A MODEL FOR THE PERFORMANCE OFAIR-BREATHING PULSE DETONATION

ENGINES

E. Wintenberger and J.E. ShepherdGraduate Aeronautical Laboratories,

California Institute of Technology, Pasadena, CA 91125

An analytical model for predicting the performance of a single-tube air-breathingpulse detonation engine has been developed. The model is based on control volumemethods and elementary gas dynamics. The pulse detonation engine considered consistsof a steady supersonic inlet, a large plenum, and a straight detonation tube (no exitnozzle). The filling process is studied in detail through numerical simulations, due to itsinfluence on the initial conditions for detonation initiation. Control volume analysis andgas dynamics are used to model the coupled flow between the plenum and the detonationtube. It is shown that the average pressure in the plenum is lower than the stagnationpressure downstream of the inlet due to the unsteadiness of the flow, and that the flowin the plenum oscillates due to valve opening and closing. Moreover, the filling processgenerates a moving flow into which the detonation has to initiate and propagate. Ourexisting single-cycle impulse model is extended to include the effect of filling velocityon detonation tube impulse. Based on this, the engine thrust is calculated using anopen-system control volume analysis. It is found to be the sum of the contributions ofdetonation tube impulse, ram momentum, and ram pressure. Performance calculationsare carried out for pulse detonation engines operating with stoichiometric hydrogen-airand JP10-air and compared to the performance of the ideal ramjet.

NomenclatureA0 effective inlet capture areaA2 plenum cross-sectional areaAV valve and detonation tube cross-sectional

areac speed of soundCP specific heat at constant pressuree internal energyf fuel-air mass ratioF thrustg Earth’s gravitational accelerationht stagnation enthalpy per unit massIdt detonation tube impulseISPF fuel-based specific impulseISPFdt detonation tube fuel-based specific im-

pulseL detonation tube lengthm mass flow ratemCin incoming mass flow ratemCout outgoing mass flow ratemf average fuel mass flow ratems mass flow rate through side surfaces of

control volumeM Mach numberMfill filling Mach numberMS Mach number of the shock wave generated

at valve opening in the burned gasesP static pressure

PR initial pressure ratio across valve in nu-merical simulations of filling process

Pt stagnation pressureq heat release per unit massR perfect gas constantt timetCJ time taken by the detonation to reach the

open end of the tube in the static case= L/UCJ

tclose valve close timetfill filling timetopen valve open timetpurge purging timeT static temperatureTt stagnation temperatureu flow velocityUCJ detonation wave velocityUfill filling velocityV detonation tube volumeVC plenum volumeZ altitudeα non-dimensional parameter correspond-

ing to time taken by first reflected char-acteristic to reach thrust surface

β non-dimensional parameter correspond-ing to pressure decay period

γ ratio of specific heatsM mass of fluid in control volumeΩ engine control volume

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π purging coefficientρ densityΣ engine control surfaceτ cycle time

Subscripts

0 freestream2 state downstream of inlet3 state behind Taylor wave during detona-

tion processC acoustic cavity (or plenum)CJ Chapman-Jouguetf state of detonation products at the end of

blowdown processi state of reactants before detonation initi-

ation at the end of filling processdt detonation tubeV valve plane∗ state at valve plane when choked during

filling processo model value during open part of cycle

Averages

X temporal average of X over a cycle

IntroductionPulse detonation engines (PDEs) are propulsion sys-

tems based on the intermittent use of detonative com-bustion. Because of the intrinsically unsteady natureof the flow field associated with the detonation pro-cess, it is difficult to evaluate the relative performanceof air-breathing PDEs with respect to conventionalsteady-flow propulsion systems.

PDE performance analysis has followed several dif-ferent approaches. Researchers started by measur-ing1–5 and modeling3,6–8 the static performance ofsingle-cycle detonation tubes. In parallel, researchershave also investigated experimentally the static multi-cycle performance of single3,9 and multiple10 detona-tion tubes.

In contrast to the numerous studies carried out onthe static performance of a PDE, very few effortshave focused on estimating the performance of an air-breathing PDE. The difficulties associated with cou-pling the inlet flow with the unsteady flow in the det-onation tubes, and the lack of common understandingabout the influence of nozzles on static PDE perfor-mance,11 are some of the difficult obstacles to suchmodeling. Wu et al.12 have presented what is so farthe most comprehensive system performance analysisfor an air-breathing hydrogen-fueled PDE, based on amodular approach including supersonic inlet dynam-ics and detonation in single and multiple tubes. Maet al.13 recently extended this work to the thrustchamber dynamics of multiple-tube PDEs and ob-tained specific impulses as high as 3800 s with a single

converging-diverging exit nozzle. However, it is notclear how the interaction and coupling of the moduleswere treated in these models.

Other propulsion performance estimates for PDEshave been based on thermodynamic cycle analysis sim-ilar to those used in conventional steady propulsionsystems, following the detonation thermodynamic cy-cle14 or a surrogate constant volume combustion cy-cle.15 Other models based on gas dynamics have beenproposed.16 Harris et al.17 showed that the model pro-posed by Talley and Coy16 offers a good approximationof the time-averaged PDE performance, whereas theHeiser and Pratt analysis14 was shown to be overlyoptimistic in its prediction of PDE performance.

Because of the inherent unsteadiness associated withthe detonation process and the pulsed operation of thePDE, the most successful performance models3,7, 16 sofar have been based on unsteady gas dynamics. Wepropose that PDEs have to be analyzed in an unsteadyfashion, using open-system control volume analysis.Our goal is to develop a simple predictive model thatcan be used for engine performance evaluation at vari-ous operating conditions. We present a fully unsteadyone-dimensional control volume analysis of a PDE,taking into account the kinetic energy terms, which arecritical for high-speed propulsion systems. We focus onthe modeling of a single-tube PDE, which includes anacoustic cavity that is large enough to minimize pres-sure transients associated with pulsed operation. Weinvestigate in detail the flow field inside the engine,including the dynamics of the detonation tube, theacoustic cavity, and the inlet response. We perform anunsteady control volume analysis of the PDE. The con-servation equations for mass, momentum, and energyare integrated over a cycle, and performance parame-ters are derived as a function of flight and operatingconditions. Finally, the performance of a single-tubeair-breathing PDE is compared with that of a conven-tional propulsion system, the ideal ramjet.

General considerationsConventional steady inlets are attractive for PDE

applications because their performance characteristicsare well understood. They can be used in an un-steady air-breathing engine such as a PDE as longas the flow downstream of the inlet is quasi-steady.There are two ways to achieve quasi-steady state atthe inlet. The first one is to design an inlet manifoldlarge enough to dampen pressure transients and bleedexcess air between detonation cycles. This approachrequires an increased engine total volume and may notbe practical. The second way is to use multiple det-onation tubes operating out of phase so that the flowupstream of the detonation tubes is decoupled fromthe unsteady flow inside the tubes, and approachesquasi-steady state. The first approach is used in ourmodeling of the single-tube PDE.

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The air-breathing PDE we consider in this paperconsists of an inlet, a large acoustic cavity (or plenum),a valve, which is assumed to open and close instanta-neously, and a detonation tube with fuel injectors. Aschematic of a single-tube PDE with a steady inlet isillustrated in Fig. 1. The inlet, operating in a steadymode, is separated from the detonation tube by anisolator (a grid or screen similar to what is used inramjets) and a large acoustic cavity, whose role is toprotect the inlet from downstream flow perturbationsdue to combustion or valve motion. The detonationtube is assumed to be straight. As there is no wideagreement on the influence of nozzles on PDE per-formance, the effect of exit nozzles on detonation tubeperformance will not be considered in this initial study.

Fig. 1 Schematic representation of a single-tubepulse detonation engine.

Detonation tube dynamicsBefore calculating the performance of a PDE, it is

instructive to study the dynamics of the detonationtube during one cycle. A cycle has three main com-ponents: detonation and blowdown of burned gases,purging of the expanded burned products, and refill-ing of the tube with fresh reactants. Experiments haveshown that purging the burned gases (usually with air)was a necessary precaution to avoid pre-ignition of thefresh mixture before the detonation is initiated. Fig. 2illustrates the complete cycle for a given detonationtube, which is described in more detail in the follow-ing sub-sections. Fuel is injected in the low-speed flowupstream of the valve plane and is assumed to mixinstantaneously with the flowing air. Total pressurelosses associated with fuel-air mixing are neglected inour idealized model. The cycle time is the sum of thevalve close time and the open time, the latter beingthe sum of the fill time and the purging time:

τ = tclose + topen = tclose + tfill + tpurge (1)

Although the unsteady flow in the detonation tubeis complex and involves many wave interactions, themain physical processes occurring during a cycle arefairly obvious from previous studies.

Detonation/blowdown process

Detonation is assumed to be instantaneously di-rectly initiated in the initial mixture at the closed endof the tube. The specific detonation tube dynamics

Fig. 2 PDE cycle schematic for a detonation tube.a) The detonation is initiated at the closed end ofthe tube. b) The detonation propagates towardsthe open end. c) The detonation diffracts outsideas a decaying shock and a reflected expansion wavepropagates to the closed end, starting the blow-down process. d) At the end of the blowdownprocess, the tube contains burned products at rest.e) The purging/filling process is triggered by theopening of the valve, sending a shock wave in theburned gases, followed by the air/products contactsurface. f) A slug of air is injected before the re-actants for purging. g) The purging air is pushedout of the tube by the reactants. h) The reactantseventually fill the tube completely and the valve isclosed.

during the detonation process were studied in detailby Wintenberger et al.7 in the static case. It wasshown that, as the detonation exits the tube, a re-flected expansion wave propagates back towards theclosed valve for hydrocarbon-air and lean and slightlyrich hydrogen-air mixtures. After interacting with theTaylor wave, this reflected expansion decreases thepressure at the closed end of the tube and acceleratesthe fluid towards the open end. The pressure insidethe tube typically decreases below the ambient pres-sure3 at the end of the blowdown process before goingback up to ambient pressure after about 20tCJ . Thissuggests that the time during which the valve is closedfor a given tube in an air-breathing PDE has to be atleast 10tCJ for maximized impulse per cycle.

In an air-breathing configuration, the flow in thedetonation tube has a slightly different behavior be-cause of the interaction of the detonation and fill-ing processes. The valve is assumed to close instan-

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taneously prior to the beginning of the detonationprocess. The valve closing sends an expansion wavethrough the tube to decelerate the flow created bythe filling process. This expansion wave decreases thepressure and density inside the tube, which will de-crease the detonation pressure, decreasing the thrust.If the detonation is assumed to be initiated as soon asthe valve closes, it will overtake the expansion wavewithin the tube and propagate into the uniform flowproduced by the filling process. The thrust for this sit-uation will be different from the case of a detonationpropagating into a stationary mixture but can be cal-culated if we assume ideal valve closing and detonationinitiation.

Purging/filling process

At the end of the detonation/blowdown process, thevalve at the upstream end of the tube, which wasclosed during the detonation/blowdown process, opensinstantaneously. The opening of the valve triggers theexpansion of the high-pressure air into the detonationtube containing burned gases at ambient pressure andelevated temperature. A shock wave is generated andpropagates into the detonation tube, followed by thecontact surface between fresh air and burned products.Fuel is typically not injected in the initial part of thefilling process because purging the burned gases withair is required in order to avoid pre-ignition of the freshmixture. An unsteady expansion wave propagates up-stream of the valve inside the acoustic cavity and setsup a steady expansion of the cavity air into the deto-nation tube. The filling process is characterized by acombination of unsteady and steady expansions.

The gas dynamics of the flow are complex and in-volve multiple wave interactions, but in the interestof simplicity, we will attempt to characterize the fill-ing process with a few key quantities. In order todo so, this problem was simulated numerically usingAmrita.18 The simulations employed the non-reactiveEuler equations in a two-dimensional domain withcylindrical symmetry, using a Kappa-MUSCL-HLLEsolver. The configuration tested, shown in Fig. 3, con-sists of a large cavity connected by a smooth areachange to a straight tube open to a half-space. Thesimulation was started with high-pressure air in thecavity at conditions given by PC/P0 = PR, TC/T0 =P

(γ−1)/γR , separated from burned gases in the tube at

pressure P0 and elevated temperature Tf = 7.69T0,representative of the temperature of the burned gasesat the end of the blowdown process. The air outsidethe detonation tube is at pressure P0 and tempera-ture T0. The problem has two contact surfaces, theinlet air-burned gas interface at the valve end, and theburned gas-outside air interface at the tube exit. Nu-merical Schlieren images of the filling process are givenin Fig. 3.

When the shock wave reaches the open end of the

Fig. 3 Numerical Schlieren images of the fillingprocess. PR = 8, Tf/T0 = 7.69, γ = 1.4.

detonation tube, it diffracts outwards, eventually be-coming a decaying spherical shock, but also generates areflected shock, due to the lower density of the burnedproducts in the detonation tube than the outside air(soft-hard interaction). The reflected shock propa-gates upstream, interacting with the expansion wavesthat propagate back into the tube from the corners.The weakened shock then interacts with the inlet air-burned gas contact surface, generating a transmittedshock and a reflected expansion wave propagating to-wards the open end of the tube. When the flow behindthe inlet air-burned gas contact surface is supersonic(for high values of PR), the transmitted shock can besteady or also be convected by the flow towards theopen end.

Although this problem is essentially multi-dimensional, the simulation results show that forpressure ratios low enough (PR < 5), the flow in-side the tube is essentially one-dimensional, exceptwithin one tube diameter of the tube exit, just

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after the exhaust of the incident shock, when themulti-dimensional corner expansion waves propagateback into the tube. These waves quickly catch upand merge with the reflected shock. The subsequentreflected shock-contact discontinuity interaction ap-pears one-dimensional. The same behavior is observedat higher pressure ratios, although two-dimensionalwaves generated by the starting process closely followthe inlet air-burned gas contact surface (Fig. 3).

Plenum/detonation tube couplingThe acoustic cavity (or plenum) is connected on one

side to the steady inlet and on the other side to the un-steady valve at the front of the detonation tube. Theconditions in the plenum and the flow in the detona-tion tube are coupled. The unsteady flow generated bythe valve opening during the filling process dependson the conditions in the plenum and in the tube atthe end of the detonation process but, in turn, affectsthe flow in the plenum. It is, therefore, critical to beable to model accurately the average conditions in theplenum.

Fig. 4 Control volume VC considered for analysisof flow in the plenum.

We assume that the cycle time is much larger thanthe characteristic acoustic transit time in the plenumso that the properties in the cavity can be modeled asspatially uniform. Assuming that the flow through theinlet diffuser is choked, the incoming mass flow rate inthe plenum mCin(t) is constant and equal to m0. Theoutgoing mass flow rate is defined as the flow rate atthe valve plane mV (t) when the valve is open. We areseeking the average conditions in the plenum, whichdetermine the average filling conditions. In order to doso, we average over a cycle the mass, momentum, andenergy conservation equations for the control volumeVC defined on Fig. 4. Although there is unsteady mass,momentum and energy variation during a cycle, therecan be no accumulation in the plenum over a cycleduring periodic operation. This yields

mV (t) = m0 (2)

mV (t)uV (t) = A2Pt2−AV PV (t)+(AV −A2)PC(t) (3)

mV (t)htV (t) = m0ht2 (4)

Based on our numerical simulations of the fillingprocess, we model the properties at the valve planeas piecewise constant functions of time. The velocityuV (t) and mass flow rate mV (t) are equal to zero whenthe valve is closed and take on values uo

V and moV when

the valve is open, as illustrated in Fig. 5. Assuming

that the volume of the plenum is much larger than thevolume of the detonation tube, the pressure inside theplenum can be approximated as constant. The pres-sure at the valve plane is equal to the average pressurein the cavity PC when the valve is closed and takes avalue P o

V when the valve is open (Fig. 5). Rewritingthe averaged energy equation in terms of the temper-ature, we obtain T o

tV = Tt2. The average conditions inthe plenum can be determined with the additional con-sideration of the flow in the detonation tube when thevalve is open. In our present geometry, in which thevalve plane corresponds to a geometrical throat, twodifferent cases must be distinguished: either the flowat the valve plane is subsonic or it is sonic (choked).

Fig. 5 Modeling of pressure and flow velocity atthe valve plane during a cycle.

Subsonic flow at the valve plane

The unsteady expansion wave generated at valveopening propagates upstream in the plenum and setsup a steady expansion through the area change be-fore decaying, becoming very weak for large area ra-tios. Assuming a large area ratio between the plenumand the valve, we ignore the weak unsteady expan-sion propagating in the plenum in our calculations.We model the flow during the filling process in thesubsonic case by a steady expansion through the areacontraction between the plenum and the detonationtube, and a shock wave propagating in the tube fol-lowed by the burned gases/fresh air contact surface, asshown in Fig. 6. We neglect the influence of the start-ing transient by assuming that the time necessary tostart the steady flow is much smaller than the timenecessary to fill the detonation tube. The interactionof the shock wave with the open end and any subse-quent reflected waves are ignored. These assumptionsare discussed with respect to the results of numericalsimulations in the next section.

Since the expansion wave is steady, the total tem-perature is constant across it and we can estimatethe average temperature in the plenum: TC ≈ TtC =

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Fig. 6 Flow configuration used to model the fillingprocess in the case of subsonic flow at the valveplane.

TtV = Tt2. Hence the average temperature inside theplenum is equal to the total temperature downstreamof the inlet.

The conditions at the valve plane are determinedfrom the average plenum conditions as a function ofthe velocity uV , using the isentropic flow relationshipsthrough a steady expansion wave.

PC =Pt2 − m0uoV

A2+

m0RTC

A2uoV

(1 − uo

V2

2CP TC

)− 1γ−1

·[1 −

(1 − uo

V2

2CP TC

) γγ−1

]

(5)

The velocity at the valve plane is determined bymatching the flow in the plenum with the downstreamconditions in the detonation tube. The pressure ra-tio across the valve at opening determines the shockwave Mach number and the velocity at the valveplane, which is also the velocity of the contact surface.Matching conditions at the interface yields19

PC = P0

1 + 2γb

γb+1 (M2S − 1)[

1 − 2(γ−1)(γb+1)2

(cf

cC

)2

(MS − 1/MS)2] γb

γb−1

(6)The velocity at the valve plane is equal to the post-shock velocity in the burned gases

uoV

cf=

2γb + 1

(MS − 1

MS

)(7)

We solve for MS by equating Eqs. 5 and 6 and replac-ing uV by the expression of Eq. 7 in Eq. 5. Once MS isknown, all the other variables of the system are calcu-lated using the relationships across the shock and theexpansion wave.

Choked flow at the valve plane

When the pressure ratio across the valve exceeds acritical value (given by PC/PV = ((γ + 1)/2)

γγ−1 ), the

velocity at the valve plane, which is also the throatin our case, becomes sonic and the flow is choked.

The flow inside the plenum is then independent of thedownstream conditions and it can be decoupled fromthe flow in the detonation tube, making the systemeasier to solve. The flow configuration is shown inFig. 7. The velocity at the valve plane is equal to thespeed of sound uo

V = c∗. Using the relationships forchoked flow at the valve plane, it is possible to directlyestimate the average plenum pressure from Eq. 3:

PC = Pt2 − m0c∗

γA2

[γ + 1 −

(γ + 1

2

) γγ−1

](8)

The properties at the valve plane are given by the stan-dard isentropic relations and the sonic condition.

Fig. 7 Flow configuration used to model the fill-ing process in the case of choked flow at the valveplane.

The flow in the detonation tube is calculated froma pressure-velocity diagram by matching conditionsacross the interface and solving for the shock waveMach number19

PC

P0=

1 + 2γb

γb+1 (M2S − 1)[√

γ+12 − γ−1

γb+1cf

cC(MS − 1/MS)

] 2γγ−1

(9)

Discussion

The coupling of the flow in the plenum with the flowin the detonation tube results in average conditions inthe plenum that are different from the stagnation con-ditions downstream of the inlet. Although the averagestagnation temperature in the plenum is equal to thestagnation temperature downstream of the inlet, theaverage plenum pressure is lower than the inlet stag-nation pressure due to the unsteadiness of the flow inthe plenum. The entropy increase associated with theunsteady waves due to valve opening and closing re-sults in losses compared to the ideal steady flow case,for which total pressure is conserved. The ratio ofthe average plenum pressure to the stagnation pres-sure downstream of the inlet is shown as a function ofthe flight Mach number in Fig. 8. Values are given onlyfor M0 ≥ 1 because we made the assumption of chokedflow in the diffuser and constant inflow in the plenum.In the case considered in Fig. 8, PC is only 2.6% lower

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than Pt2 in the worst case. However, this value canbecome significant (greater than 10%) if the ratio ofthe plenum area to the inlet capture area A2/A0 isdecreased.

M0

PC/P

t2

1 2 3 40.95

0.96

0.97

0.98

0.99

1

Fig. 8 Ratio of the average pressure in the plenumto the total pressure downstream of the inlet as afunction of the flight Mach number. P0 = 0.265 bar,T0 = 223 K, A0 = 0.004 m2, A2 = 0.04 m2, AV = 0.006m2.

In our calculations, we assumed fixed valve area AV

and valve close time; other parameters such as valveopen time and detonation tube length are then deter-mined by the periodicity of the system. This meansthat the open time was set as a free parameter deter-mined by the average mass flow rate at the valve planefrom the mass conservation equation: τm0 = topenmo

V .This makes sense only if mo

V > m0. There is a criticalvalue of the flight Mach number at which mo

V is ex-ceeded by m0, corresponding to an infinite open time.This critical value depends on the ratio of the valvearea to the inlet capture area, and increases with de-creasing AV /A0. For realistic values of this parameter,this behavior is observed at subsonic flight conditions,at which the inlet flow would be strongly affected bythe unsteady flow in the plenum, making our model in-appropriate. In practice, when mo

V approaches m0, thesystem will adjust itself to these conditions by sendingpressure waves upstream in order to modify the inletflow and to decrease the inlet mass flow rate, keepingthe open time finite.

Comparison with numerical simulationsThe predictions of our modeling for the filling pro-

cess were compared with the results of the numericalsimulations with Amrita18 described previously. Anaverage velocity uV and pressure PV at the valve planewere calculated from the two-dimensional simulationsby spatially and temporally averaging these quantitiesalong the valve plane. An average filling velocity Ufill

was calculated as the average velocity of the inlet air-burned gases contact surface between the valve planeand the tube exit. Fig. 9 shows these quantities as afunction of the initial pressure ratio PR for both modeland simulations.

PR

u/c 0

5 10 15 200

0.5

1

1.5

2

2.5

3 uV - modeluV - AmritaUfill - modelUfill - Amrita

Fig. 9 Comparison of model predictions and nu-merical simulations with Amrita18 for the velocityat the valve plane and the average filling velocity.Tf/T0 = 7.69, γ = 1.4.

According to our one-dimensional model, the flowat the valve plane is expected to become choked abovea critical value of the pressure ratio equal to 3.19 inthe case considered. For pressure ratios below thisvalue, uV = Ufill. For higher values of PR, the flow ischoked at the valve plane and an unsteady expansionaccelerates the flow to supersonic downstream of thevalve plane, so that uV 6= Ufill. The velocity at thevalve plane is predicted by the speed of sound at thethroat c∗, while the filling velocity is predicted by thevelocity of the flow behind the shock wave. The twocurves in Fig. 9 correspond to these two cases.

The model predictions for Ufill are in reasonableagreement with the results of the numerical simula-tions, with a maximum deviation of 11%, respectively.The model predictions for uV are systematically higherthan the numerical results by up to 40% near choking.The discrepancies in the velocity at the valve plane canbe attributed to the influence of the starting transient,which is ignored by our model but is characterized by alower flow velocity than in the steady expansion case,and to the effect of two-dimensional waves generatedat valve opening, which strongly affect the flow at thevalve plane. The influence of the reflected waves gen-erated at the open end was investigated by conductingsimulations with an infinitely long tube but the differ-ence observed was minimal.

The values of the flow properties at the end of thefilling process are critical to the calculation of the deto-

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nation tube impulse. The model assumes that the flowin the detonation tube is uniform and moving at a ve-locity Ufill and at a pressure equal to the post-shockpressure. In order to test the validity of this approx-imation, we plotted the pressure and velocity profilealong the centerline from our numerical simulations inFig. 10. The pressure profile indicates that the flowinside the detonation tube is relatively uniform in thefirst half of the tube but quite non-uniform in the sec-ond half, where it includes a quasi-steady left-facingshock followed by a steady expansion near the openend. The spatially averaged model pressure at the endof the filling process is between 5.8% and 22.7% higherthan the average pressure in the tube from the numer-ical simulations for values of PR between 2 and 10.These numbers are helpful to understand the influenceof our approximations on the accuracy of our predic-tions and their potential consequence on performanceparameters.

Fig. 10 Pressure profile along the centerline fromthe numerical simulations with Amrita.18 Thevalve is located at an axial distance of 100 andthe detonation tube exit is located at 200. Thedashed line shows the value of the model predic-tion. PR = 8, Tf/T0 = 7.69, γ = 1.4.

Flow in the plenumThe unsteady pressure waves generated by valve

closing and opening strongly affect the overall flow inthe plenum, which, in turn, influences the flow throughthe inlet diffuser. Since conventional steady inlets maybe sensitive to downstream pressure fluctuations, it iscritical to be able to model the unsteady flow in theplenum.

We assume that the cycle time is much larger thanthe characteristic acoustic time in the plenum, so thatthe properties in the plenum can be modeled as spa-tially uniform. We assume that the incoming flow ischoked at the diffuser. The flow downstream of theinlet diffuser has a low Mach number, so that the flow

velocity in the cavity is small and we can neglect thekinetic energy term when calculating the total energyand the total enthalpy. We model the flow into thedetonation tube with a steady expansion so that thetotal enthalpy is conserved between the cavity and thevalve plane. In order to model the unsteady flow inthe plenum, we solve the unsteady mass and energyequations for the control volume VC (Fig. 4).

VCdρC

dt= m0 − mV (t) (10)

VCρC(t)dTC

dt= γm0Tt2 − [m0 + (γ − 1)mV (t)] TC(t)

(11)This system of equations has to be solved separately

for the closed part of the cycle [0, tclose] and for theopen part of the cycle [tclose, τ ]. Assuming small vari-ations in the properties inside the plenum and becausethe flow in the detonation tube during the open partof the cycle is generated by a steady expansion, weapproximate mV (t) as constant during the open partof the cycle, as in the previous section. For sufficientlylarge supersonic flight Mach numbers, the flow at thevalve plane during the filling process is choked, andthis approximation is justified. We are seeking thelimit cycle solution to this system of equations, whichcorresponds to periodic behavior.

The density varies linearly around its average valueρC . The limit cycle solution for the temperature has toobey the averaged energy equation, which states thatthe average temperature during the open part of thecycle is equal to Tt2. The solution for the tempera-ture evolution was obtained numerically and is shownin Fig. 11. The numerical solution quickly convergestowards the limit cycle. After 10 cycles, the averagevalue of the temperature in the plenum during a cyclewas found to be within 0.35% of TC . This means thatour averaged conservation equations for the plenumare consistent with the unsteady analysis of the flowin the plenum. The characteristic acoustic time in thiscase was estimated as V

1/3C /cC and verified to be much

lower than tclose.The amplitude of the fluctuations in density and

temperature in the cavity can be determined analyti-cally:

∆ρC

ρC=

m0tclose

2VCρC(12)

The amplitude of fluctuations in density, tempera-ture, and pressure for the case shown in Fig. 11 are15.2%, 6.1%, and 22.1%, respectively. The amplitudeof the fluctuations are all controlled by the same non-dimensional parameter m0tclose/(VCρC), which repre-sents the ratio of the amount of mass added to thesystem during the closed part of the cycle to the aver-age mass in the plenum. The amplitude of oscillationsis reduced for a lower mass flow rate (correspondingto a lower flight Mach number), a lower close time,

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t (s)

TC

(K)

0 0.05 0.1 0.15 0.2300

320

340

360

380

400

420

440

460

480

500

Fig. 11 Evolution of temperature in the plenum.A0 = 0.004 m2, A2 = 0.04 m2, AV = 0.006 m2, VC = 0.02m3, m0 = 0.9915 kg/s, PC = 1.885 bar, TC = 401.4 K,tclose = 0.01 s, topen = 0.007865 s.

a higher plenum volume, or a higher average plenumdensity.

The pressure oscillations in the cavity induce an un-steady behavior of the flow in the inlet diffuser. Thisbehavior has been previously studied in the context oflongitudinal pressure fluctuations generated by com-bustion instabilities in ramjets. The effect of pressureoscillations on the inlet may be regarded as an equiv-alent loss of pressure margin that might result in inletunstart. Higher frequency oscillations tend to stabilizethe diffuser shock.20–22 The frequency of oscillationsin the cavity is given by 1/τ , which means that re-ducing the cycle time is going to benefit inlet stability.For a given inlet configuration and flight condition,the amplitude of the pressure oscillations in the cav-ity decreases with decreasing close time and increasingplenum volume. This analysis gives some general ideasabout the unsteady response of the inlet diffuser.

Control volume analysisThe performance of an air-breathing PDE is deter-

mined by performing an unsteady open-system controlvolume analysis. The control volume Ω considered,displayed in Fig. 12, is stationary with respect to theengine. The engine is attached to the vehicle througha structural support. The control surface Σ passesthrough the engine valve plane, encompasses the det-onation tube, and extends far upstream of the inletplane. The side surfaces are parallel to the freestreamvelocity. We consider the equations for mass, energy,and momentum for this control volume.

Mass conservation

Writing the general unsteady conservation equationfor mass in the control volume Ω, bounded by the sur-

Fig. 12 Control volume considered for analysis ofsingle-tube PDE.

face Σ:

dMdt

+ mV (t) − ρ0u0AV + ms = 0 (13)

Since there is no average mass storage inside the en-gine, we can integrate over a cycle the mass conser-vation equation between the inlet plane and the valveplane: ∫ τ

0

mV (t)dt = τm0 (14)

This result can be used when integrating the massconservation equation for the control volume Ω overa cycle. With the assumption of no mass storage inthe engine through a cycle during steady flight, we cancalculate the mass flow of air through the side surfacesof Ω:

ms = ρ0u0(AV − A0) (15)

Momentum conservation

The forces on the control volume consist of thepressure forces and the reaction to the thrust carriedthrough the structural support. If we assume thatthe sides of the control volume are sufficiently distantfrom the engine, then the flow crosses the side con-trol surface with an essentially undisturbed velocitycomponent in the flight direction. Applying the mo-mentum equation in the flight direction, and using theresult of Eq. 15, we obtain an expression for the in-stantaneous thrust:23

F (t) =mV (t)uV (t) − m0u0 + AV (PV (t) − P0)

+d

dt

∫Ω

ρudV(16)

The last term represents the unsteady variation of mo-mentum inside the control volume.

Energy conservation

The general unsteady conservation equation for en-ergy in the control volume Ω is, in the absence of body

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forces or heat release in the control volume (heat is re-leased only in the detonation tube, which is outsideour control volume Ω):

d

dt

∫Ω

ρ(e+u2/2)dV +mV (t)htV (t)−m0ht0 = 0 (17)

after using the result of Eq. 15. Integrating over acycle, the first term vanishes because there is no av-erage energy storage in the control volume. UsingEq. 14, the energy equation results in: ho

tV = ht0.It states that the stagnation enthalpy of the flow hasto be conserved between the freestream and the valveplane during the open part of the cycle. This is satis-fied by our model, since the stagnation enthalpy of theplenum is equal to the freestream stagnation enthalpy,and the flow into the detonation tube during the openpart is generated by a steady expansion, which con-serves stagnation enthalpy. The energy release in thedetonation is implicitly considered in the calculationof the detonation tube impulse.

Thrust calculation

The average thrust is calculated by averaging Eq. 16over a complete cycle. The unsteady term can beintegrated and corresponds to the variation in totalmomentum in the control volume during a cycle, whichvanishes during steady flight, since the total momen-tum in the control volume has a periodic behavior.During the closed part of the cycle (from 0 to tclose),the contribution of the momentum at the valve planevanishes since the valve is closed. The pressure con-tribution corresponds to the conventional detonationtube impulse Idt generated by the detonation andblowdown processes∫ tclose

0

AV (PV (t) − P0)dt = Idt (18)

The momentum and pressure contributions of the det-onation tube during the open part of the cycle (fromtclose to τ) are calculated using the model estimatesfor velocity and pressure at the valve plane during theopen part of the cycle. Using Eq. 14, the pressure andmomentum contribution for the open part of the cycleis:∫ τ

tclose

(mV (t)uV (t)dt + AV (PV (t) − P0)) dt =

τm0uoV + AV (P o

V − P0)topen

(19)

Using Eqs. 18 and 19, the average thrust can be ex-pressed as follows

F =1τ

Idt + m0(uoV − u0) +

topen

τAV (P o

V − P0) (20)

The average thrust of an air-breathing PDE is deter-mined by the contributions of detonation tube impulse,ram momentum, and ram pressure in the engine.

Influence of the purging time

The purging time has a strong influence on theoverall engine thrust since the thrust is inversely pro-portional to the cycle time, and, by definition, τ =tclose + tpurge + tfill. Since topen is determined inour model by the condition for periodicity, increasingtpurge means decreasing tfill and decreasing the massof detonable gas in the detonation tube.

For a fixed open time, the purging time relates theincoming mass flow rate to the mixture mass detonatedin the chamber. We assume that the detonation tubehas a volume equal to the volume of the slug of com-bustible gas injecteda. Assuming ideal fuel-air mixingat constant pressure and temperature, the mass bal-ance in the detonation tube shows that:

V =(

1 + f

1 + π

)τm0

ρi(21)

where π = tpurge/tfill is defined as the purge coeffi-cient. It is critical to make the distinction betweenthe air mass flow rate m0 and the average detonatedmixture mass flow rate ρiV/τ . The average fuel massflow rate is given by:

mf =ρiV f

(1 + f)τ(22)

The fuel-based specific impulse is calculated with re-spect to the fuel mass flow rate as

ISPF =F

mfg

= ISPFdt +1 + π

fg

[uo

V − u0 +AV (P o

V − P0)mo

V

](23)

The fuel-based specific impulse is the sum of threeterms representing the contributions of the detonationprocess, the ram momentum, and the ram pressure inthe engine. The first term is always positive. Thesecond term is negative because of the flow losses as-sociated with decelerating the flow through the inletand re-accelerating it unsteadily during the filling pro-cess. The third term is positive because the air injectedduring the filling process is at higher pressure thanthe outside air. However, the sum of the last twoterms is negative and corresponds to a drag term dueto flow losses and unsteadiness. The specific impulsedecreases linearly with increasing purge coefficient.

Detonation tube impulseThe static impulse of a PDE is due to the detonation

process (first term on right-hand side of Eq. 23) andhas been measured1–5 for single-cycle operation and

aThis means the length of the detonation tube is being variedwith the operating conditions in this model.

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several models have been proposed.3,6, 7 However, inpractice, the flow downstream of the detonation wavein a moving engine is not going to be at rest because ofthe filling process. This is captured in multi-cycle ex-periments3,10 but the values obtained for the impulseare still well predicted by the single-cycle estimates7

because the filling occurs at low subsonic velocity inthese tests. During supersonic flight, the large pressureratio across the valve will generate high filling veloci-ties during the filling process, which can significantlyalter the flow field and the detonation/blowdown pro-cess. We need to include this effect in our model.In an idealized case, we assume that the detonationwave is initiated immediately after valve closing. Thedetonation wave catches up with the expansion wavegenerated by the valve closing and the situation beforethe wave exits the tube corresponds to a detonationwave propagating in a flow moving in the same direc-tion at the filling velocity.

The detonation wave is followed by an expansionwave, referred to as the Taylor wave, which bringsthe products back to rest near the closed end of thetube. In the moving flow case, the energy releaseacross the wave is identical to the no flow case andthe CJ pressure, temperature, density, and speed ofsound are unchanged. However, the wave is now mov-ing at a velocity UCJ + Ufill with respect to the tube.The properties upstream of the Taylor wave, near theclosed end, are modified compared to the no flow casebecause the flow behind the detonation wave has toundergo a stronger expansion. Using the method ofcharacteristics as described in Wintenberger et al.,7

we can obtain the speed of sound and the pressurebehind the Taylor wave:

c3 =γb + 1

2cCJ − γb − 1

2(UCJ + Ufill) (24)

P3 = PCJ

(c3

cCJ

) 2γbγb−1

(25)

The pressure behind the Taylor wave decreases as thefilling velocity increases due to the additional expan-sion required to bring the flow to rest at the wall. TheTaylor wave also occupies a larger region of the tubebehind the detonation in the moving flow case.

The detonation tube impulse is calculated as theintegral of the pressure trace at the valve plane (orthrust wall):

Idt =∫ tclose

0

AV (P3(t) − P0)dt (26)

Using dimensional analysis, we idealize the pressuretrace at the thrust wall and model the pressure traceintegral in a similar fashion as described in Winten-berger et al.7 The pressure history is modeled by aconstant pressure region followed by a decay due to

gas expansion out of the tube. The detonation tubeimpulse can then be expressed as:∫ τ

topen

(P3(t)−P0)dt = ∆P3

[L

UCJ + Ufill+ (α + β)

L

c3

](27)

where ∆P3 = P3 − P0, α is a non-dimensional pa-rameter corresponding to the time taken by the firstreflected characteristic at the open end to reach thethrust wall, and β is a non-dimensional parameter cor-responding to the pressure decay period.7 The firstpart of the term in square brackets in Eq. 27 is equalto the time taken by the detonation wave to propagateto the open end. As in the no flow case, it is possibleto derive a similarity solution for the reflection of thefirst characteristic at the open end and to analyticallycalculate α. The reader is referred to Wintenbergeret al.7 for the details of the derivation in the no flowcase. For the moving flow case, the value of α is:

α =c3

UCJ + Ufill·

2(

γb − 1γb + 1

(c3 − uCJ

cCJ+

2γb − 1

))− γb+12(γb−1)

− 1

(28)

The value of β was assumed to be independent of thefilling velocity and the same value as in Wintenbergeret al.7 was used: β = 0.53.

In order to validate the model for the thrust wallpressure integration (Eq. 27), the flow was simulatednumerically using Amrita.18 The simulation solved thenon-reactive Euler equations using a Kappa-MUSCL-HLLE solver in the two-dimensional (cylindrical sym-metry) computational domain consisting of a straighttube of length L closed at the left end and open toa half-space at the right end. The moving flow wasrepresented by an idealized inviscid straight pressure-matched jet profile at constant velocity Ufill as shownon Fig. 13. The modified Taylor wave similarity so-lution was used as an initial condition, assuming thedetonation has just reached the open end of the tubewhen the simulation is started. This solution was cal-culated using a one-γ model for detonations24 for anon-dimensional energy release q/RTi = 40 across thedetonation and γ = 1.2 for reactants and products.The corresponding CJ parameters are MCJ = 5.6 andPCJ/Pi = 17.5, values representative of stoichiometrichydrocarbon-air mixtures. The initial refilling pres-sure Pi ahead of the detonation wave was taken to beequal to the pressure P0 outside the detonation tube.The configuration adopted for the moving flow is a veryelementary representation of the flow at the end of thefilling process, which will, in reality, include vorticesassociated with the unsteady flow and the unstablejet shear layers. However, the analysis of the numer-ical simulations showed that the flow in the tube is

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one-dimensional except for within one to two tube di-ameters from the open end and mainly dictated by thegas dynamic processes at the tube exit plane. Since theexit flow is choked for most of the process, the influenceof our simplified jet profile on the thrust wall pressureintegration is minimal. Fig. 14 shows the comparisonof the non-dimensionalized thrust wall pressure inte-gral as a function of the filling Mach number with thepredictions of our model. The numerical pressure in-tegration was carried out for a time equal to 20tCJ .As the filling Mach number increases, the flow expan-sion through the Taylor wave is more severe and theplateau pressure behind the Taylor wave P3 decreases.Even though P3 is lower, the blowdown process is ac-celerated due to the presence of the initial movingflow. The overall result is that the detonation tubeimpulse decreases with increasing filling Mach num-ber, as shown in Fig. 14, mainly due to the strongerexpansion through the Taylor wave as Mfill increases.The model agrees reasonably well with the results ofthe numerical simulations. It generally overpredictsthe results of the numerical simulations by as much as25% at higher filling Mach numbers. The agreementis better at lower Mach numbers (within 11% error forMfill ≤ 2 and 4% for Mfill ≤ 1).

Fig. 13 Numerical Schlieren image of the initialconfiguration for the numerical simulations of thedetonation process with moving flow. The Taylorwave is visible behind the detonation front at thetube exit.

Performance calculations

Performance calculations are carried out for a single-tube air-breathing PDE at sea level for hydrogen andJP10 fuels.

Model input parameters

The input parameters for the performance modeldescribed in Eq. 23 consist of the engine geometry, thefreestream pressure and temperature, the flight Machnumber, the specific heat ratio of air, the fuel type andstoichiometry, the valve close time, and the purgingtime.

The total pressure loss across the inlet during super-sonic flight is modeled using the military specificationMIL-E-5008B,25 which specifies the total pressure ra-tio across the inlet as a function of the flight Mach

Mfill

∫(P3-

P0)

dtc

i/VP

i

0 1 2 30

1

2

3

4

5modelAmrita

Fig. 14 Non-dimensional detonation tube impulseas a function of the filling Mach number. Com-parison of model predictions based on Eq. 27 andresults of numerical simulations with Amrita.18

number, for M0 > 1:

Pt2

Pt0= 1 − 0.075(M0 − 1)1.35 (29)

The calculations of the initial conditions for the deto-nation require solving Eqs. 5 and 6 or 8 and 9, whichrequire the knowledge of the specific heat ratio γb andthe speed of sound cf in the burned gases at the end ofthe blowdown. γb and the CJ parameters are obtainedby carrying out detonation equilibrium computationsusing realistic thermochemistry.26 The speed of soundcf is calculated assuming that the flow is isentropicallyexpanded from the CJ pressure to atmospheric pres-sure. This entire process needs to be iterated sincethe CJ parameters are obtained from Pi and Ti, whichare determined from MS , which is itself a function ofγb and cf . The solution was found by iteration untilthe prescribed values of γb and cf matched the valuesobtained at the end of the equilibrium computations.

We estimate performance for an idealized configu-ration with the detonation tube completely filled withreactants before detonation initiation. Since the fillingconditions, in particular the filling velocity and the filltime, vary with flight conditions, the length and vol-ume of the detonation tube are also varied (Eq. 21).

Conditions inside the engine

The calculation of performance parameters first re-quires solving for the conditions inside the engine, inparticular those in the plenum and the filling con-ditions in the detonation tube, including the fillingvelocity, which determines the cycle frequency. Fig. 15shows the filling velocity and the velocity at thevalve plane for a PDE operating with stoichiometrichydrogen-air flying at an altitude of 10,000 m. In this

12 of 16

particular case, the flow at the valve plane is predictedto remain subsonic up to a flight Mach number of 1.36,above which it becomes choked. For M0 > 1.36, Ufill

exceeds uV because of the additional unsteady expan-sion generated at the valve plane. The filling velocityincreases with increasing flight Mach number becauseof the increased stagnation pressure in the plenum,which generates a stronger shock wave at valve open-ing.

M0

velo

city

(m/s

)

1 2 3 4 50

500

1000

1500 Ufill

uV

Fig. 15 Filling velocity and velocity at thevalve plane as a function of flight Mach numberfor single-tube PDE operating with stoichiomet-ric hydrogen-air. Z = 10, 000 m, A0 = 0.004 m2,A2 = 0.04 m2, AV = 0.006 m2, tclose = 0.005 s.

Fig. 16 shows the pressure non-dimensionalized withthe freestream stagnation pressure at various locationsinside the engine. The inlet stagnation pressure de-creases with increasing flight Mach number relative tothe freestream stagnation pressure due to the stagna-tion pressure losses across a supersonic inlet (Eq. 29).The average plenum pressure is slightly lower thanPt2 (see Fig. 8). The pressure at the valve planeis equal to the filling pressure until the valve planebecomes choked. At higher values of M0, the ad-ditional unsteady expansion lowers even further thefilling pressure compared to the freestream total pres-sure. An important point is that, even though theaverage plenum pressure strongly increases with flightMach number, the ratio of the filling pressure to thefreestream total pressure decreases sharply, due to thesubstantial values obtained for the filling velocity (seeFig. 15).

PDE performance parameters

Variation with flight Mach numberPerformance parameters are calculated for an

air-breathing PDE operating with stoichiometrichydrogen-air and JP10-air. The specific impulse isshown in Fig. 17 as a function of the flight Mach num-

M0

Pre

ssur

era

tio

1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Pt2/Pt0

PC/Pt0

PV/Pt0

Pi/Pt0

Fig. 16 Inlet stagnation pressure, plenum pres-sure, pressure at the valve plane, and filling pres-sure non-dimensionalized with freestream totalpressure as a function of flight Mach number. Sto-ichiometric hydrogen-air, Z = 10, 000 m, A0 = 0.004m2, A2 = 0.04 m2, AV = 0.006 m2.

ber for conditions at 10,000 m altitude. The resultsshown in Fig. 17 represent the maximum values pre-dicted by the model for a given engine geometry (nopurging). We restrict ourselves to performance calcu-lations for supersonic flight because of the assumptionsmade in the derivation of the model. A data point fromthe static multi-cycle experiments of Schauer et al.10

is given as a reference point for hydrogen in the staticcase. A data point for a ballistic pendulum experi-ment27 for stoichiometric JP10-air at 100 kPa and 330K is given as a reference for the static case. Single-cycle static impulse predictions7 are also shown forconditions corresponding to an altitude of 10,000 mfor both hydrogen and JP10. Even though the presentmodel assumptions do not apply for subsonic flight,the reference values for the static case (M0 = 0) ap-parently lie on or close to a linear extrapolation of theresults obtained for supersonic flight. Our single-tubePDE generates thrust up to a flight Mach number of4.2 for hydrogen and 4 for JP10.

The specific impulse decreases almost linearly withincreasing flight Mach number from a value at M0 = 1of about 3530 s for hydrogen and 1370 s for JP10.The detonation tube impulse decreases with increas-ing flight Mach number due to the increasing fillingvelocity (see Fig. 15), as observed previously (Fig. 14).For choked flow at the valve plane, the ram momentumterm decreases linearly with M0. If we neglect the out-side pressure P0, the ram pressure term is proportionalto

√Tt0, and, therefore, increases with M0. However,

as pointed out before, the sum of these two terms isnegative and corresponds to a drag term increasingwith increasing flight Mach number, caused by the

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M0

Spe

cific

impu

lse

(s)

0 1 2 3 4 50

1000

2000

3000

4000

5000

PDE - H2

ramjet - H 2

PDE - JP10ramjet - JP10Wu et al. - H 2

Schauer et al. - H 2

impulse model - H 2

CIT- JP10impulse model - JP10

Fig. 17 Specific impulse of a single-tube air-breathing PDE compared to the ramjet operat-ing with stoichiometric hydrogen-air and JP10-air.Z = 10, 000 m, A0 = 0.004 m2, A2 = 0.04 m2,AV = 0.006 m2, π = 0. Data from multi-cycle nu-merical simulations12 by Wu et al. for M0 = 2.1at 9,300 m altitude are shown. Experimental datafrom Schauer et al.10 and Caltech27 and impulsemodel predictions7 are also given as a reference forthe static case.

stagnation pressure loss through the inlet (Eq. 29).Fig. 17 also shows a data point from the numer-

ical simulations by Wu et al.,12 who calculated theperformance of an air-breathing PDE with a straightdetonation tube flying at M0 = 2.1 at an altitude of9.3 km and operating with stoichiometric hydrogen-air. The model prediction at the same conditions, 2286s, is within 1.8% of their baseline case (2328 s12).

Variation with altitudeCalculations show that the specific impulse de-

creases with decreasing altitude. For example, the spe-cific impulse at sea level is systematically lower thanthat at 10,000 m by 150-300 s. It is possible to show us-ing scaling arguments that the different components ofthe engine specific impulse are all independent of pres-sure, but vary with outside temperature. IncreasingT0 results in a stronger shock wave at valve opening,and, therefore, in a higher filling velocity, which re-sults in a decrease in detonation tube specific impulse.The decrease in ISPFdt is the main contribution to thedecrease in engine specific impulse with decreasing al-titude.

Variation with purge coefficientAs shown earlier (Eq. 23), the effect of purging is to

decrease the specific impulse of the PDE. Increasingthe purge coefficient results in an increase of the dragterm in the specific impulse equation and, therefore,in a decrease of the overall specific impulse. At given

flight conditions, the specific impulse decreases lin-early with increasing purge coefficient. Fig. 18 showsthat the reduction in performance due to an increasein purge coefficient increases with flight Mach num-ber, because of the increasing size of the drag termat higher flight Mach numbers. The purge coefficientis found to have a substantial effect on the thrust-producing range of an air-breathing PDE. Fig. 18shows that the maximum flight Mach number for ahydrogen-fueled PDE at an altitude of 10,000 m de-creases from 4.2 at π = 0 to 3.8 at π = 0.5 and 3.5 atπ = 1.

M0

I SP

F(s

)

1 2 3 4 50

500

1000

1500

2000

2500

3000

3500

4000π = 0π = 0.1π = 0.5π = 1

Fig. 18 Specific impulse of a single-tube PDEoperating with stoichiometric hydrogen-air as afunction of flight Mach number varying the purgecoefficient. Z = 10, 000 m, A0 = 0.004 m2, A2 = 0.04m2, AV = 0.006 m2.

Issues associated with JP10Recalculating the conditions inside the engine for

JP10, it is found that the temperature of the flow atthe valve plane exceeds the auto-ignition temperatureof JP10-air (518 K28) above M0 = 3. This means thatif the fuel injection system is located at the valve plane,pre-ignition of the JP10-air mixture is expected aboveMach 3 before the detonation can be initiated, result-ing in a significant decrease in detonation tube impulsedue to potential expulsion of unburned reactants outof the detonation tube5 and reduction in thrust sur-face as part of the combustion process will take placewhile the valve is open. The filling temperature Ti

remains below the auto-ignition temperature of JP10across the entire range of flight Mach numbers.

Another issue with the use of liquid hydrocarbonfuels is related to potential condensation of the fuelin the detonation tube due to the low filling temper-ature. For the case considered here with JP10, thefilling temperature remains under 300 K as long asM0 < 2.3. The fuel injected will vaporize completely

14 of 16

as long as its vapor pressure is high enough at the tem-perature considered. It is possible that not all the fuelcorresponding to stoichiometric quantity will be ableto vaporize, and the engine may have to be run at aleaner composition depending on the flight conditionsconsidered.

Comparison with conventional propulsion systems

The performance of a single-tube air-breathing PDEis compared with that of the ideal ramjet for flight con-ditions corresponding to 10,000 m altitude in Fig. 17.Performance was calculated for both engines for hydro-gen and JP10, assuming they operate at stoichiometry.The ideal ramjet performance was calculated followingthe ideal Brayton cycle, taking into account the stag-nation pressure loss across the inlet (taken equal tothat undergone by the PDE), realistic thermochem-istry,26 and assuming thermodynamic equilibrium atevery point in the nozzle. According to our perfor-mance predictions, the single-tube air-breathing PDEin the present configuration (straight detonation tube)has a higher specific impulse than the ideal ramjet forM0 < 1.35 for both hydrogen and JP10 fuels.

The results of our performance calculations showthat PDEs in the simple configuration considered(straight detonation tube) are not competitive withconventional steady propulsion systems at high super-sonic flight Mach numbers, because of the increasedtotal pressure loss across the inlet and the decreas-ing detonation tube impulse associated with the veryhigh filling velocities and correspondingly low pressureand density in the detonation tube prior to detonationinitiation. Adding a choked converging-diverging exitnozzle has been proposed by several researchers11,12 asa means to increase the chamber pressure and decreasethe effective filling velocity of the detonation tube.The strong sensitivity of the detonation tube impulseto the filling velocity suggests a potential for improvingperformance, provided that the filling velocity can bedecreased without excessive internal flow losses. Thenumerical simulations of Wu et al.12 support this idea,showing an increase in specific impulse of up to 45%with the addition of a converging-diverging nozzle.

Uncertainty analysisSince our performance model is based on many sim-

plifying assumptions, we need to estimate the effect ofthe uncertainty on the performance parameters. Un-fortunately, due to the complexity and the unsteadi-ness of the flow in a pulse detonation engine, whichinvolves moving parts and chemical kinetics, there isno existing standard to which our model can be com-pared. It is difficult to estimate the influence of ourassumptions unless a numerical simulation of the en-tire system is conducted. At present, only Wu et al.12

and Ma et al.13 have published such computations andalthough our work agrees with their results at a single

condition, this is far from conclusive validation of ourapproach.

We know from our numerical simulations of the fill-ing process the uncertainty of the model predictions ofsome of the parameters. We carried out best case andworst case scenario calculations to estimate the modeluncertainty for a stoichiometric hydrogen-air PDE fly-ing at 10,000 m with no purging. The result of thesecalculations is given in Fig. 19. The region of uncer-tainty is the grey shaded area around the predictedspecific impulse curve. As expected, the uncertaintymargin is quite large and increases with increasingflight Mach number, due to the growing uncertainty onthe detonation tube impulse. The uncertainty on thespecific impulse at M0 = 1 is ±9.9% and at M0 = 2,it is -36.5%/+12.7%. Since the predicted detonationtube impulse overpredicts the numerical values, themagnitude of the uncertainty in the worst case sce-nario is larger than that in the best case scenario.

Fig. 19 Uncertainty associated with the cal-culation of specific impulse for a stoichiometrichydrogen-air PDE. The uncertainty region is theshaded area. Z = 10, 000 m, A0 = 0.004 m2, A2 = 0.04m2, AV = 0.006 m2, π = 0.

ConclusionsWe have developed a simple model for predicting

the performance of a single-tube air-breathing pulsedetonation engine based on gas dynamics and controlvolume methods. The model offers the possibility toevaluate in a simple way the performance of a single-tube PDE consisting of a steady supersonic inlet, alarge plenum, and a straight detonation tube. Nu-merical simulations showed that the filling process ischaracterized by a shock wave generated at valve open-ing and propagating in the detonation tube and acombination of unsteady and steady expansions be-tween the plenum and the detonation tube. The flowin the plenum is coupled to the flow in the detonation

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tube. Due to the unsteadiness of the flow, the averagepressure in the plenum is lower than the stagnationpressure downstream of the inlet. The flow in theplenum oscillates due to the opening and closing of thevalve during a cycle. An unsteady open-system con-trol volume analysis was applied to the engine. Mass,momentum and energy equations were averaged over acycle with no storage terms (steady flight conditions).The thrust of the engine is found to be the sum ofthree terms representing the detonation tube impulse,the ram momentum, and the ram pressure. Our single-cycle impulse model7 was modified to take into accountthe effect of detonation propagation into a moving flowgenerated by the filling process. The detonation tubeimpulse is found to decrease sharply with increasingfilling velocity. Performance calculations were carriedout for a PDE operating with hydrogen-air and JP10-air at 10,000 m altitude. The engine specific impulse isfound to decrease quasi-linearly with increasing flightMach number, and single-tube PDEs are found to gen-erate thrust up to a flight Mach number of about4. Comparison with conventional propulsion systemsshowed that the single-tube PDE in the configurationstudied (straight detonation tube) operating with sto-ichiometric hydrogen- or JP10-air has a higher specificimpulse than the ramjet below a flight Mach numberof 1.35.

AcknowledgmentsThis work was supported by Stanford University

Contract PY-1905 under Dept. of Navy GrantNo. N00014-02-1-0589 “Pulse Detonation Engines:Initiation, Propagation, and Performance”.

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