Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

A Performance Map for Ideal Air Breathing

Pulse Detonation Engines

NASA/TM—2001-211085

July 2001

AIAA–2001–3465
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Prepared for the

37th Joint Propulsion Conference and Exhibit

cosponsored by the AIAA, ASME, SAE, and ASEE

Salt Lake City, Utah, July 8–11, 2001

Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

A Performance Map for Ideal Air Breathing

Pulse Detonation Engines

NASA/TM—2001-211085 AIAA–2001–3465
Available from

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Available electronically at http://gltrs.grc.nasa.gov/GLTRS
1 

American Institute of Aeronautics and Astronautics 

AIAA–2001–3465 

A PERFORMANCE MAP FOR IDEAL AIR BREATHING PULSE DETONATION ENGINES

Daniel E. Paxson†

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio, USA 

Phone: 216–433–8334, Fax: 216–433–8643 

Email: Daniel.E.Paxson@grc.nasa.gov 

Abstract

The performance of an ideal, air breathing Pulse 

Detonation Engine is described in a manner that is 

useful for application studies (e.g. as a stand-alone, 

propulsion system, in combined cycles, or in hybrid 

turbomachinery cycles). It is shown that the Pulse 

Detonation Engine may be characterized by an averaged 

total pressure ratio, which is a unique function of the 

inlet temperature, the fraction of the inlet flow 

containing a reacting mixture, and the stoichiometry of 

the mixture. The inlet temperature and stoichiometry 

(equivalence ratio) may in turn be combined to form a 

non-dimensional heat addition parameter. For each 

value of this parameter, the average total enthalpy ratio 

and total pressure ratio across the device are functions 

of only the reactant fill fraction. Performance over the 

entire operating envelope can thus be presented on a 

single plot of total pressure ratio versus total enthalpy 

ratio for families of the heat addition parameter. Total 

pressure ratios are derived from thrust calculations 

obtained from an experimentally validated, reactive 

Euler code capable of computing complete Pulse 

Detonation Engine limit cycles. Results are presented 

which demonstrate the utility of the described method 

for assessing performance of the Pulse Detonation 

Engine in several potential applications. Limitations and 

assumptions of the analysis are discussed. Details of the 

particular detonative cycle used for the computations 

are described. 

Introduction

Pulse Detonation Engines (PDE’s) are receiving much 

attention as a potential means for propulsion or as part 

of a propulsion system for future aerospace vehicles. 

One reason for this seems to be the promise of high 

efficiency due to the fact that PDE cycles operate with 

near-constant volume combustion. Furthermore, it is 

often argued that PDE’s have fewer parts and are much 

less mechanically complex than other thrust producing 

systems (e.g., gas turbines). This, in turn makes them 

considerably less expensive. As yet however, there is 

very little experimental performance information in the 

literature to support the claims. What does exist has 

often been obtained from experiments that are not 

optimized for performance, or that are not self 

aspirating (i.e., air is forced through the inlet).1

Performance predictions based on such experimental 

results are not necessarily realistic. The analytical work 

is often greatly simplified and, as a result, somewhat 

optimistic.2,3 Furthermore, the analytical work tends to 

consider the PDE primarily as a stand-alone device, 

whereas many potential applications are envisioned in 

which it is a component in the propulsion system. 

Analysis of such systems requires a performance 

description of the PDE that can be combined with the 

additional components in a straightforward manner. 

Even in stand-alone applications, the PDE is typically 

seen as coupled to a nozzle (and an inlet), which is 

thermally limited. Analyses that neglect thermal limits 

and simply calculate say, specific thrust and impulse for 

a stoichiometric fuel/air mixture, are not realistic. 

This paper introduces a method for describing the 

performance of a PDE cycle that can be readily applied 

to different propulsion systems. The method utilizes 

data from an experimentally validated, quasi-onedimensional, time-accurate, reactive, CFD Euler 

solver4–6

 to provide results that are idealized due to the 

assumptions of the code, but realistic in that they 

capture the complex gas dynamics of the cycle.*

Furthermore, the use of a CFD code allows 

straightforward examination of many cycle 

modifications that can affect performance. For example, 

both partial filling and the addition of a nozzle at the 

 

*

Validation results from the code may be found in Appendix 1 of this paper. 

† Senior Member, AIAA. 

Copyright © 2001 by the American Institute of Aeronautics and

Astronautics, Inc. No copyright is asserted in the United States under

Title 17, U.S. Code. The U.S. Government has a royalty-free license to

exercise all rights under the copyright claimed herein for Governmental

Purposes. All other rights are reserved by the copyright owner. 
2 

American Institute of Aeronautics and Astronautics 

exhaust end of the PDE tube will be discussed in this 

paper. Additionally, a code can readily account for the 

fact that not all particles passing through a PDE 

necessarily go through the same thermodynamic cycle. 

A performance map is obtained from integrated 

quantities computed by the code. These are then mass 

or time-averaged to provide steady-state results. It will 

be shown that the PDE performance may be represented 

as a total pressure ratio that is a unique function of the 

total enthalpy ratio and a non-dimensional heat addition 

parameter. 

The results presented herein are primarily for selfaspirated PDE cycles with constant cross section tubes, 

and with matched inlet total and exhaust static 

pressures; however, the method is not limited to this 

configuration. Similar maps can be generated for 

differing geometries and boundary conditions. 

In the following sections, the method will be described 

and the utility of the resulting performance map will be 

demonstrated through the presentation of several 

simplified PDE system studies. The performance map 

description will be preceded by a listing of the 

assumptions and simplifications, a description of the 

particular PDE cycle under consideration, and a brief 

description of the Euler code with which performance 

data is generated. It is assumed that the reader is 

familiar with PDE cycles and numerical methods for 

computing them. 

Assumptions and Simplifications

The assumptions and simplifications used in the code 

and analysis are listed below. 

• The PDE flow is quasi-one-dimensional, inviscid and 

adiabatic. 

• The PDE flow contains a combination of only 

reactant or product. 

• Both reactant and product are calorically perfect 

gases with the same properties. 

• All reactant (air/fuel combination) entering the PDE 

is at the prescribed mixture ratio. 

• Valves on the PDE open and close instantaneously 

with no losses. 

• Detonation occurs nearly instantaneously once 

reaction commences and is assured regardless of the 

mixture composition. 

• The exhaust static pressure is identical to the inlet 

total pressure. Preliminary numerical simulations 

have shown that this boundary condition yields the 

highest performance. 

Code Description

The numerical code used to generate data has been 

described in detail elsewhere in the literature.4–6

 As 

such, only a brief description will be provided here, and 

only those aspects of the code relevant to the current 

investigation are presented. 

Governing Equations

Under the assumptions listed above, the governing 

differential equations for the PDE tube may be written 

in non-dimensional form as 

S(w, x) x

F(w)

t

w = ∂

∂

+

∂

∂ (1) 

where 

































ρ















 + ρ ρ +

γ γ

ρ

ρ

=

zA

q z A

2

u

( -1)

p

uA

A

w 0

2 (2) 

and 





































ρ















 + ρ ρ +

γ −















 + ρ

γ

ρ

=

uzA

q z A

2

u

1

p u

u A p

uA

F

0

2

2

 (3) 

The distance, x has been normalized by the combustor 

length, L. The time, t has been normalized by the 

characteristic wave transit time, L/a*

, where a* is the 

speed of sound at a chosen reference state. The 

pressure, p and density, ρ have been normalized by their 

respective reference values and the axial velocity, u has 

been normalized by a*

. The mass fraction of reactant is 

z and has a value between 0 and 1. Note that z can be 

related to, but is not the same as, actual fuel fraction. 

The ratio of specific heats is denoted by γ. For all of the 

results to be shown, the reference states are the inlet 

total conditions. The cross sectional area is A. The nondimensional heat of reaction q0 is defined as: 

( ) *

g

f

0

R T f

a 1

h

q

+ γ

∆ = (4) 

where, ∆hf is the fuel heat of reaction, a/f is the air to 

fuel ratio, Rg is the real gas constant, and T*

 is the 

reference temperature. 
3 

American Institute of Aeronautics and Astronautics 

For this non-dimensionalized form of the equations, the 

equation of state is written 

p = ρT (5) 

The speed of sound is simply √T. 

The non-dimensional source vector is written as: 

 





























−

= γ

RA

0

dx

p dA

0

S (6) (w, x)

The reaction rate, R of Eq. (6) has the form: 

( )













<

− > = ρ i ign

ign i i ign

0 0 ;T T

1 T T ;T T

R (7) K z

where K0 is the reaction rate constant (normalized by 

the reference wave transit time), and Tign is the specified 

ignition temperature. 

Numerical Integration

Equation (1) is integrated numerically with a secondorder accurate, Lax-Wendroff based scheme, utilizing 

Roe’s approximate Riemann solver to obtain flux 

estimates at the cell faces. The scheme has excellent 

shock capturing capability and has been tested 

extensively with detonative calculations. Detonation 

properties such as the wave speed, and post-detonation 

temperature and pressure are correctly computed. 

Because of the TVD properties of the scheme and the 

simplicity of the combustion model, it is found that the 

peak pressure (the von Neumann spike) is somewhat 

under-predicted; however, this is of little consequence 

in the work here. 

Boundary Conditions

The code has robust characteristic boundary condition 

routines such that subsonic, sonic, or supersonic inflow 

or outflow can occur at either end of the PDE tube. The 

routines contain logic that determines appropriate 

boundary conditions based on the current state of the 

interior cell and the imposed external pressure. Thus for 

example, the common situation in the PDE where the 

detonation wave, upon reaching the rear of the tube 

gives rise to supersonic, sonic, then subsonic flow is 

well behaved in the present code. Additionally, walltype conditions can be imposed at arbitrary times so that 

the valving process can be simulated. Application of 

boundary conditions in general are discussed in Ref. 7. 

A discussion of issues specific to PDE’s may be found 

in Ref. 8. 

Cycle Description

All of the cycles computed in this study share some 

common features. Each, as mentioned previously, has 

the imposed exhaust pressure equal to the inlet total 

pressure. This does not necessarily mean that the 

computed exit pressure is the same as that imposed. For 

sonic and supersonic flows, the exit pressure will be 

determined from the interior of the computing domain. 

Each of the cycles is a limit cycle. That is, the code has 

been run until the developed wave cycle repeats itself 

indefinitely, and the integrated flux of mass into the 

device over one cycle, matches the integrated flux of 

mass out. Unless otherwise stated, each cycle 

completely empties and fills the tube once per cycle. 

There is no so-called partial filling. Numerical 

experiments have shown (and will be presented 

subsequently) that there is little thermodynamic 

advantage to a partial-fill cycle. The frequency 

increases, of course, but the fraction of the cycle that 

generates thrust remains approximately the same. 

Furthermore, partial filling leads to a situation where a 

significant fraction of the tube wall is constantly 

exposed to hot combustion gases (it is never washed by 

the cool unburned flow), which, although not dealt with 

in this paper, may not be practical when heat loads are 

considered. Each cycle accomplishes the filling and 

emptying process in the least amount of time possible. 

Thus, the inlet opens precisely when the internal 

pressure in the tube drops below the inlet total pressure, 

and closes precisely when the volume required to fill 

the tube has entered. Detonation is initiated immediately 

after the inlet is closed. The result is a cycle that occurs 

at the highest possible frequency. Detonation is initiated 

either by rapidly adding heat to the first numerical cell 

(via a source term) until Eq. (7) is satisfied, or by 

imposing a very brief (i.e. negligible mass and 

momentum flux) high pressure, high temperature 

boundary condition at the inlet. Either method yields 

similar results. 

In order to illustrate the features described above, an 

example cycle is shown in Fig. 1. This figure shows 

contours of non-dimensional pressure, temperature, 

Mach number, and Reactant Fraction over the course of 

one PDE cycle. Thus, the horizontal dimension 

represents distance along the PDE tube. The vertical 

dimension represents time. The colors represent the 

values of the plotted quantities. Pressure and 

temperature are shown on a logarithmic scale in order to 

highlight details of the flowfield. Also shown in the 

figure, to the left of each contour are the highest and 

lowest value of the plotted quantity found in the entire 
4 

American Institute of Aeronautics and Astronautics 

x-t space. Black rectangular regions on the left of each 

contour show the portion of the cycle over which the 

inlet was closed (i.e., a wall was present). The particular 

cycle shown is one for which the value of q0=21.4, 

representative of a stoichiometric fuel/air ratio, 

hydrocarbon fueled PDE flying Mach 1.5 at an altitude 

of approximately 30,00 ft. (i.e., inlet total temperature 

approximately 600 R, ∆hf=19,000 BTU/lbm, a/f=15.5). 

The ratio of reactant flow to total flow through the inlet 

is 0.49. That is, approximately half of the PDE tube is 

filled with a detonable mixture each cycle and the other 

half is filled with pure air. The ratio of specific heats 

used for the calculation was γ=1.3. 

The path of the gas can be seen clearly in Fig. 1 by 

examining the temperature contour and the reactant 

fraction contour. The detonation path is visible in all of 

the contours, as is the transmitted shock that results 

when the detonation hits the unfueled region. Other 

features of note include the rather broad exit Mach 

number range over the course of the cycle and the 

surprisingly strong left-running shock originating on the 

right hand end after flow across the expansion fan has 

become subsonic. 

For clarification purposes, it is noted that if the cycle 

shown in Fig. 1 was a partial fill cycle, the interface 

between hot reacted gases and cool unreacted gases (or 

pure air) would not extend completely across the tube as 

shown in the temperature contour. This results in a 

portion of the tube still containing hot gas at the 

detonative portion of the cycle begins. The process is 

shown in Fig. 2 which shows a contour of the 

temperature for a partial-fill cycle with q0=28.2. The 

ratio of reactant flow to total flow through the inlet is 

0.48. The fraction of the tube filled with either reactant 

or air is 0.80. 

Performance Map Description

Consider the ideal PDE cycle described above with 

ambient boundary conditions. The mass averaged total 

enthalpy of the exhaust flow may be written as 

∫

∫

ρ

ρ

= cycle

cycle

t

0

t

0

e

( u)dt

(H u)dt

H (8) 

where

2

u H h

2

= + is the total enthalpy, and tcycle is the 

(non-dimensional) time period of one cycle. The inlet 

total enthalpy is assumed known. 

Integration of the mass energy, and species portion of 

Eq. (1) over one cycle, along with the stipulation that 

the cycle is a limit cycle will show that the ratio of mass 

averaged total exhaust enthalpy to inlet total enthalpy, 

HR, may be written as 

 















 = + γ − −

i

air

0

i

e

m

m 1 q ( 1) 1

H

H



 (9) 

Fig. 1 Contours of Log(pressure), Log(temperature), Mach number, and detonable 

mixture fraction for one ideal PDE cycle with q0=21.4, at ambient inlet conditions, 

and a purge fraction mreactant/mtotal of 0.51. The non-dimensional time spanned is 

3.74. The ratio of specific heats, γ=1.3. 

time 
5 

American Institute of Aeronautics and Astronautics 

where the ratio 

i

air

m

m



 , hereafter called the purge fraction 

is defined as 

( u )dt

( u )dt

m

m

cycle

cycle

t

0

i i

t

0

air air

i

air

∫

∫

ρ

ρ

= 

 (10) 

Here, the subscript air refers to the portion of the inlet 

flow which is pure air, and the subscript i refers to the 

total inlet flow of air or reactant (air/fuel mixture). 

Thus, the total enthalpy ratio is a function of the 

particular fuel, the ambient enthalpy or temperature, the 

stoichiometry and the purge fraction. 

The time averaged thrust of a given cycle may be 

calculated as:9

















γ = − ′ γ i

oi e

cycle

e

0i i A

p A

t

mom

p A

F (11) 

where 

∫ 





























 + ρ

γ =

cycle t

i

2 e

e e

e

e dt

A

A

u

p mom (12) 

The subscript e refers to the exit plane. The subscript i 

refers to the inlet. The (dimensional) inlet total pressure 

is p0i ′ . Note that the thrust calculated in this manner is 

greater than that which would be calculated using a 

thrust wall type integration10 such as 

∫∫

∫

−

= − + ′

dt

A

dA (p p )

t

1

(p p )dt

t

1

p A

F

i

wall 0i

cycle

t

wall 0i

0i i cycle

closed

 (13) 

where the subscript wall refers to the thrust wall of the 

PDE and the subscript closed refers to the time period 

during which the inlet end of the PDE is closed. 

The average exhaust total pressure, p for the device is 0e

then defined as the total pressure, at the total enthalpy 

defined by Eq. (9) that, when expanded through an ideal 

nozzle to the ambient pressure p0i ′ yields the same 

thrust as that calculated from Eq. (11). Thus, an average 

exit velocity u may be defined where: e

∫ρ 













 = ′ γ

cycle t

e e

i

e

cycle

e

0i i

u dt

A

A

t

u

p A

F (14) 

This may be combined with Eq. (11) to yield 

e

i

cycle oi e

e

e mf

A

t p A

mom

u

















γ −

= (15) 

where 

∫ρ 













 =

cycle t

e e

i

e

e u dt

A

A mf (16) 

With u known, and the average total enthalpy known e

from Eq. (9), the average total pressure may be obtained as 

1

e

2

e

0e 0i

2H

u 1

1

p p

γ−

γ





























−

= (17) 

 

time

Log(T)

X/L

1.083

-0.062

Fig. 2 Contour of Log(temperature) for one ideal 

partial-fill PDE cycle with q0=28.2, at ambient 

inlet conditions, and a purge fraction mreactant/mtotal

of 0.51. The non-dimensional time spanned is 3.27. 

The ratio of specific heats, γ=1.3. The fraction of

the tube filled is 0.80. 
6 

American Institute of Aeronautics and Astronautics 

With the simplifications and cycle stipulations 

described earlier it can be seen from Eqns (9) and (17) 

that the ratio PR= p is uniquely defined by q 0e p0i 0 and 

the purge fraction. Figure 3 shows PR as a function of 

the enthalpy ratio (Eq. 9) for several families of the heat 

of reaction parameter q0. For each value of q0 the 

enthalpy ratio was varied by changing the purge 

fraction. The largest value of enthalpy ratio for each 

curve represents the minimum purge possible with the 

code (in order to avoid auto-ignition) of approximately 

2%. The minimum enthalpy ratio represents an 

arbitrarily chosen maximum purge fraction of 

approximately 70%, except for the q0=7.8 case where it 

was 50%. All of the calculations were made using a 

PDE tube of uniform cross section. The ratio of specific 

heats for all calculations was γ=1.3. The number of 

numerical cells used in all of the calculations to be 

shown was 200. It can be seen that for a specified 

temperature ratio, the higher the purge fraction, the 

better the performance. Also shown in the figure are 

results from several cycles computed with constant 

q0=28.2, but with fill fractions other than 1 (i.e., partial 

fill cycles). The fill fraction is defined as the fraction 

(or percentage) of the tube filled with either air or 

detonable mixture each cycle. The points with fill 

fractions of 90, 77, and 64% were computed using 

minimum purge fractions. The point with a fill fraction 

of 80% was computed for a purge fraction of 51%. It 

can be seen that the partial fill cycles, for the most part, 

offer no performance enhancement. The one exception 

in the computed data occurs when the fill fraction is 

90%. The reason for this appears to be related to the 

timing of certain waves in the system. For this particular 

cycle a shock wave impinges on the thrust wall at 

precisely the moment that the detonation is initiated. 

More investigation is needed with regard to such tuning 

of the wave system; however, the gains, if any, appear 

quite modest. 

Figure 3 also shows two cycles computed with constant 

q0=28.2, but with reduced exit pressure. The exit 

pressures used (relative to the inlet total pressure) were 

0.80 and 0.55 as shown in the figure. It is noted that 

when calculating total pressures from Eqns. (15) and 

(17) the back pressure values were used in place of the 

term p0i. It can be seen that reducing the back pressure 

reduces overall pressure ratio (and thus performance) 

for a given enthalpy ratio. This is the reason, as stated 

earlier, that PDE cycles with exit pressure equal to inlet 

total pressure were used in the present study. 

Preliminary Geometry Variations

Because the performance data for the map of Fig. 3 

comes from a Q-1-D code, it is a relatively easy matter 

to look at geometric variations in the PDE design. For 

example, examination of Fig. 1 reveals that a substantial 

portion of the exit flow is sonic or supersonic. This 

suggests that the flow cannot fully expand within the 

tube, leading to reduced performance. An intuitive 

solution may be to provide a ‘nozzle’ section at the exit 

end of the PDE that (from a steady state perspective) 

would allow full expansion. Figure 4 shows the results 

of such an exercise. Computations were performed for 

three different geometries, all using minimum purge, 

and all with q0=21.4. The geometries are shown in 

Fig. 4. All three tubes had uniform cross sectional area 

from 0[/ )URP WKHUH WR [/  WKH\ YDULHG

smoothly from Ae/Ai=1.0 to Ae/Ai=0.75, 1.5, and 2.0. 

For the geometries examined, there does not appear to 

be any performance enhancement. These results appear 

1.0

1.5

2.0

2.5

3.0

3.5

13579

Total Enthalpy Ratio

Total Pressure Ratio, PR

q0=28.2

q0=21.4

q0=14.6

q0=7.8

q0=28.2, fill=90%

q0=28.2, fill=77%

q0=28.2, fill=80%

q0=28.2, fill=64%

pe=0.80

pe=0.55

Fig. 3 Average total pressure ratio as a function of

total enthalpy ratio for several families of q0. 

1.5

2.0

2.5

3.0

345678

Total Enthalpy Ratio

Total Pressure Ratio, PR

straight

Ae /Ai =0.75

Ae /Ai =1.5

Ae /Ai =2.0

Fig. 4 Average total pressure ratio as a function of

total enthalpy ratio for several ideal PDE tube 

geometries. In each case q0=21.4, the purge 

fraction is a minimum, and γ=1.3. 
7 

American Institute of Aeronautics and Astronautics 

consistent with similar computations made in Ref. 11 

for a rocket based PDE. They suggest that sonic or 

slightly supersonic exit flow is a fundamental attribute 

of PDE cycles and that its effects should be included in 

any idealized performance estimates. Clearly, a more 

thorough examination is needed in order to draw 

conclusive results; however, it can be seen that the use 

of a CFD-based model for performance evaluation, as 

suggested in this paper is flexible, and of great utility. 

Application

With the performance of the PDE in the form described 

above, analysis for a variety of applications is readily 

possible. 

Simple Brayton Cycle Comparison

The PDE cycle is often compared with the Brayton 

cycle, the latter being representative of conventional 

turbomachinery-based propulsion systems. When this 

comparison is made however, it is not always clear that 

the parameters held in common are meaningful. What, 

for example is the equivalent state in a PDE cycle to the 

Compressor discharge of a Brayton cycle? Figure 5 

shows a comparison in terms that are unambiguous. The 

performance of each cycle is plotted as PR versus HR 

as in Figs. 3 and 4. For the PDE cycle, all of the 

computed data has been plotted along with a simple 

curve fit defined by 

















γ−

γ

= 1

120.

PR (18) HR

This curve-fit further simplifies the computed results by 

assuming that, for the most part, varying HR by either 

q0 or purge fraction yields equivalent results. It is noted 

that the form of Eq. (18) has no theoretical basis and 

has not been tested for other values of γ. It is intended 

purely as a convenient fit for the data at hand. 

Also shown in Fig. 5 are the results from a simple 

Brayton cycle calculation12 with compressor and turbine 

adiabatic efficiencies set to 0.85 and 0.90 respectively. 

The Brayton cycle results are shown for families of 

T4/T1, which is the ratio of turbine inlet temperature to 

compressor inlet temperature. For each family, the 

compressor pressure ratio πc has been varied from 4 to 

30, which is representative of modern turbomachinery. 

Since cycle efficiency increases monotonically with PR, 

it is clear that, within the temperature range of 

conventional turbomachinery shown, the Brayton cycle 

is more efficient than the PDE cycle as computed in this 

paper. To achieve higher temperature ratios, a Brayton 

cycle with an afterburning process would be required. 

This is discussed in the next example. 

PDE With Inlet and Nozzle

As a second example of the application of the 

performance methodology, consider a PDE (or series of 

PDE tubes) with only an ideal inlet and nozzle fore and 

aft. Such a configuration may be envisioned for a high 

speed, Mach 0–5 aircraft. Standard thermodynamic 

analysis12 and Fig. 3, or more easily, Eq. (18), may be 

used to calculate the specific thrust and specific impulse 

over the Mach number regime. The results are shown in 

Figs 6 and 7. A Mach number dependent static inlet 

temperature profile was used for these calculations in 

order to account for altitude effects. The assumed 

profile was 520 R at static conditions, 400 R for 

Mach 2.0 and above, and a linear variation between 

these values. The nozzle inlet total temperature was 

chosen to be 4200 R. The ratio of specific heats 

was 1.3. For Fig. 6 the fuel heating value was 

19000 BTU/lbm. 

Also shown in the Figs. 6 and 7 are the performance 

results for a ramjet and for afterburning turbojets with 

compressor pressure ratios of 30 and 4. For the turbojet 

calculations the compressor and turbine adiabatic 

efficiencies were again 0.85 and 0.90 respectively. The 

combustor and afterburner were assumed loss free 

(i.e., constant total pressure). The turbine inlet 

temperature was 3000 R. It can be seen that the ideal 

PDE performance is fairly consistent over the Mach 

number regime and that it is comparable with a nonideal, afterburning turbojet having a compressor 

pressure ratio of 4.0. For turbojets of a more realistic 

pressure ratio, the PDE shows significantly less specific 

thrust and specific impulse. For the Ramjet, the 

performance advantages of a PDE are clear at Mach 

numbers below 3.0. Beyond this, there does not appear 

1.0

2.0

3.0

4.0

5.0

6.0

7.0

13579

Total Enthalpy Ratio

Total Pressure Ratio, PR

PDE Calculations

fit

Brayton-T4/T1=5

T4/T1=4

T4/T1=3

πc=30

πc=4

ηc=0.85

ηt

=0.90

















γ−

γ

= 1

.120

PR HR

Fig. 5 Comparison of average total pressure ratio 

as a function of total enthalpy ratio for PDE and 

Brayton cycles. The ratio of specific heats, γ=1.3. 
8 

American Institute of Aeronautics and Astronautics 

to be significant benefit. These PDE performance results 

are below what is typically listed in the literature2,3,13

however, because they represent computed results, from a 

validated code, it can be argued that they are more 

representative of an ‘idealization’ in the sense of being as 

good as can be expected. 

PDE Afterburner

As a final example of the utility of the above 

performance mapping methodology, consider a PDE as 

a replacement for a turbojet afterburner. This concept 

holds promise since the total pressure rises across a 

PDE and falls across a conventional afterburner. 

Therefore, a PDE afterburner should yield improved 

performance. With Eqn. (18) in hand, the results are 

obtained in much the same manner as those for the 

turbojet of the previous example. Figures 8 and 9 show 

the specific thrust and specific impulse over the same 

Mach number regime as Figs. 6 and 7. The same 

temperature limits and efficiencies were imposed on the 

turbomachinery and nozzle. A compressor pressure 

ratio of 30 was used. The PDE afterburner, as expected 

shows improvement in terms of increased specific thrust 

and impulse over the conventional afterburner. The 

benefits; however, are surprisingly modest. The reason 

for this is the relatively low enthalpy ratio occurring 

across the afterburner (for these calculations it was 

between 1.8 and 2.5). This, in turn, leads to a lower 

pressure ratio. It is interesting to note that even with the 

nozzle temperature restriction removed, the incoming 

temperature is relatively high. From Eqn. (4) this yields, 

even for stoichiometric mixtures, very low values of q0. 

Low values of q0 in turn lead to low enthalpy and 

pressure ratios as shown in Fig. 3. 

Discussion

Other PDE applications may be examined in a 

straightforward manner using the map of Fig. 3. 

Examples would include gas turbine topping cycles, 

bypass duct afterburners, even ejector based cycles 

(if some assumptions are made regarding the work 

transfer process). 

Of the applications that have been examined, the 

process has been made easier through the use of 

Eq. (18); however, it should be kept in mind that for a 

given application, not all values of q0 (and therefore 

enthalpy ratio are possible). This was pointed out in the 

afterburning PDE example but may appear elsewhere as 

well. For the PDE with inlet and nozzle, the inlet 

temperature at Mach 5 is 1900 R. Because of the 

4200 R thermal limit placed on the nozzle, the 

maximum enthalpy ratio was 2.21. Using a fuel heating 

value of 19,000 BTU/lbm and a stoichiometric air/fuel 

ratio of 15.5 however, would lead to a value of q0= 6.8. 

Examination of Fig. 3 shows that this heating value 

yields an enthalpy ratio in the vicinity of 2.5–3. Thus, 

the nozzle temperature restriction need not be in place, 

because the enthalpy ratio is already limited. 

Similar limitations would apply in a topping cycle 

application. For example, a small gas turbine with a 

pressure ratio of 8 has a discharge temperature at sea 

level of approximately 1000 R. If the maximum turbine 

0

20

40

60

80

100

120

140

160

012345

Flight Mach Number

Specific Thrust (lbf-s/lbm)

PDE

Ramjet

Turbojet w/ AB

Turbojet w/ AB

πc=30

πc=4

Fig. 6 Specific thrust as a function of Mach 

Number for PDE, Ramjet and Turbojet with 

Afterburner. For all systems γ=1.3, inlet and nozzle 

are ideal, and the nozzle inlet temperature is 4200 

R. For the Turbojet, adiabatic compressor and 

turbine efficiencies are 0.85 and 0.90 respectively. 

The turbine inlet temperature is 3000 R. 

0

500

1000

1500

2000

2500

3000

012345

Flight Mach Number

Specific Impulse (sec.)

PDE

Ramjet

Turbojet w/ AB

Turbojet w/ AB

πc=30

πc=4

Fig. 7 Specific Impulse as a function of Mach 

Number for PDE, Ramjet and Turbojet with 

Afterburner. For all systems γ=1.3, inlet and nozzle 

are ideal, and the nozzle inlet temperature is 4200 

R. For the Turbojet, adiabatic compressor and 

turbine efficiencies are 0.85 and 0.90 respectively. 

The turbine inlet temperature is 3000 R. The fuel 

heating value is 19,000 BTU/lbm. 
9 

American Institute of Aeronautics and Astronautics 

inlet temperature is 2400 R then the enthalpy ratio is 

2.4. This is achievable from Fig. 3 using some purge 

and/or a lean mixture. Alternately, the PDE may be 

considered as the primary source for an ejector, such 

that the PDE operates stoichiometrically with no purge 

and mixes with bypass air, the mixture being ultimately 

presented to the high-pressure turbine. Using the 

heating value and air/fuel ratio from the previous 

example however, yields a maximum q0 of 12.9. This in 

turn implies a maximum primary stream enthalpy ratio 

of only 4.8 and a maximum entrainment ratio (bypass 

flow divided by PDE flow) of 1.7. The thrust 

augmentation for an ejector system with an entrainment 

ratio of 1.7 is, from Ref. 14, between an ideal of 1.64 

and a data fit of 1.33. Using the average of these two 

and Eqns. 11–17 it can be shown that the PDE/ejector 

yields a pressure ratio of 1.61 compared to a value of 

1.58 for the PDE alone at an enthalpy ratio of 2.4. Thus, 

in this case the ejector system would offer no 

thermodynamic advantage over the PDE alone because 

of the limited q0. 

Other limits may be placed on the map of Fig. 3. It was 

noted that mixture detonability was not considered in 

the CFD model, yet detonability limits clearly exist. A 

number of fuels may be limited to mixtures near 

stoichiometric values. Material temperature limits will 

also limit the range of enthalpy ratios. For cycles with a 

purge fraction of zero, it can be expected that the peak 

uncooled wall temperature is approximately 90% of the 

mass averaged exhaust temperature. 

In closing this section of the paper, it is noted that 

during the course of developing the map, the 

performance impact of several aspects of the PDE cycle 

(purge, partial-fill, geometry, back pressure) have been, 

to a greater or lesser extent, examined. All of these 

aspects however, have been in the context of a 

particular gasdynamic cycle. It should be kept in mind 

that other cycles, such as PDE cycles with valves at the 

back (or low-loss variable back-pressure systems) may 

provide much different, and superior performance to the 

one presented. 

Conclusions

It has been demonstrated in this paper that idealized airbreathing PDE performance can be mapped onto a 

single plot of total pressure ratio versus total enthalpy 

ratio. It has further been shown that this format is useful 

in system studies since the PDE can be viewed as 

simply another ‘component’ with straightforward input 

and output. The idealized PDE performance data was 

obtained from a quasi-one-dimensional CFD code and it 

has been shown that this is a more realistic approach 

than purely analytical methods. The performance shown 

is generally below that which has been previously 

reported for so-called ‘idealized’ PDE performance but 

is still idealistic in that the losses captured are only 

those endemic to the cycle. A similar map could easily 

be generated which incorporates losses such as those 

due to heat transfer, viscous effects and valving. 

0

20

40

60

80

100

120

140

160

180

012345

Flight Mach Number

Specific Thrust (lbf-s/lbm)

Turbojet w/ PDE AB

Ramjet

Turbojet w/ AB

πc=30

Fig. 8 Specific thrust as a function of Mach 

Number for Turbojet with PDE Afterburner, 

Turbojet with conventional Afterburner, and 

Ramjet. For all systems γ=1.3, inlet and nozzle are 

ideal, and the nozzle inlet temperature is 4200 R. 

For the Turbojet, adiabatic compressor and 

turbine efficiencies are 0.85 and 0.90 respectively. 

The turbine inlet temperature is 3000 R. 

0

500

1000

1500

2000

2500

3000

012345

Flight Mach Number

Specific Impulse (sec.)

Turbojet w/ PDE AB

Ramjet

Turbojet w/ AB

πc=30

Fig. 9 Specific Impulse as a function of Mach 

Number for Turbojet with PDE Afterburner, 

Turbojet with conventional Afterburner, and 

Ramjet. For all systems γ=1.3, inlet and nozzle are 

ideal, and the nozzle inlet temperature is 4200 R. 

For the Turbojet, adiabatic compressor and 

turbine efficiencies are 0.85 and 0.90 respectively. 

The turbine inlet temperature is 3000 R. The fuel 

heating value is 19,000 BTU/lbm. 
10 

American Institute of Aeronautics and Astronautics 

References

1. Schauer, Fred, Stutrud, Jeff, Bradley, Royce, 

“Detonation Initiation Studies and Performance 

Results for Pulsed Detonation Engine Applications”

AIAA Paper 2001–1129, January 2001. 

2. Heiser, W.H and Pratt, D.T. “Thermodynamic 

Cycle Analysis of Pulse Detonation Engines,”

submitted for publication in the AIAA Journal of 

Propulsion and Power, 2001. 

3. Kentfield, J.A.C., “The Fundamentals of Idealized 

Air-Breathing Pulse-Detonation Engines,” AIAA 

Paper 2000–3469, July 2000. 

4. Nalim, R.M. and Paxson, D.E., “A Numerical 

Investigation of Premixed Combustion in Wave 

Rotors,” ASME Journal of Engineering for Gas 

Turbines and Power, Vol. 119, No. 3, 1997, 

pp. 668–675, also ASME Paper 96–GT–116, June 

1996, also, NASA TM–107242. 

5. Paxson, D.E., “Numerical Simulation of Dynamic 

Wave Rotor Performance,” AIAA Journal of 

Propulsion and Power, Vol. 12, No. 5, September 

1996, pp 949–957. 

6. Paxson, D.E., “Numerical Assessment of Four-Port 

Through-Flow Wave Rotor Cycles With Passage 

Height Variation,” AIAA paper 97–3142, July 1997. 

7. Paxson, D.E., “A General Numerical Model for 

Wave Rotor Analysis,” NASA TM–105740, 

July 1992. 

8. Wilson, J. and Paxson, D.E., “On the Exit 

Boundary Condition for One-dimensional 

Calculations of Pulsed Detonation Engine 

Performance,” to be presented at the 18th 

International Colloquium on the Dynamics of 

Explosions and Reactive Systems, July 29, 2001, 

Seattle, Washington. 

9. Foa, J.V., Elements of Flight Propulsion, John 

Wiley and Sons, 1960. 

10. Bussing, T. and Pappas, G., “An Introduction to 

Pulse Detonation Engines,” AIAA paper 94–0263, 

January 1994. 

11. Mohanraj, R. and Merkle, C.L., “A Numerical 

Study of Pulse Detonation Engine Performance,”

AIAA paper 2000–0315, January 2000. 

12. Oates, Gordon, C., Aerothermodynamics of Gas 

Turbine and Rocket Propulsion, American Institute 

of Aeronautics and Astronautics, 1997. 

13. Kailasanath, K., “A Review of PDE ResearchPerformance Estimates,” AIAA paper 2001–0474, 

January 2001. 

14. Porter, J.L., and Squyers, R.A., “A Summary/ 

Overview of Ejector Augmentor Theory and 

Performance,” Contractor Report R–91100/9CR–

47A, September 1979. 
11 

American Institute of Aeronautics and Astronautics 

Appendix 1: Code Validation 

The following two figures compare data from the code 

used in this paper to that from a direct-connect, 

Hydrogen/Air PDE experiment currently operating at 

the Air Force Research Laboratory in Dayton, Ohio.1

Although not described in this paper, the code contains 

sub-models to account for the effects of fluid viscosity, 

heat transfer, and valving. Those sub-models were 

employed in following comparison unless otherwise 

noted. Descriptions of the sub-models may be found in 

the literature.4,5

A molecular weight of M = ˆ 21 lbm/lbmole was used for 

all computations (except those for rich mixtures). The 

UDWLR RI VSHFLILF KHDWV ZDV   7KH KHDWLQJ YDOXH RI

the fuel was assumed to be 51,571 BTU/lbm. Ambient 

pressure was 14.7 psia. Inlet temperature was 

approximately 520 R. The number of numerical cells 

used in the computation was 200. 

Figure A1 shows the measured and computed thrust and 

specific impulse, Isp as a function of equivalence ratio, φ

for a 16 hz. cycle in which 100 % of the tube is filled 

with detonable mixture and 50% of the tube is filled 

with purge air each cycle (i.e., the purge air simply 

passes through the tube without any interaction with the 

detonation). Open symbols represent measured results. 

Solid symbols represent computed results. The 

agreement is fairly good. The largest disparity is 

between φ=0.8 and 1.0. For comparison, the code was 

also run at two different equivalence ratios (1.0 and 0.5) 

with heat transfer and viscous sub-models ‘turned off’. 

This data is shown as gray solid symbols. It is clear that 

the effect is quite substantial. 

Figure A2 shows the measured and computed thrust and 

Isp as a function of the fill fraction for a 16 hz. cycle 

running a stoichiometric mixture, with 50% purge air. 

Again, the agreement is quite reasonable. 

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 0.5 1.0 1.5

Fill Fraction

Thrust (lbf)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Isp (sec.)

Computed Thrust

Measured Thrust

Computed Isp

Measured Isp

Fig. A2 Measured and computed thrust and Isp as a 

function of Fill Fraction; 50% purge; 16 hz. cycle; 

φ=1.0 (one of the four tubes).

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0123

Equivalence Ratio

Thrust (lbf)

0

1000

2000

3000

4000

5000

6000

7000

Specific Impulse (sec.)

Computed Thrust

Measured Thrust

Computed Isp

Measured Isp

no heat transfer

inviscid

Fig. A1 Measured and computed thrust and Isp as a 

function of equivalence ratio for a at 100 % fill; 

50% purge; 16 hz. cycle (of one of the four tubes). 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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Pulse detonation; Computational fluid dynamics; Propulsion

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Daniel E. Paxson

A Performance Map for Ideal Air Breathing Pulse Detonation Engines

The performance of an ideal, air breathing Pulse Detonation Engine is described in a manner that is useful for application

studies (e.g., as a stand-alone, propulsion system, in combined cycles, or in hybrid turbomachinery cycles). It is shown

that the Pulse Detonation Engine may be characterized by an averaged total pressure ratio, which is a unique function of

the inlet temperature, the fraction of the inlet flow containing a reacting mixture, and the stoichiometry of the mixture.

The inlet temperature and stoichiometry (equivalence ratio) may in turn be combined to form a nondimensional heat

addition parameter. For each value of this parameter, the average total enthalpy ratio and total pressure ratio across the

device are functions of only the reactant fill fraction. Performance over the entire operating envelope can thus be presented on a single plot of total pressure ratio versus total enthalpy ratio for families of the heat addition parameter. Total

pressure ratios are derived from thrust calculations obtained from an experimentally validated, reactive Euler code capable

of computing complete Pulse Detonation Engine limit cycles. Results are presented which demonstrate the utility of the

described method for assessing performance of the Pulse Detonation Engine in several potential applications. Limitations

and assumptions of the analysis are discussed. Details of the particular detonative cycle used for the computations are

described.