213
University of Mississippi University of Mississippi eGrove eGrove Electronic Theses and Dissertations Graduate School 2014 Parametric Scramjet Analysis Parametric Scramjet Analysis Jongseong Choi University of Mississippi Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Aerospace Engineering Commons Recommended Citation Recommended Citation Choi, Jongseong, "Parametric Scramjet Analysis" (2014). Electronic Theses and Dissertations. 535. https://egrove.olemiss.edu/etd/535 This Thesis is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected].

Parametric Scramjet Analysis - eGrove

  • Upload
    others

  • View
    28

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Parametric Scramjet Analysis - eGrove

University of Mississippi University of Mississippi

eGrove eGrove

Electronic Theses and Dissertations Graduate School

2014

Parametric Scramjet Analysis Parametric Scramjet Analysis

Jongseong Choi University of Mississippi

Follow this and additional works at: https://egrove.olemiss.edu/etd

Part of the Aerospace Engineering Commons

Recommended Citation Recommended Citation Choi, Jongseong, "Parametric Scramjet Analysis" (2014). Electronic Theses and Dissertations. 535. https://egrove.olemiss.edu/etd/535

This Thesis is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected].

Page 2: Parametric Scramjet Analysis - eGrove

PARAMETRIC SCRAMJET ANALYSIS

A Thesis

presented in partial fulfillment of requirements

for the degree of Master of Science

in the Department of Mechanical Engineering

The University of Mississippi

by

JONGSEONG CHOI

May 2014

Page 3: Parametric Scramjet Analysis - eGrove

Copyright Jongseong Choi 2014

ALL RIGHT RESERVED

Page 4: Parametric Scramjet Analysis - eGrove

ii

ABSTRACT

The performance of a hypersonic flight vehicle will depend on existing materials and

fuels; this work presents the performance of the ideal scramjet engine for three different

combustion chamber materials and three different candidate fuels. Engine performance is

explored by parametric cycle analysis for the ideal scramjet as a function of material maximum

service temperature and the lower heating value of jet engine fuels. The thermodynamic analysis

is based on the Brayton cycle as similarly employed in describing the performance of the ramjet,

turbojet, and fanjet ideal engines. The objective of this work is to explore material operating

temperatures and fuel possibilities for the combustion chamber of a scramjet propulsion system

to show how they relate to scramjet performance and the seven scramjet engine parameters:

specific thrust, fuel-to-air ratio, thrust-specific fuel consumption, thermal efficiency, propulsive

efficiency, overall efficiency, and thrust flux. The information presented in this work has not

been done by others in the scientific literature. This work yields simple algebraic equations for

scramjet performance which are similar to that of the ideal ramjet, ideal turbojet and ideal

turbofan engines.

Page 5: Parametric Scramjet Analysis - eGrove

iii

LIST OF SYMBOLS

ao = freestream speed of sound, m/s

A2 = diffuser (engine inlet) exit area (Ae =A2), cm2; combustor entrance area

A*/A = area ratio

cp = specific heat at constant pressure, kJ / (kg . K)

F = thrust, N

om F = specific thrust, N / (kg / s)

gc = Newton’s constant, (kg . m) / (N . s2)

PRh = fuel lower heating value, kJ / kg

oh = freestream ambient enthalphy, kJ / kg

9h = enthalphy at engine nozzle exit, kJ / kg

2th = total enthalphy at combustor entrance, kJ / kg

4th = total enthalphy at combustor exit, kJ / kg

Mc = combustion Mach number

Mo = Mach number at freestream flight conditions

Mo* = freestream Mach number for maximum om F

Mo** = freestream Mach number for minimum S

***

oM = freestream Mach number for maximum F / A2

M9 = Mach number at engine nozzle exit (M9’ = M9’’ = M9 = Mo)

Page 6: Parametric Scramjet Analysis - eGrove

iv

om = mass flow rate of air, kg / s

fm = mass flow rate of fuel, kg / s

Po = freestream static pressure, Pa

P9 = static pressure at engine nozzle exit (Pe =P9), Pa (also P9’ and P9’’)

Pto = freestream total pressure, Pa

Pt2 = total pressure at combustor entrance, Pa (also P’’2)

Pt4 = total pressure at combustor exit, Pa

Pt9 = total pressure at engine nozzle inlet, Pa (also Pt9’ and Pt9’’)

R = gas constant for air, kJ / (kg . K)

s = entropy, (J / K)

S = thrust-specific fuel consumption, mg / (N . s)

T = temperature, K

Tmax = material temperature limit, K

T'max = burner exit total temperature, Mc < Mo, K

T''max = burner exit total temperature, Mc ≥ Mo, K

To = freestream ambient temperature, K

Tto = freestream total temperature, K

T2 = temperature at combustor entrance, K (also T’2)

Tt2 = total temperature at combustor entrance, K (also T’’2)

Tt4 = total temperature at combustor exit, K

Page 7: Parametric Scramjet Analysis - eGrove

v

T9 = temperature at engine nozzle exit, K (also T9’ and T9’’)

Tt9 = total temperature at engine nozzle inlet, K (also Tt9’ and Tt9’’)

Vo = freestream velocity

V9 = engine nozzle exit velocity (Ve = V9), m / s (also V9’ and V9’’)

netoutW = amount of work per unit time through the net area from T-s diagram, J / s

inQ = heat flow rate by combustion, J / s

γ = ratio of specific heats

f = fuel-to-air ratio

bτ = total temperature ratio across burner, 2t4t T/T , rλ /ττ

dτ = total temperature ratio across diffuser, to2t T/T

nτ = total temperature ratio across nozzle, 4t9t T/T

τr = inlet temperature ratio, Tto / To = [1 + ( γ - 1) Mo2 / 2]

τ = total temperature to freestream temperature ratio, omaxo4t T/TT/T

b = total pressure ratio across burner, 2t4t P/P

d = total pressure ratio across diffuser, to2t P/P

n = total pressure ratio across nozzle, 4t9t P/P

r = inlet pressure ratio, Pto / Po = [1 + (γ - 1) Mo2 / 2]

1/

T = thermal efficiency

Page 8: Parametric Scramjet Analysis - eGrove

vi

P = propulsive efficiency

o = overall efficiency

Page 9: Parametric Scramjet Analysis - eGrove

vii

ACKOWLEDGEMENTS

I would like to gratefully acknowledge the enthusiastic supervision of Dr. Jeffrey A.

Roux during this research. There has been his great contribution toward this thesis, which has

yielded three journal publications in the past two years. I offer my sincerest appreciation for the

learning opportunities and support provided by Dr. Arunachalam M. Rajendran. His guidance

has provided me a thoughtful and rewarding journey. I would like to thank to Dr. Chung Song

for general advice concerning scholastic life besides my research. While data collecting and

researching, Kiyun Kim spent countless hours listening to me talk about my thoughts on my

research. I am grateful to all my friends from KSA and OKCC. I have received support and

encouragement from my friend group named Sekye from Korea. Finally, I am forever indebted

to my parents for their understanding, endless patience, and encouragement when it was most

required.

Page 10: Parametric Scramjet Analysis - eGrove

viii

TABLE OF CONTENTS

ABSTRACT ........................................................................................................................... ii

LIST OF SYMBOLS ............................................................................................................... iii

ACKOWLEDGEMENTS ...................................................................................................... vii

TABLE OF CONTENTS ...................................................................................................... viii

LIST OF TABLES .................................................................................................................. xi

LIST OF FIGURES ................................................................................................................ xv

INTRODUCTION .....................................................................................................................1

1.1 Hypersonic Propulsion .............................................................................................1

1.2 Parametric Thermodynamic Cycle Analysis for Ideal Engines .................................2

1.3 Definition .................................................................................................................3

1.3.1 Ramjet ...................................................................................................................3

1.3.2 Scramjet ................................................................................................................5

1.4 Scramjet Components ...............................................................................................8

1.5 Historical Development ............................................................................................9

1.6 Materials and Fuels Candidates .............................................................................. 13

1.6.1 Materials ............................................................................................................. 14

Page 11: Parametric Scramjet Analysis - eGrove

ix

1.6.2 Fuels ................................................................................................................... 15

1.7 This Work .............................................................................................................. 17

STATEMENT OF THE PROBLEM...................................................................................... 19

2.1 Definition of the Problem ....................................................................................... 19

2.2 T-s Diagram .......................................................................................................... 20

2.2.1 Situations for Mc < Mo ......................................................................................... 21

2.2.2 Situations for Mc ≥ Mo ......................................................................................... 24

2.3 Materials and Fuels Selection ................................................................................. 25

ANALYSIS .......................................................................................................................... 27

3.1 Fundamental Assumptions for Parametric Analysis ................................................ 27

3.2 Parametric Analysis for Ideal Mass Flow Scramjet ................................................. 28

3.2.1 Summary of Equations – Ideal Mass Flow Scramjet ............................................ 36

3.3 Parametric Analysis for Non-Ideal Mass Flow Scramjet ......................................... 37

3.3.1 Summary of Equations – Non-Ideal Mass Flow Scramjet .................................... 43

3.4 Optimum Freestream Mach Number (Mo*) for Maximum Specific Thrust [22] ....... 44

3.5 Optimum Freestream Mach Number (Mo**) for Minimum Thrust-Specific Fuel

Consumption ......................................................................................................................... 45

3.5.1 Optimum Freestream Mach number (Mo**) Formulation for Mc < Mo [24] .......... 48

Page 12: Parametric Scramjet Analysis - eGrove

x

3.5.2 Optimum Freestream Mach number (Mo**) Formulation for Mc ≥ Mo [19] ........... 50

3.6 Optimum Freestream Mach Number (Mo***) for Maximum Thrust Flux [20] ......... 52

RESULTS .......................................................................................................................... 59

4.1 Performance Parameters for Ideal Mass Flow Rate ................................................. 59

4.1.1 Materials Selection .............................................................................................. 60

4.1.2 Fuel Selection .................................................................................................... 119

4.2 Performance Parameters for Non-Ideal Mass Flow Rate ....................................... 139

4.2.1 Materials Selection ............................................................................................ 140

4.2.2 Fuel Selection .................................................................................................... 171

4.3 Closing of Results ................................................................................................ 179

CONCLUSION ..................................................................................................................... 180

LIST OF REFERENCES ...................................................................................................... 183

VITA ........................................................................................................................ 187

Page 13: Parametric Scramjet Analysis - eGrove

xi

LIST OF TABLES

1-1 Worldwide Scramjet Evolution, 1955-1990 [8] ………………………………..

11

1-2 Worldwide Scramjet Evolution, 1960-2004 [8] ……………………………......

12

1-3 Important Hypersonic Technology Programs Worldwide, June 2012 [12] …….

13

2-1 Selected Materials and Maximum Service Temperatures………………………

25

2-2 Selected Fuels and Lower Heating Values……………………………………...

26

3-1 Optimum Freestream Mach number (Mo***) and Maximum Thrust Flux (F/A2)

for the Ideal Material Limited Ramjet/Scramjet Engine [20]

……………..…..……………..…..……………..…..……………..…………....

57

4-1 Optimum Freestream Mach number (Mo**) Corresponding to Minimum Thrust

Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at Various

Combustion Mach numbers………………….…..…..…..…..…..…..…..……..

71

4-2 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at

Various Combustion Mach numbers……………………..…..…..…..…..……..

73

4-3 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at

Various Combustion Mach numbers……………………..…..…..…..…..……..

75

4-4 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel consumption (S) for Ideal Mass Flow Scramjet at the

Maximum Service Temperatures of Three Candidate Materials…………….....

77

4-5 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at the

Maximum Service Temperatures of Three Candidate Materials…………….....

79

4-6 Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at Various Combustion Mach

numbers……………………………………………………………………........

102

Page 14: Parametric Scramjet Analysis - eGrove

xii

4-7 Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at Various Combustion Mach

numbers…………………………………………………………………………

104

4-8 Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at Various Combustion Mach

numbers…………………………………………………………………………

106

4-9 Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at the Maximum Service

Temperatures of Three Candidate Materials……………………………………

108

4-10 Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at the Maximum Service

Temperatures of Three Candidate Materials……………………………………

110

4-11 Scoring System for Selecting the Combustion Mach Number (Mc) Among 2.5

and 3 at the Maximum Service Temperatures of Three Candidate Materials

(Tmax = 1700 K, 2300 K, 2700 K); Mo = 8………………………….…..…..…..

117

4-12 Scoring System for Selecting the Material Among Three Maximum Service

Temperatures (Tmax) of Three Candidate Materials where 1700 K at

Combustion Mach Number (Mc) = 3, 2300 K at Mc = 2.5, 2700 K at Mc = 2.5;

Mo = 8…………………………………………………………………………..

118

4-13 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at

Various Combustion Mach numbers……………………………………………

122

4-14 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at

Various Combustion Mach numbers……………………………………………

124

4-15 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at

Various Combustion Mach numbers……………………………………………

126

4-16 Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel consumption (S) for Ideal Mass Flow Scramjet at the

Lower Heating Values of Three Candidate Fuels………………………………

128

Page 15: Parametric Scramjet Analysis - eGrove

xiii

4-17 Scoring System for Selecting the Combustion Mach Number (Mc) Among 2.5

and 3 at the Lower Hearing Values of Three Candidate Fuels (hPR = 42800

kJ/kg, 43900 kJ/kg, 120000 kJ/kg); Mo = 8, Tmax = 2700 K…………...…..…...

137

4-18 Scoring System for Selecting the Fuel Among Three Lower Heating Values

(hPR) of Three Candidate Fuels at Combustion Mach Number (Mc) = 2.5; Mo =

8, Tmax = 2700 K………………………………………………………………...

138

4-19 Optimum Freestream Mach number (Mo**) and Corresponding Minimum

Thrust-Specific Fuel Consumption (S) of Non-Ideal Mass Flow and Ideal

Mass Flow Scramjet at Various Combustion Mach numbers (Mc = 0, 1, 2, 2.5,

4) …………………………………………………………………………….....

148

4-20 Optimum Freestream Mach number (Mo**) and Corresponding Minimum

Thrust-Specific Fuel Consumption (S) of Non-Ideal Mass Flow and Ideal

Mass Flow Scramjet at the Maximum Service Temperatures of Three Selected

Materials (Tmax = 1700 K, 2300 K, 2700 K) ………………………..…..…..….

150

4-21 Optimum Freestream Mach number (Mo***) and Corresponding Maximum

Thrust Flux (F/A2) of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet at

Various Combustion Mach numbers (Mc = 0.1, 1, 2, 2.5, 3) and at Mo* = Mc ≈

3.36……………………………………………………………………………...

164

4-22 Optimum Freestream Mach number (Mo***) and Corresponding Maximum

Thrust Flux (F/A2) of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet at

the Maximum Service Temperatures of Three Selected Materials (Tmax = 1700

K, 2300 K, 2700 K) …..…..…..…..…..…..…..…..…..…..…..…..…..…..…….

166

4-23 Scoring System for Selecting the Combustion Mach Number (Mc) in the Case

of Non-Ideal Mass Flow Rate Among 2.5 and 3 at the Maximum Service

Temperatures of Three Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K);

Mo = 8………………………………………………………………..…..……...

169

4-24 Scoring System for Selecting the Material in the Case of Non-Ideal Mass

Flow Rate Among Three Maximum Service Temperatures (Tmax) of Three

Candidate Materials where 1700 K at Combustion Mach Number (Mc) = 3,

2300 K at Mc = 2.5, 2700 K at Mc = 2.5; Mo = 8…………………………….....

170

4-25 Optimum Freestream Mach number (Mo**) and Corresponding Minimum

Thrust-Specific Fuel consumption (S) of Non-Ideal Mass Flow and Ideal Mass

Flow Scramjet at the Lower Heating Values of Three Selected Fuels (hPR =

Page 16: Parametric Scramjet Analysis - eGrove

xiv

42,800 kJ/kg, 43,800 kJ/kg, 120,000 kJ/kg) …………………………...…..…..

174

4-26 Scoring System for Selecting the Fuel Among Three Lower Heating Values

(hPR) of Three Candidate Fuels at Combustion Mach Number (Mc) = 2.5; Mo =

8, Tmax = 2700 K………………………………………………………………...

178

Page 17: Parametric Scramjet Analysis - eGrove

xv

LIST OF FIGURES

1-1 Process and Components of Ramjet…………………………………………..

4

1-2 Specific Impulse (s) and Operating Regimes for Variety of Flight Systems

[8]………………………………………………………………………….......

6

1-3 Engine options as function of Mach number [8] ……………………………..

6

1-4 Boeing X-51A Attached to the Rocket Engine and Boeing B52 Stratofortress

[1] ……………………………………………………………………………..

7

1-5 Representative Scramjet Engine [10] ………………………………………....

9

1-6 Maximum Service Temperature of Various Materials [15] …………………...

16

2-1 T-s Diagram Comparison for Ramjet and Scramjet [5] ………………………

22

3-1 Heat In and Heat Out of Ideal Scramjet Combustion Chamber……………….

31

3-2 Heat In and Heat Non-Out of Ideal Scramjet Combustion Chamber…………

39

3-3 Ideal Scramjet Commensurate Freestream Mach Number vs Combustion

Mach Number for Maximum Specific Thrust (Mo*) and Minimum Thrust-

Specific Fuel Consumption (Mo**) and Maximum Thrust Flux (Mo

***) [5]...

46

3-4 Ideal Ramjet (Mc = 0) Thrust-Specific Fuel Consumption (S) vs Freestream

Mach Number (Mo) [24] ……………………………………………………...

51

3-5 Ideal Scramjet Thrust-Specific Fuel Consumption vs Freestream Mach

Number [19] …………………………………………………………………..

53

3-6 Ramjet/Scramjet Thrust Flux as a Function of Freestream Mach Number at

Different Combustion Mach Numbers (Mc), Where To = 217 K, Po = 19403

Pa (Altitude = 12 km), and Tmax = 1900 K [20] ………………………………

58

4-1 Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 1700 K) for

the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc

= 0, 1, 2, 2.5) and at Mo* = Mc ≈ 3.00 ………………………………………...

63

Page 18: Parametric Scramjet Analysis - eGrove

xvi

4-2 Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 2300 K) for

the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc

= 0, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.36………………………………………

64

4-3 Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 2700 K) for

the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc

= 0, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.55………………………………………

65

4-4. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Combustion Mach Number (Mc = 2.5) for the Cases Mc < Mo

and Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K) ………………………………….

66

4-5. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Maximum Service Temperatures of Three Candidate Materials

(Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo where

Mo*= Mc Occurs in Each Selected Material…………………………………...

67

4-6. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 1700 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.9

Corresponding to Two Values of Mo**………………………………………..

70

4-7. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.94

Corresponding to Two Values of Mo**………………………………………..

72

4-8. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2700 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97

Corresponding to Two Values of Mo** ……………………………………….

74

4-9.

Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion Mach Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Page 19: Parametric Scramjet Analysis - eGrove

xvii

Temperatures of Three Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K) 76

4-10 Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Maximum Service Temperatures of

Three Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K) for the Cases

Mc < Mo and Mc ≥ Mo where Mc’s Corresponding to Two Values of

Mo**…..…..……………………..…..……………..…..……………..……….

78

4-11. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 1700 K) for

the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc

= 0, 1, 2, 2.5) and at Mo*= Mc ≈ 3……………………………………………..

81

4-12. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 2300 K) for

the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc

= 0, 1, 2, 2.5, 3) and at Mo*= Mc ≈ 3.36……………………………………….

82

4-13. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax= 2700 K) for

the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc

= 0, 1, 2, 2.5, 3) and at Mo*= Mc ≈ 3.55……………………………………….

83

4-14. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Combustion Mach Number (Mc= 2.5) for the Cases Mc < Mo

and Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K) ………………………………….

84

4-15. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Maximum Service Temperatures of Three Selected Materials

(Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo where

Mo*= Mc Occurs in Each Candidate Material…………………………………

85

4-16. Thermal Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Three Candidate Material Maximum Service

Temperatures (1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo

at Various Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 3, 3.36) ………….

87

4-17. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax = 1700

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5) and at Mo * = Mc ≈ 3……………………………..

89

Page 20: Parametric Scramjet Analysis - eGrove

xviii

4-18. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax= 2300

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo * = Mc ≈ 3.36……………………….

90

4-19. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax = 2700

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo * = Mc ≈ 3.55……………………….

91

4-20. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Combustion Mach Number (Mc = 2.5) for the Cases

Mc < Mo and Mc ≥ Mo at the Maximum Service Temperatures of Three

Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K) ………………………

92

4-21. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥

Mo where Mo* = Mc Occurs in Each Candidate Materials…………………….

93

4-22.

Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax = 1700

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5) and at Mo * = Mc ≈ 3.00………………………….

94

4-23.

Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax = 2300

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo * = Mc ≈ 3.36……………………….

95

4-24. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax = 2700

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo * = Mc ≈ 3.55……………………….

96

4-25. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Combustion Mach Number (Mc= 2.5) for the Cases

Mc < Mo and Mc ≥ Mo at the Maximum Service Temperatures of Three

Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K) ………………………

97

Page 21: Parametric Scramjet Analysis - eGrove

xix

4-26. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥

Mo where Mo* Mc Occurs in Each Candidate Material……………………….

98

4-27. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 1700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc =

0.1, 1, 2, 2.5) and at Mo* = Mc ≈ 3.00…………………………………………

101

4-28. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 2300 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc =

0.1, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.36………………………………………

103

4-29. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 2700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc =

0.1, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.55………………………………………

105

4-30.

Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Combustion Mach Number (Mc = 2.5) for the Cases Mc < Mo and

Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K) ………………………………….

107

4-31. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Maximum Service Temperatures of Three Candidate Materials (Tmax

= 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo where Mo* =

Mc Occurs in Each Candidate Material……………………………………….

109

4-32. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Number for Maximum Specific Thrust for the Cases Mc < Mo and Mc ≥ Mo

at the Maximum Service Temperatures of Three Candidate Materials (Tmax =

1700 K, 2300 K, 2700 K) …………………………………………………….

112

4-33. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Number for Minimum Thrust Specific Fuel Consumption for the Cases Mc <

Mo and Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K) ………………………………….

113

4-34. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Page 22: Parametric Scramjet Analysis - eGrove

xx

Number for Maximum Thrust Flux for the Cases Mc < Mo and Mc ≥ Mo at the

Maximum Service Temperatures of Three Candidate Materials (Tmax = 1700

K, 2300 K, 2700 K) …………………………………………………………..

114

4-35. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Fuel Lower Heating Value (hPR =

42,800 kJ/kg) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion

Mach Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97 Corresponding to Two

Values of Mo**………………………………………………………………...

121

4-36.

Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Fuel Lower Heating Value (hPR =

43,900 kJ/kg) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion

Mach Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97 Corresponding to Two

Values of Mo**………………………………………………………………...

123

4-37. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Fuel Lower Heating Value (hPR =

120,000 kJ/kg) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion

Mach Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97 Corresponding to Two

Values of Mo**………………………………………………………………...

125

4-38. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion Mach Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Lower Heating Values of

Three Selected Fuels (hPR = 42,800 kJ/kg, 43,900 kJ/kg, 120,000 kJ/kg)…….

127

4-39.

Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Fuel Lower Heating Value (hPR = 42,800 kJ/kg) for the Cases

Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2,

2.5, 3) and at Mo* = Mc ≈ 3.55………………………………………………...

130

4-40. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Fuel Lower Heating Value (hPR = 43,900 kJ/kg) for the Cases

Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc=0, 1, 2,

2.5, 3) and at Mo* = Mc ≈ 3.36………………………………………………...

131

4-41. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Fuel Lower Heating Value (hPR = 120,000 kJ/kg) for the Cases

Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2,

2.5, 3) and at Mo* = Mc ≈ 3.55………………………………………………...

132

Page 23: Parametric Scramjet Analysis - eGrove

xxi

4-42. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Combustion Mach Number (Mc = 2.5) for the Cases Mc < Mo

and Mc ≥ Mo at the Lower Heating Values of Three Candidate Fuels (hPR =

42,800 kJ/kg, 43,900 kJ/kg, 120,000 kJ/kg) ………………………………….

133

4-43. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Number for Minimum Thrust-Specific Fuel Consumption for the Cases Mc <

Mo and Mc ≥ Mo at the Lower Heating Values of Three Candidate Fuels (hPR

= 42,800 kJ/kg, 43,900 kJ/kg, 120,000 kJ/kg) ………………………………..

135

4-44. Specific Thrust of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.36…...

143

4-45. Specific Thrust of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion Mach Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K)..

145

4-46. Thrust-Specific Fuel Consumption of Non-Ideal Mass Flow and Ideal Mass

Flow Scramjet vs Freestream Mach Number (Mo) with the Material

Maximum Service Temperature (Tmax = 2300 K) for the Cases Mc < Mo and

Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2, 4) and at Mc ≈

2.94 Corresponding to Two Values of Mo**…………………………………..

147

4-47. Thrust-Specific Fuel Consumption of Non-Ideal Mass Flow and Ideal Mass

Flow Scramjet vs Freestream Mach Number (Mo) with the Combustion

Mach Number (Mc = 2.5) for the Cases Mc < Mo and Mc ≥ Mo at the

Maximum Service Temperatures of Three Selected Materials (Tmax = 1700 K,

2300 K, 2700 K) ……………………………………………………………...

149

4-48. Fuel-to-Air Ratio of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 3) and at Mo* = Mc ≈ 3.36………..

152

4-49. Fuel-to-Air Ratio of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion March Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K)..

153

Page 24: Parametric Scramjet Analysis - eGrove

xxii

4-50. Thermal EfficiencyofNon-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Three Selected Material Maximum

Service Temperatures (1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and

Mc ≥ Mo at Various Combustion Mach Numbers (Mo = 0, 1, 2, 2.5, 3, 3.36)…

155

4-51. Propulsive Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow

Scramjet vs Freestream Mach Number (Mo) with the Material Maximum

Service Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at

Various Combustion Mach Numbers (Mc = 0, 1, 2,2.5, 3) and at Mo* = Mc ≈

3.36…………………………………………………………………………....

157

4-52. Propulsive Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow

Scramjet vs Freestream Mach Number (Mo) with the Combustion March

Number (Mc = 2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum

Service Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K,

2700 K) ……………………………………………………………………….

158

4-53. Overall Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet

vs Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.36…..

159

4-54. Overall Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet

vs Freestream Mach Number(Mo) with the Combustion March Number (Mc

= 2.5) for the Cases Mc < Mo and Mc ≥ Moat the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K)..

160

4-55. Thrust Flux of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0.1, 1, 2, 2.5, 3) and at Mo*= Mc ≈ 3.36….

163

4-56. Thrust Flux of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion March Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K)..

165

4-57.

Thrust-Specific Fuel Consumption of Non-Ideal Mass flow and Ideal Mass

Flow Scramjet vs Freestream Mach Number (Mo) with the Combustion

Mach Number (Mc = 2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Lower

Heating Values of Three Selected Fuels (hPR = 42,800 kJ/kg, 43,800 kJ/kg,

Page 25: Parametric Scramjet Analysis - eGrove

xxiii

120,000 kJ/kg) ……………………………………………………………….. 173

4-58

Fuel-to-Air Ratio of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion Mach Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Lower Heating Values of

Three Selected Fuels (hPR = 42,800 kJ/kg, 43,9 00 kJ/kg, 120,000 kJ/kg)....…

176

Page 26: Parametric Scramjet Analysis - eGrove

1

CHAPTER 1

INTRODUCTION

1.1 Hypersonic Propulsion

The achievement of hypersonic passenger transport, military operations, and space

exploration via air-breathing engine propulsion has been a significant challenge to engineers for

nearly sixty years. Turbojet engines which operate by flying at relatively low Mach numbers

employing a compressor and turbine cannot accomplish hypersonic flight. The ramjet engine has

been proposed over turbojets to produce higher thrust for hypersonic flight; theoretically ramjet

engine can reach Mach numbers as high as 5. Only scramjet propulsion has the potential to

achieve maximum Mach numbers of 15 theoretically; hence scramjet is a promising engine for a

hypersonic air-breathing flight vehicle. The recently launched X-51A project [1] proved that a

scramjet engine can achieve a powered sustained flight for about three minutes. With the

potential of reaching speeds beyond Mach number 15, the scramjet engine could make

hypersonic flight accessible in the future [1].

Significant technical challenges of developing a scramjet must be overcome for moving

it to reality. One such problem is that the scramjet cannot operate below supersonic speeds. It

must be accelerated up to supersonic speeds by another vehicle such as a ramjet, rocket, or

turbojet in order for the air to be compressed in the inlet and ignited with fuel in the combustion

chamber. Another major challenge is being able to deal with the high temperatures due to

aerodynamic heating that accompany very high flight Mach numbers. Improvements in fuel and

Page 27: Parametric Scramjet Analysis - eGrove

2

materials especially can improve the performance of the scramjet and allow for flight vehicles to

operate at very high Mach numbers, such as flight at 10 times the speed of sound or greater [1].

This work intends to complement previous research by giving a better understanding of

the performance possibilities of the scramjet engine with respect to material operating

temperature limit and fuel lower heating value (hPR). The material operating temperature limit

and the fuel heating value will be incorporated into the seven performance parameters of the

ideal scramjet engine as a function of freestream flight Mach number with combustion Mach

number as a parameter. A step-by-step mathematical description of the performance parameters

will be developed in this work.

1.2 Parametric Thermodynamic Cycle Analysis for Ideal Engines

Thermodynamic cycle analysis provides a thermodynamic description of the working

fluids such as air and combustion products in the engines. It consists of a parametric cycle

analysis and the engine performance description, which is influenced by different flight

conditions from different design choices. The parametric analysis is a valuable tool since it

provides optimum theoretical performances for determining if an engine is a good candidate for

specific mission requirements [2].

Parametric analysis to develop the mathematical expressions for specific thrust, thrust-

specific fuel consumption, fuel-to-air ratio, thermal efficiency, propulsive efficiency, overall

efficiency, and thrust flux for the ideal scramjet engine will be developed in this work. The

performance parameter development is similar to the parametric cycle analyses of the ideal

ramjet, ideal turbojet and ideal turbofan engines as documented in [2, 3]. This will allow

comparison of the characteristics of the ideal scramjet engine with the ramjet engine in a

Page 28: Parametric Scramjet Analysis - eGrove

3

fundamental way. For the ideal scramjet, this work presents a parametric description of the

scramjet engine performance which does not exist in the research literature.

1.3 Definition

A hypersonic vehicle (flight Mach number greater than 5) has to operate in the ambient

air atmosphere which applies structural heating and structural loading while delivering air flow

to the propulsion system to maintain an adequate level of net thrust for propulsion; like the ideal

ramjet, the ideal scramjet description will be based on the Brayton cycle [4]. In flight at a

velocity approximately above Mach number 3, a compressor is no longer needed to achieve

sufficient cycle pressure. The ramjet engine becomes an inefficient engine to achieve hypersonic

air-breathing vehicle for flight above Mach 5. In ramjet, air is greatly slowed down as it passes

through the inlet and enters the combustion chamber until it reaches low subsonic speeds where

heat is added in the combustor; then the gas is expanded and brought to supersonic speed via a

divergent exit nozzle. The problem with this is that the maximum aircraft speed is limited to

about Mach 5. Because of this a need was recognized for utilizing ramjet type technology, but

having combustion Mach numbers greater than one. Thus, the flow no longer needs to be slowed

to subsonic speeds, but rather only slowed to acceptable supersonic speed. This need brought the

idea of the supersonic combustion ramjet which is generally called scramjet. The fundamental

differences between the ramjet and scramjet engine is that the ramjet has combustion operating at

low subsonic speeds and scramjet has combustion operating at supersonic speeds [5].

1.3.1 Ramjet

The ramjet engine does not have a compressor or turbine and combustion occurs at low

subsonic velocities as does the turbojet engine. Ramjet is fundamentally composed of an inlet, a

Page 29: Parametric Scramjet Analysis - eGrove

4

burner, and a nozzle as shown in Fig. 1-1. During flight, air flows through the burner where it

combusts and is exhausted through the nozzle producing thrust. The ramjet is unique because

the incoming air becomes compressed by the inlet and is slowed down to low subsonic speeds

(M < 1) inside the burner. This causes the air pressure and temperature to rise entering the

burner where combustion occurs via fuel injection [2]. Therefore, at high supersonic flight

velocities the ramjet is unacceptable because the temperatures at the combustion chamber inlet

do not allow enough heat addition for high performance.

Figure 1-1. The Process and Components of Ramjet.

Very high supersonic velocities present problems for the ramjet. This is because the inlet

reduces the airflow in subsonic velocities in order to combust, and hence high gas temperatures

occur before combustion even starts. The increased temperature conditions from very high Mach

number can exceed the temperature limits of the materials of the ramjet. This gives the ramjet

limited performance at very high Mach numbers [2]. In reality, the actual supersonic vehicle

Page 30: Parametric Scramjet Analysis - eGrove

5

SR-71 Blackbird, which operates as a ramjet engine, had a cruising speed of about Mach number

3 [2].

1.3.2 Scramjet

A scramjet propulsion system is a hypersonic air-breathing engine in which heat addition

and ignition occurs in the air flow at supersonic velocities within the engine [6]. The scramjet

engine hence does not slow down the flow so much that high gas temperatures occur before

combustion starts. Supersonic combustion denotes better performance as the flight Mach

number can become greater than Mach number 5 [7]. The relationship of specific impulse

(thrust) versus freestream Mach number for different types of air-breathing engine is shown in

Fig. 1-2 [8]. It can be seen that at Mach numbers higher than approximately 6-7, the only

available propulsion systems are rockets and scramjets. Compared to rockets, scramjets have

much higher specific impulse levels; therefore, it is clear why it is advantageous to develop the

scramjet, if for this reason only. Fig. 1-2 indicates that ramjets may have a greater efficiency but

cannot operate at higher freestream Mach numbers like the scramjet engine.

Figure 1-3 displays the theoretical possible flight Mach number of various propulsion

options based on the flight Mach number [8]. The curve depicts the approximation of operating

altitude at a given flight Mach number. Also shown on this chart is the relative boundary

between the two primary fuel options for scramjets: hydrocarbons and hydrogen. Theoretically,

hydrogen can perform at higher speeds above the hydrocarbon upper limit. With current

technology and capability for a hypersonic vehicle which offers acceptable performance to about

Mach 15, only the hydrogen fueled scramjet [5]; is possible.

Page 31: Parametric Scramjet Analysis - eGrove

6

Figure 1-2. Specific Impulse (s) and Operating Regimes for Variety of Flight Systems [8].

Figure 1-3. Engine options as function of Mach number [8].

Page 32: Parametric Scramjet Analysis - eGrove

7

Since the scramjet engine functions with supersonic flow combustion, it requires

sufficient kinetic energy for hypersonic freestream velocity air to compress to operational

conditions. Thus, a scramjet air-breathing vehicle must be accelerated its operational flight

Mach number by some other propulsion, such as turbojet and ramjet [9]. In a recent scramjet-

powered aviation test, the Boeing X-51A was lifted to adequate altitude by a Boeing B52

Stratofortress and accelerated by a rocket engine to near Mach 4.5; the three aircraft are shown in

Fig. 1-4. The Boeing X-51A was carried by a B-52 and dropped off at about altitude 50,000 ft,

where an installed Army solid rocket booster turns on, carrying it nearly to velocity of Mach

number 5. After this, the Boeing X-51A would be ignited and operated until it reaches Mach

number 6 before the engine cut off [1].

Figure 1-4. Boeing X-51A Attached to the Rocket Engine and Boeing B52 Stratofortress [1].

Page 33: Parametric Scramjet Analysis - eGrove

8

Another issue for scramjet is providing an adequate mixture of jet fuel in the chamber.

The development of technology getting fuel to mix with air in an appropriate fuel-to-air ratio at

supersonic airflow conditions is still demanding and needs to be advanced for future air-

breathing propulsion to be more stable at high velocities.

1.4 Scramjet Components

Figure 1-5 illustrates the five major engine and two vehicle components: internal inlet,

isolator, combustor, internal nozzle, fuel supply system, and craft afterbody. The main

components of concern are the inlet, combustor, and nozzle. The inlet captures the atmospheric

air and compresses it for the rest of the process. When the flow compresses, the Mach number

decreases as pressure and temperature increase while traveling through the inlet. As the

compressed air travels through the combustor, fuel is added to the air at several points along the

combustor length. Air enters the combustion chamber at a Mach number greater than one which

indicates fundamentally improved idea from ramjet and the scramjet to be unique. As the gas

leaves the combustor, it enters the nozzle to produce thrust, this air is expanded ideally to the

ambient atmospheric pressure. The design of the nozzle greatly impacts the thrust production of

the system.

The design of scramjet engine is very similar to a basic ramjet engine. The main

difference is that the airflow through the engine remains supersonic. To accomplish this,

scramjet engines rely heavily on the shape of the entire aircraft. A concept known as “airframe-

integrated scramjet” became the standard for most vehicle designs and wind tunnel tests [10].

The front section of the aircraft is shaped such that the shock waves produced at the aircraft’s

Page 34: Parametric Scramjet Analysis - eGrove

9

leading edges are funneled into the engine. This concept, along with the geometry of the engine

inlets, compresses the air for combustion with the fuel.

Figure 1-5. Representative Scramjet Engine [10].

1.5 Historical Development

Research into supersonic flight began during World War II and continues into the present

day. In 1970, shock tunnel scramjet experiments were implemented Osgerby at Arnold Air

Force Station, Tennessee [10]. A thorough history of scramjet is given in [6] and will not be

repeated here; rather attention will be focused on more recent history. In 1985, the first

experiment in Australia started with the leadership of Stalker and Morgan who were residents at

The University of Queensland [6], and many scramjet experiments have been conducted at The

University of Queensland. The HySHOT team from the University of Queensland conducted

leading-edge experiments of scramjet technology and successfully tested a scramjet engine in

2001. Four practical scramjet engine tests were performed in the HySHOT program, and

supersonic combustion was achieved in the HySHOT’s II and III flights [11].

Page 35: Parametric Scramjet Analysis - eGrove

10

The first major research development of a scramjet engine in the U.S. began in the

1980’s with the National Aerospace Plane (NASP) Program conducted by NASA [1]. NASA

commissioned the initial experiment with simple components consisting of a combustion

chamber, an injector, and a thrust surface which were used to evaluate various aspects of

combustion at supersonic speeds. In March 2004, the NASA X-43A was successfully launched.

On their third attempt the NASA X-43A flight set a new speed record of Mach 9.8 which was

maintained for almost 11 second in flight later in November 2004 [12]. The worldwide history

of scramjet evolution summary is shown in Table 1-1 and 1-2 with its basic conception from

1955 through 2004 [8].

HyCAUSE which was launched in June 2007 by the Defence Science and Technology

Organization (DSTO) [11], Defence Advanced Research Agency (DARPA) and the University

of Queensland. It was launched from the ground, which was a different launching method than

the NASA X-43A and Boeing X-51A, and reached a speed of approximately Mach number 9

[11].

In May 2010, NASA and the United States Air Force successfully flew the Boeing X-

51A Waverider for approximately 200 seconds at Mach 5, setting a new world record for

duration at hypersonic airspeed. A waverider, as the name itself, is riding its own shock waves

which enhences its lift-to-drag ratio [7]. However, a second flight in June, 2011 failed after the

process engine was lit with ethylene; it failed to transition to its primary JP-7 fuel, failing to

reach full power [1].

Not only the U.S. and Australia but also Brazil, England, and France are working to

advance scramjet propulsion technology [12]. The program HIFiRE governed by Australia and

the U.S. focuses on flight testing of a scramjet using a Terrier-Orion sounding rocket to develop

Page 36: Parametric Scramjet Analysis - eGrove

11

Table 1-1. Worldwide Scramjet Evolution, 1955-1990 [8].

11

Page 37: Parametric Scramjet Analysis - eGrove

12

Table 1-2. Worldwide Scramjet Evolution, 1960-2004 [8].

12

Page 38: Parametric Scramjet Analysis - eGrove

13

and validate scramjet technologies; a second HIFiRE hypersonic test flight was implemented in

March 2010. In Brazil, 14X research on developing a Mach number 6 hypersonic UAV

propelled by a hydrogen scramjet engine was conducted intending for access to space. Program

SABRE in England employed a precooled air-breathing/rocket combined-cycle engine for Mach

number to 5. It is a ground test of subscale engine to demonstrate engine cycle for the entire

flight regime. The LEA program in France is developing a dual-mode ram/scramjet engine to fly

at Mach number 10 to 12 and scheduled to terminate in 2015 after four flight tests [12]. Table

1-3 shows currently the main hypersonic programs worldwide [12].

Table 1-3. Important Hypersonic Technology Programs Worldwide, June 2012 [12].

Nation Program Emphasis Status

Australia / US HIFiRE Flight test of a scramjet using a Terrier- Orion Sounding rocket to

develop and validate scramjet

technologies

Second HIFiRE hypersonic test flight was on March 22, 2010

Brazil 14X Mach-6 hypersonic UAV propelled by H2 scramjet engine.

Intended for access to space.

Being tested in T3 Brazilian air force hypersonic wind tunnel.

England SABRE Precooled air-breathing/rocket

combined-cycle engine for Mac 5-25 [SSTO].

Proof-of-concept. Ground test of

subscale engine to demonstrate engine cycle for entire flight

regime.

France LEA Development of experimental

vehicle propelled by dual-mode ram/scramjet engine to fly at

Mach 10 - 12.

Scheduled to terminate in 2015

after four flight tests.

U.S. X-51 Unmanned Mach-7, JP-7-fueled scramjet demonstrator. Second

attempt in 2011.

First successful flight test May 2010. Two more flights are

planned.

1.6 Materials and Fuels Candidates

The scramjet engine requires a specific material and a specific fuel to determine its

performance. The mathematical parametric description of the scramjet engine performance

Page 39: Parametric Scramjet Analysis - eGrove

14

significantly depends on choosing both a combustion chamber material and a fuel since the

maximum service temperature of a material and the lower heating value (hPR) of a fuel impact

the parametric performance characteristics. Because of this, the task is to select a material and a

fuel that showed potential for use in scramjet engines to achieve good performance.

The initial process of this work was to choose three different materials and three different

fuels that would be candidates to use in a scramjet engine, and to show the performance

differences among each of those three materials or and three fuels. The next step of the process

with each of those selections was to provide a mathematical description of the performance

parameters at various combustion Mach numbers. This work uses the seven performance

parameters to analyze and decide on a material and a fuel that enhances the performance of the

scramjet engine.

1.6.1 Materials

The main factors considered during the material selection process includes maximum

service temperatures, oxidation resistance, and fracture toughness. Also, lower thermal

conductivity, such that more heat remains in the combustion chamber and not conducted into

surrounding components, is considered for the material selection. The maximum service

temperatures allow for better performance when employed in the scramjet cycle analysis.

Oxidation resistance plays a role inside of a scramjet to protect against engine corrosion. The

fracture toughness is also an important consideration. It serves as an indicator of how well the

material can endure in such an intense environment, especially in stress fractures or cracks.

Several different material groups were explored for the scramjet application. These

materials included exotic metals, exotic high-temperature composites (such as carbon-carbon),

Page 40: Parametric Scramjet Analysis - eGrove

15

and ultra-high temperature ceramics. Each of these material groups had their own benefits, but

the material group chosen for the analysis portion of this research is ultra-high temperature

ceramics (UHTCs). UHTCs consist of borides, carbides, and nitrides of transition elements such

as zirconium, hafnium and tantalum [13]. Benefits of these materials include high operating

temperatures, high hardness, good wear resistance, good mechanical strength, and good chemical

and thermally stability; these materials can be found in several forms, which include monolith

(solid pieces of the material), matrix (composite), and coatings [14]. The maximum service

temperatures [15] for various materials are show in Fig. 1-6. The three materials selected for this

application include zirconium oxide ceramic, carbon-carbon carbide, and nickel chromium alloy

according to the given properties defined in [13, 14, 16]. The maximum service temperatures

used in this analysis are 2700 K (zirconium oxide ceramic), 2300 K (carbon-carbon carbide), and

1700 K (nickel chromium alloy).

1.6.2 Fuels

There is a limitation of fuel selection since the most hydrocarbon jet fuels in practical use

have a small range of hPR (Lower heating value) except hydrogen fuel. Hydrogen fuel has a very

high hPR values almost three times the value of the other jet fuels [17]. To address the issue

elevated stagnation temperature, an active cooling system using liquid hydrogen fuel technology

is possible. However, liquid hydrogen needs to be maintained at very low temperatures for

storage, around 20 K. Additionally, the density of liquid hydrogen is significantly lower than the

other jet fuels, which denotes liquid hydrogen fuel to be difficult for use. These two factors need

to be overcome for a better performance scramjet engine. Recently, the aircraft Ion Tiger from

the Navy showed an electric fuel cell propulsion system which is fueled by liquid hydrogen. The

Page 41: Parametric Scramjet Analysis - eGrove

16

fuel cell showed a four times higher efficiency to a comparable internal combustion engine, and

seven times higher the energy in the equivalent weight of batteries which demonstrates hydrogen

fuel technology [18]. Hydrocarbon jet fuels have very high densities, making them

comparatively easy to handle. However, in addition to their lower hPR value, they also present a

threat to the environment and to persons exposed to the hydrocarbon fuels [17].

Figure 1-6. Maximum Service Temperature of Various Materials [15].

With these considerations, the three fuels of interest for this application include liquid

hydrogen, JP-7, and JP-5 according to the given properties defined in [17]. The three hPR values

appropriate for these three fuels are 120000 kJ/kg (liquid hydrogen), 43900 kJ/kg (JP-7), and

42800 kJ/kg (JP-5). The assumption will be made that storage constraints have been overcome

for the liquid hydrogen fuels, and that proper handling has been achieved.

Page 42: Parametric Scramjet Analysis - eGrove

17

Using materials and fuels which have higher maximum service temperature and higher

hPR value allows for more work to be extracted from the scramjet cycle. This can be seen in the

T-s diagram for scramjet engines in chapter 2. Scramjet propulsion allows for greater work

extraction than ramjet propulsion due to the higher combustion temperatures. The greater work

output form this cycle is what allows scramjet to reach higher Mach numbers than ramjet.

1.7 This Work

The parametric ideal scramjet cycle analysis is fundamentally based on the Brayton cycle

[4] which is described via the temperature versus entropy (T-s) diagram. The seven performance

parameters of the ideal scramjet engine represent its performance; these are significantly

dependent on the maximum material temperature that the engine can manage, and the lower

heating value that the jet fuel can provide. Thus, this research is being done to show the different

performances of a scramjet engine for different materials that can withstand the extreme heat

created in the burner and different fuels with lower heating values that can yield high

performances.

This work studies the thermodynamic changes in an ideal scramjet engine with fixed

material limits and fuel hPR values to show comparison and impact on scramjet performance. The

performance for the ideal scramjet engine will be described mathematically the seven

performance parameters such as the specific thrust (F/ ṁo), the thrust-specific fuel consumption

(S), the fuel-to-air ratio ( f ), the thermal, propulsive, overall efficiencies (T, p, o), and the

thrust flux (F/A2). Each result figure is shown with each performance parameter versus

freestream Mach number at various combustion Mach numbers (Mc) so that the specific material

or specific fuel can be simply compared and analyzed. Also, the results will show the optimum

Page 43: Parametric Scramjet Analysis - eGrove

18

freestream Mach number for maximum specific thrust, minimum thrust-specific fuel

consumption, and maximum thrust flux via convenient algebraic expressions. Some of the

results of this work have been published in four peer-reviewed journal articles, [5], [19], [20],

and [21].

Page 44: Parametric Scramjet Analysis - eGrove

19

CHAPTER 2

STATEMENT OF THE PROBLEM

2.1 Definition of the Problem

The main objective of the present work is to investigate and compare the performances of

a scramjet engine for different materials that can withstand the temperatures created in the burner

and different fuels with heating values (hpr) that can yield high performances. This work consists

of two major portions: the first part is to define the seven parametric performance parameters of

the scramjet engine by mathematical analysis; the second part is to show the results presentation

of these parametric characteristics for the scramjet engine. Results are shown with figures

containing each performance parameter versus freestream Mach number (Mo) at various

combustion Mach numbers (Mc). This will allow a better selection for a scramjet engine by

showing the specific material and/or fuel to be simply compared and analyzed. The seven

performance relationships derived are the specific thrust (F/ om ), the thrust-specific fuel

consumption (S), the fuel-to-air ratio ( f ), the thermal, propulsive, overall efficiencies (T, p,

o), and the thrust flux (F/A2). The analysis of the performance of the scramjet engine is

fundamentally based on the Brayton cycle [2, 3, 4, 5, 21] as described in terms of the

temperature versus entropy (T-s) diagram. The different materials and fuels, which are possibly

applicable to scramjet with current technology, are employed to select high service temperatures

or high heating values. Especially for the material selection, the T-s diagram will briefly show

Page 45: Parametric Scramjet Analysis - eGrove

20

how employing different Tmax (maximum useable temperature) impacts the network output per

unit mass of the engine cycle.

This work also presents a comparison of a parametric cycle analysis for the non-ideal

mass flow rate scramjet [21] and the ideal mass flow rate scramjet [5]. By non-ideal mass flow

rate is meant that the mass flow rate of air plus the mass flow rate of fuel is not approximately

equal to the mass flow rate of the air alone,om + fm ≠

om ; whereas for the ideal mass flow rate

analyses in [2, 5] it is assumed that om + fm ≈

om . In the results, the comparison of the non-

ideal mass flow rate scramjet and the ideal mass flow rate scramjet is shown at first, and then the

result figures for comparing different materials or fuels is shown by considering only the non-

ideal mass flow rate scramjet since the non-ideal mass flow rate modeling is more realistic than

the ideal mass flow rate modeling [21].

2.2. T-s Diagram

The analysis for the ideal scramjet engine is based on the Brayton cycle [2, 3, 4, 5, 21] as

described in terms of the temperature versus entropy (T-s) diagram. The Brayton cycle is also

the basis for the parametric description of the ideal turbojet, ideal turbofan, and ideal ramjet

engines [2] and the non-ideal mass flow rate ramjet [3]. The Brayton cycle for the parametric

description of these engines consists of an isentropic inlet process, a constant pressure

combustion process, an isentropic nozzle process, and a constant pressure heat rejection process

where the nozzle exit static pressure is equal to the freestream flight ambient static pressure, Po.

The maximum useable material temperature, Tmax, is equal to the stagnation temperature

inside of a ramjet engine because the combustion Mach number, Mc, is essentially zero; this is

not the case for a scramjet. Since the scramjet burns at a constant combustion Mach number

Page 46: Parametric Scramjet Analysis - eGrove

21

greater than 1, the total burner exit stagnation temperature (defined as T'max) is not equal to the

static burner exit temperature. This condition allows additional net work per unit mass to be

achieved through supersonic combustion. The T-s diagram will show how much more work per

unit mass can be obtained, which is the main advantage of using scramjet. The T-s diagram

employed in this work contains the cases of Mc < Mo and Mc ≥ Mo which shows what is

theoretically possible [5].

2.2.1 Situations for Mc < Mo

The T-s diagrams for both the non-ideal mass flow rate and ideal mass flow rate ramjet

engines and the non-ideal and ideal scramjet engines are shown in Fig. 2-1; these illustrations are

based on the Brayton cycle [21]. The description of the scramjet operation will also be

compared to the ramjet as part of this discussion. Referring to Fig. 2-1, freestream air

approaches the engine inlet (station 0 in Fig. 2-1) at To (ambient altitude temperature), Po

(ambient altitude pressure), and Mo (freestream Mach number). For the ramjet engine, air is

isentropically brought (compressed) nearly to rest at the Tt2 (=Tto) essentially stagnation

conditions. Next a constant-total-pressure (and constant combustion Mach number, Mc ≈ 0) heat

addition (combustion) process then takes place in the combustion chamber to raise the total

temperature to Tmax; note that the total pressure and the static pressure are essentially the same

constant pressure during the ramjet combustion process since Mc ≈ 0. Isentropic expansion then

follows through the exit nozzle along the “ramjet” vertical dotted line in Fig. 2-1; next the gas

flow exits the ramjet engine nozzle exit at a static pressure equal to the freestream flight ambient

static pressure, Po, and at T9. Note that Tmax corresponds to the engine material temperature

limit.

Page 47: Parametric Scramjet Analysis - eGrove

22

Figure 2-1. T-s Diagram Comparison for Ramjet and Scramjet [5].

s

T Tto, Tt2

, Tto T2

T0

T”max

T’max

Tmax Mc 0

Mc > 0; Mc < M o

T’2

Mc = Mo

Mc > Mo

T9’’

T9’

T9

Ramjet Scramjet

Mc >> Mo

T”2

Scramjet

t9’’

t9’

t4, t9

Page 48: Parametric Scramjet Analysis - eGrove

23

For the scramjet engine, the combustion process takes place at supersonic speeds;

therefore the total (stagnation) temperature is never experienced inside the engine. However, it

should be noted, from the first law of thermodynamics, that the amount of heat addition in the

combustion process is determined in terms of the total temperature differences across the burner.

In Fig. 2-1 for the scramjet, the flow again approaches the engine inlet but is not brought

(compressed) nearly to rest as in ramjet, but is brought (slowed) to supersonic speed at T2. Next

combustion starts at temperature T2, and as with the ramjet, heat addition is assumed to occur at

a constant pressure (Brayton cycle) and constant combustion Mach number, but here Mc > 1; this

also implies a constant total pressure across the burner in order to achieve a static gas

temperature of Tmax at the exit of the scramjet burner process. Because of the aforementioned

dependence of heat addition on total temperature differences across the burner, the heat released

in the combustion chamber is dependent on Tt2 (=Tto) and T'max where T'max is defined in Eq. (7a,

25a) in Chapter 3. Finally, the gas exits the burner and is isentropically expanded from Tmax to

T9’ through the exit nozzle along the first “scramjet” vertical line in Fig. 2-1 to the ambient free-

stream pressure, Po. The net work per unit mass done by the engine is equal to the area between

the constant-total-pressure curve through station t0 or t2 and the constant-static-pressure curve

through station 9’. Thus from Fig. 2-1, it can be seen that the net work per unit mass done by a

scramjet engine is greater than that done by the ramjet engine. The same maximum material

limit temperature, Tmax, is experienced by both (ramjet and scramjet) engine materials, but in the

scramjet engine the total (stagnation) gas temperature, T'max, never affects the engine material.

The scramjet can achieve a significantly greater total temperature, and therefore greater net work

per unit mass as a result of this operation. The discussion thus far corresponds to the cases

where Mc < Mo, that is, where the combustion Mach number is less than the freestream flight

Page 49: Parametric Scramjet Analysis - eGrove

24

Mach number; these cases would be the most realistic situations for a scramjet engine and

correspond to the portion of the T-s diagram (Fig. 2-1) which is at or above the constant

pressure, Po, curve through station 0. However, it is theoretically possible for Mc ≥ Mo, that is,

where the combustion Mach number is greater than the freestream flight Mach number ; this is

considered next in this work.

2.2.2 Situations for Mc ≥ Mo

Now the situations where Mc ≥ Mo are considered which corresponds to the lower portion

of the T-s diagram (Fig. 2-1) and to the region along or below the constant pressure curve

through station 0. In this circumstance, in accordance with the Brayton cycle, the flow entering

the engine inlet is isentropically expanded (not compressed) along the dashed line to a lower

pressure and at T’2 and to a higher Mach number (velocity), Mc ≥ Mo. The flow then enters the

combustion chamber and heat addition occurs along the dashed line at a constant (low) pressure

and a constant combustion Mach number, Mc; since the pressure and combustion Mach number

are both constant, then the total pressure is again constant during the combustion process and

equal to the freesteam total pressure, as is the same for all the cases in Fig. 2-1. Again because

of the aforementioned dependence (from first law of thermodynamics) of heat addition on total

temperature differences across the burner, the heat released in the combustion chamber is now

dependent on the difference between Tt2 (=Tto) and T''max where T''max is now defined in Eq. (7b,

25b) in Chapter 3. Next the flow enters the exhaust nozzle where it is compressed (not expanded)

isentropically such that the flow must exit the nozzle at the freesteam pressure, Po, and at T9’’ and

also T9’’ must not exceed the material temperature limit, Tmax. Thus the flow for all Mc > Mo

situations must lie along the same vertical line representing the exit nozzle.

Page 50: Parametric Scramjet Analysis - eGrove

25

The case where Mc = Mo lies alone the station 0 constant pressure (bolded in Fig. 2-1)

line; here heat is added at the freestream Mach number and the freestream pressure, Po; the

combustion chamber exit stagnation temperature corresponds to T''max. Note that for all the cases

of Mc ≥ Mo, the combustion chamber exit stagnation temperature from Fig. 2-1 is given by T''max.

2.3 Materials and Fuels Selection

Shown by the T-s diagram in Fig. 2-1 are the effects that Tmax has on the net work per

unit mass, it is clear that by increasing the material limit of the combustion chamber, the scramjet

will be able to achieve more work and thus increase the performance of the aircraft engine. Thus,

for a given combustion chamber in the scramjet, using materials which have higher maximum

service temperatures Tmax allows for more work to be extracted from the scramjet cycle. For this

reason the maximum service temperature is an important constraint for screening materials to be

used in the combustion chamber. Table 2-1 displays the materials considered in this work.

Table 2-1. Selected Materials and Maximum Service Temperatures.

Material Maximum Service Temperature (K)

Zirconium Oxide Ceramic 2700

Carbon-carbon Carbide 2300

Nickel Chromium Alloy 1700

A higher combustion Mach number means less time for the fuel to properly mix with the

air and be ignited. From this reason, the fuel for the scramjet engine needs to have low energy to

be ignited so that it can properly work in supersonic speed of air. The lower heating value (hpr)

Page 51: Parametric Scramjet Analysis - eGrove

26

of a fuel offers an idea for a fuel selection and thus the fuels with higher hpr is considered for the

fuels in this work. Hydrocarbon jet fuels are currently used for various jet engines and will also

be considered in this work along with hydrogen since hydrogen has a very high hpr.

For this work, however, there is a limitation of fuel selection since the hpr for most

hydrocarbon jet fuels in practical use have a small range of hpr values as compared with

hydrogen fuel. From this reason, JP-7 and JP-5, which have hydrocarbon hpr values, which differ

the most, were selected among the various hydrocarbon jet fuels for this work.

Liquid hydrogen fuel is relatively stable in jet engines and has a very high hpr, almost

three times the value of the other jet fuels. Utilizing liquid hydrogen fuel is still demanding with

the current technology, but the higher hpr is good for this work for potential use in the future.

The assumption will be made that storage constraints have been overcome for the liquid

hydrogen fuels, and that proper handling has been achieved. The fuels selected for this work are

shown in Table 2-2 below.

Table 2-2. Selected Fuels and Lower Heating Values.

Fuel Lower Heating Value (kJ/kg)

Liquid Hydrogen 120000

JP-7 43900

JP-5 42800

With the scramjet problem having been defined via the T-s diagram in Fig. 2-1 for both

the cases of Mc < Mo and Mc ≥ Mo, the performance mathematical descriptions of these cases are

given in Chapter 3.

Page 52: Parametric Scramjet Analysis - eGrove

27

CHAPTER 3

ANALYSIS

3.1 Fundamental Assumptions for Parametric Analysis

The analysis of the performance of a scramjet engine is fundamentally based on the

Brayton cycle [2, 3, 4, 5, 21] as described in terms of the temperature versus entropy (T-s)

diagram. In order to evaluate the performance parameters of the scramjet via the T-s diagram,

the idealization of the scramjet carries several assumptions with it. First, it is assumed that the

compressions and expansions occurring in the cycle are isentropic. The T-s diagram (Fig. 2-1)

defines the isentropic compression in the inlet from state 0 to state 2 and through the nozzle from

total temperature state t9 (=t4) to ambient pressure state 9 by a vertical line on the T-s diagram;

thus

1ττ nd , and 1nd

where dτ is the total temperature ratio across the diffuser, nτ is the total temperature ratio across

the nozzle, d is the total pressure ratio across the diffuser, and n is total pressure ratio across

the nozzle. Next, it is assumed that constant total pressure combustion takes place within the

burner; thus

b =1

where b is the total pressure ratio across the burner. Next, it is assumed that the gas flow

Page 53: Parametric Scramjet Analysis - eGrove

28

through the engine behaves as an ideal gas at any given point, and the gas is calorically perfect.

Finally, it is assumed the pressure at the engine nozzle exhaust is equal to the pressure of the

ambient atmospheric air after expansion through the nozzle, which yields;

,Tch p and o9 PP

Employing these assumptions, each step for parametric analysis will now be applied to scramjet.

However, notice that step 6 in [2] which evaluates a total turbine temperature ratio will not be

included in this work since the scramjet does not have a turbine or compressor. The result of

each step mathematically developed for the seven performance parameters will illustrate the

impact of maximum material operating temperature and fuel heating value on scramjet

performance.

3.2 Parametric Analysis for Ideal Mass Flow Scramjet

The mathematical analysis presented here follows the basic notation and step-by-step

parametric cycle analysis presented in detail in [2]. Application of step 1 [2] yields the thrust as

eoe

c

ooefo A)PP(g

Vm)Vmm(F

and the assumptions for the ideal case are

om + fm ≈ om , and oe PP

since

eV = 9V , and oV = oa oM

then the ideal specific thrust becomes

o

o

9

c

o

c

oe

o

Ma

V

g

a

g

VV

m

F

(1)

Page 54: Parametric Scramjet Analysis - eGrove

29

application of step 2 [2] yields the velocity ratio, o

9

a

V

and since

oa = R oT

the velocity ratio becomes

o

2

99

2

o

9

TR

MTR

a

V

simplification yields

2

9

o

9

2

o

9 MT

T

a

V

(2)

application of step 3 [2] yields the nozzle exit Mach number, 9M ; now it is known isentropically

that

12

99t9 M2

1)(1PP

(2a)

solving for 2

9M yields

1

21

P

PM

1

9

t92

9

(2b)

also it can be written that

9

t9

P

P=

t9

t4

t4

t2

t2

to

to

o

o

9

PP

P

P

P

P

P

P

P

P

1

(2c)

or

9

t9

P

P=

t4

t9

t2

t4

to

t2

o

to

9

o

P

P

P

P

P

P

P

P

P

P=

9

o

P

Pnbdr (2d)

Page 55: Parametric Scramjet Analysis - eGrove

30

where r is toP / oP and where

1 ,1 ,1 nbd , and 9o PP

then the total pressure and static pressure ratio in Fig. 2-1 becomes

9

t9

P

P= r (2e)

thus, since r =

1

r the result becomes from Eq. (2b)

2

or

1

r

2

9 M1

2)1(

1

21)(M

(2f)

or

o999 MM M M (3)

application of step 4 [2] yields the temperature ratio, o

9

T

T

t9

t4

t4

t2

t2

to

to

o

ot9 T

T

T

T

T

T

T

T

TT

(3a)

t9Tnbdro

t4

t9

t2

t4

to

t2

o

too T

T

T

T

T

T

T

T

TT

(3b)

since

1 and ,1 nd ,

the temperature ratio at station 9 is

o

t9

T

T= r b (3c)

the isentropic temperature ratio across the nozzle in Fig 2-1 in terms of pressure is

Page 56: Parametric Scramjet Analysis - eGrove

31

r

9

9t

1

9

t9

rT

T

P

P

(3d)

thus, the temperature ratio, o

9

T

T, from Eq. (3c) and (3d)

o

9

T

T= = =

1

r

br

)(

= = (3e)

thus,

t2

t4

b

o

9

T

T

T (4)

It is in step 5 [2] that the scramjet analysis becomes different from the ramjet analysis. A

simple geometry of the burner is shown in Fig. 3-1 below. Application of the steady-state energy

equation (first law of thermodynamics) to a control volume across the combustion chamber

yields for the ideal ramjet the fuel-to-air ratio as

Figure 3-1. Heat In and Heat Out of Ideal Scramjet Combustion Chamber.

t4oPRft2o hmhmhm

thus

(A)

Page 57: Parametric Scramjet Analysis - eGrove

32

and

o

t2

T

T

o

to

T

T= rτ , rot2 TT , and

b

t2

t4 τT

T (Aa)

yields the fuel-to-air ratio for the ideal ramjet

)ττ(h

Tc1

τ

τ

h

τTc)1(τ

h

τTcr

PR

op

rPR

rop

b

PR

rop

f (B)

since

b

t2

t4

o

t2

o

t4

op

t4p

o

t4

T

T

T

T

T

T

Tc

Tc

h

h r , and

r

b (Ba)

where = t4T / oT = maxT / oT since maxt9t4 TTT

and from equation (A)

1

T

T

h

Tc

t2

max

PR

t2pf (5a)

but yields the fuel-to-air ratio for the ideal scramjet from equation (B)

f r

PR

op

h

Tc (5b)

where from Fig. 2-1 for the ideal scramjet by definition

o

max

T

for Mc < Mo or

o

max''

T

Tτ for Mc ≥ Mo (6)

but from Fig. 2-1 it is seen that

2

cmaxmax M2

1-1TT for Mc < Mo (7a)

and also from Fig. 2-1 and in light of Eq. (3) that

2

omaxmax'' M

2

1-1TT for Mc ≥ Mo (7b)

Page 58: Parametric Scramjet Analysis - eGrove

33

the ideal ramjet corresponds completely to Eq. (7a). Therefore note if Mc ≈ 0 (ramjet), then Eq.

(7a) becomes the same as for the ideal ramjet. Following the development in [2] with

defined by Eqs. (6, 7a or 7b) yields the fuel-to-air ratio for scramjet as

f r

PR

0p

h

Tc (8)

similarly, the specific thrust becomes from Eqs. (1, 4, Ba)

1

g

MaMM

T

T

g

aM

a

V

g

a

g

VV

m

Fb

c

ooo9

o

9

c

oo

o

9

c

o

c

oe

o

1g

Ma

rc

oo

or

1

g

Ma

m

F

rc

oo

o

(9)

the thrust-specific fuel consumption, S, is

S

1g

Ma

)ττ(h

Tc

m

F

f

m

F

m

m

F

m

rc

oo

r

PR

oP

oo

o

f

f

or

S =

1hMa

gTc

r

PRoo

rcop (10)

the thermal efficiency, T, where Tt9 / T9 = rτ from Eq. (3b), o

9

t2

t4

T

T

T

T from Eq. (4), and

99t

c

2

9 hh2g

V , and oto

c

2

o hh2g

V

Page 59: Parametric Scramjet Analysis - eGrove

34

9

t4

t4

t2

t4p

9

o9p

t4t2

9o

t4t2o

t2oot49o

t2t4o

c

2

o

c

2

9

o

in

netout

T

T

T

11

1T

TTc

1T

TTc

1hh

hh1

)h(hm

)h(hm)h(hm

)h(hm

2g

V

2g

Vm

Q

thus

r

T

11η

(11)

the propulsive efficiency, P, is

1V

V

2

VV

V2

2g

V

2g

Vm

)VV(g

mV

W

FVη

o

9o9

o

c

2

o

c

2

9

o

o9

c

o

o

netout

o

P

with o999 MM M M ,

o

9b

T

Tτ , and

r

defined by Eqs. (3, 4, Ba)

o

9

V

V=

2

oo

2

99

MTR

MTR

=

o

9

T

T= b

thus

1

2

1

r

b

P

(12)

and the overall efficiency, o, is

1

21

1

211ηηη

r

r

r

r

r

PTo

or

Page 60: Parametric Scramjet Analysis - eGrove

35

rr

rPTo

12ηηη

(13)

the expression for the thrust flux of a material temperature-limited ideal ramjet engine is given in

[2, 19] as

2

o

o2 A

m

m

F

A

F

(14)

where F is the engine thrust, om is the mass flow rate through the ideal engine, and A2 is the

diffuser exit area. The thrust flux yields a better [2] representation than the specific thrust of the

freestream Mach number (Mo) at which the ideal ramjet/scramjet thrust reaches a maximum

value. The expression in Eq. (14) for the specific thrust (F/ om ) is given by Eq. (9) as

1

τ

τ

g

Ma

m

F

r

λ

c

oo

o

(15)

the expression for the mass flux ( om /A2) in Eq. (14) is given as [2, 19]

3

r

0

0

2

o τT

P

A

A R,g

A

m

(16)

where

2

1

2

22

2

M2

1γ1

2

M

1

A

A

(17)

and

2

RR,g

(18)

Page 61: Parametric Scramjet Analysis - eGrove

36

where M2 is the Mach number at the diffuser exit/burner entrance. Note that Mc is the

ramjet/scramjet combustion Mach number (here M2 = Mc). Mo is the freestream flight Mach

number, r = 1 + [( -1)/2]Mo2, Tmax is the material temperature limit, To is the free-stream

temperature, ao is the freestream speed of sound and hPR is the lower heating value of the fuel.

For the ideal scramjet the values for in Eqs. (8, 9, 10, 12, and 13) are given by Eqs. (6, 7a or

7b); these equations all become identically the same as the ideal ramjet parametric equations [2]

when Mc ≈ 0, which is basically the assumption for ideal ramjet combustion. Also, notice the

thermal efficiency, T, is not a function of and thus the thermal efficiency is the same for

both the ideal ramjet and ideal scramjet.

3.2.1 Summary of Equations – Ideal Mass Flow Scramjet

Specific Thrust

1

g

Ma

m

F

rc

oo

o

(19)

Thrust-Specific Fuel Consumption

S =

1hMa

gTc

r

PRoo

rcop (20)

Fuel-to-Air Ratio

f r

PR

0p

h

Tc (21)

Thermal Efficiency

r

T

11η

(22)

Page 62: Parametric Scramjet Analysis - eGrove

37

Propulsive Efficiency

1

2

1

r

b

P

(23)

Overall Efficiency

rr

rPTo

12ηηη

(24)

Thrust Flux

Equations [14 – 18]

3.3 Parametric Analysis for Non-Ideal Mass Flow Scramjet

The mathematical analysis presented here follows the basic notation and step-by-step

parametric cycle analysis procedure presented in detail in [2], but is for the non-ideal mass flow

rate situation ( om + fm ≠ om ) and hence yields the same expressions as in [3]. Some of the

steps are similarly derived as shown for the ideal mass flow scramjet engine case in Section 3.2

and will not be repeated in here. Application of step 1 [2] yields the thrust as

eoe

c

ooefo A)PP(g

Vm)Vmm(F

and the assumptions for the non-ideal case are,

om + fm ≠ om , and oe PP

hence again

eV = 9V , oV = oa oM , and o

f

m

m

f

the specific thrust becomes

Page 63: Parametric Scramjet Analysis - eGrove

38

o

o

9

c

o

o

Ma

)V(1

g

a

m

F f

(25)

application of step 2 [2] yields the velocity ratio, o

9

a

V, as derived in the same way as Eq. (2) and

hence

2

9

o

9

2

o

9 MT

T

a

V

(26)

application of step 3 [2] yields the exit Mach number, 9M , from Eqs. (2a-2f) where again the

result is the same as Eq. (3)

o9'9'9' MMMM (27)

application of step 4 [2] yields the temperature ratio, o

9

T

T, from Eqs. (3a-3e) where again the

result is the same as Eq. (4)

o

9

bT

Tτ (28)

It is in step 5 [2] that the scramjet analysis becomes different from the ramjet analysis and

the non-ideal mass flow rate becomes different from the ideal mass flow rate case. A simple

geometry of the burner is shown in Fig. 3-2 below. Application of the steady-state energy

equation (first law of thermodynamics) to a control volume across the combustion chamber

yields for the ideal ramjet the fuel-to-air ratio as

t4foPRft2o )hmm(hmhm

Page 64: Parametric Scramjet Analysis - eGrove

39

Figure 3-2. Heat In and Heat Out of non-Ideal Scramjet Combustion Chamber.

thus

f

4t

P

PR

t2t4

t4pPR

t2pt4p

o

f

Tc

h

T-T

Tch

TcTc

m

m

(C)

where Eq. (C) is different from Eq. (A), also

o

t2

T

T

o

to

T

T= rτ , rot2 TT , and

b

t2

t4 τT

T (Ca)

which yields the fuel-to-air ratio for the non-ideal ramjet as

f

1

Tc

h-

-1

Tc

h-

T

T

T

T-1

r

orP

PRb

b

t2P

PR

t2

t4

t2

t4

(D)

since

br

t2

t4

o

t2

o

4t

op

t4p

o

t4

T

T

T

T

T

T

Tc

Tc

h

hτ , and

r

b

(Da)

where = t4T / oT = maxT / oT since maxt9t4 TTT and yields the fuel-to-air ratio for the non-

ideal scramjet as

f

oP

PR

r

rorP

PR

r

Tc

h

Tc

h

1

(29)

Page 65: Parametric Scramjet Analysis - eGrove

40

which is different from Eq. (5b) for the ideal mass flow rate case; also where from Fig. 2-1 for

the ideal scramjet by definition and shown in Eq. (30)

o

max'

T

Tτ for Mc < Mo or

o

max''

T

Tτ for Mc ≥ Mo (30)

where the given max'T and max

''T in Fig. 2-1 and light of Eqs. (7a, 7b)

2

cmaxmax' M

2

1-1TT for Mc < Mo (31a)

2

omaxmax'' M

2

1-1TT for Mc ≥ Mo (31b)

the non-ideal mass flow rate ramjet corresponds completely to Eq. (31a), therefore note if for

ramjet Mc ≈ 0, then Eq. (31a) becomes the same as for the non-ideal ramjet. Following the

development in [2] yields the fuel-to-air ratio as given by Eq. (29) with defined by Eqs. (30,

31a or 31b).

Similarly, the specific thrust from Eq. (25) [3] becomes

1τ)1(g

MaMM

T

T)1(

g

aM

a

)V(1

g

a

m

F

c

oooo

o

9

c

oo

o

9

c

o

o

bff

f

=

1

τ

τ)1(

g

Ma

c

oo

r

f

thus

1)1(

g

Ma

m

F

c

oo

o r

f

(32)

Page 66: Parametric Scramjet Analysis - eGrove

41

which is different from Eq. (A) for the ideal mass flow rate case. The thrust-specific fuel

consumption, S, is [3]

S

1g

Ma

Tc

h-

-1

m

F

m

F

m

m

F

m

c

oo

oP

PR

oo

o

f

f

r

rr

r

f

thus

S =

1MaTc

h-

g-1

oo

oP

PR

c

rrr

r (33)

which is different from Eq. (10) for the ideal mass flow rate case. The thermal efficiency, T,

where Tt9 / T9 = rτ from Eq. (3b), o

f

m

m

f ,

o

9

t2

t4

T

T

T

T , and

99t

c

2

9 hh2g

V , and oto

c

2

o hh2g

V

t2t4

o9

t2t4

ot29t4

t2t4o

c

2

o

o

c

2

9

fo

netout

Thh1

hh11

h)h(1

)h(h)h(h1

h)h(1m

2g

Vm

2g

Vmm

f

f

f

f

fQin

o

t2

t2

t4

t2p

o

9

op

T

T

11

1T

T)1(Tc

1T

T1Tc

1

f

f

thus

r

11ηT (34)

Page 67: Parametric Scramjet Analysis - eGrove

42

Equation (34) is the same as Eq. (11) for ideal mass flow rate case. The propulsive efficiency,

Pη , is

1V

V1

1V

V12

2g

V

2g

V1m

VV)1(m

V

W

FVη

2

o

9

o

9

c

2

o

c

2

9

o

o9

o

o

netout

o

p

f

f

f

fgc

since

o

9

V

V= b , and b =

r

11

112

ηP

r

r

f

f

(35)

which is different from Eq. (12) for the ideal mass flow rate case and the overall efficiency, o, is

[2]

11

1121

1ηηη PTo

r

r

rf

f

(36)

which is different from Eq. (13) for the ideal mass flow rate case.

The expression for the thrust flux of a material temperature-limited non-ideal ramjet

engine is given in [2, 5, 19]; for the non-ideal mass flow rate case the thrust flow development is

given below.

2

o

o2 A

m

m

F

A

F

(37)

Page 68: Parametric Scramjet Analysis - eGrove

43

1)1(

g

Ma

m

F

c

oo

o r

f

(38)

3

r

o

o

*

2

o τT

P

A

A R,g

A

m

(39)

2

1

2

22

2

M2

1γ1

2

M

1

A

A

(40)

2

R

γR,g

(41)

3.3.1 Summary of Equations – Non-Ideal Mass Flow Scramjet

Specific Thrust

1)1(

g

Ma

m

F

c

oo

o r

f

(42)

Thrust-Specific Fuel Consumption

S =

1MaTc

h-

g-1

oo

oP

PR

c

rrr

r (43)

Fuel-to-Air Ratio

f

oP

PR

r

rorP

PR

r

Tc

h

Tc

h

1

(44)

Thermal Efficiency

Page 69: Parametric Scramjet Analysis - eGrove

44

r

T

11η

(45)

Propulsive Efficiency

11

112

ηP

r

r

f

f

(46)

Overall Efficiency

11

1121

1ηηη PTo

r

r

rf

f

(46)

Thrust Flux

Equations [37 – 41]

3.4 Optimum Freestream Mach Number (Mo*) for Maximum Specific Thrust [22]

The results for the seven performance parameters of the scramjet will be fundamentally

defined as a function of freestream Mach number Mo. Results will show that for specific thrust,

thrust-specific fuel consumption, and thrust flux that a commensurate (maximum or minimum)

occurs at a specific freestream Mach number, oM . This work yields an analytical expression for

that commensurate freestream Mach number. Determining the freestream Mach number for

maximum specific thrust, minimum thrust-specific fuel consumption, and for maximum thrust

flux for the case of ideal mass flow rate ( om + fm om ) will be presented here.

Page 70: Parametric Scramjet Analysis - eGrove

45

For the commensurate freestream Mach number for the specific thrust, by following the

same derivation procedure as in [10] or also later in [1], the expression for the freestream Mach

number,

oM , at which the specific thrust in the case of Mc < Mo achieves a maximum is given by

2

1

3/1

o 1)1(

2M

(47)

or inserting Eqs. (6, 7a, 30, 31a) becomes

21

3/1

2

c

o

maxo 1M

2

1-1

T

T

)1(

2M

(48)

The point at which

oM = Mc (= 2.93) is shown in Fig. 3-3 from [5] and determined via Eq. (48)

by setting

oM = Mc in Eq. (48) and solving

which yields

Mc =

21

2/1

o

max

o 1T

T

)1(

2M

(49)

where again Eq. (48) becomes the same as the ramjet expression [5, 23] when Mc = 0. Figure 3-3

also demonstrates the function of Mo* with Mc as a parameter for the case of Mc ≥ Mo as a dashed

line meaning that Mo* has theoretically the same value as Mc.

3.5 Optimum Freestream Mach Number (Mo**) for Minimum Thrust-Specific Fuel

Consumption

The thrust-specific fuel consumption for the ideal mass flow rate scramjet engine is

defined in Eq. (10). This Section includes the ideal scramjet engine for both cases of Mc < Mo

Page 71: Parametric Scramjet Analysis - eGrove

46

Figure 3-3. Ideal Scramjet Commensurate Freestream Mach Number vs Combustion Mach

Number for Maximum Specific Thrust (Mo*) and Minimum Thrust-Specific

Fuel Consumption (Mo**) and Maximum Thrust Flux (Mo

***) [5].

5

6

7

8

9

1

2

3

4

5

0 1 2 3 4 5

Mo

**

*

Mo

*a

nd

Mo

**

Mc

hPR = 42,800 kJ/kgTo = 217 KTmax = 1600 Kcp = 1.004 kJ/(kg.K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo = Mc = 2.93

Mo**

Mo*

Mo***

Mo*

Mo**

2.89

Page 72: Parametric Scramjet Analysis - eGrove

47

and Mc ≥ Mo; the thrust-specific fuel consumption of the ideal scramjet engine parameter has a

different minimum based on whether Mc < Mo or Mc ≥ Mo. Investigating the commensurate

freestream Mach number for the minimum thrust-specific fuel consumption is valuable for

estimating flying time by choosing a commensurate flying condition. Thus, this work enables

the ideal scramjet engine to be designed with the commensurate amount of space and size for

storing the fuel. It is important since a high thrust-to-weight ratio is one of the essential reasons

for choosing a scramjet engine for future use.

The expression for the thrust-specific fuel consumption of a material temperature limited

ideal mass flow scramjet based on the Brayton cycle is given by Eq. (10) as

S =

1hMa

gTc

r

PRoo

rcop (50a)

or,

S =

o

rλr

PRo

cop

M

1/

ha

gTc

0

rλr

PRo

c0p

Mha

gTc (50b)

with the modification that is now defined as

o

max

T

T for Mc < Mo or

o

max''

T

T

for Mc ≥ Mo (51a, b)

where

2

cmaxmax M2

1-1TT for Mc < Mo (52a)

and also that

2

omax

''

max M2

1-1TT for Mc ≥ Mo (52b)

where the term in Eq. (51) represents the ratio between the total stagnation temperature of the

post-combustive engine condition and the freestream flight temperature condition.

Page 73: Parametric Scramjet Analysis - eGrove

48

3.5.1 Optimum Freestream Mach number (Mo**) Formulation for Mc < Mo [24]

Here in Eq. (50b), PRo

c0p

ha

gTc is independent of Mo and for mathematics purposes will be

considered as constant; S is the thrust-specific fuel consumption, Mo is the freestream Mach

number, r = 1 + [( -1)/2]Mo2, where Tmax is the material temperature limit, To is the freestream

temperature, ao is the freesteam speed of sound, PRh is the lower heating value of the fuel, and

Mc is the combustion Mach number. The purpose of this development is to present the

derivation of an algebraic expression for the value of Mo** at which S obtains a minimum value

for the material temperature limited ideal scramjet. Mo** for minimum S can be determined

numerically or graphically. Thus it would be convenient to have a closed-form algebraic

expression for the value of Mo** where S reaches a minimum, and also useful to see the explicit

functional dependency of Mo**.

To determine the Mo value at which S is a minimum, from Eq. (50b) set dS/dMo = 0;

therefore differentiation of Eq. (50b) yields after some simple algebra,

1

1

2dM

dM

r

λ

o

ro = rrλ (53)

Note that,

o

o

r 1)M(dM

d

(54)

thus multiplying Eq. (54) by Mo yields,

1)2(1)M(dM

dM r

2

o

o

ro

(55)

Page 74: Parametric Scramjet Analysis - eGrove

49

Inserting Eq. (55) into Eq. (53) with some straight forward algebraic simplification yields,

02 λ

1/2

r

3/2

r (56)

Now let z =2/1

r , hence Eq. (56) becomes,

02zz λ

3 (57a)

with

1/2

2

o

1/2

r M2

1)(1z

(57b)

or rearranging Eq. (57b) to solve for Mo** yields,

Mo** = 1)(z

1)(

2 2

(58)

where Mo** is the free-stream Mach number at which S is a minimum. Here it is seen that from

Eqs. (57a and 58) that Mo** is a function of γ and τλ only. Thus the next step is to find the

solution for z which satisfies Eq. (57a) and then inset this z value into Eq. (58). Now Eq. (57a) is

of standard form [25]; the standard form is

0baxx3 (59)

where the one real root (physically meaningful root) of Eq. (59) is given [25] by

x = A + B (60)

where,

;27

a

4

b

2

bA

1/332

1/332

27

a

4

b

2

bB

(61)

Here Eq. (59), via Eq. (57a), yields x = z, a = -2, and b = - λ in Eqs. (59, 60, 61). As an

example, the values for Tmax = 1600 K and To = 217 K from the ideal ramjet presentation of [2]

were employed in the equations above to define λ (= Tmax/To) and hence Mo** = 3.51 for the

Page 75: Parametric Scramjet Analysis - eGrove

50

minimum value of S; for Tmax = 1900 K (To = 217 K) then Mo** = 3.58; for Tmax = 2200 K (To =

217 K) then Mo** = 3.63; these values for Mo

** at which S is a minimum agree very well with the

graphical results in [2]. The curves shown in Fig. 3-4, from [24], depict the thrust-specific fuel

consumption, S, as a function of the freestream Mach number, Mo, for the ideal ramjet. The

solid symbols in Fig. 3-4 show the values from the above equations and examples. For each

material temperature-limited case (curve), S attains a minimum value at the points identified by

the solid symbols. Figure 3-4 demonstrates excellent agreement between the graphical results of

[2] and the mathematical approach presented here for determining the value of Mo** at which S is

a minimum. Also shown in Fig. 3-3 is the behavior of **oM (minimum thrust-specific fuel

consumption Mach number) from Eqs. (58, 60, and 61) as a function of Mc; the values of **oM

and

oM do not become equal at any freestream Mach number.

3.5.2 Optimum Freestream Mach number (Mo**) Formulation for Mc ≥ Mo [19]

To determine the Mo value at which S is a minimum for Mc ≥ Mo, insert the definitions of

r and in terms of Mo into Eq. (50b) which yields

o

2

o

1/2

o

max

PRo

cop

M

1M

2

1)γ(11

T

T

ha

gTcS

(62)

Now set dS/ dMo = 0; therefore differentiation of Eq. (62) with respect to Mo and setting it equal

to zero yields

1γM2

11

M

1 2

o2

o

(63)

Page 76: Parametric Scramjet Analysis - eGrove

51

Figure 3-4. Ideal Ramjet (Mc = 0) Thrust-Specific Fuel Consumption (S) vs

Freestream Mach Number (Mo) [24].

S [

mg

/(N

.s)]

Mo

Tmax = 1600 K

hPR = 42,800 kJ/kg

To = 217 K

R = 0.286 kJ/kg.K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Tmax = 1900 K

Tmax = 2200 K

Mo** = 3.63

Mo** = 3.58

Mo** = 3.51

Page 77: Parametric Scramjet Analysis - eGrove

52

or further simplifying yields

12

1)(M

2

o

(64)

or finally

236.21

2M

**

o

for = 1.4 (air) (65)

where Mo** is the freestream Mach number at which S is a minimum for Mc ≥ Mo. Here it is

explicitly seen from Eq. (65) that Mo** is a function of γ only for Mc ≥ Mo. The value of Mo

** is

shown in Fig. 3-5 for Mc ≥ Mo by the dashed line from [19] and is seen to be in excellent

agreement with the graphical results in Fig. 3-5. The parameter values employed in Fig. 3-5 are

the same as used in [2] for the ideal ramjet; this was done to facilitate easy comparison with the

ramjet results in [2].

Depicted in Fig. 3-5 is the function of Mo with Mc as a parameter; for Mc < Mo (solid-line

curve) it is seen that S increases as Mc increases and hence ideal scramjet engines will have the

deficit of operating at higher thrust-specific fuel consumption than the ideal ramjet engine. The

curve labeled Mc = 2.89 has two equal minimum values; one minimum at Mo ≈ 2.24 (Mc > Mo)

and a second minimum at Mo ≈ 3.98 (Mc < Mo); these values are marked in Fig. 3-5 [5].

3.6 Optimum Freestream Mach Number (Mo***) for Maximum Thrust Flux [20]

It would be convenient to have an algebraic expression for computing the critical value of

the freestream Mach number (Mo***) at which the thrust flux in the case of Mc < Mo achieves the

maximum values. To develop an expression for Mo***, one must differentiate Eq. (14) with

respect to Mo, then set this derivative equal to zero and then solve for Mo***; this is presented

below.

Page 78: Parametric Scramjet Analysis - eGrove

53

Figure 3-5. Ideal Scramjet Thrust-Specific Fuel Consumption vs Freestream Mach

Number [19].

40

80

120

160

0 1 2 3 4 5 6 7

S [

mg

/(N

.s)]

Mo

2Mc = 4

2.8910

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 1600 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc Mo

Mc < Mo

Mo** = 2.236

Page 79: Parametric Scramjet Analysis - eGrove

54

Inserting Eqs. (15 and 16) into Eq. (14) yields

3

r

r

λo

2

1MCA

F

(66)

from Eqs. (17 and 18)

o

o

c

o

T

P

A

A Rγ,g

g

aC

(67)

and hence it is seen that C is not a function Mo. Next differentiating Eq. (66) with respect to Mo

and setting the derivative equal to zero yields

0dM

d31M

dM

dM1C

A

F

dM

d

o

r2

r

r

λo

1/2

r

o

λ

3

ro

3

r

r

λ

2o

(68)

note that

o

3/2

r

1/2

r

o

M2

1

dM

d

(69)

since

o

o

r M1dM

d

(70)

and

2

or M2

11

(71)

Inserting Eqs. (69 and 70) into Eq. (68), next applying Eq. (71), then simplifying the algebra, and

lastly multiplying through by r1/2 yields

05667 λ

1/2

rλr

3/2

r (72)

Page 80: Parametric Scramjet Analysis - eGrove

55

Now letting 1/2

rz in Eq. (72) gives

07

5z

7

6z

7

6z λ

2

λ

3 (73)

thus from Eq. (71)

1/2

2

o

1/2

r M2

1)(1z

(74)

or squaring both sides of Eq.(74) and rearranging yields,

Mo*** = 1)(z

1)(

2 2

(75)

Equation (73) shows that z is a function of τλ (z = z( λ )) only and hence from Eq. (75) Mo*** =

Mo***( λ , ). Therefore once the appropriate root (z) of Eq. (73) is found, then the Mach

number, Mo***, at which the thrust flux is a maximum is given by Eq. (75). From [25] the

solution for the three roots of Eq. (73) is obtained in closed form; it is known from [25] that Eq.

(73) has three real unequal roots; but for the range of thermodynamic parameters investigated

here, only one root (physically meaningful) has a value such that z2 > 1 and hence is physically

meaningful when z2 is inserted into Eq. (75).

Now Eq. (73) is of standard form [25]; the standard form is

0rqypyy 23 (76a)

where the one physically meaningful root of Eq. (73) is given [25] by

y = A + B - p/3 (76b)

where

;27

a

4

b

2

bA

1/332

1/332

27

a

4

b

2

bB

(76c)

Page 81: Parametric Scramjet Analysis - eGrove

56

with

a = (3q – p2)/3, and b = (2p3 – 9pq + 27r)/27 (76d)

and the other two mathematical real [25] (non-physically meaningful) roots of Eq.(76a) are

given by

3/p2/3)BA(2/)BA(y2 (76e)

3/p2/3)BA(2/)BA(y3 (76f)

Here Eq. (76a), via Eq. (73), corresponds to y = z, p = - (6/7) λ , q = - (6/7) and r = (5/7) λ

in Eqs. (76a – 76f).

Inserting the values (To = 217K, Po = 19403Pa (altitude = 12km), Tmax = 1900K), given in

[2], associated with Fig. 3-6 into Eqs. (73 – 76d) yields the value of Mo*** (Eq. (73)) and the

corresponding maximum value (inserting Mo*** into Eq. (14)) of the thrust flux. The ideal ramjet

engine most closely corresponds to Mc = 0.1 or Mc = 0.5, and the ideal scramjet engine

corresponds to the cases for Mc ≥ 1. The values of Mo*** from Eq. (75) and the corresponding

commensurate values of F/A2 from Eq. (14) listed in Table 3-1 are in excellent agreement with

the graphical maximum values depicted in Fig. 3-6. Likewise shown in Fig. 3-3 is ***

oM

(maximum thrust flux Mach number); the maximum thrust flux [2, 20] occurs at considerably

higher values than **

oM or

oM .

Page 82: Parametric Scramjet Analysis - eGrove

57

Table 3-1. Optimum Freestream Mach number (Mo***) and Maximum Thrust Flux (F/A2)

for the Ideal Material Limited Ramjet/Scramjet Engine [20].

MC Mo*** F/A2 (N/cm2)

0.1 5.24 63

0.5 5.39 323

1.0 5.82 699

1.5 6.47 1167

2.0 7.29 1753

3.0 9.24 3335

4.0 11.4 5498

5.0 13.7 8262

6.0 16.1 11631

Note: To = 217K, Po = 19403Pa (altitude = 12km), Tmax = 1900K

Page 83: Parametric Scramjet Analysis - eGrove

58

Figure 3-6. Ramjet/Scramjet Thrust Flux as a Function of Freestream Mach Number at

Different Combustion Mach Numbers (Mc), Where To = 217 K, Po = 19403 Pa

(Altitude = 12 km), and Tmax = 1900 K [20].

0

500

1000

1500

2000

2500

3000

3500

1 2 3 4 5 6 7 8 9 10 11

F/A

2(N

/cm

2)

M0

Mc = 1.5

Mc = 2.0

Mc = 2.5

Mc = 0.5

Mc = 3.0

Mc = 0.1

Page 84: Parametric Scramjet Analysis - eGrove

59

CHAPTER 4

RESULTS

4.1 Performance Parameters for Ideal Mass Flow Rate

This chapter contains the results for the seven performance parameters that were

analytically developed in Chapter 3. The seven performance parameters will be presented in the

order of the specific thrust (F/ṁo), the thrust-specific fuel consumption (S), the fuel-to-air ratio (

), the thermal, propulsive, overall efficiencies (T, p, o), and the thrust flux (F/A2). In the

results, the freestream Mach number (Mo) is employed as the abscissa variable for each of the

seven performance functions with combustion Mach number (Mc) taken as a parameter. The

results are shown for several fixed values of Mc to illustrate how employing different values of

Mc affects the performance of scramjet. The value Mc = 0, which is the assumption correspond

to the ideal ramjet (combustion chamber flow rate very low), is employed to illustrate the

advantage of scramjet as compared to ramjet. The different materials and fuels, which are

possible candidates for scramjet with current technology, are employed to evaluate how the

maximum service temperature (Tmax) and lower heating value (hPR) affect the performance

parameters of scramjet. For all figures, the solid-line curves correspond to the cases for the

situations of Mc < Mo, and the dashed-line curves correspond to cases for situations of Mc ≥ Mo.

Both the thick-solid-line and thick-dashed-line curves correspond to certain values of Mc where

Mc = Mo * occurs for specific thrust (F/ ṁo), which depends on the specific chosen material.

Therefore, the results for each material has its own Mc satisfying the condition Mc = Mo *; only

for the thrust-specific fuel consumption S, the thick-solid-line and thick-dashed-line curves were

f

Page 85: Parametric Scramjet Analysis - eGrove

60

left out because the chosen Mc for S corresponds to where Mc has two values of Mo **. The

optimum freestream Mach numbers such as Mo *, Mo

***, and Mo ** at which respectively the

values of the F/ and F/A2 attain a maximum and S attains a minimum are found by taking the

derivative of the equation for these parameters with respect to Mo, and setting this equal to zero

as developed in Chapter 3. The accentuated dots on each line shown in the figures of F/ ṁo, S,

and thrust flux (F/A2) represent an optimum point of the parameter at the corresponding Mo. As

derived in Chapter 3, each curve drawn in the figures in this Section assumes the concept of ṁo +

ṁf ≈ ṁo; thus considering the engine to take in a very low mass of fuel as compared to the mass

of air.

4.1.1 Materials Selection

The parametric performance description of the scramjet engine is significantly

influenced by choosing an engine (combustion chamber) material since the engine maximum

service temperature (Tmax) occurs at the combustion chamber exit and is one of variables in the

mathematical parametric description. Choosing a material for the combustion chamber has a

greater impact on performance for the scramjet engine than when it is applied to the ramjet

engine. This is because the scramjet engine has Mc > 1 while for ramjet Mc ≈ 0. Thus indicating

the performance parameters of an engine operating at a higher Mc become more affected by

material selection. This Section of Chapter 4 is used to illustrate and compare the different

performance parameters among the scramjet and ramjet engines by employing different materials,

and also to show how scramjet has better performance variations than ramjet. The maximum

service temperatures used in this analysis are 1700 K (nickel chromium alloy), 2300 K (carbon-

carbon carbide), and 2700 K (zirconium oxide ceramic). The nominal properties for hPR =

om

Page 86: Parametric Scramjet Analysis - eGrove

61

42,800 kJ/kg, To = 216.7 K, Po = 19403 Pa (altitude = 12 km), cp = 1.004 kJ/(kg K), ao = 294.77

m/s, and γ = 1.4 used in this Section are the same as those employed in [2] for the ideal mass

flow rate ramjet analysis in order to facilitate the comparison with the results in [2]. When the

freestream Mach number (Mo) at which the value of the specific thrust attains a maximum is

equal to the combustion Mach number (where Mc = Mo *), then Mo

* becomes the value in the

figures; Mc = Mo * = 3 for 1700 K, Mc = Mo

* = 3.36 for 2300 K, Mc = Mo * = 3.55 for 2700 K.

4.1.1.1 Specific Thrust

Figures 4-1 to 4-5 illustrate the specific thrust for the scramjet versus Mo for various Mc

and maximum service temperature (Tmax) limits. Shown in Fig. 4-1 is the specific thrust for the

ideal mass flow scramjet (Mc ≥ 1) and the ideal mass flow ramjet (Mc ≈ 0) with a material

maximum service temperature of 1700 K. It is seen from Fig. 4-1 that the scramjet specific

thrust is superior to the ramjet. It is seen that the Mo* increases and shifts to higher freestream

Mach numbers as Mc increases. The thick-solid-line curve for Mc = 3 corresponds to the

situation where the dashed-line curve intersects the solid-line curve at the Mo = 3 value and at

which the (right-hand-side) slope of the specific thrust solid-line curve is zero. To achieve a

high value of Mo and enough thrust, Mc must be increased. At higher Mc, the possible specific

thrust is extended to higher Mo. From Fig. 4-1, for example, ramjet (Mc ≈ 0) can fly at Mo ≈ 3

while scramjet having Mc = 3 can fly with more than Mo ≈ 8. In the case of Mc ≥ Mo, each curve

having a different value of Mc does not employ Mc as a parameter for specific thrust but lies

along the dashed-line curve. Note that a combustion Mach number greater than the flight Mach

number (Mc ≥ Mo) indicates an inlet (diffuser) expansion, rather than compression for scramjet.

From Figs. 4-1 to 4-5, it may be surmised that the large amount of work per unit mass translates

Page 87: Parametric Scramjet Analysis - eGrove

62

into a large amount of specific thrust. Before the consideration of material comes into play, it

appears that greater combustion Mach numbers are desireable because of the greater specific

thrust generated. The accentuated dots also show that the maximum specific thrust increases as

Mc and Tmax increase, and normally occur at Mo ≈ 2.0 to 4.5.

Figures 4-1 to 4-3 employ nominal property values, but Tmax is varied for the materials;

these figures show how specific thrust increases as Tmax increases. When the scramjet (Mc = 3)

with 1700K flies at Mo ≈ 8, it approximately produces 610 [N/(kg/s)] of specific thrust, while the

scramjet (Mc = 3) with 2700 K flying at Mo ≈ 8 produces about 1400 [N/(kg/s)] of specific thrust.

From Fig. 4-4, which employed Mc = 2.5 for the three differnet materials, it is shown how the

increased Tmax values enhance the specific thrust. It is observed from Fig. 4-4 that a scramjet

engine flying at Mo = 8 and Tmax = 2700 K can attain almost the same amount of specific thrust

that is the maximum specific thrust of the scramjet engine with 1700 K with flying Mo = 3.

Figure 4-5 shows the specific thrust comparison with the Tmax of three selected materials where

Mc = Mo * occurs in each of the selected materials. The curves for Mc = 3 for 1700 K, Mc = 3.36

for 2300 K, Mc = 3.55 for 2700 K correspond to the situations where the dashed-line curve

intersects the solid-line curve at the Mo = 3 for 1700 K, Mo = 3.36 for 2300 K, Mo = 3.55 for

2700 K value and at which the (right-hand-side) slope of the specific thrust solid-line curve is

zero.

Page 88: Parametric Scramjet Analysis - eGrove

63

Figure 4-1. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 1700 K) for the Cases

Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2, 2.5)

and at Mo* = Mc ≈ 3.00.

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14

F/ṁ

o [N

/(k

g/s

)]

Mo

2

1

2.5

Mo * = Mc ≈ 3.00

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 1700 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Max F/ṁo

Mc = 0

Page 89: Parametric Scramjet Analysis - eGrove

64

Figure 4-2. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 2300 K) for the Cases

Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2, 2.5,

3) and at Mo* = Mc ≈ 3.36.

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14

F/ṁ

o [N

/(k

g/s

)]

Mo

3

2

1

2.5

Mo * = Mc ≈ 3.36

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Max F/ṁo

Mc = 0

Page 90: Parametric Scramjet Analysis - eGrove

65

Figure 4-3. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 2700 K) for the Cases

Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2, 2.5,

3) and at Mo* = Mc ≈ 3.55.

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14

F/ṁ

o [N

/(k

g/s

)]

Mo

3

2

1

2.5

Mo * = Mc ≈ 3.55

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2700 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Max F/ṁo

Mc = 0

Page 91: Parametric Scramjet Analysis - eGrove

66

Figure 4-4. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Combustion Mach Number (Mc = 2.5) for the Cases Mc < Mo and Mc ≥

Mo at the Maximum Service Temperatures of Three Candidate Materials (Tmax

= 1700 K, 2300 K, 2700 K).

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14

F/ṁ

o [N

/(k

g/s

)]

Mo

2700 K

1700 K

2300 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg∙K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Max F/ṁo

Page 92: Parametric Scramjet Analysis - eGrove

67

Figure 4-5. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Maximum Service Temperatures of Three Candidate Materials (Tmax =

1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo where Mo*= Mc

Occurs in Each Selected Material.

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14

F/ṁ

o[N

/(k

g/s

)]

Mo

2700 K

1700 K

2300 K

Mo * = Mc ≈ 3.55

Mo * = Mc ≈ 3.00

Mo * = Mc ≈ 3.36

Page 93: Parametric Scramjet Analysis - eGrove

68

4.1.1.2 Thrust-Specific Fuel Consumption

Depicted in Figs. 4-6 to 4-10 is the thrust-specific fuel consumption S as a function of Mo

with Mc as a parameter. Shown in Fig. 4-6 is S for the ideal mass flow scramjet (Mc ≥ 1) and

ideal mass flow ramjet (Mc = 0) with a material maximum service temperature (Tmax) 1700 K.

For Mc < Mo (solid-line curves), it is seen that S increases as Mc increases and, hence, the

scramjet engines will have the deficit of operating at a higher S compared to the ramjet engine.

This higher S yields higher specific thrust. Similar to specific thrust for the case of Mc ≥ Mo,

each dashed-line having a different value of Mc does not act as a parameter for S but follows the

dashed-line curve; note that the dashed-line curve has its own minimum value when Mo ≈ 2.22.

As seen in Fig. 4.6, the S curve labeled Mc ≈ 2.9 has two equal minimum values; one minimum

at Mo ≈ 2.22 (Mc ≥ Mo) and a second minimum at Mo ≈ 4.0 (Mc < Mo); this is also shown in

Table 4-1; these values are marked with accentuated dots in Fig. 4-6. As the cases in Figs. 4-6 to

4-10, the comparatively lower Mc corresponds to lower values of S; thus consuming less fuel to

produce a given amount of thrust. However, it is unlike the trends in the specific thrust; higher

Mc is needed to produce a given amount of specific thrust while S also increases. This

undesirable trend of S is not a problem for economical reasons since the distances the flight

vehicle travels occur in less time and thus can be compensated with the higher amount of fuel

used.

Figures 4-6 to 4-8 are for nominal input properties values but with various maximum

service temperaures of the three candidate materials. Each curve has a defined length since it is

also a function of specific thrust; however there is a termination of each curve beause of no

existing value of S when the specific thrust decreases to zero. These figures show that S

increases as the material maximum service temperature increaeses. From Fig. 4-9, for Mc = 2.5

Page 94: Parametric Scramjet Analysis - eGrove

69

for the three different materials, it is seen that the value of S increases as the engine maximum

service temperature increases. Figure 4-10 shows a comparison of S with the Tmax for the three

candidate materials where Mc values correspond to dual minimum values of Mo **. This happens

when the Mc = 2.9 for 1700 K, Mc = 2.94 for 2300 K, and Mc = 2.97 for 2700 K, respectively;

this corresponds to the situation where the dashed-line curve and the solid-line curve at which

the slope of S is zero. The exact values for minimum S and corresponding Mo ** marked with

accentuated dots are shown in the Tables 4-1 to 4-5.

Page 95: Parametric Scramjet Analysis - eGrove

70

Figure 4-6. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Material Maximum Service Temperature (Tmax =

1700 K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.9 Corresponding to Two Values of

Mo**.

40

60

80

100

120

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

2.52.9

Mc = 4

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 1700 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Min S

21

0

Page 96: Parametric Scramjet Analysis - eGrove

71

Table 4-1. Optimum Freestream Mach number (Mo**) Corresponding to Minimum Thrust

Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at Various

Combustion Mach numbers.

MC Mo** S [mg/(N∙s)]

0 3.5 42.7

1.0 3.6 45.1

2.0 3.8 51.3

2.5 3.9 55.3

4.0 4.3 68.8

2.9 2.2 , 4.0 58.7

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Tmax = 1700K

Page 97: Parametric Scramjet Analysis - eGrove

72

Figure 4-7. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Material Maximum Service Temperature (Tmax =

2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.94 Corresponding to Two Values of

Mo**.

40

60

80

100

120

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

2.52.94

Mc = 4

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Min S

2

10

Page 98: Parametric Scramjet Analysis - eGrove

73

Table 4-2. Optimum Freestream Mach number (Mo**) Corresponding to Minimum Thrust-

Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at Various

Combustion Mach numbers.

MC Mo** S [mg/(N∙s)]

0 3.7 46.8

1.0 3.8 49.6

2.0 4.0 56.8

2.5 4.1 61.3

4.0 4.5 76.9

2.94 2.2 , 4.2 65.7

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Tmax = 2300K

Page 99: Parametric Scramjet Analysis - eGrove

74

Figure 4-8. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Material Maximum Service Temperature (Tmax =

2700 K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach

Numbers (Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97 Corresponding to Two Values of

Mo**.

40

60

80

100

120

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

2.52.97

Mc = 4

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2700 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Min S

21

0

Page 100: Parametric Scramjet Analysis - eGrove

75

Table 4-3. Optimum Freestream Mach number (Mo**) Corresponding to Minimum Thrust-

Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at Various

Combustion Mach numbers.

MC Mo** S [mg/(N∙s)]

0 3.7 49.3

1.0 3.8 52.3

2.0 4.1 60.0

2.5 4.2 64.9

4.0 4.6 81.7

2.97 2.2 , 4.3 70.0

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Tmax = 2700K

Page 101: Parametric Scramjet Analysis - eGrove

76

Figure 4-9. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Combustion Mach Number (Mc = 2.5) for the

Cases Mc < Mo and Mc ≥ Mo at the Maximum Service Temperatures of Three

Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K).

40

60

80

100

120

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

2700K2300 K

1700 K

hPR = 42,800 kJ/kgTo = 217 KMc = 2.5

cp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Min S

Page 102: Parametric Scramjet Analysis - eGrove

77

Table 4-4. Optimum Freestream Mach number (Mo**) Corresponding to Minimum Thrust-

Specific Fuel consumption (S) for Ideal Mass Flow Scramjet at the Maximum

Service Temperatures of Three Candidate Materials.

Tmax Mo** S [mg/(N∙s)]

1700 K 3.8 55.3

2300 K 4.1 61.3

2700 K 4.2 64.9

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Mc = 2.5

Page 103: Parametric Scramjet Analysis - eGrove

78

Figure 4-10. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Maximum Service Temperatures of Three

Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo

and Mc ≥ Mo where Mc’s Corresponding to Two Values of Mo**.

40

50

60

70

80

90

100

110

120

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

Mc = 2.972.942.9

hPR = 42,800 kJ/kg

To = 217 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4Mc ≥ Mo

Mc < Mo

Min S2700 K

2300 K

1700 K

Page 104: Parametric Scramjet Analysis - eGrove

79

Table 4-5. Optimum Freestream Mach number (Mo**) Corresponding to Minimum Thrust-

Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at the Maximum

Service Temperatures of Three Candidate Materials.

Tmax MC Mo** S [mg/(N∙s)]

1700 K 2.9 2.2, 4.0 58.7

2300 K 2.94 2.2 , 4.2 65.7

2700 K 2.97 2.2 , 4.3 70.0

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg

Page 105: Parametric Scramjet Analysis - eGrove

80

4.1.1.3 Fuel-to-Air Ratio

Figures 4-11 to 4-15 illustrate the fuel-to-air ratio, , as a function of freestream Mach

number Mo, with the combustion Mach number Mc as a parameter. Shown in Fig. 4-11 is for

the ideal mass flow scramjet (Mc ≥ 1) and ideal mass flow ramjet (Mc = 0) with a material

maximum service temperature of 1700 K. It is seen that increases as the combustion Mach

number increases for a specific Mo , but decreases as the flight Mach number increases for Mc

< Mo. The reaches a maximum at the intersection of the dashed-line and the solid-line curves;

these are the points where Mc = Mo. Figure 4-11 demonstrates that for a scramjet engine

operating at Mo = 7 and Mc = 2.5, the would be about the same ( ≈ 0.034) as a ramjet

operating at Mo = 3.0.

To compare the fuel-to-air ratios, , of the three materials, they are presented in Figs. 4-

11 to 4-13; the impact of varying the value of Tmax is shown. These figures show that

increases as Tmax increases. For a scramjet (Mc = 3) with 1700K flying at Mo ≈ 8, Fig. 4-11

shows that ≈ 0.041 , while a scramjet (Mc = 3) with 2700 K flying Mo ≈ 8 Fig. 4-13 yields ≈

0.107. The trends of Figs. 4-11 to 4-13 are combined in Fig 4-14; this disparity is not as drastic

at lower Mc or greater Mo as the with Tmax = 2700 K decreases more rapidly with increasing

Mo than the fuel-to-air ratios of the other two materials. Figure 4-15 shows the comparison

with Tmax of the three candidate materials where Mc = Mo * occurs for each material. It follows

the trends of the specific thrust curves for Mc = 3 for 1700 K, Mc = 3.36 for 2300 K, and Mc =

3.55 for 2700 K. The maximum value corresponds to the situation where the dashed-line curve

intersects the solid-line curve.

f

f

f

f

f

f f

f

f

f f

f

f

Page 106: Parametric Scramjet Analysis - eGrove

81

Figure 4-11. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 1700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1,

2, 2.5) and at Mo*= Mc ≈ 3.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

Mo * = Mc ≈ 3

2.5

2

Mc = 0

hPR = 42,800 kJ/kgTo = 217 KTmax = 1700 Kcp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

1

Page 107: Parametric Scramjet Analysis - eGrove

82

Figure 4-12. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 2300 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1,

2, 2.5, 3) and at Mo*= Mc ≈ 3.36.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

Mo * = Mc ≈ 3.36

2.5

2

1

hPR = 42,800 kJ/kgTo = 217 KTmax = 2300 Kcp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

3

Mc = 0

Page 108: Parametric Scramjet Analysis - eGrove

83

Figure 4-13. Specific Thrust of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax= 2700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1,

2, 2.5, 3) and at Mo*= Mc ≈ 3.55.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

Mo * = Mc ≈ 3.55

2.5

2

1

hPR = 42,800 kJ/kgTo = 217 KTmax = 2700 Kcp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

3

Mc = 0

Page 109: Parametric Scramjet Analysis - eGrove

84

Figure 4-14. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Combustion Mach Number (Mc= 2.5) for the Cases Mc < Mo and

Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate Materials

(Tmax = 1700 K, 2300 K, 2700 K).

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

hPR = 42,800 kJ/kgTo = 217 KMc = 2.5cp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

2300 K

2700 K

1700 K

Page 110: Parametric Scramjet Analysis - eGrove

85

Figure 4-15. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Maximum Service Temperatures of Three Selected Materials

(Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo where

Mo*= Mc Occurs in Each Candidate Material.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

hPR = 42,800 kJ/kgTo = 217 Kcp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mo*= Mc

2700 K

2300 K

1700 K

Mo * = Mc ≈ 3.55

Mo * = Mc ≈ 3.36

Mo * = Mc ≈ 3.00

Page 111: Parametric Scramjet Analysis - eGrove

86

4.1.1.4 Thermal Efficiency

Figure 4-16 shows the thermal efficiency T of the ideal mass flow rate scramjet. The

thermal efficiency T is a function of τr only (see. Eq. (22)). The definition of τr is the ratio of

ambient altitude total temperature (Tto) to the ambient altitude static temperature (To), and it is a

function of freestream Mach number (Mo) only as derived in Chapter 3. The thermal efficiency

is the same for both the ideal ramjet and ideal scramjet, and is shown here in Fig. 4-16 as well as

in [2, 5, 19-24]; it is clear that both the ideal ramjet and the ideal scramjet exhibit higher thermal

efficiency as the flight Mach number increases. Thus, the engine material and combustion Mach

number (Mc) do not affect T. Because the ideal ramjet cannot exceed Mo ≈ 5, it cannot exceed a

T of about 80%, whereas, because the ideal scramjet can operate at much higher Mo values (Mo

≈ 8), its T can reach about 90%. It also can be seen that T rises rapidly for 0 < Mo < 5 but rises

less rapidly when Mo > 5.

Page 112: Parametric Scramjet Analysis - eGrove

87

Figure 4-16. Thermal Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Three Candidate Material Maximum Service Temperatures

(1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 3, 3.36).

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηT

(%

)

Mo

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 1700 K, 2300 K, 2700 K

Mc = 0, 1, 2, 2.5, 3, 3.36, 3.55

cp = 1.004 kJ/(kg.K)

γ = 1.4

Page 113: Parametric Scramjet Analysis - eGrove

88

4.1.1.5 Propulsive and Overall Efficiency

Figures 4-17 to 4-26 depict the propulsive efficiency P and overall efficiency o as a

function of the freestream Mach number (Mo), with the combustion Mach number Mc as a

parameter. For a given Mo, lower combustion Mach numbers (Mc) yield greater P and O. As

the Mo increases, both P and o increase for a particular combustion Mach number; for the case

of Mc < Mo, the efficiencies are seen to operate more efficiently as Mo increases; for the case of

Mc ≥ Mc, note that the propulsive efficiency is a low constant value, and it decreases as the Tmax

increases as shown in Fig. 4-20. Figures 4-17 to 4-19 show that a ramjet engine (Mc = 0)

operating at Mo = 3 would have a lower propulsive efficiency than a scramjet engine (Mc = 3)

operating at Mo = 8; this demonstrates the advantage of scramjet propulsion.

Similarly from Figs. 4-22 to 4-24, the overall efficiency for an ideal ramjet engine (Mc =

0) operating at Mo = 3 would have a significantly lower overall efficiency than an ideal scramjet

engine (Mc = 3) operating at Mo = 8; this again demonstrates the advantage of scramjet.

Figures 4-17 to 4-19 and Figs. 4-22 to 4-24 show a propulsive efficiency, P, comparison

and an overall efficiency, o, comparison for the maximum service temperaures of the three

candidate materials. From Figs. 4-20 and 4-25, for Mc = 2.5 and the three differenet materials, it

can be seen that the material with 2700 K performs less efficiently than the other two materials.

Figures 4-21 and 4-26 show a comparison of the Tmax for the three materials where Mc = Mo *

occurs for the specific thrust. The curves correspond to Mc = 3 for 1700 K, Mc = 3.36 for 2300

K, and Mc = 3.55 for 2700 K, respectively.

Page 114: Parametric Scramjet Analysis - eGrove

89

Figure 4-17. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax = 1700

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers

(Mc = 0, 1, 2, 2.5) and at Mo * = Mc ≈ 3.

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηp

(%

)

Mo

Mo* =Mc ≈ 3

2.521

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 1700 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Mc = 0

Page 115: Parametric Scramjet Analysis - eGrove

90

Figure 4-18. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax= 2300

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers

(Mc = 0, 1, 2, 2.5, 3) and at Mo * = Mc ≈ 3.36.

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηp

(%

)

Mo

Mo* =Mc ≈ 3.36

2.521

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Mc = 0 3

Page 116: Parametric Scramjet Analysis - eGrove

91

Figure 4-19. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Material Maximum Service Temperature (Tmax = 2700

K) for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers

(Mc = 0, 1, 2, 2.5, 3) and at Mo * = Mc ≈ 3.55.

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηp

(%

)

Mo

Mo* =Mc ≈ 3.55

2.521

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2700 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Mc = 0 3

Page 117: Parametric Scramjet Analysis - eGrove

92

Figure 4-20. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Combustion Mach Number (Mc = 2.5) for the Cases Mc

< Mo and Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K).

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηp

(%

)

Mo

2700 K2300 K1700 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Page 118: Parametric Scramjet Analysis - eGrove

93

Figure 4-21. Propulsive Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach

Number (Mo) with the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo

where Mo* = Mc Occurs in Each Candidate Materials.

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηp

(%

)

Mo

1700 K

hPR = 42,800 kJ/kg

To = 217 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mo*= Mc

2300 K

2700 K

Mo * = Mc ≈ 3.00

Mo * = Mc ≈ 3.55

Mo * = Mc ≈ 3.36

Page 119: Parametric Scramjet Analysis - eGrove

94

Figure 4-22. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 1700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1,

2, 2.5) and at Mo * = Mc ≈ 3.00.

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηo

(%

)

Mo

Mo * = Mc ≈ 3.00

2.51 2

hPR = 42,800 kJ/kg

To = 217K

Tmax = 1700 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Mc = 0

Page 120: Parametric Scramjet Analysis - eGrove

95

Figure 4-23. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 2300 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1,

2, 2.5, 3) and at Mo * = Mc ≈ 3.36.

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηo

(%

)

Mo

Mo * = Mc ≈ 3.36

2.51 2

hPR = 42,800 kJ/kg

To = 217K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Mc = 0 3

Page 121: Parametric Scramjet Analysis - eGrove

96

Figure 4-24. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Material Maximum Service Temperature (Tmax = 2700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1,

2, 2.5, 3) and at Mo * = Mc ≈ 3.55.

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηo

(%

)

Mo

Mo * = Mc ≈ 3.55

2.51 2

hPR = 42,800 kJ/kg

To = 217K

Tmax = 2700 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

Mc = 0 3

Page 122: Parametric Scramjet Analysis - eGrove

97

Figure 4-25. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Combustion Mach Number (Mc= 2.5) for the Cases Mc < Mo and

Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate Materials

(Tmax = 1700 K, 2300 K, 2700 K).

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηo

(%

)

Mo

2700 K2300 K1700 K

hPR = 42,800 kJ/kg

To = 217K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Mc < Mo

Page 123: Parametric Scramjet Analysis - eGrove

98

Figure 4-26. Overall Efficiency of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Maximum Service Temperatures of Three Candidate Materials

(Tmax = 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo where Mo*

Mc Occurs in Each Candidate Material.

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηo

(%

)

Mo

hPR = 42,800 kJ/kg

To = 217K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc > Mo

Mo*= Mc

2700 K

2300 K

1700 K

Mo * = Mc ≈ 3.55

Mo * = Mc ≈ 3.36

Mo * = Mc ≈ 3.00

Page 124: Parametric Scramjet Analysis - eGrove

99

4.1.1.6 Thrust Flux

Figures 4-27 to 4-31 demonstrate the thrust flux behavior as a function of Mo with Mc

held as a parameter. As in the case of specific thrust, there is an accentuated dot for each Mc at

which the maximum thrust flux occurs. The thrust flux, however, gives a more applicable

representation of Mo at which the ideal mass flow ramjet/scramjet thrust reaches a maximum

value. The thrust flux provides a better depiction [2] than the specific thrust of optimum Mo.

From Figs. 4-27 to 4-31, it is clear how rapidly the ideal scramjet thrust flux performance

increases with increasing Mc and how the maximum thrust flux occurs at a considerably higher

freestream Mach number than the maximum specific thrust. The thrust flux shows a strong

contrast between ramjet performance (Mc = 0.1) and the scramjet performance. It can be seen

that with the scramjet engine, a higher thrust flux can be achieved compared to the ramjet which

has a very low thrust flux. Thrust flux in Figs. 4-27 to 4-31 are dominated by cases where Mc <

Mo, the dashed-line curves (Mc ≥ Mo) are hardly visible in the figures.

Figures 4-27 to 4-29 show a comparison of thrust flux impact of different maximum

service temperatures (Tmax) from the three candidate materials. Similar to specific thrust, these

figures show that the thrust flux increases as material Tmax increases. The magnitude of the

increase in thrust flux for the materials, however, is greater than for the specific thrust. Note that

the scale of each figure is changed to accommodate these large differences. Figure 4-30

demonstrates the thrust flux increasing dependancy on the engine material. The accentuated dots

on each line depict the maximum point of thrust flux and the corresponding Mo. The freestream

Mach numbers Mo *** at which the value of the thrust flux attains a maximum are found by

taking the partial derivative of the thrust flux equation as developed in Chapter 3. Tables 4-6 to

4-10 provide accurate values for the maximum thrust flux and corresponding optimum

Page 125: Parametric Scramjet Analysis - eGrove

100

freestream Mach numbers Mo ***. From Tables 4-6 to 4-8, the maximum thrust flux with Mc =

2.5 on 1700K and 2700 K are 1670 [N/cm2] and 8850 [N/cm2] respectively; this indicates more

than 5 times greater thrust flux for the material maximum service temperature of 2700 K. From

Fig. 4-31, the curve for Mc = 3 for 1700 K, Mc = 3.36 for 2300 K, and Mc = 3.55 for 2700 K

correspond to the situations where the specific thrust is a maximum.

Page 126: Parametric Scramjet Analysis - eGrove

101

Figure 4-27. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 1700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0.1,

1, 2, 2.5) and at Mo* = Mc ≈ 3.00.

0

500

1000

1500

2000

2500

1 2 3 4 5 6 7 8 9 10 11

Th

ru

st F

lux

(N

/cm

2)

Mo

Mc = 1

Mc = 2

Mc = 2.5

Mc = 0.1

Mo * = Mc ≈ 3

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 1700 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Po=19403 Pa

Mc > Mo

Mc < Mo

Mo*= Mc

Max Thrust Flux

Page 127: Parametric Scramjet Analysis - eGrove

102

Table 4-6. Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at Various Combustion Mach

numbers.

MC Mo*** F/A2 [N/cm2]

0.1 4.9 42

1.0 5.5 468

2.0 6.9 1180

2.5 7.7 1670

3.0 8.7 2250

Note: To = 217K, Po = 19403Pa (altitude = 12km),hPR = 42,800 kJ/kg, Tmax = 1700K

Page 128: Parametric Scramjet Analysis - eGrove

103

Figure 4-28. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 2300 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0.1,

1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.36.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1 2 3 4 5 6 7 8 9 10 11 12 13

Th

ru

st F

lux

(N

/cm

2)

Mo

Mc = 1

Mc = 2

Mc = 2.5

Mc = 0.1

Mo * = Mc ≈ 3.36

Mc = 3

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Po=19403 Pa

Mc > Mo

Mc < Mo

Mo*= Mc

Max Thrust Flux

Page 129: Parametric Scramjet Analysis - eGrove

104

Table 4-7. Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at Various Combustion Mach

numbers.

MC Mo*** F/A2 [N/cm2]

0.1 5.9 124

1.0 6.5 1380

2.0 8.1 3450

2.5 9.1 4860

3.0 10.2 6540

3.36 11 7930

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Tmax = 2300K

Page 130: Parametric Scramjet Analysis - eGrove

105

Figure 4-29. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Material Maximum Service Temperature (Tmax = 2700 K) for the

Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0.1,

1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.55.

0

2000

4000

6000

8000

10000

12000

14000

16000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Th

ru

st F

lux

(N

/cm

2)

Mo

Mc = 1

Mc = 2

Mc = 2.5

Mc = 0.1

Mo * = Mc ≈ 3.55

Mc = 3

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2700 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Po=19403 Pa

Mc > Mo

Mc < Mo

Mo*= Mc

Max Thrust Flux

Page 131: Parametric Scramjet Analysis - eGrove

106

Table 4-8. Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at Various Combustion Mach

numbers.

MC Mo*** F/A2 [N/cm2]

0.1 6.4 220

1.0 7.1 2440

2.0 8.8 6080

2.5 9.9 8550

3.0 11.1 11500

3.55 12.5 15300

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Tmax = 2700K

Page 132: Parametric Scramjet Analysis - eGrove

107

Figure 4-30. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Combustion Mach Number (Mc = 2.5) for the Cases Mc < Mo and Mc

≥ Mo at the Maximum Service Temperatures of Three Candidate Materials

(Tmax = 1700 K, 2300 K, 2700 K).

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1 2 3 4 5 6 7 8 9 10 11 12

Th

ru

st F

lux

(N

/cm

2)

Mo

2300 K

2700 K

1700 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Po=19403 Pa

Mc > Mo

Mc < Mo

Mo*= Mc

Max Thrust Flux

Page 133: Parametric Scramjet Analysis - eGrove

108

Table 4-9. Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at the Maximum Service

Temperatures of Three Candidate Materials.

Tmax Mo*** F/A2 [N/cm2]

1700 K 7.7 1670

2300 K 9.1 4680

2700 K 9.9 8550

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Mc = 2.5

Page 134: Parametric Scramjet Analysis - eGrove

109

Figure 4-31. Thrust Flux of Ideal Mass Flow Scramjet vs Freestream Mach Number (Mo)

with the Maximum Service Temperatures of Three Candidate Materials (Tmax

= 1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc ≥ Mo where Mo* = Mc

Occurs in Each Candidate Material.

0

2000

4000

6000

8000

10000

12000

14000

16000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Th

ru

st F

lux

(N

/cm

2)

Mo

Mc = 3.55

2300 K

1700 K

3.36

2700 K

3.00

hPR = 42,800 kJ/kg

To = 217 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Po=19403 Pa

Mc > Mo

Mo*= Mc

Max Thrust Flux

Page 135: Parametric Scramjet Analysis - eGrove

110

Table 4-10. Optimum Freestream Mach number (Mo***) Corresponding to Maximum

Thrust Flux (F/A2) of Ideal Mass Flow Scramjet at the Maximum Service

Temperatures of Three Candidate Materials.

Tmax MC Mo*** F/A2 [N/cm2]

1700 K 3.0 8.7 2250

2300 K 3.36 11.0 7930

2700 K 3.55 12.5 15300

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg

Page 136: Parametric Scramjet Analysis - eGrove

111

4.1.1.7 Combustion Mach Number (Mc) and Optimum Freestream Mach Numbers (Mo *, Mo

**, Mo ***)

The optimum freestream Mach number for the specific thrust, determined via Eq. (48)

from Chapter 3, at which the specific thrust for Mc < Mo achieves a maximum, is a function of

combustion Mach number (Mc) and the maximum service temperature (Tmax). Shown in Fig. 4-

32 is the Mo *, optimum freestream Mach number for maximum specific thrust behavior as a

function of Mc with Tmax as a parameter. The points at which Mo * = Mc = 3.00, 3.36, and 3.55

are shown in Fig. 4-32 for each Tmax. The Mo * for specific thrust has a direct relation with Tmax;

as Tmax increases, Mo * increases.

Shown in Fig. 4-33 is the behavior of Mo **, optimum freestream Mach number for

minimum thrust-specific fuel consumption. For Mc = 2.9, 2.94, and 2.97, the dual minimum

values for S from Fig. 4-10 and Table 4-5 are also indicated in Fig. 4-33. As seen from Fig. 4-33,

Mo ** increases as Mc increases until it reaches a certain Mc where the dual minimum values exist.

Likewise, shown in Fig. 4-34 is Mo ***, optimum freestream Mach number for maximum thrust

flux; the maximum thrust flux occurs at considerably higher values than Mo** or Mo* values and

are not affected by the case of Mc ≥ Mo.

Page 137: Parametric Scramjet Analysis - eGrove

112

Figure 4-32. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Number for Maximum Specific Thrust for the Cases Mc < Mo and Mc ≥ Mo at

the Maximum Service Temperatures of Three Candidate Materials (Tmax =

1700 K, 2300 K, 2700 K).

2

3

4

0 1 2 3 4

Mo

*

Mc

2700 K

2300 K

1700 K

Mo*= Mc= 3.00

Mo*= Mc= 3.55

Mo*= Mc= 3.36

hPR = 42,800 kJ/kgTo = 217 K

cp = 1.004 kJ/(kg.K)

γ = 1.4Po = 19403 Pa

Mo ≥ Mc

Mo < Mc

Mo*= Mc

Page 138: Parametric Scramjet Analysis - eGrove

113

Figure 4-33. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Number for Minimum Thrust Specific Fuel Consumption for the Cases Mc <

Mo and Mc ≥ Mo at the Maximum Service Temperatures of Three Candidate

Materials (Tmax = 1700 K, 2300 K, 2700 K).

2

3

4

5

0 1 2 3 4

Mo

**

Mc

hPR = 42,800 kJ/kgTo = 217 K

cp = 1.004 kJ/(kg.K)

γ = 1.4Po = 19403 Pa

Mo ≥ Mc

Mo < Mc

Mo** Transition

2700 K

2300 K

1700 K

Page 139: Parametric Scramjet Analysis - eGrove

114

Figure 4-34. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Number for Maximum Thrust Flux for the Cases Mc < Mo and Mc ≥ Mo at the

Maximum Service Temperatures of Three Candidate Materials (Tmax = 1700

K, 2300 K, 2700 K).

4

5

6

7

8

9

10

11

12

13

14

15

0.1 1.1 2.1 3.1 4.1 5.1

Mo

**

*

Mc

hPR = 42,800 kJ/kgTo = 217 K

cp = 1.004 kJ/(kg.K)

γ = 1.4Po = 19403 Pa

Mo > Mc

1700 K

2300 K

2700 K

Page 140: Parametric Scramjet Analysis - eGrove

115

4.1.1.8 Overall Desirable Material Selection Analysis

In order to select a material that shows the overall best performance among the three

candidate materials, an overall scoring system has been adopted. The overall scoring system is

based on a given flight Mach number (Mo) and a weighted point scale for the seven performance

parameters. For scramjet propulsion, a flight Mach number of 8 is proven from the X-43A

scramjet project. To compare the performance, Mo = 8 for scramjet are used as the basis for

discussion. For the weighted point scale, thrust has been more heavily weighted since a higher

amount of thrust is considered most important for propulsion. Especially thrust flux (F/A2) is

most heavily weighted in the scoring system. Note that a lesser weight of performance is

assigned for thrust-specific fuel consumption (S) and fuel-to-air ratio ( ), and there is no weight

for thermal efficiency (T) since it turns out to be independent of material or combustion Mach

(Mc). For this work the weighted point scale for each parametric performance was chosen as 20

for specific thrust (F/ ), 10 for S, 10 for , zero for T, 10 for propulsive efficiency(P), 10

for overall efficiency (o), and 40 for F/A2, and hence overall 100 points.

Since parametric performances for scramjet vary as a function of Mc, then Mc needs to be

chosen ahead of direct material comparison. Each scramjet each parametric performance

parameter at any Mc is significantly better than ramjet. Thus Mc = 2.5 and 3 were chosen for

comparison since the other Mc values show not as good of performance. As seen in Table 4-11,

only F/ shows better performance at Mc = 3 than Mc = 2.5; whereas, the other six

performance parameters show better performance at Mc = 2.5 except the case for thrust flux with

Tmax = 1700 K as seen in Fig. 4-27. For the total score, the value of Mc = 3 is better for only Tmax

= 1700 K; Mc = 2.5 is best for Tmax = 2300 K, 2700 K, and Mc = 3 is best for Tmax = 1700 K.

From Table 4-12, the Tmax = 2700 K (zirconium oxide ceramic) with Mc = 2.5 is better

f

om f

om

Page 141: Parametric Scramjet Analysis - eGrove

116

for specific thrust than the other two materials because scramjet with the Tmax = 2700 K achieves

the highest F/ among three candidate materials. In order for the scramjet engine to operate

effectively with contemporary fuel technology, a lower S and lower are desirable. For thrust-

specific fuel consumption, the Tmax = 1700 K (nickel chromium alloy) is recommended over the

other two materials since this engine achieves the lowest value of S. Interestingly, S for Tmax =

2300 K is slightly higher than for Tmax = 1700 K while they show nearly the same value of S; this

is because Tmax = 2300 K operates at lower Mc. For fuel-to-air ratio, the Tmax = 1700 K (nickel

chromium alloy) is more appropriate than the other two candidate materials because of its lower

values of . For propulsive efficiency and overall efficiency, Tmax = 1700 K (nickel chromium

alloy) is more desirable than the other two candidate materials due to the decreasing trend of P

and o as Tmax increasing. For thrust flux, Tmax = 2700 K (zirconium oxide ceramic) shows much

better performance than the other two candidate materials as shown in Table 4-12.

Though only specific thrust and thrust flux show better performance with Tmax = 2700 K,

the total score for Tmax = 2700 K is superior to the other two candidate materials. The results of

this Section indicate the preferred material to be for Tmax = 2700 K (zirconium oxide ceramic).

om

f

f

Page 142: Parametric Scramjet Analysis - eGrove

117

Table 4-11. Scoring System for Selecting the Combustion Mach Number (Mc) Among 2.5

and 3 at the Maximum Service Temperatures of Three Candidate Materials

(Tmax = 1700 K, 2300 K, 2700 K); Mo = 8.

Tmax 1700 K 2300K 2700 K

Mc 2.5 3 2.5 3 2.5 3

F/ (20) - 20 - 20 - 20

S (10) 10 - 10 - 10 -

(10) 10 - 10 - 10 -

T (0) - - - - - -

p (10) 10 - 10 - 10 -

o (10) 10 - 10 - 10 -

F/A2 (40) - 40 40 - 40 -

Total 40 60 80 20 80 20

om

f

Page 143: Parametric Scramjet Analysis - eGrove

118

Table 4-12. Scoring System for Selecting the Material Among Three Maximum Service

Temperatures (Tmax) of Three Candidate Materials where 1700 K at

Combustion Mach Number (Mc) = 3, 2300 K at Mc = 2.5, 2700 K at Mc = 2.5;

Mo = 8.

Tmax 1700 K at Mc = 3 2300 K at Mc = 2.5 2700 K at Mc = 2.5

F/ (20) - - 20

S (10) 10 - -

(10) 10 - -

T (0) - - -

p (10) 10 - -

o (10) 10 - -

F/A2 (40) - - 40

Total 40 0 60

om

f

Page 144: Parametric Scramjet Analysis - eGrove

119

4.1.2 Fuel Selection

Similar to the material selection, choosing a fuel for combustion also significantly

influences the parametric performance description of the scramjet; the lower heating value (hPR)

is an index for a fuel indicating the amount of heat released per unit mass of fuel. This hPR value

is one of variables in the mathematical parametric description for thrust-specific fuel

consumption, S, and fuel-to-air ratio, ; the other 5 parametric performances parameters are

independent of hPR. The S and with a greater value of hPR are considered having merits of fuel

storage capability and higher energy. At the same time, the characteristics of hydrogen, such as

the lower density and lower ignition temperature, are also unattractive for propulsion because of

the handling difficulties. An assumption was made to only consider the amount of energy a fuel

can produce; the assumptions will be made that storage constraints have been overcome for the

liquid hydrogen fuel, and that proper handling has been achieved. With these assumptions, the

three candidate fuels chosen in this research are JP-5, JP-7, and liquid hydrogen with hPR values

of 42800 kJ/kg, 43900 kJ/kg, and 120000 kJ/kg respectively; they are currently used in the

propulsion field today. This section of Chapter 4 is used to illustrate and compare the different

performance parameters among the scramjet and ramjet engines by employing different fuels,

and also to show how scramjet has better performance characteristics than ramjet. The

maximum service temperature of 2700 K is chosen as the material basis for each figure to show

comparison with regard to fuel differences. Other than that, the same nominal performance

properties are employed as the previous section for material selection in order to facilitate

comparison with the results in [2].

f

f

Page 145: Parametric Scramjet Analysis - eGrove

120

4.1.2.1 Thrust-Specific Fuel Consumption

Figures 4-35 to 4-38 and Tables 4-13 to 4-16 illustrate the thrust-specific fuel consumption, S,

for scramjet versus Mo for various Mc and lower heating value (hPR) fuels. Shown in Fig. 4-35 to

4-37 are S values for the ideal mass flow scramjet (Mc ≥ 1) and the ideal mass flow ramjet (Mc ≈

0) with hPR values of 42800 kJ/kg, 43900 kJ/kg, and 120000 kJ/kg respectively. The general

behavior of S in this section is the same as depicted in Section 4.1.1.2. Figures 4-35 to 4-37 are

for nominal input properties values but with various hPR values for the three candidate fuels.

These figures show that a higher hPR value yields lower S. Especially liquid hydrogen shows

significantly lower values of S with its very high value of hPR. As seen from the Fig. 4-37, the

minimum S of the ramjet with hydrogen fuel is 17.6 [mg/(N∙s)], which is four times lower S

compared to the minimum S of the scramjet (Mc = 3) with JP-5 fuel as seen from Fig. 4-35 and

Fig. 4-37. Figure 4-38 demonstrates a direct comparison of S performance for the three fuels by

fixing Mc = 2.5. It shows that S decreases as the hPR value increases, and that liquid hydrogen

has outstanding performance among the three candidate fuels. The exact values for minimum S

and corresponding Mo ** are marked with accentuated dots and are tabulated in the Tables 4-13 to

4-16.

Page 146: Parametric Scramjet Analysis - eGrove

121

Figure 4-35. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Fuel Lower Heating Value (hPR = 42,800 kJ/kg)

for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers

(Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97 Corresponding to Two Values of Mo**.

10

30

50

70

90

110

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

10 2.5 Mc = 42.972

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2700 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4Mc ≥ Mo

Mc < Mo

Min S

Page 147: Parametric Scramjet Analysis - eGrove

122

Table 4-13. Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at Various

Combustion Mach numbers.

MC Mo** S [mg/(N∙s)]

0 3.7 49.3

1.0 3.8 42.3

2.0 4.1 60.0

2.5 4.2 64.9

4.0 4.6 81.7

2.94 2.2 , 4.3 69.9

Note: To = 217K, Po = 19403Pa (altitude = 12km), Tmax = 2700 K,hPR = 42,800 kJ/kg

Page 148: Parametric Scramjet Analysis - eGrove

123

Figure 4-36. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Fuel Lower Heating Value (hPR = 43,900 kJ/kg)

for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers

(Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97 Corresponding to Two Values of Mo**.

10

30

50

70

90

110

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

10 2.5 Mc = 42.972

hPR = 43,900 kJ/kg

To = 217 K

Tmax = 2700 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4Mc ≥ Mo

Mc < Mo

Min S

Page 149: Parametric Scramjet Analysis - eGrove

124

Table 4-14. Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at Various

Combustion Mach numbers.

MC Mo** S [mg/(N∙s)]

0 3.7 48.0

1.0 3.8 51.0

2.0 4.1 58.5

2.5 4.2 63.3

4.0 4.6 79.7

2.94 2.2 , 4.3 67.8

Note: To = 217K, Po = 19403Pa (altitude = 12km), Tmax = 2700 K, hPR = 43,900 kJ/kg

Page 150: Parametric Scramjet Analysis - eGrove

125

Figure 4-37. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Fuel Lower Heating Value (hPR = 120,000 kJ/kg)

for the Cases Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers

(Mc = 0, 1, 2, 2.5, 4) and at Mc ≈ 2.97 Corresponding to Two Values of Mo**.

10

30

50

70

90

110

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

10 2.5 Mc = 42.972

hPR = 120,000 kJ/kg

To = 217 K

Tmax = 2700 K

cp = 1.004 kJ/(kg∙K)

γ = 1.4Mc ≥ Mo

Mc < Mo

Min S

Page 151: Parametric Scramjet Analysis - eGrove

126

Table 4-15. Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel Consumption (S) for Ideal Mass Flow Scramjet at Various

Combustion Mach numbers.

MC Mo** S [mg/(N∙s)]

0 3.7 17.6

1.0 3.8 18.7

2.0 4.1 21.4

2.5 4.2 23.2

4.0 4.6 29.1

2.97 2.2 , 4.3 24.8

Note: To = 217K, Po = 19403Pa (altitude = 12km), Tmax = 2700K, hPR = 120,000 kJ/kg

Page 152: Parametric Scramjet Analysis - eGrove

127

Figure 4-38. Thrust-Specific Fuel Consumption of Ideal Mass Flow Scramjet vs Freestream

Mach Number (Mo) with the Combustion Mach Number (Mc = 2.5) for the

Cases Mc < Mo and Mc ≥ Mo at the Lower Heating Values of Three Selected

Fuels (hPR = 42,800 kJ/kg, 43,900 kJ/kg, 120,000 kJ/kg).

15

35

55

75

95

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

42,800 kJ/kg43,800120,000

Tmax = 2700 KTo = 217 KMc = 2.5

cp = 1.004 kJ/(kg∙K)Mc ≥ Mo

Mc < Mo

Min S

Page 153: Parametric Scramjet Analysis - eGrove

128

Table 4-16. Optimum Freestream Mach number (Mo**) Corresponding to Minimum

Thrust-Specific Fuel consumption (S) for Ideal Mass Flow Scramjet at the

Lower Heating Values of Three Candidate Fuels.

hPR Mo** S [mg/(N∙s)]

42,800 kJ/kg 4.2 64.9

43,800 kJ/kg 4.2 63.4

120,000 kJ/kg 4.2 23.2

Note: To = 217K, Po = 19403Pa (altitude = 12km), Tmax = 2700 K, Mc = 2.5

Page 154: Parametric Scramjet Analysis - eGrove

129

4.1.2.2 Fuel-to-Air Ratio

Figures 4-39 to 4-42 illustrate the fuel-to-air ratio, , for the scramjet versus Mo for

various Mc and for various lower heating value (hPR) fuels. Shown in Figs. 4-39 to 4-42 are

for the ideal mass flow scramjet (Mc ≥ 1) and the ideal mass flow ramjet (Mc ≈ 0) for hPR values

of 42800 kJ/kg, 43900 kJ/kg, and 120000 kJ/kg respectively. The general behavior of in this

section is the same as depicted in section 4.1.1.3. Figures 4-39 to 4-42 are for nominal input

properties values but with the various hPR values of the three candidate fuels. These figures show

that a higher hPR value yields a lower . Similarly to the behavior of thrust-specific fuel

consumption, the liquid hydrogen fuel shows a much lower value of with its superior value of

hPR. Figure. 4-42 demonstrates a direct comparison of performance for the three fuels with Mc

= 2.5; it shows that S decreases as the hPR value increases, and the liquid hydrogen fuel has the

best performance among three candidate fuels.

f

f

f

f

f

f

Page 155: Parametric Scramjet Analysis - eGrove

130

Figure 4-39. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Fuel Lower Heating Value (hPR = 42,800 kJ/kg) for the Cases Mc

< Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 3)

and at Mo* = Mc ≈ 3.55.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

Mo * = Mc ≈ 3.55

2.5

2

1

hPR = 42,800 kJ/kgTo = 217 KTmax = 2700 Kcp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

0

3

Page 156: Parametric Scramjet Analysis - eGrove

131

Figure 4-40. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Fuel Lower Heating Value (hPR = 43,900 kJ/kg) for the Cases Mc

< Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc=0, 1, 2, 2.5, 3)

and at Mo* = Mc ≈ 3.36.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

Mo * = Mc ≈ 3.55

2.5

2

1

hPR = 43,900 kJ/kgTo = 217 KTmax = 2700 Kcp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

3

Mc = 0

Page 157: Parametric Scramjet Analysis - eGrove

132

Figure 4-41. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Fuel Lower Heating Value (hPR = 120,000 kJ/kg) for the Cases

Mc < Mo and Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2,

2.5, 3) and at Mo* = Mc ≈ 3.55.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

Mo * = Mc ≈ 3.55

2.52

1

hPR = 120,000 kJ/kgTo = 217 KTmax = 2700 Kcp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

Mo*= Mc

3

Mc = 0

Page 158: Parametric Scramjet Analysis - eGrove

133

Figure 4-42. Fuel-to-Air Ratio of Ideal Mass Flow Scramjet vs Freestream Mach Number

(Mo) with the Combustion Mach Number (Mc = 2.5) for the Cases Mc < Mo and

Mc ≥ Mo at the Lower Heating Values of Three Candidate Fuels (hPR = 42,800

kJ/kg, 43,900 kJ/kg, 120,000 kJ/kg).

0.00

0.04

0.08

0.12

0 2 4 6 8 10 12

ƒ

Mo

Tmax = 2700 KTo = 217 KMc = 2.5cp = 1.004 kJ/(kg∙K)γ = 1.4

Mc ≥ Mo

Mc < Mo

43,900 kJ/kg

42,800 kJ/kg

120,000 kJ/kg

Page 159: Parametric Scramjet Analysis - eGrove

134

4.1.2.3 Combustion Mach Number (Mc) and Optimum Freestream Mach Number (Mo **)

Shown in Fig. 4-43 is the behavior of Mo **, optimum freestream Mach number for

minimum thrust-specific fuel consumption. Unlike the section on material selection, dual

minimum values for Mo** happen only at Mc = 2.97; note that the mathematical model of Mc for

dual values for Mo** is a function of material maximum service temperature Tmax, and Mc = 2.97

is for Tmax 2700 K. The dual minimum values for S occur at Mc = 2.97 from Fig. 4-35 to 4-37

and these are depicted in Table 4-13 to 4-15. As shown in Fig. 4-43, Mo** increases as Mc

increases until Mc reaches at dual minimum values for Mo**. After the dual minimum values for

Mo**, Mo

** is a constant since the Mo** now occurs for the cases where Mc ≥ Mo

Page 160: Parametric Scramjet Analysis - eGrove

135

Figure 4-43. Freestream Mach number of Ideal Mass Flow Scramjet vs Combustion Mach

Number for Minimum Thrust-Specific Fuel Consumption for the Cases Mc <

Mo and Mc ≥ Mo at the Lower Heating Values of Three Candidate Fuels (hPR =

42,800 kJ/kg, 43,900 kJ/kg, 120,000 kJ/kg).

2

3

4

5

0 1 2 3 4

Mo

**

Mc

Tmax = 2700 KTo = 217 K

cp = 1.004 kJ/(kg.K)

γ = 1.4Po = 19403 Pa

Mo ≥ Mc

Mo < Mc

Mo** Transition

42,800 kJ/kg

43,900 kJ/kg

120,000 kJ/kg

Mc = 2.97

Page 161: Parametric Scramjet Analysis - eGrove

136

4.1.2.4 Overall Desirable Fuel Selection Analysis

In order to select the fuel that shows the best overall performance among the three

candidate fuels, an overall scoring system has been adopted. Similar to the overall desirable

material selection analysis, the overall scoring system is based on a given flight Mach number

(Mo). Here Mo = 8 is chosen for scramjet as was done previously. However, unlike the overall

desirable material selection analysis, now only two parametric performance parameters will be

discussed in this section; note that only the thrust-specific fuel consumption, S, and the fuel to-

to-air ratio, , are functions of hPR. Therefore, there will be no graduated weight scale in this

analysis; the scale for the two parametric performances is assigned as 50 for S, 50 for , and

hence overall 100 points.

Since the parametric performance parameters for scramjet vary as a function of Mc, then

Mc needs to be chosen ahead for direct material comparison. Similar to the overall desirable

material selection analysis, Mc = 2.5 and 3 were chosen for comparison. Since this fuel selection

analysis takes into account only S and , the other five parametric performances will be

neglected though they still have their own results in Mc comparison in Table 4-17. As shown in

the Table 4-17, Mc = 2.5 shows lower S and than Mc = 3 for the three candidate fuels. Simply,

from the total score, the value of Mc = 2.5 is the more desirable for all three candidate fuels.

In order for the scramjet engine to operate effectively with current fuel technology, a

lower S and lower are desirable. For both thrust-specific fuel consumption and fuel-to-air

ratio, the hPR = 120000 kJ/kg (liquid hydrogen) is recommended over the other two fuels as

determined from the score in Table 4-18. The results of this section indicate the preferred fuel to

be for hPR = 120000 kJ/kg (hydrogen fuel).

f

f

f

f

f

Page 162: Parametric Scramjet Analysis - eGrove

137

Table 4-17. Scoring System for Selecting the Combustion Mach Number (Mc) Among 2.5

and 3 at the Lower Hearing Values of Three Candidate Fuels (hPR = 42800

kJ/kg, 43900 kJ/kg, 120000 kJ/kg); Mo = 8, Tmax = 2700 K.

hPR 42800 kJ/kg 43900 kJ/kg 120000 kJ/kg

Mc 2.5 3 2.5 3 2.5 3

F/ (0) - - - - - -

S (50) 50 - 50 - 50 -

(50) 50 - 50 - 50 -

T (0) - - - - - -

p (0) - - - - - -

o (0) - - - - - -

F/A2 (0) - - - - - -

Total 100 0 100 0 100 0

om

f

Page 163: Parametric Scramjet Analysis - eGrove

138

Table 4-18. Scoring System for Selecting the Fuel Among Three Lower Heating Values

(hPR) of Three Candidate Fuels at Combustion Mach Number (Mc) = 2.5; Mo =

8, Tmax = 2700 K.

hPR 42800 kJ/kg

at Mc = 2.5

43900 kJ/kg

at Mc = 2.5

120000 kJ/kg

at Mc = 2.5

F/ (0) - - -

S (50) - - 50

(50) - - 50

T (0) - - -

p (0) - - -

o (0) - - -

F/A2 (0) - - -

Total 0 0 100

om

f

Page 164: Parametric Scramjet Analysis - eGrove

139

4.2 Performance Parameters for Non-Ideal Mass Flow Rate

By non-ideal mass flow rate is meant that the mass flow rate of air plus the mass flow

rate of fuel is not approximately equal to solely the mass flow rate of the air, ṁo + ṁf ≈ ṁo;

whereas for the ideal mass flow rate analysis it was assumed that ṁo + ṁf ≠ ṁo. This expanded

parametric performance mathematical modeling gives more realistic results since the mass flow

rate of fuel is no longer considered negligible for the scramjet. This non-ideal mass flow rate

assumption surprisingly shows significant parametric performance improvement, especially for

the scramjets having higher combustion Mach numbers. To illustrate the performance

improvement between the different mass flow rate assumptions, in this section a comparison for

the case of non-ideal mass flow rate versus ideal mass flow rate at various Mc is shown in each

figure. Then the results figures for comparing different materials or fuels is shown at Mc = 2.5,

which is the chosen Mc basis. Similar to Section 4.1 Performance Parameters for Ideal Mass

Flow rate, the seven performance parameters will be presented in the order of the specific thrust

(F/ṁo), the thrust-specific fuel consumption (S), the fuel-to-air ratio ( ), the thermal, propulsive,

overall efficiencies (T, p, o), and the thrust flux (F/A2). The results shown with several fixed

values of Mc are to illustrate how employing different values of Mc impacts the performance of

scramjet. The value Mc = 0, which is the assumption correspond to the ideal ramjet (combustion

chamber mass flow rate is very low), is employed to illustrate the advantages of scramjet.

Similarly to the previous sections, the same three candidate materials and candidate fuels are

employed. For all figures, both the thick-solid-line and thick-dashed-line curves correspond to

the case of non-ideal mass flow rate, and the thin-solid-line and thin-dashed-line curves

correspond to the case of ideal mass flow that were shown previously. The solid-line curves

correspond to the cases for the situations of Mc < Mo, and the dashed-line curves correspond to

f

Page 165: Parametric Scramjet Analysis - eGrove

140

cases for situations of Mc ≥ Mo. The optimum freestream Mach numbers such as Mo *, Mo

***,

and Mo ** are respectively the values of the F/ , and F/A2 to attain a maximum and S to attain

a minimum; the accentuated dots were left out of the figures in this section. Again, as derived in

Chapter 3, each curve drawn in the figures in this section corresponds to both the concept of ṁo

+ ṁf ≈ ṁo and ṁo + ṁf ≠ ṁo.

4.2.1 Materials Selection

This section of Chapter 4 is used to illustrate and compare the different parametric

performance parameters among the scramjet and ramjet engines by employing different materials,

and also to show how the scramjet with the non-ideal mass flow basis has better parametric

performance than with the ideal mass flow rate basis. The employed nominal property values are

the same as previous section, but the case of non-ideal mass flow (ṁo + ṁf ≠ ṁo) was assumed

and added to the figures; these figures show how the case of non-ideal mass flow rate performs

in the parametric performance parameters for the materials candidates. In the materials analysis,

it is seen that the magnitude of the parametric performance change between the case of non-ideal

and ideal mass flow rates is not uniform. The parametric performance description of the

scramjet engine with the non-ideal mass flow rate basis is more significantly influenced by

choosing an engine (combustion chamber) material; the non-ideal mass flow rate basis illustrates

that there are stronger increases in parametric performance for scramjet than the ideal mass flow

rate basis as the employed material maximum service temperature, Tmax, increases. Thus, the

assumption of non-ideal mass flow rate depends more significantly on scramjet parametric

performance parameters as the scramjet operates at both higher Mc and higher Tmax. Since the

initial purpose of this section is to show the differences in the parametric performance

om

Page 166: Parametric Scramjet Analysis - eGrove

141

parameters between the two different mass flow rate assumptions; hence each figure is for a

specific candidate material, with Mc variation as specified in Section 4.1.1. All three candidate

materials with Mc variation are not repeated in this section; Tmax = 2300 K is chosen in each

figure for Mc variation; Mc = 2.5 is chosen for material variation will be shown next.

4.2.1.1 Specific Thrust

Figures 4-44 and 4-45 illustrate the specific thrust for scramjet versus Mo for various Mc

and maximum service temperature (Tmax) limits. Shown in Fig. 4-44 is the specific thrust for the

non-ideal mass flow ramjet and scramjet (ṁo + ṁf ≠ ṁo), and the ideal mass flow ramjet and

scramjet (ṁo + ṁf ≈ ṁo) with a material maximum service temperature of 2300 K. It is seen

from Fig. 4-44 that the specific thrust for the case of non-ideal mass flow rate is superior to the

case of ideal mass flow rate. Also, it is clear that the non-ideal mass flow assumption is not as

influential for ramjet as scramjet; the scramjet with Mc = 3 has a difference of approximately 500

N/(kg/s) maximum (peak) specific thrust difference between non-ideal and ideal mass flow rates,

whereas the ramjet has only about a 50 N/(kg/s) maximum specific thrust difference. For the

scramjet flying at Mo = 8 with Mc = 2.5, the non-ideal mass flow rate showed approximately 30

% greater specific thrust than the ideal mass flow rate. It is seen that the Mo* increases and shifts

to higher freestream Mach numbers, Mo, as Mc increases; the Mo* is a function of Mc and thus

the Mo* is fixed regardless of the mass flow rate assumption. For the case of non-ideal mass

flow rate, the possible specific thrust is extended to higher Mc than the case of ideal mass flow

rate. From Fig. 4-44, for example, ramjet (Mc ≈ 0) with ideal mass flow rate can fly at Mo ≈ 3,

while scramjet having Mc = 2.5 can fly at Mo ≈ 7.6 with about the same amount of specific

thrust; however, ramjet (Mc ≈ 0) with non-ideal mass flow rate can fly at Mo ≈ 3 while scramjet

Page 167: Parametric Scramjet Analysis - eGrove

142

having Mc = 2.5 can fly with Mo ≈ 8.2. This proves the scramjet with the non-ideal mass flow

rate assumption analytically shows a higher specific thrust than the scramjet with the ideal mass

flow rate assumption, and this is because the non-ideal and ideal mass flow rate assumptions

show higher disparity at higher Mc. In the inlet expansion case (Mc ≥ Mo), the slope for the case

of non-ideal mass flow rate is steeper than the case of ideal mass flow. For both non-ideal and

ideal mass flow rates, a higher Mc is essentially employed to achieve a high value of Mo and

enough thrust.

Figures 4-45 employs nominal property values with Mc = 2.5, but Tmax is varied for the

candidate materials; this figure shows how specific thrust increases as Tmax increases for both

non-ideal and ideal mass flow cases. In Fig. 4-45, the increased Tmax values non-uniformly

enhance the specific thrust disparity between the two different mass flow rate assumptions, and

the disparity is more notable at Tmax = 2700 K than Tmax = 1700 K. It is observed from Fig. 4-45

that a scramjet engine flying at Mo = 8 and Tmax = 2700 K shows a nearly 1500 N/(kg/s) specific

thrust differnce between non-ideal and ideal mass flow rate; whereas Tmax = 1700 K shows an

approximately 50 N/(kg/s) difference. This is a huge disparity, and shows how scramjet engines

should employ the case of non-ideal mass flow rate analysis for the scramjet parametric

performance evaluation.

Page 168: Parametric Scramjet Analysis - eGrove

143

Figure 4-44. Specific Thrust of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.36.

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14

F/

ṁo

[N/(

kg

/s)]

Mo

Mo * = Mc ≈ 3.36

2

Mc = 0

3

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

1

2.5

Page 169: Parametric Scramjet Analysis - eGrove

144

Figure 4-45. Specific Thrust of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion Mach Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K).

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12 14

F/

ṁo

[N/(

kg

/s)]

Mo

2700 K

2300 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

1700 K

Page 170: Parametric Scramjet Analysis - eGrove

145

4.2.1.2 Thrust-Specific Fuel Consumption

Depicted in Figs. 4-46 and 4-47 is the thrust-specific fuel consumption, S, as a function

of Mo for various Mc and maximum service temperature (Tmax) limits. Shown in Fig. 4-46 is S

for the non-ideal mass flow ramjet and scramjet (ṁo + ṁf ≠ ṁo) and the ideal mass flow ramjet

and scramjet (ṁo + ṁf ≈ ṁo) for a material maximum service temperature of 2300 K. It is clear

that the non-ideal mass flow assumption is not as influential for ramjet as for scramjet; the

scramjet with Mc = 4 shows greater S difference than the S difference for ramjet (Mc ≈ 0)

between the non-ideal and ideal mass flow rate bases. The S increases as Mc increases and,

hence, the case of non-ideal mass flow rate engines will have the advantage of operating at a

higher Mo compared to the case of ideal mass flow rate engine. For the scramjet flying at Mo = 8

with Mc = 2.5, non-ideal mass flow rate showed approximately 14 % lower thrust-specific fuel

consumption than ideal mass flow rate. The minimum S occurs at higher Mo** in the case of

non-ideal mass flow rate than the case of ideal mass flow rate, and Table 4-19 shows this

numerically. The dashed-line curve has its own minimum value when Mo** ≈ 2.22 no matter

what Mc and mass flow rate are; note that this is the case for inlet expansion where Mc ≥ Mo. It

is seen that the Mo** increases as Mc increases; at higher Mc, the minimum S shifts to higher Mo

**,

and this characteristic is the same for both non-ideal and ideal mass flow cases.

Figures 4-47 is for nominal input properties values with Mc = 2.5, but with various Tmax

for the three candidate materials; the figure shows how S increases as Tmax increases for the both

non-ideal and ideal mass flow cases. In Fig. 4-47, the increased Tmax values non-uniformly

enhance the S disparity from between the two different mass flow rate assumptions, and the

disparity is more notable at Tmax = 2700 K than Tmax = 1700 K. The minimum values of S are

shown in Table 4-20 numerically.

Page 171: Parametric Scramjet Analysis - eGrove

146

The lower S for the non-ideal mass flow rate case than the ideal mass flow rate case

denotes the advantage of applying non-ideal mass flow rate basis for scramjet parametric

perforamcne analysis. The assumption of the non-ideal mass flow is more realistic for

improving fuel consumption and shows better perfomance on specific thrust, both of which are

desirable. The increased specific thrust, more realistic, and less thrust-specific fuel consumption

are the main advantage of applying the assumption of non-ideal mass flow.

Page 172: Parametric Scramjet Analysis - eGrove

147

Figure 4-46. Thrust-Specific Fuel Consumption of Non-Ideal Mass Flow and Ideal Mass

Flow Scramjet vs Freestream Mach Number (Mo) with the Material

Maximum Service Temperature (Tmax = 2300 K) for the Cases Mc < Mo and

Mc ≥ Mo at Various Combustion Mach Numbers (Mc = 0, 1, 2, 4) and at Mc ≈

2.94 Corresponding to Two Values of Mo**.

40

60

80

100

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

Mc = 42.510 2

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 173: Parametric Scramjet Analysis - eGrove

148

Table 4-19. Optimum Freestream Mach number (Mo**) and Corresponding Minimum

Thrust-Specific Fuel Consumption (S) of Non-Ideal Mass Flow and Ideal

Mass Flow Scramjet at Various Combustion Mach numbers (Mc = 0, 1, 2, 2.5,

4).

MC Mo**

Non-Ideal Mass Flow

S [mg/(N∙s)] Mo

** Ideal Mass Flow

S [mg/(N∙s)]

0 3.7 45.4 3.6 46.8

1.0 3.8 47.9 3.7 49.6

2.0 4.1 54.3 3.9 56.8

2.5 4.3 58.2 4.0 61.3

4.0 5.1 71.2 4.5 76.9

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Tmax = 2300K

Page 174: Parametric Scramjet Analysis - eGrove

149

Figure 4-47. Thrust-Specific Fuel Consumption of Non-Ideal Mass Flow and Ideal Mass

Flow Scramjet vs Freestream Mach Number (Mo) with the Combustion Mach

Number (Mc = 2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum

Service Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K,

2700 K).

40

60

80

100

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

1700 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

2700 K

2300 K

Page 175: Parametric Scramjet Analysis - eGrove

150

Table 4-20. Optimum Freestream Mach number (Mo**) and Corresponding Minimum

Thrust-Specific Fuel Consumption (S) of Non-Ideal Mass Flow and Ideal

Mass Flow Scramjet at the Maximum Service Temperatures of Three

Selected Materials (Tmax = 1700 K, 2300 K, 2700 K).

Tmax Mo**

Non-Ideal Mass Flow

S [mg/(N∙s)] Mo

** Ideal Mass Flow

S [mg/(N∙s)]

1700 K 4.0 52.9 3.8 55.3

2300 K 4.3 58.2 4.0 61.3

2700 K 4.4 61.3 4.1 64.9

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Mc = 2.5

Page 176: Parametric Scramjet Analysis - eGrove

151

4.2.1.3 Fuel-to-Air Ratio

Figures 4-48 and 4-49 illustrate the fuel-to-air ratio, , for the scramjet versus Mo for

various Mc and maximum service temperature (Tmax) limits. Shown in Fig. 4-48 is the fuel-to-air

ratio for the non-ideal mass flow ramjet and scramjet (ṁo + ṁf ≠ ṁo) and the ideal mass flow

ramjet and scramjet (ṁo + ṁf ≈ ṁo) with a material maximum service temperature of 2300 K.

Similar to previous figures, Fig. 4-48 shows that the fuel-to-air ratio for the case of non-ideal

mass flow rate is higher than the case of ideal mass flow rate. Also, it is clear that the non-ideal

mass flow assumption is not as influential for ramjet as scramjet; higher Mc has a larger

difference for the non-ideal than the ideal mass flow rate assumption. For both non-ideal and

ideal mass flow rate, a higher Mc is employed to achieve a high value of Mo and hence a higher

fuel-to-air ratio. For the scramjet flying at Mo = 8 with Mc = 2.5, non-ideal mass flow rate

showed approximately 3 % greater fuel-to-air ratio than ideal mass flow rate.

To compare of the three candidate materials, they are presented in Figs. 4-49 with

fixed Mc = 2.5; the impact of varying the value of Tmax is shown. Figure 4-49 shows that

increases as Tmax increases, and the difference for the mass flow rate assumptions is more

pronounced at higher Tmax. The difference of for different mass flow rate assumptions is not

as strong at lower Tmax or at higher Mo.

f

f

f

f

f

f

Page 177: Parametric Scramjet Analysis - eGrove

152

Figure 4-48. Fuel-to-Air Ratio of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 3) and at Mo* = Mc ≈ 3.36.

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

3

Mc = 3.36

2

1

Mc = 0

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

2.5

Page 178: Parametric Scramjet Analysis - eGrove

153

Figure 4-49. Fuel-to-Air Ratio of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion March Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K).

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

2300 K

2700 K

1700 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 179: Parametric Scramjet Analysis - eGrove

154

4.2.1.4 Thermal Efficiency

Figure 4-50 shows the thermal efficiency T of the non-ideal and ideal mass flow rates

scramjet. The thermal efficiency T is a function of τr only (see. Eq. (22)). The definition of τr is

the ratio of ambient altitude total temperature (Tto) to the ambient altitude static temperature (To),

and it is a function of freestream Mach number (Mo) only as derived in Chapter 3. The thermal

efficiency is the same for both non-ideal and ideal mass flow rates, and is shown here in Fig. 4-

50; it is clear that both the ideal ramjet and the ideal scramjet exhibit higher thermal efficiency as

the flight Mach number increases. Thus, the engine material and combustion Mach number (Mc)

do not affect T. Because the ideal ramjet cannot exceed Mo ≈ 5, it cannot exceed a T of about

80%; whereas, because the ideal scramjet can operate at much higher Mo values (Mo ≈ 8), its T

can reach about 90%. It also can be seen that T rises rapidly for 0 < Mo < 5 but rises less

rapidly when Mo > 5.

Page 180: Parametric Scramjet Analysis - eGrove

155

Figure 4-50. Thermal EfficiencyofNon-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Three Selected Material Maximum

Service Temperatures (1700 K, 2300 K, 2700 K) for the Cases Mc < Mo and Mc

≥ Mo at Various Combustion Mach Numbers (Mo = 0, 1, 2, 2.5, 3, 3.36).

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηT

(%

)

Mo

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

Mc = 0, 1, 2, 2.5, 3, 3.36

cp = 1.004 kJ/(kg.K)

γ = 1.4

Page 181: Parametric Scramjet Analysis - eGrove

156

4.2.1.5 Propulsive and Overall Efficiency

Figures 4-51 to 4-54 depict the propulsive efficiency P and overall efficiency o as a

function of the freestream Mach number, Mo, with the combustion Mach number, Mc, as a

parameter. Shown from Figs. 4-51 to 4-54 are for the non-ideal mass flow ramjet and scramjet

(ṁo + ṁf ≠ ṁo) and the ideal mass flow ramjet and scramjet (ṁo + ṁf ≈ ṁo) for a material

maximum service temperature of 2300 K. For the scramjet flying at Mo = 8 with Mc = 2.5, non-

ideal mass flow rate showed approximately 12 % greater propulsive and overall efficiencies than

ideal mass flow rate. It is seen from Figs. 4-51 and 4-53 that both efficiencies for the case of

non-ideal mass flow rate is greater than the case of ideal mass flow rate. The disparity of the

efficiencies between non-ideal and ideal mass flow assumptions in ramjet is fairly small;

however, it shows significant differences when the scramjet employs Mc = 3. In the Fig. 4-51,

the scramjet with Mc = 3 flying at Mo = 8 has P ≈ 90 % for the non-ideal mass flow rate basis;

whereas, P ≈ 80 % for ideal mass flow rate basis. This demonstrates the advantage of the

parametric performance analysis for the non-ideal mass flow rate in scramjet propulsion. For P,

in the inlet expansion case (Mc ≥ Mo), there is an increasing inclined dashed line for the case of

non-ideal mass flow rate, whereas a flat line is shown in the case of ideal mass flow rate.

Figures 4-52 and 4-54 are for P and o comparison respectively by varying Tmax. In

these figures, it is shown how the increased Tmax values contribute to both effciencies for the

different mass flow rate assumptions, and it is more significant at higher Tmax. It is observed

from Fig. 4-54 that a scramjet engine flying at Mo = 8 and Tmax = 2700 K shows nearly 10 %

difference of o between the non-ideal and ideal mass flow rate basis; whereas Tmax = 1700 K

shows about a 2 % difference.

Page 182: Parametric Scramjet Analysis - eGrove

157

Figure 4-51. Propulsive Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet

vs Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2,2.5, 3) and at Mo* = Mc ≈ 3.36.

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηp

(%

)

Mo

Mc = 3.36

3

1

2

0

2.5

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 183: Parametric Scramjet Analysis - eGrove

158

Figure 4-52. Propulsive Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet

vs Freestream Mach Number (Mo) with the Combustion March Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K).

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηp

(%

)

Mo

2700 K

1700 K

2300 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 184: Parametric Scramjet Analysis - eGrove

159

Figure 4-53. Overall Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0, 1, 2, 2.5, 3) and at Mo* = Mc ≈ 3.36.

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηo

(%

)

Mo

2

2.5

Mc = 0

3.36

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 185: Parametric Scramjet Analysis - eGrove

160

Figure 4-54. Overall Efficiency of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number(Mo) with the Combustion March Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Moat the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K).

0%

20%

40%

60%

80%

100%

0 2 4 6 8 10 12 14

ηo

(%

)

Mo

2700 K

2300 K

1700 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 186: Parametric Scramjet Analysis - eGrove

161

4.2.1.6 Thrust Flux

Figures 4-55 and 4-56 demonstrate the thrust flux behavior as a function of Mo with Mc

held as a parameter. Shown in Fig. 4-55 is the thrust flux for the non-ideal mass flow ramjet and

scramjet (ṁo + ṁf ≠ ṁo), and the ideal mass flow ramjet and scramjet (ṁo + ṁf ≈ ṁo) with a

material maximum service temperature of 2300 K. From Fig. 4-55, it is seen how rapidly the

thrust flux performance increases with increasing Mc and how the maximum thrust flux

considerably increases in the case of non-ideal mass flow rate as compared to the case of ideal

mass flow rate. Also, the thrust flux shows a stronger contrast between ramjet performance (Mc

= 0.1) and the scramjet performance in the case of non-ideal mass flow rate. Similar to the

specific thrust, Fig. 4-55 shows that the thrust flux for the case of non-ideal mass flow rate is

superior to the case of ideal mass flow rate; the scramjet with Mc = 3 and flying at Mo = 8 has a

peak thrust flux of approximately 4300 N/cm2 in the case of non-ideal mass flow rate, whereas a

peak thrust flux of about 3000 N/cm2 in the case of ideal mass flow rate. For the scramjet flying

at Mo = 8 with Mc = 2.5, the non-ideal mass flow rate showed approximately 32 % greater thrust

flux than the ideal mass flow rate. However, it is clear that the non-ideal mass flow assumption

is not as much influenced for ramjet as for scramjet. It is seen that the Mo*** increases and shifts

to higher freestream Mach numbers, Mo, as Mc increase in both mass flow rate cases; exact

values for the maximum thrust flux and Mo*** are shown in Table 4-21 and 4-22 numerically.

Figure 4-56 employs nominal property values with Mc = 2.5, but Tmax is varied for the

materials; this figure shows how thrust flux increases as Tmax increases for both non-ideal and

ideal mass flow cases. In Fig. 4-56, the thrust flux difference between the mass flow

assumptions is more pronounced at Tmax = 2700 K than at Tmax = 1700 K. It is observed from

Fig. 4-56 that a scramjet engine flying at Mo = 8 and Tmax = 2700 K shows a thrust flux of nearly

Page 187: Parametric Scramjet Analysis - eGrove

162

1500 N/cm2 difference between the non-ideal and ideal mass flow rate assumptions, whereas at

Tmax = 1700 K the thrust flux difference is approximately 250 N/cm2; this is a significant

disparity, and shows why the scramjet engine analysis should employ non-ideal mass flow rate

for the scramjet parametric performance evaluation.

Page 188: Parametric Scramjet Analysis - eGrove

163

Figure 4-55. Thrust Flux of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Material Maximum Service

Temperature (Tmax = 2300 K) for the Cases Mc < Mo and Mc ≥ Mo at Various

Combustion Mach Numbers (Mc = 0.1, 1, 2, 2.5, 3) and at Mo*= Mc ≈ 3.36.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Th

rust

Flu

x (

N/c

m2)

M0

hPR = 42,800 kJ/kg

To = 217 K

Tmax = 2300 K

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Mc = 3

Mo * = Mc ≈ 3.36

Mc = 2.5

Mc = 0

Mc = 1

Mc = 2

Page 189: Parametric Scramjet Analysis - eGrove

164

Table 4-21. Optimum Freestream Mach number (Mo***) and Corresponding Maximum

Thrust Flux (F/A2) of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet at

Various Combustion Mach numbers (Mc = 0.1, 1, 2, 2.5, 3) and at Mo* = Mc ≈

3.36.

MC Mo***

Non-Ideal Mass Flow

F/A2 [N/cm2]

Ideal Mass Flow

F/A2 [N/cm2]

0.1 5.9 137 124

1.0 6.5 1560 1380

2.0 8.1 4140 3450

2.5 9.1 6100 4860

3.0 10.2 8700 6540

3.36 11.0 11073 7930

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Tmax = 2300K

Page 190: Parametric Scramjet Analysis - eGrove

165

Figure 4-56. Thrust Flux of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion March Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Maximum Service

Temperatures of Three Selected Materials (Tmax = 1700 K, 2300 K, 2700 K).

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

1 2 3 4 5 6 7 8 9 10 11 12

Th

rust

Flu

x (

N/c

m2)

M0

1700 K

2700 K

2300 K

hPR = 42,800 kJ/kg

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 191: Parametric Scramjet Analysis - eGrove

166

Table 4-22. Optimum Freestream Mach number (Mo***) and Corresponding Maximum

Thrust Flux (F/A2) of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet at

the Maximum Service Temperatures of Three Selected Materials (Tmax = 1700

K, 2300 K, 2700 K).

Tmax Mo***

Non-Ideal Mass Flow

F/A2 [N/cm2]

Ideal Mass Flow

F/A2 [N/cm2]

1700 K 7.8 1970 1670

2300 K 9.1 6100 4860

2700 K 9.9 11200 8550

Note: To = 217K, Po = 19403Pa (altitude = 12km), hPR = 42,800 kJ/kg, Mc = 2.5

Page 192: Parametric Scramjet Analysis - eGrove

167

4.2.1.7 Overall Desirable Material Selection Analysis

This section is to select the best performance material among the three candidate

materials based on non-ideal mass flow rate results. This is analogous to the overall desirable

material selection analysis for the case of ideal mass flow rate; the same method and weighted

point system are employed in this section. Similarly, in Table 4-23, scramjet engines of Mc = 2.5

and 3 are compared through the seven parametric performance parameters; both scramjets of Mc

= 2.5 and 3 were proven from previous results that they showed better parametric performance

than ramjet engine when the scramjet flying at Mo = 8 and ramjet flying at Mo = 3 are assumed.

As shown in Table 4-23, only F/ṁo shows better performance at Mc = 3 than Mc = 2.5; whereas

the other six performance parameters show better performance at Mc = 2.5 except the case for

thrust flux with Tmax = 1700 K as seen in Fig. 4-27. For the total score, the value of Mc = 3 is

better for only Tmax = 1700 K; Mc = 2.5 is best for Tmax = 2300 K and 2700 K.

From Table 4-23, the Tmax = 2700 K (zirconium oxide ceramic) is better for specific

thrust than the other two materials. In order for the scramjet engine to operate effectively with

contemporary fuel technology, a lower S and lower are desirable. For thrust-specific fuel

consumption, the Tmax = 1700 K (nickel chromium alloy) is recommended over the other two

materials since this engine achieves the lowest value of S. For fuel-to-air ratio, the Tmax = 1700

K (nickel chromium alloy) is more appropriate than the other two candidate materials because of

its lower values of . For propulsive efficiency and overall efficiency, Tmax = 1700 K (nickel

chromium alloy) is more desirable than the other two candidate materials due to the decreasing

trend of P and o as Tmax increasing. For thrust flux, Tmax = 2700 K (zirconium oxide ceramic)

shows much better performance than the other two candidate materials as shown in Table 4-24.

Though only specific thrust and thrust flux show better performance with Tmax = 2700 K,

f

f

Page 193: Parametric Scramjet Analysis - eGrove

168

the total score for Tmax = 2700 K is superior to the other two candidate materials. Similarly to the

case of ideal mass flow rate, the results of this section indicate the preferred material to be for

Tmax = 2700 K (zirconium oxide ceramic), and this is the same result for the case of ideal mass

flow rate.

Page 194: Parametric Scramjet Analysis - eGrove

169

Table 4-23. Scoring System for Selecting the Combustion Mach Number (Mc) in the Case of

Non-Ideal Mass Flow Rate Among 2.5 and 3 at the Maximum Service

Temperatures of Three Candidate Materials (Tmax = 1700 K, 2300 K, 2700 K);

Mo = 8.

Tmax 1700 K 2300K 2700 K

Mc 2.5 3 2.5 3 2.5 3

F/ (20) - 20 - 20 - 20

S (10) 10 - 10 - 10 -

(10) 10 - 10 - 10 -

T (0) - - - - - -

p (10) 10 - 10 - 10 -

o (10) 10 - 10 - 10 -

F/A2 (40) - 40 40 - 40 -

Total 40 60 80 20 80 20

om

f

Page 195: Parametric Scramjet Analysis - eGrove

170

Table 4-24. Scoring System for Selecting the Material in the Case of Non-Ideal Mass Flow

Rate Among Three Maximum Service Temperatures (Tmax) of Three

Candidate Materials where 1700 K at Combustion Mach Number (Mc) = 3,

2300 K at Mc = 2.5, 2700 K at Mc = 2.5; Mo = 8.

Tmax 1700 K at Mc = 3 2300 K at Mc = 2.5 2700 K at Mc = 2.5

F/ (20) - - 20

S (10) 10 - -

(10) 10 - -

T (0) - - -

p (10) 10 - -

o (10) 10 - -

F/A2 (40) - - 40

Total 40 0 60

om

f

Page 196: Parametric Scramjet Analysis - eGrove

171

4.2.2 Fuel Selection

This section illustrates the impact on the parametric performance parameters by

employing different fuels based on the non-ideal mass flow rate assumption. A comparison of

the non-ideal with the ideal mass flow rate is also described in the figures. As mentioned in the

previous section of fuel selection (ideal mass flow rate assumption), the fuel lower heating value

(hPR) is an index for a fuel indicating the amount of heat released per unit mass of fuel. This hPR

value is one of variables in the mathematical parametric description for thrust-specific fuel

consumption, S, and fuel-to-air ratio, ; the other 5 parametric performance parameters are

independent of hPR. The same assumptions were established for this analysis as already made

previously; namely storage constraints have been overcome for the liquid hydrogen fuel, and that

proper fuel handling has been achieved. The three candidate fuels chosen are JP-5, JP-7, and

liquid hydrogen with hPR values of 42800 kJ/kg, 43900 kJ/kg, and 120000 kJ/kg respectively.

The maximum service temperature of 2300 K is chosen as the material basis for each figure to

show comparison with regard to fuel differences. Other than that, the same nominal properties

are employed as previously for material selection in order to facilitate comparison with the

results in [2].

Since the behavior of employing different combustion Mach number (Mc) on a material

basis has been shown in the fuel selection section already in the case of ideal mass flow rate, the

Mc variation for this section will be omitted; the results for this section will reveal the

comparison of the three candidate fuels with a chosen Mc basis and material basis, these three

fuels are explored with both the cases of non-ideal and ideal mass flow rate results.

f

Page 197: Parametric Scramjet Analysis - eGrove

172

4.2.2.1 Thrust-Specific Fuel Consumption

Figure 4-57 and Table 4-25 illustrate the thrust-specific fuel consumption, S, for scramjet

versus Mo for the lower heating value (hPR) of the candidate fuels; the hPR values of 42800 kJ/kg,

43900 kJ/kg, and 120000 kJ/kg respectively. Shown in Fig. 4-57 is S for the non-ideal mass

flow ramjet and scramjet (ṁo + ṁf ≠ ṁo), and the ideal mass flow ramjet and scramjet (ṁo + ṁf ≈

ṁo) for a material maximum service temperature of 2300 K. Figure 4-57 is for nominal input

properties values with Mc = 2.5, but with various hPR for the three candidate fuels; the figure

shows how S decreases as hPR increases for the both non-ideal and ideal mass flow cases. Figure

4-57 shows that a higher hPR value yields a lower S. Especially liquid hydrogen shows

significantly lower values of S with its very high value of hPR, and that liquid hydrogen has the

best performance among the three candidate fuels. The lower value of S for the non-ideal mass

flow rate case than the ideal mass flow rate case denotes the advantage of applying non-ideal

mass flow rate basis for scramjet parametric performance analysis. As shown in Fig. 4-57, the

difference values of S with liquid hydrogen between the case of non-ideal and ideal mass flow

rates is greater than the other two fuels. The non-ideal mass flow assumption enhances the

advantage when it is applied to using liquid hydrogen fuel for scramjet. The exact values for

minimum S and the corresponding Mo ** are tabulated in the Table 4-25. The dashed-line curve

in Fig. 4-57 has its own minimum value of Mo** ≈ 2.22 no matter what Mc and mass flow rate

are; note that this is the inlet expansion case where Mc ≥ Mo.

Page 198: Parametric Scramjet Analysis - eGrove

173

Figure 4-57. Thrust-Specific Fuel Consumption of Non-Ideal Mass flow and Ideal Mass

Flow Scramjet vs Freestream Mach Number (Mo) with the Combustion

Mach Number (Mc = 2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Lower

Heating Values of Three Selected Fuels (hPR = 42,800 kJ/kg, 43,800 kJ/kg,

120,000 kJ/kg).

15

35

55

75

95

0 2 4 6 8 10 12 14

S [

mg

/(N

∙s)]

Mo

42,800 120,000 kJ/kg 43,800

Tmax = 2300 K

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 199: Parametric Scramjet Analysis - eGrove

174

Table 4-25. Optimum Freestream Mach number (Mo**) and Corresponding Minimum

Thrust-Specific Fuel consumption (S) of Non-Ideal Mass Flow and Ideal Mass

Flow Scramjet at the Lower Heating Values of Three Selected Fuels (hPR =

42,800 kJ/kg, 43,800 kJ/kg, 120,000 kJ/kg).

hPR Mo**

Non-Ideal Mass Flow

S [mg/(N∙s)] Mo

** Ideal Mass Flow

S [mg/(N∙s)]

42,800 kJ/kg 4.3 58.2 4.0 61.3

43,800 kJ/kg 4.3 56.7 4.0 59.9

120,000 kJ/kg 4.3 19.1 4.0 21.87

Note: To = 217K, Po = 19403Pa (altitude = 12km), Tmax = 2300 K, Mc = 2.5

Page 200: Parametric Scramjet Analysis - eGrove

175

4.2.2.2 Fuel-to-Air Ratio

Figure 4-58 illustrates the fuel-to-air ratio, , for the scramjet versus Mo for various Mc

and for various lower heating value (hPR) fuels. Shown in Fig. 4-58 is for the non-ideal mass

flow ramjet and scramjet (ṁo + ṁf ≠ ṁo) and the ideal mass flow ramjet and scramjet (ṁo + ṁf ≈

ṁo) for a material maximum service temperature of 2300 K for hPR values of 42800 kJ/kg, 43900

kJ/kg, and 120000 kJ/kg respectively. Figure 4-58 demonstrates a direct comparison of

performance for the three fuels with Mc = 2.5; it shows that S decreases as the hPR value

increases, and the liquid hydrogen fuel has the best performance among three candidate fuels.

Note that a lower value of , as with the favorable value for S, is desirable for a scramjet engine.

The general behavior of in this section is the same as shown previously. Figure 4-58 shows

that a higher hPR value yields a lower . Especially liquid hydrogen shows significantly lower

values of with its very high value of hPR, and that liquid hydrogen has outstanding

performance among the three candidate fuels. The slightly higher value of for the non-ideal

mass flow rate case than the ideal mass flow rate case denotes that applying the non-ideal mass

flow rate assumption has a deficit on ; however, the differences between the two assumptions

are very small as compared to the other parametric performances. It is seen from Fig. 4-58 that

the two curves for with liquid hydrogen fuel in the two different mass flow rate assumptions

are almost the same with non-distinguishable disparites.

f

f

f

f

f

f

f

f

f

f

Page 201: Parametric Scramjet Analysis - eGrove

176

Figure 4-58. Fuel-to-Air Ratio of Non-Ideal Mass Flow and Ideal Mass Flow Scramjet vs

Freestream Mach Number (Mo) with the Combustion Mach Number (Mc =

2.5) for the Cases Mc < Mo and Mc ≥ Mo at the Lower Heating Values of Three

Selected Fuels (hPR = 42,800 kJ/kg, 43,9 00 kJ/kg, 120,000 kJ/kg).

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0 2 4 6 8 10 12 14

ƒ

Mo

43,900

42,800 kJ/kg

120,000

Tmax = 2300 K

To = 217 K

Mc = 2.5

cp = 1.004 kJ/(kg.K)

γ = 1.4

Mc ≥ Mo

Non-Ideal Mass Flow

Ideal Mass Flow

Page 202: Parametric Scramjet Analysis - eGrove

177

4.2.2.3 Overall Desirable Fuel Selection Analysis

In order to select the fuel that shows the best overall performance among the three

candidate fuels for the non-ideal mass flow rate, an overall scoring system has been adopted.

Similar to the previous overall desirable selection analysis, the overall scoring system is based on

a given flight Mach number (Mo). Here Mo = 8 was chosen for scramjet, as was done previously.

However, unlike the overall desirable material selection analysis, now only two parametric

performance parameters will be discussed in this section; note that only the thrust-specific fuel

consumption, S, and the fuel to-to-air ratio, , are functions of hPR. Therefore, there will be no

graduated weight scale in this analysis; the scale for the two parametric performances is assigned

as 50 for S, 50 for , and hence overall 100 points.

Since the behavior of S and for scramjet in the case of non-ideal mass flow rate shows

very similar results to the case of ideal mass flow, the comparison process of Mc is skipped in

this section; therefore following Mc = 2.5 from the overall desirable fuel selection analysis in the

case of ideal mass flow rate.

Again, in order for the scramjet engine to operate effectively with current fuel

technology, a lower S and lower are desirable. For both thrust-specific fuel consumption and

fuel-to-air ratio, the hPR = 120000 kJ/kg (liquid hydrogen) is recommended over the other two

fuels as determined from the score in Table 4-26. The results of this section indicate the

preferred fuel to be for hPR = 120000 kJ/kg (hydrogen fuel).

f

f

f

f

Page 203: Parametric Scramjet Analysis - eGrove

178

Table 4-26. Scoring System for Selecting the Fuel Among Three Lower Heating Values

(hPR) of Three Candidate Fuels at Combustion Mach Number (Mc) = 2.5; Mo =

8, Tmax = 2700 K.

hPR 42800 kJ/kg

at Mc = 2.5

43900 kJ/kg

at Mc = 2.5

120000 kJ/kg

at Mc = 2.5

F/ (0) - - -

S (50) - - 50

(50) - - 50

T (0) - - -

p (0) - - -

o (0) - - -

F/A2 (0) - - -

Total 0 0 100

om

f

Page 204: Parametric Scramjet Analysis - eGrove

179

4.3 Closing of Results

Having completed the discussion of the performance parameters for both the ideal mass

flow rate and non-ideal mass flow rate, and for the selection of an engine material and a

combustion fuel, the key conclusions of this work will be given next in Chapter 5.

Page 205: Parametric Scramjet Analysis - eGrove

180

CHAPTER 5

CONCLUSION

In this study, the seven parametric performance parameters of ramjet and scramjet

propulsion have been investigated and compared based on Brayton-cycle analysis. The results

for specific thrust (F/ṁo), thrust-specific fuel consumption (S), fuel-to-air ratio ( ), thermal,

propulsive, overall efficiencies (T, p, o), and thrust flux (F/A2) have been shown for

parametric performance analysis for understanding scramjet engine application. These

parameters were modeled using simple algebraic expressions similar to Ref [2]. The results

obtained were in excellent agreement with Mattingly’s expressions for ramjet [2], except that

with the ideal scramjet the expressions for τ are given by Eqs. (6) and (7). The results

presented in this study can be useful to predict the parametric performances of scramjet when

employing different engine materials and fuels. The results demonstrate that the non-ideal mass

flow rate analysis plays a significant role for scramjet propulsion especially at high combustion

Mach numbers. Non-ideal mass flow rate analysis also provides guidance to determine the

appropriate combustion Mach number to achieve better engine performance as the flight velocity

(freestream Mach number) changes. The main findings in this work which were specifically

shown in detail in the Results chapter are as follows:

Ramjet and scramjet comparison (employing different combustion Mach numbers)

Three different candidate materials and fuels comparison

The assumptions of ideal and non-ideal mass flow rate cases comparison

f

Page 206: Parametric Scramjet Analysis - eGrove

181

The main conclusions of this work are summarized below.

The parametric description expressions of the ideal scramjet are the same as ramjet

expression except the ramjet engine takes the combustion Mach number as zero, thus

only τ is changed. This provides a simple and effective means for understanding the

ideal performance of an ideal scramjet engine similar to the ideal ramjet. Thus, the

scramjet engine analysis, which prior to this work did not exist in the research literature,

was derived to include scramjet propulsion behavior via simple algebraic equations and

figures similar to the ramjet engine analysis in [2]. This study showed that scramjet

propulsion can achieve higher freestream velocity by achieving greater specific thrust and

thrust flux.

Among the three candidate engine materials, zirconium oxide ceramic (Tmax = 2700 K)

was chosen as the best material. The higher Tmax value for an engine material is

favorable for the achieving greater thrust for a scramjet engine, though there is a

disadvantage for some other parametric performance parameters. For the scramjet flying

at Mo = 8 with Mc = 2.5, zirconium oxide ceramic for Tmax = 2700 K shows about 340 %

and 223 % greater performance than nickel chromium alloy for Tmax = 1700 for specific

thrust and thrust flux respectively. This study employed a weighted point system, and

thrust was more heavily weighted since a higher amount of thrust was considered most

important for propulsion. Among the three candidate fuels, liquid hydrogen (hPR =

120,000 kJ/kg) was selected as the best fuel. Only the two parametric expressions of

thrust-specific fuel consumption and fuel-to-air ratio are functions of hPR, and a much

higher hPR value for liquid hydrogen helps to reduce the thrust-specific fuel consumption

and fuel-to-air ratio significantly. For a scramjet flying at Mo = 8 with Mc = 2.5, the

Page 207: Parametric Scramjet Analysis - eGrove

182

liquid hydrogen fuel with hPR = 120,000 kJ/kg shows an approximate 67 % and 60 %

lower performance than JP-5 with hPR = 42,800 kJ/kg for thrust-specific fuel

consumption and fuel-to-air ratio respectively. It is assumed here that the storage

constraints have been overcome for the liquid hydrogen fuel, and that proper handling has

been achieved.

The major advantage of modeling the non-ideal mass flow rate case is that it is more

realistic and shows better performance for the scramjet engine especially at higher

combustion Mach numbers. For scramjet propulsion, which consumes a significant

amount of fuel, the mass of fuel cannot be ignored by assuming ṁo + ṁf ≈ ṁo. For a

scramjet flying at Mo = 8 with Mc = 2.5 with Tmax = 2300 K, the non-ideal mass flow rate

showed approximately 32 % greater thrust flux than for the ideal mass flow rate.

Similarly, the non-ideal mass flow rate shows an improvement of about 30 % for specific

thrust, 14 % lower for thrust-specific fuel consumption, 3 % greater for fuel-to air ratio,

12 % greater for propulsive and overall efficiency than the ideal mass flow rate

assumption. These quantitative results show the non-ideal mass flow rate to be superior

and more realistic than the ideal mass flow rate performances.

This study basically focused on deriving parametric performance expressions for scramjet

propulsion for various parameters; hence, now a useful model exists for understanding and

predicting scramjet engine performance. Until now no research has been done on how

employing different materials and fuels can impact the performance of scramjet. The results

presented here are hoped to provide useful guidance to improve and predict the performance

of a scramjet engine as well as determining an adequate material and fuel for scramjet

propulsion.

Page 208: Parametric Scramjet Analysis - eGrove

183

LIST OF REFERENCES

Page 209: Parametric Scramjet Analysis - eGrove

184

[1] Lewis, M., “X-51 Scrams into the Future”, Aerospace America, AIAA, 2010, pp. 26-31.

[2] Mattingly, J. D., and Ohain, H., Elements of Propulsion: Gas Turbines and Rockets,

AIAA Education Series, AIAA, Reston, VA, 2006, pp. 266-277.

[3] Hill, P. G., and Peterson, C. R., Mechanics and Thermodynamics of propulsion, Addison-

Wesley Publishing Co., New York, 1992, pp. 155-164.

[4] Cengel, Y. A., and Boles, M. A., Thermodynamics: An Engineering Approach, 3rd ed.,

McGraw-Hill, New York, 1998, pp. 472-477.

[5] Roux, J. A., Shakya, N., and Choi, J., “Revised Parametric Ideal Scramjet Cycle

Analysis”, Journal of Thermophysics and Heat Transfer, Vol. 27, No. 1, 2013, pp. 178-

183.

[6] Curran, E. T., and Murthy, S. N. B., Scramjet Propulsion, Progress in Astronautics and

Aeronautics, edited by P. Zarchan, Vol. 189, AIAA, Reston, VA, 2000, pp. 369-414.

[7] Williamson, M., “Hypersonic Transport”, Aerospace America, AIAA, 2012, pp. 40-45.

[8] Ronald, S. F., "A Century of Ramjet Propulsion Technology Evolution", Journal of

Propulsion and Power, Vol. 20, No. 1, 2004, pp. 27-58.

[9] Ingenito, A., and Bruno, C., "Physics and Regimes in Supersonic Combustion", AIAA

Journal, Vol. 48, No 3, 2010.

[10] Rogers, R. C., Capriotti, D. P., and Guy, R. W., “Experimental Supersonic Combustion

Research at NASA Langley”, 20th AIAA Advanced Measurement and Ground Testing

Technology Conference, AIAA, 1998.

[11] Smart, M. K., Hass, N. E., and Paull, A., “Flight Data Analysis of Hyshot 2 Scramjet

Flight Experiment”, AIAA journal, Vol. 44, No. 10, 2006, pp. 2366-2375.

Page 210: Parametric Scramjet Analysis - eGrove

185

[12] Musielak, D., “National Hypersonic Centers: Fast Track to Truly Fast Flight”, Aerospace

America, AIAA, 2012, pp. 40-45.

[13] Johnson, S. M., “Ultra High Temperature Ceramics: Application, Issues and Prospects”,

2nd Ceramic Leadership Summit, Baltimore, MD, 2011.

[14] Ohlhorst, W. C., Vaughn, L. W., Ransone, O. P., and Tsou H., “Thermal Conductivity

Database of Various Structural Carbon-Carbon Composite Materials”, NASA Technical

Memorandum 4787, 1997.

[15] "Strength – Max. Service Temperature”

http://www-materials.eng.cam.ac.uk/mpsite/interactive_charts/strength-

temp/NS6Chart.html

[16] Peter, B., Kelly, J., and Wilson, J., “Materials Selection Considerations for Thermal

Process Equipment”, Industrial Technologies Program Energy Efficiency and Renewable

Energy, U.S. Department of Energy, 2004.

[17] Edwards, T., “Liquid Fuels and Propellants for Aerospace Propulsion: 1903-2003”,

Journal of Propulsion and Power, Vol. 19 No.6, 2003, pp. 1089-1107.

[18] Benchergui, D., Flamm, J., and Wahls, R., “Air-Breathing Propulsion Systems

Integration”, Aerospace America, AIAA, 2010, pp. 56-57.

[19] Roux, J. A., Shakya, N., and Choi, J., “Scramjet: Minimum Thrust-Specific Fuel

Consumption with Material Limit”, Journal of Thermophysics and Heat Transfer, Vol.

27, No. 2, 2013, pp. 367–368.

[20] Roux, J. A., “Optimum Freestream Mach Number for Maximum Thrust Flux for Ideal

Ramjet/Scramjet”, Journal of Thermophysics and Heat Transfer, Vol. 26, No. 2, 2011, pp.

390–392.

Page 211: Parametric Scramjet Analysis - eGrove

186

[21] Roux, J. A., Choi, J., and Shakya, N., “Parametric Scramjet Cycle Analysis for Non-Ideal

Mass Flow Rate”, Journal of Thermophysics and Heat Transfer, Vol. ??, No. ?, 2014,

pp. ???–???. (published online but will appear later in print)

[22] Roux, J. A., “Parametric Ideal Scramjet Cycle Analysis”, Journal of Thermophysics and

Heat Transfer, Vol. 25, No. 4, 2011, pp. 581-585.

[23] Roux, J. A., “Ideal Ramjet: Optimum M for Fuel Limit and Material Limit”, Journal of

Spacecraft and Rockets, Vol. 19, No. 3, 1982, pp. 286-287.

[24] Roux, J. A., “Ideal Ramjet: Optimum Freestream Mach number for Minimum Thrust-

Specific Fuel Consumption”, Journal of Thermophysics and Heat Transfer, Vol. 25, No.

4, 2011, pp. 626-627.

[25] Weast, R. C., and Selby, S. M., CRC Handbook of Tables for Mathematics, The

Chemical Rubber Company, Cleveland, OH, 1975, pp. 129-130.

Page 212: Parametric Scramjet Analysis - eGrove

187

VITA

JONGSEONG CHOI [email protected]

Education

May, 2014

the University of Mississippi, Oxford, MS

Master of Science in Engineering,

Mechanical Engineering

May, 2012

the University of Mississippi, Oxford, MS

Bachelor of Science in Engineering,

Mechanical Engineering

Publication Roux, J. A., Choi, J., and Shakya, N., “Parametric Scramjet Cycle Analysis for Non-Ideal Mass Flow Rate”,

Journal of Thermophysics and Heat Transfer, Vol. ??, No. ?, 2014, pp. ???–???. (published online, later in print)

Roux, J. A., Shakya, N., and Choi, J., “Scramjet: Minimum Thrust-Specific Fuel Consumption with Material

Limit”, Journal of Thermophysics and Heat Transfer, Vol. 27, No. 2, 2013, pp. 367–368.

Roux, J. A., Shakya, N., and Choi, J., “Revised Parametric Ideal Scramjet Cycle Analysis”, Journal of

Thermophysics and Heat Transfer, Vol. 27, No. 1, 2013, pp. 178-183.

Professional Employment

Teaching Assistant, the University of Mississippi, Oxford, MS, Aug 2012 – Current

Resident Assistant, Office of International Program, Oxford, MS, Aug 2012 – May 2013

Assist residents of 45 undergraduate male international students

Vitalise UK Disabled People Care Center, Southport, Merseyside, UK

Caring disabled people as a volunteer staff member, Sep 2009 – Aug 2010

Operated specially designed electronics for disabled people.

Was promoted to staff from volunteer member

Korea Army 603 Automobile Crops, Inje-gun, Kangwon-do, South Korea

Checked and repaired military use vehicles as an auto mechanic, Oct 2006 – Aug 2008

Assembled and disassembled engine and suspension parts of military use vehicles

Assisting the National Self-made Hybrid Car Competition for College Students

The Korea University of Technology and Education NURI foundation

Took part in 1st, 2nd and 5th competitions, assisted in Standard Inspection Team, April 2005, 2006, 2009

Test and inspect participants’ vehicles for the competition regulations

Page 213: Parametric Scramjet Analysis - eGrove

188

Honors and Awards

- Vice president; Korean Student Association (KSA), the University of Mississippi (2013- current)

- Student member, ASME

- Graduate assistant scholarship, the University of Mississippi (2012- current)

- Graduate assistant Non-Resident scholarship, the University of Mississippi (2012 – current)

- Adler Engineering scholarship, the University of Mississippi (2011-2012)

- Dean’s Honor Roll, the University of Mississippi (2012)

- Chancellor’s Honor Roll, the University of Mississippi (2011)

- Academic Excellence scholarship, Korea University of Technology and Education (2005, 2006, 2008)

- Achievement award of acting festival, Cheonan Youth Community (2008)