= 6 4.779E-08 4.862E-08 5.266E-08 5.844E-08 6.663E-08 IY= 4 3.871E-08 3.968E-08 4.188E-08 4.492E-08 4.918E-08 IY= 2 2.191E-08 2.585E-08 2.832E-08 3.038E-08 3.270E-08 IX= 51 53 55 57 59 IY= 40 7.957E-06 6.432E-06 5.395E-06 2.107E-06 2.777E-06 IY= 38 1.089E-10 1.832E-10 7.417E-11 6.765E-11 1.997E-10 IY= 36 2.799E-06 2.435E-06 3.454E-07 3.097E-07 1.003E-06 IY= 34 3.240E-06 4.342E-06 4.301E-06 2.408E-06 1.715E-06 IY= 32 2.833E-06 2.939E-06 3.632E-06 2.478E-06 1.924E-06 IY= 30 2.055E-06 1.931E-06 3.021E-06 2.462E-06 1.124E-06 IY= 28 1.437E-06 1.470E-06 2.583E-06 2.389E-06 7.989E-07 IY= 26 1.097E-06 1.545E-06 1.282E-06 2.511E-06 1.108E-06 1Y= 24 1.586E-06 1.826E-06 1.565E-06 7.627E-07 1.167E-06 IY= 22 2.504E-06 2.783E-06 1.720E-06 6.391E-07 1.386E-06 IY= 20 2.160E-07 2.546E-07 2.122E-07 1.634E-07 1.085E-07 IY= 18 1.193E-06 1.163E-06 1.373E-06 1.158E-06 9.794E-07 IY= 16 5.754E-07 5.009E-07 4.954E-07 6.332E-07 7.120E-07 IY= 14 2.634E-07 2.642E-07 2.683E-07 3.689E-07 4.405E-07 IY= 12 1.584E-07 1.759E-07 2.300E-07 3.063E-07 3.444E-07 IY= 10 1.225E-07 1.294E-07 1.612E-07 2.859E-07 2.745E-07 IY= 8 9.828E-08 1.114E-07 1.227E-07 2.066E-07 2.494E-07 IY= 6 7.566E-08 9.181E-08 1.115E-07 1.407E-07 1.940E-07 IY= 4 5.494E-08 6.364E-08 7.918E-08 1.023E-07 1.214E-07 IY= 2 3.590E-08 4.012E-08 4.503E-08 5.132E-08 5.964E-08 IX= 61 63 65 67 69 FIELD V A L U E S O F R H O l 2.703E+01 2.739E+01 2.803E+01 IY= 40 2.635E+01 2.740E+01
232
Appendix A
IY= 38 2.608E+01 IY= 36 2.577E+01 IY= 34 2.555E+01 IY= 32 2.541E+01 IY= 30 2.514E+01 IY= 28 2.493E+01 IY= 26 2.487E+01 IY= 24 2.495E+01 IY= 22 2.516E+01 IY= 20 O.OOOE+00 IY= 18 1.397E+02 IY= 16 1.405E+02 IY= 14 1.413E+02 1Y= 12 1.419E+02 IY= 10 1.425E+02 IY= 8 1.430E+02 IY= 6 1.434E+02 IY= 4 1.437E+02 IY= 2 1.441E+02 1 3 K= IY= 40 2.846E+01 IY= 38 2.555E+01 IY= 36 2.549E+01 IY= 34 2.550E+01 IY= 32 2.550E+01 IY= 30 2.549E+01 IY= 28 2.548E+01 IY= 26 2.548E+01 IY= 24 2.551E+01 IY= 22 2.559E+01 IY= 20 0.0O0E+00 IY= 18 4.995E+01 IY= 16 5.006E+01 IY= 14 5.016E+01 IY= 12 5.025E+01 IY= 10 5.033E+01 IY= 8 5.039E+01 IY= 6 5.044E+01 IY= 4 5.048E+01 IY= 2 5.051E+01 13 IX= 11 IY= 40 3.095E+01 IY= 38 2.806E+01 IY= 36 2.679E+01 IY= 34 2.650E+01 IY= 32 2.649E+01 IY= 30 2.648E+01 1Y= 28 2.648E+01 IY= 26 2.649E+01 IY= 24 2.650E+01 IY= 22 2.653E+01 IY= 20 2.910E+01 IY= 18 3.619E+01 IY= 16 3.592E+01 IY= 14 3.581E+01 IY= 12 3.578E+01 IY= 10 3.580E+01
2.503E+01 2.467E+01 2.486E+01 2.455E+01 2.480E+01 2.453E+01 2.473E+01 2.455E+01 2.467E+01 2.461E+01 2.471E+01 2.470E+01 2.482E+01 2.482E+01 2.501E+01 2.498E+01 2.524E+01 2.517E+01 0.OOOE+O0 O.OOOE+00 8.186E+01 6.830E+01 8.225E+01 6.850E+01 8.268E+01 6.870E+01 8.314E+01 6.891E+01 8.362E+01 6.914E+01 8.409E+01 6.939E+01 8.454E+01 8.496E+01 6.965E+01 8.533E+01 6.991E+01 5 77.016E+01 9 2.876E+01 2.912E+01 2.574E+01 2.570E+01 2.608E+01 2.571E+01 2.593E+01 2.572E+01 2.593E+01 2.571E+01 2.593E+01 2.570E+01 2.592E+01 2.570E+01 2.592E+01 2.573E+01 2.592E+01 2.580E+01 2.595E+01 O.OOOE+00 2.602E+01 4.614E+01 O.OOOE+00 4.624E+01 4.291E+01 4.634E+01 4.300E+01 4.642E+01 4.309E+01 4.650E+01 4.317E+01 4.657E+01 4.325E+01 4.663E+01 4.332E+01 4.668E+01 4.339E+01 4.673E+01 4.345E+01 15 4.350E+01 3.145E+01 17 19 2.939E+01 3.182E+01 2.789E+01 3.043E+01 2.697E+01 2.909E+01 2.663E+01 2.662E+01 2.801E+01 2.662E+01 2.705E+01 2.663E+01 2.675E+01 2.666E+01 2.676E+01 2.673E+01 2.677E+01 2.924E+01 2.680E+01 3.672E+01 2.684E+01 3.576E+01 2.860E+01 3.499E+01 3.656E+01 3.450E+01 3.617E+01 3.423E+01 3.529E+01 3 441E+01 3 369E+01
2.488E+01 2.480E+01 2.482E+01 2.487E+01 2.491E+01 2.495E+01 2.501E+01 2.511E+01 2.524E+01 O.OOOE+00 6.035E+01 6.050E+01 6.064E+01 6.076E+01 6.087E+01 6.098E+01 6.109E+01 6.120E+01 6.130E+01
2.529E+01 2.521E+01 2.523E+01 2.525E+01 2.525E+01 2.524E+01 2.525E+01 2.530E+01 2.540E+01 O.OOOE+00 5.456E+01 5.468E+01 5.479E+01 5.489E+01 5.497E+01 5.504E+01 5.509E+01 5.513E+01 5.517E+01
2.990E+01 2.658E+01 2.616E+01 2.614E+01 2.614E+01 2.613E+01 2.613E+01 2.613E+01 2.616E+01 2.621E+01 O.OOOE+00 4.011E+01 4.020E+01 4.028E+01 4.036E+01 4.043E+01 4.050E+01 4.056E+01 4.063E+01 4.068E+01
3.051E+01 2.721E+01 2.638E+01 2.633E+01 2.632E+01 2.632E+01 2.632E+01 2.632E+01 2.634E+01 2.638E+01 O.OOOE+00 3.768E+01 3.776E+01 3.782E+01 3.789E+01 3.795E+01 3.801E+01 3.807E+01 3.813E+01 3.818E+01
3.213E+01 3.127E+01 3.031E+01 2.946E+01 2.859E+01 2.762E+01 2.694E+01 2.695E+01 2.696E+01 2.697E+01 2.784E+01 3.567E+01 3.637E+01 3.573E+01 3.492E+01 3.406E+01
3.245E+01 3.210E+01 3.160E+01 3.117E+01 3.082E+01 3.048E+01 3.009E+01 2.946E+01 2.817E+01 2.753E+01 2.755E+01 3.421E+01 3.642E+01 3.617E+01 3.554E+01 3.483E+01
233
Appendix A
IY= 8 3.584E+01 1Y= 6 3.589E+01 IY= 4 3.594E+01 IY= 2 3.600E+01 23 IX= <21 IY= 40 3.268E+01 IY= 38 3.254E+01 IY= 36 3.216E+01 IY= 34 3.169E+01 IY= 32 3.122E+01 IY= 30 3.077E+01 IY= 28 3.033E+O1 IY= 26 2.986E+01 IY= 24 2.933E+01 IY= 22 2.847E+01 IY= 20 2.779E+01 IY= 18 3.254E+01 IY= 16 3.613E+01 IY= 14 3.671E+01 IY= 12 3.632E+01 IY= 10 3.582E+01 IY= 8 3.529E+01 IY= 6 3.476E+01 IY= 4 3.427E+01 IY= 2 3.389E+01 33 IX= 31 IY= 40 3.441E+01 IY= 38 3.458E+01 IY= 36 3.425E+01 IY= 34 3.385E+01 IY= 32 3.340E+01 IY= 30 3.298E+01 IY= 28 3.265E+01 IY= 26 3.241E+01 IY= 24 3.220E+01 IY= 22 3.197E+01 IY= 20 3.170E+01 IY= 18 3.191E+01 IY= 16 3.512E+01 IY= 14 3.882E+01 IY= 12 4.069E+01 IY= 10 4.145E+01 IY= 8 4.182E+01 IY= 6 4.206E+01 IY= 4 4.220E+01 IY= 2 4.228E+01
rx=
41 43 3.388E+01 IY= 40 IY= 38 3.400E+01 IY= 36 3.417E+01 IY= 34 3.424E+01 IY= 32 3.425E+01 IY= 30 3.429E+01 IY= 28 3.438E+01 IY= 26 3.437E+01 IY= 24 3.430E+01 IY= 22 3.422E+01
3.411E+01 3.408E+01 3.408E+01 3.410E+01
25 3.279E+01 3.269E+01 3.227E+01 3.178E+01 3.129E+01 3.086E+01 3.046E+01 3.006E+01 2.963E+01 2.900E+01 2.811E+01 3.107E+01 3.551E+01 3.721E+01 3.730E+01 3.703E+01 3.673E+01 3.644E+01 3.619E+01 3.602E+01
35 3.627E+01 3.628E+01 3.597E+01 3.559E+01 3.508E+01 3.457E+01 3.423E+01 3.401E+01 3.384E+01 3.393E+01 3.433E+01 3.535E+01 3.818E+01 4.073E+01 4.188E+01 4.224E+01 4.235E+01 4.241E+01 4.245E+01 4.247E+01
45 3.455E+01 3.505E+01 3.526E+01 3.530E+01 3.525E+01 3.535E+01 3.549E+01 3.537E+01 3.516E+01 3.520E+01
3.318E+01 3.287E+01 3.269E+01 3.259E+01 27 29 3.292E+01 3.287E+01 3.252E+01 3.210E+01 3.167E+01 3.128E+01 3.094E+01 3.063E+01 3.028E+01 2.988E+01 2.831E+01 3.019E+01 3.484E+01 3.759E+01 3.833E+01 3.832E+01 3.819E+01 3.8O4E+01 3.790E+01 3.782E+01 37 39 3.887E+01 3.845E+01 3.820E+01 3.797E+01 3.761E+01 3.696E+01 3.611E+01 3.603E+01 3.634E+01 3.664E+01 3.701E+01 3.896E+01 4.140E+01 4.325E+01 4.404E+01 4.424E+01 4.426E+01 4.426E+01 4.425E+01 4.424E+01 47 49 3.400E+01 3.480E+01 3.499E+01 3.498E+01 3.492E+01 3.492E+01 3.492E+01 3.492E+01 3.497E+01 3.512E+01
3.327E+01 3.262E+01 3.215E+01 3.184E+01
3.408E+01 3.336E+01 3.274E+01 3.227E+01
3.315E+01 3.318E+01 3.292E+01 3.257E+01 3.218E+01 3.179E+01 3.146E+01 3.119E+01 3.090E+01 3.054E+01 2.999E+01 2.999E+01 3.441E+01 3.789E+01 3.924E+01 3.947E+01 3.942E+01 3.929E+01 3.914E+01 3.904E+01
3.314E+01 3.334E+01 3.321E+01 3.297E+01 3.265E+01 3.229E+01 3.195E+01 3.167E+01 3.137E+01 3.097E+01 3.074E+01 3.034E+01 3.438E+01 3.845E+01 4.016E+01 4.054E+01 4.053E+01 4.040E+01 4.024E+01 4.012E+01
3.843E+01 3.781E+01 3.749E+01 3.737E+01 3.734E+01 3.733E+01 3.725E+01 3.687E+01 3.592E+01 3.579E+01 3.626E+01 3.846E+01 4.087E+01 4.267E+01 4.349E+01 4.371E+01 4.376E+01 4.377E+01 4.378E+01 4.378E+01
3.654E+01 3.609E+01 3.567E+01 3.545E+01 3.535E+01 3.531E+01 3.533E+01 3.537E+01 3.542E+01 3.548E+01 3.460E+01 3.628E+01 3.870E+01 4.065E+01 4.163E+01 4.197E+01 4.208E+01 4.212E+01 4.215E+01 4.217E+01
3.496E+01 3.547E+01 3.566E+01 3.576E+01 3.586E+01 3.593E+01 3.577E+01 3.568E+01 3.566E+01 3.575E+01
3.467E+01 3.584E+01 3.609E+01 3.616E+01 3.622E+01 3.629E+01 3.639E+01 3.654E+01 3.635E+01 3.622E+01
234
Appendix A
IY= 20 3.399E+01 3.539E+01 3.547E+01 3.609E+01 3.657E+01 IY= 18 3.579E+01 3.761E+01 3.780E+01 3.819E+01 3.858E+01 IY= 16 3.798E+01 3.968E+01 3.967E+01 3.999E+01 4.049E+01 IY= 14 3.967E+01 4.124E+01 4.108E+01 4.143E+01 4.212E+01 1Y= 12 4.051E+01 4.202E+01 4.180E+01 4.221E+01 4.312E+01 IY= 10 4.078E+01 4.229E+01 4.207E+01 4.248E+01 4.356E+01 IY= 8 4.086E+01 4.237E+01 4.217E+01 4.254E+01 4.370E+01 IY= 6 4.089E+01 4.240E+01 4.222E+01 4.254E+01 4.375E+01 IY= 4 4.091E+01 4.241E+01 4.226E+01 4.252E+01 4.377E+01 IY= 2 4.092E+01 4.242E+01 4.228E+01 4.251E+01 4.378E+01 DC= 51 53 55 57 59 IY= 40 3.458E+01 3.648E+01 3.912E+01 4.253E+01 4.229E+01 IY= 38 3.589E+01 3.721E+01 3.945E+01 4.251E+01 4.215E+01 IY= 36 3.600E+01 3.735E+01 3.957E+01 4.256E+01 4.218E+01 IY= 34 3.605E+01 3.745E+01 3.964E+01 4.261E+01 4.223E+01 IY= 32 3.610E+01 3.749E+01 3.969E+01 4.267E+01 4.229E+01 IY= 30 3.611E+01 3.749E+01 3.973E+01 4.273E+01 4.236E+01 IY= 28 3.612E+01 3.748E+01 3.975E+01 4.280E+01 4.245E+01 IY= 26 3.611E+01 3.747E+01 3.978E+01 4.287E+01 4.259E+01 IY= 24 3.612E+01 3.749E+01 3.984E+01 4.298E+01 4.279E+01 IY= 22 3.624E+01 3.760E+01 3.998E+01 4.316E+01 4.313E+01 IY= 20 3.665E+01 3.799E+01 4.036E+01 4.367E+01 4.399E+01 IY= 18 3.892E+01 4.023E+01 4.273E+01 4.640E+01 4.686E+01 IY= 16 4 067E+01 4.195E+01 4.456E+01 4.826E+01 4.844E+01 IY= 14 4 218E+01 4.344E+01 4.621E+01 4.996E+01 4.982E+01 IY= 12 4.313E+01 4.439E+01 4.735E+01 5.116E+01 5.075E+01 IY= 10 4.358E+01 4.482E+01 4.793E+01 5.179E+01 5.127E+01 IY= 8 4 376E+01 4.495E+01 4.816E+01 5.206E+01 5.150E+01 IY= 6 4 384E+01 4.498E+01 4.824E+01 5.216E+01 5.161E+01 IY= 4 4 390E+01 4.499E+01 4.826E+01 5.221E+01 5.166E+01 IY= 2 4.394E+01 4.498E+01 4.827E+01 5.224E+01 5.170E+01 K= 61 63 65 67 69 Whole-field residual sum(s) before solution Resref values determined by E A R T H resfac = 1.000E-02 variable resref (res sum)/resref Pl 7.487E-07 1.453E+02 U l 5.657E-05 3.084E+03 VI 1.311E-05 4.715E+03 K E 3.801E-07 1.295E+05 E P 2.067E-02 3.750E+04 HI 2.760E-02 1.776E+02 Net source of Rl at patch named: INL1 Net source of Rl at patch named: INL2
=9.260E-04 = 2 048E-04
MNVAJ> 5 39TC+05 V.275E+021.923^01 1.161E-02 1.7»I*01 « i Y v i r - ^K7«Bf05 i 968E+02 2.098E+O1 4.189E-02 U 2 6 E + 0 2
S
t
VARIABLE
S
S
2.020E+O) 2.486E-02 5.472E+0.
HI 235
Appendix A
M I N V A L = 6.093E+04 M A X V A L = 8.740E+04 C E L L A V = 7.257E+04 1.00+....U..P.+....+....+H...+....+.V.H+....+....+.H..E
V 0.90+ .P
U U P V P H U
U H
0.80+ V .U 0.70 H V
K PV PK E V . +
V
U
P KE K E KF H V K E V KE E H V
0.60 + 0.50 P
H P V
0.40+ U 0.30 +
P K K EP K E K H H P
0.20+
. H U KE 0.10 +
HUE + E U
HV
+ +
P
. EH E H V
+
V. UH
U U U
K E
+
P
H . H
U U.
0.O0E....P....+....+..P.+U...+....+.H.V+....+....+....+ 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 the abscissa is ISWP. min= l.OOE+00 max= 9.51E+02 ************************************************************ ************************************************************
residuals vs sweep or iteration number VARIABLE Pl Ul VI KE EP MINVAL= 4.670E+00 7.916E+00 8.297E+00 1.096E+01 9.277E+00 M A X V A L = 5.298E+00 8.759E+0O 9.020E+00 1.170E+01 1.086E+01 VARIABLE HI MINVAL= 5.134E+00 M A X V A L = 1.254E+01 1.00H....+....+....+....+....+V..U+....+....+....+.K..+ 0.90 V K U K K
0.80+ P 0.70+ U 0.60 + U 0.50+
U V
UK K K U V K . K VK E+ K K PE KP U P E U K VU K V V E V VP K E U E
0.40K
.V 0.30+ K
E U P E E E . PE V +
UP VEEEE U
0.20 + V E .P 0.10+ H E
U U . U +
P H
H
V P+ V . H U +
236
Appendix A
. H
HH
H
HHH.
0.00+.££..H.+H...+....+H..H+....+.H.H+....+....H.V..H 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 the abscissa is I S W P . min= l.OOE+00 m a x = 9.51E+02 ************************************************************ ************************************************************ SATLIT RUN NUMBER = 1; LIBRARY REF.= 0 RUN COMPLETED AT 14:02:36 ON MONDAY, 18 SEPTEMBER 1995 ************************************************************
237
APPENDIX B
B.l Additional Experimental Results
Not all the results of experiments carried out for the present thesis were presente in the body of the thesis. Here the results of a number of these experiments are presented in Figures B. 1 to B.4.
238
Appendix B
029
1 0.28-
.©
0.27-
{g
0.26-
•g
0.25 -
>—«v J ^ v . /
N.
/
\ V.
0)
g
0.24 -
|
0.23-
~
0.22 -
w
0.21 0.20-|—I-T—i—i——I—'—I—•—i—'—l—•—l—•— 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Nozzle Position [X/D21 Figure B.l Relation Between Entrainment Ratio and Nozzle Position Performance Conditions: Tg = 66.5±0.2°C, T e = 12.5±fl.2°C {Ejector 2.a (see Section 3.7.1)}
239
Appendix B
0.190.18-
o -.s
0.17-
c v E c
0.15-
'«
0.12-
b c
0.11 -
0.16-
0.140.13-
0.100.0
•
I
0.5
' — I
1.0
"
1
1
1.5
1 — •
2.0
1 —
2.5
3.0
Nozzle Position [X/D2] Figure B.2 Relation Between Entrainment Ratio and Nozzle Position Performance Conditions: Tg = 66.5±0.3°C, Te = 5.0±0.1°C, {Ejector 2.b (see Section 3.7.2)}
ed
OS C 9t
B c 'ea u •M
e Ed
3.0
3.5
4.0
Nozzle Position [X/D2] Figure B.3 Relation Between Entrainment Ratio and Nozzle Position Performance Conditions: Tg = 73.0±0.2°C, Te = 12.5±0.2°C, {Ejector 2.b (see Section 3.7.2)}
240
Appendix B
B
1000
Tc=30.6°C es Om
Tcr=28.1°C Tc=24.2°C
Ix
s
0 20 40 60 Distance From Nozzle Exit [mm]
80
100
Figure B.4 Pressure Distribution Along the Supersonic Ejector Tc = Condenser Temperature[°C]; Tcr = Critical Condenser Temperature[°C] Performance Conditions: Tg = 66.5±0.3°C, Te = 5.0±0.1°C, {Ejector 2.b (see Section 3.7.2)}
241
Appendix B
B.2 Pressure Transducers The pressure change with d.c. Voltage for one of the pressure transducers was used in the present work is shown in Figure B.5.
Pressure Transducer [No. 1]
3000
ea PCJ IN
3 Vi Vi
2 ft01
o Vi
<
-i—•—i—'—r40 50 60
-r~ 70
80
Voltage [ m V ]
Figure B.5 Absolute Pressure Versus the Voltage for the Pressure Transducer
242
APPENDIX C
J(.
< i
C.1 Simulation Analysis Program for Secondary Choking Theory The simulation program for the secondary choking theory using one-dimensional analysis is presented on the following pages.
243
Appendix C
C****************** y a p o u r Jet Refrigeration System *********************** C This program simulates the vapour jet refrigeration system based on C the secondary choking theory which was introduced by Munday and Bagster (1977) C .using one-dimensional analysis of the ejector refrigeration system. C Variable definitions C TCOND C C C C C C
- Operating temperature of condenser P C O N D - Operating pressure of condenser V G C O N D - Specific volume of saturated vapour at condenser V F C O N D - Specific volume of saturated liquid at condenser L A T E N T C O N D - Latent heat of vaporisation at condenser temperature H F C O N D - Enthalpy of saturated liquid at condenser temperature H G C O N D - Enthalpy of saturated vapour at condenser temperature
C TEVAP C C C C C C
- Operating temperature of evaporator P E V A P - Operating pressure of evaporator V G E V A P - Specific volume of saturated vapour at evaporaor V F E V A P - Specific volume of saturated liquid at evaporator L A T E N T E V A P - Latent heat of vaporisation at evaporator temperature H F E V A P - Enthalpy of saturated liquid at evaporator temperature H G E V A P - Enthalpy of saturated vapour at evaporator temperature
C TBOFL C C C C C C
- Operating temperature of generator P B O I L - Operating pressure of generator V G B O I L - Specific volume of saturated vapour at generator V F B O I L - Specific volume of saturated liquid at generator L A T E N T B O I L - Latent heat of vaporisation at generator temperature H F B O I L - Enthalpy of saturated liquid at generator temperature H G B O I L - Enthalpy of saturated vapour at generator temperature
C T - Dummy temperature variable C TI - Initial guess of temperature P - D u m m y pressure variable C V - D u m m y specific volume variable C C VI - Initial guess of specific volume C E N T H A L P Y - D u m m y enthalpy variable S - D u m m y entropy variable C C H F - D u m m y enthalpy of liquid variable H G - D u m m y enthalpy of vapour variable C S G - Entropy of saturated vapour C C S F G - Entropy of C V F - D u m m y specific volume of liquid variable V G - D u m m y specific volume of vapour variable C C L A T E N T - D u m m y latent heat of vaporization variable Q - D u m m y heat exchange variable C C KA - Specific heat ratio at the motive nozzle inlet K B - Specific heat ratio at the secondary nozzle inlet C C A R A T I O - Area ratio, A2/At C P 0 3 - Pressure at the outlet of ejector C V E L 2 - Stream velocity at the entrance of diffuser C E N T R A I N M E N T - Entrainment ratio C C O M P R A T I O - Pressure ratio of condenser to evaporator C C P - Heat capacity at constant pressure C V - Heat capacity at constant volume C
244
Appendix C
C C C C C C C
EFFD-Diffuser efficiency EFFN - Nozzle efficiency T O A - Temperature at the inlet of motive nozzle POA - Pressure at the inlet of motive nozzle V O A - Specific volume at the inlet of motive nozzle TOB - Temperature at the inlet of secondary nozzle POB - Pressure at the inlet of secondary nozzle
C DIATHROAT - Diameter of motive nozzle throat DIAA - Diameter of motive nozzle exit C C DIA2 - Diameter of constant area tube C ME - Mass flow rate of refrigerant through evaporator C M B - Massflowrate of refrigerant through generator C M C O N D - Massflowrate ofrefrigerantthrough condenser C QEVAP - Heat absorbed through evaporator QBOIL - Heat input through generator C C Q C O N D - Heat rejected through condenser WPUMP-Work of the pump C C BETA - System coefficient of performance C C A R N O T - Carnot coefficient of performance of system C H0,H2,H3,H4,H5,H6,H7 - Enthalpies of each state in the cycle C P0,P2,P3,P4,P5,P6,P7 - Pressures of each state in the cycle C T0,T2,T3,T4,T5,T6,T7 - Temperatures of each state in the cycle C S2 - Entropy at 2 C VO - Specific volume at 0 C X - Quality C Y - Dummy variable for program re-run p*******************************************************************^***** REAL P,P0,P2,P3,P4>P5,P6,P7,PCONDl,PEVAP,PBOIL,POA,POB,PO3,PPl 1 REALT,TI,T0>T2,T3,T4,T5,T6,T7,TOA,TOB,TCONDl,TBOIL,TEVAP,TCOND REAL V,VI,VG,VF,VGCOND,VGEVAP,VGBOIL,VFBOIL,VFEVAP,VFCOND,VOA REAL LATENT,LATENTBOIL,LATENTEVAP,LATENTCOND REALHG,HF,HGCOND,HGEVAP,HGBOIL,HFCOND,HFEVAP,HFBOIL REAL H)H0,H2,H3,H4,H5,H6,H7,ENTHALPY REAL S,S2,SG,SFG REALKA,KB,MBJ^>lEVAP,MCOND,ARATIO,ENTRAINMENT,COMPRATIO RFAL CP CV REALDIATHROAT)DIAA,DIA2,DIAYYA,DIATOTAL,RATBATN,RATBAYYA REAL ATHROAT,A2,AYYA,ATB,ATN,ATOTAL,AA REALQ,QEVAP,QBOIL,QCOND,WPUMP,BETA,CARNOT REAL EFFN,EFFD,VEL2 REAL PCRIT,TCRIT REALX,Y REAL TSBOIL,T0S,TSEVAP,T4S,VGSEVAP,VGSBOIL,PSBOIL>PSEVAP)AC,PERFORM)DROPP C CONDENSER DATA
245
Appendix C
1
WRITE(*,*)-ENTER THE CONDENSER TEMP (C)' READ(*,*) T TCONDl=T
CALL SATPRESS(T,P) PCONDl=P C EVAPORATOR DATA WRTTE(*,*) -ENTER THE EVAPORATOR TEMPERATURE (CV READ(*,*) T TEVAP=T CALL SATPRESS(T,P) PEVAP=P PSEVAP=P DROPP=0. PSEVAP=PSEVAP-DROPP CALL SVG(T,P,VG) VGEVAP=VG CALL SVF(T,VF) VFEVAP=VF CALL SLATENT(VG,VF,T,P,LATENT) LATENTEVAP=LATENT CALL SHG(T,VG,P,HG) HGEVAP=HG CALL SHF(HG,LATENT,HF) HFEVAP=HF C SUPERHEATING EVAPORATOR DATA WRTTE(*,*) ENTER SUPERHEATING EVAPORATOR TEMPERATURE(C)' READ(*,*) T TSEVAP=T P=PSEVAP CALL SVG(T,P,VG) VGSEVAP=VG C GENERATOR DATA WRTTE(*,*) "ENTER THE GENERATOR TEMP (C)' READ(*,*) T TBOIL=T CALL SATPRESS(T,P) PBOIL=P PSBOD>P DROPP=0. PSBOIL=PSBOIL-DROPP CALL SVG(T,P,VG) VGBOD>VG CALL SVF(T,VF) VFBOD>VF CALL SLATENT(VG,VF,T,PJ.ATENT) LATENTBOIL=LATENT CALL SHG(T,VG,P,HG)
246
Appendix C
HGBOIL=HG CALL SHF(HG,LATENT,HF) HFBODL=HF C SUPERHEATING GENERATOR DATA WRJTE(*,*) -ENTER SUPERHEATING GENERATOR TEPERATURE(C)' READ(*,*)T ' TSBOD>T P=PSBOIL CALL SVGCTJWG) VGSBOIL=VG C NOZZLE AND DIFFUSER DATA WRITE(*,*) -ENTER THE NOZZLE EFFICIENCY' READ(*,*) EFFN WRITE(*,*) ENTER THE DIFFUSER EFFICIENCY' READ(*,*) EFFD C STATES PO=PBOIL WRITE(*,*) 'PO IN MATN=',PO C P3=PCOND P4=PEVAP C P5=PCOND P6=PEVAP P7=PBOIL T0=TBOrL T0S=TSBOIL T4=TEVAP T4S=TSEVAP C T5=TCOND T6=TEVAP C T7=TCOND H0=HGBOIL H4=HGEVAP C H5=HFCOND C H6=HFCOND C H7=HFCOND C MASS FLOW RATE THROUGH THE EVAPORATOR WRITE(*,*) -ENTER THE COOLING CAPACITY (kW)' READ(*,*) Q QEVAP=Q C ME=QEVAP/(H4-H6) C MEVAP=ME C EJECTOR DATA C Inlet to the motive nozzle
247
Appendix C
T=T0S TOA=T0 P=PO POA=PO WRITE(*,*) 'POA IN MAIN= ',POA V=VGSBOJL VOA=VGSBOIL WRTTE(*,*) 'VOA IN MATN= ',VOA CALL HEATCV(T,V,CV) CALL HEATCP(T,V,CV,CP) KA=CP/CV WRTTE(*,*) 'KA=',KA C Inlet to the secondary nozzle T=T4S TOB=T4 P=P4 POB=P4 V=VGSEVAP CALL HEATCV(T,V,CV) CALL HEATCP(T,V,CV,CP) KB=CP/CV WRITE(*,*) -KB=',KB C Outiet from ejector P03=PCONDl C EJECTOR DESIGN BASED ON THE THEORY OF SECONDARY CHOKING CALLCHOKENTRAIN(H4,KA,KB,TOA,TOB,POA,POB,P3,EFFN,EFFD,QEVAP, 1 ARATIO,ENTRATNMENT,COMPRATIO,ME)MB,PYY)VFCOND)T5,H5,H6,VEL2,P2) 1 PCRIT,TCRIT,VOA,DIATHROAT,DIAYYA,DIATOTAL,DIA2)
C CALL EJECTAREA(MB,MEVAP,ARATIO,POA,POB,PYY,KA>KB>TOA,TOB)VOAX>IATHROAT,D IAYYA, 1 DIATOTAL,DIA2,RATBATN,RATBAYYA) C H2=l./(MB+ME)*(MB*H0*1000.+ME*H4*1000.)-(VEL2**2)/2. H2=H2/(1.E3) H=H2
248
Appendix C
P=P2 CALL SATTEMP(P,T) CALL SVG(T,P,VG) CALL SHG(T,VG,P,HG) IF(H2.GE.HG)THEN TI=T VI=VG CALL INVENTHAL(H,P,TI,VI,T,V) T2=T CALL SS(T,V,S) S2=S ELSE T2=T CALL SVF(T,VF) CALL SLATENT(VG,VF,T,P,LATENT) CALL SHF(HG,LATENT,HF) X=(H2-HF)/LATENT WRTTE(*,*) -X='^ V=VG CALL SSOW.S) SG=S SFG=LATENT/(T2+273.15) S2=X*SFG+(SG-SFG) S=S2 END IF WRITE(*,*) 'S2=',S2 P=P3 P5=P3 WRITE(*,*) 'P3 IN MATN= \P3 T7=T5 CALL ENTrTER(P,S,T,V) CALL SUPERENTH(T,V,P,ENTHALPY) H3=ENTHALPY H=ENTHALPY TI=T VI=V
249
Appendix C
CALL INVENTHAL(H,P,TI,VI(T,V) T3=T H3=(H3-H2)/EFFD+H2 C CALCULATION OF MASS FLOW RATE THROUGH CONDENSER MCOND=MB+ME WRTTE(*,*) -ME IN MATN= ',ME WRITE(*,*) -MB= 'MB C COOLING CAPACITY = QEVAP QEVAP=QEVAP*1000. C PUMP WORK WPUMP=MB*VFCOND*(PBOIL-PCOND)*1000 H7=WPUMP/(1000.*MB)+H5 C HEAT INPUT QBOIL=MB *(H0-H7)* 1000. WRTTE(*,*) -HO=,H7= -,HO,H7 C HEAT REJECTED CONDENSER QCOND=MCOND*QT3-H5)* 1000. C COEFFICIENT OF PERFORMANCE BETA=QEVAP/(QBOJL+WPUMP) C CARNOT COEFFICIENT OF PERFORMANCE TCOND=T5 CARNOT=(TBOIL-TCOND)/(TBOIL+273.15)*(TEVAP+273.15)/(TCOND-TEVAP) C EJECTOR PERFORMANCE=(QEVAP)/(MB) [KJ/KG] PERFORM=QEVAP/MB C PRINT OUTS WRITE(*,*)'' WRTTE(*,*)'' WRTTE(*,*) '(1) EJECTOR SIMULATION' WRTTE(*,*)'' WRrTE(*,*)' THE PRESSURE AT YY= *,PYY,'kPa' WRTTE(*,*) ' THE PRESSURE AT 2 = -,P2,' kPa" WRITE(*,*)' THE TEMPERATURE AT 2 =',T2,' C WRTTE(*,*)' THE ENTHALPY AT 2 = \H2,' kJ/kg' WRTTE(*,*)- THEA2/At= ',ARATIO WRrTE(*,*)' THE ENTRAINMENT RATIO =',ENTRAINMENT WRITE(*,*)' THE COMPRESSION RATIO =',COMPRATTO WRTTE(*,*)'' WRTTE(*,*)'' WRITE(*,*) '(2) EJECTOR GEOMETRY' WRTTE(*,*)''
250
Appendix C
WRITE(*,*)' DIAMETER AT THROAT = 'JDIATHROAT,' mm' WRTTE(*,*)' DIAMETER OF EXPANDED MOTIVE STREAM AT SECONDARY CHOKING SECTION=-,DIAYYA,' mm' WRITER,*)- DIAMETER AT CONSTANT AREA MIXING TUBE = \DIA2,' mm' WRTTE(*,*) ' DIAMETER OF EJECTOR AT SECONDARY CHOKING SECTION=',DIATOTAL,' mm' WRJTE(*,*)'' WRITE(*,*)'' WRTTE(*,*) '(3) EJECTOR PERFORMANCE ' WRITE(*,*)' ' WRiTE(*,*)' EJECTOR PERFORMANCE=QEVAP/MB ='.PERFORM,' KJ/KG WRTTE(*,*) '(4) SYSTEM SIMULATION' WRTTE(*,*)'' WRITE(*,*)' THE COOLING CAPACITY =',QEVAP,' W WRJTE(*,*)' THE HEAT INPUT = \QBOIL,' W WRITE(*,*)' THE PUMP W O R K = \WPUMP,' W WRITE(*,*)' THE HEAT REJECTED = '.QCOND/ W WRnE(*,*)' COPQ3.R. SYSTEM) = ',BETA WRITE(*,*)' COP(CARNOT) = *,CARNOT WRTTE(*,*) - • WRrTE(*,20)T0,PO,H0,T3)P3)H3,T4)P4,H4,T5,P5,H5)T6,P6)H6,T7,P7,H7
20
FORMAT(3(/),3X,'POINT 1
2 3 4 5 6 7
TEMPERATURE
PRESSURE
ENTHALPY',/,
•****#***********************************************'
,/,5X,,0,,10X,F4.0,9X,F7.0,6X,F7.2,/) 5X,'3\10X,F4.0,9X,F7.0,6X,F7.2,/, 5X,'4',10X,F4.0,9X,F7.0,6X,F7.2,/, 5X,*5,,10X,F4.0,9X,F7.0,6X,F7.2,/, 5X,'6,,10X,F4.0,9X,F7.0,6X,F7.2)/, 5X,7,,10X,F4.0,9X,F7.0,6X,F7.2)
WRTTE(*,*)'' WRITE(*,*)'' WRITE(*,*) '(5) CRITICAL CONDITION' WRrTE(*,*)'' WRTTE(* * ) ' T H E CRITICAL PRESSURE = '.PCRTT,' kPa WRTTE(*,*)' T H E CRITICAL TEMP. (SAT.) = '.TCRIT,' C C WRTTEC*,*) '(6) THE OTHER INFORMATION ' WRTTE(*,*)' C WRTTE(*,*)'' WRITE(* * ) ' ' WRTTE(*!*)' D O Y O U W I S H T O R U N A G A I N (Y=l)?' READ(*,*) Y WRTTE(*,*)'' IF(Y=1) GOTO 1 STOP END
251
Appendix C
Q ****************** SUBROUTINES *************************************** £*********************************************************************** C SUBROUTINE TO CALCULATE THE OPTIMUM EJECTOR ENTRAINMENT RATIO C AND HENCE DETERMINE OPTIMUM PERFORMANCE CHARACTERISTICS OF THE C EJECTOR SYSTEM FOR THE REFRIGERATION CYCLE CHOSEN. C KA - Specific heat ratio at point 0 K B - Specific heat ratio at point 4 C C T O A - Stagnation temperature at point 0 C T O B - Stagnation temperature at point 4 C P O A - Stagnation pressure at point 0 C P O B - Stagnation pressure at point 4 C P 0 3 - Stagnation pressure at point 3 C P E V A P - Evaporator pressure P A - Pressure at point 0 C C P B - Pressure at point 4 C E F F D - Diffuser efficiency C E F F N - Nozzle efficiency W - Molecular weight C R - Gas constant C Pl - Pressure at point 1 C C P Y Y - Optimum pressure at point 1 C P2 - Pressure at point 2 C A R A T I O - Area ratio of section 2 to throat C E N T R A I N - Entrainment ratio C E N T R A I N M E N T - Optimum entrainment ratio C C B C A R A T I O - Critical speed of sound ratio of point 4 to 0 C C 2 C A R A T I O - Critical speed of sound ratio of point 2 to 0 C M I A - Dimensionless velocity of motive vapour at point 1 M 1 B - Dimensionless velocity of secondary vapour at point 1 C M 2 - Dimensionless velocity or Mach number at point 2 C C X M A X - Maximum entrainment ratio C K 2 - Specific heat ratio at point 2 C T 0 2 T O A R A T I O - Temperature ratio of point 2 to 0 C C O M P R A T T O - Pressure ratio of point 3 to 4 C B - Component of Eq.(3.11) C M B - Mass flow rate through evaporator M E - Mass flow rate through generator C C2-Crirical speed of sound at point 2 C C C A - Critical speed of sound at point 0 C V E L 2 - Velocity at point 2 C I M A X - D u m m y variable for calculation of maximum entrainment ratio C N - Number of iteration for calculation of entrainment ratio C PART1,PART2,PART3 - Part of Eq.(3.16) PART4.PART5 - Part of Eq.(3.13) C C P A R T 6 - Part of Eq.25 [11] C PART7,PART8,PART9 - Part of Eq.(3.11) *5T FBROTITINE
CHOKENTRAIN(H4,KA,KB,TOA,TOB ) POA,POB ) P3,EI^FN,EFFD,QEVAP,ARATIO, ENTRAINMENT,COMPRATIOMEMB,PYY)WCO>ro,T5,H5,H6,VEL2,P2,PCRIT,TCR^ 1 VOADIATHROAT,DIAYYA,DIATOTAL,DIA2)
252
Appendix C
REAL KA,KB,TOATOB,W,POA,POB,R,EFFN,EFFD,PEVAP P T Y REAL CBCARATI0,ARATI0,MYA,MYB,H4 REAL XMAX,PYY,EOTRAI>JMEOT,EmTlAIN,E^^ REAL P03,COMPRATIO,P2PlRATIO>IB,ME>P2)C2,CA,VEL2,PCRrT,TCRrT,TCOND REAL PART1,PART2,PART3,PART4,PART5,PART6,PART7,PART8,PART9,PART10,PART11 REAL PYB,PYA,HFCOND,HGCOND,VFCOND,VGCOND,T5,P3,PO,P4,H5,H6,PCOND REAL MACH,ROWO,VOA,VTHROAT,ROWTHROAT,A2,ATHROAT,DIA2>DIATHROAT,AA,AA TN REAL ATB,AYYAD,ATOTALD,DIAYYA,DTOTAL,RATBATN,RATBDATN,RATBAYYA,RATBD AYYAD REALDIATOTAL(50),DIAA(50),DIFDIA,ATOTAL(50),AYYA(50),ATBD INTEGER I
Constants from [13] W=120.91 R=8.3144*1000.AV TOA=TOA+273.15 TOB=TOB+273.15 WRTTE(*,*) TOA= ',TOA WRITE(*,*) TOB= ',TOB 1
WRTTE(*,*)'ENTER THE AREA RATIO' READ(*,*) ARATIO
C THE FOLLOWING P R O G R A M IS CALCULATION OF THE SECONDARY STREAM PRESSURE C FOR SECONDARY CHOKING P H E N O M E N O N IN EJECTOR AT CHOKING SECTION. BASED O N C EQUILIBRIUM PRESSURE OF PRIMARY A N D SECONDARY VAPOUR IN START OF
MDONG, C THE PRESSURE OF THE EXPANDED MOTIVE STREAM IS EQUAL TO THE SECONDARY C STREAM PRESSURE AT SECONDARY CHOKING SECTION.THEN THE ENTRAINMENT RATIO C IS CALCULATED BY SUBSTITUTION OF THAT PRESSURE INTO Eq.(3.16). C IN THE FOLLOWING CHOKING EQUATION FOR SECONDARY STREAM THE (POB) IS THE
253
Appendix C
C C C
INITIAL STAGNATION PRESSURE FOR SECONDARY STREAM: P/POB=(2/(KB+l))**(KB/(KB-l)) POB=P(STATIC)+ROW*V**2/2(VELOCITY PRESSURE)
P=POB*((270CB+1.))**(KB/(KB-1.))) PYA=P PYB=P WRITE(*,*) 'PYA= ',PYA C
These equations are for constant area mixing. [11]
C Equation 22 CBCARATIO=SQRT((KB*(KA+l .)*W*TOB)/((KB+l .)*KA*W*TOA)) C DO 15 I=1,N C Equation 20 MYA=(EFFN*(KA+1 .)/(KA-l .)*(1 .-(PYA/POA)**((KA-l .)/KA)))**0.5 WRTTE(*,*) *MYA= ',MYA WRITE(*,*) 'EFFN= '.EFFN WRTTE(*,*) 'KA= ',KA WRITE(*,*) 'POA= ',POA C Equation 21 MYB=((KB+1.)/(KB-1.)*(1.-(PYB/POB)**((KB-1.)/KB)))**0.5 C Equation 18 PARTl=POB/POA*W/W*TOA/TOB *MYB/MYA*CBCARATIO PART2=((PYB/POB)**(l./KB))/((PYA/POA)**(l./KA)) PART3=ARATIO*MYA*(PYA^OA)**(l ./KA)*((KA+1 .)/2.)** 1 (W(KA-1.))-1. ENTRAIN=PART1 *PART2*PART3 WRTTE(*,*) 'PART1= ',PART1 WRITE(*,*) 'POB= ',POB WRTTE(*,*) 'PART2= ',PART2 WRTTE(*,*) 'PART3= \PART3 POP=POA/PYA ROWO=1.A^OA C Mach No. at nozzle exit Eq(3.27) MACH=SQRT(2./(KA-l.)*(POP**((KA-l.)/KA)-l.)) WRTTE(*,*) ,MACH=',MACH
IF(MACH>1.)THEN C Flow is supersonic C Area ratio of nozzle exit to throat Eq(3.26)
254
Appendix C
AA=l./MACH*(27(KA+l.)*(l.+(KA-l.)/2.*MACH**2))**((KA+l.)/(2 *(KA-1 WRTTE(*,*)'AA= \AA C
Density at the throat
ROWTHROAT=ROWO*(2./(KA+l.))**(l./(KA-l.)) C Velocity at the throat VTHROAT=SQRT(2.*KA*R*TOA/(KA+l.)) ELSE WRTTE(*,*) 'SUBSONIC END IF C Equation (3.13) ENTRAINMENT=ENTRAIN PART4=KA/(KA-1.)+KB/(KB-1.)*ENTRAINMENT*W/W PART5=17(KA-1.)+1./(KB-1.)*ENTRAINMENT*WAV K2=PART4/PART5 C WRITE(*,*) "K2=',K2 C Equation 25 PART6=KB/(KB-1.)*ENTRAINMENT*WAV T02TOARATIO=(KA/(KA-1 .)+PART6*TOB/TOA)/(KA/(KA-l .)+PART6) C WRTTE(*,*) T2TA=',T02TOARATIO C Equation 26 C2CARATIO=SQRT(K2/(K2+l.)*(KA+l.)/KA*WAV*T02TOARATIO) CA=(2.*KA/(KA+1.)*R*TOA)**0.5 C2=C2CARATIO*CA C Equation 11 PART7=MYA+MYB*ENTRAINMENT*CBCARATIO PART8=17KA*ARATIO*PYA/POA*((KA+l .)/2.)**(KA/(KA-l.)) PART9=(K2+1 .)/K2*(l .+ENTRAINMENT)*C2CARATIO B=(PART7+PART8)/PART9 WRTTE(*,*)' B=',B D?(B.LE.1.)THEN WRTTE(*,*) 'Because of the value of B<1, the dimentionless velocity at 1 mixing section is undefined. Besides, it is not able to find 1 the critical condenser conditions.Therefore.design an ejector under those conditions is impossible.' 1 G O T O 50
255
Appendix C
ELSE M2=B-(B**2-1.)**0.5
PART10=MYA+MYB*ENTRATNMENT*CBCARATIOM2*(l.+ENTRAINMENT)*C2CARATIO PART11=1./KA*ARATI0*PYA/P0A*((KA+1.)/2.)**(KA/(KA-1.)) P2P1RATIO=1.+PART10/PART11 P2=P2P1RATI0*PYA VEL2=M2*C2 C
Equation 15
M2=((27(K2+1.)*M2**2)/(1.-(K2-1.)/(K2+1.)*M2**2))**0.5 C P22=P03/((1.+(K2-1.)/2.*(M2**2))**(K2/(K2-1.))) C WRTTE(*,*) "P2 (WHEN EFFD=1)= ',P22 P3=P2*((1-KK2-1.)/2.*(M2**2)*EFFD)**(K2/(K2-1.))) P=P3 CALL SATTEMP(P,T) TCOND=T T5=TCOND CALL SVG(T,P,VG) VGCOND=VG CALL SVF(T,VF) VFCOND=VF CALL SLATENT(VG,VF,T,P,LATENT) LATENTCOND=LATENT CALL SHG(T,VG,P,HG) HGCOND=HG CALL SHF(HG,LATENT,HF) HFCOND=HF H5=HFCOND H6=HFCOND WRrTE(*,*) 'M2=',M2 PCRIT=P2PlRATIO*PYA*((l.+(K2-l.)/2.*(M2*n)*EFro)**(K2/(K2-l.))) END IF P=PCRIT CALL SATTEMP(P,T) TCRTT=T WRTTE(*,*) 'PCRrT=',PCRIT,'KPa' WRTTE(*,*) TCRrT=',TCRIT,' C WRTTE(*,*) TCOND=\TCOND,' C WRTTE(* * ) ' ' C WRTTE(*',*) "DO Y O U WISH TO C H A N G E THE AREA RATIO (Y=l) ?' C READ(*,*) Y WRITE(*,*)
256
Appendix C
IF(Y=l)GOTO 1
COMPRATIO=P3/POB ME=QEVAP/(H4-H6) MB=ME/ENTRAIN WRTTE(*,*) "ME IN SUB= ',ME WRITE(*,*) 'MB IN SUB= ",MB PYY=PYA C AREA OF THE THROAT B Y USE OF CONSERVATION OF MASS. ATHROAT=MB/(ROWTHROAT*VTHROAT) DIATHROAT=(SQRT(4./3.14159*ATHROAT))*1000. ATN=ATHROAT WRTTE(*,*) 'ATHROAT=ATN \ATHROAT WRITE(*,*) 'ATN=ATHROAT '.ATN C Area at the end of the nozzle C When the secondary choking phenomenon occures in some part of ejector, C the necessary area around that choking section for expanded motive stream C is called [AYYA]. A=ATHROAT*AA DAA=(SQRT(4./3.14159*A))*1000. AYYAD=A DIAYYA=DAA WRITE(*,*)'A=AYYAD \A WRTTE(*,*) 'AYYAD=A '.AYYAD WRITE(*,*) 'DAA=DIAYYA \DAA WRITE(*,*) T)IAYYA=DAA '.DIAYYA C AreaA2 A2=ARATIO*ATHROAT DIA2=(SQRT(4./3.14159*A2))*1000. WRITE(*,*) 'A2= \A2 WRITE(*,*) T)IA2= ",DIA2 C Calculation of the assumed throat area for secondary vapour in choking C section of secondary stream in ejector[ATB]. ATBD=(ME/(POB*1000.))*(SQRT(R*TOB/KB))*((KB+1 .)/2.)**((KB+l .)/(2.*(KB-l.))) WRTTE(*,*) 'ATBD= ',ATBD C The total required area section of ejector in choking section of secondary stream[ATOTAL]. C ATOTALD=ATBD+AYYAD WRJTE(*,*)'ATOTALD= '.ATOTALD DTOTAL=(SQRT(4./3.14159*ATOTALD))*1000. WRJTE(*,*) T>TOTAL= ",DTOTAL C Ratio of ATBD and ATN=RATBDATN RATBDATN=ATBD/ATN WRITE(*,*) 'RATBDATN=ATBD/ATN \RATBDATN RATBDAYYAD=ATBD/AYYAD
257
Appendix C
WRJTE(*,*) 'AREA RATIO OF Alb/Ala=ATBD/AYYAD=RATBDAYYAD '.RATBDAYYAD C ENTRAIN2=ENTRAIN WRTTE(*,*) 'lstE.R. F R O M EQ.(3.16)= '.ENTRAIN ENTRAIN2=(MYB/MYA)*((POB/PYY)**((KB-l)/KB))*((PYY/POA)**(aCAD/KA)) 1 *RATBDAYYAD*CBCARATIO D O 121=1,5 ENTRAINMENT=ENTRAIN2 WRITE(*,*) "ENTRAINMENT IN ITERATION=',ENTRAINMENT C Equation (3.13) PART4=KA/0*A-l.)+KB/(KB-l.)*ENTRAINMENT*W/w PART5=17(KA-1.)+U(KB-1.)*ENTRAINMENT*WAV K2=PART4/PART5 C WRTTE(*,*) •K2=',K2 C Equation 25 PART6=KB/(KB-1 .)*ENTRAINMENT*W/W T02TOARATIO=(KA/(KA-l.)+PART6*TOB/TOA)/(KA/(KA-l.)+PART6) C WRITE(*,*) 'T2TA=',T02TOARATIO C Equation 26 C2CARATIO=SQRT(K2/(K2+l.)*(KA+l.)/KA*WAV*T02TOARATIO) CA=(2.*KA/(KA+1.)*R*TOA)**0.5 C2=C2CARATIO*CA C Equation 11 PART7=MYA+MYB*ENTRAINMENT*CBCARATIO PART8=l./KA*ARATIO*PYA/POA*((KA+l .)/2.)**(KA/(KA-l.)) PART9=(K2+1 .)/K2*(l .+ENTRAINMENT)*C2CARATIO B=(PART7+PART8)/PART9 WRITE(*,*)' B=',B IF(B.LE.1.)THEN WRITE(*,*) 'Because of the value of B
258
Appendix C
C
Equation 15 M2=((27(JC2+1.)*M2**2)/(1.-(K2-1.)/(K2+1.)*M2**2))**0.5 C P22=PO3/((l.-KK2-l.)/2.*(M2**2))**0C2/(K2-l.))) C WRTTE(*,*) 'P2 (WHEN EFFD=1)= ',P22 P3=P2*((1+(K2-1.)/2.*(M2**2)*EFFD)**(K2/(K2-1.))) P=P3 CALL SATTEMP(P,T) TCOND=T T5=TC0ND CALL SVG(T,P,VG) VGCOND=VG CALL SVF(T,VF) VFCOND=VF CALL SLATENT(VG,VF,T,P,LATENT) LATENTCOND=LATENT CALL SHG(T,VG,P,HG) HGCOND=HG CALL SHF(HG,LATENT,HF) HFCOND=HF H5=HFC0ND H6=HFC0ND WRTTE(*,*) 'M2=',M2 PCRIT=P2PlRATIO*PYA*((l .+(K2-1 .)/2.*(M2**2)*EFFD)**(K2/(K2-l.))) END IF P=PCRIT CALL SATTEMP(P,T) TCRTT=T WRTTE(*,*) •PCRIT=,,PCRrT,'KPa' WRITE(*,*) TCRIT=',TCRIT,' C WRrTE(*,*) TCOND^.TCOND,' C WRITE(*,*)'' WRTTE(*,*) D O Y O U WISH TO CHANGE THE AREA RATIO (Y=l) ?' READ(*,*) Y WRITE(*,*) IF(Y=l)GOTO 1 C COMPRATIO=P3/POB ME=QEVAP/(H4-H6) C C MB=ME/ENTRAIN WRITE(*,*) 'ME EST SUB= '.ME C C WRITE(*,*) 'MB IN SUB= ',MB C PYY=PYA TOA=TOA-273.15 C C TOB=TOB-273.15 C50 RETURN C END
259
Appendix C
C*************************»**********************************M#+t##„ C
S U B R O U T I N E T O C A L C U L A T E T H E DIMENSIONS O F A N O P T I M U M EJECTOR
C This subroutine uses the equations for compressible C isentropic flow. C POA - Stagnation pressure at point 0 P Y Y - Optimum pressure at point 1 C C P T H R O A T - Pressure at throat C P O P - Pressure ratio of point 0 to 1 or point 0 to throat T OA - Stagnation temperature at 0 C C T T H R O A T - Temperature at throat C T O T - Temperature ratio of point 0 to throat or point 0 to 1 C M A C H - Mach number at point 1 M T H R O A T - Mach number at throat C C V T H R O A T - Velocity at throat C V O A - Specific volume at point 0 C ROWO-Density at point 0 C ROWTHROAT-Density at throat C R O W O R O W - Density ratio of point 0 to throat or point 0 to 1 K A - Specific heat ratio at point 0 C C R - Gas constant A R A T T O - Area ratio of section 2 to throat C M B - Massflowrate through generator C C A 2 - Area at section 2 C A - Motive nozzle area at 1 C A T H R O A T - Area at throat A A - Area ratio of motive nozzle exit to throat C C DIA2 - Diameter at section 2 C D I A T H R O A T - Diameter at throat C D I A A - Diameter at motive nozzle exit C SUBROUTINE EJECTAREA(MB,MEVAP,ARATIO,POA,POB ) PYY,KA > KB,TOA,TOB,VOA,DIATHROAT, C 1 DIAYYADIATOTAL,DIA2,RATBATN,RATBAYYA) C REAL MACH.ROWO, VOA C REAL VTHROAT,ROWTHROAT C REAL A2,ARATIO,ATHROAT C REAL DIA2,DIATHROAT,AA,A C REAL ATN>1EVAP,ATB,AYYA,ATOTAL,DIAYYA,DIATOTAL,RATBATN,RATBAYYA C R=8.3144*10007120.91 C Pressure ratio across nozzle
C
Inlet temp to nozzle and evaporator
C TOA=TOA+273.15 TOB=TOB+273.15 C C Inlet density to nozzle
260
Appendix C
C
D O 121=1,50 DIATOTAL(l)=DTOTAL DIATOTALa+l)=DIATOTAL(I)-0.1 DIFDIA=(DIATOTAL(I)-DIATOTAL(I+l))/2. ATOTALa+l)=((DIATOTALa+l)**2)*3.1415)/4. DIAA(1)=DAA DIAA(I+l)=DIAAa>+DIFDIA AYYA(I+l)=((DIAAa+l)**2)*3.1415)/4. ATB=ATOTALa+l)-AYYA(I+l) C Ratio of ATB and ATN=RATB ATN RATBATN=ATB/(ATN*1000000) WRTTE(*,*) 'RATBATN=ATB/ATN '.RATBATN RATBAYYA=ATB/AYYA(I+1) WRrTE(*,*) 'AREA RATIO OF Alb/Ala=ATB/AYYA0>RATBAYYA '.RATBAYYA WRTTE(*,*) DIATOTALa+l)J>laa+l),Alb,Ala',DIATOTAL(I+l),DIAA(I+l),ATB,AYYAa+l) C TO FIND THE E.R. BY USE OF ATB/AYYA(A lb/A la), WHEN ATB+AYYA IS NOT EQUAL A2 ENTRAIN2=(MYB/MYA)*((POB/PYY)**((KB-l)/KB))*((PYY/POA)**((KA-l)/KA)) 1 *RATBAYYA*CBCARATIO WRITE(*,*) "E.R. WHEN Alb+Ala IS NOT EQUAL A2 = \ENTRAIN2 C PRECISION=0.001 C IF ((ABS(FJ«miAIN2-ENTRAINMENT)).GE.PRECISION) GO TO 100 12 CONTINUE IF (ATOTALD.GT.A2) THEN WRTTE(*,*) THE SECONDARY CHOKING PHENOMENON OCCURS BEFORE THE END OF 1 THE CONVERGING DUCT OF EJECTOR.THERE IS ALSO A PRESSURE 1 RECOVERY AFTER THE SECONDARY CHOKING BECAUSE OF THE 1 DECELERATION OF SUPERSONIC AFTER THAT CHOKTNG.IN THAT CASE 1 Alb+Ala>A2.' ELSE WRITER,*) THE SECONDARY CHOKING PHENOMENON OCCURS AT THE END OF THE 1 CONVERGING DUCT(N0 PRESSURE RECOVERY) AND JUST AT THE 1 BEGINNING OF THE CONSTANT AREA MIXING TUBE. IN THAT CASE 1 Alb+Ala=A2.' TOA=TOA-273.15 END IF TOB=TOB-273.15 50 RETURN END
261
Appendix C
C.2 Subroutines for Calculation of R12 Thermo-Physical Properties (Kanashige, 1992) Subroutines for calculation of R12 thermo-physical properties are listed on the following pages.
262
Appendix C
p*********************************************************************** C S U B R O U T I N E T O C A C U L A T E T H E S A T U R A T E D PRESSURE F R O M A GIVEN TEMPERATURE C T - Temperature P - Pressure C A,B,C J) - Constants C SUBROUTINE SATPRESS(T,P) REALT.P R E A L A,B,C,D T=T+273.15 Constants used from "vapour pressure equation" [16] A=37.5386535459 B=-1909.240126 C=-12.47152228 D=8.5147963956E-3 Calculation of the pressure P=10**(A+B/T+C*LOG10(T)+D*T) T=T-273.15 RETURN
END
263
Appendix C
C S U B R O U T I N E T O C A L C U L A T E T H E S A T U R A T E D V A P O U R SPECIFIC V O L U M E C T - Temperature C V - Specific volume C V G - Specific volume of saturated vapour C P - Pressure C F V G - Equation of state equatedtozero C D F V G - Differential of equation of state with respect to V G V G U E S S - Guess value of specific volume C C E R R O R - Error of Newton method C A,B,CJ)3J7,G,H,IJ - Constants C E X P - Function of temperature S U B R O U T I N E SVGCT,P,VG) REAL T,VG,V,P,FVG,DFVG,ERROR, VGUESS R E A L AB,C,D,E,F,G,H,I,J,EXP T=T+273.15 C The following IF THEN ELSE statement finds an initial specific C volume guess from the temperature input. C The calculation is based on a regression analysis of V G vs T. C The calculation IS split into two sections (a) for temperatures C below 60 C and (b) temperatures above 60 C. IF(T<333.)THEN VGUESS=114.81-1.7826*T+1.1071E-2*(T**2)-3.4338E-5*(T**3) VGUESS=VGUESS+5.3146E-8*T**4-3.2817E-11*T**5 ELSE VGUESS=-17.143+.19825*T-8.5524E-4*T**2+1.6337E-6*T**3 VGUESS=VGUESS-1.167E-9*T**4 END IF Constants from "equation of state" [16] A=6.87405915574E-2 B=-9.1614480001E-2 C=7.71055926593E-5 D=-1.5208370561 E=1.01039343781E-4 F=-5.67481547006E-8 G=2.19960358497E-3 H=-5.74583589914E-8 I=4.08154331713E-14 J=-1.66291319877E-10 EXP=2.71828**((-1.42146257032E-2)*T) VGUESS=VGUESS-4.06368111521E-4
264
Appendix C
C C
This F V G is the actual function with specific volume equated to zero.
20 FVG=(A*T)/VGUESS+(B+C*T+D*EXP)/(VGUESS**2) FVG=FVG+(E+F*T+G*EXP)/(VGUESS**3) FVG=FVG+H/(VGUESS**4)+(I*T+J*EXP)/(VGUESS**5) FVG=FVG-P
D F V G is the derivative of F V G withrespectto V G
C
DFVG=-2*(B+C*T+D*EXP)/(VGUESS**3)-3*(E+F*T+G*EXP)/(VGUESS**4) DFVG=DFVG-4*H/(VGUESS**5)-5*(I*T+J*EXP)/(VGUESS**6) DFVG=DFVG-1*(A*T)/(VGUESS**2) C Iteration by means of Newton Method V=VGUESS-07VG/DFVG) C Calculation of the error between guess and iterated step. If the error is greater than 1 % then the process repeats C C with an improved guess ERROR=(V-VGUESS)/VGUESS* 100. ERROR=ABS(ERROR) IF(ERROR>=l.)THEN C Improved guess VGUESS=V G O T O 20 END IF T=T-273.15 VG=V+4.06368111521E-4 RETURN
END
265
Appendix C
C S U B R O U T I N E T O C A L C U L A T E T H E S A T U R A T E D LIQUID SPECIFIC V O L U M E C F R O M T H E DENSITY E Q U A T I O N C T - Temperature C T T - D u m m y temperature C V F - Specific volume of saturated liquid C DENSITY - Density of saturated liquid C AB.C.D - Constants SUBROUTINE SVF(T,VF) REAL T,VF,DENS1TY,TT R E A L A,B,C,D,E T=T+273.15 C Constants from "liquid density equation" [16] A=558.0831464 B=0.77734382688 C=17.9433033024 D=l 17.435807269 E=-.000340228688079 TT=385.166-T C Calculation of the liquid density DENSITY=A+B*TT+C*(TT**0.5)+D*(TT**0.3333)+E*(TT**2) C Calculation of the saturated liquid specific volume by inverting C the density. VF=1-/DENSITY T=T-273.15 RETURN
END
266
Appendix C
C************************************************************ C S U B R O U T I N E T O C A L C U L A T E T H E L A T E N T H E A T O F VAPOURISATION C T - Temperature (C) C TF - Temperature (R) C V G - Specific volume of saturated vapour C V F - Specific volume of saturated liquid C P - Pressure C L A T E N T - Latent heat of vaporization C A B , C A E - Constants S U B R O U T I N E SLATENT(VG,VF,T,P,LATENT) REAL T,TF,VG,VF,P .LATENT REAL A,B,CAE Converting SI units into Imperial units T=T*975+32 TF=T+459.7 VG=VG* 16.0179 VF=VF* 16.0179 P=P*1000.*(1.450E-4) Constants from "latent heat of vaporization" [17] A=2.302585093 B=-3436.632228 C=-12.47152223 D=0.004730442442 E=0.185053 Calculation of latent heat of vaporization LATENT=(VG-VF)*P*A*(-(B/TF)+C/A+D*TF)*E Converting the results from Imperial units into SI units T=579*(T-32.) LATENT=0.996*4.1868/1.8*LATENT VG=VG/16.0179 VF=VF/16.0179 P=P/(1.450E-1)
RETURN END
267
Appendix C
C SUBROUTINE T O CALCULATE THE SATURATED VAPOUR ENTHALPY C T - Temperature (C) C T F - Temperature (R) C V G - Specific volume of saturated vapour C V R - D u m m y specific volume C P - Pressure C H G - Enthalpy of saturated vapour H I ,H2,H3,H4 - Part of enthalpy equation C C A,B,C,D,E,F,G,H,I,J,K,L,M - Constants C E X P - Function of temperature SUBROUTINE SHG(T,VG,P,HG) REAL T,TF,VR,VG,H1,H2,H3,H4,HG,P R E A L A,B,C,D,E,F,G)H,U)K,L,M,EXP C Converting SI units into Imperial units T=9.*T/5+32. TF=T+459.7 VG=VG* 16.0179 VR=VG-0.0065093886 P=P*1000.*(1.450E-4) C Constants from "enthalpy of vapour equation" [17] A=0.0080945 B=3.32662E-4 C=-2.413896E-7 D=6.72363E-11 E=-3.409727134 F=0.06023944654 G=-0.000548737007 H=-56.7627671 1=1.311399084 J=0.185053 K=5.475 L=-2.54390678E-5 M=39.55655122 EXP=2.71828**(-K*TF/693.3) C Calculation of enthalpy Hl=A*TF+B*(TF**2)/2.+C*(TF**3)/3.+D*(TF**4)/4. H2=J*P*VG H3=(FVVR+F/(2.*VR**2)+G/(3.*VR**3)) H4=(H/VR+I/(2.*VR**2)+L/(4.*VR**4)) HG=Hl+H2+J*H3+J*H4*EXP*(l.+K*TF/693.3)+M C Converting the results from Imperial units into SI units
268
Appendix C
T=5./9*(T-32.) HG=1.0077*4.1868/1.8*(HG-15.5)+200. VG=VG/16.0179 P=P/(1.450E-1) RETURN END
p*********************************************************************** C SUBROUTINE T O CALCULATE THE S ATURETED LIQUID ENTHALPY C HF - Enthalpy of saturated liquid C H G - Enthalpy of saturated vapour C LATENT - Latent heat of vaporization SUBROUTINE SHF(HGXATENT,HF) REAL HF,HG,LATENT HF=HG-LATENT RETURN END
269
Appendix C
p********************************************************************** C SUBROUTINE T O CALCULATE THE SATURATED V A P O U R ENTROPY C T - Temperature (C) C TF - Temperature (R) C V - Specific volume C V G - D u m m y specific volume C S - Entropy C S 1,S2,S3,S4 - Part of entropy equation SUBROUTINE SS(T,V,S) REAL T,TF,V,VG,S,S1,S2,S3,S4 R E A L A,B,CAE,F,G,H,U,K,L,M,N,EXP C Converting SI units into Imperial units T=9.*T/5+32. TF=T+459.7 V=V*16.0179 VG=V-0.0065093886 C Constants from "entropy of vapour equation" [17] A=0.0080945 B=3.32662E-4 C=-2.413896E-7 D=6.72363E-11 E=0.088734 F=0.00159434848 G=-1.879618431E-5 H=3.468834E-9 I=-56.7627671 J=0.185053 K=5.475 L=1.311399084 M=-2.54390678E-5 N=-0.0165379361 EXP=2.71828**(-K*TF/693.3) C Calculation of entropy Sl=A*LOG(TF)+B*TF+C*(TF**2)/2.+D*(TF**3)/3. S2=J*E*LOG(VG) S3=(FA^G+G/(2.*VG**2)+H/(4.*VG**4)) S4=(77VG+L/(2.*VG**2)+M/(4.*VG**4)) S=Sl+S2-J*S3+J*S4*EXP*K/693.3+N C Converting the results from Imperial units into SI units T=579*(T-32.) S=1.0078*4.1868*(S-0.033905)+l. V=V/16.0179 RETURN
END
270
Appendix C
P***************************************************!,.***********^^^,,.^* *** C SUBROUTINE T O CALCULATE THE SPECIFIC V O L U M E A N D TEMP F R O M PRESS AND CENTROPY C This subroutine will determine the temperature and me specific C volume of the refrigerant knowing the pressure and the entropy. It utilises the Newton Method in two dimensions. C C P - Pressure C S - Entropy T - Temperature (C) C C T F - Temperature (R) V - Specific volume C C V G - D u m m y specific volume C S V T - Entropy equation equated to zero C D S T - Differential of entropy equation with respect to temperature C D S V - Differential of entropy equation with respect to specific volume C C P V T - Equation of state equated to zero C D P T - Differential of equation of state with respect to temperature C D P V - Differential of equation of state with respect to specific volume C C S V T 1 ,SVT2,SVT3.S V T 4 - Part of entropy equation C A N T - Inverse matrix used in Newton method C F U N C T I O N - Matrix of entropy and pressure functions I T E R A T I O N - Matrix for improved guess used in Newton method C C N E W - Matrix for new guess used in Newton method C G U E S S - Matrix for guess for iteration used in Newton method C M U L T - Multiplier for inverse matrix A N T A,B,CAE,F,G,H,I,J,K,L)M,N. Constants C C AA,BB,CC,DD,EE,FF,GG,HH,II,JJ - Constants C E X P - Function of temperature SUBROUTINE ENTITER(P,S,T,V) REALP,S,T,TF,V,VG,SVT,PVT,DSV,DST,DPV,DPT R E A L SVT1 SVT2 SVT3 SVT4
REALA,B,CAE,F',G,H,U,K,LAI,N ) EXP,AA,BB,CC,DD,EEJT,GG,HH,IUJ REALANT(2,2),FUNCTION(2,l),ITERATION(2,l),NEW(2,l),GUESS(2,l),MULT C The iteration process needs an initial guess for both the temperature C and the specific volume. C T=70 (C) C V=0.01 (M3/KG) C Converting SI units into Imperial units T=9.*70/5+32. TF=T+459.7 V=0.01*16.0179 VG=V-0.0065093886 S=(S-1.)/4.1868+0.033905
271
Appendix C
P=(1.450E-1)*P C Constants from "entropy of vapour equation" [17] A=0.0080945 B=3.32662D-4 C=-2.413896D-7 D=6.72363D-11 E=0.088734 F=0.00159434848 G=-1.879618431D-5 H=3.468834D-9 I=-56.7627671 J=0.185053 K=5.475 L=1.311399084 M=-2.54390678D-5 N=-0.0165379361 C The following Do loop is used for the iteration process. DO 20 IT=1,30 EXP=2.71828**(-K*TF/693.3) C SVT is the entropy function with respect to specific volume and C temperature equated to zero. SVTl=A*DLOG(TF)+B*TF+C*(TF**2)/2.+D*(TF**3)/3. SVT2=J*E*DLOG(VG) SVT3=07/VG+G/(2.*VG**2)+H/(4.*VG**4)) SVT4=(WG+L/(2.*VG**2)+M/(4. *VG**4)) SVT=SVTl+SVT2-J*SVT3+J*SVT4*EXP*K/693.3+N-S C DST is the entropy function differentiated with respect to C temperature DST=A/TF+B+C*TF+D*TF**2 DST=DST+J*EXP*(I/VG+L/(2.*VG**2)+M/(4.*VG**4))*K/693.3*(-K/693.3) C DSV is the entropy function differentiated with respect to C specific volume. DSV=J*E/VG-J*(-F/(VG**2)-G/(VG**3)-H/(VG**5)) DSV=DSV+J*EXP*K/693.3*(-I/(VG**2)-L/(VG**3)-M/(VG**5)) C Constants from "equation of state" [17] AA=0.088734 BB=-3.40972713 CC=1.59434848D-3 DD=-56.7627671 EE=0.0602394465 FF=-1.87961843D-5 GG=1.31139908 HH=-5.4873701D-4 H=3.468834D-9
272
Appendix C
JJ=-2.54390678D-5 C PVT is the pressure function with respect to specific volume C and temperature equated to zero. PVT=AA*TF/VG+(BB+CC*TF+DD*EXP)/(VG**2) PVT=PVT-KEE+FF*TF+GG*EXP)/(VG**3) PVT=PVT+HH/(VG**4)+(II*TF+JJ*EXP)/(VG**5)-P C DPT is the pressure function differentiated with respect to C temperature DPT=AA/VG+(CC-K/693.3*DD*EXP)/(VG**2) DPT=DPT-KFF-K/693.3*GG*EXP)/(VG**3) DPT=DPT-KII-K/693.3*JJ*EXP)/(VG**5) C DPV is the pressure function differentiated with respect to C specific volume. DPV=-AA*TF/(VG**2)-2.*(BB+CC*TF+DD*EXP)/(VG**3) DPV=DPV-3.*(EE+FF*TF+GG*EXP)/(VG**4) DPV=DPV-4.*HH/(VG**5)-5.*(II*TF+JJ*EXP)/(VG**6) C The following is the matrix setup for the Newton iteration C method in two dimensions. MULT=1/(DST*DPV-DPT*DSV) ANT(1,1)=DPV*MULT ANT(1,2)=-DS V * M U L T ANT(2,1)=-DPT*MULT ANT(2,2)=DST*MULT FUNCTION(l,l)=SVT FUNCTION(2,l)=PVT NEW(l,l)=ANT(l,l)*FUNCTION(l,l)+ANT(l,2)*FUNCTION(2,l) NEW(2,l)=ANT(2,l)*FUNCTION(l,l)+ANT(2,2)*FUNCTION(2,l) GUESS(1,1)=TF GUESS(2,1)=VG ITERATION(l,l)=GUESS(l,l)-NEW(l,l) ITERATION(2,l)=GUESS(2,1)-NEW(2,1) C The new guess for me next iteration. TF=ITERATION(l,l) VG=rTERATION(2,l) 20 CONTINUE C Converting the results from Imperial units into SI units T=(TF-459.7-32.)*5./9 V=(VG+0.0065093886)/16.0179 S=4.1868*(S-0.033905)+l. P=P/(1.450E-1) RETURN
END
273
Appendix C
C S U B R O U T I N E T O C A L C U L A T E T H E E N T H A L P Y IN T H E S U P E R H E A T R E G I O N C T - Temperature (C) C T F - Temperature (R) C V - Specific volume C V R - D u m m y specific volume C P - Pressure C E N T H A L P Y - Superheat enthalpy C H I ,H2,H3,H4 - Part of enthalpy equation C A,B,C AE,F,G,H,I,J,K,L,M - Constants C E X P - Function of temperature SUBROUTINE SUPERENTH(T,V,P,ENTHALPY) REAL T,TF,V,VR,H1,H2,H3,H4,ENTHALPY,P R E A L A,B,CAE,F,G,H,I,J,K,L,M,EXP C Converting SI units into Imperial units T=9.*T/5+32. TF=T+459.7 V=V*16.0179 VR=V-0.0065093886 P=P*1000.*(1.450E-4) C Constants from "enthalpy of vapour equation" [17] A=0.0080945 B=3.32662E-4 C=-2.413896E-7 D=6.72363E-11 E=-3.409727134 F=0.06023944654 G=-0.000548737007 H=-56.7627671 1=1.311399084 J=0.185053 K=5.475 L=-2.54390678E-5 M=39.55655122 EXP=2.71828**(-K*TF/693.3) C Calculation of enthalpy Hl=A*TF+B*(TF**2)/2.+C*(TF**3)/3.+D*(TF**4)/4. H2=J*P*V H3=(F7VR+F/(2.*VR**2)+G/(3.*VR**3)) H4=(H/VR+I/(2.*VR**2)+L/(4.*VR**4)) ENTHALPY=Hl+H2+J*H3+J*H4*EXP*(l.+K*TF/693.3)+M C Converting the results from Imperial units into SI units T_579*(T-32) ENTHALPY=1.0077*4.1868/1.8*(ENTHALPY-15.5)+200. V=V/16.0179 P=P/(1.450E-1) RETURN
END
274
Appendix C
p*********************************************************************** C SUBROUTINE T O CALCULATE THE SATURATED TEMP F R O M A SATURATED PRESSURE SUBROUTINE SATTEMP(P,T) REAL P,T,TP,TG,DTP,A,B,CAE,F,ERROR C Constants from regression analysis (order 5) A=37.5386535459 B=-1909.240126 C=-12.47152228 D=8.5147963956E-3 C Initial guess E=260.55 F=0.0403 TG=E+F*P C Calculation of the temperature 10 TP=A+B/TG+C*(LOG10(TG))+D*TG-LOG10(P) DTP=-B/(TG**2)+C/(TG*LOG(10.))+D T=TG-(TP/DTP) ERROR=(T-TG)/TG* 100. ERROR=ABS(ERROR) IF(ERROR.GE.l.)THEN TG=T
GO TO 10 END IF T=T-273.15 RETURN END
275
Appendix C
p***********************^*^*^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ C SUBROUTINE TO CALCULATE THE SPECIFIC VOLUME AND TEMPERATURE FROM C PRESSURE A N D ENTHALPY C This subroutine will determine the temperature and the specific volume. It utilises the Newton Method in two C C dimensions. C P - Pressure H-Enthalpy C C T - Temperature (C) C T F - Temperature (R) C V - Specific volume C V G - D u m m y specific volume C H V T - Enthalpy equation equated to zero C D H T - Differential of enthalpy equation withrespectto temperature C D H V - Differential of enthalpy equation with respect to specific C volume C P V T - Equation of state equated to zero C D P T - Differential of equation of state with respect to temperature C D P V - Differential of equation of state with respect to specific C volume C HVT1,HVT2,HVT3,HVT4 - Part of enthalpy equation C A N T - Inverse matrix used in Newton method C F U N C T I O N - Matrix of enthalpy and pressure functions C ITERATION - Matrix for improved guess used in Newton method C N E W - Matrix for new guess used in Newton method C G U E S S - Matrix for guess for iteration used in Newton method C M U L T - Multiplier for inverse matrix A N T A,B,CAE^,G,HHH,U,K,L,M - Constants C C AA,BB,CC,DD,EE,FF,GG,HH,II,JJ - Constants C E X P - Function of temperature S U B R O U T I N E JNVENTHAL(H,P,TI,VI,T,V) REAL P,H,T,TI,TF,V,VI,VG,HVT,PVT,DHV,DHT,DPV A^ R E A L HVT1,HVT2,HVT3,HVT4,MULT R E A L A,B,CAE,F,G,HHH)I,J,K,L,M,EXP R E A L AA,BB,CC,DD,EE,FF,GG,HH,II,JJ REALANT(2,2),FUNCTION(2,l),ITERATION(2,l),NEW(2,l),GUESS(2,l) C The iteration process needs an initial guess for both the C temperature and the specific volume. C T=TI C V=VI C Converting SI units into Imperial units T=9.*TI/5.+32. TF=T+459.7 V=VI*16.0179 VG=V-0.0065093886
276
Appendix C
H=(H-200.)* 1.8/4.1868+15.50 P=(1.450E-1)*P Constants from "enthalpy of vapour equation" [17] A=0.0080945 B=3.32662D-4 C=-2.413896D-7 D=6.72363D-11 E=-3.409727134 F=0.06023944654 G=-0.000548737007 HHH=-56.7627671 1=1.311399084 J=0.185053 K=5.475 L=-2.54390678D-5 M=39.55655122 C The following Do loop is used for die iteration process. DO 20 IT=1,40 EXP=2.71828**(-K*TF/693.3) C HVT is the enthalpy function with respect to specific volume C and temperature equated to zero. HVT1=A*TF+B *(TF* *2)/2.+C*(TF* *3)/3 .+D*(TF**4)/4. HVT2=J*P*V HVT3=Qi/VG+F/(2.*VG**2)+G/(3.*VG**3)) HVT4=(HHH/VG+y(2.*VG**2)+L/(4.*VG**4)) HVT=HVTl+HVT2+J*HVT3+J*HVT4*EXP*(l.+K*TF/693.3)+M-H C DHT is the enthalpy function differentiated with respect to C temperature. DHT=A+B*TF+C*TF**2+D*TF**3 DHT=DHT+J*(HHH/VG+I/(2.*VG**2)+L/(4.*VG**4))*(-K/693.3*EXP) 1 *(l.+K*TF/693.3) C DHV is the enthalpy function differentiated with respect to C specific volume. DHV=J*P+J*(-E/(VG**2)-F/(VG**3)-G/(VG**4)) DHV=DHV+J*EXP*(l.+K*TF/693.3)*(-HHH/(VG**2)-I/(VG**3)-L/(VG**5)) C Constants from "equation of state" [17] AA=0.088734 BB=-3.40972713 CC=1.59434848D-3 DD=-56.7627671 EE=0.0602394465 FF=-1.87961843D-5
277
Appendix C
GG=1.31139908 HH=-5.4873701D-4 n=3.468834D-9 JJ=-2.54390678D-5 C PVT is the pressure function with respect to specific volume C and temperature equated to zero. PVT=AA*TF/VG+(BB+CC*TF+DD*EXP)/(VG**2) PVT=PVT+(EE+FF*TF+GG*EXP)/(VG**3) PVT=PVT+HH/(VG**4)+(II*TF+JJ*EXP)/(VG**5)-P C DPT is the pressure function differentiated with respect to C temperature DPT=AA/VG+(CC-K/693.3*DD*EXP)/(VG**2) DPT=DPT-KFF-K/693.3*GG*EXP)/(VG**3) DPT=DPT+ai-K/693.3*JJ*EXP)/(VG**5) C D P V is the pressure function differentiated with respect to C specific volume. DPV=-AA*TF/(VG**2)-2.*(BB+CC*TF+DD*EXP)/(VG**3) DPV=DPV-3.*(EE+FF*TF+GG*EXP)/(VG**4) DPV=DPV-4.*HH/(VG**5)-5.*(II*TF+JJ*EXP)/(VG**6) C The following is the matrix setup for the Newton iteration C method in two dimensions. MULT=1/(DHT*DPV-DPT*DHV) ANT(1,1)=DPV*MULT ANT(1,2)=-DHV*MULT ANT(2,1)=-DPT*MULT ANT(2,2)=DHT*MULT FUNCTION(l,l)=HVT FUNCTION(2,l)=PVT NEW(l,l)=ANT(l,l)*FUNCTION(l,l)+ANT(l,2)*FUNCTION(2)l) NEW(2,l)=ANT(2,l)*FUNCTION(l(l)+ANT(2,2)*FUNCTION(2,l) GUESS(1,1)=TF GUESS(2,1)=VG ITERATION(l,l)=GUESS(l,l)-NEW(l,l) ITERATION(2,l)=GUESS(2)l)-NEW(2>l) C The new guess for the next iteration. TF=ITERATION(l,l) VG=rTERATION(2,l) 20 CONTINUE C Converting the results from Imperial units into SI units T=(TF-459.7-32.)*5./9 V=(VG+O.0065093886)/16.0179 H=4.1868/1.8*(H-15.50)+200. P=P/(1.450E-1) RETURN
END
278
Appendix C
p********************************************************************* C S U B R O U T I N E T O D E T E R M I N E T H E Cv C CV - Heat capacity at consyant volume C V - Specific volume C V G - D u m m y specific volume C T - Temperature (C) C TF - Temperature (R) AB,CAEJ 7 ,G,H,I - Constants C C E X P - Function of temperature S U B R O U T I N E HEATCV(T,V,CV) REAL CV,V,VG,T,TF R E A L A,B,CAE,F,G,H,I,EXP Converting SI units into Imperial units
C
T=T*9./5+32. TF=T+459.7 V=V*16.0179 VG=V-0.0065093886 C Constants from "heat capacity at constant volume C equation" [17] A=0.0080945 B=3.32662E-4 C=-2.413896E-7 D=6.72363E-11 E=5.475 F=693.3 G=-56.7627671 H=1.311399084 I=-2.54390678E-5 EXP=2.718282**(-E*TF/F) C Calculation of heat capacity CV=A+B*TF+C*TF**2+D*TF**3-(0.185053*E**2*TF*EXP)/(F**2) *(GWG+H/(2.*VG**2)+I/(4.*VG**4)) 1 C Converting the results from Imperial units into SI units T=5./9*(T-32.) V=V/16.0179 CV=CV*4.1868 RETURN
END
279
Appendix C
P********************************************************************** C S U B R O U T I N E T O D E T E R M I N E Cp C CP - Heat capacity at constant pressure C T - Temperature (C) C TF - Temperature (R) C V - Specific volume V G - D u m m y specific volume C PART1.PART2 - Part of heat capacity equation C C A,B,CAEJ7,G,H,IJ,KJL - Constants C E X P - Function of temperature SUBROUTINE HEATCP(T,V,CV,CP) REAL CP,CV,T,TF,V,VG,PART1,PART2 R E A L AB,CAE,F,G,H,U,K,L,EXP C Converting SI units into Imperial units T=T*9./5+32. TF=T+459.7 V=V*16.0179 VG=V-0.0065093886 CV=CV/4.1868 C Constants from "heat capacity at constant pressure C equation" [17] A=0.088734 B=-3.409727134 C=0.00159434848 D=-56.7627671 E=0.06023944654 F=-1.879618431E-5 G=1.311399084 H=-0.000548737007 I=3.468834E-9 J=-2.54390678E-5 K=5.475 L=693.3 EXP=2.71828**(-K*TF/L) C Calculation of heat capacity PARTl=-A*TF/(VG**2)-2.*(B+C*TF+D*EXP)/(VG**3)-3. *(E+F*TF+G*EXP)/(VG**4) 1 2 -4.*H/(VG**5)-5.*(I*TF+J*EXP)/(VG**6) PART2=AArG+(C-K*D*EXP/L)/(VG**2)+(F-K*G*EXP/L)/(VG**3) 1 +(I-K*J*EXP/L)/(VG**5) CP=CV-0.185053*TF*(PART2**2)/PART1 C Converting the results from Imperial units into SI units T=5./9*(T-32.) V=V/16.0179 CP=CP*4.1868 CV=CV*4.1868 RETURN
END 280
Appendix C
P******************************+*+*+++#+^++++++++++++++++++++++++#+](I1)t!|t](t)|tj|t!)1 C S U B R O U T I N E T O C A L C U L A T E T H E SPECIFIC V O L U M E IN T H E S U P E R H E A T REGION SUBROUTINE SV(T,P.V) REAL TG,VG,V,P,FVG,DFVG,ERROR R E A L A,B,CAE,F,G,H,U,K,EXP C Calculation of an initial guess for the Newton Method. C This guess is actually the saturated temperature of the known C pressure E=226.56 F=.18612 G=-1.5409E-4 H=7.2344E-8 J=-1.6282E-11 K=1.382E-15 TG=E+F*P+G*P**2+H*P**3+J*P**4+K*P**5 C The following IF THEN ELSE statement finds an initial specific C volume guess from the temperature guess. The calculation is based C on a regression analysis of V G vs T. The calculation IS split into two sections (a) for temperatures below 60 C and tb) C C temperatures above 60 C. IF(TG<333.)THEN VG=114.81-1.7826*TG+1.1071E-2*(TG**2)-3.4338E-5*(TG**3) VG=VG+5.3146E-8*TG**4-3.2817E-11*TG**5 ELSE VG=1.10027-8.479E-3*TG+2.2158E-5*TG**2-1.9638E-8*TG**3 END IF T=T+273.15 Constants from "equation of state" [16] A=6.87405915574E-2 B=-9.1614480001E-2 C=7.71055926593E-5 D=-l.5208370561 E=1.01039343781E-4 F=-5.67481547006E-8 G=2.19960358497E-3 H=-5.74583589914E-8 I=4.08154331713E-14 J=-1.66291319877E-10 EXP=2.71828**((-1.42146257032E-2)*T)
281
Appendix C
VG=VG-4.06368111521E-4 C This FVG is the actual function with specific volume equated to C zero. 20 FVG=(A*T)/VG+03+C*T+D*EXP)/(VG**2)+(E+F*T+G*EXP)/(VG**3) FVG=FVG+H/(VG**4)+(I*T+J*EXP)/(VG**5) FVG=FVG-P C DFVG is the derivative of FVG with respect to VG. DFVG=-2*(B+C*T+D*EXP)/(VG**3)-3*(E+F*T+G*EXP)/(VG**4) DFVG=DFVG-4*H/(VG**5)-5*(I*T+J*EXP)/(VG**6)-1*(A*T)/(VG**2) C Iteration by means of Newton Method V=VG-(FVG/DFVG) C Calculation of the error between guess and iterated step. C If the error is greater than 1 % then the process repeats C with an improved guess ERROR=(V-VG)/VG* 100 ERROR=ABS(ERROR) D?(ERROR>=l.)THEN C Improved guess VG=V G O T O 20 END IF V=Vt4.06368111521E-4 T=T-273.15 RETURN
END
282
Appendix C
C3 Simulation Program for Two-Fluid VJRS Ejectors The simulation program for two-fluid VJRS ejectors using one-dimensional analysis is listed on the following pages.
283
Appendix C
C**********Two-Fluid Vapour Jet Refrigeration System (VJRS) Ejectors******** C C C C C
This program simulates the ejector refrigeration system using the constant area analysis in [11]. In mis program two dissimilar molecular weights of workingfluids(Rll & R12) are used which are seperated in condenser. Rll is the motive vapour and R 1 2 is the secondary vapour.
C Variable definitions C TCOND C C C C C C
- Operating temperature of condenser P C O N D - Operating pressure of condenser V G C O N D - Specific volume of saturated vapour at condenser V F C O N D - Specific volume of saturated liquid at condenser L A T E N T C O N D - Latent heat of vaporisation at condenser temperature H F C O N D - Enthalpy of saturated liquid at condenser temperature H G C O N D - Enthalpy of saturated vapour at condenser temperature
C TEVAP C C C C C C
- Operating temperature of evaporator P E V A P - Operating pressure of evaporator V G E V A P - Specific volume of saturated vapour at evaporaor V F E V A P - Specific volume of saturated liquid at evaporator L A T E N T E V A P - Latent heat of vaporisation at evaporator temperature H F E V A P - Enthalpy of saturated liquid at evaporator temperature H G E V A P - Enthalpy of saturated vapour at evaporator temperature
C
- Operating temperature of generator P B O J T L - Operating pressure of generator V G B O r L - Specific volume of saturated vapour at generator V F B O I L - Specific volume of saturated liquid at generator L A T E N T B O I L - Latent heat of vaporisation at generator temperature H F B O I L - Enthalpy of saturated liquid at generator temperature H G B O J T L - Enthalpy of saturated vapour at generator temperature
TBOJTL
C C C C C C
C T - Dummy temperature variable C TI - Initial guess of temperature C P - D u m m y pressure variable C V - D u m m y specific volume variable C VI - Initial guess of specific volume C E N T H A L P Y - D u m m y enthalpy variable C S - D u m m y entropy variable C H F - D u m m y enthalpy of liquid variable C H G - D u m m y enthalpy of vapour variable C S G - Entropy of saturated vapour C S F G - Entropy of C V F - D u m m y specific volume of liquid variable C V G - D u m m y specific volume of vapour variable C L A T E N T - D u m m y latent heat of vaporization variable C Q - D u m m y heat exchange variable C KA - Specific heat ratio at the motive nozzle inlet C K B - Specific heat ratio at the secondary nozzle inlet C A R A T I O - Area ratio, A2/At C P 0 3 - Pressure at the oudet of ejector C V E L 2 - Stream velocity at the entrance of diffuser C E N T R A I N M E N T - Entrainment ratio C C O M P R A T I O - Pressure ratio of condenser to evaporator
284
Appendix C
C C C C C C C C C
C P - Heat capacity at constant pressure C V - Heat capacity at constant volume E F F D - Diffuser efficiency E F F N - Nozzle efficiency T O A - Temperature at the inlet of motive nozzle POA-Pressure at the inlet of motive nozzle V O A - Specific volume at the inlet of motive nozzle T O B - Temperature at the inlet of secondary nozzle P O B - Pressure at the inlet of secondary nozzle
C DIATHROAT - Diameter of motive nozzle throat D I A A - Diameter of motive nozzle exit C C DIA2 - Diameter of constant area tube C ME - Mass flow rate of refrigerant through evaporator M B - Massflowrate of refrigerant through generator C C M C O N D - Massflowrate of refrigerant through condenser C QEVAP C C C C C
- Heat absorbed through evaporator Q B O D L - Heat input through generator Q C O N D - Heat rejected through condenser W P U M P - Work of the pump B E T A - System coefficient of performance C A R N O T - Carnot coefficient of performance of system
C H0,H2,H3,H4,H5,H6,H7 - Enthalpies of each state in the cycle C P0,P2,P3,P4,P5,P6,P7 - Pressures of each state in the cycle TO,T2,T3,T4,T5,T6,T7 - Temperatures of each state in the cycle C S2-Entropy at 2 C C V O - Specific volume at 0 C X - Quality Y - D u m m y variable for program re-run C
P************************************************************************* REAL PP0,P2,P3,P4,P5,P6,P7>PCOND,PEVAP,PBOIL,POA,POB,PO3,PPl 1 REALT,TI,T0,T2,T3,T4)T5,T6>T7>TOA)TOB,TCOND,TBOE.,TEVAP REAL V V L V G V F VGCOND,VGEVAP,VGBOrL,VFBOIL,VFEVAP,VFCOND,VOA ' REALLATENT,LATENTBOIL,LATENTEVAP,LATENTCOND R E A L HG,HF,HGCOND,HGEVAP,HGBOIL,HFCOND,HFEVAP,HFBOIL R E A L H,H0,H2,H3,H4,H5,H6,H7,ENTHALPY R E A L S,S2,SG,SFG REAL KA,KB>IB,MEMCOND,ARATIO,ENTRAINMENT,COMPRATIO
REAL CP CV REAL DIATHROAT,DIAA,DIA2 REALQ,QEVAP,QBOIL,QCOND,WPUMP,BETA,CARNOT REAL EFFN,EFFD,VEL2 REAL PCRTT,TCRIT REALX.Y
285
Appendix C
C CONDENSER DATA 1 WRJTE(*,*) "ENTER THE CONDENSER TEMP (C)' READ(*,*) T TCOND=T c WRITE(*,*) W I E R THE CONDENSER PRESSURE (Rl 1 & R12) [KPa]' c READ(*,*) PC c PCOND=PC c WRITE(*,*) "ENTER THE PRESSURE OF R12 IN CONDENSER [KPa]' c READ(*,*) P CALL SVG(T,P,VG) VGCOND=VG CALL SVF(T,VF) VFCOND=VF CALL SLATENT(VG,VF,T,P,LATENT) LATENTCOND=LATENT CALL SHG(T,VG,P,HG) HGCOND=HG CALL SHF(HG,LATENT,HF) HFCOND=HF C WRITE(*, *) "ENTER THE PARTIAL PRESSURE OF R11 IN CONDENSER [KPa]' C READ(*,*) P C EVAPORATOR DATA WRTTE(*,*) 'ENTER THE EVAPORATOR SATURATED TEMPERATURE (C)[R12]" READ(*,*) T TEVAP=T CALL SATPRESS(T,P) PEVAP=P CALL SVG(T,P,VG) VGEVAP=VG CALL SVF(T,VF) VFEVAP=VF CALL SLATENT(VG,VF,T,P,LATENT) LATENTEVAP=LATENT CALL SHG(T,VG,P,HG) HGEVAP=HG CALL SHF(HG,LATENT,HF) HFEVAP=HF C GENERATOR DATA WRITE(*,*) 'ENTER THE GENERATOR SATURATED TEMP (C)[R11]' READ(*,*) T TBOIL=T WRTTE(*,*) "ENTER THE GENERATOR SATURATED PRESSURE (KPa)[Rll]' READ(*,*) P
286
Appendix C
C
CALL SATPRESS(T,P) PBOD>P WRTTE(*,*) "ENTER THE Vg(SATURATED)[L/Kg][Rll] IN THE GENERATOR' REAEX.*,*) VG C CALL SVG(T,P,VG) VGBOIL=VG WRTTE(*,*) "ENTER THE Vf(SATURATED)[L/Kg][Rl 1] IN THE GENERATOR' READ(*,*) VF C CALL SVF(T,VF) VFBOOVF C CALL SLATENT(VG,VF,T,P,LATENT) C LATENTBOIL=LATENT WRTTE(*,*) 'ENTER HG[KJ/Kg] [Rl 1] IN THE GENERATOR' READ(*,*) HG C CALL SHG
CALL HEATCV(T,V,CV) CALL HEATCP(T,V,CV,CP)
KA=1.6 WRTTE(*,*) 'KA=',KA C Inlet to the secondary nozzle T=T4 TOB=T4 P=P4 POB=P4 V=VGEVAP CALL HEATCV(T,V,CV) CALL HEATCP(T,V,CV,CP) KB=CP/CV WRITE(*,*) 'KB=',KB C Oudet from ejector P03=PCOND C EJECTOR DESIGN EJECTOR(ME,KA,O.TOA,TOB,K)A,POB,P3,EFFN>EFFD,PEVAP,TCOND,ARATIO) ENTRAI>JMENT)COMPRATIO,MB)PPll)VEL2,P2,PCRIT)TCRIT) CALL EJECTAREA(MB)ARATIO,POA,PPl l,KA,TOA,VOA,DIATHROAT,
288
Appendix C
1
DIAA.DIA2)
HT2=MB*HO+ME*H4-(MB+ME)*(VEL2**2)/2000. H2=HT2/(MB+ME) HT3=HT2+(MB+ME)*(VEL2**2)/2000 H3=HT3/(MB+ME) WRrTE(*,*) ,MB=,,MB,' ME=',ME WRTTE(*,*) 'HO=",H0,' H4=",H4,"VEL2=",VEL2 WRTTE(*,*) 'H2=,H3= ',H2,H3 CALLSOLVE(AN,B,M,DET,MB,ME,HT2,HT3,VEL2,Xl,X2,X3,X4) HR11A2=X1 HR11ATN3=X2 HR12B2=X3 HR12BIN3=X4 H=X3 WRITE(*,*) "HR11A2,HR11AIN3,HR12B2,HR12BIN3= ',X1,X2,X3,X4 P=P2/3 CALL SATTEMP(P,T) CALL SVG(T,P,VG) CALL SHG(T,VG,P,HG) WRITE(*,*) "HR12B2,HG= ",HR12B2,HG IF (HR12B2.GE.HG) THEN TI=T VI=VG CALL ESTVENTHAL(H,P,TI,VI,T,V) T2=T CALL SS(T,V,S) S2=S ELSE T2=T CALL SVF(T VF) CALL SLATENT(VG,VF,T,P,LATENT) CALL SHF(HG,LATENT,HF) X=(HR12B2-HF)/LATENT WRITE(*,*) "X=',X V=VG CALL SS(T,V,S) SG=S SFG=LATENT/(T2+273.15)
289
Appendix C
S2=X*SFG+(SG-SFG) S=S2 END IF WRTTE(*,*) "S2=',S2 P=P3/3 CALL ENnTER(P,S,T,V) CALL SUPERENTH(T,V,P,ENTHALPY) H3R12=ENTHALPY WRITE(*,*) "HR12BIN3 FROM SUB.(SOLVE)=,H3R12 FROM S2=",X4,H3R12 H=ENTHALPY TI=T VI=V CALL INVENTHAL(H,P,TI,VI,T)V) T3=T C H3=(H3-H2)/EFFD+H2 C CALCULATION OF MASS FLOW RATE THROUGH CONDENSER MCOND=MB+ME C COOLING CAPACITY = QEVAP QEVAP=QEVAP* 1000. C PUMP WORK C WPUMP=MB*VFCOND*(PBOIL-PCOND)*1000. C H7=WPUMP/(1000.*MB)+H5 WRITE(*,*) "ENTER THE HF[R11](KJ/Kg) AT CONDENSER TEMP.' READ(*,*) H7 C HEAT INPUT QBOIL=MB*(H0-H7)*1000. C HEAT REJECTED CONDENSER C QCOND=MCOND*(H3-H5)*1000. QCOND=ME*(H4+25-H5)*1000+MB*170*1000 C COEFFICIENT OF PERFORMANCE BETA=QEVAP/(QBOIL) C CARNOT COEFFICIENT OF PERFORMANCE
290
Appendix C
CARNOT=(TBOIL-TCOND)/(TBOIL+273.15)*(TEVAP+273.15)/(TC0NDTEVAP) C PRINT OUTS WRITE(*,*)"' WRTTE(*,*)'' WRTTE(*,*) '(1) EJECTOR SIMULATION' WRITE(*,*)'' WRITE(*,*)' THE PRESSURE AT 1 = ',PP11,' kPa' WRTTE(*,*)' THE PRESSURE AT 2= ",P2,'kPa" WRTTE(*,*)' THE TEMPERATURE AT 2 =',T2," C WRTTE(*,*)' THE ENTHALPY AT 2= ',H2,"kJ/kg" WRTrE(*,*)' THEA2/At= ",ARATIO WRITE(*,*)' THE ENTRAINMENT RATIO =',ENTRAINMENT WRrTE(*,*)' THE COMPRESSION RATIO =',COMPRATIO WRrTE(*,*) WRITE(*,*) WRrTE(*,*) (2) EJECTOR GEOMETRY' WRTTE(*,*) DIAMETER AT THROAT = '.DIATHROAT,' mm' WRITE(*,*) DIAMETER AT NOZZLE EXIT =',DIAA,' mm' WRTTE(*,*) DIAMETER AT 2 = ',DIA2,' mm" WRTTE(*,*) WRITE(*,*) WRTTE(*,*) WRTTE(*,*) (3) SYSTEM SIMULATION' WRrTE(*,*) THE COOLING CAPACITY =',QEVAP,' W WRITE(*,*) THE HEAT INPUT = \QBOIL,' W WRTTE(*,*) THE PUMP W O R K = ',WPUMP,'W WRTTE(*,*) THE HEAT REJECTED = *,QCOND,' W WRITER,*) COP(E.R. SYSTEM) = ',BETA WRTTE(*,*) COP(CARNOT) = \CARNOT WRITE(*,*) WRTTE(*,*)'' WRrTE(*,20)T0,P0,H0,T3,P3,H3,T4,P4,H4,T5,P5,H5)T6,P6,H6,T7)P7,H7
20
FORMAT(3(/),3X,"POINT TEMPERATURE ,
2 3 4 5 6 7
PRESSURE ENTHALPY",/,
i * * , ! , , ) , * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
,
,/,5X, 0',10X,F4.0)9X,F7.0,6X,F7.2,/, 5X,"3,,10X,F4.0,9X)F7.0,6X,F7.2,/, 5X,'4",10X,F4.0,9X,F7.0,6X,F7.2,/, 5X,"5",10X,F4.0,9X,F7.0,6X,F7.2,/! 5X,"6,,10X,F4.0,9X,F7.0,6X,F7.2,/, 5X,"7",10X,F4.0)9X,F7.0,6X,F7.2)
WRITE(*,*)'' WRITE(*,*)"' WRTTE(*,*) "(4) C R I T I C A L C O N D I T I O N ' WRTTE(*,*)''
291
1
Appendix C
WRTTE(*,*)' WRITE(*,*)'
THE CRITICAL PRESSURE = \PCRJT,' kPa' THE CRITICAL TEMP. (SAT.) = '.TCRTT,' C
WRTTE(*,*)'' WRITE(*,*)'' WRITE(*,*) " D O Y O U WISH TO R U N AGAIN (Y=l)?' READ(*,*) Y WRITE(*,*)'' IF(Y=1) GOTO 1 STOP END
292
Appendix C
p ****************** SUBROUTINES *************************************** p*********************************************************************** C SUBROUTINE TO CALCULATE THE OPTIMUM EJECTOR ENTRAINMENT RATIO C AND HENCE DETERMINE OPTIMUM PERFORMANCE CHARACTERISTICS OF THE C EJECTOR SYSTEM FOR THE REFRIGERATION CYCLE CHOSEN. C KA - Specific heat ratio at point 0 K B - Specific heat ratio at point 4 C C T O A - Stagnation temperature at point 0 C T O B - Stagnation temperature at point 4 C P O A - Stagnation pressure at point 0 C P O B - Stagnation pressure at point 4 C P 0 3 - Stagnation pressure at point 3 C P E V A P - Evaporator pressure P A - Pressure at point 0 C C P B - Pressure at point 4 C E F F D - Diffuser efficiency C E F F N - Nozzle efficiency W - Molecular weight C R - Gas constant C Pl - Pressure at point 1 C C PP11 - Optimum pressure at point 1 C P2 - Pressure at point 2 C A R A T T O - Area ratio of section 2 to throat C E N T R A I N - Entrainment ratio C E N T R A I N M E N T - Optimum entrainment ratio C C B C A R A T I O - Critical speed of sound ratio of point 4 to 0 C C 2 C A R A T I O - Critical speed of sound ratio of point 2 to 0 C M I A - Dimensionless velocity of motive vapour at point 1 C M 1 B - Dimensionless velocity of secondary vapour at point 1 C M 2 - Dimensionless velocity or Mach number at point 2 C X M A X - Maximum entrainment ratio C K 2 - Specific heat ratio at point 2 C T 0 2 T O A R A T I O - Temperature ratio of point 2 to 0 C C O M P R A T I O - Pressure ratio of point 3to4 C B - Component of Eq.(3.11) C M B - Mass flow rate through evaporator M E - Mass flow rate through generator C C 2 - Crirical speed of sound at point 2 C C CA-Critical speed of sound at point 0 C VEL2-Velocity at point 2 C I M A X - D u m m y variable for calculation of maximum entrainment ratio C N - Number of iteration for calculation of entrainment ratio C PART1,PART2,PART3 - Part of Eq.(3.16) PART4.PART5 - Part of Eq.(3.13) C C P A R T 6 - Part of Eq.25 [11] C PART7,PART8,PART9 - Part of Eq.(3.11) ^ _ SUBROUTINE EJECTOR(ME,KA,KB,TOA,TOB,POA,POB,P3,EFFN,EFFD,PEVAP,TCOND, ARATIO,ENTRAINMENr,COMPRATIO,MB)PPll,VEL2,P2,PCRIT,TCRrT)
293
Appendix C
REAL KA,KB,T0AT0B,W,P0A,P0B,R,EFFN3FFD,PEVAP,P,T,Y,MAC1A>IAC1B RF^LPl(200),CBCARATIO,ARATIO,FJSrrRAIN(200)>IlA(200)>IlB(200) R E A L XMAX,PP113NTRAINMENT,K2 ) T02TOARATIO,C2CARATIO,B>12 REALP03,C0MPRATI0,P2P1RATI0,MB,ME,P2,C2,CAVEL2,PCRIT,TCRIT REAL PART1,PART2,PART3,PART4,PART5,PART6,PART7,PART8,PART9,PART10,PART11 INTEGER M A X . N C Constants from [13] WB=120.91 WA=137 R=8.3144*1000./121 TOA=TOA+273.15 TOB=TOB+273.15 1 WRnE(*,*) TiNTER THE AREA RATIO' READ(*,*) ARATIO C WRITE(*,*) "ENTER T H E C O N D E N S E R PRESSURE(KPa)' C READ(*,*) P C O N D C This DO LOOP is to vary Pl from 5kPa to die evaporator C pressure. Tis is so that a maximum entrainment ratio can C be found. N=PEVAP/5. Pl(l)=5. DO 10 I=1,N-1 Pia+l)=PKJ)+5. 10 CONTINUE C These equations are for constant area mixing. [11] C Equation 22 CBCARATIO=SQRT((KB*(KA+1.)*WA*TOB)/((KB+1.)*KA*WB*TOA)) DO 15 I=1,N C Equation 20 MlAa)=(EFFN*(KA+l.)/(KA-l.)*(l.-(Pl(D/POA)**((KA-l.)/KA)))**0.5 C Equation 21 M1B(D=((KB+1.)/(KB-1.)*(1.-(P1(I)/POB)**((KB-1.)/KB)))**0.5 C Equation 18 PARTl=POB/POA*WBAVA*TOAA,OB*MlBa)/MlA(I)*CBCARATIO PART2=((Pl(I)/POB)**(l ./KB))/((Pl(I)/POA)**(l ./KA))
294
Appendix C
PART3=ARATIO*MlA(I)*0>KD/POA)**(l./KA)*((KA+l.)/2.)** 1 (U(KA-1.))-1. ENTRATNa)=PARTl *PART2*PART3 15 CONTINUE C This routine is to find the maximum (optimum) entrainment C ratio and the corresponding value of Pl. XMAX=ENTRAIN(1) IMAX=1 DO 20 I=2JST IF(ENTRAIN(r).GT.XMAX)THEN XMAX=ENTRAINa) IMAX=I END IF 20 CONTINUE FJsTTRAINnVlENT=ENTRAIN(IMAX) WRJTE(*,*) 'ISt ENTRArNMENT=',ENTRAINMENT PR1=P1(IMAX) WRITE(*,*) 'ISt GUESS OF PR1=P1(IMAX)=',P1(IMAX) C100 PRl=PRl+3 C WRITE(*,*) 'PR1 IN THE L00P=',PR1 C Equation 20 MAC1A=(EFFN*(KA+1.)/(KA-1.)*(1.-(PR1/POA)**((KA-1.)/KA)))**0.5 IF(MAC1A.LE.1) G O TO 50 C Equation 21 MAC1B=((KB+1.)/(KB-1.)*(1.-(PR1/POB)**((KB-1.)/KB)))**0.5 C WRJTE(*,*) •KB=,PRl=,POB=',KB,PRl,POB C Equation 18 PARTl=POB/POA*WBAVA*TOA/TOB*MAClB/MAClA*CBCARATIO PART2=((PRl/POB)**(l ./KB))/((PRl/POA)**(l ./KA)) PART3=ARATIO*MAClA*(PRl/POA)**(l./KA)*((KA+l.)/2.)** 1 (l./(KA-l.))-l. ENTRAINMENT=PART1*PART2*PART3 WRITE(*,*)'M1A,M1B,PART1=',MAC1A,MAC1B,PART1 WRITE(*,*) •PART2=,PART3=,,PART2,PART3 WRJTE(*,*) "ENTRAINMENT IN THE LOOP=",ENTRAINMENT C Equation (3.13) PART4=KA/(KA-1.)+KB/(KB-1 .)*ENTRAINMENT*WA/WB PART5=U(KA-1.)+1 ./(KB-1.)*ENTRAINMENT*WA/WB K2=PART4/PART5
295
Appendix C
C
WRITE(*,*) 'K2=',K2
C Equation before (3.15) PART6=KB/(KB-1.)*ENTRAINMENT*WA/WB TO2TOARATIO=(KA/(KA-l.)+PART6*TOB/TOA)/aCA/aCA-l.)+PART6) C WRrTE(*,*) T2TA=',T02TOARATIO C Equation (3.14) W2=(WA*(1+ENTRAI>MENT))/(1+ENTRAINMENT*WA/WB) C2CARATIO=SQRT(K2/(K2+l .)*(KA+1 .)/KA*WA/W2*T02TOARATIO) CA=(2.*KA/(KA+1.)*R*TOA)**0.5 C2=C2CARATIO*CA C Equation 11 PART7=MAC1A+MAC1B*ENTRAINMENT*CBCARATI0 PART8=l./KA*ARATIO*PRl/POA*((KA+l .)/2.)**(KA/(KA-l.)) PART9=(K2+1 .)/K2*(l .+ENTRAINMENT)*C2C ARATIO B=(PART7+PART8)/PART9 WRITER,*)" B=",B D?(B.LE.1.)THEN WRTTE(*,*) "NO EJECTOR AVAILABLE' G O T O 50 ELSE M2=B-(B**2-1.)**0.5 PART10=MAC1A+MAC1B*ENTRAINMENT*CBCARATIOM2*(l.+ENTRAINMENT)*C2CARATIO PARTll=l./KA*ARATIO*PRl/POA*((KA+l.)/2.)**(KA/(KA-l.)) P2P1RATIO=1.+PART10/PART11 P2=PR1*P2P1RATI0
VEL2=M2*C2 C Equation 15 M2=((27(K2+1.)*M2**2)/(1.-(K2-1.)/(K2+1.)*M2**2))**0.5 C P2=P03/((1.+(K2-1.)/2.*(M2**2)*EFFD)**(K2/(K2-1.)))
WRTTE(*,*) "M2=",M2 PCRTT=P2P1RATI0*PR1*((1.+(K2-1.)/2.*(M2**2)*EFFD)**(K2/(K2-1.)))
296
Appendix C
WRTTE(*,*) "PCRIT= ".PCRIT JF(ABS(PCOND-PCRIT).GT.10) G O TO 100 E N D IF P=PCRJT P3=PCRTT CALL SATTEMP(P,T) TCRJT=T WRTTE(*,*)'' WRITE(*,*) ,TCRrT=,,TCRIT," C WRTTE(*,*) TCOND=',TCOND," C WRTTE(*,*)'' WRrTE(*,*) "DO Y O U WISH TO CHANGE THE AREA RATIO READ(*,*) Y WRITE(*,*) IF(Y=l)GOTO 1
C
COMPRATIO=P03/POB
MB=ME/ENTRAINMENT WRTTE(* *) "MB=,,MB,*ME=,,ME PP11=PR1 TOA=TOA-273.15 TOB=TOB-273.15 50 RETURN END
297
Appendix C
P********************************************************************** C SUBROUTINE T O C A L C U L A T E T H E DIMENSIONS O F A N O P T I M U M EJECTOR C This subroutine uses the equations for compressible C isentropic flow. C PO A • Stagnation pressure at point 0 PP11 - Optimum pressure at point 1 C C P T H R O A T - Pressure at throat C P O P - Pressure ratio of point 0 to 1 or point 0 to throat C T O A - Stagnation temperature at 0 C T T H R O A T - Temperature at throat C T O T - Temperature ratio of point 0 to throat or point 0 to 1 C M A C H - Mach number at point 1 C M T H R O A T - Mach number at throat C V T H R O A T - Velocity at throat C V O A - Specific volume at point 0 C R O W O - Density at point 0 C R O W T H R O A T - Density at throat C R O W O R O W - Density ratio of point 0 to throat or point 0 to 1 C K A - Specific heat ratio at point 0 C R - Gas constant ARATIO-Area ratio of section 2 to throat C C M B - Massflowrate through generator A 2 - Area at section 2 C C A - Motive nozzle area at 1 C A T H R O A T - Area at throat C A A - Area ratio of motive nozzle exit to throat C DIA2 - Diameter at section 2 C D I A T H R O A T - Diameter at throat C DIAA - Diameter at motive nozzle exit SUBROUTINE EJECTAREA(MB,ARAnO,POA,PPll,KA,TOA,VOA,DIATHROAT, I DIAA.DIA2) REAL POA,PPl l,POP,KA,MACH,TOA,ROWO,VOA,R REAL VTHROAT,ROWTHROAT R E A L A2,ARATIO,MB,ATHROAT R E A L DIA2 ATAA.DIATHROAT.AA.A R=8.3144*1000./137 C Pressure ratio across nozzle POP=POA/PPll C Inlet temp to nozzle TOA=TOA+273.15 C Inlet density to nozzle ROWO=1./VOA
298
Appendix C
C Mach No. at nozzle exit MACH=SQRT(27(KA-l.)*(POP**((KA-l.)/KA)-l.)) WRITE(*,*) "MACH=",MACH
IF(MACH>1.)THEN C Flow is supersonic C Area ratio of nozzle exit to throat AA=1./MACH*(2./GCA+1.)*(1 .+(KA-1 .)/2.*MACH**2))**(0CA+l .)/(2.*(KA-l.)))
C Density at the throat ROWTHROAT=ROWO*(2./(KA+l .))**(1 ./(KA-1.)) C Velocity at the throat VTHROAT=SQRT(2.*KA*R*TOA/(KA+l.)) ELSE WRITE(*,*) 'SUBSONIC END IF C Area of the throat using conservation of mass. ATHROAT=MB/(ROWTHROAT*VTHROAT) DIATHROAT=(SQRT(4./3.14159*ATHROAT))*1000. C Area at the end of the nozzle A=ATHROAT*AA DIAA=(SQRT(473.14159*A))*1000. C AreaA2 A2=ARATIO*ATHROAT DIA2=(SQRT(4./3.14159*A2))* 1000.
TOA=TOA-273.15 RETURN
END
299
Appendix C
C.4 Derivation of the Governing Equation for COPcarnot for a Vapour Jet Refrigeration System (VJRS)
Assuming the pump work is negligible, then the following relation can be writte for entropy change in a V J R S :
ASg + ASe + ASC = 0 In the above equation S, g, e and c denote entropy, generator, evaporator and condenser,respectively.This equation can be expressed as:
%Qe.Qc=0 Tg Te Tc where Qg and Qe are heat added to the generator and evaporator, respectively and Q c is heat rejected from the condenser. According to the first law of thermodynamics, it can be written:
Qg + Qe = Qc then:
Qg , Qejg+Qejg Tg Te
Tc
, Qe
Tc Tc
or: 1 1_ j l_ =Q8 \Te Tc) \Jc'T8)
G,
Thus, COPcarnot can be defined as follows:
QP Tg-Tc Te COPcarnot- n ~ * ^ T T T K? g c le
300
