37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference...

AIAA 2001-3811AN ANALYTICAL MODEL FOR THEIMPULSE OF A SINGLE-CYCLEPULSE DETONATION ENGINEE. Wintenberger, J.M. Austin, M. Cooper, S. Jackson,and J.E. ShepherdGraduate Aeronautical Laboratories, California Instituteof Technology, Pasadena, CA 91125

37th AIAA/ASME/SAE/ASEEJoint Propulsion Conference and Exhibit

July 8–11, 2001, Salt Lake City, UTFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

AN ANALYTICAL MODEL FOR THEIMPULSE OF A SINGLE-CYCLE PULSE

DETONATION ENGINE

E. Wintenberger, J.M. Austin, M. Cooper, S. Jackson, and J.E. ShepherdGraduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125

An analytical model for the impulse of a single-cycle pulse detonation engine has beendeveloped and validated against experimental data. The model analyzes the pressuredifferential at the thrust surface of the detonation tube. The pressure inside the tubeis modeled with a constant pressure region and a blowdown process. A careful study ofthe gas dynamics inside the tube enables the derivation of a similarity solution for theconstant pressure part. The blowdown process is modeled using dimensional analysis andempirical observations. The model predictions are validated against direct experimentalmeasurements in terms of impulse per unit volume, specific impulse, and thrust. Impulseper unit volume and specific impulse calculations are carried out for a wide range of fuel-oxygen-nitrogen mixtures (including aviation fuels) varying initial pressure, equivalenceratio, and nitrogen dilution. The effect of the initial temperature is also investigated.Comparisons with numerical simulation estimates of the specific impulse are given.

Nomenclature

A cross-sectional area of detonation tubec1 sound speed of reactantsc2 sound speed of burned gases just behind det-

onation wavec3 sound speed of burned gases behind Taylor

waved inner diameter of detonation tubef cycle repetition frequencyg gravity accelerationI single-cycle impulseIV impulse per unit volumeIsp mixture-based specific impulseIspf fuel-based specific impulseK proportionality coefficientL length of detonation tubeMCJ Chapman-Jouguet Mach numberP pressureP0 pressure outside detonation tubeP1 initial pressure of reactantsP2 Chapman-Jouguet pressure peakP3 pressure of burned gases behind Taylor wavet timet1 time taken by the detonation wave to reach

the open end of the tubet2 time taken by the first reflected characteristic

to reach back the thrust surfacet3 time associated with blowdown processT thrustT1 initial temperature of reactantsu flow velocity

Copyright c⃝ 2001 by California Institute of Technology. Pub-lished by the American Institute of Aeronautics and Astronautics,Inc. with permission.

u2 flow velocity just behind detonation waveUCJ Chapman-Jouguet detonation velocityV inner volume of detonation tubeXF fuel mass fractionα non-dimensional parameter corresponding to

time (t1 + t2)β non-dimensional parameter corresponding to

blowdown process∆P pressure differential∆P3 pressure differential at the thrust surfaceη similarity variableγ ratio of specific heatsλ cell sizeφ equivalence ratioρ1 initial density of reactants

Introduction

A key issue1–5 in evaluating pulse detonation en-gine propulsion concepts is reliable estimates of

the performance as a function of operating conditionsand fuel types. The purpose of the present work isto develop a simple analytical model for performancebased on elementary gas dynamics and dimensionalanalysis. This model is based on experimental obser-vations and validated against measurements reportedin our companion paper, Cooper et al.6 In developingour model we have considered only the simplest con-figuration of a detonation engine, a tube open at oneend and closed at the other, and single-cycle operation.We realize that there are significant issues3 associatedwith inlets, valves, exits and multicycle operation thatare not addressed in our approach. Our goals here areto develop and validate a simple model that can pro-vide an upper bound on the impulse expected from thesimplest configuration possible.

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American Institute of Aeronautics and Astronautics Paper 2001-3811

In developing our model, we considered the follow-ing idealized representation of the processes in a singlecycle of pulse detonation operation. A pulse detona-tion engine goes through four major steps during onecycle: the filling of the device with a combustible mix-ture, the initiation of the detonation, the propagationof the detonation down the tube and finally the ex-haust of the products in the atmosphere through ablowdown process. A schematic of the cycle is shownin Figure 1. The pressure differential created by thedetonation wave generates an unsteady thrust. Therepetition of this cycle at a high frequency (ideally afew hundred Hz) creates a quasi-steady thrust.

One of the most important performance parametersof a pulsed detonation engine is the impulse deliveredduring one cycle. The knowledge of the impulse al-lows the determination of the thrust as soon as thecycle frequency is known. However, this fundamentalparameter has been estimated only through numeri-cal simulations, and the results of these computationsare subject to wide variation.5 We are motivated todevelop an analytical model in order to be able topredict trends and to better understand the influenceof fuel type, initial conditions, and tube size withoutcarrying out a large number of numerical simulations.This paper presents an analytical model that was de-veloped and validated against direct experimental im-pulse measurements.6 The model provides estimatesfor the impulse per unit volume and specific impulseof a single-cycle pulse detonation engine for a widerange of fuels (including aviation fuels) and initial con-ditions.

Analytical model for impulsecalculations

The impulse of a single-cycle pulse detonation en-gine has been determined for several mixtures using amodel involving gas dynamics calculations and exper-imental results.

Single-cycle pulse detonation engine

During a cycle, thrust is being generated as longas the pressure inside the tube on the thrust surface(where the detonation was initiated) is greater thanthe atmospheric pressure. A closer look at the gasdynamics inside the tube can lead us to define fourdifferent stages in a pulse detonation engine cycle byconsidering the reflected wave off the interface at theopen end of the tube, as depicted in Figure 1.

Boundary conditions at the open end of the tube

The analysis of the gas dynamics inside the tube re-quires the knowledge of what is happening when theincident detonation wave reaches the open end of thetube. The flow behind the detonation wave is subsonicand has a Mach number M2 = u2/c2 of approximately0.8 for typical hydrocarbon mixtures. Hence when thedetonation wave reaches the open end, a disturbance

∗

1. The detonation is initiatedat the thrust wall.

Fuel/oxidizer UCJ

2. The detonation, followed bythe Taylor wave, propagates tothe open end of the tube.

3. An expansion wave is reflectedoff of the interface and immediatelyinteracts with the Taylor wave whilethe products start to exhaust fromthe tube.

4. The first characteristic of thereflected expansion wave reachesthe thrust wall and decreases thepressure at the wall.

Non-simple region

Fig. 1 Pulse detonation engine cycle.

propagates back into the tube in the form of a re-flected wave. The interface at the open end of thetube can be modeled in one-dimension as a contactsurface. When the detonation wave is incident on thiscontact surface, a transmitted wave will propagate outof the tube while a second wave is reflected back to-wards the thrust surface. A contact surface separatesthe gas processed by the incident and reflected wavesfrom that processed by the transmitted wave. Thereflected wave can be either a shock or an expansionwave. A simple way to determine the nature of thereflected wave is to use a pressure-velocity diagram asthe pressure and velocity must be matched across thecontact surface after the interaction. In the case ofa detonation wave exiting into air at one atmosphere,the transmitted wave will always be a blast wave; thelocus of solutions (the shock adiabat) is shown in Fig-ures 2 and 3. The shock adiabat is given by the shockjump conditions for an ideal gas:

∆u

c1=

∆P/P1

γ!

1 + γ+12γ

∆PP1

"12

(1)

where u is the flow velocity, c1 is the sound speed inair, P1 is the initial pressure, and γ is the ratio of thespecific heats.

The reflected wave initially propagates back into theburnt products behind the detonation wave, gas whichis assumed to be at the Chapman-Jouguet (CJ) state.The CJ states for various fuels at various equivalenceratios are also shown in Figures 2 and 3. If the CJpoint is below the shock adiabat, the reflected wavemust increase the pressure to match that behind thetransmitted blast wave and is therefore a shock. Al-ternatively, if the CJ state is above the shock adiabat,a pressure decrease is required and the reflected waveis an expansion.

Hydrocarbon fuels all produce a reflected expansionwave at the open end for any stoichiometry. However,a reflected shock is obtained for hydrogen-oxygen at anequivalence ratio φ > 0.8 and for very rich hydrogen-air mixtures (φ > 2.2). All the reflected waves forstoichiometric fuel-oxygen and fuel-air mixtures areexpansion waves with the exception of stoichiomet-

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∆u (m/s)

∆P(bar)

0 500 1000 1500 20000

10

20

30

40

50

60Shock adiabatC2H4/O2C3H8/O2C2H2/O2H2/O2Jet A/O2JP10/O2

Reflected shock

Reflected expansion

stoichiometric points

Fig. 2 Open end boundary conditions for fuel-oxygen mixtures.

∆u (m/s)

∆P(bar)

0 500 1000 15000

5

10

15

20

25 Shock adiabatC2H4/airC3H8/airC2H2/airH2/airJet A/airJP10/air

Reflected expansion

Reflected shockstoichiometric points

Fig. 3 Open end boundary conditions for fuel-airmixtures.

ric hydrogen-oxygen for which the reflected wave isa shock. Ultimately, following the initial interactionof the detonation wave with the contact surface, thepressure at the exit of the tube will drop as the trans-mitted blast wave propagates outward from the tubeexit. This will produce expansion waves that will prop-agate back up into the tube, eventually reducing thepressure inside the tube to that of the surrounding at-mosphere.

Analysis of the gas dynamics in the detonationtube

The gas dynamics inside an idealized tube (with-out any obstacles) can be analyzed using a distance-time (x-t) diagram shown in Figure 4. The x-t dia-gram displays the detonation wave propagating at the

Chapman-Jouguet velocity UCJ followed by the Tay-lor expansion wave. The first reflected characteristicfrom the mixture-air interface at the open end of thetube is also shown. This characteristic has an initialslope determined by the conditions at the interface,which is then modified by the interaction with theTaylor wave. Once it has passed through the Taylorwave, this first characteristic of the reflected expansionwave propagates at the sound speed of the medium c3.The region lying behind this first characteristic is non-simple, as the reflected expansion wave interacts withthe incoming Taylor wave. Two characteristic timescan be defined: t1 corresponding to the reflection of thedetonation wave at the interface, and t2 correspondingto the time necessary for the first reflected character-istic to reach the thrust surface.

Thrust wallL

Open end

Non-simple region

x

t

-c3

c3 UCJ

t1

t1+t2

1

2

3

Contact surface

detonation wave

Taylor wave

transmittedshock

first reflectedcharacteristic

Fig. 4 x-t diagram for a detonation wave propa-gation and interaction with the tube open end.

The pressure differential at the thrust surface gen-erates the single-cycle impulse. Therefore, it is inter-esting to focus on the pressure history at the thrustsurface. When the detonation is initiated, a pres-sure spike is recorded (the Chapman-Jouguet pressurepeak) before the pressure decreases to P3 by the pas-sage of the Taylor wave. The pressure at the thrustsurface remains constant until the first reflected char-acteristic reaches the wall. The reflected expansionwave then decreases the pressure to the atmosphericvalue. (In some cases, the expansion wave may over-expand the flow so that the pressure is decreased belowatmospheric for a period of time.)

The pressure-time trace at the thrust surface hasbeen modeled as shown in Figure 5. The Chapman-Jouguet pressure peak is considered to occur during aninfinitely short time. The pressure stays constant fora total time t1 + t2 at pressure P3, which can be foundfrom an isentropic flow calculation through the Taylorexpansion wave. Then the pressure is affected by the

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P

t

t1 t2 t3

P1

P3

P2

Ignition

Fig. 5 Ideal modeling of the evolution of the pres-sure at the thrust surface with time.

reflected expansion and decreases to atmospheric. Forclarity, the interaction of the transmitted wave andthe contact surface with the Taylor wave is omitted onFigure 4.

Parameters for impulse calculation

The impulse of a single-cycle pulse detonation en-gine can be computed the following way:

I = A

# ∞

0∆P (t) dt (2)

where A is the area of the cross-section of the tube and∆P is the pressure differential over the thrust surface.Ignition is assumed to occur at t = 0. Idealizing thepressure-time trace, the impulse can be decomposedinto three terms:

I = A

$

∆P3(t1 + t2) +# ∞

t2

∆P (t)dt

%

(3)

where t1 is the time necessary for the detonation toreach the end of the tube of length L: t1 = L/UCJ , andt2 is the time necessary for the first reflected character-istic to reach the thrust surface. The time t2 dependsprimarily on the length of the tube and the charac-teristic velocity behind the Taylor wave, which is thesound speed c3. Thus, it will be modeled by intro-ducing a non-dimensional parameter α: t2 = αL/c3.The last part of the pressure-time integral will be non-dimensionalized with respect to the sound speed c3 asit is the characteristic velocity of the medium wherethe reflected wave propagates, and the pressure differ-ential ∆P3. The portion of the integral between t2 and∞ is modeled by introducing another time t3, whichis expressed in terms of a non-dimensional parameterβ.

# ∞

t2

∆P (t)dt = ∆P3t3 = ∆P3βL

c3(4)

The ideal model of the pressure trace is shown onFigure 5. The hatched zone represents the equivalentarea of the decaying part of the pressure-time trace, t> t2. Finally, the impulse can be written as:

I = A∆P3

$

L

UCJ+ (α + β)

L

c3

%

(5)

Determination of α

The parameter α is determined by the interactionof the reflected wave and the Taylor wave. A simi-larity solution can be derived to compute the time atwhich the first reflected characteristic arrives at thethrust surface.7 The assumption is that the reflectedwave at the interface is an expansion wave, which isthe case for hydrocarbon mixtures. The flow param-eters behind the detonation front (state 2), referredto as the Chapman-Jouguet (CJ) parameters, weredetermined through equilibrium calculations carriedout using the program STANJAN.8 The pressure P2,the sound speed just behind the detonation wave c2,the detonation velocity UCJ , and the flow velocity be-hind the detonation wave u2 were calculated. State 3parameters were calculated using characteristics andisentropic flow conditions across the Taylor wave. Theform of the equations inside the expansion fan suggeststhe introduction of a similarity variable η = x/c2t. Anordinary differential equation can then be derived forη:

tdη

dt+

2(γ − 1)γ + 1

$

η − u2

c2+

2γ − 1

%

= 0 (6)

The solution to this equation with the appropriateinitial conditions gives the time at which the firstreflected characteristic exits the Taylor wave. It sub-sequently propagates at the sound speed c3 of medium3. The time t2 at which it reaches the thrust surfacecan therefore be computed, giving the following resultfor α:

α =c3

UCJ

&

2'

γ − 1γ + 1

$

c3 − u2

c2+

2γ − 1

%(− γ+12(γ−1)

− 1

)

(7)The quantities involved in this expression essentially

depend on two non-dimensional parameters: γ and thedetonation Mach number MCJ = UCJ/c1. The pa-rameters behind the shock wave (labeled 2) can becomputed analytically using the ideal gas model for aCJ detonation. The resulting expression7 for α in theideal gas case is:

α(γ,MCJ ) =12

'

1 +1

M2CJ

(

·'

2$

γ − 1γ + 1

'

γ + 32

+2

γ − 1− (γ + 1)2

2·

M2CJ

1 + γM2CJ

(%− γ+12(γ−1)

− 1

*

(8)

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Determination of β

The region lying to the right of the first reflectedcharacteristic in Figure 4 is a non-simple region cre-ated by the interaction of the reflected expansion wavewith the Taylor wave. This makes the derivation of ananalytical solution for the parameter β difficult. It is,however, possible to rely on experimental data to cal-culate β. We considered data from our experiments6as well as Zitoun et al.9 who carried out experimentsaimed at measuring the impulse of pulse detonationengines using tubes of different length. The impulsewas calculated for stoichiometric ethylene-oxygen mix-tures by integrating the pressure differential at thethrust surface. Zitoun showed that the impulse scaleswith the length of the tube. The analysis of thepressure-time traces showed that the overpressure, af-ter being roughly constant for a certain period, de-creases and becomes negative before coming back tozero. The presence of this negative overpressure is at-tributed to an over-expansion of the flow coming outof the tube. The integration of the decaying part ofthe pressure-time trace had to be carried out up to atime late enough (typically greater than 20t1) to en-sure that the overpressure is back to zero after theover-expansion. The result gives the following valuefor β:

β = 0.53 (9)

Validation of the model

The model was validated against experimental data.Comparisons were made in terms of impulse per unitvolume and specific impulse. The impulse per unitvolume is IV = I/V where I is the single-cycle impulseand V is the volume of the tube. The mixture-basedspecific impulse Isp is defined as the ratio of the single-cycle impulse to the product of the mixture mass ρ1Vand earth gravitational acceleration g:

Isp =I

ρ1V g=

IV

ρ1g(10)

The fuel-based specific impulse Ispf is defined withrespect to the fuel mass instead of the mixture mass:

Ispf =I

ρ1XF V g=

IV

XF(11)

where XF is the fuel mass fraction.

Impulse per unit volume comparisonsThe analytical model predictions were compared in

terms of impulse per unit volume with extensive datafrom Cooper et al.,6 who carried out direct experimen-tal impulse measurements using a ballistic pendulumtechnique. In these experiments, detonation initiationwas obtained via deflagration-to-detonation transition(DDT). Obstacles were mounted inside the detonationtube in some of the experiments in order to enhanceDDT. A correlation plot showing the impulse per unit

volume obtained with the model versus the experi-mental values is displayed in Figure 6. The valuesdisplayed here cover experiments with 4 different fu-els (hydrogen, acetylene, ethylene and propane) overa wide range of initial conditions (equivalence ratio,initial pressure, and nitrogen dilution variation). Thesolid line represents perfect correlation between the ex-perimental data and the model. The full symbols rep-resent the data for unobstructed tubes, while the opensymbols correspond to cases for which obstacles wereused in the detonation tube. The analytical model pre-dictions were close to the experimental values of theimpulse, especially at high pressure and zero-dilution.The model assumes direct initiation of detonation, soit does not take into account any phenomenon associ-ated with DDT. The agreement is better for cases withhigh initial pressure and no nitrogen dilution, since theDDT time (time it takes the initial flame to transitionto a detonation) is the shortest for these mixtures. Ingeneral, the model values fall within 15% of the ex-perimentally determined values, except for two cases,one where the model underpredicts and one where itoverpredicts the impulse by about 25%. However, themodel systematically underpredicts the values for theunobstructed tube experiments by 5% to 15%, exceptfor the acetylene case, where it is about 25% too low.When obstacles are used, the experimental values aremuch lower and the model overpredicts them by asmuch as 25%, although in general the error is within15%. The lower experimental values for cases with ob-stacles are caused by the additional drag created bythe obstacles.

Model impulse (kg/m2s)

Experim

entalimpulse(kg/m

2 s)

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500H2 - no obstaclesC2H2 - no obstaclesC2H4 - no obstaclesH2 - obstaclesC2H4 - obstaclesC3H8 - obstaclesImodel=Iexp

Fig. 6 Model predictions versus experimental datafor the impulse per unit volume. Filled symbolsrepresent data for unobstructed tubes, whereasopen symbols show data for cases in which obsta-cles were used.

The value of the parameter β was checked againstvalues obtained from the ballistic pendulum experi-

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ments.6 For stoichiometric ethylene-oxygen mixturesat 100 and 80 kPa initial pressure, the values foundwere 0.5 and 0.73. The first value is fairly close to theone considered in the model. The higher discrepancywith the second value is due to the DDT initiation usedin the experiments. The DDT time is much shorter at100 kPa and the process is more similar to the directinitiation assumed in the model.

The value of the parameter α was verified with theexperimental pressure traces obtained at the thrustsurface. The time at which the pressure starts to de-crease after its constant part is compared to the timeprediction using the model. The agreement is in gen-eral very good. A careful study of an experimentaltrace6 revealed that the time between detonation ini-tiation and the overpressure decrease corresponding tothe blowdown process for a mixture of stoichiometricethylene-air at 100 kPa initial pressure was 1.43 msin a tube 1.016 m long. The corresponding calculatedtime (depending on α) was 1.39 ms, within 3% of theexperimental estimate. Similarly, comparing with thevalue9 for a tube of length 0.225 m, excellent agree-ment is obtained between the value of our model (313µs) and Zitoun’s data (315 µs), within 1% error.

The model parameters are relatively constant,1.07 < α < 1.13 for all the mixtures studied here andβ = 0.53. A reasonable estimate for α is a value of 1.1.The ratio UCJ/c3 for fuel-oxygen-nitrogen mixtures isapproximately 2. For quick estimates of the impulse,the following approximate formula can be used:

I = 4.3∆P3

UCJAL = 4.3

∆P3

UCJV (12)

where V = AL is the volume of the detonation tube.The approximate formula reproduces the exact expres-sions within 2.5%.

Specific impulse and thrust comparisonsThe model predictions were compared versus spe-

cific impulse and thrust measurements from Schaueret al.10,11 Results are given in terms of fuel-basedspecific impulse for hydrogen-air10 and propane-air11mixtures varying the equivalence ratio. Schauer etal.10 conducted experiments in a 50.8 mm diameterby 914.4 mm long tube using a damped thrust stand.They collected data during multi-cycle steady stateoperation and the thrust was averaged over many cy-cles. Specific impulse comparison plots presented inFigure 7 for hydrogen-air and in Figure 8 for propane-air show that the impulse model predictions are fairlyclose to the experimental data. Two sets of data11 aregiven for propane, corresponding to their most sta-ble cases (in which the peak wave velocity was about80% of the Chapman-Jouguet value) and most unsta-ble, intermittent detonations. Figure 7 also includesexperimental hydrogen-oxygen data from our own ex-periments.7 The model gives higher values than the

experiments because of the drag due to the obstaclesin the DDT-initiated experiments (the model assumesdirect detonation initiation). The dashed line on Fig-ure 7 shows the boundary after which a reflected shockis obtained for hydrogen-air. The fact that the modelstill correctly predicts the impulse confirms the ideathat the reflected shock is weak and the result in termsof pressure integration is similar to the one obtainedassuming a reflected expansion wave.

The dropoff in the experimental data at low equiva-lence ratio is certainly due to cell size effects in the caseof propane as the cell size at the lower equivalence ra-tios gets close to the usual limit of π times the diameterof the tube for detonation propagation; the cell size atφ = 0.74 is λ=152 mm for propane-air.12 In the case ofhydrogen-air, the cell size at φ = 0.75 is 21 mm, whichis much smaller. A possible explanation according tothe work of Dorofeev et al.13 may be the effect of theexpansion ratio of the mixture. However, calculationsfor lean hydrogen-air showed that the expansion ratiois always higher than the critical value defined13 forhydrogen mixtures. Another reason may be related tothe transition distance of the mixtures. Dorofeev etal.14 studied the effect of scale on the onset of deto-nations and validated the L = 7λ criterion, where L isthe characteristic geometrical size and λ the cell sizeof the mixture. They defined the criterion L > 7λas a necessary condition for transition to detonationwhere the characteristic geometrical size L is carefullydefined in the presence of obstacles. Schauer et al.10used a 4.8 mm diameter, 45.7 mm pitch Schelkin spiralin their pulse detonation tube to initiate detonations.Applying Dorofeev’s definition results in a characteris-tic geometrical size of 257 mm. The cell size increaseswith decreasing equivalence ratio for lean mixtures andtakes a value of 31 mm for hydrogen-air at an equiva-lence ratio φ = 0.67.12 The value of 7λ is comparableto the characteristic geometrical size of the system forthis equivalence ratio, which corresponds to the firstexperimental specific impulse value significantly lowerthan the model prediction. For higher values of 7λ,i.e. lower equivalence ratios, this criterion14 predictsno transition to detonation, which could explain thelower specific impulses obtained experimentally.10

Thrust can be calculated from the impulse modelpredictions, assuming a very simple pulse detonationengine model. The thrust T obtained is equal to theproduct of the single-cycle impulse per unit volume IV

with the volume of the tube V and the cycle repetitionfrequency f : T = IV V f . Schauer et al.10 measuredthe thrust delivered by a hydrogen-air pulse detona-tion engine operated at a frequency of 16 Hz. Thecorresponding thrust calculation was carried out usingthe analytical model and is compared with Schauer’sdata on Figure 9. The computation of the thrust withthe model shows good agreement with the experimen-tal data, except at low equivalence ratios because of

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Equivalence ratio

Ispf(s)

0 0.5 1 1.5 2 2.5 30

1000

2000

3000

4000

5000

6000

7000

8000

H2/O2 - modelH2/air - modelSchauer - H2/airCIT - H2/O2

P1=100 kPa, T1=300 K

reflected shock

Fig. 7 Specific impulse comparison betweenmodel predictions and experimental data7,10 forhydrogen-air varying equivalence ratio.

Equivalence ratio

Ispf(s)

0 0.5 1 1.5 2 2.5 30

500

1000

1500

2000

2500

3000

modelSchauer - detSchauer - no det?CIT

P1=100 kPa, T1=300 K

Fig. 8 Specific impulse comparison between modelpredictions and experimental data7,11 for propane-air varying equivalence ratio.

the increasing transition distance discussed above.

Impulse calculationsImpulse per unit volume

Impulse calculations were carried out using themodel for different mixtures including hydrocarbonfuels and hydrogen, and for a wide range of initial pa-rameters including equivalence ratio, initial pressureand nitrogen dilution. The results were expressed interms of impulse per unit volume of the tube, which issize-independent. The input required by the modelconsists of the detonation velocity UCJ , the soundspeed behind the detonation front c2, the Chapman-Jouguet pressure P2, and the ratio of the specific heats

Equivalence ratio

Thrust(lbf)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

CIT modelSchauer

Fig. 9 Thrust calculation for a 50.8 mm diameterby 914.4 mm long hydrogen-air pulse detonationengine operated at 16 Hz. Comparison with ex-perimental data.10

of the products γ. All these parameters were com-puted using the element-potential method for chemicalequilibrium analysis implemented in the interactiveprogram STANJAN.8 The results are presented forthe following fuels: ethylene, propane, acetylene, hy-drogen, Jet A, and JP10 varying the initial pressure(Figure 10), the equivalence ratio (Figure 11) and thenitrogen dilution (Figure 12). Two sets of calcula-tions are presented for the liquid fuels (Jet A andJP10). These fuels are in the liquid state at ambi-ent temperature but may be heated and vaporized inthe detonation tube. We have therefore examined thesituations of liquid and vapor fuel separately for thesecases. Two thermodynamic files were written as in-put to STANJAN, one defining these fuels as liquidsand including a value for the density, and the otherdescribing them as gases. These calculations define auseful range for the impulse of jet fuels and avoid hav-ing to deal with liquid fuel combustion. Regardinghydrogen mixtures, the results for hydrogen-oxygenare strictly valid for φ up to 0.8 and for hydrogen-air up to 2.2. For higher values of the equivalenceratio, a reflected shock is generated at the interface.However, the strength of this shock is small, whichimplies that the model may give a good approxima-tion. Indeed, a ballistic pendulum experiment7 carriedout with hydrogen-oxygen resulted in the directly mea-sured impulse being within 10% of the value predictedby the model. In these cases, the calculations are prob-ably reasonable estimates but the reader has to keep inmind that the underlying physical assumption is notjustified any more.

The initial pressure variation is characterized by alinear dependence of the impulse (see Figure 10) on

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Initial pressure (bar)

Impulseperunitvolum

e(kg/m

2 s)

0 0.5 1 1.5 20

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000 C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A-liquid/O2Jet A-gas/O2JP10-liquid/O2JP10-gas/O2C2H4/airC3H8/airC2H2/airH2/airJet A-liquid/airJet A-gas/airJP10-liquid/airJP10-gas/air

stoichiometric fuel/O2, T1=300 K

Fig. 10 Impulse per unit volume varying with ini-tial pressure.

initial pressure. The impulse values for stoichiometrichydrocarbon fuel-air mixtures are very similar at allpressures. However, the computed impulse values forhydrocarbon fuel-oxygen mixtures vary over a widerrange. The hydrogen cases are very different than thehydrocarbons: the impulse per unit volume is muchlower due to the lower molecular mass of hydrogen,resulting in smaller density and CJ pressure. Anotherinteresting feature is that the impulse of hydrogen-air mixtures is actually higher than the impulse ofhydrogen-oxygen mixtures. The addition of nitrogenincreases the molecular mass of the mixture whichcompensates the dilution effect, resulting in similar CJpressures. At the same time, the dilution produces adecrease in the CJ detonation velocity and the soundspeed c3, resulting in an increase in the time of appli-cation of the pressure differential at the thrust surfaceand, therefore, in a higher impulse.

The plots showing the impulse per unit volumeagainst the equivalence ratio are shown in Figure 11and present a maximum on the rich side at a value de-pendent on the type of fuel, except for hydrogen. Thisshift of the maximum impulse on the rich side is cor-related to a shift of the detonation velocity maximumand is caused by phenomena of dissociation similar tothe deflagration case. In the case of hydrogen, thisdissociation effect is not as prominent, so the maxi-mum should occur closer to 1, which is the case forhydrogen-air. The hydrogen-oxygen curve decreasesfrom the lean side to the rich side. Unlike hydrocarbonfuels, which have a molecular mass comparable to orhigher than oxygen and air, hydrogen has a much lowermolecular mass. Thus increasing the equivalence ratiocauses a decrease in the mixture density. The CJ pres-sure variation is not as pronounced as for other fuels,while the detonation velocity UCJ and the sound speed

Equivalence ratio

Impulseperunitvolum

e(kg/m

2 s)

0 0.5 1 1.5 2 2.5 30

500

1000

1500

2000

2500

3000

3500

4000

4500

5000 C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A-liquid/O2Jet A-gas/O2JP10-liquid/O2JP10-gas/O2C2H4/airC3H8/airC2H2/airH2/airJet A-liquid/airJet A-gas/airJP10-liquid/airJP10-gas/air

P1=100 kPa, T1=300 K

Fig. 11 Impulse per unit volume varying withequivalence ratio.

Nitrogen dilution (%)

Impulseperunitvolum

e(kg/m

2 s)

0 25 50 75 1000

500

1000

1500

2000

2500

3000C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A-liquid/O2Jet A-gas/O2JP10-liquid/O2JP10-gas/O2


Fig. 12 Impulse per unit volume varying with ni-trogen dilution.

c3 vary significantly, increasing with φ. In particular,on the lean side, the density (and accordingly the pres-sure P3) is high and UCJ and c3 are low, which leads toa high impulse. This dominant effect of the velocitiesexplains the monotonically decreasing behavior of theimpulse with increasing equivalence ratio in the case ofhydrogen-oxygen. The nitrogen dilution for hydrogen-air mixtures reduces the variation of the characteristicvelocities and enhances the influence of the pressureP3, since the molecular mass of the mixture does notvary as much due to the presence of nitrogen.

The impulse per unit volume generated by the differ-ent fuels studied with oxygen can be ranked in all casesas follows from lowest to highest: hydrogen, acetylene,ethylene, propane, Jet A and JP10. The general trendis that the fuels with a lower hydrogen to carbon ratio

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generate a higher impulse, the exception being acety-lene. The case of acetylene can be explained by thelower H/C ratio and presence of a triple bond betweenthe two carbon atoms. Reactions involving hydrocar-bons with a higher carbon to hydrogen ratio will createcomparatively more carbon dioxide and therefore havea higher heat of combustion. The hydrogen to car-bon ratio is the following for the fuels studied here:2 for ethylene, 2.67 for propane, 1.8 for Jet A, 1.6for JP10, and 1 for acetylene. The combustion of hy-drogen produces only water, which results in a lowerheat of combustion. The results obtained for the im-pulse per unit volume versus the equivalence ratio arepresented for an equivalence ratio range from 0.4 to2.6. Calculations at higher equivalence ratios were car-ried out; however, the results obtained were unreliablebecause carbon production, which is very difficult tomodel, occurs for very rich mixtures, in particular forJet A and JP10.

The nitrogen dilution calculations (see Figure 12)show that the impulse decreases with increasing nitro-gen dilution for hydrocarbon fuels. However, as thedilution increases, the values of the impulse for thedifferent fuels get closer to each other. The presenceof the diluent masks the effect of the hydrogen to car-bon ratio. The hydrogen curve is much lower due tothe lower CJ pressures caused by the lower molecularmass and heat of combustion of hydrogen. Unlike forhydrocarbons, this curve has a maximum. The pres-ence of this maximum can be explained by the twocompeting effects of nitrogen addition: one is to di-lute the mixture, which is dominant at high dilution,while the other is to increase the molecular mass of themixture, which is dominant at low dilution. Note thatthe highest value of the impulse is obtained around50% dilution, which is close to the case of air (55.6%dilution).

Specific impulse calculations

The specific impulse was calculated for the differ-ent cases considered based on mixture mass and onfuel mass. The specific impulse was deduced from theimpulse per unit volume by dividing by the productof the mixture or fuel density and earth gravitationalacceleration g. The mixture-based specific impulseIsp is plotted versus initial pressure, equivalence ra-tio, and nitrogen dilution in Figures 13, 14, and 15respectively. Only one curve was shown for the jetfuels (Jet A and JP10) corresponding to the gaseouscase. The Isp curve for hydrocarbon fuels versus ini-tial pressure decreases steeply at low pressures due tothe importance of endothermic dissociation phenom-ena with decreasing pressure. With increasing pres-sure, recombination occurs and all the curves seem totend towards a high-pressure limit. At high pressures,∆P3/P0 ∝ P2/P0 and the Chapman-Jouguet pressurevaries linearly with initial pressure. The parameters α


Isp(s)

0 0.5 1 1.5 20

25

50

75

100

125

150

175

200

C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A/O2JP10/O2C2H4/airC3H8/airC2H2/airH2/airJet A/airJP10/air


fuel/O2

fuel/air

Fig. 13 Mixture-based specific impulse varyinginitial pressure.

Equivalence ratio

Isp(s)

0 0.5 1 1.5 2 2.5 30

25

50

75

100

125

150

175

200

225

250 C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A/O2JP10/O2C2H4/airC3H8/airC2H2/airH2/airJet A/airJP10/air

P1=100 kPa, T1=300 K

fuel/air

H2/air

fuel/O2

H2/O2

Fig. 14 Mixture-based specific impulse varyingequivalence ratio.

and β are almost constant, and the characteristic ve-locities vary little with pressure (less than 5% between0.2 and 2 bar). From Equation 5, the mixture-basedspecific impulse therefore tends to a limiting value.

The Isp plots for hydrocarbon fuels varying theequivalence ratio are very similar to the ones obtainedfor the impulse per unit volume. This is expected, asthe only difference is due to the density, which does notvary much for hydrocarbon fuels, which have a masscomparable to the oxidizer mass, resulting in littlemixture density variation with the equivalence ratio.However, this effect is important in the case of hydro-gen; the mixture density decreases significantly as theequivalence ratio increases. This accounts clearly forthe shape of the monotonically increasing hydrogen-oxygen curve. The mixture density effect is masked be-

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Isp(s)

0 25 50 75 1000

25

50

75

100

125

150

175

200

C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A/O2JP10/O2

stoichiometric fuel/O2, P1=100 kPa, T1=300 K

Fig. 15 Mixture-based specific impulse varyingnitrogen dilution.

cause of the nitrogen dilution in the case of hydrogen-air, which explains the nearly constant portion of thecurve on the rich side.

The variation of the Isp with nitrogen dilutionis the same for all fuels including hydrogen. Themixture-based specific impulse decreases as the ni-trogen amount in the mixture increases. The slightincrease in the impulse per unit volume for hydrogen iscompensated for by the molecular mass effect. Addingnitrogen to the mixture increases the molecular massas nitrogen is a much heavier compound than hydro-gen. This effect is dominant especially at low nitrogendilution.

The fuel-based specific impulse Ispf is plotted versusinitial pressure, equivalence ratio and nitrogen dilutionin Figures 16, 17, and 18 respectively. The Ispf curvesfor initial pressure variation are very similar to thecorresponding Isp curves. The curves are individuallyshifted by a factor equal to the fuel mass fraction. Notethe obvious shift of the hydrogen curves because of thevery low mass fraction of hydrogen. The fuel-basedspecific impulse is about 3 times higher for hydrogenthan for other fuels.

The equivalence ratio plots show a monotonicallydecreasing Ispf with increasing equivalence ratio. Thisis due to the predominant influence of the fuel massfraction, which goes from very low on the lean side tohigh on the rich side, whereas the mixture-based spe-cific impulse Isp does not vary as much. The hydrogenmixtures again give much higher values compared tothe hydrocarbon fuels due to the difference in massfraction.

Similarly, the nitrogen dilution plots exhibit a mono-tonically increasing behavior with increasing nitrogendilution. This behavior is once again mainly dictatedby the decreasing fuel mass fraction as the nitrogen


Ispf(s)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A/O2JP10/O2C2H4/airC3H8/airC2H2/airH2/airJet A/airJP10/air

fuel/O2

fuel/airH2/O2

H2/air

Fig. 16 Fuel-based specific impulse varying initialpressure.

Equivalence ratio

Ispf(s)

0 0.5 1 1.5 2 2.5 30

1000

2000

3000

4000

5000

6000C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A/O2JP10/O2C2H4/airC3H8/airC2H2/airH2/airJet A/airJP10/air

fuel/O2

fuel/air

H2/O2

H2/air

Fig. 17 Fuel-based specific impulse varying equiv-alence ratio.

dilution increases.

Influence of initial temperature

Temperature is another initial parameter that maysignificantly affect the impulse of a given pulse det-onation engine. Up to now, temperature has beenassumed constant at a value of 300 K in the modelcalculations. However, it is interesting to examine theeffect of varying the temperature, especially to highervalues that become of interest for flight performanceor doing experiments in a heated detonation facility.Calculations were carried out using an initial mixtureof stoichiometric JP10-air, which is the most interest-ing mixture in terms of practical applications. Theimpulse per unit volume (shown in Figure 19) and themixture-based specific impulse (shown in Figure 20)

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Ispf(s)

0 25 50 75 1000

1000

2000

3000

4000

5000

6000 C2H4/O2C3H8/O2C2H2/O2H2/O2Jet A/O2JP10/O2

stoichiometric fuel/O2, P1=100 kPa, T1=300 K

Fig. 18 Fuel-based specific impulse varying nitro-gen dilution.

Initial temperature (K)

Impulseperunitvolum

e(kg/m

2 s)

300 400 500 6000

1000

2000

3000

4000

5000

6000

7000

8000

P0=P1=0.5 barP0=P1=1 barP0=P1=2 barP0=P1=3 barP0=P1=4 barP0=P1=5 bar

Fig. 19 Impulse per unit volume varying initialtemperature for different values of the stagnationpressure.

were calculated as a function of the initial temperaturefor different values of the engine stagnation pressure P1

assumed to be equal to the outside stagnation pressureP0.

The curves showing the impulse per unit volume ver-sus the initial temperature are decreasing hyperbolae.So increasing the temperature plays a negative role forthe impulse per unit volume. This is due to the factthat the impulse per unit volume scales with the initialdensity. The pressure differential at the thrust surface∆P3 scales with P2. The Chapman-Jouguet pressureP2 scales as ρ1U2

CJ . Given that the detonation veloc-ity UCJ and the sound speed of the products c3 do notchange significantly with initial density, the impulseper unit volume scales directly with the initial density

Initial temperature (K)

Specificimpulse(s)

300 400 500 6000

50

100

150

Po=P1=0.5 barP0=P1=1 barP0=P1=2 barP0=P1=3 barP0=P1=4 barP0=P1=5 bar

Fig. 20 Mixture-based specific impulse varyinginitial temperature for different values of the stag-nation pressure.

ρ1. The temperature variation is made while keepingP1 constant, so the initial density is proportional tothe inverse of the temperature. This explains why aset of hyperbolae is obtained. At the same time, theCJ pressure also scales directly with the initial pres-sure when the temperature is kept constant, so thatthe specific impulse plot should be essentially inde-pendent of initial temperature, verified in Figure 20.The specific impulse is calculated by dividing the im-pulse per unit volume by the initial density. Clearlyspecific impulse is the most useful parameter for es-timating performance since it almost independent ofinitial pressure and temperature.

Pulse detonation engine performance

The specific impulse is a parameter that is used tocompare the performance of different propulsion sys-tems. In particular, rocket performance is usuallydescribed in terms of specific impulse. The specificimpulse calculations can therefore be compared withother pulse detonation engine studies and the perfor-mance of this type of engine can be evaluated withrespect to other propulsion systems.

Zitoun et al.9 calculated the specific impulse ofa multi-cycle pulse detonation engine for various re-active mixtures based on a formula developed fromtheir experimental data for ethylene-oxygen mixtures:Isp = K∆P3/(gρ1UCJ). The coefficient K is esti-mated to be 5.4 in their study, whereas we obtainedan estimate of 4.3. This accounts for the differencein the specific impulse results presented in Table 1.The analytical model impulse is about 20% lower thanZitoun’s predictions.

It is also interesting to make comparisons with nu-merical computation estimates for the specific impulse.Numerical computations are very sensitive to the spec-

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Mixture Model Isp Zitoun et al.9

C2H4+3O2 151.1 200C2H4+3(O2+3.76N2) 117.3 142

C2H2+2.5O2 150.9 203C2H2+2.5(O2+3.76N2) 120.6 147

H2+0.5O2 172.9 226H2+0.5(O2+3.76N2) 123.7 149

Table 1 Comparison of the model predictions forthe mixture-based specific impulse.

ification of the outflow boundary condition at the openend, and the numerical results are very different follow-ing the type of boundary condition adopted. Sterlinget al.1 obtained an average value of 5151 s for the fuel-based specific impulse of a stoichiometric hydrogen-airmixture in a multi-cycle simulation using a constantpressure boundary condition. Bussing et al.3 obtaineda range of values of 7500-8000 s. Other predictions byCambier and Tegner,4 including a correction for the ef-fect of the initiation process, gave values between 3000and 3800 s. More recently, Kailasanath and Patnaik5

tried to reconcile these different studies for hydrogen-air by highlighting the effect of the outflow boundarycondition. They varied the pressure relaxation rate atthe exit and obtained a range of values from 4850 s(constant pressure case) to 7930 s (gradual relaxationcase). Our analytical model predicts 4335 s for the spe-cific impulse of stoichiometric hydrogen-air, and theexperimental value of Schauer et al.10 is 4024 s.

The performance of pulse detonation engines can becompared using the analytical model predictions to theperformance of other types of engines. Liquid propel-lant rockets using LO2-LH2 generally have a specificimpulse of 450 s.15 More specifically, the J-2 Apollorocket engine produced a specific impulse of 426 s,15while the space shuttle main engine gave a value of363 s.16 For comparison, the predicted performanceof a hydrogen-oxygen single-cycle pulse detonation en-gine is 172.9 s. A significant portion of the rocketmotor specific impulse is derived from conversion ofthermal energy to kinetic energy in the nozzle. With-out the nozzle, the thrust chamber specific impulsealone is around 240 s for hydrocarbon-oxygen rocketmotors. The H-1 Saturn C-1 booster propelled byRP1-O2 has an impulse of 257 s and the S-4 Atlassustainer, also using RP1-O2, has an impulse of 222s.16 The pulse detonation engine model specific im-pulse for hydrocarbon-oxygen mixtures is about 150 s.Ammonium-nitrate-based solid rockets produce spe-cific impulses of 192 s while ammonium perchloratesolid rockets result in a higher impulse of about 260s. As an example, the first-stage Minuteman missilemotor has a specific impulse at sea level of 214 s.All these values are given here to illustrate the per-formance of an idealized single-cycle pulse detonationengine. The specific impulse is lower for this kind of

engine than for the existing rocket engines. However,the preceding argument compares the impulse of pulsedetonation engines, which are unsteady devices, withthat of rocket engines, which are steady devices. Thethrust delivered by a pulse detonation engine is di-rectly proportional to the cycle repetition rate, which,therefore, plays a significant role in performance esti-mates.17 No comparison is made with air-breathingpropulsion systems since they involve a more detailedsystem analysis including inlets.

The advantages presented in terms of hardware andcycle efficiency are apparently counterbalanced by alower specific impulse. However, care should be takenin analyzing these results. First, the specific impulseshould not be the only parameter to be considered. Asshown in the different impulse calculations, the resultis very sensitive to the mixture molecular mass. Forexample, rich hydrogen-oxygen mixtures have muchhigher specific impulses than other mixtures, so thereader should be aware of the mass ratios when consid-ering the fuel-based specific impulse. Based on specificimpulse values, hydrogen looks like a much more effi-cient fuel, but a comparison of the impulse per unitvolume shows that hydrogen does not produce as highan impulse as the other fuels, and, therefore, is notas efficient a propellant. This means that one param-eter is not sufficient to characterize the performanceof a pulse detonation engine. The specific impulses(mixture- and fuel-based) should be taken into accountas well as the impulse per unit volume, which is finallythe most direct size-independent characteristic param-eter.

AcknowledgementsThis work was supported by the Office of Naval

Research Multidisciplinary University Research Ini-tiative Multidisciplinary Study of Pulse DetonationEngine (grant 00014-99-1-0744, sub-contract 1686-ONR-0744), and General Electric contract GE-POA02 81655 under DABT-63-0-0001. We thank FredSchauer at the AFRL for sharing his data with us.

References1Sterling, J., Ghorbanian, K., Humphrey, J., Sobota, T.,

and Pratt, D., “Numerical Investigations of Pulse DetonationWave Engines,” 31st AIAA/ASME/SAE/ASEE Joint Propul-sion Conference and Exhibit, July 10–12, 1995, San Diego, CA,AIAA 95–2479.

2Bussing, T. R. A. and Pappas, G., “Pulse DetonationEngine Theory and Concepts,” Progress in Aeronautics and As-tronautics, Vol. 165, 1996.

3Bussing, T. R. A., Bratkovich, T. E., and Hinkey, J. B.,“Practical Implementation of Pulse Detonation Engines,” 33rdAIAA/ASME/SAE/ASEE Joint Propulsion Conference and Ex-hibit, July 6–9, 1997, Seattle, WA, AIAA 97-2748.

4Cambier, J. L. and Tegner, J. K., “Strategies forPulsed Detonation Engine Performance Optimization,” 33rdAIAA/ASME/SAE/ASEE Joint Propulsion Conference and Ex-hibit, July 6–9, 1997, Seattle, WA, AIAA 97-2743.

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5Kailasanath, K. and Patnaik, G., “Performance Estimatesof Pulsed Detonation Engines,” Proceedings of the 28th Interna-tional Symposium on Combustion, The Combustion Institute,July 2000, Edinburgh, Scotland.

6Cooper, M., Austin, J., Jackson, S., Wintenberger, E., andShepherd, J. E., “Direct experimental impulse measurements fordeflagrations and detonations,” 37th AIAA/ASME/SAE/ASEEJoint Propulsion Conference, July 8–11, 2001, Salt Lake City,UT, AIAA 2001-3812.

7Wintenberger, E., Austin, J., Cooper, M., Jackson, S., andShepherd, J. E., “Impulse of a pulse detonation engine: single-cycle model,” GALCIT Report FM00-8, Pasadena, CA 91125,2001.

8Reynolds, W., “The Element Potential Method for Chem-ical Equilibrium Analysis: Implementation in the InteractiveProgram STANJAN,” Technical Report, Mechanical Engineer-ing Department, Stanford University, 1986.

9Zitoun, R. and Desbordes, D., “Propulsive Performancesof Pulsed Detonations,” Comb. Sci. Tech., 1999, Vol. 144, pp.93–114.

10Schauer, F., Stutrud, J., and Bradley, R., “Detonation Ini-tiation Studies and Performance Results for Pulsed DetonationEngines,” 39th AIAA Aerospace Sciences Meeting and Exhibit,January 8–11, 2001, Reno, NV, AIAA 2001-1129.

11Schauer, F., Private communication, 2001.12Shepherd, J. E. and Kaneshige, M., “Detonation

Database,” www.galcit.caltech.edu/∼jeshep/detn db/html/.13Dorofeev, S., Kuznetzov, M. S., Alekseev, V. I., Efimenko,

A. A., and Breitung, W., “Evaluation of limits for effectiveflame acceleration in hydrogen mixtures,” Russian ResearchCenter Kurchatov Institute-Forschungszentrum Karlsruhe Ger-many, Report IAE-6150/3 FZKA-6349, 1999.

14Dorofeev, S., Sidorov, V. P., Kuznetzov, M. S., Matsukov,I. D., and Alekseev, V. I., “Effect of scale on the onset of deto-nations,” Shock Waves, Vol. 10, 2000, pp. 137–149.

15Hill, P. G. and Peterson, C. R., “Mechanics and Thermody-namics of Propulsion,” Addison-Wesley, Second Edition, ISBN0-201-14659-2, 1992.

16Sutton, G. P., “Rocket Propulsion Elements,” Wiley-Interscience, Fifth Edition, ISBN 0-471-80027-9, 1986.

17Chao, T., Wintenberger, E., and Shepherd, J. E., “On thedesign of pulse detonation engines,” GALCIT Report FM00-7, Graduate Aeronautical Laboratories, California Institute ofTechnology, Pasadena, CA 91125, 2001.

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