MODELING AND LASER-BASED SENSING OF PULSED ...

MODELING AND LASER-BASED

SENSING OF PULSED DETONATION

ENGINES

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Ethan A. Barbour

April 2009

ii

© Copyright by Ethan A. Barbour 2009

All Rights Reserved

iii

This dissertation is dedicated to my beautiful wife Shengmei. Her love, patience and

encouragement have enabled me to achieve more than what I could have ever imagined.

v

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in

scope and quality as a dissertation for the degree of Doctor of Philosophy.

________________________________

(Ronald K. Hanson) Principal Adviser



________________________________

(Craig. T. Bowman)



________________________________

(Christopher M. Brophy)

Approved for the Stanford University Committee on Graduate Studies.

vii

Abstract This work is concerned with two major aspects of pulse detonation engines (PDE)

research: modeling and laser-based sensing. The modeling addresses both ideal and real

considerations relevant to PDE design. First, an ideal nozzle model is developed which

provides a tool for choosing area ratios for fixed-geometry converging, diverging, or

converging-diverging nozzles. Next, losses associated with finite-rate chemistry are

investigated. It was found that PDEs can experience up to ~ 10% reduction in specific

impulse from this effect if O2 is used as the oxidizer, whereas the losses are negligible for

air-breathing applications. Next, heat transfer and friction losses were investigated and

found to be greater than the losses from simple straight-tube PDEs. These losses are

most pronounced (~ 15%) when converging nozzles are used.

The second portion of this work focuses on laser-based absorption sensing for PDEs.

The mid-infrared was chosen as the best way to address the challenges of signal-to-noise

ratio, sensitivity, robustness, and sensor bandwidth. A water vapor sensor was developed

and applied to the PDE at the Naval Postgraduate School. This sensor provided

improvements in temperature accuracy, and it revealed that water (generated by the

vitiator) inhibited performance of the engine. Next, a JP-10 absorption sensor was

developed and applied to the same engine. This sensor provided thermometry data at a

higher temporal resolution than the water sensor. The sensor also provided crucial

information on equivalence ratio and fuel arrival time which enabled the engine to be

successfully operated on JP-10 and air for the first time.

ix

Acknowledgements First and foremost, I would like to thank my advisor, Prof. Ronald Hanson. His

guidance, encouragement, and wisdom have shaped me greatly over the past several

years. I have grown not only as a scientifically minded individual, but also as an

academic, an engineer, an educator, an entrepreneur, and as a manager. I have also

grown personally, learning from his examples on how to best balance my professional

and family lives. His guidance has more than prepared me for my life post-Stanford.

I would also like to acknowledge the other members of my reading committee, Prof.

C. T. Bowman, and Prof. Christopher Brophy. Prof. Bowman has had a large influence

on my abilities as an engineer through his lucid explanations of the many concepts he has

covered in the classroom setting. Prof. Brophy has been a wonderful friend and

colleague ever since my first measurement campaign to Monterey.

Of course, my research would have been much more difficult without the help of Dr.

Dave Davidson and Dr. Jay Jeffries. These two gentlemen were always willing and able

to help me through any kind of problem, while still having time for the dozens of other

students in need of their assistance. A special thanks to Dave for our many long and

fascinating discussions on the subtleties of the spirit.

So many others deserve thanks. My parents, Brian and Anne, raised me to pursue

what interests me, and I therefore owe my time at Stanford to them. My brothers, Lyall

and Jesse, have always been a source of levity. My parents-in-law, Lidong Zhang and

Yan Zhao, have provided much encouragement and great enthusiasm. I owe all of my

machining knowledge to Scott, Lakhbir and Bill. My many friends have made the tough

times easy: Rob (for teaching me about the joys of apathy), Greg (for always being able

to lighten the office mood), Zach (for showing me that scientists can be hip), Dan H. (for

his shared interest in the strange), Pedram (for his neverending randomness) and

Giancarlo (our gigs were some of the best). Of course I can’t forget Matei, Venky, Dave

R., Dave J., Chico (a-Hoy-Hoy), Franz, Ben G., Matt O., Matt B., Dan M. (2x), Subith,

Megan, Genny, Zekai, Brian C., Shannon, Eric, Ben W. (BN), Aamir (Salaam 'Alaykum),

Jason, Brian L., Kent, Andrew, John, Dan S., Mike and Gabe.

xi

Table of Contents Abstract ........................................................................................................................ vii

Acknowledgements ....................................................................................................... ix

Table of Contents .......................................................................................................... xi

List of Tables ............................................................................................................. xvii

List of Figures ............................................................................................................. xix

Glossary ..................................................................................................................... xxv

Nomenclature ........................................................................................................... xxvii

Chapter 1 : Introduction ................................................................................................. 1

1.1 Motivation....................................................................................................... 1

1.2 Theory ............................................................................................................. 3

1.2.1 Chapman-Jouget Detonations ................................................................. 3

1.2.2 Taylor Expansion Wave.......................................................................... 4

1.2.3 Real Detonations..................................................................................... 5

1.3 Overview of PDE Operation........................................................................... 7

1.4 Thesis Breakdown........................................................................................... 8

1.4.1 Modeling of PDEs................................................................................... 9

1.4.2 Laser-Based Sensing in PDEs............................................................... 10

Chapter 2 : Idealized Model for Quasi-1D PDE Nozzles ............................................ 13

2.1 Introduction................................................................................................... 13

2.2 Configuration of Detonation Tube with Diverging Nozzle .......................... 15

2.3 Equation of State........................................................................................... 16

2.4 Numerical Model .......................................................................................... 18

2.5 Time-Varying Thrust .................................................................................... 19

2.6 Steady Choked State ..................................................................................... 23

2.6.1 Derivation ............................................................................................. 23

2.6.2 Choked State Validation ....................................................................... 26

2.7 Impulse Model .............................................................................................. 28

2.7.1 Straight-Tube Model............................................................................. 28

xii

2.7.2 Nozzle Model........................................................................................ 29

Nozzle Force .................................................................................................... 29

Impulse from Taylor Wave and Steady Flow in Nozzle ................................... 31

Impulse from Nozzle Blowdown ....................................................................... 32

Nozzle Impulse ................................................................................................. 34

Impulse of Straight-Tube + Nozzle .................................................................. 35

2.8 Model Validation .......................................................................................... 36

2.8.1 Steady State Model ............................................................................... 37

2.8.2 Unsteady Model .................................................................................... 37

2.9 Nozzle Design............................................................................................... 38

2.10 Converging and Converging-Diverging Nozzles.......................................... 41

2.10.1 Converging Nozzle ............................................................................... 44

2.10.2 Performance with Converging Nozzle.................................................. 45

2.10.3 Converging-Diverging Nozzle.............................................................. 47

2.10.4 Steady Nozzle Stagnation Pressure....................................................... 48

2.11 Summary ....................................................................................................... 50

Chapter 3 : Finite-Rate Chemistry Effects on PDE Performance ................................ 53

3.1 Introduction................................................................................................... 53

3.2 Background................................................................................................... 54

3.3 CEF and CFF in Steady Nozzles .................................................................. 56

3.4 Computations of PDE with Nozzle............................................................... 59

3.4.1 Chemical Mechanisms .......................................................................... 59

3.4.2 Modeling Chemical Equilibrium Flow ................................................. 60

3.4.3 Problem Setup....................................................................................... 62

3.5 Performance Results ..................................................................................... 63

3.6 Summary ....................................................................................................... 66

Chapter 4 : Heat Transfer and Friction Effects on PDEs with Nozzles ....................... 69

4.1 Introduction................................................................................................... 69

4.2 Model Description ........................................................................................ 70

4.2.1 Heat Conduction Loss Model ............................................................... 70

4.2.2 Heat Convection Loss Model................................................................ 71

xiii

4.2.3 Hybrid Heat Loss Model....................................................................... 73

4.2.4 Wall Temperature ................................................................................. 73

4.2.5 Friction Model ...................................................................................... 73

4.2.6 St and Cf Coefficients............................................................................ 74

4.2.7 Additional Modeling Information......................................................... 75

4.3 Thrust and Impulse Breakdown.................................................................... 75

4.4 Straight-Tube PDEs ...................................................................................... 76

4.5 Model Validation .......................................................................................... 77

4.6 Straight-Tube with Converging Nozzles ...................................................... 78

4.6.1 Varying Geometry ................................................................................ 79

4.6.2 Varying Mixture.................................................................................... 80

4.7 Straight-Tube with Diverging Nozzles ......................................................... 82

4.7.1 Problem Setup....................................................................................... 83

4.7.2 Effect of Nozzle Losses on Impulse ..................................................... 83

4.8 Summary ....................................................................................................... 88

Chapter 5 : Laser-Based Mid-IR H2O Sensing ............................................................ 89

5.1 Introduction................................................................................................... 89

5.2 Infrared Water Spectrum............................................................................... 90

5.3 Sensor Theory ............................................................................................... 92

5.3.1 Beer’s Law............................................................................................ 92

5.3.2 Lineshape .............................................................................................. 93

5.3.3 2-Wavelength Temperature Sensing..................................................... 95

5.3.4 Mole Fraction Sensing .......................................................................... 97

5.4 Wavelength Selection ................................................................................... 98

5.4.1 Interfering Species ................................................................................ 98

5.4.2 Isolation, Strength and Sensitivity ........................................................ 99

5.5 Spectroscopic Measurements...................................................................... 102

5.5.1 Experimental Setup............................................................................. 102

5.5.2 Pure Water Measurements .................................................................. 104

5.5.3 Water/Air Mixture Measurements ...................................................... 107

5.6 Sensor Setup................................................................................................ 108

xiv

5.6.1 Sensor Hardware................................................................................. 109

5.6.2 Time Multiplexing .............................................................................. 109

5.7 Uncertainty Analysis................................................................................... 110

5.7.1 Temperature ........................................................................................ 111

5.7.2 Mole Fraction...................................................................................... 113

5.8 Sensor Validation........................................................................................ 114

5.9 NPS Campaign............................................................................................ 116

5.9.1 Sensor Setup........................................................................................ 116

5.9.2 Cooling by Injected Fuel..................................................................... 117

5.9.3 Vitiator Impact on Ignitor Performance.............................................. 118

5.10 Summary ..................................................................................................... 119

Chapter 6 : Laser-Based Mid-IR JP-10 Sensing ........................................................ 121

6.1 Introduction................................................................................................. 121

6.2 JP-10 Description........................................................................................ 122

6.3 Sensor Theory ............................................................................................. 123

6.4 JP-10 Spectrum ........................................................................................... 123


6.4.2 Results................................................................................................. 124

6.5 Tunable Mid-IR Laser................................................................................. 127

6.6 Sensor Hardware......................................................................................... 128

6.7 Wavelength Selection ................................................................................. 129

6.7.1 Temperature Sensor ............................................................................ 129

6.7.2 Fuel Sensor.......................................................................................... 129

6.8 Temperature Sensor Bandwidth.................................................................. 130

6.9 Sensor Validation........................................................................................ 131

6.10 NPS Campaign............................................................................................ 132


6.10.2 Equivalence Ratio Measurements....................................................... 134

6.10.3 Temperature Measurements................................................................ 138

6.11 Summary ..................................................................................................... 140

Chapter 7 : Conclusions & Future Work ................................................................... 141

xv

7.1 Conclusions................................................................................................. 141

7.1.1 Modeling............................................................................................. 141

Ideal PDE Nozzles ......................................................................................... 141

Finite-Rate Chemistry in PDE Nozzles .......................................................... 142

Heat Transfer due to PDE Nozzles ................................................................ 142

7.1.2 Laser-Based Sensing........................................................................... 143

Water Sensing ................................................................................................ 143

JP-10 Sensing................................................................................................. 143

7.2 Future Work ................................................................................................ 144

Appendix A : NASA Polynomials for Achieving Constant-γ Equation of State....... 147

A.1 Single Species ............................................................................................. 148

A.2 Mixture........................................................................................................ 148

A.2.1 Derivation ........................................................................................... 149

A.2.2 Example .............................................................................................. 151

Appendix B : Derivation Details for Quasi-1D Nozzle Model.................................. 153

B.1 Deriving Eq. (11) ........................................................................................ 153

B.2 Deriving Eq. (14) ........................................................................................ 155

B.3 Ω1, Ω2, and Π .............................................................................................. 156

Appendix C : Isentropic Relations for Chemical Equilibrium Flow.......................... 161

C.1 Isentropic Relations .................................................................................... 162

C.2 Stagnation and Choke States....................................................................... 164

C.3 Area Relation .............................................................................................. 165

C.4 Discussion on Temperature......................................................................... 167

C.5 Conclusions................................................................................................. 169

Appendix D : Chemical Mechanisms ........................................................................ 171

D.1 H2 / O2 (frozen N2)...................................................................................... 171

D.2 Fully-Reversible Varatharajan (frozen N2)................................................. 173

Appendix E : Laser Collimation ................................................................................ 175

Bibliography............................................................................................................... 181

xvii

List of Tables Table 1. Breakdown of impulse. C2H4 + 3O2, P1 = 1 atm, T1 = T∞ = 298 K, γ = 1.14,

P∞ = 0.01 atm, ε = 40, Ls = 1 m, Ln = 0.2 m, Ds = 50 mm. ....................................... 22

Table 2. Gasdynamic properties of two choked states. C2H4 + 3O2, P1 = 1 atm,

T1 = 298 K................................................................................................................. 26

Table 3. Taylor wave/steady flow impulse factors, Δ1 & Δ2, taken from simulations of

straight-tube without nozzle. T1 = 298 K. ‘air’ represents O2 + 3.76N2. All γ ’s

correspond to chemical equilibrium evaluated at the CJ state. Thermo properties

used to calculate P3 and γ taken from Ref. 48 ......................................................... 32

Table 4: Summary of mechanisms used for computations. See Appendix D for details. 60

Table 5: Impulse and losses for three mixtures. T1 = 500 K, P1 = 1 atm. εd = 100. ....... 64

Table 6: Constants used in heat conduction model. ‘air’ represents O2 + 3.76N2.

P1 = 1 atm; T1 = 500 K.............................................................................................. 71

Table 7: Summary of impulse breakdown for PDE with diverging nozzle. Ls /Ds = 50,

Ls = 0.5 m, εd = 100. P∞ = 155 Pa. All values in N·ms. .......................................... 87

Table 8: Summary of spectroscopic parameters for H2O sensor. See Ref. 90 for

HITRAN 2004. ....................................................................................................... 102

Table 9: Laser collimation strategies tested.................................................................... 177

xix

List of Figures Figure 1: Steady 1-dimensional combustion wave. ............................................................ 3

Figure 2: Detonation wave with Taylor expansion wave and plateau region in a tube with

a closed end................................................................................................................. 4

Figure 3. Configuration for straight-tube with diverging nozzle. † States 1 and ∞ are

initial conditions. ‡ State 4 occurs only during steady nozzle flow......................... 16

Figure 4: Equilibrium and frozen isentropes for CJ products of C2H4 + 3O2 (P1 = 1 atm;

T1 = 298 K) using Eq. (2) (solid lines). γ evaluated at CJ state: γfrozen = 1.24;

γequil = 1.14. Real values (circles) obtained from STANJAN. ................................. 17

Figure 5: Breakdown of forces. ........................................................................................ 19

Figure 6. Sample histories of end-wall and nozzle force. a) entire cycle; b) early times.

P1 = 1 atm, T1 = T∞ = 298 K, γ = 1.14, P∞ = 0.01 atm, ε = 40, Ls = 1 m, Ln = 0.2 m.

................................................................................................................................... 21

Figure 7. Schematic of characteristics used for finding steady choked state.................... 23

Figure 8. Validating state 4 model. C2H4 + 3O2, T1 = 298 K, γ = 1.14. ■ current

measurement. Uncertainty bars represent extent of pressure change due to heat

transfer. ..................................................................................................................... 27

Figure 9. Geometric factors required to calculate nozzle thrust. ...................................... 30

Figure 10: Subdividing nozzle force history. Same conditions as Figure 6. ................... 30

Figure 11: Decay of straight-tube exit pressure, comparing full solution with CV

blowdown, Eq. (14). C2H4 + 3O2, P1 = 1 atm, T1 = 298 K, P∞ = 0.01 atm, γ = 1.14,

P4 = 4.15 bar, c4 = 1128 m/s, Ls = 1 m...................................................................... 33

Figure 12. Comparison of experimental data with models. ■ and steady model (choked

state 3) from Ref. 41. “2D CFD” model from Ref. 42.

C2H4 + 3O2, P1 = 80 kPa, T1 = T∞ = 298 K, γ = 1.14, ε = 6.5................................... 36

Figure 13. Performance and area ratio for optimized nozzle. C2H4 + 3O2, P1 = 1 atm,

T1 = 298 K, γ = 1.14. ................................................................................................. 39

xx

Figure 14. Optimized area ratio vs. (a) ambient/fill pressure ratio and (b) ambient/nozzle

stagnation pressure ratio. Steady nozzle calculation also shown in panel (b).

P1 = 1 atm, T1 = 298 K.............................................................................................. 40

Figure 15: End-wall static pressure vs. time for straight-tube (a) without converging

nozzle and (b) with converging nozzle. Pressure losses inside the tube are neglected.

................................................................................................................................... 43

Figure 16: Configuration for straight-tube with converging nozzle. † States 1 and ∞ are

initial conditions. ‡ State 4 occurs only during steady nozzle flow......................... 44

Figure 17: Time-varying thrust for straight-tube with converging nozzle. C2H4 + 3O2,

P1 = 1 atm, P∞ = 0.1 atm, T1 = T∞ = 298 K, γ = 1.14, εc = 1.62, Ds = 50 mm, Ls = 1 m,

θc = 12°. .................................................................................................................... 45

Figure 18: Impulse for straight-tube with converging nozzle. C2H4 + 3O2, P1 = 1 atm,

T1 = T∞ = 298 K, γ = 1.14, Ds = 50 mm, Ls = 1 m, θc = 12°. .................................... 46

Figure 19: Configuration for straight-tube with converging-diverging nozzle. † States 1

and ∞ are initial conditions. ‡ State 4 occurs only during steady nozzle flow. ....... 47

Figure 20: Impulse of a straight-tube with converging-diverging nozzle is estimated by

equating it to a straight-tube with diverging nozzle. The straight-tube area, As, and

diverging area ratio, εd, are the same in both configurations. The exit area, An,e, is

larger in the modeled system than in the real system. .............................................. 48

Figure 21: Stagnation pressure ratio vs. refresh Mach number. ....................................... 50

Figure 22: Pressure ratio vs. area ratio showing the difference between CEF and CFF.

Inlet state is taken as state 4 of H2 + 0.5O2, P1 = 1 atm, T1 = 500 K. ....................... 57

Figure 23: Specific impulse vs. area ratio for a pressure-matched steady nozzle. Inlet

state is taken as state 4 of H2 + 0.5O2, P1 = 1 atm, T1 = 500 K. ............................... 58

Figure 24: Determining tchem using steady flow through a conical diverging nozzle

(Di = 50 mm; θd = 12°). Temperature is compared for true equilibrium, true finite-

rate chemistry, and finite-rate chemistry with extended reaction time, tchem. Inlet

state corresponds to state 4 of H2 + 0.5O2 (T1 = 500 K; P1 = 1 atm). Chemical

mechanism: H2. ......................................................................................................... 61

xxi

Figure 25: Time-varying forces corresponding to equilibrium and finite-rate chemistry

flow. Straight-tube experienced CEF in both cases. Ls = 0.5 m, Ds = 10 mm.

θd = 12º, εd = 100, P∞ = 155 Pa. Mechanism: H2..................................................... 64

Figure 26: Static temperature in steady nozzle for H2 + 0.5O2 (red) and H2 + 0.5air

(black). Equilibrium solution shown with solid lines; frozen solution shown with

dashed lines. Inlet state taken as state 4. T1 = 500 K, P1 = 1 atm. .......................... 66

Figure 27: Breakdown of forces in PDE with nozzle. Pressure forces and shear forces are

included. Dark blue: straight-tube pressure force; red: nozzle pressure force; green:

straight-tube shear force; light blue: nozzle shear force. .......................................... 76

Figure 28: Summary of straight-tube PDE dimensions. Imbedded values indicate L / D.

Solid symbols: single-cycle engines; open symbols: multi-cycle engines. ■ [77];

■ [71]; ■ [36]; ▼[78]; ● [79]; □ [6]; ○ [33, Fig. 7]; ○ [51, initiator]; ○ [80]........... 77

Figure 29: Impulse efficiency as a function of Ls /Ds and Ls, (a) Ls = 0.5 m, (b) Ls = 1 m.

H2 + 0.5O2; P∞ = 1 atm. Tw = 500 K. ....................................................................... 80

Figure 30: Impulse efficiency for three different mixtures. Ls/Ds = 50; Ls = 0.5 m.

Tw = 500 K. ............................................................................................................... 81

Figure 31: End-wall force for three mixtures with and without losses. (a) Force,

(b) normalized force. Straight-tube geometry (Ls = 0.5 m; Ds = 10 mm). Shear

forces not shown. T1 = Tw = 500 K, P1 = P∞ = 1 atm............................................... 81

Figure 32: Forces in diverging nozzle for (i) ideal tube & ideal nozzle (red), (ii) tube with

losses & ideal nozzle (green), and (iii) tube with losses & nozzle with losses (blue).

Ls /Ds = 50, Ls = 0.5 m; εd = 100; θd = 12º; H2 + 0.5O2; T1 = Tw = T∞ = 500 K,

P1 = 1 atm; P∞ = 155 Pa............................................................................................ 84

Figure 33: Static pressure ratio for Fanno flow and Rayleigh flow. Arrows indicate

trajectories due to friction (Fanno) and heat loss (Rayleigh). The case of heat

addition is not considered. ........................................................................................ 85

Figure 34: Steady nozzle with no losses, with friction losses, and with friction & heat

losses. Inlet is at state 4 for H2 + 0.5O2, T1 = 500 K, P1 = 1 atm. Di = 10 mm;

θd = 2º........................................................................................................................ 86

Figure 35: Infrared absorption spectrum of water. T = 520 K; P = 1 atm; XH2O = 4%;

L = 7.3 cm. (Source: HITRAN 2004 [90].) ............................................................. 91

xxii

Figure 36: Sample absorption feature. .............................................................................. 93

Figure 37: ν1 and ν3 bands of H2O spectrum with interfering CO2 spectrum. T = 520 K;

P = 1 atm; XH2O = XCO2 = 4%; L = 7.3 cm. (Source: HITRAN 2004 [90].)............. 99

Figure 38: Water transition near 3982 cm-1 used for current water sensor. T = 520 K;

P = 1 atm; XH2O = 4%; L = 7.3 cm. (Source: HITRAN 2004 [90]) ........................ 100

Figure 39: Candidate transitions for 2-wavelength water sensor. (First transition of

sensor shown in Figure 37.) E″ shown over each transition. T = 520 K; P = 1 atm;

XH2O = 4%; L = 7.3 cm. (Source: HITRAN 2004 [90]) ......................................... 101

Figure 40: Experimental setup for water spectrum measurements. LP is the low-pressure

transducer (100 torr); HP is the high-pressure transducer (1,000 torr); T is the

thermocouple readout. The path length, L, is 76 cm for characterizing the transition

at 3982.06 cm-1, and 9.9 cm for the transition at 3920.09 cm-1. ............................. 104

Figure 41: Sample absorbance plot of pure water with best fit using Voigt lineshape.

T = 874 K; P = 15.1 torr; ν0 = 3920.09 cm-1........................................................... 105

Figure 42: Linestrength and self-broadening at various pressures. T = 874 K;

ν0 = 3920.09 cm-1.................................................................................................... 105

Figure 43: Linestrength vs. T for (a) ν0 = 3982.06 cm-1 and (b) ν0 = 3920.09 cm-1....... 106

Figure 44: Broadening coefficients vs. T for (a) ν0 = 3982.06 cm-1 and

(b) ν0 = 3920.09 cm-1. ............................................................................................. 107

Figure 45: Sample absorbance plot of water/air mixture with best fit using Voigt

lineshape. T = 825 K; P = 759 torr; XH2O = 1.3%; ν0 = 3920.09 cm-1. .................. 108

Figure 46: General setup of fiber-coupled water sensor. Red lines indicate freespace

beams. L: plano-convex lens; BS: beam splitter (wedged); W: window (wedged); F:

filter; D: InSb detector; PF: pitch fiber; CF: catch fiber. ....................................... 109

Figure 47: Injection current to lasers #1 and #2. ............................................................ 110

Figure 48: Schematic depicting two sources of error for temperature sensor. Values do

not reflect actual sensor. ......................................................................................... 111

Figure 49: Validating temperature sensor....................................................................... 115

Figure 50: Validating mole fraction sensor. ................................................................... 115

xxiii

Figure 51: Water sensor installed on NPS PDE. Red line represents freespace beam. See

Figure 45 for details on fiber coupling and de-coupling. “PZT” represents a piezo-

electric pressure transducer..................................................................................... 117

Figure 52: Periodic cooling by injected C2H4 in NPS PDE using near-IR [89, pg. 85] and

current mid-IR (2008-06-04-7) water sensors. Both sensors scanned at 5 kHz. ... 118

Figure 53: Average water mole fraction and peak pressure vs. engine cycle. Engine

frequency is 30 Hz. ν0 = 3920.09 cm-1. ................................................................. 119

Figure 54: Structure of JP-10.......................................................................................... 123

Figure 55: Experimental setup for JP-10 spectrum measurements. LP is the low-pressure

transducer (100 torr); HP is the high-pressure transducer (50 psia); the thermocouple

is mounted inside the optical cell, where T represents the thermocouple readout.. 124

Figure 56: Cross-section of the fundamental band for the C-H stretch of JP-10.

(a) fundamental and (b) first overtone. T = 302 K. The uncertainty bar applies only

to the 1st overtone band. .......................................................................................... 125

Figure 57: Cross-section of the fundamental band for the C-H stretch of JP-10 at various

T. Also shown are HeNe laser data taken from Klingbeil et al. [106]................... 126

Figure 58: Integrated cross-section of the fundamental band for the C-H stretch of JP-10.

................................................................................................................................. 126

Figure 59: Tunable mid-IR DFG laser, operating in 2-color mode. Notice that the

modulation signals for lasers B and C are out of phase.......................................... 128

Figure 60: General setup of fiber-coupled water sensor. Red lines indicate freespace

beams. L: plano-convex lens; W: window (wedged); F: filter; D: InSb detector; PF:

pitch fiber; CF: catch fiber...................................................................................... 128

Figure 61: Validating fuel and temperature sensors. For fuel sensor T = 374 K. For

temperature sensor XJP-10 = 0.1%. P = 1 atm in both bases.................................... 132

Figure 62: NPS PDE engine with JP-10 sensors. Red lines represent freespace beam. D1:

InAs detector for HeNe; C: mechanical chopper. See Figure 59 for DFG catch

optics. ...................................................................................................................... 134

Figure 63: JP-10 measured in unfired NPS PDE: (a) transmitted intensity and

(b) equivalence ratio. T = 477 K; P = 2.6 atm(abs); m& = 0.25 kg/s; engine

frequency = 20 Hz................................................................................................... 135

xxiv

Figure 64: Comparing unfired cycle and fired cycle from same run. T = 478 K;

P = 1.45 atm(abs); m& = 0.11 kg/s; engine frequency = 10 Hz................................ 136

Figure 65: Comparison of reactant temperature measured by thermocouple and DFG in

unfired NPS PDE. P = 1.8 ~ 2.1 atm(abs); m& = 0.5 kg/s; engine frequency = 30 Hz.

................................................................................................................................. 138

Figure 66: Time-varying temperature for representative cold (Tmean = 450 K) and hot

(Tmean = 603 K) runs. Sensor bandwidth = 100 kHz. ............................................. 139

Figure 67: Shock factor................................................................................................... 157

Figure 68: Comparison of predicted and actual density as a function of pressure.

Predicted values obtained using Eq. (66), γ = 1.14. Reference state taken as CJ of

C2H4 + 3O2 detonation, P1 = 1 atm, T1 = 298 K. .................................................... 168

Figure 69: Comparison of predicted and actual temperature as a function of pressure.

Predicted values obtained using Eq. (76), γ = 1.14. Reference state taken as CJ of

C2H4 + 3O2 detonation, P1 = 1 atm, T1 = 298 K. .................................................... 169

Figure 70: Collimating a diode laser using a lens (top) or mirror (bottom). .................. 176

Figure 71: Defining f/# for the diode laser: f/# ≡ y/x. ..................................................... 176

Figure 72: Setup for measuring laser beam profile......................................................... 177

Figure 73: Beam profile using mirror. (a) vertical axis, (b) horizontal axis. ................. 178

Figure 74: Beam profile using lens #1............................................................................ 178



xxv

Glossary air = O2 + 3.76N2

CEF = chemically equilibrated flow

CFD = computational fluid dynamics

CFF = chemically frozen flow

CJ = Chapman / Jouguet

CV = constant volume

DA = direct absorption

DDT = deflagration-to-detonation transition

DFB = distributed feedback (laser)

DFG = difference frequency generation

FRCF = finite-rate chemistry flow

FRV = fully reversible Varatharajan (mechanism)

FTIR = Fourier transform infrared (spectrometer)

H2 = H2/O2 mechanism with updated high pressure rate for H + O2 + M

IR = infrared

MOC = method of characteristics

NPS = Naval Postgraduate School

PDE = pulsed detonation engine

PZT = piezo-electric pressure transducer

rms = root-mean-square

SNR = signal-to-noise ratio

STP = standard temperature and pressure, i.e. P = 1 atm, T = 298 K

telecom = refers to equipment normally used in the telecommunications industry

Wintenberger/Cooper model

= performance model for straight-tube without nozzle, taken from

Refs. [9] and [32]

ZND = Zel’dovich / von Neumann / Döring

xxvii

Nomenclature Roman Characters A = cross-sectional area

A = affinity

B = ratio of nozzle exit pressure to nozzle inlet pressure

Cf = friction coefficient

c = static sound speed

cp = specific heat capacity at constant pressure per unit mass [J/kg·K]

D = diameter

E″ = lower state energy [cm-1]

F = force

G1 = geometric factor corresponding to P3

G2 = geometric factor corresponding to P∞

g = gravitational acceleration

H = integrated absorbance

h = mixture enthalpy per unit mass [J/kg]

I = impulse

I0 = incident laser intensity

Isp = specific impulse

It = transmitted laser intensity

Kn,1 = proportionality constant corresponding to P3

Kn,2 = proportionality constant corresponding to P∞

Ks = straight-tube model proportionality constant

L = length

M = Mach number

Mrefresh = refresh Mach number

m = mass

m& = mass flow rate

nair = power law exponent for air-broadening

xxviii

nself = power law exponent for self-broadening

P = static pressure

Pr = Prandtl number

Q = partition function

q ′′& = heat flux [W/m2]

R = ratio of absorbances (JP-10 sensor) or integrated absorbances (water

sensor)

R = gas constant

S = temperature-dependent linestrength [1/atm/cm2]

St = Stanton number

T = static temperature

T = net thrust

t = time

t0 = time from ignition to beginning of nozzle blowdown

t1 = time from ignition to detonation wave at straight-tube exit

t4 = time from detonation wave at straight-tube exit until nozzle force

reaches steady state

t5 = time duration of steady flow in nozzle

t6 = time from end of steady nozzle flow to appearance of shock at nozzle

exit

tchem = portion of computational step allotted to chemistry

tcycle = cycle time (time between ignition and when end-wall reaches P∞)

tgd = portion of computational step allotted to gasdynamics

UCJ = Chapman-Jouguet wave speed

u = gas velocity

V = volume

W = molar mass

w = uncertainty

X = mole fraction

x = spatial coordinate

Y = mass fraction

xxix

Greek Characters αν = spectral absorbance

γ = polytropic exponent

2γair = air-broadening coefficient

2γself = self-broadening coefficient

Δ1 = Taylor wave/steady flow impulse factor corresponding to P3

Δ2 = Taylor wave/steady flow impulse factor corresponding to P∞

ΔνC = full-width at half maximum by collisional broadening [cm-1]

ε = area ratio

ζ = sensitivity of temperature sensor, ζ ≡ (dR/R ) / (dT/T )

η = extent of reaction

θ = conical nozzle half-angle

Π = shock factor

ρ = static density

τ = non-dimensional time, τ ≡ tc3/L

τw = wall shear stress [N/m2]

Φ = equivalence ratio

φ = lineshape

Ω1 = blowdown impulse factor corresponding to P3

Ω2 = blowdown impulse factor corresponding to P∞

ω1 = first factor for Wintenberger/Cooper model (denoted α in their model)

ω2 = second factor for Wintenberger/Cooper model (denoted β in their

model)

xxx

Superscript * = choked state

0 = stagnation state

^ = per unit mole

~ = optimized

¯ = (overbar) boundary between Taylor wave and plateau region

press = pressure force

shear = shear force

Subscripts ∞ = ambient state

1 = reactant state

2 = CJ state

3 = plateau state behind Taylor wave

4 = straight-tube exit state (or nozzle inlet state) during steady flow

aw = conditions at wall for adiabatic flow

c = converging nozzle

cond = conduction

conv = convection

d = diverging nozzle

e = exit

equil = equilibrium

i = inlet

n = nozzle

q = quiescent

prop = propellants

r = reference

s = straight-tube

throat = location in nozzle where area is minimum

w = wall

1

Chapter 1: Introduction

1.1 Motivation The pulsed detonation engine (PDE) is a device which has, over the past several years,

gained widespread interest by propulsion engineers and scientists [1]. The source of this

interest is threefold: 1) PDEs attempt to offer improvements in overall energy extraction

efficiencies over traditional technologies; 2) PDEs are inherently simple mechanically,

and therefore robust and inexpensive; and 3) by addressing questions on PDEs, a demand

is created for fundamental research into detonations.

Improvements in energy conversion efficiencies are thought to be possible because of

the PDE’s mechanism for transforming chemical into thermal energy. This mechanism is

a supersonic detonation wave, which is much faster than the subsonic flames which

convert energy in traditional devices such as gas turbine engines, ramjets and rockets.

For example, an ethylene/air detonation wave travels at speeds close to 2 km/s, whereas

an ethylene/air flame typically has speeds much lower than 10 m/s. This high rate of

burning provided by the detonation wave suggests that the energy conversion process in a

PDE can be approximated as constant volume (CV). When compared to a Brayton cycle

of the same compression ratio, the constant-volume process provides additional work

output for the same heat input.

Problems with this over-simplified comparison to CV heat addition exist, but have

largely been overcome. One such problem was considered by Wintenberger and

Shepherd [2] who showed that entropy generated by a detonation wave arises from both

the leading shock and the combustion chemistry, while the (chemically reacting) Brayton

cycle only generates entropy via the combustion chemistry. This makes comparison

between a detonation-based cycle and the Brayton cycle problematic. Another issue

concerns the popular approach of assuming the PDE to be a steady device [3], which may

or may not be appropriate depending on the details of the PDE design in question [4].

Fortunately, time-accurate computations with finite-rate chemistry [5,6,7,8] address both

of these problems, yielding results which can easily be compared to measurements and

2

predictions of gas turbine, ramjet and rocket performance, without unnecessary

assumptions. Furthermore, unsteady models based partially on experimental calibration

have also proved successful at predicting PDE performance [9,10].

Several studies have shown that PDEs offer advantages over traditional propulsion

technologies. For example, Morris [7] used time-accurate CFD to show that, when fitted

with an appropriate nozzle, pulse detonation rocket engines outperform their steady

rocket counterparts over a wide range of ambient pressures. Wintenberger and

Shephered [10] modeled a particular design for an air-breathing PDE, the results from

which suggest better performance than ramjets at relatively low flight Mach numbers.

Ma et al. [8] used both computational and analytical approaches to show that their PDE

design can outperform ramjets over a wide range of flight Mach numbers.

Recent efforts have also been made to develop fully functioning PDEs. Goldmeer

et al. [11] considered the novel idea of replacing the combustor of a high pressure turbine

with an array of PDEs, thereby providing additional pressurization to the combustion

products before they enter the turbine. Bussing et al. [12] designed and built a PDE

based on a rotary valve concept. The purpose of the valve was to allow high flowrates of

air, while also periodically providing a thrust surface for the detonation products to act

upon. Brophy has developed a valveless PDE which has been operated on C2H4, C3H8

and JP-10 [13,14,15]. By avoiding air valves, the design exploits the simplicity made

possible by the PDE concept. Kasahara et al. [16] recently constructed a rocket powered

by a PDE and tested it on a horizontal rail system. Lastly, an aircraft powered by a PDE

designed by Schauer et al. [17] was flown during a proof-of-concept test. While the

vehicle required gas-turbine assist during takeoff, it was fully powered by a PDE during

cruise. This test demonstrated the feasibility of PDE technology and in so doing

motivates further research.

3

1.2 Theory

1.2.1 Chapman-Jouget Detonations A steady 1-dimensional combustion wave traveling through a reactive mixture

(labeled state 1) will leave behind products (labeled state 2), having converted chemical

energy into sensible and bulk kinetic energy during the combustion process (see Figure 1).

While it is always the case that T2 > T1, there exist some solutions which allow

supersonic compressive combustion waves (uwave > c1; P2 > P1) and some solutions which

allow subsonic rarefaction combustion waves (uwave < c1; P2 < P1). The compression type

is also called a detonation and is of interest here. A special subset of possible detonation

waves is called Chapman-Jouget (CJ). The CJ wave is a detonation wave in which the

products are choked in the wave frame (u2 – uwave = c2). For this special case we define

UCJ ≡ uwave. Only CJ detonation waves are considered in this work.

Figure 1: Steady 1-dimensional combustion wave.

The CJ state is prescribed uniquely by the reactant mixture and state, and can be

determined easily using a chemical equilibrium solver, e.g. STANJAN [18] or CEA [19].

Further details of CJ detonation theory can be found in Ref. 20 . The CJ state is

synonymous with state 2 throughout this work.

uwave u2 - uwave

P1

T1

Xi1

c1

P2

T2

Xi2

c2

4

1.2.2 Taylor Expansion Wave The previous section described the CJ point immediately behind a 1-dimensional

steady CJ detonation wave. If the wave travels down an infinitely long tube, the CJ state

would persist at all points downstream of the wave. However, detonations in the context

of PDEs travel down tubes which are closed at one end (see Figure 2). The end-wall

causes the product gases to stagnate, thereby altering the thermodynamic state. The

resulting plateau state (labeled state 3) is especially important to PDE research because

the pressure, P3, contributes in a large way to the engine thrust. An unsteady expansion

fan must exist between the CJ state (state 2) and the plateau state (state 3) and is called

the Taylor expansion wave. Figure 2 shows states 1, 2 and 3, as well as the Taylor wave,

in a detonation tube. The figure also shows the ambient state ∞ which represents the

state outside of the tube (e.g. air at STP).

Figure 2: Detonation wave with Taylor expansion wave and plateau region in a tube with a closed end.

closed end open end

UCJ

x

P

state 1

state 2 (CJ)

state 3

Taylor wave

ignition point

ambient (P∞)

L

state ∞

end-wall

UCJ

5

It is possible to take advantage of the power law relationship which commonly exists

between pressure and density (i.e. P ∝ ρ γ) in order to obtain an analytic relationship

between P3 and P2 (first discovered by Taylor [ 21 ] and presented here using

Wintenberger’s formulation [22, § 1.1.4]):

1

2

2

323

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

γγ

cc

PP (1)

where the sound speed in the plateau region, c3, is related to the CJ sound speed, c2, and

the CJ wave speed, UCJ, using:

CJUcc2

12

123

−−

+=

γγ

This theory will be revisited in Chapter 2 when investigating flow at the open end of the

PDE.

1.2.3 Real Detonations The 1-dimensional steady CJ wave of § 1.2.1 is very useful for understanding the

thermochemistry and gasdynamics of detonations. However, real detonations are highly

unsteady and 3-dimensional, thereby making predictions much more difficult. The 3-

dimensional structure manifests as a collection of longitudinal waves, transverse waves

and shear layers, all three of which meet at triple points. Triple point trajectories have

been measured by inserting metal foil inside detonation tubes and covering them with

soot. As the triple points move across the soot, their paths are etched and a cellular

pattern emerges. A thorough review of the gasdynamics and chemistry of real

detonations is covered by Fickett and Davis [20, § 7] and a more recent description

containing updated research and references can be found in Ref. 23.

For the purposes of propulsion, it is the pressure which is of utmost interest, since the

pressure acts on surfaces of the PDE to produce thrust. Although the CJ and plateau

6

pressures are easily computed, an important question to ask is whether this simple

1-dimensional steady theory is a sufficient predictor of the pressure inside real

detonations and their associated Taylor waves.

Fickett and Davis [20, § 3A] compared the CJ theory with experiments and concluded

that the measured speed of the leading shock (UCJ) in a real detonation wave is captured

by the CJ theory to within 2%. However, CJ pressure is not as easily measured owing to

the fact that pressure is highly non-steady, and the Taylor wave expands the gas

immediately after the reaction zone has passed the measurement location. Computations

are used instead to investigate the unsteady nature of detonations. Fickett and Wood [24]

showed that pressure oscillations are possible behind the leading shock even when the

flow is only 1-dimensional. They showed that the average value of pressure taken over

many oscillations was equal to the CJ pressure to within 1%. As computational power

increased over the years, the same case covered by Fickett and Wood was extended to

two dimensions by Abouseif and Toong [25], who found the same result regarding

pressure.

The aforementioned computations on unsteady detonation waves were performed on

mixtures with somewhat contrived chemical mechanisms (i.e. single-reaction chemistry).

Owens [26, § 5] performed 2-dimensional unsteady computations using full finite-rate

chemistry and showed once again that the average pressure after the leading shock

matches CJ theory. In addition, he also showed the same is true for the plateau pressure,

i.e. oscillations in the plateau region have an average value equal to P3 predicted by

Eq. (1). His simulations included losses due to transverse shocks, but these were not

enough to incur a noticeable penalty in P3 or engine performance.

In light of the enormous computational expense of resolving the full 3-dimensional

flow of a real detonation wave, and in light of the previous work which showed that

1-dimensional theory is sufficient for studying engine performance, all computations

performed in this work will use 1-dimemsional models for performance predictions.

7

1.3 Overview of PDE Operation The prototypical PDE consists of a straight-tube of length L which is closed at one end

and open at the other, as depicted in Figure 2. A nozzle may or may not be included, and

for now no nozzle is assumed. (Nozzles will be dealt with in later chapters.) There are

four basic steps to the PDE cycle:

Detonation: The cycle begins with the tube filled with a reactant mixture at

temperature T1 and pressure P1. The ambient pressure is designated P∞ . The

mixture is ignited at the closed end. The resulting flame travels for some distance

and then goes through deflagration-to-detonation transition (DDT), after which a CJ

detonation wave travels down the tube at UCJ and exits into the atmosphere.

Meanwhile, P3 is acting on the end-wall of cross-sectional area A, thereby

producing a thrust equal to T = (P3 – P∞)A.

Blowdown: After the detonation wave exits the tube, expansion waves move

upstream towards the end-wall. When these waves reach the end-wall, the pressure

there begins to drop below P3. Thrust is still being produced, but is decreasing.

Eventually the end-wall pressure drops low enough that purge gases can be

admitted.

Purge: The exhaust gases are at low pressure, but are still at relatively high

temperature. In order to prevent the next cycle’s charge from pre-igniting, a slug of

purge gas is typically injected. For air-breathing engines this gas is simply air. For

rockets, this gas can be the oxidizer, the fuel, or any other inert gas which is carried

on board.

Fill: With the hot products removed, the next cycle is prepared by injecting a fresh

mixture of reactants at T1 and P1.

The most important quantity during this cycle is the engine thrust, T . However, since

the thrust is unsteady it is difficult to use as a performance metric. The cycle impulse, I,

is used instead. It is defined as:

8

( )dttIcyclet∫≡ T

where tcycle is the overall cycle time. Finally, in order to account for differences in fuel,

equivalence ratio and engine size, the impulse is normalized by the propellant weight to

obtain the specific impulse, Isp:

gm

IIprop

sp ≡

where g is the gravitational acceleration. For rockets, mprop is the mass of the fuel and the

oxidizer. For air-breathing engines, mprop is the mass of the fuel only. Since both the

cycle impulse, I, and the mass of propellants, mprop, approximately increase linearly with

each cycle, the specific impulse, Isp, is approximately the same for both single-cycle and

multi-cycle operation.

The work presented here will deal with the impulse generated during the first two

segments of PDE operation: detonation and blowdown. These two segments generate the

largest fractions of cycle impulse. Furthermore, the detonation passage and subsequent

blowdown are the least well understood aspects of PDE operation, whereas the purge and

fill are reasonably modeled using standard steady flow techniques. We take the cycle,

therefore, to cover only the detonation and blowdown. Specifically, the cycle begins at

ignition, and ends when the end-wall pressure reaches the ambient pressure, P∞ .

1.4 Thesis Breakdown This thesis investigates two important aspects of PDE research: 1) modeling and

2) laser-based sensing. Modeling is invaluable to the success of this technology because

it enables various phenomena to be highlighted by artificially removing other unwanted

physical processes during a simulation. Experimental work is equally valuable because it

can both validate the models and provide insight into phenomena which models are not

capable of capturing. While many experimental techniques exist, laser-based absorption

9

sensing is preferred in many circumstances because it is non-intrusive, versatile, and

offers the fast time response needed in unsteady devices such as the PDE.

1.4.1 Modeling of PDEs Research into PDEs via modeling has taken many different forms. Most modeling

approaches have been of a 1-dimensional nature. For example, Cheatham and

Kailasanath have used 1-dimensional CFD to investigate droplet evaporation and

subsequent detonation [27,28], Morris has used quasi-1-dimensional CFD to predict

nozzle contributions to PDE impulse [7], and Radulescu and Hanson used the method of

characteristics (unsteady, 1-dimensional) to show that convective heat transfer can be an

important loss mechanism [29]. As mentioned earlier, 2-dimensional solutions have also

been employed [26] but are ultimately not required to capture thrust for simple engine

geometries. Complicated geometries, however, do require 2- and 3-dimensional

solutions. For example, Ma et al. [6] used multi-dimensional CFD to capture the

gasdynamics inside the PDE developed at the Naval Postgraduate School (NPS), despite

the highly complicated geometry involved.

The aforementioned research efforts have all been based on numerical strategies

(i.e. CFD or discretized method of characteristics). Others have used analytic or semi-

analytic modeling approaches in order to devise scaling laws and limiting behavior of

idealized PDEs. Talley and Coy [30] used an unsteady, 0-dimensional, constant-γ

approach to compare the limiting performance of devices which incorporate constant-

volume or constant-pressure heat addition. Heiser and Pratt [3] used thermodynamic

cycle analysis to compare their idealized PDE cycle with the Brayton cycle. One of the

most successful and widely used semi-analytic models for predicting PDE performance

was developed by Wintenberger et al. [9,31] and extended by Cooper and Shepherd [32].

This model exploits analytic relationships to predict impulse during early times within the

cycle, and is calibrated using experiments to predict impulse during late times within the

cycle. Their expression for specific impulse takes the following simple form:

10

( )CJ

sp

gUPPKI

13

1ρ∞−=

where the calibration parameter, K, is dependent on the pressure ratio P3/P∞ . The

remaining quantities can be obtained knowing the state of the reactants. We also notice

that Isp is independent of engine volume, as expected since impulse and propellant weight

both scale with volume. Also notice that low density fuels, such as H2, are expected to

have higher Isp. This model will henceforth be referred to as the “Wintenberger/Cooper

model”.

The modeling work presented herein will first incorporate the analytic constant-γ

strategies used for the Wintenberger/Cooper model in order to extend performance

predictions to PDEs with ideal nozzles (Chapter 2). This will focus on the commonly

used reactants of C2H4 and O2. The reactant conditions (T1 = 298 K, P1 = 1 atm) were

chosen in order to mimic the conditions of the experiments used for model validation.

After the work on ideal nozzles, various nozzle loss mechanisms will be investigated

using a higher-level CFD approach: finite-rate chemistry effects are covered in Chapter 3;

heat transfer and friction effects are covered in Chapter 4. Because chemical mechanisms

will be incorporated, a mixture of H2 and O2 will be the focus in order to save on the

computational expense of solving the C2H4/O2 system. Nevertheless, some attention will

also be given to H2/air and C2H4/O2 mixtures. Reactant conditions which are more

relevant to multi-cycle PDEs will be favored (T1 = 500 K, due to wall-heating by the hot

combustion products), rather than the room temperature condition used in Chapter 2.

1.4.2 Laser-Based Sensing in PDEs The sensors discussed in this work are based on the direct absorption (DA) of laser

power by an absorbing species. Direct absorption sensing is an important branch of

experimental techniques available for PDEs because it offers a non-intrusive means of

ascertaining various aspects of the gasdynamics and chemistry inside the engine.

Furthermore, by utilizing optical fibers to deliver laser power to the engine, the sensors

can be incorporated with engines which translate, vibrate and even fly. Like any

11

experimental strategy, laser-based techniques also have their drawbacks, such as

contending with beam steering, emission, interference from condensed phases, etc.

However, these can be overcome with careful optical engineering and spectroscopic

approaches.

The simplest type of DA sensor for PDEs is a “time of arrival” sensor. For example,

as a plug of fuel is convected downstream to prepare the engine for ignition and

detonation, it is valuable to know exactly when the fuel has properly filled the engine in

order to prevent fuel from exiting the tube before the ignitor discharges. Klingbeil et

al. [33] used a 3.39 μm HeNe laser to detect ethylene and propane arrival in two different

PDEs, enabling the engine developers to adjust their injection and ignition timing

appropriately. A similar strategy was employed separately by Mattison et al. [34] and

later by Ma et al. [35], their strategies using a tunable diode laser at 1.62 μm. However,

not only was arrival of ethylene detected, but this measurement was also used in an active

control scheme. By applying control, flame holding ‡ was greatly reduced and the

engines’ thrust improved dramatically.

While fuel sensing is extremely important to PDE development, it is only one of many

possible applications of laser-based sensing. Measurements of combustion products can

also be extremely valuable, for example to help understand the unsteady motion of the

exhaust gases in the engine and their effect on the subsequent cycle’s reactants. Water is

a common species of interest, not only because of its abundance in the products, but also

because it often exists in substantial quantities in the reactant stream by virtue of a

vitiator which is sometimes employed to heat the incoming flow. Furthermore, due to the

overlap of its spectrum with widely available telecommunications (telecom) lasers,

quantitative water spectroscopy is practiced widely.

Finally, temperature is also a valuable measurement because it can be used to validate

models which address the very high temperatures of detonation products [36].

‡ Flame holding is an undesired effect which occurs when the hot combustion products of the previous

engine cycle prematurely ignite the reactants of the current engine cycle. This can be mitigated using

active control. See Ref. 34 for more details.

12

The current work will extend all three of the above measurement types (fuel, water,

temperature). First, simultaneous water/temperature sensing will be discussed in Chapter

5. The current work is distinct from the previous water sensing studies mentioned above

by its use of the mid-IR (~ 2.5 μm) water spectrum. The benefit of this wavelength lies

in the enhanced absorption and increased signal-to-noise ratio (SNR) over previous work,

which were based on the relatively weak near-IR (~ 1.4 μm) water spectrum. Next, fuel

and temperature sensing will be discussed in Chapter 6. This fuel sensing work

represents the first measurements of the JP-10 spectrum and the application of a tunable

mid-IR (~ 3.4 μm) laser towards quantifying the unsteady equivalence ratio of JP-10/air

inside a multi-cycle PDE.

13

Chapter 2: Idealized Model for Quasi-1D

PDE Nozzles

2.1 Introduction This chapter is concerned with the development of a simple performance model for

idealized PDEs with nozzles. The model is beneficial in three ways: 1) via its

development, it provides insight into the behavior of the unsteady flow experienced by

PDE nozzles; 2) it provides design tools for easy calculation of appropriate area ratios;

and 3) it acts as a guide for work in Chapter 3 and Chapter 4 which address various loss

mechanisms and their effects on PDEs with nozzles.

Various types of nozzles have been considered, the most common of which are

diverging and converging-diverging. (Straight nozzles have also received attention [37],

but are not discussed further here.) In general, diverging and converging-diverging

nozzles have been shown to improve single-cycle performance. Cambier and

Tegnér [38] numerically simulated the thrust from a straight detonation tube fitted with a

diverging nozzle. Both the tube and the nozzle were filled with a reactive mixture. By

increasing the exit area they were able to increase impulse. Fuel-based specific impulse

was also observed to increase for sufficiently large exit area, despite the added fuel mass

which resulted from the larger nozzle volumes. An optimum exit area ratio was not

identified, nor was the method of choosing the range of tested area ratios discussed.

Eidelman and Yang [39] also performed numerical simulations on a straight detonation

tube fitted with a bell-shaped diverging nozzle. The exit/tube area ratio, ε, was fixed at 5

and was based on the plateau pressure, P3. Their design, too, was able to increase

impulse over the straight-tube result. Morris [7] numerically simulated the effect of

converging-diverging nozzles on PDE single-cycle impulse over a wide range of area

ratios and ambient pressures. He showed that an optimum diverging area ratio exists for

each ambient pressure. He also showed that the relative gain of an added nozzle becomes

14

more pronounced as the ambient pressure is decreased. Owens and Hanson [40] also

investigated the question of optimum area ratio by numerically simulating a PDE fitted

with nozzles having different area ratios. The optimum area ratio was found to be well

predicted by assuming a steady nozzle operating with a stagnation pressure equal to the

time-averaged pressure acting at the end-wall of the straight-tube. The model did not,

however, quantify impulse. They also established that both the diverging and

converging-diverging nozzles are choked during a significant portion of the blowdown by

experimentally observing expansion fans at the throat. Although a converging section is

required to choke the steady refresh flow between engine cycles, Ref. 40 showed that a

converging section is not needed to choke the combustion products during blowdown.

Cooper and Shepherd [41] experimentally studied the effects nozzles have on straight-

tube, single-cycle impulse. The area ratio and ambient pressure were varied. The nozzle

contained quiescent air at ambient conditions. In all cases, the addition of a nozzle

increased specific impulse. The authors divided all cases into two broad groups: quasi-

steady and unsteady. Quasi-steady blowdown was attributed to cases which had a low

mass of air initially inside the nozzle, and thus short nozzle startup times. Unsteady

blowdown corresponded to large air mass initially inside the nozzle, and thus long startup

times. A simple model was developed to attempt to capture performance under quasi-

steady operation. Complex phenomena such as transient nozzle separation were expected

to be present for the unsteady cases because ambient pressures were relatively high.

Morris [42] attempted to reproduce Cooper’s measurements using a chemically reacting

2-dimensional Euler-based code. Agreement was best at low ambient pressures where

separation effects were expected to be at a minimum. Since the Euler-code was not able

to capture separation, agreement at high ambient pressures was worse.

The majority of nozzle models proposed so far have been CFD-based and are therefore

computationally expensive. This is especially true if the computational approach is used

to systematically model a wide range of nozzle geometries by brute force in order to

identify the optimum. Furthermore, the resulting design would be specific to a particular

straight-tube length, ambient pressure, initial pressure, fuel, oxidizer and equivalence

ratio. Therefore, a need exists for both a physical understanding of the transient flow

15

through nozzles, as well as any resulting scaling laws, which could then be applied to

simplify the nozzle design methodology.

This chapter addresses this need using a constant-γ approach, and is formulated in

such a way that it can interface with the constant-γ Wintenberger/Cooper straight-tube

model. The majority of the model development focuses on purely diverging nozzles,

with a shorter section on converging and converging-diverging nozzles afterwards. The

model assumes quasi-steady flow within the diverging nozzle and the nozzle’s impulse is

calibrated using results from a numerical simulation. The model is compared to available

diverging nozzle experimental results. Finally, a nozzle design strategy is developed by

exploiting the model’s ability to predict Isp at different area ratios and ambient pressures.

The mixture of choice is C2H4 + 3O2 because of its ability to easily detonate, which

makes it quite relevant to PDE research. In order to compare simulations with available

measurements, the reactants state is chosen as T1 = 298 K and P1 = 1 atm.

2.2 Configuration of Detonation Tube with Diverging

Nozzle The configuration considered here (see Figure 3) was chosen to mimic the

configuration used by Cooper and Shepherd [32], thus making direct comparison to their

experiments possible. The straight-tube section has a length Ls and cross-sectional area

As, subscript ‘s’ representing ‘straight-tube’. The nozzle is conical, and thus has a

linearly increasing diameter. The exit area is denoted as An,e, subscript ‘n’ representing

‘nozzle’ and ‘e’ representing ‘exit’. The nozzle area ratio is ε ≡ An,e /As. The length of

the nozzle is Ln. The interface between the straight-tube and the nozzle is referred to as

the throat, despite the lack of a converging section. An imaginary diaphragm is placed

between the straight-tube and nozzle. The initial condition consists of quiescent reactants

at initial pressure P1 and initial temperature T1 to the left of the diaphragm, and quiescent

air at ambient pressure P∞ and ambient temperature T∞ to the right of the diaphragm.

Unless otherwise stated, the following baseline case is implied throughout this chapter:

C2H4 + 3O2, P1 = 1 atm, T1 = T∞ = 298 K, Ds = 50 mm and Ls = 1 m. Ethylene is the

16

chosen fuel because of its prevalence in fundamental PDE research and because of its use

as an initiator for more practical fuels such as JP-10. The ignition source is located at the

end-wall. Kiyanda et al. [43] showed that DDT has a minimal effect on the impulse of

straight-tubes so we simplify matters by assuming the ignition event leads immediately to

a CJ detonation wave at t = 0.

Figure 3. Configuration for straight-tube with diverging nozzle. † States 1 and ∞ are initial conditions. ‡ State 4 occurs only during steady nozzle flow.

2.3 Equation of State The equation of state is assumed to be

const=γρP § (2)

Figure 4 compares the density taken from this equation of state (solid lines) to the actual

density calculated from STANJAN [18] (solid circles) over a range of pressures.

§ The architecture of the numerical solver was originally designed to solve full finite-rate chemistry,

meaning equations of state are represented by the NASA polynomials, rather than Eq. (2). See Appendix A

for details on implementing the equation of state (2) into the numerical solver.

Ls

End-wall

State 1† State ∞†

x

Ln

Diaphragm

State 4 ‡ θ

17

1E-3 0.01 0.1 1

0.01

0.1

1

10

Pre

ssur

e [b

ar]

Density [kg/m3]

Frozen Equilibrium

CJ

Figure 4: Equilibrium and frozen isentropes for CJ products of C2H4 + 3O2 (P1 = 1 atm; T1 = 298 K) using Eq. (2) (solid lines). γ evaluated at CJ state: γfrozen = 1.24; γequil = 1.14. Real values (circles) obtained from STANJAN.

The CJ state is chosen as the reference state and the two γ ’s used are also evaluated at

the CJ state. Agreement between the solid lines and circles is quite good over the range

considered.

For chemically frozen flow**, γ is the ratio of specific heats at CJ:

24.1==CJv

Pfrozen c

cγ

For chemically equilibrated flow, γ is taken from the CJ equilibrium sound speed, cequil,

calculated using STANJAN:

** Only the effects of chemistry are considered here, so equilibrium is assumed to exist between

vibrational modes.

18

14.122

===CJ

equil

CJ

equilequil T

cP

c R

ργ

where R is the mass-based gas constant. Note that γequil is not equal to the ratio of

specific heats:

v

Pequil c

c≠γ

For the purposes of this chapter, the flow is assumed to be in chemical equilibrium (a

condition which will be relaxed in Chapter 3), so γequil is used. This choice is made in

light of work by Mattison et al. [36] who showed that for a mixture of C2H4/O2 the

chemistry in the straight-tube is effectively in equilibrium. The subscript ‘equil’ is

dropped and γ will henceforth represent the equilibrium value. Most properties (such as

pressure, density and sound speed) can be calculated directly using the analytic approach

presented in this chapter. However, unlike for frozen flow, temperature must be obtained

from a numerical equilibrium solver such as STANJAN (see Appendix C).

2.4 Numerical Model In addition to the analytical model developed herein, a numerical model is employed

to help understand transient flow within the nozzle, to guide nozzle design and to

calibrate the analytic model. The numerical model is described fully in Ref. 26. The

model is quasi-1-dimensional, adiabatic, and inviscid. Both end-wall thrust and nozzle

thrust are determined by integrating pressure over the appropriate thrust surfaces at each

instant in time.

Numerical simulations are initiated by imposing the constant-γ Taylor wave between

the end-wall and the diaphragm using the following expression for pressure [22, § 1.1.4]

19

3

12

33

)(:

1111)(:

PxPxx

cU

LxPxPxx CJ

s

=<

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

−=>−γγ

γγ

(3)

where x is the boundary between the plateau region and the Taylor wave, and is given

by ( ) CJs UcLx 3= . The preceding equation corresponds to the detonation wave at the

tube exit, i.e. t1 = Ls/UCJ. For t between 0 and t1 (i.e. as the detonation wave travels down

the straight-tube) state 3 persists at the end-wall.

The grid resolution was 0.5 mm in all cases.

2.5 Time-Varying Thrust Figure 5 shows the engine configuration with the relevant forces. The pressure forces

for this ideal system are the straight-tube force, denoted Fs (blue arrows), and the nozzle

force, denoted Fn (red arrows).

Figure 5: Breakdown of forces.

We write the overall force as the sum of these two

)()()( tFtFt ns +=T

Each force can also be integrated to obtain an impulse, thus we have the impulse from the

straight-tube, Is, and the impulse from the nozzle, In. We also have the total impulse:

Straight-tube

Nozzle

20

ns III +=

Figure 6 shows the time-varying force histories obtained via numerical simulations. The

ambient pressure, P∞ , was 0.01 atm. The area ratio, ε, was 40 which is close to the

optimum area ratio for these conditions. The simulation was terminated when the end-

wall thrust reached zero. The figure is broken down into panels (a) and (b), the only

difference being the time scale in each. Figure 6(a) shows the entire cycle until the

end-wall thrust has reached zero. Figure 6(b) shows detail at early times. After ignition

(t = 0), the detonation wave travels at CJ speed towards the diaphragm. The plateau

pressure, P3, persists at the thrust wall during the detonation wave traversal of the tube

and impulse begins to accumulate. Before the detonation wave reaches the diaphragm,

the end-wall force is (P3 – P1)As. After the diaphragm breaks (t = t1), the end-wall force

suddenly jumps to (P3 – P∞)As because there is no longer a diaphragm acting as a

negative thrust surface. The CJ wave transmits a shock wave into the nozzle and the

force from the nozzle rapidly builds as this shock wave traverses the nozzle. The peak

nozzle force corresponds to the shock wave reaching the exit of the nozzle (see Figure

6(b)). After the shock has left the nozzle, the nozzle force starts to decay. The dynamics

of this decay are governed by passage of the Taylor wave through the nozzle, as well as

the expansion waves which begin to enter the nozzle. These expansion waves later reach

the end-wall at which point the end-wall force begins its decay. Meanwhile, the nozzle

force is steady, having begun at time t1 + t4. (t2 and t3 were defined in the

Wintenberger/Cooper model, and so have not been repeated here.) At time t1 + t4 + t5,

the nozzle begins to blow down.

Several important conclusions can be drawn from this figure. First: a nozzle which is

choked (i.e. Mthroat = 1, see Figure 6(a)) need not be steady. As the pressure and burnt

gas velocity within the straight-tube decay, the pressure and velocity at the nozzle

entrance also decay, resulting in nozzle thrust decay. All the while, however, the Mach

number throughout the nozzle remains steady, determined uniquely at each location by

the local area ratio. The flow is in fact quasi-steady, meaning flow parameters

everywhere within the nozzle are changing with time, but the steady nozzle equations can

21

still be applied. That is to say, all unsteady terms appearing in the mass, momentum and

energy equations are negligible. Second: a converging section is not required for nozzle

choking, as previously observed by Owens and Hanson [40]. This will be discussed in

more detail in the next section. Third: the time for the transmitted shock to traverse the

nozzle is very short. Fourth: since the nozzle is choked for nearly the entire event, the

straight-tube is practically unaffected by the presence of the nozzle. Only after

disturbances from the environment reach the throat is the straight-tube flowfield affected

by the nozzle. However, this happens so late that the end-wall force is essentially

unaffected by the presence of the nozzle. This independence of the straight-tube from the

nozzle will be foundational for the current analytical model. Fifth: breaking down the

nozzle impulse into its components – viz. shock & Taylor wave passage (t4), steady flow

(t5), nozzle blowdown (t6), shock moving upstream – we see that the first three

components are important and the fourth is negligible. Table 1 quantifies these

contributions.

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5 (a)

Nozzle blowdown

Steady nozzle flow

t5t6

Shock appears at nozzle exit

Nozzle End-wall

Forc

e [k

N]

Time [ms]

End of cycle

Nozzle choked (Mthroat = 1)

0.00 0.25 0.50 0.75 1.000

1

2

3

4

5

6

Fs = (P3 - P1)As∞Fs = (P3 - P )As

Steady

Shock wave at nozzle exit

Taylor wavetraversing nozzle

t1t4

Nozzle End-wall

Forc

e [k

N]

Time [ms]

Shock wavetraversing nozzle

(b)

Figure 6. Sample histories of end-wall and nozzle force. a) entire cycle; b) early times. P1 = 1 atm, T1 = T∞ = 298 K, γ = 1.14, P∞ = 0.01 atm, ε = 40, Ls = 1 m,

Ln = 0.2 m.

22

Table 1. Breakdown of impulse. C2H4 + 3O2, P1 = 1 atm, T1 = T∞ = 298 K, γ = 1.14, P∞ = 0.01 atm, ε = 40, Ls = 1 m, Ln = 0.2 m, Ds = 50 mm.

Impulse Descriptor Relevant Time

Impulse [N·s]

Fraction of Overall Impulse

[%]

Straight-Tube entire cycle 5.23 67

Shock & Taylor wave

passage t4 1.03 13

Steady t5 1.02 13

Blowdown t6 0.58 7

Noz

zle

Remainder remainder -0.066 < 1

Overall n/a 7.78 100

These conclusions are used to lay out the strategy for developing the analytical model.

First of all, the mass of air initially residing in the nozzle will be considered negligible.

This is true for small values of P∞ /P1 and/or small values of Ln/Ls. The important

implication is that the nozzle is instantaneously choked after the CJ wave has passed,

allowing us to segregate straight-tube and nozzle flowfields from each other. Thus, the

straight-tube without nozzle will be simulated and its time-varying exit state will be

tracked in order to calibrate the model. Nozzle impulse will be determined by integrating

force over t4 + t5 + t6, i.e. from the arrival of the CJ wave at the nozzle entrance to the

appearance of a shock at the nozzle exit near the end of the cycle. Additional impulse

produced by the nozzle while this shock wave moves upstream is negligible

(viz. -0.066 N·s in Table 1).

Before beginning the model development, we will first discuss the steady flow regime

evident in Figure 6 in more detail. This steady state can be derived analytically and will

be useful later in modeling the nozzle blowdown, as well as guiding simulations in

Chapter 3 and Chapter 4.

23

2.6 Steady Choked State

2.6.1 Derivation The steady force shown in Figure 6 can be derived analytically by the method of

characteristics (MOC). Instead of considering the nozzle configuration with a Taylor

wave profile, consider for the moment a straight-tube of length L with pressurized

quiescent gases: Pq > P∞ and uq = 0 everywhere, where q represents ‘quiescent’. As

before, there exists a diaphragm at x = L. At time zero the diaphragm breaks and gases

are allowed to escape, expanding and cooling isentropically. As the gases escape they

accelerate, increasing the Mach number at the exit. In fact, if the pressure differential

between the exit and ambient is large enough, the exit Mach number will continue to

increase until the ambient ceases to communicate with the tube flow, i.e. when the exit

becomes choked. We will assume that the startup time for choking is instantaneous, so at

time t = 0+ the exit Mach number is unity. Figure 7 is a schematic of the characteristics

in such a flow field. The left-running characteristics are denoted C- ; the right-running

are denoted C+. Also shown is the speed (cq) of the leading C- characteristic moving

away from the diaphragm.

Figure 7. Schematic of characteristics used for finding steady choked state.

Some textbooks (e.g. Ref. 44 , pp. 186-198) deal with the blowdown of a

1-dimensional tube using the method of characteristics, but the exit state itself is rarely

given any special attention. To proceed we follow Thompson [45, pg. 175]. Recognizing

that the flow is homentropic (meaning each particle experiences isentropic changes, and

x = 0 x = L

t

u = 0

M = 1

C +

C –

cq

24

all particles have the same entropy), constant area and without body forces, it can be

shown that the Riemann variables are conserved along characteristics. In our case we are

interested in the right-moving characteristics:

constant1

2=

−+≡+ cuJ

γ (4)

Thus, finding the exit state is simply a matter of evaluating the Riemann invariant at a

known point on the characteristic (i.e. u = 0 and c = cq) and setting ue = ce by virtue of the

choked exit condition. As such, we obtain:

qe cc1

2+

=γ

(5)

We now abandon the quiescent scenario and adjust the initial state profile to include

the plateau region and Taylor wave. We expect to capture the same steady state at the

exit because the Taylor wave is followed by its own quiescent region. In other words,

although the characteristics are now much more complicated than in the wholly quiescent

case, the Riemann invariant is still conserved from a stagnant region all the way to the

choke point. The difference now is that the steady flow at the exit is preceded by a

period of adjustment arising from the Taylor wave. The quiescent region behind the

Taylor wave has traditionally been labeled state 3. We extend this nomenclature by

introducing state 4 to represent the steady flow at the exit. Using Eq. (5) we get

34 12 cc+

=γ

(6)

Using the well-known isentropic relations, we can obtain an expression for the static and

stagnation pressures:

3

12

4 12 PP

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=γ

γ

γ (7)

25

03

10

4 12 PP

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=γγ

γ (8)

This is the state that persists between t1 + t4 < t < t1 + t4 + t5 in Figure 6.

Recalling that the addition of a diverging nozzle changes nothing in regard to state 4,

we make the important conclusion that the stagnation pressure which exists in a diverging

nozzle during steady flow is not equal to the stagnation pressure at the end-wall: P40 ≠ P3

0.

In fact, for γ = 1.14, the nozzle stagnation pressure is 57% that of the end-wall pressure.

This can also be shown by considering the behavior of a fluid particle’s stagnation

enthalpy as the particle accelerates from near the end-wall to the end of the straight-tube.

For adiabatic, inviscid flow, the material derivative of stagnation enthalpy is given by

[46, pg. 179]:

tP

DtDh

∂∂

=ρ10

Thus, since the static pressure in the detonation tube is everywhere decreasing with time,

the stagnation enthalpy of a particle must decrease as that particle is accelerated towards

the exit. Likewise, the stagnation pressure must also decrease.

By using the following equation of state for isentropic flow (see § C.1, Eq. (67)):

constant1

2

=−

−γch

the change in stagnation enthalpy from state 3 to state 4 can be found explicitly:

1

230

304 +

−=γchh

We see immediately that h40 < h3

0.

26

2.6.2 Choked State Validation Next, state 4 was validated experimentally. A second state was also considered for the

sake of comparison. This second state consisted of accelerating state 3 to the sonic

condition, while conserving stagnation enthalpy. This case will be referred to as “choked

state 3”. This has been a popular approach in previous studies, for example in Ref. 39

where state 3 was used to design the nozzle area ratio and in Ref. 41 where state 3 was

used to model nozzle performance. In order to best identify the correct state, various

gasdynamic properties were calculated for both choked state 3 and state 4. These are

listed in Table 2. STANJAN was used, so no constant-γ assumption was required. We

see that static pressure is most sensitive to the choice of state.

Table 2. Gasdynamic properties of two choked states. C2H4 + 3O2, P1 = 1 atm, T1 = 298 K.

Choked State 3 State 4

h0 [MJ/kg] 1.06 0.368

P [bar] 7.26 4.15

T [K] 3462 3319

u [m/s] 1169 1128

H2O 0.257 0.272

OH 0.104 0.0955

CO 0.238 0.230

Mol

e Fr

actio

n

CO2 0.146 0.161

The validation was performed using Stanford’s detonation tube. The experimental

setup is described in Ref. 47, and is repeated here for convenience. The facility consists

of a stainless steel tube with a length of 1.6 m and an inner diameter of 3.81 cm. A

0.001″ Mylar diaphragm was installed at the open end, and the tube was evacuated. A

mixture of stoichiometric C2H4/O2 was prepared in a separate mixing tank. Once the

mixture was sufficiently mixed, the detonation tube was charged to an initial pressure P1,

27

which was varied between 0.4 and 2.1 atm. The initial temperature was 298 K, and the

ambient pressure was 1 atm. The mixture was ignited near the closed end using a 75 mJ

spark and deflagration-to-detonation transition took place approximately 30 cm from the

end-wall. Three piezo-electric pressure transducers were mounted 6.83, 16 and 28 cm

from the open end, and covered with thermal insulation. These three measurements were

extrapolated to the exit plane in order to obtain time-varying exit plane static pressure.

The error incurred by this extrapolation was found to be negligible compared to heat

transfer effects, which are discussed in the next paragraph. Pressure at the end-wall was

recorded by a fourth piezo-electric transducer, also covered with thermal insulation.

Because heat transfer losses are important in a detonation tube with such a large L/D

ratio [29], P3 was somewhat lower than what adiabatic theory predicts using Eq. (1).

Thus, the models used to predict exit pressure were based on the measured P3, rather than

the theoretical P3. Furthermore, since heat losses caused end-wall and exit pressures to

drop somewhat over time, the reported values correspond to the time-averaged pressure

at the respective locations. Results are shown in Figure 8. State 4 matches theory,

whereas choked state 3 greatly overpredicts the steady exit pressure.

0 5 10 15 20 250

2

4

6

8

10

12

14

Predicted (state 4)

Predicted (choked state 3)

Exi

t Pre

ssur

e [a

tm]

Measured End-Wall Pressure [atm]

Figure 8. Validating state 4 model. C2H4 + 3O2, T1 = 298 K, γ = 1.14. ■ current measurement. Uncertainty bars represent extent of pressure change due to heat

transfer.

28

2.7 Impulse Model The strategy used here is to evaluate impulses from the straight-tube and nozzle

separately. The straight-tube impulse, Is, is obtained from the Wintenberger/Cooper

model, which required experimental calibration. The nozzle impulse, In, is derived herein,

and will be calibrated using numerical simulations.

2.7.1 Straight-Tube Model The impulse of the straight-tube is taken directly from the model developed by

Wintenberger et al. [9]. Their model was then modified by Cooper and Shepherd [32] to

account for reduced ambient pressures. The impulse was expressed as:

CJ

sss U

VPPKI )( 3 ∞−= (9)

where

3

21113

13 )(1

cU

PPPPPP

K CJs ωω ++

−−

≡∞

Here, Is is the impulse of the straight-tube in [N·s]. ω1 and ω2 †† are dimensionless

parameters which are nearly constants across all reactant states of interest. Ultimately, Ks

was determined experimentally by Cooper and Shepherd [32] who reported it as a

function of P∞ /P1.

†† The original nomenclature for these constants used in Ref. 9 was α and β, respectively.

29

2.7.2 Nozzle Model

Nozzle Force To begin the derivation of the nozzle impulse, In, we first require the time-varying

nozzle force, Fn. For quasi-steady flow we have:

)()()( ,,,,,, ∞∞ −−−+−= PPAPPAuumF inineneninennn & (10)

As before, subscript ‘n’ represents ‘nozzle’ and ‘e’ represents ‘exit’. We have introduced

‘i’ to represent ‘inlet’. We next recognize that the nozzle inlet is given by the exit of the

straight-tube, i.e. Pn,i = Ps,e and un,i = us,e. Since the nozzle is choked, we can obtain Ps,e

and us,e by simulating the straight-tube without a nozzle. It can be shown (see § B.1) that

the force reduces to:

( )21,, )()( GPGtPAtF esinn ∞−= (11)

where G1 and G2 are geometric factors which depend on ε and γ, defined here:

( ) ( ) ( ) 1, 12

,11

,,1 −−+≡ +−

+−

− γεεεγγε γγ

γγ

enenen MMMG

( ) 12 −≡ εεG

and Mn,e is determined from ε and γ using the well-known area/Mach number relation (see

Appendix C, specifically § C.3):

)1(2

1

21

2,2

1

,

11 −+

+

−

⎥⎦

⎤⎢⎣

⎡ +=

γγ

γ

γ

ε en

en

MM

(12)

30

Figure 9 shows G1 and G2. Two values of γ are shown, where γ = 1.14 corresponds to

equilibrium chemistry and γ = 1.24 corresponds to frozen chemistry, both for C2H4 + 3O2

(reactants at STP).

1 10 1000.01

0.1

1

10

100

G2

G1 (γ = 1.24)

Geo

met

rical

Fac

tor

Exit Area Ratio, ε

G1 (γ = 1.14)

Figure 9. Geometric factors required to calculate nozzle thrust.

Next we need to integrate Fn over time. This is done by subdividing the nozzle force

history into distinct flow regimes. These regimes were briefly discussed in the context of

Figure 6, but are shown explicitly in Figure 10.

0 1 2 3 4

-1

0

1

2

3

4

5

t1t4 t6

t5

Time [ms]

Ω1, Ω2Δ1, Δ2

steady nozzle flow

Area under entire curve: In

nozzle blowdownTaylor wave passage

Noz

zle

Forc

e, F

n [kN

]

Figure 10: Subdividing nozzle force history. Same conditions as Figure 6.

31

There are three distinct flow regimes: 1) passage of the detonation and Taylor waves

through the nozzle; 2) steady nozzle flow (i.e. state 4); and 3) nozzle blowdown. The

combined area under all three regimes is the desired nozzle impulse, In. The simplest

way to proceed is to derive expressions for the integrated force for the various regimes.

The Taylor wave passage and the steady nozzle flow will be grouped together (blue

hatching), and the nozzle blowdown (green cross-hatching) will be treated separately.

The derivation will result in four dimensionsless constants: Δ1 and Δ2 for the combined

Taylor/steady regime; and Ω1 and Ω2 for the blowdown regime. These constants are

indicated in Figure 10.

Impulse from Taylor Wave and Steady Flow in Nozzle The nozzle force (Eq. (11)) is integrated in time throughout the Taylor wave passage

and the steady flow:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−=

=

∫∫

∫++

∞

++

++

541

1

541

1

541

1

23

,13

3

,/,

/,

τττ

τ

τττ

τττ

τdGPd

PP

GPc

LAI

dtFI

essinsteadyTaylorn

ttt

t nsteadyTaylorn

and the desired expression becomes

( )221133

/, Δ−Δ= ∞GPGPcVI s

steadyTaylorn (13)

where

( )

542

3

,1

541

1

541

1

τττ

ττ

τττ

τ

τττ

τ

+=≡Δ

≡Δ

∫

∫++

++

d

dP

P es

We have introduced c3 and Ls due to the non-dimensionalizing of time (τ ≡ tc3/Ls),

following Ref. 9. The straight-tube volume is denoted Vs. Because the flow is self-

similar, Δ1 and Δ2 are completely independent of geometry and ambient pressure, and

32

depend only on the state of the reactants. Unfortunately, there is no closed-form solution

for Δ1 and Δ2, so these were evaluated by numerically simulating a straight-tube (without

nozzle) and monitoring the exit pressure. Results are shown in Table 3 for various

mixtures and initial pressures. As expected, these quantities depend only weakly on the

initial state.

Table 3. Taylor wave/steady flow impulse factors, Δ1 & Δ2, taken from simulations of straight-tube without nozzle. T1 = 298 K. ‘air’ represents O2 + 3.76N2. All γ ’s correspond to chemical equilibrium evaluated at the CJ state. Thermo properties

used to calculate P3 and γ taken from Ref. 48.

Reactants P1 [atm]

P3 [atm] γ Δ1 Δ2

H2 + ½O2 1.0 7.12 1.13 0.84 1.80 H2 + ½air 1.0 5.93 1.16 0.81 1.79

C2H4 + 3O2 1.0 12.3 1.14 0.84 1.80 C2H4 + 3O2 2.0 25.3 1.14 0.84 1.80 C2H4 + 3O2 3.0 38.4 1.15 0.85 1.81 C2H4 + 3air 1.0 6.91 1.16 0.87 1.92

JP-10 + 14O2 1.0 14.5 1.14 0.90 1.92 JP-10 + 14O2 2.0 29.7 1.14 0.85 1.81 JP-10 + 14O2 3.0 45.1 1.14 0.84 1.80 JP-10 + 14air 1.0 7.01 1.16 0.88 1.93

Impulse from Nozzle Blowdown The portion of nozzle impulse from the nozzle blowdown flow regime is somewhat

more complicated than for the Taylor wave/steady flow impulse. The reason is that the

integration time now depends on P∞ , whereas Δ1 and Δ2 were completely independent of

P∞ . This obstacle is overcome by making a key assumption which actually leads to a

closed form solution. We assume that by the time the nozzle blowdown begins, the

contents of the straight-tube are spatially uniform. This reduces the problem to that of an

33

isentropic constant-volume (CV) blowdown with an initial state equal to state 4. Using

this assumption, the pressure at the exit of the straight-tube is given by:

1

2

04

4

, )(2

11)( −

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+=

γγ

γ

s

es

Lttc

PtP

(14)

where t0 is the time at which nozzle blowdown begins (see § B.2 for derivation). Figure

11 compares the true straight-tube exit pressure to the one predicted by Eq. (14).

Agreement is more than adequate to capture the desired impulse. (Recall from Table 1

that the contribution from this regime is only 7% of total engine impulse, so

imperfections in modeling the nozzle blowdown can be tolerated.)

0 1 2 3 40

1

2

3

4

CV blowdown

Stra

ight

-Tub

e E

xit P

ress

ure

[bar

]

Time After Blowdown Starts [ms]

Full solution

Figure 11: Decay of straight-tube exit pressure, comparing full solution with CV blowdown, Eq. (14). C2H4 + 3O2, P1 = 1 atm, T1 = 298 K, P∞ = 0.01 atm, γ = 1.14,

P4 = 4.15 bar, c4 = 1128 m/s, Ls = 1 m.

Next, the thrust is obtained from Eq. (11) and then integrated from the beginning of

nozzle blowdown to the time at which a shock enters the diverging nozzle

(t = t1 + t4 + t5 + t6 see Figure 6 or Figure 10). At this point in the blowdown, the nozzle

force is small enough that additional impulse is negligible. The time t6 has a closed form

34

solution, and is discussed in § B.3. For clarity, though, only the final result for the nozzle

impulse during blowdown is reproduced here:

( )221133

, Ω−Ω= ∞GPGPcVI s

blowdownn (15)

which is analogous to Eq. (13) and shows the dimensionless constants Ω1 and Ω2

anticipated by Figure 10. Unlike Δ1 and Δ2 (which were computed numerically), Ω1 and

Ω2 have closed form solutions thanks to the assumption of CV blowdown:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛Π−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

≡Ω

+

∞− γ

γγ

γ

γ

21

3

12

1 11

2PP (16)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛Π

−+

≡Ω

−−

∞ 111 2

1

32

γγ

γγ

PP (17)

The dimensionless parameter, Π, is called the shock factor and has the expression

( ) ( )12

12

1, 2,

12

,

12

+−+

⎟⎠⎞

⎜⎝⎛ +

=Π +−

γγγεγγε γ

γγγ

enen M

M

See § B.3 for details.

Nozzle Impulse The expression for the nozzle impulse is

blowdownnsteadyTaylornn III ,/, +=

and can be rewritten as

35

( )CJ

snnn U

VKPKPI 2,1,3 ∞−= (18)

where

( )1113

1, Ω+Δ≡ Gc

UK CJn

( )2223

2, Ω+Δ≡ Gc

UK CJn

Equation (18) is analogous to Eq. (9) which was derived for the straight-tube (without

nozzle). Note that both In and Is are proportional to the straight-tube volume, and

inversely proportional to the CJ wave speed. Both impulses are also proportional to a

pressure difference, modified by constants: Is depends directly on Ks(P3 – P∞), whereas In

involves constants which cannot be factored out because the pressure difference across

the nozzle wall changes as we move from the throat to the nozzle exit.

Impulse of Straight-Tube + Nozzle Finally, the overall impulse and specific impulse of the straight-tube + nozzle are

given (respectively) by:

ns III += (19)

gV

IIs

sp

1ρ= (20)

These relations can now be used to find the specific impulse for any straight-tube

geometry, nozzle area ratio, reactant state and ambient pressure. The procedure is as

follows:

Table 3 is used to determine Δ1 and Δ2, which depend only on the reactants

Eqs. (16) and (17) are used to determine Ω1 and Ω2, which depend on the reactants

and the nozzle geometry

Eq. (18) is used to determine the nozzle impulse, In

36

The straight-tube impulse, Is, is calculated using Eq. (9), its calibration coefficient

Ks being obtained from Ref. 32.

The overall impulse is obtained from Eq. (19), and the overall specific impulse from

Eq. (20)

2.8 Model Validation There is a scarcity of experimental data available in the literature for single-cycle

impulse from detonation tubes with nozzles. Cooper and Shepherd [41] performed a

comprehensive study where they measured impulse using a ballistic pendulum over a

wide range of ambient pressures. The initial mixture was stoichiometric C2H4/O2 at

80 kPa and 298 K. Results from their nozzle with ε = 6.5 are shown in Figure 12 along

with several models.

0.01 0.1 1 100

100

200

300

400

Steady model (state 4)

Steady model (choked state 3)

2D CFD

Spe

cific

Impu

lse,

Isp [s

]

Ambient Pressure, P [bar]

Current model

Figure 12. Comparison of experimental data with models. ■ and steady model (choked state 3) from Ref. 41. “2D CFD” model from Ref. 42.

C2H4 + 3O2, P1 = 80 kPa, T1 = T∞ = 298 K, γ = 1.14, ε = 6.5.

37

2.8.1 Steady State Model The two green lines in Figure 12 represent model results which follow Cooper and

Shepherd’s steady state strategy [41], in which they assumed that the nozzle is steady and

pressure-matched to the ambient. This leads to the following expression for specific

impulse:

g

uI ensp ,= (21)

Thus, by specifying the nozzle area ratio and the inlet state we can determine the outlet

state, and thereby obtain Isp. The question remains as to which inlet state to use. Cooper

and Shepherd chose to use the state which corresponded to choked flow with the same

stagnation enthalpy as state 3. This state was encountered in § 2.6.2 and was labeled

“choked state 3”. The second green line also represents Eq. (21), but uses state 4 as the

nozzle’s inlet state. This correction enables the steady state model to perform better, but

it still greatly overpredicts Isp at low P∞ . This is due to the pressure-matched assumption:

an actual steady nozzle operating at these low pressures would be underexpanded,

resulting in an Isp below what is predicted by Eq. (21).

2.8.2 Unsteady Model The unsteady model developed in § 2.7 is shown as the solid red line. Agreement is

best at low ambient pressures. As ambient pressure increases the model fails to predict

Isp. There are two reasons for this. First, nozzle startup time is prolonged which results

in significant impulse generated as the transmitted shock passes through the nozzle.

Second, flow separation in the nozzle tends to increase impulse above that predicted

when separation is ignored. This is because the pressure is not allowed to stay sub-

atmospheric downstream of the separation point. (See Refs. 41 and 49, pp. 41–68, for

further discussion on separation.)

The increased impulse due to delayed startup time can be accounted for by performing

a full CFD computation. This computation was performed by Morris [42] who attempted

38

to reproduce Cooper and Shepherd’s results, and is labeled “2D CFD” in Figure 12. The

CFD model approaches the experimental data at high ambient pressure yet still falls short

due to the unmodeled flow separation. At low ambient pressure the CFD model

overpredicts the data. This is likely due to losses such as heat transfer and friction which

the CFD model neglects. The current model, on the other hand, is able to capture these

effects because it relies on the parameter Ks. This parameter was obtained experimentally

and therefore inherently subsumes non-ideal effects.

2.9 Nozzle Design The current model can now be used to design diverging nozzles for PDEs with an

optimal area ratio for a given reactant state and ambient pressure. Since Is does not

depend on ε, only In need be considered in this optimization procedure. After choosing

the reactant and ambient states, the most straightforward and accurate way to proceed is

to vary ε in Eq. (18) until In is maximized. This gives the optimized area ratio, denoted

ε~ , and its corresponding nozzle impulse. The straight-tube impulse is then added to get

overall impulse. The result is normalized by reactant weight to obtain specific impulse.

The results are plotted in Figure 13 for the baseline case (solid blue line). Also included

is specific impulse from a straight-tube (dashed blue line). This shows the level of

increase in specific impulse by the addition of an optimized diverging nozzle. The

optimized area ratio is also shown (solid red line).

39

0.01 0.1 10

50

100

150

200

250

300

1

10

100

Optimized area ratio

Isp for straight tube + optimized nozzle

Spe

cific

Impu

lse,

Isp [s

]

Ambient Pressure, P [bar]

Isp for straight tube ~

Opt

imiz

ed A

rea

Rat

io, ε

Figure 13. Performance and area ratio for optimized nozzle. C2H4 + 3O2, P1 = 1 atm, T1 = 298 K, γ = 1.14.

Figure 14(a) shows ε~ for various fuels and oxidizers plotted against P∞ /P1, the

important “ambient-to-fill” pressure ratio. Furthermore, since ε~ is commonly computed

using the “ambient-to-stagnation” pressure ratio in the context of steady nozzles,

panel (b) shows the same data as panel (a), plotted against P∞ /P40. (The nozzle

stagnation pressure P40 is obtained using Eq. (8).) We find that by plotting ε~ against

P∞ /P40 instead of P∞ /P1 the data come close to collapsing onto a single line. This is also

true for steady nozzles, whose optimized area ratios depend primarily on P∞ /P0.

40

0.01 0.1 11

10

100~

Opt

imiz

ed A

rea

Rat

io, ε

(a)

∞Ambient Pressure / Fill Pressure (P / P1)

H2 + 0.5O2 H2 + 0.5air C2H4 + 3O2 C2H4 + 3air JP10 + 14O2 JP10 + 14air

1E-3 0.01 0.11

10

100

~

Steady nozzle(γ = 1.15)

∞

Opt

imiz

ed A

rea

Rat

io, ε

Ambient Pressure / Nozzle Stagnation Pressure (P / P04)

H2 + 0.5O2 H2 + 0.5air C2H4 + 3O2 C2H4 + 3air JP10 + 14O2 JP10 + 14air

Curve fit

(b)

Figure 14. Optimized area ratio vs. (a) ambient/fill pressure ratio and (b) ambient/nozzle stagnation pressure ratio. Steady nozzle calculation also shown

in panel (b). P1 = 1 atm, T1 = 298 K.

A 2nd order least-squares fit to these data yields a convenient design tool for sizing the

optimized exit area ratio:

( ) 004

1

2

04

2 lnln~ln aPP

aPP

a +⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= ∞∞ε (22)

where a2 = 0.02808, a1 = -0.4618 and a0 = -0.3482. Owens and Hanson [40] numerically

simulated stoichiometric C2H4/O2 with P1 = 1 atm (P40 = 7.11 atm) and found that the

specific impulse was optimized for ε~ = 1.8. This agrees well with Eq. (22) which

predicts ε~ = 1.9. Eidelman and Yang [39] numerically simulated stoichiometric C2H2/air

with P1 = 1 atm (P40 = 4.1 atm). They chose ε = 5 and found that the flow was

overexpanded. In light of the current work this overexpansion is expected since Eq. (22)

suggests a more reasonable area ratio for this mixture to be ε~ = 1.4. Li et al. [50]

performed experimental measurements of Isp using a multi-cycle straight-tube PDE fitted

with diverging nozzles ranging from ε = 1 to 4.24. The mixture was kerosene/O2,

41

P1 = 1 atm, Φ = 1.2 (P40 = 10 atm‡‡). Their experimental results indicated an optimum

area ratio between 2 and 3, which agrees well with the predicted value of ε~ = 2.4 using

Eq. (22). Finally, it is interesting to consider the nozzle used for the experimental

validation data of Figure 12. Recall, the mixture was C2H4 + 3O2, P1 = 80 kPa

(P40 = 5.6 atm) and a particular with ε = 6.5 was used. According to Eq. (22), the

appropriate ambient pressure for this nozzle and reactant state is P∞ = 0.12 bar. Recall

that the current model’s Isp predictions diverged from the validation data at high P∞ .

However, note that the current model’s Isp prediction does very well at P∞ = 0.12 bar, the

pressure at which this particular nozzle would ultimately be used.

Also included in Figure 14(b) is the optimum area ratio for a pressure-matched steady

nozzle using the well known isentropic area relation (see § C.3). An average value of the

γ ’s from Table 3 was chosen: γ = 1.15. Although this relation seems to overpredict the

required area ratio at low P∞ , the optimized PDE nozzle and the optimized steady nozzle

are strikingly close in their area ratios for a given P∞ /P40. This implies that a diverging

nozzle for a PDE can be thought of as essentially a steady nozzle with a stagnation

pressure equal to P40, obtained using Eq. (8).

2.10 Converging and Converging-Diverging Nozzles The preceding work has focused on the analysis and design of purely diverging

nozzles for PDEs. A PDE with this type of nozzle is easily modeled because the

end-wall impulse can be predicted using the Wintenberger/Cooper model. However, a

purely diverging nozzle may, in some cases, be impractical. For example, when

considering the purge/fill segments of the cycle (i.e. beyond the detonation/blowdown

segments considered heretofore; see § 1.3) it may sometimes be necessary for a PDE to

maintain steady choked flow during purge and fill. This steady flow during purge and fill

‡‡ This assumes the surrogate n-C12H26 for modeling the thermodynamic properties of kerosene. At CJ,

this yields P2 = 45.9 bar, γ = 1.14. Properties taken from Ref. 48.

42

is sometimes called ‘refresh’ (see, e.g., Ref. 51). This is depicted using end-wall static

pressure in Figure 15 for the cases of no converging nozzle (panel (a)) and with a

converging nozzle (panel (b)). The end-wall is closed during detonate/blowdown, and

open during purge/fill (refresh). Notice the purge/fill (refresh) pressure is equal to the

ambient pressure when no nozzle is used, and is elevated when a converging nozzle is

added. (Pressure losses inside the tube are neglected.) This refresh period has a unique

Mach number which is denoted Mrefresh.

Since the flow is steady during the refresh, a purely diverging nozzle will not be able

to maintain choked flow. In order to choke this steady refresh flow, a converging nozzle

is required. This section discusses the requisite area ratio for the converging section.

43

Figure 15: End-wall static pressure vs. time for straight-tube (a) without converging nozzle and (b) with converging nozzle. Pressure losses inside the tube are neglected.

End-

wal

l sta

tic p

ress

ure

Time

Ambient pressure, P∞

Cycle #1 Cycle #2

Purge/fill (‘refresh’) En

d-w

all s

tatic

pre

ssur

e

Time

Ambient pressure, P∞

Detonate/ blowdown

Purge/fill (‘refresh’)

Detonate/ blowdown

Purge/fill pressure

(b)

(a)

Purge/fill pressure

44

2.10.1 Converging Nozzle The setup, consisting of a straight-tube and a converging nozzle, is depicted in

Figure 16.

Figure 16: Configuration for straight-tube with converging nozzle. † States 1 and ∞ are initial conditions. ‡ State 4 occurs only during steady nozzle flow.

The configuration is identical to Figure 3, except the conical diverging nozzle is

replaced with a conical converging nozzle, and the throat is now referring to the nozzle

exit. The nomenclature is expanded to accommodate the new nozzle type: subscript ‘c’

represents ‘converging’, e.g. Ln,c is the length of the converging nozzle, εc is the area ratio

of the converging section, i.e. εc ≡ As/Athroat, and θc is the converging nozzle’s half-angle.

The major complication with trying to model the single-cycle performance of this

system arises from the pressure waves which reflect off the converging section and send

this information back to the end-wall. Figure 17 shows simulation results of the end-wall

and nozzle thrust for the configuration in Figure 16 and a converging area ratio of 1.62.

Ls

End-wall

State 1† State ∞†

x

Ln,c

Diaphragm

State 4 ‡

Throat

θc

45

0 2 4 6 8 10 12

-2

-1

0

1

2

3

steady flowIn,c = -1.95 Ns

Straight-tube Nozzle

Forc

e [k

N]

Time [ms]

Is = 7.02 Ns

Figure 17: Time-varying thrust for straight-tube with converging nozzle. C2H4 + 3O2, P1 = 1 atm, P∞ = 0.1 atm, T1 = T∞ = 298 K, γ = 1.14, εc = 1.62,

Ds = 50 mm, Ls = 1 m, θc = 12°.

Note the series of shocks which start at the nozzle, reflect, and then reach the end-wall.

This behavior was observed by Owens [52] using a velocity diagnostic. Also note that

the negative impulse from the converging section is not negligible compared with the

impulse from the end-wall.

2.10.2 Performance with Converging Nozzle Although the converging nozzle complicates modeling, the overall impulse of the tube

and nozzle can be obtained by full simulation and integrating the thrust history. This was

performed for several values of εc over a range of ambient pressures for C2H4 + 3O2 at

46

STP. The results are shown in Figure 18. The Wintenberger/Cooper model for a

straight- tube is also shown§§.

0.01 0.1 13.0

3.5

4.0

4.5

5.0

5.5

6.0

Numerical Computations(straight tube)

∞

Stra

ight

Tub

e +

Con

verg

ing

Noz

zle

Impu

lse

[Ns]

Ambient Pressure, P [atm]

εc = 1.00 ε

c = 1.10

εc = 1.62

Wintenberger/Cooper model (no nozzle)

Figure 18: Impulse for straight-tube with converging nozzle. C2H4 + 3O2, P1 = 1 atm, T1 = T∞ = 298 K, γ = 1.14, Ds = 50 mm, Ls = 1 m, θc = 12°.

The important implication of Figure 18 is that the impulse from a straight-tube with a

converging nozzle is nearly independent of the converging area ratio, εc. (All differences

in impulse between the various geometries, at a particular pressure, are less than 1%.)

This is because the negative impulse produced by the compressive shocks striking the

converging section are cancelled by the positive impulse produced by the same

compressive shocks later striking the end-wall. This ultimately means that the specific

§§ Since the Wintenberger/Cooper model appealed to experiments for calibration, losses were present in

their values of Ks. For the purposes of comparison here, however, the model needed to be re-calibrated for

this lossless context. Impulse from a straight-tube configuration was therefore computed over a wide range

of ambient pressures. A constant value of Ks = 5.0 was found to work well at all ambient pressures.

47

impulse of a straight-tube with a converging nozzle can be predicted by simply using the

Wintenberger/Cooper model.

2.10.3 Converging-Diverging Nozzle Finally, a diverging section is added to the system, as shown in Figure 19. A subscript

‘d’ identifies quantities associated with the diverging section of the nozzle, such as θd,

Ln,d and εd ≡ Ae/Athroat.

Figure 19: Configuration for straight-tube with converging-diverging nozzle. † States 1 and ∞ are initial conditions. ‡ State 4 occurs only during steady nozzle

flow.

The impulse of this system can be modeled by invoking the Wintenberger/Cooper

model for the straight-tube + converging section, and using the nozzle model derived

earlier in the present chapter for the diverging section. In other words, an equivalent

system consisting of a straight-tube and diverging nozzle, but without a converging

section, is used in order to exploit the Wintenberger/Cooper model and the current

diverging nozzle model. This equivalency is depicted in Figure 20. The straight-tube

cross-sectional area, As, and the diverging area ratio, εd, must be the same in both

configurations. This implies that the nozzle exit area will be larger in the modeled

system than in the real system.

Ls

End-wall

State 1†

x

Ln,c

Diaphragm

State 4‡

Ln,d

θd

Throat

State ∞†

48

Figure 20: Impulse of a straight-tube with converging-diverging nozzle is estimated by equating it to a straight-tube with diverging nozzle. The straight-tube area, As,

and diverging area ratio, εd, are the same in both configurations. The exit area, An,e, is larger in the modeled system than in the real system.

2.10.4 Steady Nozzle Stagnation Pressure Just as was the case with the purely diverging nozzle, we expect the converging nozzle

to experience a period of steady flow. (This is indicated in Figure 17 between t = 1 ms

and 2 ms.) As was also the case for purely diverging nozzles, the conditions inside the

nozzle during this period can be determined analytically. Despite the fact that most of the

blowdown is in fact unsteady, it will be instructive to determine this steady thrust. As

before, the corresponding gasdynamic state is labeled ‘state 4’ and refers to steady flow

at the interface between the straight-tube and the nozzle (see Figure 16 and Figure 19).

The derivation of state 4 is straightforward once we recognize that the only effect the

converging nozzle has is to smoothly alter the Mach number from a value of M4 (at the

nozzle inlet) to unity (at the throat). Recall Eq. (4):

constant1

2=

−+≡+ cuJ

γ

This equation is valid along all C+ characteristics everywhere in the straight-tube. We

equate the Riemann invariant at states 3 and 4:

4433 12

12 cucu

−+=

−+

γγ

=

Real system Modeled system

49

and set u3 to zero. Previously, u4 was set equal to c4 in the absence of a converging

nozzle. This is generalized to include a converging nozzle by setting u4 equal to M4c4.

This leads to the final result:

( ) 34

4 212 cM

c+−

=γ

(23)

Equation (6), which addressed the case of no converging section, is a special case of

Eq. (23) with M4 = 1. As was done for the purely diverging nozzle, the static and

stagnation pressures can also be computed:

( ) 3

12

44 21

2 PM

P−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=γ

γ

γ

( )0

3

1

242

1

242

10

41

1P

M

MP

−

−

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

+=

γγ

γ

γ

(24)

If we assume that γ4 = γrefresh, we can make the important observation that M4 = Mrefresh.

This is because whichever converging area ratio is chosen to achieve Mrefresh, the nozzle

inlet Mach number will reach the same value whether the flow consists of combustion

products expanding during blowdown, or reactants during steady engine refresh. (The

pressure, however, will be different during unsteady blowdown and steady fill.)

Equation (24) is implemented as follows:

εc is determined from the design Mrefresh target

M4 is set equal to Mrefresh

Eq. (24) is used to calculate the nozzle stagnation pressure during steady nozzle

flow

Figure 21 shows Eq. (24) graphically versus Mrefresh. In the case of no converging

section, where the refresh Mach number is unity, the disparity between the end-wall

stagnation pressure (P30) and the nozzle stagnation pressure (P4

0) is greatest. At lower

50

refresh Mach numbers, a converging section is required and the nozzle stagnation

pressure moves closer to the end-wall stagnation pressure.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0γ

4 = 1.14

Sta

gnat

ion

Pre

ssur

e R

atio

, P0 4 /

P0 3

Inlet Mach Number of Converging Nozzle, M4

purely diverging nozzle

Figure 21: Stagnation pressure ratio vs. refresh Mach number.

If we relax the condition that γ4 = γrefresh this results in M4 ≠ Mrefresh, but the

calculations are still straight-forward. Using this 2-γ approach, the refresh value (γrefresh)

is used to obtain the required εc, while the combustion product value (γ4) is used to

calculate P40 using Eq. (24) or Figure 21.

A converging-diverging nozzle can now be designed in much the same way that a

purely diverging nozzle was designed earlier. First, the converging area ratio, εc, must be

chosen (likely based on requirements for Mrefresh). Next, the stagnation pressure of the

nozzle, P40, is computed. Finally, the diverging area ratio, εd, is calculated using Eq. (22).

2.11 Summary This chapter explored the potential to predict PDE nozzle performance using a simple

constant-γ approach. By exploiting the fact that the flow is choked at the inlet of the

diverging nozzle, the straight-tube and nozzle were segregated. By applying the

51

Wintenberger/Cooper model, the straight-tube performance is known, and by solving the

time-varying state at the tube exit, the nozzle performance is also known. A simple fit of

optimized area ratio at various ambient pressures was provided for nozzle design

purposes.

Converging and converging-diverging nozzles were also addressed. It was discovered

that the specific impulse of a straight-tube with a converging nozzle is independent of the

contraction area ratio. This fact allows the Wintenberger/Cooper model to be used on

PDEs with converging nozzles, thereby expanding the capabilities of their model as well

as the current diverging nozzle model to a wider range of PDE configurations.

In the process of the above model development, the steady stagnation state of a PDE

nozzle was obtained. The result showed that the stagnation pressure of a PDE nozzle

(named ‘state 4’) is significantly lower than the stagnation pressure at the end-wall. It

was found that if the flow is assumed steady, state 4 is capable of reasonable predictions

for optimized area ratio using simply the standard Mach #-area relation.

53

Chapter 3: Finite-Rate Chemistry Effects

on PDE Performance

3.1 Introduction Chapter 2 introduced the PDE nozzle and investigated the benefits which an ideal

lossless nozzle provides to engine Isp. The natural progression is to now investigate the

effect of losses on PDE nozzles. This chapter focuses specifically on the losses incurred

from finite-rate chemistry, which can be found in diverging nozzle flows.

Finite-rate chemistry in steady diverging nozzles has been studied for many years.

Wegener [53] showed that the chemistry departs from equilibrium for a system of N2O4

diluted in N2. This was done by experimentally measuring the concentration of NO2,

which is the product of N2O4 decomposition. Zonars [54] and Duffy [55] both studied

the blowdown of air and observed a departure from equilibrium using static pressure

measurements. In order to gain a clear picture of the impact of finite-rate chemistry on

nozzle performance, Olson [56] compiled the research of others who had focused on

measurements of nozzle thrust and Isp for a variety of fuels and equivalence ratios. He

found that, while in all cases considered the frozen and equilibrium solutions are quite

disparate, only for a limited set of mixtures does finite-rate chemistry lead to significant

penalty in thrust and Isp.

Other researchers (e.g. Anderson [ 57 ], Scofield and Hoffman [ 58 ], and Rizkalla

et al. [59]) have addressed the effects of finite-rate chemistry by devising sophisticated

methods for nozzle shape design.

A PDE with a purely diverging nozzle is studied here in order to determine the effect

finite-rate chemistry has on its performance. Cooper [60, § 6] addressed part of this

54

problem by considering a simple straight-tube PDE (without nozzle) and showed that the

characteristic reaction times for C2H4 + 3air*** are several orders of magnitude longer

than those for C2H4 + 3O2. This suggests that fuel/air mixtures are much more

susceptible to chemistry-related losses than fuel/O2 mixtures. However, she did not

quantify any performance metrics and so we can not yet conclude whether these

prolonged timescales of fuel/air mixtures will have a practical impact.

The purpose of this chapter is to establish the extent that finite-rate chemistry has on

the performance of PDEs operating with diverging nozzles. Whereas Chapter 2 focused

on the mixture C2H4 + 3O2 (298 K; 1 atm), the current chapter focuses on H2 + 0.5O2

because of the computational savings afforded by the relatively simple H/O chemistry.

Furthermore, the reactant temperature is chosen as T1 = 500 K in order to better mimic

conditions found in real multi-cycle PDEs. (The initial pressure is again chosen as

P1 = 1 atm.)

3.2 Background The temperature, pressure and species mole fractions of a flow are influenced both by

how quickly the gases expand (i.e. gasdynamic effects) and how quickly the gases react

(i.e. chemical effects). Two extreme cases can be conceptualized: 1) chemically

equilibrated flow (CEF) and 2) chemically frozen flow (CFF). In CEF, the chemical

reactions are sufficiently fast that the gas mixture remains chemically equilibrated as the

thermodynamic state changes. This type of flow can occur when the chemical reaction

rates are very high (e.g. because of high temperature and/or pressure), or when the

gasdynamic rates of change of temperature and pressure are very low (e.g. very gradual

area change in a duct). In CFF, on the other hand, chemical reactions are so slow

compared to changes in temperature and pressure that chemistry can literally be thought

of as frozen. Being the opposite of CEF, CFF can therefore take place when chemical

*** ‘air’ represents O2 + 3.76N2

55

reaction rates are slow (e.g. with low temperature and/or pressure), or when gasdynamic

rates of change of temperature and pressure are very high (e.g. sudden area change in a

duct).

In general, the thermodynamic state is determined by specifying two intensive

properties and the species mole fractions. For example, pressure P can be determined

from density ρ, specific entropy s, and the mole fractions Xi’s:

( )iXsfP ,,ρ=

At this point, it is important to note that both CEF and CFF are internally reversible flows.

If we further assume zero heat transfer, we have s = const. Finally, for both CEF and

CFF, the mole fractions are determined uniquely by the mixture at a reference state.

Therefore, we have

( )ρCEFfP =

( )ρCFFfP =

This functionality was depicted in Figure 4 (Chapter 2) when it was necessary to evaluate

γ for the idealized nozzle model. However, the constant-γ assumption is abandoned

henceforth in order to accurately capture the low pressures which exist in large area-ratio

nozzles. Thus, the equations of state are left symbolically as fCEF and fCFF.

The regime between CEF and CFF is known as finite-rate chemistry flow (FRCF).

Here, chemical reactions are fast enough that species are not frozen, but not so fast that

chemical equilibrium is achieved. Unlike CEF and CFF (which are both described using

simple equations of state), FRCF depends strongly on the details of the gasdynamic and

chemical histories. Thus, the pressure is not related to density via a simple equation of

state, nor is the flow reversible or isentropic. The equations of mass, momentum, energy

and species must be fully integrated in time.

This chapter ultimately covers the effects of CEF, CFF and FRCF in unsteady PDEs.

Before moving on to the more complicated case of unsteady flow, we begin by focusing

on the special case of steady flow, viz. steady flow through a diverging nozzle.

56

3.3 CEF and CFF in Steady Nozzles The static pressure distribution through a steady nozzle directly determines the

nozzle’s thrust. This pressure distribution is obtained by recognizing that the entropy,

stagnation enthalpy and mass flow rate are maintained in an adiabatic and internally

reversible steady nozzle. If quasi-1D flow is assumed, all flow variables become unique

functions of the local cross-sectional area once a specific inlet state for the nozzle is

chosen. The most straightforward way to evaluate the nozzle flow is to use the static

pressure, P, as the independent variable. Then all other variables, including cross-

sectional area, can be explicitly calculated.

Start by specifying the nozzle inlet state, which is assumed to be choked (represented

by * ). This consists of the stagnation enthalpy, h0, the static enthalpy, h, the static

pressure, P, the static density, ρ, and the sound speed, c. The gas velocity, u, is equal to c.

Next, choose a lower static pressure representing a point downstream of the inlet. From

this, calculate ρ and h using STANJAN (which can be applied to both equilibrium and

frozen processes):

);(Pf=ρ )(Pfh =

Next, calculate the gas velocity using the definition of stagnation enthalpy:

( )hhu −= 02

Finally, knowing ρ and u, the local area ratio is obtained from the conservation of mass:

uu

AA

ρρ **

* =

This procedure is repeated for all values of A/A* which are of interest for a particular

nozzle. The result is the desired relationship between P and A/A*. While the nozzle’s

shape will be important in determining whether a flow is ultimately CEF or CFF, notice

that the equilibrium and frozen solutions themselves do not depend on nozzle shape.

57

The pressure-area relation is calculated for CEF and CFF and plotted in Figure 22.

The reactants are H2 + 0.5O2 (500 K, 1 atm) and the nozzle inlet state is chosen as state 4

of this mixture as discussed in § 2.6.1. The equilibrium polytropic exponent (γ = 1.13)

was used to evaluate P4 from P2, in light of the work by Mattison et al. [36] who showed

that species inside the straight-tube are in chemical equilibrium. Equation (7) is used to

calculate P* = P4 = 1.47 bar. Next, STANJAN is used to calculate the sound speed

(c* = c4 = 1374 m/s), the static enthalpy (h* = h4 = -0.648 MJ/kg) and the static density

(ρ* = ρ4 = 0.0866 kg/m3). The stagnation enthalpy, h0 = 0.297 MJ/kg, is obtained from

2

24

404

chh +=

With the inlet state established, the remaining points are obtained using the procedure

outlined above.

1 10 1001E-4

1E-3

0.01

0.1

1

Pre

ssur

e R

atio

, P/P

*

Area Ratio, A/A*

Equilibrium Frozen

Figure 22: Pressure ratio vs. area ratio showing the difference between CEF and CFF. Inlet state is taken as state 4 of H2 + 0.5O2, P1 = 1 atm, T1 = 500 K.

This figure shows that at a given axial location in a nozzle, the local static pressure

will depend on whether the propellants have been experiencing CEF (leading to higher P)

58

or CFF (leading to lower P). The integration of this static pressure along the nozzle wall

leads to thrust, thus leading to a reduction in thrust if the flow is frozen.

The above results apply to flow inside the nozzle, regardless of the particular ambient

pressure existing outside the nozzle. If the ambient pressure is specified, Isp can be

obtained. To simplify matters, we assume that the nozzle is pressure-matched to the

ambient, meaning that the exit area ratio is such that the resulting exit pressure is equal to

the ambient pressure. For a steady pressure-matched nozzle, the Isp is directly

proportional to the gas velocity at the exit plane, ue:

gu

I esp = (25)

Using this relation the Isp is plotted in Figure 23 for CEF and CFF over a range of exit

area ratios.

1 10 1000

50

100

150

200

250

300

350

400

450

500

Spe

cific

Impu

lse,

Isp [s

]

Exit Area Ratio, Ae/A*

Equilibrium Frozen

Figure 23: Specific impulse vs. area ratio for a pressure-matched steady nozzle. Inlet state is taken as state 4 of H2 + 0.5O2, P1 = 1 atm, T1 = 500 K.

This figure shows that for a steady nozzle with a sufficiently large exit area ratio, the

losses can be important. For example, the space shuttle main engine has an area ratio of

59

78, which according to Figure 23 can potentially incur a penalty of 8% − sufficient

enough to warrant serious investigation.

3.4 Computations of PDE with Nozzle Having introduced the problem of finite-rate chemistry flow (FRCF) using steady

nozzles, the question of its effect on the performance of a PDE with a nozzle will now be

addressed. This section begins by describing the chemical mechanisms used for

modeling FRCF, and then describes how the CEF solution is obtained. Next, the details

of the problem are described, and finally results are presented.

3.4.1 Chemical Mechanisms Computations performed in this work rely on integrating chemical reaction rates. The

chemical mechanisms which contain these rates are discussed in this section. All H2/O2

cases are handled by the H/O sub-mechanism of GRI-Mech 3.0 [61] with the H + O2 + M

reaction rate updated for high pressures using Ref. 62. This mechanism contains 8

species and 25 reactions. In comparison, the H/O/N sub-mechanism of GRI-Mech 3.0

contains 18 species and 67 reactions. While the number of reactions does not impact the

computational expense very much, the number of species does. Thus, although

molecular nitrogen will ultimately be incorporated into the system when air is used as an

oxidizer, nitrogen chemistry will be ignored in order to save on computational expense.

This reduces the H/O/N mechanism from 18 to 9 species, a very large savings. This

mechanism (without nitrogen chemistry) is reproduced in § D.1 and is labeled “H2”.

The reduced mechanism by Varatharajan and Williams [63] was used for all C2H4/O2

cases. Their work specifically targeted ethylene chemistry for use in detonation

computations, and thus directly applies to the current simulations. This reduced

mechanism has also been used previously on 1-dimensional and 2-dimensional

detonation problems by Owens et al. [40,52], Mattison et al. [36] and Tangirala et al [64].

Because comparisons will be made between FRCF and CEF, care has been taken to

ensure that the proper equilibrium state is recovered with this mechanism. This was done

by making all reactions of Varatharajan’s mechanism reversible, a strategy also employed

60

in Refs. 52 and 36. The resulting mechanism is reproduced in § D.2. Henceforth, this

Fully-Reversible Varatharajan mechanism will be designated “FRV”.

Table 4 summarizes the chemical mechanisms used in the computations.

Table 4: Summary of mechanisms used for computations. See Appendix D for

details. Mechanism

Label References Number of species

Number of reactions Notes

H2 [61,62] 9 25 N2 chemistry frozen

FRV [63] 21 33 All reactions made

fully reversible N2 chemistry frozen

3.4.2 Modeling Chemical Equilibrium Flow The finite-rate chemistry solution to the PDE blowdown is obtained by directly

implementing the numerical solver as described in Ref. 26. The chemical equilibrium

solution, however, requires a slightly different approach to achieve. Two methods of

achieving the equilibrium solution were considered. First, the reaction rates contained in

the chemical mechanisms were artificially increased. The intended result was for

equilibrium to be reached nearly instantaneously via the inflated reaction rates. This

method proved unreliable, however, since the high reaction rates led to a very stiff set of

equations, and often resulted in the code crashing.

Before discussing the second strategy, we first review the operator-splitting [65]

approach employed by the numerical solver. This approach is designed to efficiently

handle gasdynamics and chemistry separately in order to minimize computational cost.

Each time step is subdivided into a gasdynamic sub-step and a chemistry sub-step. First,

the chemistry is frozen and the gasdynamics are allowed to proceed for a time tgd

according to the equations of motion. Then the chemistry is allowed to proceed for a

time tchem along a constant-U,V path. This operator splitting approach has been used

successfully by others [26,66,67].

61

In light of this operator-splitting scheme, the second method for achieving chemical

equilibrium is to simply extend the time allotted to the chemical reaction sub-step, i.e.

tchem is artificially increased. If the chemistry is allowed to proceed sufficiently long

enough, chemical equilibrium is reached. This approach was tested by computing a flow

known to be in chemical non-equilibrium with various values for tchem and comparing the

results to the true CEF solution obtained using the method outlined in § 3.3. Temperature

was chosen as the metric for evaluating this procedure since temperature is more

sensitive to the effects of finite-rate chemistry than are pressure or density. Non-

equilibrium flow was achieved using a conical nozzle with an inlet diameter of 50 mm

and a diverging half-angle of 12°. The results are shown in Figure 24.

1 10 100

1000

1500

2000

2500

3000

3500

True equilibrium

True finite-rate chemistry

Reaction Time (tchem)

Sta

tic T

empe

ratu

re, T

[K]

Area Ratio, A/A*

101 sec 10-1 sec 10-3 sec 10-5 sec

Figure 24: Determining tchem using steady flow through a conical diverging nozzle (Di = 50 mm; θd = 12°). Temperature is compared for true equilibrium, true finite-rate chemistry, and finite-rate chemistry with extended reaction time, tchem. Inlet

state corresponds to state 4 of H2 + 0.5O2 (T1 = 500 K; P1 = 1 atm). Chemical mechanism: H2.

We first see that the equilibrium solution (black squares) diverges quickly from the

finite-rate solution (dotted line). Furthermore, we see that as tchem is increased, the

resulting temperature distribution approaches the equilibrium solution. For tchem > 0.1 sec,

62

the equilibrium solution is obtained. This method proved perfectly robust and was

therefore chosen for achieving CEF in the subsequent simulations. A reaction time of

tchem = 10 sec was used in all cases.

3.4.3 Problem Setup The reactant state was chosen as P1 = 1 atm, T1 = 500 K which is based on realistic

PDE reactant conditions. Additionally, a high T1 prevents water from condensing on the

walls, an effect found to be quite a large contributor to Isp loss.[71] The straight-tube

dimensions are Ds = 10 mm and Ls = 0.5 m. The flow is adiabatic and inviscid. The

constant-γ assumption used in Chapter 2 is abandoned here for the sake of accurately

predicting static pressure inside the nozzle. This means that the flow could not be

initiated using Eq. (3), as was done in Chapter 2. Instead, the reactants were ignited by

imposing a high temperature (3000 K) / high pressure (30 atm) region in the first 2 mm of

the tube. A detonation wave soon forms from this energetic region and travels down the

tube at CJ speed. A grid resolution of 0.1 mm was needed to produce the required CJ

state. After the detonation wave has left the straight-tube, such a fine grid is no longer

needed so the number of grid points is reduced in order to expedite calculations.

All nozzles are diverging and conical with a 12° half-angle. The area ratio is εd = 100.

Nozzles are attached to the end of the straight-tube, as in Figure 3. The straight-tube and

nozzle are separated by a diaphragm. To the left of the diaphragm are the reactants; to

the right is air at an ambient temperature, T∞ , and ambient pressure, P∞ . The value for

T∞ was fixed at 500 K in order to match the initial reactant temperature, T1. (This is

reasonable since, for multi-cycle operation, we expect the reactants in the straight-tube to

be of similar temperature to the buffer gas in the nozzle before ignition.) The effects of

chemistry in the nozzle are of interest, so two runs for each mixture are executed: one

with equilibrium flow in the nozzle, and one with finite-rate chemistry flow in the nozzle.

The straight-tube is made to experience equilibrium chemistry at all times.

Three different reactant mixtures are investigated: H2 + 0.5O2, C2H4 + 3O2 and

H2 + 0.5air. These two fuels (H2 and C2H4) were chosen for their relevance as research

and sensitizing fuels (i.e. they are relatively easy to detonate), and because their chemical

63

mechanisms are very well known. The two oxidizers (O2 and air) were chosen because

of their relevance to rocket and air-breathing applications, respectively. All mixtures are

stoichiometric. Integration of thrust starts at ignition and ends when the end-wall force

reaches zero.

3.5 Performance Results Figure 25 shows the time-varying end-wall and nozzle forces for H2 + 0.5O2. The

ambient pressure, P∞, was 155 Pa which was obtained by assuming the proper design

pressure for the chosen nozzle of εd = 100 (see Eq. (22)). Data qualitatively similar to

these have already been seen for ideal nozzles (Figure 6). Here, however, both CEF and

FRCF in the nozzle have been calculated. The difference in integrated area for these two

curves represents the impulse loss: 18.2 − 13.2 = 5.0 N·ms. This represents 27% of the

nozzle impulse, or 11% of the overall straight-tube + nozzle impulse. The overall system

loss (11%) is close to the expected result for steady nozzles (see Figure 23, ε = 100)

which predicts a loss of 8.5%. The PDE and the steady nozzle, however, cannot be

strictly compared, so some discrepancy is expected. The PDE results are summarized in

Table 5.

64

0 1 2 3 4

0

15

30

45

60

ImpulseFo

rce

[N]

Time [ms]

Straight-tube (CEF) Nozzle (CEF) Nozzle (FRCF)

Straight-tube (CEF): 29.4 N-msNozzle (CEF): 18.2 N-msNozzle (FRCF): 13.2 N-ms

Figure 25: Time-varying forces corresponding to equilibrium and finite-rate chemistry flow. Straight-tube experienced CEF in both cases. Ls = 0.5 m,

Ds = 10 mm. θd = 12º, εd = 100, P∞ = 155 Pa. Mechanism: H2.

Next, C2H4 + 3O2 and H2 + 0.5air were tested and their results are also reproduced in

Table 5. Each nozzle had an area ratio of 100. The ambient pressures suited for each

case were re-calculated using Eq. (22) and are listed in Table 5.

Table 5: Impulse and losses for three mixtures. T1 = 500 K, P1 = 1 atm. εd = 100.

Straight-tube Nozzle Total

P∞ [Pa] CEF

[N·ms] CEF

[N·ms]FRCF [N·ms]

CEF [N·ms]

FRCF *

[N·ms] Loss [%]

)a( )b( )c( =)d(

)b()a( + =)e(

)c()a( + )d()e()d( −

H2 + 0.5O2 155 29.4 18.2 13.2 47.6 42.6 11 C2H4 + 3O2 265 62.3 38.8 30.0 101 92.3 8.6 H2 + 0.5air 132 34.5 14.3 13.5 48.8 48.0 1.7

* FRCF in nozzle; CEF in straight-tube

65

In all cases there is a reduction in impulse arising from finite-rate chemistry. The

reason for this can be understood in terms of heat release. At typical CJ conditions there

exists large amounts of O, H, H2, OH and (for hydrocarbons) CO. All of these species

recombine when they cool to the more familiar combustion products H2O and CO2. This

recombination releases heat, raising the temperature and pressure, and consequently

thrust and specific impulse. If expansion takes place quickly enough so that these species

remain frozen, this additional heat is not released and performance suffers. Further

details on the theory can be found in Refs. 68 (§ 8) or 69 (§ 3).

These results clearly show that mixtures consisting of O2 are more likely to incur

losses from finite-rate chemistry than are mixtures which consist of air. This may seem

unexpected since the longer chemical timescales in the air mixtures would suggest finite-

rate chemistry losses would be more prevalent with those mixtures. This is reconciled by

considering a steady nozzle having the same geometry as the PDE nozzle. State 4

(obtained by equilibrium expansion from state 2) is taken as the inlet state. Using the

technique outlined in § 3.3, the equilibrium and frozen solutions are calculated for

H2 + 0.5O2 and H2 + 0.5air in a steady nozzle and the results for static temperature are

shown in Figure 26. Since the inlet temperature differs for the O2 and air mixtures, the

temperature is normalized by T * for the sake of comparison.

The figure reveals that the equilibrium and frozen solutions for the O2 mixture are very

disparate, while for the air mixture the solutions are similar. This is because the air

mixture is heavily diluted with inert N2, which contributes nothing to chemical

recombination. Therefore, despite the fact that chemistry proceeds at a much slower rate

for H2 + 0.5air than it does for H2 + 0.5O2, the effect on performance is neglibible. While

studying finite-rate chemistry effects in scramjets, Sangiovanni et al. [ 70 ,Table 3

(Cs = 0.988)] also observed only very small thrust losses (~1%) associated with chemical

recombination of the H2/air products.

66

1 10 1000.0

0.2

0.4

0.6

0.8

1.0 EquilibriumFrozen

Solid line:Dashed line:

H2 + 0.5air

Sta

tic T

empe

ratu

re R

atio

, T/T

*

Area Ratio, A/A*

H2 + 0.5O2

Figure 26: Static temperature in steady nozzle for H2 + 0.5O2 (red) and H2 + 0.5air (black). Equilibrium solution shown with solid lines; frozen solution shown with

dashed lines. Inlet state taken as state 4. T1 = 500 K, P1 = 1 atm.

3.6 Summary This chapter dealt with finite-rate chemistry effects, a loss mechanism commonly

studied for steady nozzles. As gases expand through the straight-tube and nozzle, there is

a potential for chemically freezing the flow, thereby inhibiting recombination and extra

heat release. This is especially true at low pressures and temperatures, where the

chemical time scales are long.

Finite-rate chemistry was explored by simulating the unsteady flowfield using a

numerical solver which incorporates chemical mechanisms. A PDE with a straight-tube

length of 0.5 m was simulated, using H2 and C2H4 as fuels, and O2 and air as oxidizers.

Chemical equilibrium solutions were also obtained for each set of conditions. This was

achieved by allowing chemical reactions to proceed to equilibrium during each

computational step.

67

Results showed that for the H2/O2 and C2H4/O2 mixtures, finite-rate chemistry can

have a large effect (~ 10%) on impulse, while for H2/air the effect is much smaller (~ 1%).

The low loss for the air mixture was attributed to the smaller amount of chemical energy

available during recombination due to nitrogen dilution.

69

Chapter 4: Heat Transfer and Friction

Effects on PDEs with Nozzles

4.1 Introduction Chapter 3 built on Chapter 2 by incorporating a particular type of loss mechanism

(finite-rate chemistry) into performance predictions. This chapter adds yet another layer

by exploring losses due to heat transfer and friction.

Recent work in straight-tube PDEs have revealed that heat transfer and friction can be

major causes of performance loss. Radulescu and Hanson [29] showed that convective

heat losses increase as the ratio of tube length to tube diameter, L/D, increases. The

reason is that the energy lost due to heat transfer scales with the tube wetted area and

cycle time (Eout ∝ LD ·tcycle ∝ L2D), while the chemical energy input scales with the tube

volume (Ein ∝ LD2). The ratio of energy lost to energy input therefore scales with L/D,

indicating that this ratio is extremely important in evaluating PDE losses. Their model

quantified heat loss using Reynolds’ analogy:

2

fCSt = (26)

where the Stanton number, St, is used to quantify heat flux at the wall, and the friction

coefficient, Cf, is used to quantify shear at the wall. Reynolds’ analogy is useful since it

enables a prediction of heat flux if information on Cf is available, or a prediction of wall

shear if St is available.

This convective model was limited in that it did not account for heat losses which

occur in the stagnant region between the end-wall and the Taylor wave. The convective

model was therefore extended by Owens and Hanson [ 71 ] who incorporated heat

conduction. In so doing, they showed that, depending on the particular values for L and

D, conduction heat losses can be as important as convection heat losses. This “hybrid”

70

model combined the aforementioned convective losses with conduction losses predicted

by Du et al [72]. The heat loss rates obtained using this 1-dimensional approach matched

quite well with 2-dimensional numerical simulations at low pressure (where boundary

layers could be resolved numerically), as well as with experimental data at high pressure

(where boundary layers could not be resolved numerically).

The aforementioned heat transfer studies were all performed on a straight-tube

geometry. This chapter uses Owens’ hybrid model [71] to investigate the effects that

heat and friction losses have on single-cycle PDE performance when nozzles are included.

4.2 Model Description A full description of the heat transfer/friction model can be found in Ref. 26 (§ 6).

The important characteristics are repeated as here.

4.2.1 Heat Conduction Loss Model Conduction losses occur in the absence of a flow field, and follow Fourier’s

conduction law which states that the local heat flux is proportional to the local

temperature gradient. This particular loss mechanism becomes important in the stagnant

region between the Taylor expansion wave and the end-wall. The heat loss was obtained

analytically by Du et al. [72] using self-similarity transformations. The solution was

applied to the stagnant region by Owens for predicting heat flux by conduction:

( )

t

UBq CJ

condξ

μρξ 112

1 ⋅−=′′& (27)

where condq ′′& is the heat flux by conduction in [W/m2], UCJ is the detonation wave speed, ρ1

and μ1 are the reactant density and dynamic viscosity, respectively, and t is the time after

ignition. The self-similarity variable ξ is defined as ξ ≡ 1 – x/xCJ where xCJ is the location

of the detonation wave, i.e. xCJ = tUCJ. The overbar denotes the end of the Taylor

71

expansion wave, so⎯ξ = 1 – c3/UCJ. The non-dimensional constant B1 in Eq. (27) is a

function of ξ and the particular reactants in question (due to the thermal properties of

their respective combustion products). The values of B1(ξ ) used in this study are listed

in Table 6. Notice that Eq. (27) is consistent with the expected behavior of transient heat

conduction, viz. condq ′′& ∝ t – 1/2.

Table 6: Constants used in heat conduction model. ‘air’ represents O2 + 3.76N2.

P1 = 1 atm; T1 = 500 K.

H2 + 0.5O2 C2H4 + 3O2 H2 + 0.5air ξ 0.475 0.486 0.470

B1(ξ ) 2.76 2.20 1.94

4.2.2 Heat Convection Loss Model Convection losses occur when there is non-zero axial flow. In the case of heat transfer,

the axial flow enhances losses by providing a faster mechanism for removing heat than

what is capable by pure conduction. For turbulent flow, this removal of heat is further

enhanced due to the added radial motion which promotes the transport of hot gases from

the core to the wall.

As with conduction, heat flux is forced by a temperature difference. However, the

heat flux is no longer simply proportional to the temperature gradient. A common way to

express convective heat loss in compressible flow is

( )wpconv TTcuStq −⋅⋅⋅−=′′ 0ρ& (28)

where St is the Stanton number, Tw is the wall temperature, T 0 is the stagnation

temperature, and all properties are evaluated in the freestream. A recovery factor of unity

is assumed in the above expression.

The problem with formulating convective heat loss using Eq. (28) is that we are

restricted to choosing a single value for cp. Since detonation waves of fuels burning in

72

pure O2 typically surpass 3000 K, choosing a single cp represents a gross assumption. In

order to circumvent this, a difference in enthalpy, rather than temperature, is commonly

used to evaluate heat flux:

( )wconv hhuStq −⋅⋅⋅−=′′ 0ρ&

where hw is the static enthalpy of the gas evaluated at Tw. This approach, which

circumvents evaluating cp altogether, has been used by widely used, for example by

Khvostov et al. [73] to study heat transfer from a diverging nozzle, and by Keener et

al. [74] to study heat transfer from a flat plate. See also the comprehensive review of

heat transfer form nozzles by Boldman and Graham [75] for further examples.

Next the assumption of unity recovery factor is relaxed, which means the enthalpy

which would exist at the wall for adiabatic flow, haw, is lower than h0. A commonly used

recovery factor for turbulent flow is Pr1/3, which leads to the following expression

023/1

21 huPrhhaw <+=

and the heat flux is obtained using

( )wawconv hhuStq −⋅⋅⋅−=′′ ρ& (29)

This “Δh” model has previously been used to successfully reproduce gas velocity [52],

temperature [36] and OH mole fraction [36] in PDEs. However, this model cannot

successfully capture impulse losses because it neglects heat conduction. The conduction

and convection heat loss models are combined in the next section.

73

4.2.3 Hybrid Heat Loss Model In order to account for both conduction and convection heat losses, Owens combined

the above two models in the following way. Since heat loss is predominantly convection-

driven in the Taylor expansion wave, Eq. (29) is used there. Since the gas velocity is

zero in the plateau region between the end-wall and the Taylor wave, Eq. (27) is used

there. For times soon after the detonation wave has left the tube and expansion waves

enter, the larger of the two (conduction vs. convection) is applied at each point in the flow.

When these expansion waves reach the end-wall, the velocity is everywhere non-zero and

the conduction model is turned off entirely.

For the current work on nozzles, an additional restriction will be added that only

convection losses occur in the nozzle due to their high gas velocities.

4.2.4 Wall Temperature Both conduction and convection heat models require the wall temperature, Tw, to be

specified. The simplest approach is to assume a constant Tw. In the case of a room

temperature single-cycle device (i.e. a detonation tube), the thermal mass of the tube’s

wall is large enough to absorb the heat lost by the hot gases so as to stay constant during

such a short experiment (tcycle ~ 10 ms). In the case of a real multi-cycle engine, again the

high temperatures of the detonation products exist for only a small fraction of the overall

cycle, and so the engine wall temperature remains constant (albeit at a higher value than

room temperature). As was done in Chapter 3, simulations in this chapter will assume

Tw = 500 K.

4.2.5 Friction Model When a non-zero bulk flow is present, friction losses are also produced alongside heat

losses. Friction results from shear stress at the wall, τw, which itself arises due to a

velocity gradient across the tube. These losses are quantified using

74

uuC f

w ⋅⋅⋅= ρτ2

(30)

Friction losses directly impact impulse by generating negative forces along the side

walls of the tube and nozzle. Heat losses, on the other hand, have an indirect effect on

impulse since they act to lower the local pressure, which sends expansion waves to the

various thrust surfaces (e.g. end-wall, nozzle walls), thereby lowering the forces on those

surfaces.

Owens and Hanson [71] found that the effects of both heat and friction on Isp are of

similar magnitude. For example, for their test case of H2 + 0.5O2 (STP) in a tube with

L/D = 10, the ideal Isp was calculated to be 193 s. Heat losses lowered this value by 5 s,

while friction losses lowered it by an additional 4 s. Therefore, both heat and friction

losses are considered in the current work.

4.2.6 St and Cf Coefficients Equations (28) and (30) are useful only with knowledge of St and Cf. The friction

coefficient for flow behind a detonation wave was measured by Edwards et al. [76] who

obtained Cf = 0.0062. This value has since been used successfully by other

researchers [29,52,36] who studied heat and friction losses from PDEs and is therefore

adopted here.

The value for St is obtained by appealing to Reynolds’ analogy, which takes advantage

of the symmetry observed by Reynolds between the energy and momentum equations.

This relation was already presented as Eq. (26), but is repeated here with a Prandtl

number adjustment commonly used in turbulent flows:

3/2

12 Pr

CSt f=

where the Prandtl number is evaluated at the conditions of the core. The friction

coefficient and the Stanton number are expected to be, at most, functions of Reynolds’

75

number for laminar flow, and nearly constants for turbulent flow. For a C2H4 + 3O2

detonation wave (reactants at STP) travelling through a tube of diameter 50 mm, the

Reynolds number immediately behind the wave is higher than 106. Additionally, the true

3-dimensional nature of the detonation flowfield is expected to further enhance

turbulence. Boldman and Graham [75] presented a comprehensive review of heat

transfer from nozzles and found that the Stanton number in turbulent flow scales as

St ∝ Re–1/5 – a rather weak dependence. Therefore, a constant St and Cf are assumed here.

4.2.7 Additional Modeling Information The numerical model used for the gasdynamics and chemistry is the same as was used

in Chapter 2 and Chapter 3. The mixture is ignited in the same way as in § 3.4.3, and the

grid size is 0.1 mm while the detonation wave traverses the straight-tube, after which the

grid size is increased. The chemistry is forced to follow equilibrium trajectories using the

method described in § 3.4.2. A diaphragm is placed between the straight-tube and the

nozzle. The nozzle is initially filled with air. Unless otherwise stated, the following is

adopted throughout this chapter: T1 = T∞ = Tw = 500 K, P1 = 1 atm.

4.3 Thrust and Impulse Breakdown In Chapter 2 and Chapter 3 the thrust was broken down into two time-varying

components. This breakdown is repeated in Figure 27: a force from pressure acting on

the straight-tube end-wall (dark blue), presssF , and a force from pressure acting on the

nozzle walls (red), pressnF . Once again, subscript ‘s’ represents ‘straight-tube’ and

subscript ‘n’ represents nozzle. Superscript ‘press’ has been introduced in order to

emphasize that these forces are due to pressure.

76

Figure 27: Breakdown of forces in PDE with nozzle. Pressure forces and shear forces are included. Dark blue: straight-tube pressure force; red: nozzle pressure

force; green: straight-tube shear force; light blue: nozzle shear force.

Figure 27 also shows two additional forces: the shear force acting along the straight-

tube wall (green), shearsF , and the shear force acting along the nozzle wall (light blue),

shearnF . Here the superscript ‘shear’ has been introduced to emphasize that these are

shear forces. All four forces must be taken into account to evaluate the net thrust:

shearn

pressn

shears

presss FFFF +++≡T

The net thrust is the quantity of interest and which is captured by a thrust stand or

ballistic pendulum.

In the same way that the two forces encountered in earlier chapters had associated

impulses, likewise the two new shear forces will also have their own associated impulses.

Thus, the overall impulse consists of four components:

shearn

pressn

shears

presss IIIII +++≡

4.4 Straight-Tube PDEs Before investigating the effects that nozzles have on heat and friction losses, it is

worth reviewing the dimensions of straight-tube PDEs encountered in practice. Figure 28

shows the range of dimensions for straight-tube engines studied by other researchers.

Straight-tube Nozzle

77

The L/D ratio is indicated by straight lines on this log-log plot. While this list is not

necessarily exhaustive, we see that there is a wide range of dimensions, resulting in L/D

ratios which range from 6 to 53. In terms of performance, consider Owens’ simulated

case [71] as a reference point, indicated by the blue square. This straight-tube PDE had

an L/D ratio of 10 and an associated Isp loss of 5%. Since this reference case has a

relatively low L/D ratio compared to other engines, we expect losses to be even more

pronounced for the majority of studies depicted in Figure 28. Using the figure as a guide,

the range of the Ls/Ds ratio (for the straight-tube) studied in this chapter was chosen as 10

to 50.

0.1 1 100.01

0.1

1

Isp loss

= 5%

heat & friction more important

L / D = 1 2 5 10

20

100

Tube

Dia

met

er, D

[m]

Tube Length, L [m]

50

heat & friction less important

Figure 28: Summary of straight-tube PDE dimensions. Imbedded values indicate L / D. Solid symbols: single-cycle engines; open symbols: multi-cycle engines.

■ [77]; ■ [71]; ■ [36]; ▼[78]; ● [79]; □ [6]; ○ [33, Fig. 7]; ○ [51, initiator]; ○ [80].

4.5 Model Validation The aforementioned reference point from Ref. 71 was also used to validate the current

model. The specific conditions were reproduced (H2 + 0.5O2; T1 = 298K; P1 = 1 atm;

L = 0.2 m; D = 20 cm) and the end-wall and friction contributions to net Isp were

calculated. With losses, the end-wall produced an Isp of 188 s, and the shear stress

78

reduced this by 4 s, resulting in a net Isp of 184 s. All three values matched the results in

Ref. 71 perfectly.

4.6 Straight-Tube with Converging Nozzles First, an engine geometry consisting of a straight-tube with converging nozzle (see

Figure 16) was studied. This problem is of importance because the restricted outlet area

caused by the converging nozzle leads to prolonged blowdown times, which in turn leads

to increased losses beyond the simple straight-tube case. For example, Cooper and

Shepherd [41] measured specific impulse using a ballistic pendulum and observed a 27%

drop in Isp when a converging section was added. They attributed some of this to

enhanced heat/friction losses resulting from the prolonged blowdown. Since converging

nozzles tend to be short, we should not expect there to be much additional losses from the

nozzle walls themselves. Rather, the converging nozzle is expected to increase the losses

at the side walls of the straight-tube by increasing the residence time of hot moving gases

in the straight-tube.

A diverging section is ignored for now in order to first ascertain the effect that the

converging section has on heat/friction losses. Since the flow is choked, adding a

diverging section to the present system would not alter the results presented in this

section.

The nozzle has a converging area ratio, εc, of 3.05 which corresponds to Mrefresh = 0.2.

This area ratio is also very close to the value found in well known rocket engines, such as

the RL-10 and the SSME (see Ref. 81, Table 11.4). The ratio of the nozzle length to

straight-tube diameter, Ln,c/Ds, has a value of 2 for all configurations so that there are

enough grid points in the nozzle. First, a mixture of H2 + 0.5O2 will be used to

investigate the effect of geometry, i.e. Ls/Ds, and Ls. Then two other mixtures

(C2H4 + 3O2 and H2 + 0.5air) will be used to investigate the effect of oxidizer and fuel.

In what follows, both the single-cycle specific impulse in the absence of losses, Ispideal,

and in the presence of losses, Isp, will be calculated. Since Ispideal is in general different

for each geometry and mixture configuration, a more lucid presentation of the results is

achieved by plotting the ratio Isp/Ispideal, which is called “impulse efficiency” in this work.

79

Thus, an impulse efficiency of 1 represents no loss, and an impulse efficiency less than 1

indicates losses are present.

4.6.1 Varying Geometry We begin by varying Ls/Ds and Ls. For the case of a straight-tube with no heat

conduction, the solution is self-similar in Ls/Ds [29]. However, by including heat

conduction the problem ceases to be self-similar, so we can also expect an additional

dependence on Ls. Furthermore, by adding the converging nozzle, the effects of Ls/Ds

and Ls are completely unknown and need to be investigated.

Figure 29 shows the impulse efficiency over a range of Ls/Ds for εc = 1 (straight-tube)

and εc = 3.05 (Mrefresh = 0.2). The length of the straight-tube, Ls, is 0.5 m in panel (a) and

1 m in panel (b). Calculation results are represented by solid lines. These are

extrapolated down to Ls/Ds = 0 with dotted lines.

For the straight-tube (black lines), we see that the losses increase linearly as Ls/Ds

increases, as was also observed by Radulescu and Hanson [29]. By adding the nozzle

(red lines), more losses are experienced at all values of Ls/Ds, and the line shifts down.

This added loss is approximately 5% in all cases, which means that for Ls/Ds ~ 10 the

losses are doubled when the nozzle is added.

Notice that the impulse efficiency is much more sensitive to Ls/Ds than it is to Ls. This

is true for both the straight-tube and the straight-tube with converging nozzle

configurations. Thus, while the problem is not self-similar in general, the aspect ratio

Ls/Ds should be considered the main determinant of performance.

80

0 10 20 30 40 50 600.7

0.8

0.9

1.0

penalty incurred by adding nozzle

Impu

lse

Effi

cien

cy, I

sp /

Ispid

eal

Straight-Tube Aspect Ratio, Ls / Ds

Straight tube Straight tube w/converging nozzle (ε

c = 3.05)

no losses (a)

0 10 20 30 40 50 600.7

0.8

0.9

1.0(b)

penalty incurred by adding nozzle

Impu

lse

Effi

cien

cy, I

sp /

Ispid

eal

Straight-Tube Aspect Ratio, Ls / Ds

Straight tube Straight tube w/converging nozzle (ε

c = 3.05)

no losses

Figure 29: Impulse efficiency as a function of Ls /Ds and Ls, (a) Ls = 0.5 m, (b) Ls = 1 m. H2 + 0.5O2; P∞ = 1 atm. Tw = 500 K.

4.6.2 Varying Mixture Figure 30 shows the impulse efficiency for the mixture covered in the last section

(H2 + 0.5O2), as well as two other mixtures: C2H4 + 3O2 and H2 + 0.5air. We see that the

losses with a mixture of C2H4/O2 is somewhat less than – although still very close to – the

mixture of H2/O2. On the other hand, the mixture of H2/air leads to noticeably more

losses (e.g. when the nozzle is added the impulse efficiency drops by an additional 14%,

as compared to 5% for H2/O2). We explain this by focusing on the straight-tube case

(orange bars), and the conclusions can equally be used to explain the observed losses for

the tube with converging nozzle (blue bars). Figure 31 shows the time-varying pressure

force, pressnF , for the three mixtures both with and without losses. Panel (a) shows force

in Newtons, and panel (b) shows force normalized by the plateau force.

81

0.0

0.2

0.4

0.6

0.8

1.0

H2 + 0.5airC2H4 + 3O2

Impu

lse

Effi

cien

cy, I

sp /

Ispid

eal

Straight-tube Straight-tube w/converging nozzle (εc = 3.05)

no losses

H2 + 0.5O2

Figure 30: Impulse efficiency for three different mixtures. Ls/Ds = 50; Ls = 0.5 m. Tw = 500 K.

0.0 0.5 1.0 1.50

15

30

45

60

Solid: without lossesDash: with losses

(a)

Stra

ight

-Tub

e P

ress

ure

Forc

e, F

pres

sn

[N]

Time [ms]

H2 + 0.5O2 C2H4 + 3O2 H2 + 0.5air

blowdown begins(C2H4/O2)

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

For

ce

Time [ms]

H2 + 0.5O2 C2H4 + 3O2 H2 + 0.5air

(b)

Solid: without lossesDash: with losses

Figure 31: End-wall force for three mixtures with and without losses. (a) Force, (b) normalized force. Straight-tube geometry (Ls = 0.5 m; Ds = 10 mm). Shear

forces not shown. T1 = Tw = 500 K, P1 = P∞ = 1 atm.

Panel (a) shows the plateau force, followed by the blowdown, both with and without

losses. Although it is difficult to predict the plateau force decay rate (in Newtons per

second) between ignition and blowdown due to the multiple effects of convective heat

loss, conductive heat loss, and friction, we see that in this case the decay rates are quite

similar. However, the time histories differ for the various mixtures in two important

82

respects: 1) the plateau force (without losses) is very different for each mixture, and

2) the time at which blowdown begins is also different. The plateau force comes directly

from the plateau pressure, P3. For a particular decay rate of end-wall pressure (in Pascal

per second) induced by losses, having a lower P3 results in effectively magnifying this

decay rate when the pressure (or force) is normalized, as in panel (b). This has the effect

of enhancing the relative reduction in integrated area. The second important difference

between mixtures is the time at which blowdown begins. (This time is pointed out in

panel (a) for the C2H4/O2 mixture.) The onset of blowdown is dictated by the sound

speed of the gases: the higher the sound speed, the sooner blowdown begins. Of the three

mixtures considered, H2/air has the lowest sound speed (due to the low temperatures),

and H2/O2 has the highest (due to high temperatures and low molecular weight). C2H4/O2

has a moderate sound speed. This trend is reflected in Figure 31, where we see H2/O2

beginning blowdown first, followed by C2H4/O2, and finally H2/air. The effect this has

on losses is to prolong the decay, thereby exacerbating the amount of overall pressure

drop for mixtures with lower sound speeds. This leads to a further reduction in impulse.

4.7 Straight-Tube with Diverging Nozzles We now abandon the converging nozzle and consider a straight-tube with a purely

diverging nozzle (see Figure 3). The mechanism by which heat and friction impact

engine performance is very different for PDEs with diverging nozzles than it was for

PDEs with converging nozzles. In the case of converging nozzles, it was the influence of

the converging section on the straight-tube flowfield which led to enhanced losses. For

the diverging nozzle, however, the straight-tube flowfield is essentially unchanged.

Rather, diverging nozzles can potentially experience noticeable losses themselves owing

to their large surface area. Furthermore, since τw ∝ u2 the shear force on the diverging

nozzle walls is enhanced by the faster moving fluid. (It is also true that q& ′′ ∝ u, so heat

transfer will be enhanced as well, albeit not as much as friction.)

83

4.7.1 Problem Setup Having already reviewed various mixtures in § 4.6.2, only the mixture H2 + 0.5O2 will

be considered here. The dimensions of the straight-tube section are Ls/Ds = 50 and

Ls = 0.5 m. The nozzle area ratio was εd = 100. Using Eq. (22) the corresponding

ambient pressure for this mixture was chosen as P∞ = 155 Pa.

4.7.2 Effect of Nozzle Losses on Impulse

Figure 32 shows the time-varying forces in the diverging nozzle: pressnF is the force

due to pressure, and shearnF is the force due to shear. The divergence half-angle is 12º, a

common value for conical nozzles. Three different scenarios are shown:

(i) ideal tube; ideal nozzle (red),

(ii) tube with losses; ideal nozzle (green), and

(iii) tube with losses; nozzle with losses (blue).

84

0 1 20.1

1

10

100

shear force

(Fshearn )

Noz

zle

Forc

es [N

]

Time [ms]

(i) ideal tube; ideal nozzle (ii) tube w/losses; ideal nozzle

(iii) tube w/losses; nozzle w/losses

pressure force

(F press n )

Figure 32: Forces in diverging nozzle for (i) ideal tube & ideal nozzle (red), (ii) tube with losses & ideal nozzle (green), and (iii) tube with losses & nozzle with

losses (blue). Ls /Ds = 50, Ls = 0.5 m; εd = 100; θd = 12º; H2 + 0.5O2; T1 = Tw = T∞ = 500 K, P1 = 1 atm; P∞ = 155 Pa.

The purpose of multiple scenarios is to distinguish the losses arising from the straight-

tube from the losses arising from the nozzle. The first curve (i) shows qualitatively the

same results as those achieved using the constant-γ approach of Chapter 2 (see Figure 6).

For (i), the shear force is zero.

For scenario (ii), the tube has losses, and the nozzle remains idealized. This results in

a drop in stagnation pressure throughout the tube, and since the tube feeds the nozzle, the

nozzle’s stagnation pressure drops as well. This ultimately results in a drop in the nozzle

pressure force, pressnF . Again, the nozzle shear force is zero.

For scenario (iii), the nozzle is allowed to have losses. This has two effects. First, the

nozzle now has a non-zero shear force resulting from friction (dashed blue line). The

second effect is to alter the force due to pressure (solid blue line). Whereas we may have

expected a large drop in this nozzle pressure force owing to losses, we notice that in fact

the effect is extremely small (i.e. the green and blue solid lines are nearly identical). This

can be explained by appealing to the theories of Fanno and Rayleigh flow.

85

Recall that Fanno flow is adiabatic duct flow with friction, and Rayleigh flow is

frictionless duct flow with heat transfer, both applied to a constant area duct [82]. Figure

33 shows the static pressure behavior for Fanno flow and Rayleigh flow. For Fanno flow,

the effect of friction is to drive the Mach number towards unity, both for supersonic and

subsonic flow. By contrast, for Rayleigh flow the effect of heat loss is to drive the Mach

number away from unity, again for both supersonic and subsonic flow. This is indicated

by the arrows in the figure. (The case of Rayleigh flow with heat addition is not

considered.)

0.1 1 100.01

0.1

1

10

sonic point

Sta

tic P

ress

ure

Rat

io, P

/P*

Mach Number, M

Fanno Rayleigh

γ = 1.4

Figure 33: Static pressure ratio for Fanno flow and Rayleigh flow. Arrows indicate trajectories due to friction (Fanno) and heat loss (Rayleigh). The case of heat

addition is not considered.

Thus, there are competing effects between Fanno and Rayleigh flows. As a result, the

static pressure, P, may go up or it may go down when both heat loss and friction are

present. The practical implication is that pressnF may also go up or go down, since this

force is simply a spatial integration of P – P∞ . The net nozzle thrust, shearn

pressn FF + ,

however must always drop in the presence of heat and friction losses as dictated by the

2nd law of thermodynamics. This is achieved by the nozzle shear force, shearnF , created by

the friction.

86

The relative effect of friction and heat loss in generalized nozzle flow (area change +

Fanno flow + Rayleigh flow) is therefore important. Figure 34 shows the static pressure

in a diverging conical nozzle for ideal flow (Cf = 0, St = 0), flow with friction loss (Cf > 0,

St = 0), and flow with friction and heat losses (Cf > 0, St > 0). The flow is steady. The

divergence half-angle is quite small (θd = 2º) in order to increase the nozzle’s surface area

and enhance the losses so they become noticeable.

1 10 100

0.01

0.1

1

Sta

tic P

ress

ure,

P [b

ar]

Area Ratio, A/A*

No Losses w/Friction Losses w/Friction and Heat Losses

pressnF

6.2 N8.7 N15.8 N

-10.6 N-10.8 N0 N

16.8 N19.5 N15.8 N

w/Friction & Heat Losses

w/Friction Losses

No Losses

6.2 N8.7 N15.8 N

-10.6 N-10.8 N0 N

16.8 N19.5 N15.8 N

w/Friction & Heat Losses

w/Friction Losses

No Losses

shearnF

shearn

pressn FF +

Figure 34: Steady nozzle with no losses, with friction losses, and with friction & heat losses. Inlet is at state 4 for H2 + 0.5O2, T1 = 500 K, P1 = 1 atm. Di = 10 mm; θd = 2º.

The black curve represents ideal flow. Next, by including friction losses, the pressure

rises (blue curve), consistent with Fanno flow theory (Figure 33). Then, by also

including heat loss, the pressure drops (red curve), consistent with Fanno flow theory

(Figure 33). These competing effects between heat loss and friction loss is the reason

that the nozzle pressure force is nearly unchanged in Figure 32. As expected, the

tabulated forces in Figure 34 show that the net nozzle thrust gets progressively lower as

losses are added.

87

Now we return to the unsteady flow relevant to PDEs and, in so doing, we turn our

attention to impulse, rather than force. Table 7 tabulates the contributions to straight-tube

& nozzle impulse for scenarios (i), (ii) and (iii) for a PDE with two different nozzles:

θd = 5º and θd = 12º.

By comparing scenarios (i) and (ii), we see that losses from the straight-tube reduce

the total impulse of the overall system (straight-tube + nozzle) performance. For the 5º

nozzle, this is a drop from 48.3 to 38.9 N·ms, which implies a 19% loss in Isp. Next, by

comparing scenarios (ii) and (iii), we can quantify the effect that nozzle losses have on

the overall system (straight-tube + nozzle) performance. Again for the 5º nozzle, the

nozzle losses reduce the system impulse from 38.9 N·ms to 34.7 N·ms. This implies an

overall loss of 28%, with 19% coming from the straight-tube and 9% coming from the

nozzle. This nozzle loss of 9% is large enough to motivate special care during nozzle

design. For example, this loss can be mitigated by choosing a larger value for θd: using

the data in Table 7 for θd = 12º, we find that the nozzle loss in this case is only 3%, a

significant improvement over the previous 9% loss.

Table 7: Summary of impulse breakdown for PDE with diverging nozzle.

Ls /Ds = 50, Ls = 0.5 m, εd = 100. P∞ = 155 Pa. All values in N·ms.

(i) Ideal (ii) Losses in Tube (iii) Losses in Tube & Nozzle

θd = 5º press

sI 29.8 27.7 27.7 shearsI 0 -2.4 -2.4 press

nI 18.5 13.5 14.1 shearnI 0 0 -4.77

Total 48.3 38.9 34.7

θd = 12º (Figure 32) press

sI 29.69 27.7 27.7 shearsI 0 -2.4 -2.4 press

nI 18.1 13.2 13.5 shearnI 0 0 -2.1

Total 47.8 38.5 36.6

88

Comparison of these results with other findings for PDEs remains difficult due to the

lack of literature, but it is worth considering the approximate penalty incurred by these

losses by appealing to steady nozzle studies. Back et al. [83] measured thrust in a nozzle

with a 15º divergence half-angle. They then compared this to the thrust predicted by

ideal theory. The losses in thrust were between 3 and 4%, in good agreement with the

current results for the 12º nozzle.

4.8 Summary This chapter focused on losses arising from heat transfer and friction. Results were

obtained using the same heat/friction model which was proven on straight-tube

geometries in the past. Two different mechanisms for heat/friction loss were identified.

The first arises when a converging nozzle is attached which results in restricted outflow,

thereby leading to enhanced losses from the straight-tube. Losses from the nozzle itself

contribute very little to the overall losses. It was found that despite the tube + nozzle

configuration being non-self-similar, the losses were still primarily dependent on a single

variable, viz. Ls/Ds. The nozzle resulted in an additional ~ 5% decrease in Isp for the

H2/O2 and C2H4/O2 mixures, and 14% for the H2/air mixture. The larger losses for the air

mixture were attributed to a prolonged blowdown time and a lower plateau pressure, P3.

Diverging nozzles can also lead to losses. It was shown that for a nozzle with a small

divergence angle (and therefore a large surface area), the losses can be approximately

10%. These losses are reduced as the divergence angle is increased.

89

Chapter 5: Laser-Based Mid-IR H2O

Sensing

5.1 Introduction The previous chapters have covered modeling of PDEs, with a focus on those fitted

with nozzles. Modeling complements – and is complemented by – laboratory

measurements. The following two chapters detail measurement strategies employed on

PDEs as well as the insight obtained through such measurements. Since a strong demand

currently exists for sensors designed for real air-breathing PDEs, the sensors described

herein address this particular need.

Laser-based water sensing by direct absorption (DA) has had a long history because of

the ubiquity of water transitions throughout the electromagnetic spectrum, and because of

the importance of water in chemistry, atmospheric sciences, and engineering. Water

sensing in engines has recently focused on wavelengths near 1.4 μm because of the

wavelength overlap with most telecom semiconductor lasers, fibers and detectors. This

class of hardware is attractive because of its low-cost and robustness and has been used in

the context of internal-combustion engines [84,85], scramjets [86], and combustion

control [87], to name a few.

PDEs have also received some attention in the field of water sensing. Water is a major

combustion product, and it also appears in the reactant stream when a vitiator is used to

heat the engine’s incoming flow. Fiber-coupled lasers are employed to overcome the

challenge of contending with the motion of a vibrating/translating engine. Sanders et

al. [88] applied fiber-coupled telecom lasers between 1.3 and 1.8 μm on the PDE at the

Naval Postgraduate School (NPS) in Monterey, CA to demonstrate their potential for

water sensing in this type of engine. Mattison [89, § 4] later extended this work by using

similar lasers to identify specific combustion modes in the PDE and how they lead to

successful detonations or misfires.

90

While telecom lasers provide easy to use tools for water sensing, the wavelengths

inherent to these lasers provide only minimal absorption levels. This chapter discusses a

DA water sensor for PDEs which is based on mid-IR wavelengths, and thereby offers

significantly increased absorption and signal-to-noise ratio (SNR) than its near-IR

counterpart. The sensor employs a scanned-wavelength technique in order to circumvent

lineshape dependence on pressure and temperature. A 2-wavelength approach is

employed in order to extract both temperature and mole fraction. The sensor is applied to

the NPS PDE and is used to 1) demonstrate improved signal-to-noise ratio (SNR), and

2) to establish that the engine’s vitiator has a detrimental impact on ignitor performance.

5.2 Infrared Water Spectrum Water has three vibrational modes: symmetric stretch (ν1 = 3651 cm-1); symmetric

bend (ν2 = 1595 cm-1); asymmetric stretch (ν3 = 3756 cm-1). In addition to these

fundamental bands, many overtone and combination bands are also possible. Figure 35

shows the infrared spectrum of water with each fundamental band labeled ν1, ν2 and ν3.

Notice that the ν1 and ν3 bands overlap. Also shown are the weaker overtone band (2ν2)

and combination bands (ν1 + ν2, ν3 + ν2, ν1 + ν3). The temperature, pressure, mole

fraction and path length represent nominal operating conditions of the NPS PDE.

91

1000 2000 3000 4000 5000 6000 7000 80000

20

40

60

80

100

All DFBs

10 3.5 2.5 2.0

ν1+ν

3ν

2+ν

3ν

1+ν

2

ν2

2ν2

ν3ν1

Abso

rptio

n [%

]

Frequency, ν [cm-1]

Fiber-coupled DFBs

1.5

Wavelength, λ [μm]

Figure 35: Infrared absorption spectrum of water. T = 520 K; P = 1 atm; XH2O = 4%; L = 7.3 cm. (Source: HITRAN 2004 [90].)

The wavelength range of fiber-coupled distributed feedback (DFB) lasers is indicated

by the blue bar. These lasers can easily access the ν1 + ν3 combination band of water, as

was done in the previously mentioned near-IR water sensing work by Sanders et al. [88].

We see that the strengths of the combination bands are significantly lower than those of

the fundamental bands, so sensors utilizing lasers near 2.7 μm would provide much better

sensitivity than their near-IR counterparts. By utilizing exotic materials, laser developers

have been able to construct tunable DFB lasers which reach these coveted wavelengths

(see, e.g., Refs. 91 and 92). Despite lacking the advantage of having built-in fiber-

coupling, these lasers were chosen for the current work because of the enhanced

absorption they offer.

92

5.3 Sensor Theory

5.3.1 Beer’s Law Direct absorption spectroscopy of gases is based on the principle that certain

molecules absorb light at certain wavelengths. The amount of light absorbed is governed

by Beer’s law, which for IR water sensing is typically cast in the following form:

( ) ( )LPXTXPTSII

iit ),,()(expexp0

ννν

φα −=−=⎟⎟⎠

⎞ (31)

where I0 is the incident intensity of (monochromatic) light, It is the transmitted intensity,

αν is the absorbance, S is the temperature-dependent linestrength, φν is the lineshape, P is

the static pressure, XH2O is the mole fraction of water, and L is the path length. The

lineshape is defined such that its integrated area is unity, i.e.

1=∫∞

∞−νφν d (32)

By combining Eqs. (31) and (32) we find that the integrated absorbance, H, is not

influenced by the lineshape:

LPXTSdLPXTSdH ii )()( ==≡ ∫∫∞

∞−

∞

∞−νφνα νν (33)

A sample absorption feature is shown in Figure 36. Notice that the lineshape, φν,

results in absorption taking place at frequencies other than the center frequency, ν0. The

mechanisms which cause this so-called “line broadening” are discussed next.

93

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

full-width @half max

Abs

orba

nce,

α

ν - ν0 [cm-1]

integrated absorbnce, H

Figure 36: Sample absorption feature.

5.3.2 Lineshape Although the lineshape, φν, is ultimately unimportant for the scanned wavelength

strategy used here, there is value in quantifiably understanding the lineshape in order to

know how broad a transition will be under the conditions of interest, and to increase

confidence in measurements by comparing the measured lineshape parameters with those

contained in HITRAN.

The lineshape is dictated by the type and extent of the broadening mechanism(s)

present. The three most common types of broadening encountered in gas sensing are

Doppler, collisional and Voigt.

Doppler broadening arises due to the distribution of molecular velocities present at a

given temperature. As some molecules move towards the source of photons, and some

molecules move away, the photon frequency (νp) at which the molecules absorb is shifted

by the Doppler effect. This means that absorption will occur not only at νp = ν0, but also

at νp < ν0 and νp > ν0, leading to a broadening of the lineshape. Since the velocity

distribution of molecules is only a function of temperature, Doppler broadening is also

only a function of temperature. This type of broadening becomes more important as the

94

temperature increases. Doppler broadening is captured very well by theory and need not

be quantified empirically.

Collisional broadening arises from the Heisenberg uncertainty principle. This

principle states that particles which have a finite lifetime in a particular energy level have

an inherent uncertainty in their precise energy. Furthermore, as the lifetime decreases,

the uncertainty in energy increases. This uncertainty in energy leads to a wider range of

νp over which absorption can take place. Higher pressures leads to higher collisions rates,

which lead to shorter lifetimes, which lead to more uncertainty in energy, and finally a

wider range of νp for absorption to take place. Thus, collisional broadening is dominated

by pressure.

Collisional broadening theory is usually supported by empirical measurements of the

collisional full-width at half maximum, ΔνC. This quantity is proportional to pressure

(ΔνC = 2γP), and the so-called broadening coefficient, 2γ, is typically reported. In

addition, different collision partners yield different values for 2γ, where the partner is

indicated by a subscript. The effects of all partners are assumed to scale linearly with

partial pressure, resulting in the following empirical law

∑=Δi

iiC XP γν 2

Typically, self-broadening (2γself) and air-broadening (2γair) are the most important.

These values are measured and reported in this chapter, and are compared to values

contained in HITRAN 2004. The collisional broadening coefficients are functions of

temperature and are expected to follow power-law relationships:

( )( )

selfnr

rself

self

TT

TT

⎟⎠⎞

⎜⎝⎛=

γγ

22

(34)

95

( )( )

airnr

rair

air

TT

TT

⎟⎠⎞

⎜⎝⎛=

γγ

22 (35)

where subscript r denotes a references state†††.

Finally, the Voigt lineshape is an amalgamation of the two previously mentioned

broadening mechanisms, Doppler and collisional. The Voigt lineshape is necessary when

effects of temperature (Doppler) and pressure (collisional) are both important. Typical

conditions in PDEs require use of the Voigt lineshape, so measurements of lineshape will

be fit with a Voigt profile in order to extract the pertinent spectroscopic parameters.

5.3.3 2-Wavelength Temperature Sensing Temperature sensing takes advantage of Eq. (33) which shows a very simple

relationship between integrated absorbance, temperature, and partial pressure. If two

different water absorption features are scanned to obtain two integrated absorbances (H1

and H2), then their ratio (R) depends only on temperature:

)(2

1

2

1

2

1 TfSS

LPXSLPXS

HHR

i

i ===≡ (36)

The temperature sensor discussed herein consists of two measurements of integrated

absorbance for two separate water features. The integrated absorbance ratio is then used

to directly determine temperature. This dependence can be found explicitly by first

expressing the linestrength in terms of temperature:

††† Tr = 296 K in this work and in HITRAN 2004 [90]

96

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′−

⎟⎠⎞

⎜⎝⎛ ′′

−=

rr

rr

r

kThckT

hc

TE

khc

TE

khc

TQTQ

TT

TSTS

0

0

exp1

exp1

exp

exp

)()(

)()(

ν

ν

(37)

where r denotes the reference state ‡‡‡ , Q is the partition function (evaluated using

Ref. 93), E″ is the lower state energy in cm-1, ν0 is the transition’s center frequency, and

hc/k has a value of 1.4388 K/cm-1. Each of the two transitions has its own S(Tr), E″ and

ν0, all of which are compiled in the HITRAN 2004 database [90]. The expression for

temperature as a function of R is then found to be

( )

( ) ( )rr

r

TEE

khc

TSTSR

EEkhc

T21

1

2

21

)()(lnln

′′−′′+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

′′−′′= (38)

With this strategy in hand, we can now ask how sensitive R is to T. The relative

sensitivity, ζ, is defined as:

)(// TfTdTRdR

=≡ζ (39)

and is, like R, only a function of temperature and the choice of transitions. For example,

if a particular line pair and temperature yield ζ = 2, this means that a 1% increase in

temperature will result in a 2% increase in R. However, a less sensitive line pair with,

say, ζ = 0.5 implies a 1% increase in temperature will result in only a 0.5% increase in R.

Thus, we desire a line pair which has a large value of ζ at the temperature of interest.

Differentiating Eq. (38) according to Eq. (39) yields the sensitivity, ζ:

‡‡‡ Tr = 296 K in this work and in HITRAN 2004 [90]

97

( )T

EEkhc 21 ′′−′′

=ζ

where subscripts 1 and 2 denote the first and second transition, respectively. This

expression indicates that for a particular temperature, the sensitivity can only be made

larger by increasing the difference between the two lower state energies. Thus, we desire

water transitions which have a sufficiently large separation in E″. There are, however,

other important criteria for choosing transitions such as isolation, strength and interfering

species. All of these criteria are discussed in § 5.4.

5.3.4 Mole Fraction Sensing Once temperature is measured using the 2-wavelength approach, it is straightforward

to extract the partial pressure of water, PXH2O, using Eq. (33)

LTS

HPX)(H2O =

If a separate measurement of static pressure is also made, the mole fraction, XH2O, is

obtained. Thus, by using this 2-wavelength approach, both temperature and mole

fraction are measured simultaneously.

As with the temperature sensor, the sensitivity of the mole fraction sensor should be

taken into account. The sensitivity of XH2O to the measured quantity H is unity, i.e.

1// H2OH2O =

HdHXdX

regardless of the chosen transition. However, the sensitivity of XH2O to temperature does

depend on the choice of transition via S(T). Although temperature will be measured, it is

98

useful to choose a transition for which S is as insensitive to temperature as possible. This

will minimize the uncertainty in inferred XH2O.

5.4 Wavelength Selection As was mentioned earlier, there are several criteria for properly selecting the two

transitions which will be used to measure T and XH2O: 1) interfering species, 2) isolation

from neighboring transitions, 3) strength, and 4) sensitivity.

5.4.1 Interfering Species The chosen band for the current water sensor is centered near 2.7 μm. It is important

to consider other species which will also absorb near this wavelength. As the sensor is to

be used in the presence of reactants, the species present (other than H2O) include O2, N2

and fuel (typically a hydrocarbon). Since O2 and N2 do not have infrared spectra and

hydrocarbons tend to absorb at 3.4 μm, interfering species in the reactants do not pose a

problem. Nevertheless, in anticipation that CO2 from the previous cycle’s products could

mix with the reactants, the spectrum of CO2 has been simulated and is plotted alongside

H2O in Figure 37.

99

3200 3400 3600 3800 4000 42000.0

0.5

1.0

1.5

2.0

2.5

3.03100 3000 2900 2800 2700 2600 2500 2400

Abs

orba

nce,

α


H2O CO2

Wavelength [nm]

Figure 37: ν1 and ν3 bands of H2O spectrum with interfering CO2 spectrum.

T = 520 K; P = 1 atm; XH2O = XCO2 = 4%; L = 7.3 cm. (Source: HITRAN 2004 [90].)

Clearly CO2 would interfere with an H2O diagnostic over a large fraction of the

spectrum. Therefore, the current H2O sensor has been restricted to wavelengths shorter

than 2.63 μm (frequencies higher than 3800 cm-1).

5.4.2 Isolation, Strength and Sensitivity The three criteria of isolation, strength and sensitivity must be addressed

simultaneously. To start, one of the two transitions was chosen based on the experience

of Farooq et al. [94] These authors used a pair of transitions near 2.5 μm to measure

temperature and water mole fraction in a flame. One of their transitions (viz.

ν0 = 3982.06 cm-1) was chosen for the current work. This transition is shown in Figure

38 at various temperatures. Notice the strength of the feature strongly depends on

temperature. The other transition used by Farooq et al. [94] is not amenable to the

100

temperatures investigated here because of its high E″. A second transition is therefore

needed which will provide sufficient isolation, strength and sensitivity.

3981 3982 39830.00

0.05

0.10

0.15

0.202512.0 2511.5 2511.0 2510.5

440 K 520 K 600 K

Abs

orba

nce,

α


Chosen transition(E" = 1581 cm-1)

Wavelength [nm]

Figure 38: Water transition near 3982 cm-1 used for current water sensor. T = 520 K; P = 1 atm; XH2O = 4%; L = 7.3 cm. (Source: HITRAN 2004 [90])

Since the first transition is sensitive to temperature, we desire a second transition

which is insensitive to temperature. This second transition will then be used to infer XH2O.

For sufficiently large ν0, Eq. (37) implies that E″ uniquely determines the temperature at

which dS / dT = 0. For the nominal temperature of 520 K, the required E″ is ~ 800 cm-1.

Three candidate temperature-insensitive transitions are shown in Figure 39, each one

identified by its respective E″. Notice that each candidate has an E″ which lies between

700 and 760 cm-1. (The reason that the E″’s for temperature-insensitivity are different

than the expected value of 800 cm-1 is due to the temperature dependence of the lineshape,

φν.)

Panels (a) and (b) show transitions which are strongly affected by neighbors. In order

to take advantage of lineshape integration (Eq. (32)), interference from neighboring lines

should be avoided. The transition in panel (c), on the other hand, is very well isolated

101

from its neighbors. Therefore, this transition is chosen for the current water sensor. The

difference in E″ for the two transitions is 1581 – 704 = 877 cm-1, resulting in a sensitivity

for the temperature sensor of ζ = 2.4 at 520 K.

The separation between the two transitions (~ 60 cm-1) is too large to be covered by a

single DFB laser. Thus, two lasers have been obtained (Nanosystems and Technologies

GmbH), one for each transition. Relevant spectroscopic data for the two chosen

transitions are listed in Table 8. The table includes both values from HITRAN as well as

values measured in the laboratory. These measurements are detailed in the next section.

3904 3905 39060.0

0.1

0.2

0.3

0.4

0.52561.5 2561.0 2560.5 2560.0

(a) 600 K 520 K 440 K

Abs

orba

nce,

α


E" = 758 cm-1

Wavelength [nm]

3870.5 3871.0 3871.5 3872.0 3872.50.0

0.1

0.2

0.3

0.4

0.52583.5 2583.0 2582.5

600 K 520 K 440 K

Abs

orba

nce,

α


E" = 742 cm-1

(b)

Wavelength [nm]

3919 3920 39210.0

0.2

0.4

0.6

0.8

1.0

1.2

1.42552.0 2551.5 2551.0 2550.5

(c) 440 K 520 K 600 K

Abs

orba

nce,

α


E" = 704 cm-1

Wavelength [nm]

Figure 39: Candidate transitions for 2-wavelength water sensor. (First transition of sensor shown in Figure 38.) E″ shown over each transition. T = 520 K; P = 1 atm;

XH2O = 4%; L = 7.3 cm. (Source: HITRAN 2004 [90])

102

Table 8: Summary of spectroscopic parameters for H2O sensor. See Ref. 90 for HITRAN 2004.

Transition #1 Transition #2

ν0 [cm-1] 3982.06 3920.09 λ0 [nm] 2511.26 2550.96

E″ [cm-1] 1581 704.2

HITRAN 2004 8.84×10-3 0.639 S @ 296 K [1/cm2/atm] Measured 8.90×10-3

(±0.23×10-3) 0.640

(±2.8×10-3)

HITRAN 2004 0.776 0.794 2γself @ 296 K [cm-1/atm] Measured 0.602 0.815

HITRAN 2004 0.100 0.132 2γair @ 296 K [cm-1/atm] Measured 0.103 0.133

nself Measured 0.68 0.76 HITRAN 2004 0.41 0.53

nair Measured 0.52 0.55

5.5 Spectroscopic Measurements Having chosen the two transitions for the water sensor, it was important to confirm the

spectroscopic data contained within the HITRAN database [90]. This was accomplished

by making high-resolution measurements of each transition in a static cell, heated within

a furnace.

5.5.1 Experimental Setup The setup is shown in Figure 40. Each transition is characterized separately, so the

figure shows only one laser. The laser power is collimated using a small lens (focal

length: 1.2 mm) in order to make the beam as narrow as possible (see Appendix E for

details on the laser collimation). All lenses were plano-convex, with the curved surface

on the side of the collimated beam in order to minimize spherical aberrations [95, § 7].

103

The laser is modulated in power and wavelength by using a sawtooth injection current

signal. The modulation frequency is 1 kHz and 50 scans are averaged together. The

beam is split by a wedged ZnSe beam splitter: ~ 50% to a reference detector; ~ 50%

through the furnace. The purpose of the reference detector is to track the time-varying

wavelength resulting from modulation. This is done using a Ge etalon (free spectral

range: 0.0161 cm-1). After the cell, the beam passes through a narrow-pass spectral filter,

then a sapphire lens (focal length: 25 mm), and finally onto the second detector. Both

detectors (bandwidth: 1 MHz; noise: 0.5 mVrms) are liquid-N2 cooled InSb, and are

matched during fabrication. The freespace beam path is purged with nitrogen to remove

ambient water.

The cell is made of quartz with IR-grade fused silica windows. The cell has three

zones and therefore 4 windows, all of which are wedged. The outer two zones remain

evacuated during the experiment in order to reduce absorption from water in the ambient.

Experiments were performed on both pure distilled water and mixtures of distilled

water in air. The pure water measurements yield information on linestrength and self-

broadening. The water/air mixture measurements yield information on air-broadening.

104

Figure 40: Experimental setup for water spectrum measurements. LP is the low-

pressure transducer (100 torr); HP is the high-pressure transducer (1,000 torr); T is the thermocouple readout. The path length, L, is 76 cm for characterizing the

transition at 3982.06 cm-1, and 9.9 cm for the transition at 3920.09 cm-1.

5.5.2 Pure Water Measurements Each laser was scanned at 1 kHz. An example of absorbance versus frequency is

shown for ν0 = 3920.09 cm-1 in Figure 41. The best fit using the Voigt lineshape is

included and the peak-normalized residual between the fit and the data is shown at the

bottom. The resulting integrated absorbance, H, and collisional full-width, ΔνC, are

shown in the figure.

detector

L

T

dry

air

mix

ing

tank

distilled water

HP

LP to vacuum

to vacuum

detector

lens

etalon

beam splitter

modulation laser diode

collimating lens

filte

rle

ns

furnace

105

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3919.95 3920.00 3920.05 3920.10 3920.15 3920.20

-202

Abs

orba

nce,

α

Experiment Voigt fitH = 7.75 x 10-2 cm-1

ΔνC = 7.36 x 10-3 cm-1

(2008-10-25-1051)

Res

idua

l [%

]


Figure 41: Sample absorbance plot of pure water with best fit using Voigt lineshape. T = 874 K; P = 15.1 torr; ν0 = 3920.09 cm-1.

The measurements are repeated for a fixed temperature over a range of pressures.

Each measurement yields SP and the collisional full-width (@ half max), ΔνC. (The

quantity ΔνC is equal to 2γself P.) Both SP and 2γself P are plotted in Figure 42.

0 5 10 15 20 25 30 350.000

0.005

0.010

0.015

0.00

0.01

0.02

0.03

0.04

Measurement Linear fit

Line

stre

ngth

x P

ress

ure,

SP

[cm

-2]

Pressure, P [torr]

(2008-10-24)

Sel

f-bro

aden

ing

coef

ficie

nt x

Pre

ssur

e, 2

γ self P

[cm

-1]

Figure 42: Linestrength and self-broadening at various pressures. T = 874 K; ν0 = 3920.09 cm-1.

106

The slopes of the two linear fits in Figure 42 represent the linestrength, S, and the self-

broadening coefficient, 2γself. Since both S and 2γself depend on temperature, the

measurements described hitherto were repeated over a range of temperatures. The

linestrength results are shown in Figure 43 and the broadening results are shown in

Figure 44. Finally, a best fit of Eq. (37) is used to infer the free variable S(Tr) and a best

fit of Eq. (34) is used to infer the free variables 2γself (Tr) and nself. The rms deviation

between the measured linestrength and the fit are used to quantify the uncertainty in S(Tr).

This uncertainty is approximately 2.5% for ν0 = 3982.06 cm-1 and 1% for

ν0 = 3920.09 cm-1. Both HITRAN and measured values are listed in Table 8. We see

that agreement between the measurement and HITRAN is quite good for S(Tr); this is due

to the fact that HITRAN values for S(Tr) are themselves obtained experimentally.

Agreement is somewhat worse for 2γself(Tr) and nself, likely due to the HITRAN values

typically being estimates based on theoretical predictions.

200 400 600 800 1000 1200 14000.00

0.02

0.04

0.06

0.08

0.10

Line

stre

ngth

, S [c

m-2/a

tm]

Temperature, T [K]

Measurement Best fit

(a)

200 400 600 800 1000 1200 14000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Line

stre

ngth

, S [c

m-2/a

tm]

Temperature, T [K]


(b)

Figure 43: Linestrength vs. T for (a) ν0 = 3982.06 cm-1 and (b) ν0 = 3920.09 cm-1.

107

200 400 600 800 1000120014000.01

0.1

1

air-broadening

Bro

aden

ing

Coe

ffici

ent,

2γ [

cm-1/a

tm]

Temperature, T [K]


(a)self-broadening

200 400 600 800 1000 120014000.01

0.1

1

air-broadening

Bro

aden

ing

Coe

ffici

ent,

2γ [

cm-1/a

tm]

Temperature, T [K]


(b)self-broadening

Figure 44: Broadening coefficients vs. T for (a) ν0 = 3982.06 cm-1 and

(b) ν0 = 3920.09 cm-1.

5.5.3 Water/Air Mixture Measurements Broadening by species other than water is important because most gas systems include

many different species, each with its own collisional cross-section and broadening

coefficient. Furthermore, the self-broadening coefficient is usually quite different from

other species, so separate measurements must be made in order to accurately predict the

lineshape for the conditions of the NPS PDE. Air is a commonly studied collision partner

and is included in the HITRAN database. In order to make proper comparisons, therefore,

the measurements made here will be of water/air mixtures.

A mixture of water and air was prepared in a separate mixing tank by first admitting

~ 16 torr of water, and then ~ 800 torr of air. The mixture is mixed for 15 minutes.

Figure 45 shows a sample absorption feature with significant air broadening. The

linestrength is known from the pure water measurements of the previous section, so the

mole fraction, XH2O, is measured spectroscopically. (This is necessary since a prediction

of XH2O using partial pressures during mixture preparation is not trustworthy due to

substantial water adhesion to the walls of the mixing tank, plumbing and cell.)

108

0.0

0.1

0.2

0.3

0.4

0.5

3919.8 3920.0 3920.2 3920.4

-202

(2008-06-23-1101)

Abs

orba

nce,

α

Experiment Voigt fitΔνC = 7.78 x 10-2 cm-1

Res

idua

l [%

]


Figure 45: Sample absorbance plot of water/air mixture with best fit using Voigt lineshape. T = 825 K; P = 759 torr; XH2O = 1.3%; ν0 = 3920.09 cm-1.

As was done for pure water, data on 2γair were extracted by repeating the

measurements over a range of pressures and temperatures. The results are shown along

with self-broadening in Figure 44. The best-fit parameters, 2γair(Tr) and nair, are listed in

Table 8.

5.6 Sensor Setup With the spectroscopic database in hand, incorporation of the sensor’s fibers is now

discussed. Then a short discussion on how the two water transitions are measured on a

single detector. Finally, the sensor is validated in a controlled environment.

109

5.6.1 Sensor Hardware Since the chosen wavelengths of 2.5 μm are too far in the infrared to be compatible with

common Si-based fibers, fluoride glass fibers were selected for fiber coupling. Like Si

fibers, these fibers have low loss, but are more expensive and less robust. The bending

radius must stay above ~ 30 cm, so the fibers were fitted with rigid jackets by the

manufacturer (FiberLabs, Inc.). Two fibers were used, one for pitch (160 μm core dia.;

0.27 numerical aperture) and the other for catch (480 μm core dia.; 0.28 numerical

aperture). The catch fiber needed to be larger than the pitch fiber because the beam

spreads somewhat along its freespace path. Two spectral filters were used, each having a

pass width of ~ 100 nm. The filters are rotated in order to shift the pass region towards

shorter wavelengths [96]. By twisting the filters by different amounts, the overall pass

width of the pair is reduced, thereby improving the emission rejection capability of the

sensor.

Figure 46 shows the general setup.

Figure 46: General setup of fiber-coupled water sensor. Red lines indicate freespace beams. L: plano-convex lens; BS: beam splitter (wedged); W: window (wedged); F:

filter; D: InSb detector; PF: pitch fiber; CF: catch fiber.

5.6.2 Time Multiplexing As depicted in Figure 46, the sensor consists of two lasers and one detector. It was

possible to discern the signals from the two lasers by adopting a strategy of time

laser #1

L

L

L

L L D

F F CF

PF

W

W

engine or furnace

laser #2

BS

110

multiplexing. Specifically, this means that at any given moment, only one of the two

lasers is turned on. One half cycle later, the first laser is turned off and the second laser is

turned on. In addition to this switching, each laser is also ramped with its own sawtooth.

Figure 47 shows a schematic of the injection current delivered to each laser. At any

given moment, only one laser at most is turned on. For a short period during each cycle,

both lasers are turned off in order to record the background signal. The lasers are

modulated at 5 kHz, a rate limited by the lasers but which has proven successful at

making time-varying measurements in the NPS PDE [89, § 4].

0.0 0.1 0.2 0.3 0.4

0

20

40

60

80

100

0.0 0.1 0.2 0.3 0.4

0

20

40

60

80

100

Lase

r #1

Time [ms]

bothlasers offIn

ject

ion

Cur

rent

[mA

]

Lase

r #2

Figure 47: Injection current to lasers #1 and #2.

5.7 Uncertainty Analysis The uncertainty analysis follows the standard methodology found in most engineering

texts (see, e.g., Ref. 97, § 7). If a dependent variable, p, is a function of multiple

independent variables, z1, z2, z3, …

,...),,( 321 zzzfp =

111

then the uncertainty in p, wp, will have contributions from uncertainties in each of the

independent variables, namely wz1, wz2, wz3, … The relationship between wp and wzi is

commonly taken as

...2

3

2

2

2

1321

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= zzzp wzpw

zpw

zpw (40)

5.7.1 Temperature Temperature is a unique function of the integrated absorbance ratio, T = f(R), given by

Eq. (38). Therefore, when R is being measured with the intension of extracting T,

uncertainty in R will lead to uncertainty in T. Furthermore, the actual function linking T

and R is imperfectly known because of uncertainties in S1(Tr) and S2(Tr). Figure 48

schematically depicts these two sources of error of the temperature sensor. This figure is

purely illustrative and is not meant to quantifiably reflect the actual sensor.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

200

400

600

800

1000

(wT)1 (wT)2

T = f (R) assuming perfect knowledge of S1(Tr ) and S2(Tr )

wR uncertainty inmeasured R

T = f (R) assuming imperfect knowledge of S1(Tr ) and S2(Tr )

Tem

pera

ture

, T [K

]

Ratio of Linestrengths, R

Figure 48: Schematic depicting two sources of error for temperature sensor. Values do not reflect actual sensor.

112

First, consider the red curve. This represents Eq. (38) constructed using measured

values of S1(Tr) and S2(Tr). When the sensor is applied to a real gas system (e.g. a PDE),

this curve transforms the measured value of R into T. For example, the dark blue line

shows that a measured value of R = 1.5 corresponds to T = 500 K. The first source of

uncertainty in T, (wT)1, stems from the fact that the measurement of R in imperfect. This

is because laser noise, fiber noise and detector noise will lead to some level of noise

recorded by the data acquisition system, leading to uncertainties in ανi, Hi, R and finally T.

This uncertainty in R is depicted by the green bar in Figure 48, and the resulting

temperature uncertainty, (wT)1, is obtained by following the dotted green lines. Using

Eq. (40), the relationship between wR and (wT)1 is

( )

Rw

Tw RT

ζ11 =

Notice that an increase in ζ leads to a decrease in (wT)1, a fact which helped motivate the

choice of transitions. For the conditions of interest, the detector noise is negligible. The

laser noise was measured to be 0.1%rms (for both lasers) and a vibrating fiber generates a

gross noise level of 0.5%rms. This leads to wR/R ~ 2%. For conditions expected in the

NPS PDE (T = 520 K; ζ = 2.4), (wT)1 is therefore approximately 5 K.

Another source of uncertainty stems from the fact that uncertainties in the measured

values of S1(Tr) and S2(Tr) lead to imperfections in the functional relationship between R

and T. This is depicted by the dotted black lines in Figure 48. Assuming a perfect

measurement of R is made in the PDE, an uncertainty in temperature, (wT)2, results. This

is indicated by the dotted light blue lines. Recognizing that from this perspective

T = f(S1(Tr), S2(Tr)), the expression for (wT)2 becomes

( ) 2

2

)(2

1

)(2

)()(1 21

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

r

TS

r

TST

TSw

TSw

Tw rr

ζ

113

Once again, an increase in ζ leads to a decrease in wT. It was mentioned in § 5.5.2 that

)(/ 1)(1 rTS TSwr

~ 2.5% and )(/ 2)(2 rTS TSwr

~ 1%. Plugging into the above equation (again

for T = 520 K; ζ = 2.4) gives a value for (wT)2 of approximately 10 K.

The two sources of temperature uncertainty are combined using

( )[ ] ( )[ ]22

21 TTT www +=

for an overall uncertainty of 11 K. This value is used for the error bars during sensor

validation (§ 5.8).

5.7.2 Mole Fraction Using Eq. (40) in conjunction with the expression for mole fraction

SPLHX =H2O

leads to

222

H2O

H2O ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

Pw

Sw

Hw

Xw

PSHX

where the uncertainty in path length has been neglected. The uncertainty in S stems

uniquely from uncertainty in T. Therefore

222

H2O

H2O ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

Pw

Sw

dTdS

Hw

Xw

PTHX (41)

The expression for S(T) is not easily differentiated because of the partition function, Q.

Instead, the relevant value of dTdS

S1 for the transition at 3920.09 cm-1 is calculated

numerically and found to be -1.2 × 10-3 K-1 at 520 K. (As a matter of interest, the

114

corresponding value for the sensor’s other water transition at 3982.06 cm-1 is

3.4 × 10-3 K-1, which makes the chosen transition almost 3× less sensitive to uncertainties

in T. This was the reason for using 3920.09cm-1 to extract XH2O, as discussed in § 5.4.2.)

Equation (41) now becomes

( )2

232

H2O

102.1H2O ⎟⎠⎞

⎜⎝⎛+×+⎟

⎠⎞

⎜⎝⎛= −

Pww

Hw

Xw

PT

HX

where wT is in Kelvin. As was the case for the temperature sensor, the uncertainty in H is

limited by fiber noise, which leads to wH / H ~ 1%. The uncertainty in temperature is

taken from the previous section, wT ~ 11 K. The uncertainty in pressure is determined by

the gauge used in the PDE, and is assumed to be 1%. This leads to H2OH2OXwX ~ 2%.

This value is used for the error bars during sensor validation (§ 5.8).

5.8 Sensor Validation The fiber-coupled sensor, modulated at 5 kHz, was validated using the furnace

described in § 5.5. The fibers were shaken in order to induce the levels of noise

anticipated during actual PDE tests. A mixture of water and air was prepared and

measured at various temperatures. The pressure was set to 1 atm in each case. The

detector’s signal was de-multiplexed to obtain absorbance for each transition. These

absorbance measurements were then integrated to obtain H1 and H2. The ratio, R, was

then used to extract temperature. The results are compared to the furnace’s thermocouple

readings in Figure 49. See § 5.7.1 for calculation of the error bars in Figure 49.

With temperature known, measurements of the transition at ν0 = 3920.09 cm-1 were

used to extract water mole fraction. Since water is known to condense on the walls of the

mixing tank, plumbing and cell, a comparison cannot be made with the mole fraction

expected from partial pressures. Rather, the sensor’s precision was confirmed by

establishing that the same mole fraction is obtained at each temperature when the same

115

mixture is used. These results are included for two different mixtures in Figure 50. See

§ 5.7.2 for calculation of the error bars in Figure 50.

450 500 550 600

450

500

550

600

Lase

r Sen

sor T

empe

ratu

re [K

]

Thermocouple Temperature [K]

Figure 49: Validating temperature sensor.

450 500 550 6000.00

0.25

0.50

0.75

1.00

1.25

Mixture #1 Mixture #2

Lase

r Sen

sor M

ole

Frac

tion

[%]

Thermocouple Temperature [K]

average of measurements

average of measurements

Figure 50: Validating mole fraction sensor.

116

5.9 NPS Campaign The sensor was next transported to NPS in Monterey, CA for measurements on a

multi-cycle, air-breathing PDE. A general description of the engine can be found in

Ref. 13. The engine is fed with air which is heated to temperatures above 500 K by an

H2/O2 vitiator located upstream of the inlet. It is water from this vitiator which is

measured by the current sensor. The sensor was used to capture the temperature

fluctuations which occur due to the cooling of injected fuel, and to identify the effect

water has on the performance of the engine’s ignitor.

5.9.1 Sensor Setup A schematic of the engine and sensor is shown in Figure 51. Steady vitiated air enters

from the left. The flow is diverted through 4 branches (2 shown), each of which includes

a C2H4 injector. The injectors operate at frequencies between 20 and 40 Hz. Next the

flow moves past the ignitor, followed by the optical section (2 wedged sapphire

windows), and finally into the “detonator” which includes multiple obstacles to promote

deflagration-to-detonation transition (DDT). A piezo-electric pressure transducer (PZT)

is mounted in the detonator and is used to determine whether a particular cycle has either

detonated, burned without detonation, or misfired. Additional static temperature and

pressure gauges are mounted near the optical section. The ignitor is based on transient

plasma physics and was developed by Gundersen et al. [98]. The pitch and catch optics

are stored in neighboring rooms to protect this delicate hardware from heat and vibration.

117

Figure 51: Water sensor installed on NPS PDE. Red line represents freespace beam.

See Figure 46 for details on fiber coupling and de-coupling. “PZT” represents a piezo-electric pressure transducer.

5.9.2 Cooling by Injected Fuel Previous measurements of water in the NPS PDE were made by Mattison [89, § 4].

The sensor was based on wavelengths near 1.4 μm so its accuracy was aversely affected

by the low level of absorbance at these wavelengths (see Figure 35). Nevertheless, those

measurements revealed periodic cooling by the injected C2H4 during unfired operation.

The same measurements were made with the current sensor, and the results are compared

in Figure 52. No filtering or cycle-averging has been performed. The average

temperature was slightly different in both cases, so the 1.4 μm sensor results were scaled

for comparison.

static P and T gauges

detonator

vitiated air

wall

- 2 lasers - 1 beam splitter - 1 lens

ignitor C2H4 injector

PZT

- 2 lenses - 2 filters - 1 detector

Pitch Optics

Catch Optics

branch

118

0 1 2 3 4 5 6 7 8350

400

450

500

550

600

vitiator onfuel injected (unfired)

Tem

pera

ture

[K]

Engine Cycle #

1.4 μm sensor 2.5 μm sensor Thermocouple

Trms (1.4 μm): 13 KTrms (2.5 μm): 4.9 K

B

Scatter of data set B

Figure 52: Periodic cooling by injected C2H4 in NPS PDE using near-IR [89, pg. 85] and current mid-IR (2008-06-04-7) water sensors. Both sensors scanned at 5 kHz.

We see that both sensors reveal the periodic cooling due to the injected C2H4. The

current sensor, however, has noticeably lower scatter. This scatter is quantified for a

particular subset of data, as indicated in the figure. The near-IR sensor has a standard

deviation of 13 K, while the current mid-IR sensor has a standard deviation of 4.9 K

(which is close to the sensor’s uncertainty due to fluctuations in R, (wT)1 ~ 5 K, as

estimated in § 5.7.1).

5.9.3 Vitiator Impact on Ignitor Performance Next the sensor was employed to investigate the effect the vitiator has on the ignitor

performance. Previous engine studies had shown that the reactants were more likely to

burn and successfully detonate if the vitiator was turned off. However, it was unclear

how quickly or slowly water from the vitiator was purged by the continuous air flow after

119

the vitiator was turned off, so to truly determine whether water had an effect on ignitor

performance was possible only with an in situ water measurement.

Figure 53 shows average water mole fraction for each cycle during a run when the

vitiator was shut off at a prescribed time. Fuel was injected and the ignitor activated,

both at 30 Hz. Also included is the cycle peak pressure recorded by the PZT located in

the detonator. There is a clear correlation between water content and engine

performance: before vitiator shut-off, the presence of water suppresses combustion and

even leads to misfires for more than 50% of the cycles. After the water is pushed out of

the engine, ignitor performance improves greatly, both in terms of increased peak

pressure and the total absence of misfires.

0 20 40 60 80 1000

1

2

3

4

5

6

7

0

2

4

6

8

Ave

rage

Wat

er M

ole

Frac

tion

[%]

Cycle #

Vitiator shutoff

Pea

k P

ress

ure

[atm

, g]

2008-06-04-7

Figure 53: Average water mole fraction and peak pressure vs. engine cycle. Engine frequency is 30 Hz. ν0 = 3920.09 cm-1.

5.10 Summary This chapter covered the development of a novel 2.5 μm water sensor for the purpose

of measuring XH2O and T in a real, multi-cycle PDE. The sensor used scanned direct

absorption of two water features. A fiber-coupling strategy was implemented in order to

easily deliver laser power to the engine, and return it to the detector. A time multiplexing

120

scheme was used to track two wavelengths with a single detector. The sensor was

characterized and then validated on a heated static cell at Stanford.

The sensor was employed at the Naval Postgraduate School on a real multi-cycle PDE.

The sensor successfully reproduced temperature data previously achieved using a 1.4 μm

sensor and proved to have a higher SNR due to the stronger absorption of water at 2.5 μm.

The sensor was also used to probe the vitiator dynamics of the PDE. The results showed

a direct link between the presence of water and the occurrence of misfires. When the

vitiator was turned off and the water removed, misfires were totally eliminated.

121

Chapter 6: Laser-Based Mid-IR JP-10

Sensing

6.1 Introduction Fuel sensing is a crucial aspect of PDE research. Real time in situ DA sensing enables

Isp to be quantified, provides information on the fraction of fuel burned, and reveals the

precise distribution of fuel throughout the engine. PDEs have been historically studied

using lighter, more easily detonated fuels such as H2, C2H2, C2H4, and C3H8. However,

the importance of practical fuels such as JP-8 and JP-10 has led to a demand for

specialized engine hardware and modeling, which have in turn led to a demand for

specialized fuel diagnostics.

Diagnostics for PDEs call for, in general, time-resolved strategies due to the inherently

unsteady nature of the engine. This is especially true for fuel sensing since fuel injection

profiles can often be imperfect and because arrival time of fuel at the ignitor must be

accurately known for ignition scheduling. Furthermore, sensing should be done in situ in

order to avoid disturbing the flow. Laser-based strategies meet both of these needs.

Previous time-resolved, DA, in situ fuel sensing for PDEs has focused on research

fuels, such as C2H4 [34,35,99] and C3H8 [99]. Quantitative measurements of C2H4 in a

PDE was first done by Ma et al. [35] at 1625 nm using the first overtone of the C-H

stretch. These authors measured equivalence ratio, Φ, in the Stanford PDE in order to

ascertain fuel arrival time and the unsteady equivalence ratio. Measurements of

temperature, T, using C2H4 were also made in order to validate simple models. The

choice of a near-IR wavelength enabled the sensor to make use of readily available

telecom lasers, silicon fibers, and uncooled InGaAs detectors.

A more common approach to hydrocarbon sensing is to take advantage of the HeNe

laser transition at 3392 nm. This wavelength overlaps fortuitously with the fundamental

C-H stretch transition of all hydrocarbons, which is ~100× stronger than the first overtone,

as used in Ref. 35. Klingbeil et al. [99] took advantage of the HeNe wavelength to

122

measure C2H4 in the NPS PDE and C3H8 in the General Electric PDE. Although the

3392 nm laser was simple to operate, thermo-electrically cooled InAs detectors and

exotic fluoride glass fibers were needed at these longer wavelengths.

Similar strategies could, in principal, be applied to more practical fuels such as JP-8

and JP-10. However, in order to avoid the low signal-to-noise ratio (SNR) which results

from the weak overtone band in the near-IR, attention will be focused on the much

stronger mid-IR fundamental band. This would suggest using the HeNe at the

fundamental band. However, whereas the absorption of the overtone band of JP-10 is too

weak, absorption at the HeNe wavelength is too strong, leading to prohibitively excessive

detector noise and bit noise. Rather, a tunable laser near 3.4 μm is necessary to make

accurate measurements. This chapter details the design of a mid-IR, DA, in situ, time-

resolved JP-10 sensor utilizing such a tunable laser. Similar to Chapter 5, the current

sensor uses two wavelengths in order to extract temperature. Unlike Chapter 5, however,

absorption features are not scanned, and so the JP-10 sensor uses a fixed-wavelength

approach. The sensor is applied to the NPS PDE for measuring equivalence ratio and

temperature.

6.2 JP-10 Description The fuel studied here is “jet propellant 10”, or JP-10. It is also known as

tricyclodecane and is a single-component cycloalkane with the chemical formula C10H16.

(Known impurities exist and are not expected to significantly affect sensor design or

engine measurements.) Its structure is depicted in Figure 54. As a single-component fuel

it is attractive from the standpoint of having properties which do not vary from batch to

batch. Osmont et al. [100] review some of the important properties. In addition, JP-10 is

known to be resistant to cracking [101], making it attractive to applications which require

high reliability and prolonged down time, such as air-breathing missiles [102].

123

Figure 54: Structure of JP-10.

6.3 Sensor Theory The fundamentals of the sensor theory are the same as for the direct absorption water

sensor used in Chapter 5. For example, Beer’s law is used to obtain absorbance, and

temperature sensing is achieved by using two different wavelengths. What differentiates

JP-10 from water is that the former’s transitions cannot be isolated and scanned. This

implies that the lineshape function, φν, and Eq. (31) are not practical when dealing with

JP-10 spectroscopy. In light of this, a form of Beer’s law more appropriate for JP-10 is

used:

⎟⎠⎞

⎜⎝⎛−=⋅⋅−=−=⎟⎟

⎠

⎞ − LT

PXLnII

R10JP

10-JP exp)exp()exp(0

νννν

σσα (42)

where σν is the cross-section and nJP-10 is the number density of JP-10. By comparing

Eqs. (31) and (42), we see that σν can in general be a function of T and P.

6.4 JP-10 Spectrum

6.4.1 Experimental Setup The spectral cross-section of JP-10 was measured using a Fourier Transform Infra-Red

(FTIR) spectrometer with a resolution of 0.06 cm-1. The experimental setup is shown in

Figure 55. The JP-10 was stored in a flask and purified by freezing with liquid nitrogen,

evacuating, and then thawing. This purification process was repeated several times. The

124

mixture consisted of JP-10 diluted with nitrogen and was prepared in a heated mixing

tank. Partial pressures were used to establish mole fraction. It was determined that a

mixing time of 1 hour was required for full mixing. The mixture was then admitted into

the optical cell, which was contained inside a furnace. The cell’s path length was 15 cm.

Measurements using pure JP-10 were also performed.

Figure 55: Experimental setup for JP-10 spectrum measurements. LP is the low-pressure transducer (100 torr); HP is the high-pressure transducer (50 psia); the

thermocouple is mounted inside the optical cell, where T represents the thermocouple readout.

6.4.2 Results Beer’s law was used to calculate the frequency-dependent cross-section, σν. In

general the cross-section can be a function of P and T. Figure 56(a) shows the

fundamental band cross-section of the C-H stretch of JP-10 for two pressures: 1 atm

(JP-10 diluted with N2) and 1.12×10-3 atm (pure JP-10). The temperature in each case is

302 K. The largest disparity between the cross-section at low and high pressure occurs at

the peak near 3380 nm and is less than 2%. At wavelengths which will be of interest –

FTIR

detector

HP

T

furnace

optical cell

beam

N2 JP-10

LP

to vacuum

heated mixing tank w/ stirrer

125

which are far from the peak – the effect of pressure is much lower. This simplifies the

sensor design because the cross-section is now known to be a function of only T.

2800 2850 2900 2950 3000 3050

0

50

100

150

3550 3500 3450 3400 3350 3300

0.85 torr; pure (2007-07-08-3)

Cro

ss-s

ectio

n, σ

ν [mol

e/m

2 ]


1 atm; 0.036% (2007-07-12-4)

(a)

Wavelength, λ [nm]

5500 5600 5700 5800 5900 60000.0

0.1

0.2

0.3

0.4

0.51800 1760 1720 1680

Cro

ss-s

ectio

n, σ

ν [mol

e/m

2 ]


1 atm (2005-09-29) (b)

Wavelength, λ [nm]

Figure 56: Cross-section of the fundamental band for the C-H stretch of JP-10. (a) fundamental and (b) first overtone. T = 302 K. The uncertainty bar applies only

to the 1st overtone band.

Figure 56(b) shows the first overtone of the C-H stretch of JP-10 near 1.7 μm.

Because the absorption levels are so low at these wavelengths, noise from the FTIR was

quite large and the data were smoothed, leading to the uncertainty bar. Nevertheless, we

see that the first overtone of JP-10 is 500× weaker than the fundamental. This implies

that a JP-10 sensor based on the first overtone band (e.g. to make use of telecom lasers,

fibers and detectors) is ultimately impractical.

The absorption cross-section was also measured at various temperatures, as shown in

Figure 57. Also included are data at the HeNe wavelength taken from Ref. 106.

Agreement is within the measurement uncertainty. Figure 57 shows a strong dependence

on T. An increase in T results in a drop in σν at some wavelengths, and a rise in σν at

other wavelengths. This behavior was observed by Klingbeil for many other

hydrocarbons [103,104] and is expected because the integrated cross-section (denoted

here by Σ is independent of T [105]. That is,

)(Tfdband

≠≡Σ ∫ νσν

126

The integrated cross-section is plotted in Figure 58. The low scatter in Σ confirms the

accuracy of the cross-section data in Figure 57.

2800 2850 2900 2950 3000 30500

25

50

75

100

125

150

3550 3500 3450 3400 3350 3300

400 K500 K600 K700 K

Cro

ss-s

ectio

n, σ

ν [m2 /m

ole]


from Klingbeil et al.

2007-05-27~2007-05-29

Wavelength, λ [nm]

Figure 57: Cross-section of the fundamental band for the C-H stretch of JP-10 at

various T. Also shown are HeNe laser data taken from Klingbeil et al. [106].

350 400 450 500 550 600 650 700 7500

1,000

2,000

3,000

4,000

5,000

6,000

Inte

grat

ed c

ross

-sec

tion,

Σ [

cm-1m

2 /mol

e]

Temperature, T [K]

mean value: 5,647 cm-1cm2/mole(scatter: 1.4%rms)

Figure 58: Integrated cross-section of the fundamental band for the C-H stretch of

JP-10.

127

6.5 Tunable Mid-IR Laser This section describes the tunable mid-IR laser used for JP-10 sensing. The laser

makes use of non-linear frequency mixing of two near-IR fiber-coupled lasers to produce

mid-IR light. Specifically, the laser uses difference frequency generation (DFG) to

produce power with an optical frequency equal to the difference of the two input optical

frequencies: νout = νA – νB, where νout is the mid-IR frequency near 2980 cm-1 (3.35 μm),

νA is input laser A’s frequency near 9430 cm-1 (1.06 μm) and νB is input laser B’s

frequency near 6450 cm-1 (1.55 μm). A periodically-poled lithium niobate (PPLN)

crystal is used for the non-linear mixing. See Ref. 107 for more details on this

technology.

The laser system was developed by Novawave and is depicted schematically in Figure

59. The Novawave system consists of two near-IR fiber-coupled lasers, labeled A and B.

Laser A has a fixed wavelength at 1.064 μm. Laser B is modular and can be selected

from a set of lasers, each tunable with a central wavelength near 1.5 μm. Laser B’s

power is amplified to approximately 1 W and, along with laser A’s power, delivered to

the PPLN crystal via a fiber. The low efficiency of the crystal means that the laser

system’s ultimate power output is 200 μW at 3.35 μm. The system’s overall mid-IR

wavelength selection range (limited by the PPLN crystal) is 3337 to 3548 nm which

covers the entire JP-10 spectrum.

The Novawave system was expanded by incorporating a third laser C which is

multiplexed with laser B before being amplified. By switching between lasers B and C

(i.e. laser B is turned off while laser C is turned on, and vice versa) at a high rate, a time-

multiplexed 2-color mid-IR beam is produced. This multi-wavelength strategy enables

the potential for more information extraction, such as temperature or multi-species

mixtures [108]. Furthermore, by turning both lasers B and C off for a fraction of each

cycle, background emission can be tracked along with the two mid-IR signals. The

modulation between laser B, laser C, and the “off” period was done at 100 kHz. Finally,

since the PPLN crystal’s temperature needs to be adjusted for a particular wavelength, the

crystal temperature cannot be optimized for both lasers B and C simultaneously. It was

128

found that a maximum separation of 20 cm-1 between the two mid-IR frequencies can be

tolerated.

Figure 59: Tunable mid-IR DFG laser, operating in 2-color mode. Notice that the modulation signals for lasers B and C are out of phase.

6.6 Sensor Hardware The sensor setup is shown schematically in Figure 60 and follows closely the setup

used for the water sensor (§ 5.6.1). As with the water sensor, the detector has a

bandwidth of 1 MHz and a noise level of 0.5 mVrms. The DFG laser noise was

measured at less than 0.1%rms.

Figure 60: General setup of fiber-coupled water sensor. Red lines indicate freespace beams. L: plano-convex lens; W: window (wedged); F: filter; D: InSb detector; PF:

pitch fiber; CF: catch fiber.

laser A

laser B

laser C

fiber-amplifier

PPLN200 μW 2-color mid-IR

Novawave system

modulation signals

DFG laser

L

L

L

L L D

F F CF

PF

W

W

engine or furnace

129

6.7 Wavelength Selection

6.7.1 Temperature Sensor Temperature sensing is possible because the ratio of the absorbance at two

wavelengths is only a function of temperature:

)()()(

2

1

2

1

10JP

10JP

2

2 TfTT

LnLn

R ==⋅⋅

⋅⋅=≡

−

−

ν

ν

ν

ν

σσ

σσ

αα

(43)

This definition of R differs slightly from the one used with water sensing in that for

JP-10 there is no integrating of absorbance. Thus, R is defined in terms of α, rather than

H. The last equality of Eq. (43) illustrates the importance of establishing that σν is not a

function of pressure (see Figure 56(a)). The two frequencies ν1 and ν2 need to be chosen

so that each has an acceptable level of absorbance. In addition, the choice of frequencies

is governed by the sensitivity ζ which is expressed as:

)(11// 2

2

1

1

TfdT

ddT

dT

TdTRdR

=⎥⎥⎦

⎤

⎢⎢⎣

⎡−=≡ ν

ν

ν

ν

σσ

σσ

ζ

Notice that, like σν and R, the sensitivity ζ is only a function of T. The sensitivity

should be made as large as possible. Finally, recall that as the separation between ν1 and

ν2 is increased, the efficiency of the PPLN crystal is reduced, thereby reducing the output

power of the DFG laser. With all of this in mind, the frequencies (wavelengths) for the

temperature sensor were chosen as 2969 cm-1 (3368 nm) and 2983 cm-1 (3352 nm).

6.7.2 Fuel Sensor The fuel sensor’s frequency is chosen to minimize uncertainty in fuel mole fraction,

XJP-10. Recall that the water sensor uncertainty was minimized in part by choosing a

130

wavelength which was insensitive to T. For JP-10, however, 10JP−Xw is dominated by

uncertainty in absorbance, wα /α, rather than uncertainty in temperature, wT. Furthermore,

since the relative uncertainty in XJP-10 is equal to the relative uncertainty in α, we

therefore desire to minimize wα /α. This quantity is related to uncertainty in transmission

by the relation

0

01II

ww II⋅=αα

α (44)

which results by applying Eq. (40) to Beer’s law. For the case of small α (e.g. using the

first overtone near 1.7 μm) the uncertainty in transmission is determined by laser and

fiber noise. For the case of large α (e.g. using the HeNe laser) the uncertainty in

transmission is determined by detector and bit noise. Both cases lead to large relative

uncertainties in α, and in turn uncertainties in XJP-10. Thus, we desire a wavelength which

yields a moderate value of α. Knowing the amount of noise present in the system and the

nominal detector signal (~ 1V), the absorbance which minimizes wα /α is α ~ 1.2. The

conditions in the NPS PDE for tests with JP-10 are nominally: T ~ 500 K, P ~ 2 atm,

XJP-10 ~ 1.5% (Φ ~ 1) and L = 7.3 cm. This leads to a required σν ~ 24 m2/mole, which

implies a frequency of 2976 cm-1.

6.8 Temperature Sensor Bandwidth As mentioned already, the 2-wavelength strategy alternates between the two colors at

100 kHz, enabling temperature data to be acquired at this rate. It is worth constrasting

this with the water-based temperature sensor described in Chapter 5. Recall that the

modulation rate of the water-based sensor was much lower than the JP-10-based sensor,

i.e. 5 kHz vs. 100 kHz. The reason for this disparity in bandwidth is because the water-

based temperature sensor necessitated entire transitions to be scanned, whereas the JP-10-

based temperature sensor is essentially operated in a fixed wavelength mode.

131

From the perspective of bandwidth, a fixed wavelength scheme is advantageous to a

scanning scheme. However, if a fixed wavelength strategy were adopted for a water-

based temperature sensor, serious problems would arise because water’s cross-section, σν,

is not only temperature-dependent, but also pressure-dependent:

Water: ( )( ) ),(

,,

2

1

2

1

22

1OH PTf

PTPT

LnLn

R ==⋅⋅

⋅⋅=≡

ν

ν

ν

ν

σσ

σσ

αα

Thus, uncertainties in pressure would corrupt the inferred temperature measurement.

JP-10, on the other hand, has a pressure-independent cross-section:

JP-10: ( )( ) )(

2

1

2

1

2

110JP Tf

TT

LnLn

R ==⋅⋅

⋅⋅=≡−

ν

ν

ν

ν

σσ

σσ

αα

This makes JP-10 a perfect candidate for high-bandwidth temperature sensing. This fact

will be exploited for measurements on a real PDE, discussed later.

6.9 Sensor Validation Both the fuel and temperature sensors were validated in a static cell. The setup of

§ 6.4.1 was used, with the FTIR replaced by the DFG laser, fibers and detection optics.

As with the water sensor validation, the fibers were shaken. Since the mixing tank is not

hot enough to achieve the same mole fraction existing in the PDE (viz. 1.5%), the laser

frequencies used for validation were altered to increase accuracy at the low mole

fractions inside the optical cell: 2959 cm-1 for fuel sensing; ν1 = 2959 cm-1 and

ν2 = 2950 cm-1 for temperature sensing. The DFG was modulated at 100 kHz: on / off for

the fuel sensor and ν1 / ν2 / off for the temperature sensor.

For the fuel sensor multiple mixtures were prepared and admitted into the cell at a

constant temperature. For the temperature sensor a single mixture was prepared and

measured at multiple temperatures. Unlike the system used for water sensor validation,

132

the mixing tank and plumbing are fully heated, making it possible to reliably know the

mole fraction of fuel via partial pressures during mixture preparation. The laser-derived

results were compared with expected values of XJP-10 and T. The results are shown in

Figure 61 for fuel on the left and bottom axes, and temperature on the right and top axes.

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

400 500 600 700 800

400

500

600

700

800

Mole fraction Temperature

Mol

e fra

ctio

n fro

m D

FG [%

]

Mole Fraction from partial pressures [%]

2007-06-17 / 2007-06-18

Tem

pera

ture

from

DFG

[K]

Temperature from thermocouple [K]

Figure 61: Validating fuel and temperature sensors. For fuel sensor T = 374 K. For temperature sensor XJP-10 = 0.1%. P = 1 atm in both bases.

6.10 NPS Campaign As with the water sensor, the JP-10 sensor was put to use on the NPS PDE. In the

present study, the sensor was motivated by the need to know 1) accurate arrival time of

fuel at the engine’s ignitor; 2) the quantity of fuel delivered by the injectors; and 3) the

residence time of fuel at the ignitor. In addition, varying amounts of mixing associated

with liquid vaporization was observed using the temperature sensor.

133

6.10.1 Experimental Setup The NPS PDE, along with the sensor, is shown in Figure 62. The general setup is

similar to that of the water sensor. A HeNe laser is also included for the sake of

comparison against the DFG sensor. The HeNe power is modulated using a mechanical

chopper with a chop rate of 6.4 kHz. Because of the low bandwidth requirements for

collecting the HeNe signal, a thermo electrically (TE) cooled InAs detector is used for the

HeNe rather than a liquid nitrogen-cooled InSb detector. Being TE-cooled means the

detector could be mounted directly on the engine (see D1 in Figure 62). The HeNe

detector is preceded by a narrow HeNe spectral filter and a focusing lens (not shown).

The HeNe optical station is located 6.35 cm downstream of the DFG optical station.

The engine operates in the same way as for the water measurement campaign, except

the JP-10 injectors are mounted further upstream than the C2H4 injectors discussed in the

context of water sensing. This was done in order to facilitate the mixing and evaporation

of JP-10, which is injected as a liquid. Full vaporization is reasonably assumed because

the vaporization time for 10 μm droplet produced by the injectors is estimated to be 3 ms

(see Ref. 109 for evaporation model and Refs. 110 and 111 for thermodynamic/transport

properties of JP-10), while the convection time for a droplet traveling from the injector to

the laser sensor is on the order of 30 ms (for an air flow rate of 0.25 kg/s). A static T

gauge (mounted near the optical stations) is required to establish σν and a static P gauge

(mounted at the same location) is required to extract XJP-10.

134

Figure 62: NPS PDE engine with JP-10 sensors. Red lines represent freespace beam.

D1: InAs detector for HeNe; C: mechanical chopper. See Figure 60 for DFG catch optics.

6.10.2 Equivalence Ratio Measurements Figure 63(a) shows the transmitted laser intensity for both DFG and HeNe sensors and

Figure 63(b) shows the resulting equivalence ratio measurements. The engine repetition

rate was 20 Hz, with two injectors opened simultaneously. The engine was not fired.

Because the absorption coefficient of the HeNe is 4.3× greater than that of the DFG

(tuned to 2976 cm-1), the HeNe-based sensor cannot accurately measure absorption levels

at these equivalence ratios where the weak transmitted signal is dominated by detector

and bit noise. These measurements clearly demonstrate the utility of a tunable mid-IR

laser.

These time-varying equivalence ratio data were used to set the ignition timing (relative

to the injector opening time), as well to evaluate how much time is required for an

individual fuel plug to pass. For example, with an air flow of 0.25 kg/s as in Figure 63,

the fuel completely disappears in 52 ms. This sets an upper limit on repetition rate of

approximately 20 Hz. However, some fuel overlap between cycles may be tolerable and

static P and T gauges

D1 detonator (contains obstacles)

HeNe

DFG

ignitor electrode

C

vitiated

air and JP-10

wall

- 2 lenses - 2 filters - 1 detector

Catch Optics

135

will depend on how much thrust can be gained by increasing repetition rate relative to

how much fuel is wasted and at what point flame holding becomes a problem [112].

0 10 20 30 40 50 60 70

0.0

0.2

0.4

0.6

0.8

1.02006-09-20-21

HeNe

DFG(2976 cm-1)

Lase

r Tra

nsm

issi

on

Time [ms]

(a)

0 10 20 30 40 50 60 700.0

0.2

0.4

0.6

0.8

1.0

1.22006-09-20-21

DFG(2976 cm-1)

Equ

ilvan

ece

Rat

io, Φ

Time [ms]

HeNe

(b)

Figure 63: JP-10 measured in unfired NPS PDE: (a) transmitted intensity and (b) equivalence ratio. T = 477 K; P = 2.6 atm(abs); m& = 0.25 kg/s; engine

frequency = 20 Hz.

With equivalence ratio and engine filling time known, the ignition timing and

repetition rates were set. The engine was successfully fired (88% of cycles burned) with

a repetition rate of 20 Hz and an air mass flow rate of 0.20 kg/s. This represents the first

time a valveless PDE was successfully operated on JP-10 and air without any sensitizing

fuels or oxidizers.

The equivalence ratio can be measured during such a fired run in order to better

understand the dynamics of fuel flow in this harsh environment. Figure 64 shows two

engine cycles from the same run, one in which the mixture was burned and the other in

which no ignition occurred.

136

0 20 40 60 80 100

0.0

0.5

1.0

1.5

fired

Equi

vale

nce

Rat

io, Φ

Time [ms]

2006-09-21-36

unfired

Figure 64: Comparing unfired cycle and fired cycle from same run. T = 478 K; P = 1.45 atm(abs); m& = 0.11 kg/s; engine frequency = 10 Hz.

The details of Figure 64 reveal the internal gasdynamics of the PDE. The ignitor,

which is located upstream of the optical station, is discharged at t = 30 ms. Two

combustion waves then emerge: wave A moves downstream relative to the bulk air flow,

and wave B moves upstream relative to the bulk air flow. Knowing the mass flow rate of

air, as well as the static temperature and pressure, the bulk air velocity is calculated to be

~ 25 m/s. The arrival of wave A at the optical station at 35 ms allows the calculation of

flame A’s speed in the lab frame: ~ 30 m/s. This implies an average flame speed of

~ 5 m/s during this early stage of combustion and is consistent with expected turbulent

flame speeds for typical hydrocarbons. Assuming wave B is also moving at 5 m/s

relative to the bulk air flow, we find that wave B is not fast enough to move upstream in

the lab frame: it, too, moves downstream towards the optical station. This allows time for

the tail end of the fuel plug to move past the optical station before being consumed. This

motion of the tail end of the fuel plug is observed by the laser sensor, appearing between

50 and 80 ms in Figure 64 and was also observed by Klingbeil et al. in other PDEs [99].

The implication of the fuel being burned downstream (as opposed to upstream) of the

ignitor is that it could possibly escape the engine before being consumed, thereby

137

reducing specific impulse. This type of loss is prevented by the long detonator section

depicted in Figure 62.

The amount of fuel passing the optical station can be estimated as follows. The mass

of fuel is given by

∫ −− = dtmm 10JP10JP & (45)

where the fuel mass flow rate, 10JP−m& , is obtained knowing the fuel mass fraction, YJP-10,

and using the relation

( )airtotal mmYmYm &&&& +== −−−− 10JP10JP10JP10JP

Rearranging yields

airmY

Ym &&

10JP

10JP10JP 1 −

−− −

= (46)

The mass fraction is related to the molar fraction

totalW

WXY 10JP

10JP10JP−

−− = (47)

where Wi is the molecular weight of species i. We will assume that Wtotal = Wair.

Equations (46) and (47) allow the mass flow rate of fuel to be calculated knowing the

time-varying mole fraction, XJP-10. Next, Eq. (45) is used to find the total fuel mass

which has passed the optical station during a specific window of time. Using the

unburned data from Figure 64 (where airm& = 0.11 kg/s), the total mass of fuel injected is

236 mg. Knowing that two fuel injectors were used simultaneously, this agrees well the

manufacturer’s specification of 121 mg per injector. Now, using the burned data from

Figure 64, the amount of fuel passing the optical station between 50 and 80 ms is found

to be 13.5 mg. Thus, if this amount of fuel were allowed to escape the engine before

being burned, this would represent a 6% waste of fuel, and thus a 6% penalty in Isp. By

138

virtue of being able to capture the dynamics of the time-varying fuel load in the PDE, the

fuel sensor could be used to extract detailed information about the fraction of fuel which

is burned or unburned.

6.10.3 Temperature Measurements The static temperature of the unburned, vitiated reactants was measured using the

2-color strategy described previously. The engine was operated at 30 Hz, unfired. Figure

65 compares these results with those measured by the thermocouple. No filtering or

cycle-averaging has been performed. Agreement between the DFG sensor and the

thermocouple is good. The error bars represent the rms fluctuations of temperature

occurring over 5 ms, which is the time during which sufficient JP-10 is present to make a

reasonable measurement of T. (The data between 500 and 550 K should be considered

outliers, since for these runs the laser power had dropped as a result of contaminants

which had infiltrated the DFG laser.)

400 450 500 550 600 650400

450

500

550

600

650

Tem

pera

ture

from

2-c

olor

DFG

[K]

Temperature from thermocouple [K]

outliers2006-09-22

Figure 65: Comparison of reactant temperature measured by thermocouple and DFG in unfired NPS PDE. P = 1.8 ~ 2.1 atm(abs); m& = 0.5 kg/s; engine

frequency = 30 Hz.

139

Ignoring the two outliers, we see that the rms fluctuations tend to decrease as the

temperature increases. Figure 66 shows details of the temperature history for the coldest

(450 K) and hottest (603 K) runs of Figure 65. The high bandwidth of this JP-10-based

temperature sensor makes it possible to capture the fluctuations seen in Figure 66.

Recall that JP-10 is injected as a liquid upstream of the optical station. Although there

is enough residence time for the droplets to vaporize before they reach the optical station,

the droplets which experience higher temperatures are vaporized more quickly and

therefore have more time to fully mix with the air. These hot mixtures are consequently

more uniform when flowing past the optical station. The colder mixtures, on the other

hand, show evidence of temperature fluctuations and non-uniformities due to imperfect

mixing. This type of information could be used to infer the extent of mixing and

uniformity of the reactants. Such a measurement would not have been possible with the

water-based temperature sensor scanned at 5 kHz (Chapter 5).

18 19 20 21 22 23

0.6

0.8

1.0

1.2

1.4

Nor

mal

ized

Tem

pera

ture

, T /

T mea

n

Time [ms]

Tmean = 450 K Tmean = 603 K

Figure 66: Time-varying temperature for representative cold (Tmean = 450 K) and

hot (Tmean = 603 K) runs. Sensor bandwidth = 100 kHz.

140

6.11 Summary This chapter described the development of a laser-based sensor for JP-10. JP-10 is a

common jet fuel, but cannot be easily measured using the standard HeNe-based approach

because of its high level of absorption at this wavelength. In order to overcome this

obstacle, a novel tunable mid-IR laser was employed to access a weaker portion of the

JP-10 spectrum. Two strategies were implemented, one for sensing fuel (single

wavelength) and another for sensing temperature (two wavelengths). As with the

previous water sensor, the JP-10 sensor was fiber-coupled.

Measurements were made on the multi-cycle PDE at NPS. Time-varying equivalence

ratio measurements were used to determine the proper engine operating parameters

(number of injectors, pulse width, ignitor discharge). The engine was successfully run on

JP-10/air without any sensitizing fuels or oxidizers. During fired operation, the sensor

was able to resolve the motion of unburned fuel before being consumed downstream of

the optical station. The sensor revealed the average flame speed to be less than the bulk

gas speed, indicating all combustion took place downstream of the ignitor.

The temperature sensor was used to reproduce the temperature obtained by the

thermocouple. Temperature fluctuations were resolved by virtue of the sensor’s high

bandwidth. These fluctuations were observed to decrease as the temperature increased

and were attributed to incomplete mixing and non-uniformities.

141

Chapter 7: Conclusions & Future Work

7.1 Conclusions This work spanned two important areas of PDE research: modeling and laser-based

sensing. The modeling work was motivated by the need to better understand PDE

nozzles and their losses. The laser-based sensing work built on past sensor development

efforts in order to provide more accurate measurements of water and JP-10 in a real PDE.

7.1.1 Modeling One of the most valuable tools for driving PDE research forward is modeling. The

first part of this thesis attempted to use this approach to address performance-related

issues.

Ideal PDE Nozzles The unsteady flowfield of PDE nozzles was generalized with the goal of predicting

engine performance over a range of reactant states, mixtures, geometries and ambient

pressures. The results are valuable to engine designers who require a simple means of

establishing an acceptable nozzle area ratio.

In addition, the steady state flow which persists in a PDE nozzle was derived (labeled

state 4). In should also be noted that, while this thermodynamic state was not directly

used in predicting PDE performance, it nevertheless became extremely valuable at

several points throughout the modeling work of Chapter 2, Chapter 3 and Chapter 4. For

example, for both diverging and converging-diverging nozzles, state 4 can be used to

roughly choose the diverging an area ratio if the ambient pressure is known. (See dashed

line in Figure 14(b).)

But in addition to being a helpful design tool, state 4 proved to be valuable in

assessing the losses encountered in later chapters. In the case of finite-rate chemistry,

state 4 provided the appropriate nozzle inlet state for assessing the extreme cases of

142

chemically equilibrated flow and chemically frozen flow (Figure 22, Figure 23 and

Figure 26). In much the same way, state 4 provided the appropriate nozzle inlet state

when investigating Fanno and Rayleigh flow in nozzles (Figure 33). State 4 was also

used when determining the appropriate tchem for achieving chemically equilibrated flow

(Figure 24).

Finite-Rate Chemistry in PDE Nozzles Depending on the mixture and area ratio used, the effect of finite-rate chemistry can

have a major impact on overall PDE performance. A large area ratio nozzle was chosen

to test the effects of finite-rate chemistry by exploiting the low pressures present during

high altitude flight. For mixtures using O2 as the oxidizer, the associated losses can be on

the order of 10%. For air-breathing applications, though, finite-rate chemistry need not

be of concern since the losses are typically ~ 1%, owing to the small amount of chemical

recombination possible in air.

Heat Transfer due to PDE Nozzles By adding a converging nozzle to a straight-tube PDE the blowdown is retarded and

losses from the straight-tube are enhanced. It was found that for mixtures of fuel with O2

an additional ~ 5% penalty will be paid in specific impulse by adding a converging

nozzle (see Figure 30). If the mixture is fuel with air, the added penalty can be much

greater (~ 14%), owing to the longer blowdown times and lower plateau pressures for air

mixtures.

By adding a diverging nozzle, the straight-tube losses are unaffected. For this type of

nozzle, it is the large surface area of the nozzle which becomes important; meaning that if

the nozzle has a small area ratio and/or a large divergence angle, the lossess will be small.

It was found that an additional 9% loss can be expected by adding a diverging nozzle

with ε = 100 and θd = 5º. For a more realistic angle of 12º, this loss drops to 3%.

143

7.1.2 Laser-Based Sensing The second part of this thesis attempted to demonstrate the importance of laser-based

sensing to PDEs and to offer improvements in sensing strategy with the ultimate goal of

achieving proper engine operation. Specifically, alternative sensor designs using the

mid-IR were explorerd and the benefits they offered were showcased by measurements

on the NPS PDE.

Water Sensing In the case of water, the fundamental mid-IR band is at least 10× stronger than the

combination bands commonly accessed with telecom lasers and therefore offers

improved sensitivity and SNR. The strategy was realized by the development of a 2-

wavelength scanned direct absorption water sensor based at 2.5 μm. This sensor

consisted of 2 freespace DFB lasers, a pitch fiber, a catch fiber, detector and other optical

components. The sensor was implemented on the NPS PDE, which has a high moisture

content in the reactant stream due to a vitiator. The sensor provided improved SNR in

measuring transient temperature during unsteady engine fueling. The sensor was also

used to ascertain the amount of time needed to purge water from the engine after the

vitiator was shut-off. This information showed a clear link between water and poor

performance of the engine’s ignitor. With water removed, the ignitor performs as

designed and zero misfires were observed. Based on these results, the next generation

vitiator is being designed at NPS to produce as little water as possible.

JP-10 Sensing The JP-10 sensor was also based on mid-IR wavelengths. In order to improve SNR

beyond what was possible with a HeNe laser, a tunable laser was employed which

enabled the optimal level of absorbance to be achieved. Furthermore, by adopting a

strategy of switching between various lasers, a 2-wavelength temperature sensor with a

bandwidth of 100 kHz was developed. Like the water sensor, the JP-10 sensor was fiber-

144

coupled and implemented on the NPS PDE. Results showed significant improvement in

quantitative measurements of JP-10 over what was achieved using the HeNe. The sensor

was used to guide engine operation, and as a result the PDE was run successfully on

JP-10/air for the first time. Temperature was also measured. The sensor’s high

bandwidth was able to resolve high frequency fluctuations, which became amplified as

the engine’s temperature was lowered. This was attributed to poor mixing of fuel and air

at these low temperatures.

These two sensors demonstrated that laser-based sensing is a powerful tool in the

context of PDE development. Their capacity to perform in situ sensing with high

bandwidth and high SNR makes them the ideal solution for diagnosing the harsh

environment found inside PDEs.

7.2 Future Work The modeling work presented herein focused on research type fuels, such as H2 and

C2H4. A natural extension, therefore, would be to re-evaluate the losses (finite-rate

chemistry, heat transfer, friction) for more practical fuels such as JP-10 or JP-8. Studying

finite-rate chemistry would be especially interesting because the chemical timescales of

large hydrocarbons can be significantly larger than those of H2 or C2H4.

It was found that both finite-rate chemistry and heat transfer can be important to

nozzle design. We should note that finite-rate chemistry can be mitigated by extending

the nozzle length (thereby providing more time for chemical recombination), whereas

heat/friction losses can be mitigated by shortening the nozzle length (thereby reducing the

surface area available for these types of losses). This suggests an optimum nozzle length

should exist, which could be explored in the future.

The model results of losses induced by a converging nozzle presented here could be

checked experimentally by fitting a straight-tube PDE with a nozzle and measuring the

resulting change in integrated thrust. This test could be repeated for a range of Ls/Ds

ratios, thereby experimentally generating Figure 29. (Experimentally producing a

detonation wave which is not affected by water condensation has been a problem in the

145

past [71]. This could possibly be overcome by detonating a mixture of CO and H2. The

CO combustion produces no water, but is very difficult to detonate. A small amount of

H2 sensitizes the CO to make it more detonable [113]. If this can be done without

excessive amounts of H2, water condensation should be mitigated.)

A clear extension of fuel sensing would involve JP-8. This fuel is becoming more

relevant to PDE researchers as the engine moves to fill the needs of aircraft propulsion,

rather than missile propulsion. A multi-wavelength scheme will likely be required

because JP-8 is a multi-component fuel and so the sensor will need to be sensitive to the

range of compositions that exist from batch to batch.

147

Appendix A: NASA Polynomials for

Achieving Constant-γ Equation of State It is often valuable to simulate an isentropic gasdynamic process which follows the

commonly-used equation of state:

const=γρP (48)

where P is the static pressure, ρ is the static density, and γ is the polytropic exponent

(assumed to be constant).

The above equation of state can be indirectly incorporated into a numerical solver

designed to solve chemical mechanisms. This can be accomplished by contriving the

NASA polynomials found in the thermo file that accompanies the chemical mechanism.

Recall the thermo file contains coefficients for the following polynomials [114, § 2.3.4]:

45

34

2321ˆ

ˆTaTaTaTaa

c p ++++=R

(49)

Ta

Ta

Ta

Ta

Ta

aTh 64534232

1 5432ˆˆ

+++++=R

(50)

74

53

42

321 )ln(ˆˆ

aTaTaTaTaTasr +++++=R

(51)

where T is the temperature, pc is the specific heat at constant pressure in J/mole·K, R is

the universal gas constant in J/mole·K, h is the enthalpy in J/mole, and rs is the entropy

at reference pressure in J/mole·K. (Each species of the thermo files actually contains two

sets of a1 through a7, one for the low temperature range and one for the high temperature

range.) By properly choosing the coefficients a1 through a7, polynomials can be obtained

which result in the equation of state (48).

148

A.1 Single Species If the problem consists of a single species, the solution is quite straightforward. Since

γ is to be constant, pc must also be a constant. Thus, a2 = a3 = a4 = a5 = 0. Now, we use

the well-known relation between pc and γ :

1ˆ

ˆ

−=

γγ

Rpc

which directly yields:

11 −

=γ

γa

Coefficient a6 and a7 are now found from Eqs. (50) and (51):

Taha 16 ˆˆ

−=R

)ln(ˆˆ

17 Tas

a r −=R

So, once the state is specified (in terms of h , γ, T and rs ), the polynomial coefficients

can be determined. If the desired state has a pressure different than the reference pressure,

the desired entropy must be transformed to the reference state. Furthermore, since we

desire temperature-independence, the coefficients for the low and high ranges should be

identical.

A.2 Mixture In addition to defining the polytropic exponent γ, it may also be necessary to specify

the molecular weight, W. In the case of a single species, e.g. N2, the molecular weight is

specified directly in the thermo file by indicating that two nitrogen atoms make up this

149

particular species. The CHEMKIN interpreter, having information on the weight of the

nitrogen atom, then is able to calculate the molecular weight of this species, N2.

However, when a mixture is to be modeled, it is unlikely that the mixture molecular

weight can be matched by a single species. For example, if we are trying to capture the

gasdynamics behind a C2H4/O2 detonation wave where the mixture molecular weight is

22.647 kg/kmol, we cannot obtain this molecular weight with a single species. One

solution is to use two species with disparate molecular weights and ‘tune’ their respective

mole fractions in order to match the desired mixture molecular weight. The first species

is designated by a pre-subscript 1 and the second species by a pre-superscript 2. Thus,

there are fourteen coefficients to determine: 1a1 through 1a7, and 2a1 through 2a7. As with

the pure species case, the low and high temperature ranges will have identical coefficients.

A.2.1 Derivation

As was the case for the pure species, coefficients 2 through 5 are set to zero. Since h

and pc are both linear combinations of h and pc of species 1 and 2, we can treat

coefficients a1 and a6 equally for both species, i.e.

111211 −

===γ

γaaa (52)

Tahaaa 166261 ˆˆ

−===R

(53)

However, since entropy is not a linear combination of the entropies of species 1 and 2,

this takes more care. The mixture entropy is calculated as follows:

( )∑∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−== k

rkrkkk X

PPXsXss lnˆlnˆˆˆˆ , RR

150

where s is the mixture entropy in J/mole·K, ks is the entropy of pure species k in

J/mole·K, rks ,ˆ is the entropy of pure species k at the reference pressure, Xk is the mole

fraction of species k, R is the universal gas constant, P is the pressure and Pr is the

reference pressure. For the CHEMKIN interpreter, Pr = 1 atm.

For our case of two species,

( ) ( ) 22,211,1 lnˆlnˆˆlnˆlnˆˆˆ XPPXsX

PPXss

rr

rr ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−= RRRR

Rearranging,

( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++−+=

rrr P

PXXXXXsXss lnlnlnˆˆˆˆ 22112,11,1 R

Now use Eq. (51) to eliminate rs ,1ˆ and rs ,2ˆ

[ ] [ ] ( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++−+++=

rPPXXXXXaTaXaTas lnlnln)ln()ln(ˆ

ˆ221127211711R

recalling that 1a1 = 2a1 = a1. The only unknowns now are 1a7, 2a7, X1 and X2. The relation

between 1a7 and 2a7 an be made arbitrarily, so we make them equal: 1a7 = 2a7 = a7.

Solving for a7:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+++−=

rPPXXXXTasa lnlnln)ln(ˆ

ˆ221117 R

(54)

Now all that remains is to find X1 and X2. This is done by using the desired mixture

molecular weight, W:

2211 WXWXW +=

Recognizing that the mole fractions must sum to unity, we get:

151

21

21 WW

WWX−−

= (55)

12 1 XX −= (56)

A.2.2 Example As an example, let us take the CJ state of a C2H4 + 3O2 detonation (reactants at 298 K

and 1 atm). This yields (using values in terms of kg rather than moles):

h = 2.4225 MJ/kg

T = 3938.38 K

P = 33.502 atm

W = 22.647 kg/kmol

s = 1.1701×104 J/kg·K

γ = 1.14

where γ is the equilibrium isentropic polytropic exponent (obtained from the equilibrium

sound speed), not the ratio of specific heats.

We choose atomic hydrogen, H, for species 1 and atomic argon, Ar, for species 2.

This is done in order to have a wide range of ‘tunability’ when specifying mixture

molecular weight.

Using Eqs. (55) and (56):

X1 = 0.4443

X2 = 0.5557

Using Eqs. (52), (53) and (54):

a1 = 8.1429

a6 = -2.5471×104

a7 = -32.713

152

All other coefficients are zero. This result is implemented as follows:

The thermo file must contain two species (H and Ar) with the above coefficients.

The low and high temperature ranges must be identical.

The chemical mechanism must contain no reactions.

The code which uses the above thermo and mechanism files must contain a mixture

of H and Ar with the calculated X1 and X2

This strategy offers a huge computational saving because 1) there are no reaction rates

to compute and 2) there are only two species to solve for. This represents a large saving

over running a fully reacting simulation involving many species if we know in advance

that the mixture can be modeled by Eq. (48).

153

Appendix B: Derivation Details for

Quasi-1D Nozzle Model

B.1 Deriving Eq. (11) Starting with the nozzle force equation:

)()()( ,,,,,, ∞∞ −−−+−= PPAPPAuumF inineneninennn &

we recast the mass-flow rate as An,iρn,iun,i and recognize that un,i = cn,i and that

un,e = Mn,ecn,e:

)()()( ,,,,,,,,,, ∞∞ −−−+−= PPAPPAccMcAF ininenenineneninininn ρ

Next, factor out An,i:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−= ∞∞ PPPP

cc

McAF inenin

eneninininn ,,

,

,,

2,,, 1 ερ

We wish to replace all densities and sound speeds by pressures. Using the isentropic

relations (see § C.1):

1

2

333

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

γγγ

ρρ

cc

PP

we obtain:

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= ∞∞

−−

PPPPPP

MPP

cPP

AF inenin

enen

inininn ,,

21

,

,,

1

3

,23

1

3

,3, 1 ερ

γγ

γγ

γ (57)

154

The parameter B is defined as the ratio of static pressure at the nozzle exit to the nozzle

inlet:

( ) 12

,

1

2,2

12

1

,

,

1+

−−

−

+

=⎟⎟⎠

⎞⎜⎜⎝

⎛

+=≡ γ

γγγ

γ

γ

ε enenin

en MMP

PB (58)

Next, substitute B into Eq. (57) and eliminate ρ3 and c3 using P3:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∞∞

−

PPPBPBMPP

PAF ininenin

inn ,,2

1

,3

,3, 1 εγ γ

γ

Group the expression into a Pn,i term and a P∞ term:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−= ∞

−

11121

,,, εεγ γγ

PBBMPAF enininn

Eliminate B with Eq. (58) and replace Pn,i with Ps,e:

( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ −−⎟

⎠⎞

⎜⎝⎛ −+−= ∞+

−+−

− 1112

,11

,,,, εεεγεγ γγ

γγ

PMMMPAF enenenesinn

By defining G1 and G2:

( ) ( ) 112

,11

,,1 −−+≡ +−

+−

− γεεεγ γγ

γγ

enenen MMMG

12 −≡ εG we recover Eq. (11):

( )21,, GPGPAF esinn ∞−=

155

B.2 Deriving Eq. (14) Start with conservation of mass:

mdtdm

&−=

On the right side, recast the mass-flow rate as Asρs,eus,e and recognize that us,e = cs,e. On

the left side, express m using volume and density:

esesses

s cAdt

dV ,,

, ρρ

−=

Replace Vs/As with Ls and express all thermodynamic variables in terms of state 4:

444

,

4

,

4

,4 c

cc

dtdL eseses

s ρρρ

ρρ

ρ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

Eliminate sound speed using the isentropic relations derived in § C.1:

4

21

4

,

4

, cdtdL eses

s

+

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛γ

ρρ

ρρ

Rearrange:

dtLcd

s

eses 4

4

,2

1

4

, −=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

ρρ

ρρ

γ

Since the initial density is ρ4 we integrate ρs,e/ρ4 from 1 to ρs,e/ρ4 and t from t0 to t:

1

2

04

4

, )(2

11−

−

⎥⎦

⎤⎢⎣

⎡ −−+=

γγρρ

s

es

Lttc

156

Express as pressure using isentropic relation to recover Eq. (14):

1

2

04

4

, )(2

11−

−

⎥⎦

⎤⎢⎣

⎡ −−+=

γγ

γ

s

es

Lttc

PP

B.3 Ω1, Ω2, and Π Impulse from the nozzle blowdown is obtained by integrating nozzle force from the

start of blowdown until the time when a shock appears at the nozzle exit

(t = t1 + t4 + t5 + t6). This occurs when

12

12,

, +−+

= ∞ γγγ

enen M

PP

Given the expression for Ps,e (Eq. (14))

1

2

04

4

, )(2

11−

−

⎥⎦

⎤⎢⎣

⎡ −−+=

γγ

γ

s

es

Lttc

PP

it is straightforward to show that t6 is given by

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛Π

−=

−−

∞ 11

2 21

346

γγ

γ PP

cLt s (59)

where

( ) ( )12

12

1, 21

212

,

,

,

4

3

+−+

⎟⎠⎞

⎜⎝⎛ +

=≡Π +−

∞ γγγεγγε γ

γγγ

nene

en

en

in

MM

PP

PP

PP

157

and where the ratio of nozzle exit to inlet pressures (denoted B) was used:

( ) 12

,

1

2,2

12

1

,

,

1+

−−

−

+

=⎟⎟⎠

⎞⎜⎜⎝

⎛

+=≡ γ

γγγ

γ

γ

ε enenin

en MMP

PB

The parameter Π is called the shock factor and is plotted in Figure 67.

1 10 1001

10

100

γ = 1.24

Sho

ck F

acto

r, Π

Exit Area Ratio, ε

γ = 1.14

Figure 67: Shock factor.

With t6 in hand, we can now integrate the nozzle force to obtain nozzle impulse

(corresponding to nozzle blowdown):

∫+

= 60

0

tt

t nn dtFI

Using the expressions for pressure at the straight-tube exit (Eq. (14)) and nozzle force

(Eq. (11)):

158

1

2

04

4

, )(2

11−

−

⎥⎦

⎤⎢⎣

⎡ −−+=

γγ

γ

s

es

Lttc

PP

( )21,, GPGPAF esinn ∞−=

we obtain

∫+

∞

−−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−⎥

⎦

⎤⎢⎣

⎡ −−+= 60

0214

12

04,

)(2

11tt

ts

inn dtGPGPL

ttcAIγ

γ

γ

Evaluate the integral:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡ −+−

+= ∞

−+

−

62

11

64

414, 2

1111

2 tGPLtc

cLGPAI

s

sinn

γγ

γγ

Multiply by Ls/c3 and express P4 using P3:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡ −+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

= ∞

−+

−−

ss

sinn L

ctGPLtc

ccGP

cLA

I 362

11

64

4

31

2

133

,

2111

12

12 γ

γγ

γ

γγγ

(60)

Focus on the P3 term of Eq. (60). Define Ω1:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡ −+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

≡Ω−+

−− 1

1

64

4

31

2

1 2111

12

12 γ

γγ

γ

γγγ sL

tccc

Substituting the expression for t6, Eq. (59), we obtain:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛Π+−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

=Ω

−+

−−−

∞−

11

21

34

31

2

1 1111

21

2γγ

γγ

γγ

γγ PP

cc

159

which simplifies to

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛Π−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=Ω

+

∞− γ

γγ

γ

γ

21

3

12

1 11

2PP

which is the form of Ω1 as it was defined in Eq. (16). Now focus on the P∞ term of

Eq. (60), specifically the quantity t6c3/Ls:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛Π

−=

−−

∞ 11

2 21

34

336γ

γ

γ PP

cc

Lct

s

Eliminate c3/c4 using Eq. (6) and define the result as Ω2:

2

21

3

36 111

Ω≡⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛Π

−+

=

−−

∞γ

γ

γγ

PP

Lct

s

Rewrite Eq. (60) with Ω1 and Ω2, and replace An,iLs with Vs:

( )221133

Ω−Ω= ∞GPGPcVI s

n

which is Eq. (15), the nozzle impulse from blowdown.

161

Appendix C: Isentropic Relations for

Chemical Equilibrium Flow

The quasi-1D isentropic models discussed in Chapter 2 rely on simple analytic

expressions when the equation of state is assumed to be

const=γρP (61)

It is well known that for chemically frozen flow, this equation of state leads to

the isentropic relations,

rrrr P

PTT

cc

ρργγγ

=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−1

11

12

, ( )22

11

rr cchh −−

=−γ

the stagnation states,

21

20

211 ⎥⎦

⎤⎢⎣⎡ −

+= Mcc γ , ⎥⎦

⎤⎢⎣⎡ −

+= 20

211 M

TT γ ,

120

211

−

⎥⎦⎤

⎢⎣⎡ −

+=γγ

γ MP

P , 1

1

20

211

−

⎥⎦⎤

⎢⎣⎡ −

+=γγ

ρρ M

the choke states,

21

0

*

12

⎥⎦

⎤⎢⎣

⎡+

=γc

c , ⎥⎦

⎤⎢⎣

⎡+

=1

20

*

γTT ,

1

0

*

12 −

⎥⎦

⎤⎢⎣

⎡+

=γγ

γPP ,

11

0

*

12 −

⎥⎦

⎤⎢⎣

⎡+

=γ

γρρ

162

and

the area relation,

( )121

2

*

21

2111

−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

=

γγ

γ

γ M

MAA

It will be shown here that for chemically equilibrated flow, all of the above relations hold

except for those containing temperature.

C.1 Isentropic Relations Start with the conservation of stagnation enthalpy:

22

22

2

21

1uhuh +=+ (62)

Appeal to the Gibbs equation for a mixture which may include chemical reactions

[115, § 8-2]:

ηρ

ddPTdsdh Α++=

where A is the affinity and η is the extent of reaction. Recognizing that the flow is

isentropic (ds = 0) and either frozen (dη = 0) or in chemical equilibrium (A = 0)

[115, §10-6], we obtain:

ρ

dPdh = (63)

Next we appeal to the definition of the sound speed:

163

ρρ d

dPPcs

=∂∂

≡2 (64)

where the differentiation is along either a frozen or equilibrium isentrope, depending on

the type of flow. Combining (61) and (64) we get:

2cP=

ργ (65)

Combining (61) and (65) we get:

rrr P

Pcc

ρργγ

=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −1

12

(66)

where r denotes a reference state.

These isentropic relations allow us to find an additional relation between h and c.

First, find the derivative of P with respect to c:

1

2−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

γγ

rr cc

PP

1

121

2

11

2 −−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= γγγ

γ

γγ c

cP

dcdP

rr

11

2

12 −

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= cccP

dcdP

rr

γγ

γγ

1

12 −

−= c

PPP

dcdP

rr γ

γ

cP

dcdP

12−

=γ

γ

164

Now use Eq. (63) to eliminate dP:

cP

dcdh

12−

=γ

γρ

Use Eq. (65) to eliminate ρ:

cP

dcdh

cP

12

2 −=

γγγ

cdcdh1

2−

=γ

Integrating:

( )22

11

rr cchh −−

=−γ

(67)

where again r denotes a reference state. Equations (66) and (67) are the desired

isentropic relations. Notice that temperature has not appeared in the derivation and its

isentropic relations to c, ρ, and P have been purposely left out. See § C.4 for a discussion.

C.2 Stagnation and Choke States With this equation of state (Eq. (67)), eliminate h from the energy equation (Eq. (62)):

( ) ( )21

121

1 2222

2

2122

1ucchucch rrrr +−

−+=+−

−+

γγ

22

22

21

21 2

12

1 ucuc −+=

−+

γγ (68)

Equation (68) can be used to find both the stagnation and choke states. As with frozen

flow, the stagnation state for equilibrium flow is simply the state that exists if the flow is

decelerated isentropically. Since Eq. (68) incorporates the isentropic assumption, it can

be applied for both frozen and equilibrium flow. Setting u1 = 0, we get:

165

21

20

211 ⎥⎦

⎤⎢⎣⎡ −

+= Mcc γ (69)

where 0 denotes the stagnation state. Similarly for pressure and density:

12

0

211

−

⎥⎦⎤

⎢⎣⎡ −

+=γγ

γ MP

P (70)

1

1

20

211

−

⎥⎦⎤

⎢⎣⎡ −

+=γγ

ρρ M (71)

The same strategy can be applied for the choke state. Once again using Eq. (68) and

setting u2 = c2 we obtain and u1 = 0:

21

0

*

12

⎥⎦

⎤⎢⎣

⎡+

=γc

c (72)

1

0

*

12 −

⎥⎦

⎤⎢⎣

⎡+

=γγ

γPP (73)

1

1

0

*

12 −

⎥⎦

⎤⎢⎣

⎡+

=γ

γρρ (74)

where * represents the choke state. Equations (69) – (74) are identical to the stagnation

and choke expressions derived for frozen flow. Notice that temperature, once again, has

been purposely left out. See § C.4 for a discussion.

C.3 Area Relation We begin by combining Eqs. (69) and (72) to obtain c*/c:

166

21

2*

21

211

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

=γ

γ M

cc

Next we use the definition of Mach number on the left-hand side:

21

2

*

*

21

211

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

= γ

γ M

uM

Mu

Next we set M * = 1 and eliminate u and u* using conservation of mass:

21

2

**

21

211

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

= γ

γ

ρρ M

MAA

Now solve for A/A*

21

2*

*

21

2111

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

= γ

γ

ρρ M

MAA

Eliminate ρ*/ρ using Eqs. (71) and (74):

21

211

2

*

21

2111

21

211

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

=

−

γ

γ

γ

γ γM

M

M

AA

Simplify:

167

( )121

2

*

21

2111

−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−+

=

γγ

γ

γ M

MAA (75)

Once again, we recover the standard result normally derived for frozen flow.

C.4 Discussion on Temperature Notice that the specific heat cp, temperature T, and mass-based gas constant R were

never used in the above derivation. This is paramount in being able to treat chemical

equilibrium flow because the mixture-based gas constant R is constantly changing in

this type of flow. This is why neither static nor stagnation temperature was mentioned in

the preceding sections. The changing R in equilibrium flow would lead to severe errors

in temperature if the equation of state ( ) γγ 1−∝ PT is used.

For example, consider the chemical equilibrium expansion from the CJ state of

C2H4 + 3O2 (starting at STP). The polytropic exponent at this state derived from Eq. (65)

is γ = 1.14. The density through the expansion can be calculated using STANJAN [18]

for various pressures, and then compared to the predicted value obtained from Eq. (66).

We see from Figure 68 that the prediction is quite good. Any deviation between the

predicted and the actual density is due to small changes in γ over the range of pressures

considered.

168

0.01 0.1 1 101E-3

0.01

0.1

1

Actual Predicted

CJD

ensi

ty [k

g/m

3 ]

Pressure [bar]

Figure 68: Comparison of predicted and actual density as a function of pressure. Predicted values obtained using Eq. (66), γ = 1.14. Reference state taken as CJ of

C2H4 + 3O2 detonation, P1 = 1 atm, T1 = 298 K.

If the same analysis is done to predict temperature from pressure in the same

chemically equilibrated expansion using the well-known isentropic relation derived for

frozen flow:

γ

γ 1−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

rr PP

TT (76)

we obtain vastly diverging results, as shown in Figure 69. We see that temperature must

be treated very carefully when dealing with chemically equilibrated flow. Using an

equilibrium solver (e.g. STANJAN) is the safest way to treat temperature.

169

0.01 0.1 1 101000

2000

3000

4000

5000

Actual Predicted

CJ

Tem

pera

ture

[K]

Pressure [bar]

Figure 69: Comparison of predicted and actual temperature as a function of pressure. Predicted values obtained using Eq. (76), γ = 1.14. Reference state taken

as CJ of C2H4 + 3O2 detonation, P1 = 1 atm, T1 = 298 K.

C.5 Conclusions If we can find an exponent γ which satisfies Eq. (61), we can develop this simple

isentropic theory for either frozen or equilibrium flow. In either case, γ, c, and M must be

treated properly (viz. as frozen or equilibrium) and must be consistent with each other.

Pressure, density and sound speed are computed directly. Temperature must be

calculated via an equilibrium solver, such as STANJAN [18] or CEA [19].

Finally, it should be pointed out that although special attention was given to the

stagnation state, the choked state, and the area relation (all of which are heavily used in

steady nozzle calculations), the equations used for deriving these relations are also the

basis for more sophisticated time-resolved isentropic constant-γ models (e.g. method of

characteristics). So, the above conclusions and discussion apply not only to steady

analytic nozzle calculations, but also to complex unsteady quasi-1D flows.

171

Appendix D: Chemical Mechanisms

D.1 H2 / O2 (frozen N2) This mechanism is taken from GRI-Mech 3.0 [61] with the H + O2 + M reaction rate

updated for high pressures using Ref. 62. Molecular nitrogen is included, but nitrogen

chemistry is left out. This mechanism is referred to as “H2” throughout this work

!David Davidson Stanford 650-725-2072 [email protected]

!This is GRI-Mech v3.0 with only the H/O reactions included.

!Note that we now have accurate values for H+O2+M=HO2+M

!in particular some pressure dependence for this reaction

!which is included here (but not in GRI-Mech v3.0)

!June 14, 2006

ELEMENTS

O H N

END

SPECIES

H2 H O O2 OH

H2O HO2 H2O2 N2

END

REACTIONS

2O+M<=>O2+M 1.200E+17 -1.000 .00

H2/ 2.40/ H2O/15.40/

O+H+M<=>OH+M 5.000E+17 -1.000 .00

H2/2.00/ H2O/6.00/

O+H2<=>H+OH 3.870E+04 2.700 6260.00

O+HO2<=>OH+O2 2.000E+13 .000 .00

O+H2O2<=>OH+HO2 9.630E+06 2.000 4000.00

!

! note pressure dependence here based on Bates et al. 2001 (+Ar) rates

H+O2(+M)<=>HO2(+M) 9.040E+12 -.200 .00

LOW / 6.800E+18 -1.200 00.00/

TROE/ 0.3 100000.00 0.01 100000.00 /

H2/2.600/ H2O/20.00/ N2/ 3.3/ O2/ 1.25/

!

!Below is old GRI-Mech v3.0

!H+O2+M<=>HO2+M 2.800E+18 -.860 .00

172

!O2/ .00/ H2O/ .00/ N2/ .00/

!H+2O2<=>HO2+O2 2.080E+19 -1.240 .00

!H+O2+H2O<=>HO2+H2O 11.26E+18 -.760 .00

!H+O2+N2<=>HO2+N2 2.600E+19 -1.240 .00

!H+O2+AR<=>HO2+AR 7.000E+17 -.800 .00

!

H+O2<=>O+OH 2.650E+16 -.6707 17041.00

2H+M<=>H2+M 1.000E+18 -1.000 .00

H2/ .00/ H2O/ .00/

2H+H2<=>2H2 9.000E+16 -.600 .00

2H+H2O<=>H2+H2O 6.000E+19 -1.250 .00

H+OH+M<=>H2O+M 2.200E+22 -2.000 .00

H2/ .73/ H2O/3.65/

H+HO2<=>O+H2O 3.970E+12 .000 671.00

H+HO2<=>O2+H2 4.480E+13 .000 1068.00

H+HO2<=>2OH 0.840E+14 .000 635.00

H+H2O2<=>HO2+H2 1.210E+07 2.000 5200.00

H+H2O2<=>OH+H2O 1.000E+13 .000 3600.00

OH+H2<=>H+H2O 2.160E+08 1.510 3430.00

2OH(+M)<=>H2O2(+M) 7.400E+13 -.370 .00

LOW / 2.300E+18 -.900 -1700.00/

TROE/ .7346 94.00 1756.00 5182.00 /

H2/2.00/ H2O/6.00/

2OH<=>O+H2O 3.570E+04 2.400 -2110.00

!

OH+HO2<=>O2+H2O 1.450E+13 .000 -500.00

DUPLICATE

OH+HO2<=>O2+H2O 0.500E+16 .000 17330.00

DUPLICATE

!

OH+H2O2<=>HO2+H2O 2.000E+12 .000 427.00

DUPLICATE

OH+H2O2<=>HO2+H2O 1.700E+18 .000 29410.00

DUPLICATE

!

2HO2<=>O2+H2O2 1.300E+11 .000 -1630.00

DUPLICATE

2HO2<=>O2+H2O2 4.200E+14 .000 12000.00

DUPLICATE

END

173

D.2 Fully-Reversible Varatharajan (frozen N2) This mechanism is taken from Ref. 63. All reactions are made reversible. As a result,

the following reactions become redundant and are removed: #2, #5, #13, #15 and #31

(where numbers correspond to the convention of Ref. 63). Molecular nitrogen is

included, but nitrogen chemistry is left out. This mechanism is referred to as “FRV”

throughout this work.

! Center for Energy Research Mechanism 02/09/2001

! Thermodata and Transport data file attached

! Questions and comments email [email protected]

! TROE Parameters in the order a, T***, T*, T**

ELEMENTS

O H C N

END

SPECIES

C2H4 O2 OH H2O O H HO2 H2O2 CO CO2

HCO CH2O CH3 H2 C2H5 C2H2 C2H3 CH2CHO CH2CO

C2H4O N2

END

REACTIONS

H+O2<=>OH+O 3.520e+16 -0.700 17053.6

H2+OH<=>H2O+H 1.170e+09 1.300 3630.46

H2O+O<=>2OH 7.600e+00 3.840 12778.26

H+O2+M<=>HO2+M 2.600e+19 -1.200 0.00

N2/1.00/ O2/0.30/ H2O/7.00/ CO/0.75/ CO2/1.50/

HO2+H<=>2OH 1.700e+14 0.000 883.73

HO2+H<=>H2+O2 4.280e+13 0.000 1409.2

HO2+OH<=>H2O+O2 2.890e+13 0.000 -501.58

2HO2<=>H2O2+O2 3.020e+12 0.000 1385.30

H2O2(+M)<=>2OH(+M) 2.550e+20 -1.680 52331.14

N2/1.00/ H2/2.00/ H2O/6.00/ CO/1.50/ CO2/2.00/

LOW / 7.940e+24 -2.210 50635.33/

TROE/ 0.735 94.0 1756.0 5182.0 /

H+OH+M<=>H2O+M 2.200e+22 -2.000 0.00

N2/1.00/ O2/0.30/ H2O/7.00/ CO/0.75/ CO2/1.50/

CO+OH<=>CO2+H 4.400e+06 1.500 -740.42

C2H4+O2<=>C2H3+HO2 4.220e+13 0.000 57561.86

C2H4+OH<=>C2H3+H2O 5.530e+05 2.310 2961.689

174

C2H4+O<=>CH3+HCO 2.250e+06 2.080 0.00

C2H4+O<=>CH2CHO+H 1.210e+06 2.080 0.00

C2H4+HO2<=>C2H4O + OH 2.230e+12 0.000 17173.02

C2H4+H<=>C2H3+H2 4.490e+07 2.120 13351.49

C2H4+H(+M)<=>C2H5(+M) 1.080e+12 0.450 1815.23

LOW / 1.900e+35 -5.57 5039.648 /

TROE/ 0.832 1e-50 1203.0 1e50/

C2H4+M<=>C2H3+H+M 2.600e+17 0.000 96493.74

C2H3+H<=>C2H2+H2 1.210e+13 0.000 0.00

C2H3+O2<=>CH2O+HCO 1.700e+29 -5.31 6496.61

C2H3+O2<=>CH2CHO+O 7.000e+14 -0.611 5254.61

CH3+O2<=>CH2O+OH 3.300e+11 0.000 8932.84

CH3+O<=>CH2O+H 8.430e+13 0.000 0.00

CH2CHO<=>CH2CO+H 1.05e+37 -7.19 44425.34

C2H5+O2<=>C2H4+HO2 2.000e+12 0.000 4991.88

CH2CO+H<=>CH3+CO 1.110e+07 2.000 2006.31

CH2O+OH<=>HCO+H2O 3.900e+10 0.890 406.04

HCO+M<=>CO+H+M 1.860e+17 -1.000 16981.94

H2/1.90/ H2O/12.00/ CO/2.50/ CO2/2.50/

HCO+O2<=>CO+HO2 3.000e+12 0.000 0.00

HCO+H<=>CO+H2 1.000e+14 0.000 0.00

C2H2+OH<=>CH2CO+H 1.900e+07 1.700 1003.15

C2H4O+HO2<=>CH3+CO+H2O2 4.000e+12 0.000 17005.83

END

175

Appendix E: Laser Collimation The lasers used for the water sensor (Chapter 5) were fabricated without a collimating

lens mounted. A lens for each laser therefore had to be chosen and aligned. This

appendix describes the options which were investigated.

Two general strategies for collimating are considered, i.e. using a lens or a mirror, as

depicted in see Figure 70. Notice that the lens/mirror must be placed at a distance from

the laser equal to its focal length, f. Also, the laser emits with a particular solid angle,

which means that the laser power will only be completely captured by the lens/mirror if

the lens/mirror has a sufficiently large diameter, D. This is commonly referred to as

“matching the f/# ” of the laser. The f/# of a lens/mirror is defined as f/# ≡ f/D. The

diode laser also has an f/# which is uniquely determined by the solid angle of its

diverging beam. The laser f/# is defined in Figure 71. If the f/# of the laser is smaller

than that of the lens or mirror, some power is lost. If the f/# of the laser is equal to or

larger than that of the lens or mirror, all of the power is collected and collimated. A

typical diode laser f/# is less than 1. The two following guidelines are therefore

recommended:

In order to collect all of the laser power, we desire a lens/mirror with an f/# no

greater than 1.

Since the diameter of the resulting collimated beam is proportional to the focal

length, we desire a small f.

176

Figure 70: Collimating a diode laser using a lens (top) or mirror (bottom).

Figure 71: Defining f/# for the diode laser: f/# ≡ y/x.

One mirror and three lenses were tested in order to find the best scheme. In addition

to requiring a small f/# and small f, the lens/mirror needed to have high

transmission/reflectivity. The four tested components tested are listed in Table 9.

ray at edge

of beam

y

x

ray collected by lens

ray not collected by lens

lens

mirror

f

f

ray not collected by mirror

ray collected by mirror

177

Table 9: Laser collimation strategies tested.

Label Material Transmission/ reflectivity @

2.5 μm Shape f

[mm] D

[mm] f/# P/N

Mirror Silver >90% Parabolic 25 ~25* ~1 (unknown)

Lens #1 BK7 60% Bi-

convex 4.5 5 0.9 Melles Griot 01 LDX 401

Lens #2 CaF2 95% Plano-

convex 12 12 1 (unknown)

Lens #3 LaSFN9 80% Plano-

convex 1.2 2 0.6 Melles Griot 01 LPX 407

* The mirror has an elliptical outline so the diameter is an average of the major and minor axes.

Three metrics are used to choose the collimator: 1) quality (i.e. shape) of the beam;

2) width of the beam; and 3) power in collimated beam. The first two metrics were

measured using the setup shown in Figure 72. A chopper is used as a rotating knife-edge

in order to determine the profile of the beam [116]. The chopper is moved to different

locations between the laser and the detector in order to determine how the beam profile

evolves.

The third metric listed above (power in collimated beam) is addressed by recording the

unchopped detector signal. This signal is reported here in volts.

Figure 72: Setup for measuring laser beam profile.

chopper Lchopper

~ 45 cm

lens/mirror being tested

178

The results are shown in Figure 73 through Figure 76. The legends show Lchopper in

centimeters. The relative power is included inside each figure and labeled ‘Signal’. Two

panels are included in Figure 73 for the mirror because this mirror is elliptic in shape and

therefore generates a non-axisymmetric beam.

-15 -10 -5 0 5 10 15

Signal: 4 V

7.5 cm 25 cm 42.5 cm

Nor

mal

ized

Pow

er

x [mm]

(a)

-60 -50 -40 -30 -20 -10 0 10 20 30

(b) 7.5 cm 25 cm 42.5

Nor

mal

ized

Pow

er

x [mm]

Signal: 4 V

Figure 73: Beam profile using mirror. (a) vertical axis, (b) horizontal axis.

-5 -4 -3 -2 -1 0 1 2 3 4 5

42.5 cm 25 cm 5 cm

Nor

mal

ized

Pow

er

x [mm]

Signal: 1.5 V

Figure 74: Beam profile using lens #1.

179

-5 -4 -3 -2 -1 0 1 2 3 4 5

42.5 cm 25 cm 5 cm

Nor

mal

ized

Pow

er

x [mm]

Signal: 2.0 V


-5 -4 -3 -2 -1 0 1 2 3 4 5

40 cm 25 cm 5 cm

Nor

mal

ized

Pw

er

x [mm]

Signal: 2.9 V


The mirror provides high signal because of its high reflectivity. However, it yields a

very wide and divergent beam, and has therefore been rejected (notice the larger abscissa

scale used for the mirror versus the lenses). Of the lenses, lens #3 performs the best with

its narrow beam, Gaussian profile, and relatively high power output. Recall from Table 9

that lens #3 has an extremely small focal length (leading to the narrow beam) and a small

f/# (leading to high power output). The small f/# also means the edges of the lens do not

influence the profile much which leads to a Gaussian beam, whereas lenses #2 and #3

180

clip the beam edges which leads to beam distortion. Therefore, lens #3 was chosen for

the water sensor of Chapter 5.

181

Bibliography

[1] Roy, G.D., Frolov, S.M., Borisov, A.A., and Netzer, D.W., “Pulse detonation

propulsion: challenges, current status, and future perspective”, Progress in Energy and

Combustion Science, 30(6):545-672, 2004.

[2] Wintenberger, E., and Shepherd, J.E., “Stagnation Hugoniot Analysis for Steady

Combustion Waves in Propulsion Systems”, Journal of Propulsion and Power,

22(4):835-844, 2006.

[3] Heiser, W.H., and Pratt, D.T., “Thermodynamic Cycle Analysis of Pulse

Detonation Engines”, Journal of Propulsion and Power, 18(1):68-76, 2002.

[4] Wintenberger, E., Austin, J.M., Cooper, M., Jackson, S., and Shepherd, J.E.,

“Reply to Comment on “Analytical Model for the Impulse of Single-Cycle Pulse

Detonation Tube” by W. H. Heiser and D. T. Pratt”, Journal of Propulsion and Power,

20(1):189-191, 2004.

[5] Kailasanath, K., and Patnaik, G., “Performance Estimates of Pulsed Detonation

Engines”, 28th International Symposium on Combustion, Edinburgh, 28:595-601, 2000.

[6] Ma, F., Choi, J.-Y., and Yang, V., “Internal Flow Dynamics in a Valveless

Airbreathing Pulse Detonation Engine”, Journal of Propulsion and Power, 24(3):479-490,

2008.

[7] Morris, C.I., “Numerical Modeling of Single-Pulse Gasdynamics and Performance

of Pulse Detonation Rocket Engines”, Journal of Propulsion and Power, 21(3):527-538,

2005.

[8] Ma, F., Choi, J.-Y., and Yang, V, “Propulsive Performance of Airbreathing Pulse

Detonation Engines”, Journal of Propulsion and Power, 22(6):1188-1203, 2006.

[9] Wintenberger, E., Austin, J.M., Cooper, M., Jackson, S., and Shepherd, J.E.,

“Analytical Model for the Impulse of Single-Cycle Pulse Detonation Tube”, Journal of

Propulsion and Power, 19(1):22-38, 2003.

182

[10] Wintenberger, E., and Shepherd, J.E., “Model for the Performance of

Airbreathing Pulse-Detonation Engines”, Journal of Propulsion and Power, 22(3):593-

603, 2006.

[11] Goldmeer, J., Tangirala, V., and Dean, A., “System-Level Performance

Estimation of a Pulse Detonation Based Hybrid Engine”, Journal of Engineering for Gas

Turbines and Power, 130:011201, 2008.

[12] Hinkey, J.B., Williams, J.T., Henderson, S.E., and Bussing, T.R.A., “Rotary-

valved, multiple-cycle, pulse detonation engine experimental demonstration”,

AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Seattle, AIAA-1997-

2746, 1997.

[13] Brophy, C.M., and Hanson, R.K., “Fuel Distribution Effects on Pulse Detonation

Engine Operation and Performance”, Journal of Propulsion and Power, 22:1155-1161,

2006.

[14] Brophy, C.M., Sinibaldi, J.O., Netzer, D.W., Johnson, R.G., “Operation of a

JP-10/Air Pulse Detonation Engine”, 36th AIAA/ASME/SAE/ASEE Joint Propulsion

Conference, Huntsville, AIAA 2000-3591, 2000.

[15] Brophy, C.M., Sinibaldi, J.O., Ma, L., Klingbeil, A.E., “Effects of Non-Uniform

Mixture Distributions on Pulse Detonation Engine Performance”, 43rd AIAA Aerospace

Sciences Meeting and Exhibit, Reno, AIAA 2005-1304, 2005.

[16] Kasahara, J., Hasegawa, A., Nemoto, T., Yamaguchi, H., Yajima, T., and Kojima,

T., “Performance Validation of a Single-Tube Pulse Detonation Rocket System”, Journal

of Propulsion and Power, 25(1):173-180, 2009.

[17] Barr, L., “Pulsed detonation engine flies into history”, Air Force Material

Command news, http://www.afmc.af.mil/news/story.asp?id=123098900, May 16, 2008.

[18] Reynolds, W.C., “STANJAN: Interactive Computer Program for Chemical

Equilibrium Analysis” Tech. rep. no. SUMET-8108, Dept. of Mechanical Engineering,

Stanford University, Stanford, CA, January 1986.

183

[19] Gordon, S., and McBride, B.J., “Computer Program for Calculation of Complex

Chemical Equilibrium Compositions and Applications, I. Analysis”, NASA reference

publication #1311, 1994.

[20] Fickett, W., and Davis, W.C., Detonation: Theory and Experiment, Dover, New

York, 2000.

[21] Taylor, G., “The Dynamics of the Combustion Products Behind Plane and

Spherical Detonation Fronts in Explosives”, Proceedings of the Royal Society of

London A, 200(1061):235-247, 1950.

[22] Wintenberger, E. “Application of Steady and Unsteady Detonation Waves to

Propulsion”, Ph.D. Dissertation, California Institute of Technology, Pasadena, CA, 2004.

[23] Austin, J.M., “The Role of Instability in Gaseous Detonation”, Ph.D. Dissertation,

California Institute of Technology, Pasadena, CA, 2003.

[24] Fickett, W., and Wood, W.W., “Flow Calculations for Pulsating One-

Dimensional Detonations”, The Physics of Fluids, 9(5):903-916, 1966.

[25] Abouseif, G.E., and Toong, T.-Y., “Theory of Unstable Two-Dimensional

Detonations: Genesis of the Transverse Waves”, Combustion and Flame, 63:191-207,

1986.

[26] Owens, Z., “Flowfield Characterization and Model Development in Detonation

Tubes”, Ph.D. Dissertation, Stanford University, Stanford, CA, 2008.

[27] Cheatham, S., and Kailasanath, K., “Single-Cycle Performance of Idealized

Liquid-Fueled Pulse Detonation Engines”, AIAA Journal, 43(6):1276-1283, 2005.

[28] Cheatham, S., and Kailasanath, K., “Numerical modelling of liquid-fuelled

detonations in tubes”, Combustion Theory and Modelling, 9(1):23-48, 2005.

[29] Radulescu, M.I., and Hanson, R.K., “Effect of Heat Loss on Pulse-Detonation-

Engine Flow Fields and Performance”, Journal of Propulsion and Power, 21(2):274-285,

2005.

[30] Talley, D.G., and Coy, E.B., “Constant Volume Limit of Pulsed Propulsion for a

Constant γ Ideal Gas”, Journal of Propulsion and Power, 18(2):400-406, 2002.

184

[31] Wintenberger, E., Austin, J.M, Cooper, M., Jackson, S., and Shepherd, J.E.,

“Analytical Model for the Impulse of Single-Cycle Pulse Detonation Tube (Erratum)”,

Journal of Propulsion and Power, 20(4): 765-767, 2004.

[32] Cooper, M., and Shepherd, J.E., “Detonation Tube Impulse in Subatmospheric

Environments”, Journal of Propulsion and Power, 22(4): 845-851, 2006.

[33] Klingbeil, A.E., Jeffries, J.B., and Hanson, R.K., “Design of a Fiber-Coupled

Mid-IR Fuel Sensor for Pulse Detonation Engines”, AIAA Journal, 45(4):772-778, 2007.

[34] Mattison, D.W., Brophy, C.M., Sanders, S.T., Ma, L., et al., “Pulse Detonation

Engine Characterization and Control Using Tunable Diode-Laser Sensors”, Journal of


[35] Ma, L., Sanders, S.T., Jeffries, J.B., and Hanson, R.K., “Monitoring and Control

of a Pulsed Detonation Engine using a Diode-Laser Fuel Concentration and Temperature

Sensor”, Proceedings of the Combustion Institute, Sapporo, 29:161-166, 2002.

[36] Mattison, D.W., Oehlschlaeger, M.A., Morris, C.I., Owens, Z.C., Barbour, E.A.,

Jeffries, J.B., and Hanson, R.K., “Evaluation of Pulse Detonation Engine Modeling Using

Laser-Based Temperature and OH Concentration Measurements”, Proceedings of the

Combustion Institute, Chicago, IL, 30:2799-2807, 2005.

[37] Cooper, M., Shepherd, J.E., and Schauer, F., “Impulse Correlation for Partially

Filled Detonation Tubes”, Journal of Propulsion and Power, 20(5):947-950, 2004.

[38] Cambier, J.-L., and Tégner, J.K., “Strategies for Pulsed Detonation Engine

Performance Optimization”, Journal of Propulsion and Power, 14(4):489-498, 1998.

[39] Eidelman, S. and Yang, X., “Analysis of the Pulse Detonation Engine Efficiency”,

34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Cleveland, AIAA-1998-3877,

1998.

[40] Owens, Z.C., and Hanson, R.K. “Single-Cycle Unsteady Nozzle Phenomena in

Pulse Detonation Engines”, Journal of Propulsion and Power, 23(2):325-337, 2007.

[41] Cooper, M. and Shepherd, J.E., “Single-cycle impulse from detonation tubes with

nozzles”, Journal of Propulsion and Power, 24(1):81-87, 2008.

185

[42] Morris, C.I., “Axisymmetric Modeling of Pulse Detonation Rocket Engines”, 41st

AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Tucson, AIAA-2005-3508, 2005.

[43] Kiyanda, C.B., Tanguay, V., and Higgins, A.J., “Effect of Transient Gasdynamic

Processes on the Impulse of Pulse Detonation Engines”, Journal of Propulsion,

18(5):1124-1126, 2002.

[44] Rudinger, G., Nonsteady Duct Flow: Wave-Diagram Analysis, Dover, New York,

1969.

[45] Thompson, P.A., Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972.

[46] Anderson, J.D., Jr., Modern Compressible Flow with Historical Perspective,

McGraw-Hill, New York, 1982.

[47] Barbour, E.A., Hanson, R.K., Morris, C.I., and Radulescu, M.I., “A Pulsed

Detonation Tube with a Converging-Diverging Nozzle Operating at Different Pressure

Ratios,” 43rd AIAA Aerospace Sciences Meeting, Reno, AIAA-2005-1307, 2005.

[48] Burcat, A., http://garfield.chem.elte.hu/Burcat/burcat.html, 2009.

[49] Sutton, G.P., Rocket Propulsion Elements, John Wiley and Sons, Inc., New York,

sixth ed., 1992.

[50] Li, Q., Fan, W., Yan, C., and Hu, C., and Ye, B., “Experimental Study of

Kerosene-Fueled Pulse Detonation Rocket Engine”, 45th AIAA Aerospace Sciences

Meeting, Reno, AIAA 2007-238, 2007.

[51] Hutcheson, P.D., Brophy, C.M., and Sinibaldi, J.O., “Design, Modeling and

Evaluation of an Initiator Unit for a Split-Path JP-10/Air Pulse Detonation Combustor”,

45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, AIAA 2007-233, 2007.

[52] Owens, Z.C., Mattison, D.W., Morris, C.I., and Hanson, R.K., “Flowfield

characterization and simulation validation of multiple-geometry PDEs using cesium-

based velocimetry”, Proceedings of the Combustion Institute, Chicago, IL, 30:2791-2798,

2005.

[53] Wegener, P.P., “Supersonic Nozzle Flow with a Reacting Gas Mixture”, The

Physics of Fluids, 2(3):264-275, 1959.

186

[54] Zonars, D., “Nonequilibrium Regime of Airflows in Contoured Nozzles: Theory

and Experiments”, AIAA Journal, 5(1):57-63, 1967.

[55] Duffy, R.E., “Experimental Study of Nonequilibrium Expanding Flows”, AIAA

Journal, 3(2):237-244, 1965.

[56] Olson, W.T., “Recombination and Condensation Processes in High Area Ratio

Nozzles”, Journal of the American Rocket Society, 32(5):672-680, 1962.

[57] Anderson, J.D., Jr., “A Time-Dependent Analysis for Vibrational and Chemical

Nonequilibrium Nozzle Flows”, AIAA Journal, 8(3):545-550, 1970.

[58] Scofield, M.P., and Hoffman, J.D., “Maximum Thrust Nozzles for

Nonequilibrium Simple Dissociating Gas Flows”, AIAA Journal, 9(9):1824-1832, 1971.

[59] Rizkalla, Q., Chinitz, W., and Erdos, J.I., “Calculated Chemical and Vibrational

Nonequilibrium Effects in Hypersonic Nozzles”, Journal of Propulsion and Power,

6(1):50-57, 1990.

[60] Cooper, M. “Impulse Generation by Detonation Tubes”, Ph.D. Dissertation,

California Institute of Technology, Pasadena, CA, 2004.

[61] Smith, G.P., Golden, D.M., Frenklach, M., Moriarty, N.W., Eiteneer, B.,

Goldenberg, M., Bowman, C.T., Hanson, R.K., Song, S., Gardiner, W.C., Jr., Lissianski,

V.V., and Qin, Z., http://www.me.berkeley.edu/gri_mech/, 1999.

[62] Bates, R.W., Golden, D.M., Hanson, R.K., and Bowman, C.T., “Experimental

Study and Modeling of the Reaction H + O2 + M HO2 + M (M = Ar, N2, H2O) at

Elevated Pressures and Temperatures between 1050 and 1200K”, Phys. Chem. Chem.

Phys., 3:2337-2342, 2001.

[63] Varatharajan, B. and Williams, F.A., “Ethylene Ignition and Detonation

Chemistry, Part 1: Detailed Modeling and Experimental Comparison”, Journal of


[64] Tangirala, V.E., Dean, A.J., Chapin, D.M., Pinard, P.F., and Varatharajan, B.

“Pulsed Detonation Engine Processes: Experiments and Simulations”, Combustion

Science and Technology, 176:1779-1808, 2004.

187

[65] Strang, G., “On the Construction and Comparison of Difference Schemes”, SIAM

Journal on Numerical Analysis, 5(3):506-517, 1968.

[66] Li, H., Owens, Z.C., Davidson, D.F., and Hanson, R.K., “A Simple Reactive

Gasdynamic Model for the Computation of Gas Temperature and Species Concentrations

behind Reflected Shock Waves”, International Journal of Chemical Kinetics, 40(4):189-

198, 2008.

[67] Fedkiw, R.P., Merriman, B., and Osher, S., “High Accuracy Numerical Methods

for Thermally Perfect Gas Flows with Chemistry”, Journal of Computational Physics,

132:175-190, 1997.

[68] Vincenti, W.G., and Kruger, C.H. Jr., Introduction to Physical Gas Dynamics,

Krieger Publishing Co., Malabar, 1965.

[69] Penner, S.S., “Chemical Reactions during Adiabatic Expansion through a

DeLaval Nozzle”, Introduction to the Study of Chemical Reactions in Flow Systems,

Butterworths Scientific Publications, London, 1955.

[70] Sangiovanni, J.J., Barber, T.J. and Syed, S.A. “Role of hydrogen/air chemistry in

nozzle performance for a hypersonic propulsion system”, Journal of Propulsion and

Power, 9(1):134-138, 1993.

[71] Owens, Z.C., and Hanson, R.K., “The Influence of Wall Heat Transfer, Friction

and Condensation on Detonation Tube Performance”, submitted to Combustion, Science

& Technology for review. See also Ref. 26.

[72] Du, X., Liu, W.S., and Glass, I.I., “Laminar Boundary Layers Behind Blast and

Detonation Waves”, UTIAS report #259, 1982.

[73] Khvostov, N.I., Chekalin, V.E., Sukhobokov, A.D., and Skirda, K.N., “Heat

Transfer in Aerodynamic-Tube Nozzles with High-Temperature Gas Flow”, Journal of

Engineering Physics and Thermophysics, 35(6):1466-1470 (English ed.), 1072-1077

(Russian ed.), 1978.

188

[74] Keener, E.R., and Polek, T.E., “Measurements of Reynolds Analogy for a

Hypersonic Turbulent Boundary Layer on a Nonadiabatic Flate Plate”, AIAA Journal,

10(6):845-846, 1972.

[75] Boldman, D.R., and Graham, R.W., “Heat Transfer and Boundary Layer in

Conical Nozzles”, NASA Technical Note, TN D-6594, 1972.

[76] Edwards, D.H., Brown, D.R., Hooper, G., and Jones, A.T., “The influence of wall

heat transfer on the expansion following a C-J detonation wave”, Journal of Physics D,

3(3):365-376, 1970.

[77] Cooper, M., Jackson, S., Austin, J., Wintenberger, E., and Shepherd, J.E., “Direct

Experimental Impulse Measurements for Detonations and Deflagrations”, Journal of


[78] Laviolette, J.-P., Kiyanda, C.B., and Higgins, A.J., “The Effect of Friction and

Heat Transfer on Impulse in a Detonation Tube”, Proceedings of the Combustion

Institute (Canadian Section), Windsor, 2002.

[79] Harris, P.G., Farinaccio, R., Stowe, R.A., Higgins, A.J., Thibault, P.A., and

Laviolette, J.-P., “The Effect of DDT Distance on Impulse in a Detonation Tube”, 37th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Salt Lake City, AIAA

2001-3467, 2001.

[80] Schauer, F., Stutrud, J., and Bradley, R., “Detonation Initiation Studies and

Performance Results for Pulsed Detonation Engine Applications”, 39th Aerospace

Sciences Meeting and Exhibit, Reno, AIAA-2001-1129, 2001.

[81] Hill, P.G., Peterson, C.R., Mechanics and Thermodynamics of Propulsion,

Addison-Wesley Publishing Co., Reading, MA, 2nd ed., 1992.

[82] Oosthuizen, P.H., and Carscallen, W.E., Compressible Fluid Flow, McGraw-Hill,

New York, 1997.

[83] Back, L.H., Massier, P.F., and Gier, H.L., “Comparisons of Experimental with

Predicted Wall Static-Pressure Distributions in Conical Supersonic Nozzles”, Jet

Propulsion Laboratory, report #32-654, 1964.

189

[84] Rieker, G.B., Li, H., Liu, X., Liu, J.T.C., Jeffries, J.B., Hanson, R.K., et al.,

“Rapid measurements of temperature and H2O concentration in IC engines with a spark

plug-mounted diode laser sensor”, Proceedings of the Combustion Institute, 31:3041-

3049, 2007.

[85] Mattison, D.W., Jeffries, J.B., Hanson, R.K., Steeper, R.R., et al., “In-cylinder

gas temperature and water concentration measurements in HCCI engines using a

multiplexed-wavelength diode-laser system: Sensor development and initial

demonstration”, Proceedings of the Combustion Institute, 31:791-798, 2007.

[86] Rieker, G., Jeffries, J.B., Hanson, R.K., Mathur, T., Gruber, M.R., and Carter,

C.D., “Diode laserbased detection of combustor instabilities with application to a

scramjet engine”, Proceedings of the Combustion Institute (in press).

[87] Docquier, N., and Candel, S., “Combustion control and sensors: a review”,

Progress in Energy and Combustion Science, 28(2):107-150, 2002.

[88] Sanders, S.T., Baldwin, J.A., Jenkins, T.P., Baer, D.S., and Hanson, R.K.,

“Diode-Laser Sensor for Monitoring Multiple Combustion Parameters in Pulse

Detonation Engines”, Proceedings of the Combustion Institute, 28:587-594, 2000.

[89] Mattison, D., “Development and Application of Laser-Based Sensors for Harsh

Combustion Environments”, PhD dissertation, Stanford University, 2006.

[90] Rothman, L.S., Jacquemart, D., Barbe, A., Benner, D.C., et al., “The HITRAN

2004 molecular spectroscopic database”, Journal of Quantitative Spectroscopy &

Radiative Transfer, 96:139-204, 2005.

[91] Hümmer, M., Rößner, K., Benkert, A., and Forchel, A., “GaInAsSb–AlGaAsSb

Distributed Feedback Lasers Emitting Near 2.4 μm”, IEEE Photonics Technology Letters,

16:380-382, 2004.

[92] Rossner, K., Hummer, M., Benkert, A., and Forchel, A., “Long-wavelength

GaInAsSb/AlGaAsSb DFB lasers emitting near 2.6 μm”, Physica E: Low-dimensional

Systems and Nanostructures, 30:159-163, 2005.

190

[93] Gamache, R.R., Hawkins, R.L., and Rothman, L.S., “Total Internal Partition

Sums in the Temperature Range 70 - 3000K: Atmospheric Linear Molecules”, Journal of

Molecular Spectroscopy, 142:205-219, 1990.

[94] Farooq, A., Jeffries, J.B, and Hanson, R.K., “In situ combustion measurements of

H2O and temperature near 2.5 μm using tunable diode laser absorption”, Measurement

Science and Technology, 19:075604, 2008.

[95] Mahajan, V.N., Aberration Theory Made Simple, SPIE Optical Engineering Press,

Bellingham, 1991.

[96] Saurel, J.M., and Roig, J., “Proprietes des Filtres Interferentiels Type Perot-Fabry

Eclaires en Incidence Oblique”, Journal of Optics, 10(4), pp. 179-193, 1979.

[97] Wheeler, A.J., and Ganji, A.R., Introduction to Engineering Experimentation,

Prentice-Hall, Englewood Cliffs, 1996.

[98] Wang, F., Liu, J.B., Brophy, C., Kuthi, A., Jiang, C., Ronney, P., and Gundersen,

M.A., “Transient Plasma Ignition of Quiescent and Flowing Air/Fuel Mixtures”, IEEE

Transactions on Plasma Science, 33:844-849, 2005.

[99] Klingbeil, A.E., Jeffries, J.B., and Hanson, R.K., “Design of a Fiber-Coupled

Mid-Infrared Fuel Sensor for Pulse Detonation Engines”, AIAA Journal, 45(4):772-778,

2007.

[100] Osmont, A., Gökalp, I., and Catoire, L., “Evaluating Missile Fuels”, Propellants,

Explosives, Pyrotechnics, 31(5):343-354, 2006.

[101] Striebich, R.C., and Lawrence, J., “Thermal Decomposition of High-Energy

Density Materials at High Pressure and Temperature”, Journal of Analytical and Applied

Pyrolysis, 70: 339-352, 2003.

[102] Edwards, T., “Liquid Fuels and Propellants for Aerospace Propulsion: 1903–

2003”, Journal of Propulsion and Power, 19(6):1089-1107, 2003.

[103] Klingbeil, A.E., Jeffries, J.B., and Hanson, R.K., “Temperature- and

composition-dependent mid-infrared absorption spectrum of gas-phase gasoline: Model

and measurements”, Fuel, 87:3600-3609, 2008.

191

[104] Klingbeil, A.E., “Mid-IR Laser Absorption Diagnostics For Hydrocarbon Vapor

Sensing In Harsh Environments”, PhD dissertation, Stanford University, 2007.

[105] Edwards, D.K., and Menard, W.A., “Comparison of Models for Correlation of

Total Band Absorption”, Applied Optics, 3(5):621-625, 1964.

[106] Klingbeil, A.E., Jeffries, J.B., and Hanson, R.K., “Temperature- and pressure-

dependent absorption cross sections of gaseous hydrocarbons at 3.39 μm”, Measurement

Science and Technology, 17:1950-1957, 2006.

[107] Weibring, P., Richter, D., Fried, A., Walega, J.G., and Dyroff, C., “Ultra-high-

precision mid-IR spectrometer II: system description and spectroscopic performance”,

Applied Physics B, 85:207-218, 2006.

[108] Klingbeil, A.E., Jeffries, J.B., and Hanson, R.K., “Tunable Mid-IR Laser

Absorption Sensor for Time-Resolved Hydrocarbon Fuel Measurements”, 31st

Symposium (International) on Combustion, Heidelberg, 2006.

[109] Sirignano, W.A., Fluid Dynamics and Transport of Droplets and Sprays,

Cambridge University Press, Cambridge, Chap. 2, 1999.

[110] Szekely, G.A., “Experimental Evaluation of a Carbon Slurry Droplet

Combustion Model,” Ph.D. Dissertation, Mechanical Engineering Dept., Penn State

Univ., University Park, 1982.

[111] Anonymous, “Handbook of Aviation Fuel Properties,” Coordinating Research

Council, CRC 530, 1983.

[112] Mattison, D.W., Liu, J.T.C., Jeffries, J.B., Hanson, R.K., Brophy, C.M., and

Sinibaldi, J.O., “Tunable Diode-Laser Temperature Sensor for Evaluation of a Valveless

Pulse Detonation Engine”, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno,

AIAA 2005-224, 2005.

[113] Austin, J.M., and Shepherd, J.E., “Detonations in Hydrocarbon Fuel Blends”,

Combustion and Flame, 132(1-2):73-90, 2003.

[114] Chemkin Software, Theory Manual, Reaction Design, release 4.1, 2006.

192

[115] Wark, K., Jr., Advanced Thermodynamics for Engineers, McGraw-Hill, New

York, 1995.

[116] Suzaki, Y., and Tachibana, A., “Measurement of the μm sized radius of

Gaussian laser beam using the scanning knife-edge”, Applied Optics, 14(12):2809-2810,

1975.

MODELING AND LASER-BASED SENSING OF PULSED ...

Documents

Transcript of MODELING AND LASER-BASED SENSING OF PULSED ...