# Gas ejector modeling for design and analysis

x TABLE OF CONTENTS

Page ABSTRACT.............................................................................................................. iii DEDICATION.......................................................................................................... v ACKNOWLEDGEMENTS...................................................................................... vi NOMENCLATURE.................................................................................................. vii TABLE OF CONTENTS.......................................................................................... x LIST OF FIGURES................................................................................................... xiii LIST OF TABLES.................................................................................................... xix CHAPTER I INTRODUCTION AND LITERATURE REVIEW............................ 1 Introduction.................................................................................... 1 Gas Ejectors Literature Review...................................................... 5 Dissertation Organization............................................................... 10 II THEORETICAL BACKGROUND AND NOZZLES......................... 12 Introduction.................................................................................... 12 Theoretical Background................................................................. 13 Primary Nozzle of Ejector.............................................................. 23 Summary........................................................................................ 25 III GAS EJECTOR 1-D ANALYTICAL MODELS................................ 26 Introduction.................................................................................... 26 Supersonic Primary Nozzle............................................................ 27 Constant-Area Mixing Model........................................................ 30 Constant-Pressure Mixing Model................................................... 41 Subsonic Diffuser........................................................................... 48 Parametric Analysis........................................................................ 48 Ejector Design Using 1-D Models................................................. 70

xi CHAPTER Page Summary and Discussion.............................................................. 74 IV SINGLE PHASE GAS EJECTOR CFD SIMULATIONS.................. 77 Introduction.................................................................................... 77 FLUENT and Simulation Aspects.................................................. 78 Benchmark..................................................................................... 80 Ejector Optimization...................................................................... 85 Flow Field Details of Gas Ejector.................................................. 92 Ejector Design for Scalable PEM Fuel System.............................. 98 Conclusions.................................................................................... 104 V A NOVEL GENERALIZED EJECTOR MODEL.............................. 106 Introduction.................................................................................... 106 Generalized Ejector Model............................................................. 107 General Features of Generalized Ejector Model............................ 128 Optimal Ejector Performance......................................................... 134 Links among Various Ejector Models............................................ 138 Limitations on Ejector Design and Operation................................ 143 Conclusions.................................................................................... 153 VI EJECTOR WORKING WITH TWO-PHASE FLOW......................... 155 Introduction.................................................................................... 155 Gas-Liquid Mixture Model............................................................ 156 Parametric Analysis of Mixture Properties.................................... 159 Ejector Working with Two-Phase Mixture.................................... 161 Conclusions.................................................................................... 173 VII SUMMARY AND CONCLUSIONS................................................... 174 Summary........................................................................................ 174 Conclusions.................................................................................... 176 Recommendations.......................................................................... 179 REFERENCES.......................................................................................................... 181 APPENDIX A........................................................................................................... 185 APPENDIX B........................................................................................................... 195

xii Page APPENDIX C........................................................................................................... 212 APPENDIX D........................................................................................................... 236 VITA......................................................................................................................... 250

xiii LIST OF FIGURES

Page Fig. 1.1 Cross sectional view of a typical liquid jet pump............................. 2

Fig. 1.2 Cross sectional view of a typical gas ejector..................................... 2

Fig. 1.3 Schematic of the PEM fuel cell system test bed............................... 4

Fig. 1.4 Three-dimensional ejector solution surface, Addy & Dutton et al... 8

Fig. 2.1 Control volume for 1-D flow............................................................ 13

Fig. 2.2 Pressure profile for isentropic flow in a converging-diverging nozzle............................................................. 17

Fig. 2.3 Maximum mass flow rate vs. stagnation pressure for various throat diameters, with air at 0 300 T K = .......................... 20

Fig. 2.4 Nozzle and diffuser shapes for subsonic flow and supersonic flow........................................................................... 22

Fig. 2.5 Property ratios at the nozzle exit for given Mach numbers............... 24

Fig. 3.1 Supersonic primary nozzle................................................................ 28

Fig. 3.2 Schematic of constant-area ejector model......................................... 31

Fig. 3.3 Control volume for derivation of constant-area mixing model......... 32

Fig. 3.4 Control volume for analysis of initial interaction region.................. 38

Fig. 3.5 Constant-pressure ejector flow model............................................... 41

Fig. 3.6 Control volume of constant-pressure mixing chamber..................... 43

Fig. 3.7 Area ratio of nozzle exit to its throat vs. 1p M for various n η .......... 50

Fig. 3.8 Pressure ratio of nozzle exit to its throat vs. 1p M for various n η ...................................................................... 51

xiv Page Fig. 3.9 1p M calculated by using least square data fit................................... 53

Fig. 3.10 Constant-area ejector ER vs. area ratio for various primary nozzles............................................................... 55

Fig. 3.11 Constant-area ejector ER vs. diameter ratio for various primary nozzles............................................................... 56

Fig. 3.12 Constant-area ejector ER curve for various 1 P values...................... 57

Fig. 3.13 Constant-area ejector ER vs. AR for various 0 0 s p T T ...................... 58

Fig. 3.14 Constant-area ejector ER vs. DR for various 0 0 s p T T ...................... 59

Fig. 3.15 Impact of specific heat ratio on the constant-area ejector performance........................................................................... 60

Fig. 3.16 Relationship of 1 1 s p A A with constant-area ejector ER..................... 61

Fig. 3.17 Relationship of mixed flow Mach number with constant-area ejector ER.................................................................... 62

Fig. 3.18 Plot of CR vs. ER in semi-log scale for constant-area ejector.......... 63

Fig. 3.19 Plot of CR vs. ER in higher ER region for constant-area ejector...... 64

Fig. 3.20 Relationship of ER with AR for constant-pressure ejector with various 1 P ...................................................................... 66

Fig. 3.21 ER curves for constant-area ejector and constant-pressure ejector................................................................... 67

Fig. 3.22 Plot of 2m M against 2m t A A for constant-pressure ejector............... 68

Fig. 3.23 Relationship between CR and ER for constant-pressure ejector....... 69

Fig. 3.24 Plot of CR against 2m M for constant-pressure ejector...................... 70

Fig. 3.25 Typical ejector configuration used by ESDUpac A9242.................. 72

xv Page Fig. 3.26 Ejector design procedure for using ESDUpac A9242....................... 73

Fig. 4.1 2-D axis-symmetric ejector design.................................................... 79

Fig. 4.2 Schematic of ejector performance test system.................................. 80

Fig. 4.3 ITP ejector structure.......................................................................... 81

Fig. 4.4 ITP ejector performance curve.......................................................... 82

Fig. 4.5 Ejector geometry configuration simulated by FLUENT................... 83

Fig. 4.6 Primary flow rate vs. gas supply pressure......................................... 83

Fig. 4.7 ER vs. driving pressure..................................................................... 85

Fig. 4.8 Ejector geometry used for optimization............................................ 86

Fig. 4.9 Ejector performance against s L ........................................................ 88

Fig. 4.10 Influence of diameter ratio m t D D on ejector performance............ 89

Fig. 4.11 Influence of mixing tube length on ejector performance.................. 90

Fig. 4.12 Influence of diffuser expansion angle on ejector performance......... 91

Fig. 4.13 Contours of static pressure in gas ejector.......................................... 94

Fig. 4.14 Contours of axial velocity in gas ejector........................................... 94

Fig. 4.15 Contours of static temperature in gas ejector.................................... 95

Fig. 4.16 Plot of gas density along the centerline of gas ejector...................... 96

Fig. 4.17 Plot of Mach number along the centerline of gas ejector.................. 97

Fig. 4.18 Plot of static temperature along the centerline of gas ejector............ 97

Fig. 4.19 Schematic of PEM fuel system......................................................... 98

Fig. 4.20 ER sensitivity.................................................................................... 99

xvi Page Fig. 4.21 Primary flow rate vs. gas supply pressure......................................... 100

Fig. 4.22 Interpolation of ejector performance curve....................................... 101

Fig. 4.23 Geometry details of designed gas ejector.......................................... 102

Fig. 4.24 Performance curve of designed gas ejector....................................... 103

Fig. 5.1 Schematic of generalized ejector model............................................ 108

Fig. 5.2 Control volume of mixing chamber for generalized ejector model.................................................................. 114

Fig. 5.3 Plot of total entropy change with respect to µ ................................. 124

Fig. 5.4 Plots of the total entropy change and its two components................ 130

Fig. 5.5 Plot of entrainment ratio against variable µ .................................... 132

Fig. 5.6 Plot of the mixing chamber exit Mach number against µ ................ 133

Fig. 5.7 Plot of τ against µ .......................................................................... 133

Fig. 5.8 Plots of 1p M , 1 s M and normalized ER............................................. 136

Fig. 5.9 Plots of optimal ER against κ for various θ ................................... 137

Fig. 5.10 Plots of optimal CR against κ for various θ ................................... 138

Fig. 5.11 Plots of τ against µ for various θ .................................................. 140

Fig. 5.12 Plots of τ against µ for various κ .................................................. 141

Fig. 5.13 Plots of ER against µ for various θ ................................................ 142

Fig. 5.14 Upper limits of ER for ejectors......................................................... 145

Fig. 5.15 s∆ surface for ejectors working at the optimal point....................... 146

Fig. 5.16 Top-downward view of s ∆ surface

xvii Page intercepted by 0s ∆ = plane.............................................................. 147

Fig. 5.17 s ∆ surface for constant-area mixing ejector..................................... 149

Fig. 5.18 s ∆ surface of ejectors with various θ .............................................. 150

Fig. 5.19 s∆ surface of ejectors with various θ , intercepted by 0 s ∆ = plane.............................................................. 150

Fig. 5.20 Performance surface plotted against ejector boundary pressures................................................................ 152

Fig. 5.21 Operable range of pressure ratio of outlet flow to secondary supply.................................................... 153

Fig. 6.1 Plot of mixture gas constant against the liquid quality x ................. 160

Fig. 6.2 Plot of mixture specific heat ratio against the liquid quality x ........ 161

Fig. 6.3 3-D plot of operational 0b s P P against 0 0p s P P and x , two-phase mixture supplied as the primary stream........................... 163

Fig. 6.4 2-D plot of operational 0b s P P against 0 0p s P P for various x , two-phase mixture supplied as the primary stream........................... 165

Fig. 6.5 Optimal ER surface against κ and x , two-phase mixture supplied as the primary stream........................... 166

Fig. 6.6 Optimal ER surface against θ and x , two-phase mixture supplied as the primary stream........................... 166

Fig. 6.7 3-D plot of 0b s P P vs. 0 0p s P P and x , mixture supplied at suction port........................................................ 168

Fig. 6.8 2-D plot of 0b s P P vs. 0 0p s P P for various x , mixture supplied at suction port........................................................ 168

Fig. 6.9 Optimal ER surface vs. κ and x , mixture supplied at suction port........................................................ 169

xviii Page Fig. 6.10 Optimal ER surface vs. θ and x , mixture supplied at suction port........................................................ 169

Fig. 6.11 Top-down view of s ∆ surface intercepted by 0 s ∆ = plane, mixture supplied as primary stream, 1 θ = ........................................ 171

Fig. 6.12 Top-down view of s ∆ surface intercepted by 0s ∆ = plane, mixture supplied as primary stream, 100 κ = ................................... 172

Fig. 6.13 Top-down view of s ∆ surface intercepted by 0 s ∆ = plane, mixture supplied at suction port, 1 θ = ............................................. 172

Fig. 6.14 Top-down view of s ∆ surface intercepted by 0 s ∆ = plane, mixture supplied at suction port, 100 κ = ......................................... 173

xix LIST OF TABLES

Page Table 4.1 Geometry parameters of gas ejector with D t =0.005”........................ 102 Table 4.2 PEM fuel system operating parameters............................................. 104 Table 5.1 Geometry and operating parameters used for the new model investigation.............................................................. 128

1 CHAPTER I INTRODUCTION AND LITERATURE REVIEW

INTRODUCTION

Ejectors are devices used to induce a secondary fluid by momentum and energy transfer from a high velocity primary jet. Ejectors can be operated with incompressible fluids (liquids), and in this application are normally referred to as jet pumps or eductors. On the other hand when ejectors are operated with compressible fluids (gases and vapors) the terms ejector and injector are generally employed. A major difference between the two, besides the working fluid states, is the supersonic, choked flow nozzle of the gas ejector system. The supersonic approach allows a greater conversion of primary fluid energy to secondary fluid pressure head increase. However, this occurs with the penalty of considerable thermodynamic complexity in the mixing and diffusion sections. Fig. 1.1 and Fig. 1.2 are typical cross sectional views of liquid jet pumps and gas ejectors.

The working process in a liquid jet pump or in a gas ejector is the same. A high-pressure fluid with very low velocity at the primary inlet is accelerated to high velocity jet through a converging nozzle for the liquid jet pump or a converging-diverging supersonic nozzle for the gas ejector. The supply pressure at the inlet is partly converted

______ This dissertation follows the style of Journal of Fluids Engineering.

2 to be the jet momentum at the nozzle exit according to the Bernoulli equation. The high velocity, low static pressure primary jet induces a secondary flow from the suction port and accelerates it in the direction of the driving jet. The two streams then combine in the mixing section, and ideally the process is complete by the end of this section. A diffuser is usually installed at mixing chamber exit to lift the static pressure of mixed flow.

Fig. 1.1 Cross sectional view of a typical liquid jet pump

Fig. 1.2 Cross sectional view of a typical gas ejector

3 Ejectors have simple geometry and no moving parts. Their operation does not require electrical or mechanical shaft energy input. This greatly reduces equipment mass and increases reliability. Ejectors have found wide use in power plant, aerospace, propulsion and refrigeration application because of the above mentioned features. Liquid Jet pumps have very good resistance to cavitation compared to other types of pumps. Thus, jet pumping may be an attractive method for waste heat transport in new generations of spacecraft. The Inter-phase Transport Phenomenon (ITP) Laboratory of Texas A&M University [1] proposed a Rankine Cycle System, in which a liquid jet pump is used to increase pressure recovery for power generation systems of future spacecraft. However, the research and development of jet pump technology for incompressible fluids (liquid) is more mature than that for ejectors used for compressible fluids (gases). This dissertation focuses on gas ejector and discussion on liquid jet pump is limited.

Gas ejectors are also found in high-altitude aircraft, thrust augmentation of aircraft and hydrogen fuel cells, etc. Lee et al. [2] studied the ejector systems used for a hydrogen fuel cell. The Center of Space Power (CSP) at Texas A&M University is designing a gas-liquid separation system for a Proton Exchange Membrane (PEM) fuel cell. In order to pump the exhaust from a PEM fuel cell, a two-phase gas ejector is integrated into the system. Fig. 1.3 is the schematic of PEM fuel cell system test bed. A gas tank supplies high pressure gas to the ejector primary nozzle inlet and the PEM fuel cell exhaust line is connected to the ejector secondary inlet (suction) port. PEM cell exhaust is sucked into the ejector and mixed with the motive stream; the mixed flow then is directed into the

4 vortex separator. Gas and liquid are split by the separator; liquid is discharged and the gas is directed to the PEM inlet port.

gas supply PEM cell gas ejector Liquid discharge vortex separator gas outlet gas supply PEM cell gas ejector Liquid discharge vortex separator gas outlet

Fig. 1.3 Schematic of the PEM fuel cell system test bed

The goal of the dissertation is to perform a comprehensive theoretical study of ejector energy and momentum transfer, to develop a universal ejector model which embodies all existing ejector models and provide answers to the problems left unsolved by these existing models, and to investigate the feasibility as well as limitations of using gas ejector to work for gas-liquid two-phase flow.

The theoretical background of ejector working mechanism, the fundamental concepts and governing equations of aerodynamics are introduced as the first step. Equations of the constant-pressure ejector model and constant-area ejector model are then derived in great details. Parametric analysis is performed for these 1-D analytical models for gas ejector with single-phase flow. The general gas ejector design procedures and steps

5 using the 1-D analytical models are discussed. ESDUpac A9242, a computer program with comprehensive 1-D analytical models for gas ejector design, is introduced briefly.

In order to gain a better understanding of the mixing process in the ejector, the commercial Computational Fluid Dynamics (CFD) software FLUENT is used to model the gas ejector. It will be shown that there is good agreement between the CFD simulation results and experimental data. Based on this benchmark, FLUENT is employed to optimize the ejector geometric configuration.

Confidence in the ejector working mechanism is gained from the above-mentioned studies and simulations. A generalized, comprehensive ejector model is successfully developed based on the understanding of momentum and energy transfer between the primary and secondary streams. This novel model provides answers to the unsolved problems of existing models. Since the motivation of this research is to design a working ejector for the PEM fuel system, the new model was applied to study the limitations of design and operation of ejector working with a gas-liquid two-phase mixture.

GAS EJECTORS LITERATURE REVIEW

The ejector was introduced as an engineering device in the early 20th century. At the same time, researchers started to investigate its working mechanism. In a paper published in 1950, Keenan and Neumann [ 3 ] presented the first comprehensive

6 theoretical and experimental analysis of the ejector problem. The constant-pressure mixing model and the constant-area mixing model developed by Keenan and Neumann [3] became the basis of ejector design and performance analysis since then. Based on these 1-D analytical approaches, much research effort has been devoted to the improvement of ejector design methods and hundreds of papers relating to supersonic ejectors are published. In a review carried out by Bonnington and King [4], 413 ejector references about ejector dating prior to 1976 were cited; 1/3 of them are about gas ejectors. More recently, a comprehensive review of published research related to the design and application of supersonic ejectors is given by D.W. Sun and I.W. Eames (1995) [5].

In the derivation of constant-pressure mixing model, Keenan and Neumann [3] assumed that the fluids of the primary flow and secondary flow were the same gas. Keenan and Neumann [3] also neglected nozzle and diffuser efficiencies and frictional effects. This method is not very accurate, but it avoids the complicated expressions of thermodynamic properties for mixed flow as well as the use of experimentally determined constants. In general, experimental values of pressure-rise and flow entrainment are found to be approximately 85% of the calculated values.

DeFrate and Hoerl [6] modified the constant-pressure mixing model by taking the ideal gas law with molecular weight into their derivation, since gas constant is determined by the molecular weight and universal gas constant by the relationship of R R W= .

7 Different heat ratios of the primary and secondary fluids were also included in their method. Based on constant-pressure mixing and in terms of enthalpy, Emanuel [7] developed a simple analytical model for optimizing the steady-state performance of an ejector. Rice and Dandachi [8] derived equations for steam ejectors to predict the primary flow rate by including the friction and mixing losses which have usually been neglected. Huang et al. [9] assumed constant-pressure mixing to occur inside the constant-area section of the ejector with the entrained flow in a choking condition. Huang et al. [9] determined various loss factor coefficients in their approach by matching the test data with the analytical results.

Another 1-D analytical model, the constant-area mixing ejector, was also considered by Keenan and Neumann [3,10]. The mixing in a constant-area ejector operates in two distinct regimes, depending on whether the flow characteristics of the ejector are dependent or independent of the back pressure b P . Fabri and Siestrunck [11] introduced the concept of “aerodynamic throat” and included the primary nozzle wall thickness in their constant-area mixing model. They referred to the b P -dependent regime as the “mixed” regime (MR), and to the b P -independent regime as the “supersonic” regime (SR) and/or the “saturated-supersonic” regime (SSR). For a better understanding of the mixing mechanism, Addy and Chow [12,13], Dutton and Carroll [14] showed these regimes on a three-dimensional solution surface as shown in Fig. 1.4. This solution surface is based on three variables: primary-to-secondary stagnation pressure ratio 0 0 / p s P P , the static-to-secondary stagnation pressure-ratio 0 / b s P P and entrainment ratio

8 ω . A constant-area ejector is constrained to operate at some point on the solution surface. The entrainment ratio ω is predicted by the other two variables, 0 0 / p s P P and 0 / b s P P .

Fig. 1.4 Three-dimensional ejector solution surface, Addy & Dutton et al. [12, 13, 14]

The conditions required by 1-D models are not as restrictive as they may appear. As long as the flow can be simplified to 1-D or suitable mean values can be used, these models can be applied for non-uniform flow conditions. However, for turbulent flow through very short sections this is not the case. A 1-D analysis can be utilized for the purposes of engineering design because, despite its relative simplicity, it has been shown to give

9 consistent and reasonably accurate results within its limitations. Significant progress has been made for these widely used analytical models since 1950, but some problems are still unsolved. There is no way to determine the optimum shape of the mixing section for constant-pressure ejector. Also, nobody has yet established a definite link between the performance of constant-area and constant-pressure ejectors. It is highly desired to develop a new ejector model which would unify all ejector models and provide answers for the unsolved problems of existing models.

Many researchers have used various 2-D models in order to gain a better understanding of the flow process in ejectors, particularly within the mixing section. Goff and Coogan [15] were the first to consider the two-dimensional aspects of ejector performance. Mikhail [16] assumed various velocity profiles at each stage of the mixing process in a constant-area tube. The predictions of 2-D analytical models could be more accurate than those of 1-D models, but a number of disadvantages are associated with them. The 2-D models are inherently more complicated, and generally require more specialist knowledge to implement. Fruthermore, there are usually empirical coefficients or constants to be determined from experiment, and they are usually applicable only for certain particular ejectors.

Hedges and Hill [17] took a first step into Computational Fluid Dynamics (CFD) and developed a finite-difference scheme to model the flow process within a gas ejector. Gilbert and Hill [18] further refined this method and their theoretical results were

10 generally in good agreement with experimental results. Commercial CFD software programs now widely available enable researchers to look at the complicated supersonic flow and mixing problems in the gas phase. Neve [19] studied the flow in the diffuser section of an ejector. Riffat et al. [20] used CFD to analyze the performance of ejectors. A commercial code, FLUENT, was employed by Riffat et al. [20] to investigate the impact of the primary nozzle exit position on the performance of the ejector. In the CFD modeling, Riffat et al. [20] have adopted the standard k ε − model and Renormalization Group (RNG) k ε − model to solve the turbulence problems. Bartosiewicz et al. [21,22] also used FLUENT to simulate ejectors, but they concluded that the shear stress transport (SST) version of the k ω − turbulence model agrees best with the test data.

There is no study of the flow in a whole ejector channel, let alone using CFD to optimize the ejector design. Also, there are few comparisons between CFD predictions and data. In order to validate the CFD simulations, it is necessary to carry out more experimental- CFD and analytical-CFD comparisons. The feasibility of whether CFD methods are capable of dealing adequately with supersonic flow, particularly shock and expansion wave in ejectors, has not been previously investigated.

DISSERTATION ORGANIZATION

The theoretical background and some important concepts used in the derivation of analytical models for gas ejectors will be introduced in Chapter II. Chapter III will

11 introduce the existing analytical models for gas ejectors with single phase flow; the Engineering Science Data Unit (ESDU) program ESDUpac A9242 Version 2 also will be introduced and analyzed. Chapter IV consists of sing-phase gas ejector simulations using the CFD software FLUENT. This chapter also includes the introduction of simulation, benchmark, analysis of performance characteristics and flow field details of gas ejectors, and optimization of geometric configurations. Chapter V derives a novel generalized ejector model for the ejector design and performance analysis. Chapter VI studies the general features and limitations of design and operation of ejector working with gas-liquid two-phase flow. Chapter VII summarizes present work, address existing problems and suggest possible direction for future work.

12 CHAPTER II THEORETICAL BACKGROUND AND NOZZLES

INTRODUCTION

The theoretical background used in the derivation of 1-D analytical models for gas ejectors is introduced in this chapter. All 1-D analysis of compressible gas streams in the ejector are made by application of conservation equations of continuity, momentum, and energy, as well as the ideal-gas law. For compressible flow, an important dimensionless parameter, Mach number, usually is utilized to represent the velocity. Isentropic expansion is an important assumption during the derivations, though some researchers also use loss factor coefficients obtained from experimental data to represent the friction or other loss. To achieve better performance, modern gas ejectors are normally operated in a supersonic condition at the exit of primary nozzle. Therefore, it is necessary to introduce the choking phenomena occurs at the throat of primary nozzle.

The basic idea of a gas ejector is to accelerate the motive flow to supersonic by a converging-diverging nozzle, primary flow exit at the suction chamber where secondary flow is induced by this high-velocity, depressurized flow. In most cases, there is also a diffuser installed at the exit of the mixing section to induce pressure recovery. The nozzles used to accelerate/decelerate or pressurize/depressurize the compressible flows are introduced in this chapter.

13 THEORETICAL BACKGROUND

Conservation and Ideal Gas Law

The conservation equations and idea gas law for steady 1-D compressible flow in an arbitrary variable-area control volume as sketched in Fig. 2.1 are given below. The definitions of termininologies can be found in the nomencluatre section.

Fig. 2.1 Control volume for 1-D flow

Continuity equation a a a b b b m V A V A ρ ρ = = (2.1)

14 Momentum equation b a A a a a a b b b b A P A m V PdA P A mV+ + = + ∫ (2.2) Energy equation 2 2 2 2 a b a b V V h h+ = + (2.3) Ideal gas law P R T ρ = (2.4) Where R is the gas constant with unit of ( ) .J kg K . R is related to its molecular weight by following equation: R R W = (2.5) In the above equation, R is the universal gas constant with unit of ( ) . J kmol K and W is the molecular weight with unit of ( ) kg kmol .

15 Mach Number

Mach number, M , is a very important dimensionless parameter for compressible flow, specially, for supersonic flow. Mach number is defined as the ratio of the fluid velocity to the local sonic speed. local fluid velocity local sonic speed V M c = = (2.6) The local sound speed c in a medium with temperature T is given by: c RTγ= (2.7)