ANALYSIS OF SUPERSONIC SHOCK FREE EJECTORS
26
ANALYSIS OF SUPERSONIC SHOCK FREE EJECTORS VIRENDRA KUMAR DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI JANUARY 2018
Transcript of ANALYSIS OF SUPERSONIC SHOCK FREE EJECTORS
ANALYSIS OF SUPERSONIC SHOCK FREE
EJECTORS
VIRENDRA KUMAR
DEPARTMENT OF MECHANICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY DELHI
JANUARY 2018
© Indian Institute of Technology Delhi (IITD), New Delhi, 2018
ANALYSIS OF SUPERSONIC SHOCK FREE
EJECTORS
by
VIRENDRA KUMAR
Department of Mechanical Engineering
Submitted
in fulfillment of the requirements of the degree of Doctor of Philosophy
to the
INDIAN INSTITUTE OF TECHNOLOGY DELHI
NEW DELHI, INDIA
JANUARY 2018
CERTIFICATE
The thesis entitled “Analysis of Supersonic Shock Free Ejectors” being submitted by
Mr. Virendra Kumar to the Indian Institute of Technology Delhi for award of the degree of
Doctor of Philosophy is a record of original bonafide research work carried out by him. He
has worked under my guidance and supervision, and has fulfilled the requirements for the
submission of this thesis, which has attained the standard required for a Ph.D degree of this
institute. The results presented in this thesis have not been submitted elsewhere for the award
of any degree or diploma.
Dr. P.M.V. Subbarao,
Professor,
Department of Mechanical Engineering,
Indian Institute of Technology Delhi
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ACKNOWLEDGEMENTS
During my study at Indian Institute of Technology in the Department of Mechanical
Engineering, I received a lot of support and help from many people, who I would like to thank.
The first is my supervisor Professor P.M.V. Subbarao, who introduced the Ejector system to
me and guided me in the right way from the very first. His effort, encouragement, and guidance
were generous throughout. He created an excellent research environment from which I
benefited. The second person I wish to thank is Dr. Gaurav Singhal, who gave me his generous
expert advice on ejector technology and always suggested required improvements.
I wish to thank Turbo-machinery lab staffs Mr. Raj Kumar and Mr. Rajendra Singh for
their technical assistance over whole period of experimentation and development of test rig
I also would like to express my sincere gratitude to my Government, which provided
me assistantship for part of my time at IIT Delhi.
Finally, I wish to express my sincere thanks to my family and friends, particularly my
elder brother Er. Surendra Kumar and my wife for their encouragement and support to achieve
my goal.
Virendra Kumar
Jan., 2018
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ABSTRACT
The focus of present study is to develop novel design principles for construction of
supersonic ejectors. An ejector is an assembly of primary flow nozzle, secondary flow passage,
mixing section and diffuser section. Current design theories use modification of geometric
profiles of individual components to improve the performance of an ejector. Entrainment ratio
is the most important parameter used for ejector performance. Any design principle is aimed
to enhance entrainment ratio of an ejector used in a given application. Recently physics based
design principle was introduced in this area. This principle is known as Constant Rate of
Momentum Change (CRMC) principle. This is an important step towards development of
efficient ejector designs. However, this principle was used only for the design of diffuser
section and isentropic gas dynamic analysis was used to implement the CRMC theory. In the
present work CRMC principle of design is implemented using frictional gas dynamic analysis
and also used for design of all the components of the ejectors. An exclusive design package is
developed using MATLAB. A typical ejector was designed, fabricated and tested to evaluate
the merits of CRMC theory. CFD analysis was also carried out to validate the gas dynamic
model and also reveal complete supersonic flow physic occurring in various parts of an ejector.
A general purpose ejector test rig was constructed as a part of this research work.
It is also felt that the ultimate phenomena occurring in an ejector is mutual exchange of
enthalpy into kinetic energy and vice versa to entrain a fluid from low pressure reservoir and
push the same into high pressure reservoir or atmosphere. Based on this observation another
physics based theory, namely Constant Rate of Kinetic Energy Change (CRKEC) was
developed and implemented to develop all the parts of an ejector. This CRKEC design was
applied for some selected capacity and complete design was created. This design was used to
Abstract
iii
fabricate a model CRKEC ejector and both experimental and computational investigations
were carried out to understand supersonic fluid flow with shocks. A comparative study of both
CRMC and CRKEC ejectors was carried out in terms of performance and flow physics. Both
the designs are found to be better than conventional models at design point. An off-design
analysis was also carried out test the applicability of these theories in development of field
level ejectors.
It is finally found that these design theories are valid and will have high acceptance in
future industrial use of ejectors. Major outcome of this research include; a physics based one
dimensional gas dynamic design package, A test bench and A customized CFD package in
ANSYS-Fluent 12.0.
सार
वर्तमान अध्ययन का उदेस्य पराध्वननक बेदखलदार के ननमातण के ललए लसद्ाांर्ों को ववकलसर् करना है। एक बेदखलदार प्राथलमक प्रवाह नोजल, माध्यलमक प्रवाह मार्त, लमश्रण अनुभार् और ववसारक
खांड की एक ववधानसभा है। मौजूदा युक्तर् लसद्ाांर्ों ने इजेतटर के प्रदर्तन को बेहर्र बनाने के ललए व्यक्तर्र्र् घटकों के ज्यालमर्ीय आकृनर् मे सांर्ोधन का उपयोर् ककया है। लमश्रण अनुपार् बेदखलदार
प्रदर्तन के ललए सबसे महत्वपूणत मापदण्ड है। ककसी भी लसद्ाांर् का उदे्दश्य ककसी ददए र्ए सांबांध में उपयोर् ककए र्ए बेदखलदार के लमश्रण अनुपार् को बढाने के ललए है। हाल ही में भौनर्की आधाररर्
लसद्ाांर् इस क्षेत्र में पेर् ककया र्या था। इस लसद्ाांर् को र्नर् पररवर्तन (सी आर एम सी) के लसद्ाांर् के रूप
में जाना जार्ा है। यह कुर्ल बेदखलदार डडजाइनों के ववकास की ददर्ा में एक महत्वपूणत कदम है। हालाांकक,
इस लसद्ाांर् का उपयोर् केवल ववसारक खांड के ललए ककया र्या था और सीआरएमसी लसद्ाांर् को लार्ू करन े
के ललए र्ैस र्नर्र्ील ववश्लेषण का उपयोर् ककया र्या था। वर्तमान कायत में सीआरएमसी उपाय के लसद्ाांर्
का उपयोर् घषतण र्ैस र्नर्र्ील ववश्लेषण के उपयोर् म ेऔर बेदखलदार के सभी घटकों के ललए भी उपयोर्
ककया र्या है। मैट लैब का उपयोर् करके एक अनन्य युक्तर् ववकलसर् ककया र्या है। सीआरएमसी लसद्ाांर्
के र्ुणों का मूलयाांकन करने के ललए एक ववलर्ष्ट बेदखलदार का ननमातण, और परीक्षण ककया र्या है। सीएफडी ववश्लेषण का उपयोर् युक्तर् को मान्य करने के ललए भी ककया र्या है और यह बेदखलदार के
ववलभनन् भार्ों में होन ेवाली पूणत पराध्वननक प्रवाह भौनर्क को प्रकट करर्ा है। इस र्ोध कायत के परीक्षण के
ललए एक सामान्य उदे्दश्य बेदखलदार का ननमातण ककया र्या था।
यह भी महसूस ककया जार्ा है, कक बेदखलदार में होन ेवाली पारस्पररक आदान-प्रदान को ऊजात में उत्थान
के और इसके ववपरीर् कम दबाव पैदा करने और उच्च दबाव वायुमांडल में धकेलने के ललए ककया जार्ा है। इस अवलोकन के आधार पर एक अन्य भौनर्की आधाररर् लसद्ाांर्, अथातर् ्काइनेदटक एनजी चेंज (सी आर के ईसी) ववकलसर् की र्ई और इस ेएक बेदखलदार के सभी दहस्सों के ववकास के ललए ववकलसर् ककया र्या । यह सी आर के ईसी डडजाइन लसद्ाांर् कुछ चयननर् क्षमर्ा के ललए लार्ू ककया र्या था और पूरा डडजाइन
बनाया र्या था। इस लसद्ाांर् सी आर के ई सी को बेदखलदार बनाने और पराध्वननक के द्रव प्रवाह, झटके को समझने के ललए प्रयोर्ात्मक और कम्प्यूटेर्नल जाांच ककए र्ए। सी आर एम सी और सी आर के ई सी दोनों के र्ुलनात्मक अध्ययन को प्रदर्तन और प्रवाह भौनर्की के सांदभत में ककया र्या। दोनों युक्तर्
लसद्ाांर् बब ांद ुपर पारांपररक र्ुलना में बेहर्र पाए जार्े हैं। ववकास के इन लसद्ाांर्ों के प्रयोज्यर्ा का एक लसद्ाांर्
बब ांद ुके अलावा ववश्लेषण भी ककया र्या था।
अांर् में यह पाया र्या है कक, ये लसद्ाांर् मान्य हैं और भववष्य के औद्योगर्क उपयोर्ों में उच्च स्वीकृनर् प्रा्र्
होर्ी इस र्ोध में प्रमुख पररणाम र्ालमल हैं; एक भौनर्क ववज्ञान आधाररर् एक आयामी र्ैस र्नर्र्ील
लसद्ाांर्, एक परीक्षण बेंच और ANSYS-Fluent 12.0 में एक अनुकूललर् सीएफडी लसद्ाांर्।
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TABLE OF CONTENTS
Acknowledgments i
Abstract ii-iii
Table of contents iv-xi
List of Figures xii-xvi
List of Tables xvii-xviii
Nomenclature xix-xx
CHAPTER 1 INTRODUCTION 1-9
1.1 Introduction and Motivation 1
1.2 Working principle of ejectors 3
1.3 Brief History of Ejector Development 5
1.4 Ejector Performance 6
1.5 Focus of this Thesis 7
1.6 Organization of Thesis 8
1.7 Conclusions 9
CHAPTER 2 LITERATURE REVIEW 10-33
Table of contents
v
2.1 Introduction 10
2.2 Ejector design approaches 10
2.3 Geometrical parameters and their effect 13
2.3.1 Primary nozzle geometry 13
2.3.2 Mixing chamber geometry 15
2.3.3 Diffuser section geometry 15
2.3.4 Effect of ejector throat (constant area) length and area ratio 16
2.4 Effect of operating parameters 17
2.4.1 Primary flow pressure and temperature 18
2.4.2 Secondary flow pressure and temperature 18
2.4.3 Back/exit pressure 19
2.5 Computational Fluid Dynamic (CFD) Studies on Ejector 19
2.5.1 Turbulence model selection 20
2.5.2 Computational Mesh 20
2.6 Irreversibility in Ejector system 21
2.7 Working fluids 22
2.8 Ejector applications 23
2.8.1 Vacuum ejector 23
Table of contents
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2.8.2 Thermo-compressor 24
2.8.3 Thrust Augmentation 25
2.8.4 Evacuation chute deployment 25
2.8.5 Refrigeration and air-conditioning 25
2.9 Concluding remarks on the past work 28
2.9.1 Work already completed 28
2.9.2 Work partially completed 30
2.9.3 Work not attempted 31
2.10 Objectives of research work 31
2.11 Execution of the research work 32
2.12 Conclusions 33
CHAPTER 3 DESIGN OF EJECTOR SYSTEM USING ONE-
DIMENSIONAL GAS DYNAMIC THEORY 34-58
3.1 Introduction 34
3.2 One-dimensional ejector design theory 34
3.2.1 Assumptions made in one-dimensional theory 36
3.3 The CRMC and CRKEC Ejector Design Theory 36
3.3.1 Design of Primary Flow Nozzle. 37
Table of contents
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3.3.1.1 Computation of Primary Nozzle Throat 37
3.3.1.2 CRMC Nozzle Design Theory 38
3.3.1.3 CRKEC Nozzle Design Theory 39
3.3.2 Mixing Section Design Theory 41
3.3.2.1 Definition of the Virtual Entrance to Mixing Section Design 42
3.3.2.2 Computation of Equilibrium Properties at the Exit of Mixing
Section 42
3.3.2.3 Selection of Length of Mixing section 43
3.3.2.4 CRMC Design of Mixing Section Profile 44
3.3.2.5 CRKEC Mixing Section Design 45
3.3.3 Design of Diffuser Section 46
3.3.3.1 Design of CRMC Diffuser Section 47
3.3.3.2 Design of CRKEC Diffuser Section 48
3.4 Method of Solution and Computation of Geometrical Details 49
3.4.1 Selection of Design conditions. 50
3.4.2 Computation of Nozzle Section Profile. 51
3.4.2.1 Computation of Converging Section Profile. 52
3.4.2.1 Computation of Diverging Section Profile. 53
3.4.2 Mixing Section. 54
Table of contents
viii
3.4.3 Diffuser Section. 56
3.5 Conclusions 58
CHAPTER 4 COMPUTATIONAL FLUID DYNAMIC (CFD)
ANALYSIS 59-76
4.1 Introduction 59
4.2 Governing equations and turbulence model 60
4.3 Turbulence modeling 61
4.4 CFD Analysis 62
4.4.1 Boundary conditions and convergence criterion 63
4.4.2 Grid independency study 65
4.4.3 Selection of Turbulence model 67
4.5 Detailed Validation of CRMC/CRKED Gas Dynamics Models 70
4.5.1 Validation of Nozzle Models 70
4.5.2 Combined Validation of CRMC and CRKEC Mixing Section and
Diffuser Models 72
4.6 Conclusions 75
CHAPTER 5 DEVELOPMENT AND COMMISSIONING OF
EJECTOR TEST RIG 77-94
Table of contents
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5.1 Introduction 77
5.2 Design and Manufacturing of Test Rig. 77
5.2.1 Nozzle manufacturing 78
5.2.2 Mixing section manufacturing 80
5.2.3 Diffuser section manufacturing 82
5.2.4 Suction chamber manufacturing 84
5.2.5 Nozzle connecting pipe 85
5.3 Ejector assembly 86
5.4 Ejector Test Rig Accessories 87
5.4.1 The primary and secondary flow storage tank 87
5.4.2 The primary and secondary flow measuring instruments 88
5.4.3 Temperature Measurement 88
5.4.4 Pressure scanner panel 89
5.5 The Ejector Test Rig 89
5.6 Commissioning of the Test Rig 91
5.7 Experimental Procedure 91
5.8 Experimental Test Rig Troubleshooting 92
5.9 Uncertainty analysis 93
Table of contents
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5.10 Conclusions 93
CHAPTER 6 ANALYSIS OF FLOW PHYSICS IN EJECTORS
USING EXPERIMENTAL AND
COMPUTATIONAL RESULTS 95-117
6.1 Introduction 95
6.2 Description of Performance and Wall Static Pressure Profiles
at Design Conditions 95
6.2.1 Entrainment ratio (ω) 96
6.2.2 Axial Variation of Static Pressure 97
6.3 Description of Performance and Wall Static Pressure Profiles at off-Design. 98
6.3.1 Effect of Nozzle Exit Position (NXP) on Ejector Performance 98
6.3.2 Effect of Primary Flow Total Pressure on Ejector Performance 105
6.3.3 Effect of Secondary Flow Total Pressure on Ejector Performance 111
6.4 Conclusions 116
CHAPTER 7 SUMMARY, CONCLUSIONS AND
RECOMMENDATIONS 118-125
7.1 Summary 118
7.2 Conclusions 122
7.3 Recommendations for scope of future work 124
Table of contents
xi
REFERENCES 126-136
APPENDICES 137-146
Appendix A. CRMC and CRKEC nozzle, mixing and diffuser section dimensions 137
Appendix B. CFD code setup 146
LIST OF PUBLICATIONS 156
BRIEF BIO-DATA OF THE AUTHOR 157
xii
LIST OF FIGURES
Description of Figures Page
no.
Figure 1.1. A schematic diagram of ejector system 3
Figure 1.2. Henri Giffard’s ejector from 1864 with integrated spindle valve for
control of motive flow rate
5
Figure 2.1. The Constant Area Mixing Ejector 11
Figure 2.2. The Constant Pressures Mixing Ejector 11
Figure 2.3. The flow characteristic in the ejector 12
Figure 2.4. The variation in passage radius with axial distance through conventional
and CRMC diffusers
16
Figure 2.5. Cut-view of the single vacuum ejector 24
Figure 2.6. Thermo-compressor 24
Figure 2.7. Steam jet refrigeration cycle 26
Figure. 2.9. CRMC and CRKEC ejector study execution flow chart 32
Figure 3.1. Discrete Modeling of Convergent-divergent (CD) nozzle 37
Figure 3.2. Mixing section 41
Figure 3.3. Diffuser section 47
Figure 3.4. Design flow chart for CRMC and CRKEC nozzles 51
Figure 3.5. Execution flow chart for the 1D gas dynamic theory for CRMC and
CRKEC mixing section
54
Figure 3.6. Execution flow chart for the 1D gas dynamic theory for CRMC and
CRKEC mixing section
56
Figure 3.7. CRMC ejector profile 58
List of Figures
xiii
Figure 3.8. CRKEC ejector profile 58
Figure 4.1. CRMC ejector computational domain with mesh 63
Figure 4.2. CRKEC ejector computational domain with mesh 63
Figure 4.3. Boundary conditions used in computational domain 64
Figure 4.4. Mach number contours for CRMC ejector for various turbulence models 68
Figure 4.5. Mach number contours of CRKEC ejector for various turbulence models 69
Figure 4.6. Predicted Mach number contour within the mixing section 70
Figure 4.7. The Mach number variation along CRMC and CRKEC nozzles 71
Figure 4.8. Variation of ratio of static and total pressure along the CRMC and
CRKEC nozzles
71
Figure 4.9. Static temperature variation along CRMC and CRKEC nozzles 72
Figure 4.10. Mach number variation along CRMC and CRKEC mixing and diffuser
section
73
Figure 4.11. Static pressure variation along CRMC and CRKEC mixing and diffuser
sections
74
Figure 4.12. Total pressure variation along CRMC and CRKEC mixing and diffuser
sections
74
Figure 4.13. Static temperature variation along CRMC and CRKEC mixing and
diffuser section
75
Figure 5.1. Profile of the cutting tool and photograph of the CRMC nozzle section
with CAD model
79
Figure 5.2. Profile of the cutting tool and photograph of the CRKEC nozzle section
with CAD model
80
Figure 5.3. Profile of the cutting tool and photograph of the CRMC mixing section
with CAD model
81
List of Figures
xiv
Figure 5.4. Profile of the cutting tool and photograph of the CRKEC mixing section
with CAD model
82
Figure 5.5. Profile of the cutting tool and photograph of the CRMC diffuser section
with CAD model
83
Figure 5.6. Profile of the cutting tool and photograph of the CRKEC diffuser section
with CAD model
84
Figure 5.7. Drawing of suction chamber and photograph 85
Figure 5.8. Nozzle connecting pipe 85
Figure 5.9. Schematic diagram of ejector assembly 86
Figure 5.10. Photograph of ejector assembly 86
Figure 5.11. The primary and secondary flow storage tank 87
Figure 5.12. The primary and secondary flow measuring instruments 88
Figure 5.13. Pressure scanner panel 89
Figure 5.14. Schematic diagram of a CRMC/CRKEC ejector test rig 90
Figure 5.15. Photograph of Experimental Test Rig 90
Figure 6.1. Entrainment ratio (ω) at on-design CRMC ejector 96
Figure 6.2. Entrainment ratio (ω) at on-design CRKEC ejector 96
Figure 6.3. Axial variation of wall static pressure along CRMC mixing and diffuser
sections
97
Figure 6.4. Axial variation of wall static pressure along CRKEC mixing and diffuser
sections
98
Figure 6.5. Effect of NXP on entrainment ratio (ω): Experimental Results 99
Figure 6.6. Effect of NXP on entrainment ratio (ω): CFD Results 100
Figure 6.7. CFD predictions of Mach number contours in CRMC ejector at various
NXP
101
List of Figures
xv
Figure 6.8. CFD predictions of Mach number contours in CRMC ejector at various
NXP
102
Figure 6.9a. Experimentally measured static pressure variation along CRMC mixing
and diffuser section at various NXP
103
Figure 6.9b. CFD predicted static wall pressure variation along CRMC mixing and
diffuser sections at various NXP
104
Figure 6.10a. Experimentally measured static pressure variation along CRKEC
mixing and diffusers section at various NXP
104
Figure 6.10b. CFD static wall pressure variation along CRKEC mixing and diffuser
sections at various NXP
105
Figures 6.11. Effect of primary flow total pressure on entrainment ratio at NXP=0:
Experimental Results
106
Figures 6.12. Effect of primary flow total pressure on entrainment ratio at NXP=0:
CFD Results
106
Figure 6.13. Effect of primary flow total pressure on experimentally calculated
entrainment ratio with various NXP values
107
Figure 6.14. Effects of primary flow total pressures on CFD predicted entrainment
ratio at various NXP values
107
Figure 6.15a. Effect of primary flow total pressures on wall static pressure variation:
CRMC ejector: Experimental Results
108
Figure 6.15b. Effect of primary flow total pressures on wall static pressure variation:
CRMC ejector: Computational results
109
Figure 6.16a. Effect of primary flow total pressures on wall static pressure variation:
CRKEC ejector: Experimental results
109
List of Figures
xvi
Figure 6.16b. Effect of primary flow total pressures on wall static pressure variation:
CRKEC ejector: Computational results
110
Figure 6.17. Mach number contours for CRMC ejectors at various primary flow total
pressures
110
Figure 6.18. Mach number contours for CRKEC ejectors at various primary flow
total pressures
111
Figure 6.19. Variation in entrainment ratio (ω) with secondary flow total pressure 112
Figure 6.20. Variation of entrainment ratio with various secondary flow total
pressure and NXP
113
Figure 6.21. Mach number contours along CRMC mixing sections at various
secondary flow total pressure
113
Figure 6.22. Mach number contours along CRKEC mixing sections at various
secondary flow total pressure
114
Figure 6.23a. Experimental static pressure variations along CRMC mixing and
diffuser sections with secondary flow total pressures
115
Figure 6.23b. CFD wall static pressure variations along CRMC mixing and diffuser
sections with secondary flow total pressures
115
Figure 6.24a. Experimental static pressure variations along CRKEC mixing and
diffuser sections with secondary flow total pressures
116
Figure 6.24b. CFD wall static pressure variations along CRKEC mixing and diffuser
sections with secondary flow total pressures
116
xvii
LIST OF TABLES
Table description
Page
no.
Table. 3.1. Input parameter for computation of CRMC and CRKEC geometry 50
Table 3.2. Profiles and flow parameters of converging section for various values
for 𝜆𝑛,𝑐/∅𝑛,𝑐 .
52
Table 3.3. Profiles and flow parameters of diverging section for various values for
𝜆𝑛,𝑑/∅𝑛,𝑑
53
Table 3.4. The specifications of Selected Designs of CRMC & CRKEC nozzles 54
Table 3.5. Mixing section flow parameters for various 𝜆𝑚/∅𝑚 55
Table 3.6. The specification of selected designs of CRMC and CRKEC mixing
section
55
Table 3.7. Diffuser section flow parameters for various 𝜆𝑑/∅𝑑 57
Table 3.8. The specification of CRMC and CRKEC diffuser section 57
Table 4.1. The complete mesh used for CRMC ejector system 66
Table 4.2. The complete mesh used for CRKEC ejector system 66
Table 4.3. Selection of Turbulence model for CRMC ejector 67
Table 4.4. Selection of Turbulent model for CRKEC ejector 68
Table 7.1. Effect of NXP on entrainment ratio (ω) for CRMC ejector at on/off
design conditions
120
Table 7.2. Effect of NXP on entrainment ratio (ω) for CRKEC ejector at on/off
design conditions
121
List of Tables
xviii
Table 7.3. Effect of primary flow total pressure on entrainment ratio (ω) at on/off
design operating conditions
121
Table 7.4. Effect of secondary flow total pressure on entrainment ratio at on/off
design operating conditions
122
xix
NOMENCLATURE
Q heat (kJ/kg)
h enthalpy (kJ/kg. K)
A area (m2)
V, v velocity (ms-1)
D diameter (m)
L Length
𝑀∙ the momentum of the stream (kg m s-1)
M Mach number (-)
m mass flow rate (kg s-1)
P static pressure (Pa)
ω entrainment ratio )/( m pms
T temperature (K)
K wall roughness(µm)
x axial distance (m)
R individual gas constant (J kg-1 K-1)=287
μ dynamic viscosity (N-s m-2)
λ CRMC constant (kg/s2)
φ CRKEC constant (kg.m/s3)
K.E. Kinetic Energy
ρ density (kg m-3)
γ ratio of specific heat values(-)= 1.4
f Fanning coefficient of friction
θ divergence angle
t Time (s), throat
Re Reynolds number
k Turbulence kinetic energy (J.m3/kg)
Gk generation of turbulence kinetic energy due to the mean
velocity gradients (J)
Gb generation of turbulence kinetic energy due to buoyancy
force (J)
ε rate of dissipation rate (J.m3/kg.s)
YM contribution of the fluctuating dilation in compressible
turbulence to the overall dissipation (J)
C1ε model constant=1.44
C2ε model constant=1.92
σk turbulent Prandtl number for k = 1.0
σε turbulent Prandtl number for ε = 1.3
Sk user defined source term for k (J)
Nomenclature
xx
Sε user-defined source term for ε (J)
Subscripts
n nozzle
d diffuser, divergent
p primary flow, pump
s secondary flow, suction
o stagnation condition
x the axial distance
mix primary mass flow rate(𝑚𝑝∙ )+secondary mass flow rate(𝑚𝑠̇ )
e exit, evaporator
t Throat, turbulence
b boiler
c condenser, convergent
f fluid
g vapor
r recovery
