Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Tom Kaemming

Innovative Scientific Solutions Inc., Dayton, Ohio

The Influence of Unsteadiness on the Analysis of 

Pressure Gain Combustion Devices

NASA/TM—2013-217831

January 2013

AIAA–2013–0280
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Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Tom Kaemming

Innovative Scientific Solutions Inc., Dayton, Ohio

The Influence of Unsteadiness on the Analysis of 

Pressure Gain Combustion Devices

NASA/TM—2013-217831

January 2013

AIAA–2013–0280

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2013-217831 1

The Influence of Unsteadiness on the Analysis of Pressure Gain 

Combustion Devices

Daniel E. Paxson 

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135 

Tom Kaemming

Innovative Scientific Solutions Inc.

Dayton, Ohio 45440 

Abstract

Pressure gain combustion (PGC) has been the object of scientific study for over a century due to its 

promise of improved thermodynamic efficiency. In many recent application concepts PGC is utilized as a 

component in an otherwise continuous, normally steady flow system, such as a gas turbine or ram jet 

engine. However, PGC is inherently unsteady. Failure to account for the effects of this periodic 

unsteadiness can lead to misunderstanding and errors in performance calculations. This paper seeks to 

provide some clarity by presenting a consistent method of thermodynamic cycle analysis for a device 

utilizing PGC technology. The incorporation of the unsteady PGC process into the conservation equations 

for a continuous flow device is presented. Most importantly, the appropriate method for computing the 

conservation of momentum is presented. It will be shown that proper, consistent analysis of cyclic 

conservation principles produces representative performance predictions.

Introduction

Pressure gain combustion (PGC) refers to a process whereby there is a rise in the averaged total 

pressure across the device in which the combustion is taking place. Typically, the device has fixed 

volume (i.e., no pistons). The pressure rise is in contrast to the more typical pressure loss associated with 

conventional, steady combustion utilized in aircraft propulsion and electric power generation systems. 

The pressure rise can be used with a nozzle to produce increased thrust in a ram propulsion application, or 

to provide increased availability to the downstream turbine in a gas turbine application. This enhanced

performance potential of PGC over conventional combustion has been the impetus for many experimental 

and analytical investigations in the aerospace community over at least the past decade (Refs. 1 to 16). 

Pressure gain combustion may be achieved with detonative or deflagrative chemical reactions. In 

either case however, the complete PGC process is periodic (i.e., pulsed), and therefore fundamentally 

unsteady. Referring to Figure 1, the working fluid within a given chamber undergoes a cyclic process that 

roughly consists of three phases: 1) chamber filling; 2) rapid, confined combustion or chemical heat 

release; 3) chamber exhaust or blowdown. The combustion chambers, of which many are usually 

envisioned to comprise a full combustor unit, may be fixed or rotating. The combustion chambers are 

typically valved at one or both ends. Valves may be of the aero (i.e., fluidic) or mechanical variety.

In a propulsion or power system, PGC devices may be coupled to continuous flow, normally steady 

components such as a turbines and nozzles. When analyzing the potential performance of such systems, 

particularly at the fundamental (i.e., preliminary) level, this coupling of seemingly disparate operational 

modes can lead to confusion. However, careful observance, integration and application of traditional 

control volume and control mass thermodynamic cycle equations can provide representative performance 

estimates. This paper outlines a methodology for doing so. A relatively simple mathematical model for 

the PGC device is presented. The appropriate integrals for assessing periodic mass, momentum, and 

energy flux are derived, and the implications and interpretations of their forms are discussed. 
NASA/TM—2013-217831 2

Figure 1.—Schematic of a basic, thrust producing PGC with cycle stages. Colors represent qualitative 

temperature.

Notional PGC Process

In order to estimate limits of performance, and to understand the fundamental operation of PGC

devices, it is often useful to develop low order models(Refs. 2 to 4, 6, and 16). However, if such models 

are to be meaningful, they must account for the impact of fundamental unsteadiness. This will be 

illustrated using the basic, thrust-producing PGC engine concept, as illustrated schematically in Figure 1. 

This engine is represented by the Atkinson cycle. In many recent publications, this has also been referred 

to as the Humphrey cycle. As illustrated in the T-s diagram of Figure 2, the idealized version of this cycle 

is described by the following phases: a) isentropic (ram) compression (stations 0 to 3), b) constant volume 

heat addition (stations 3 to 4), c) isentropic expansion in the combustion chamber and nozzle (stations 4 to 

9), and d) isobaric heat rejection (stations 9 to 0). It is assumed that the ram compression is steady and 

provides high pressure air for a multitude of combustion chambers. Each chamber is assumed to contain a 

converging/diverging exhaust nozzle which serves to limit the chamber Mach number (including the refill 

period) to a negligibly small value, and to expand (accelerate) the post-combustion fluid to ambient 

pressure. Each chamber also contains an ideal inlet valve which opens to allow refilling and closes to 

allow combustion and blowdown. In order to simplify the subsequent analysis, the classical air standard 

cycle analysis assumptions are also used: the fluid properties in each vessel are spatially uniform; the 

working fluid is a thermally and calorically perfect gas; the mass of the fuel is negligible. Finally, in order 

to remove the complexity of the notional PGC device (i.e., the need for an ideal exhaust valve), the 

chemical combustion reaction is assumed instantaneous. 

Along with the Atkinson cycle, Figure 2 also shows the well-known idealized ram Brayton cycle for 

the same specific heat addition parameter and flight Mach number. The heat addition parameter is defined 

as follows.

( )

0

0

Tc ˆ

afh q

p

∆ v = (1) 

Figure 2 is helpful in that all of the fluid states comprising each cycle can be visualized, and the 

reduced entropy associated with constant volume combustion (compared to constant pressure 

combustion) is clear. However, for the case of unsteady PGC integrated into a ram propulsive device, the 

figure can be misleading for several reasons. 

The first reason involves the types of thermodynamic cycle analyses used to predict cycle 

performance (i.e., control mass or control volume), and their traditional associations with Brayton and 

Atkinson cycles. By presenting only the thermodynamic states and paths in Figure 2, it is not possible to 

identify where work is done by or on the working fluid. For example, in a control mass analysis (i.e., a 

fixed mass with changing volume, such as a piston/cylinder) work is done by the fluid during both 

expansion and constant pressure heat addition phases. Similarly, work is done on the fluid during 

compression and constant pressure heat rejection phases. In a control volume analysis (i.e., a fixed 

volume with surfaces across which fluid flows) work is done by the fluid only during the expansion 

phase, and is done on the fluid only during the compression phase. 

Slow chamber fill with 

reactant

Instantaneous reaction at 

constant volume

Isentropic chamber 

blowdown to fill pressure

Ideal valve

Ram air

Ideal Nozzle Ambient

pressure

Low High

Temperature
NASA/TM—2013-217831 3

Figure 2.—T-s diagram of control volume Atkinson and Brayton cycles with the 

same ram compression and specific heat addition.

The Brayton cycle is nearly always analyzed using a steady, control volume analysis because the gas 

turbine and ramjet components are, in fact, steady flow, control volumes. The classical Atkinson cycle 

diagram is typically analyzed with a control mass analysis, since it was (and is still) primarily 

implemented in piston-based engines. If these respective analyses are used in interpreting the Brayton and 

Atkinson cycles shown in Figure 2, there is no convenient way to visually compare the net work, and 

therefore thermodynamic efficiency obtained from each. 

In this paper, the Atkinson cycle shown in Figure 2 is performed by a PGC device with continuous 

flow components fore and aft. Heretofore referred to as the PGC Atkinson cycle, it requires a control 

volume form of analysis, and can therefore be compared in many ways directly with the Brayton cycle. 

However, there is a critical distinction, which leads to the second somewhat misleading aspect of 

Figure 2, namely, the interpretation of the states shown.

In the context of the ram device considered here, all the Brayton cycle states are fixed in time and 

space, as previously explained. Thus, each point on the Brayton cycle can be directly related to a specific 

physical location in the engine, and the properties at that point do not change. In a classical piston-based 

Atkinson cycle each point on the T-s diagram can be related to a different moment in time; however, the 

mass is fixed. Therefore, since network and other performance parameters calculable from a T-s diagram 

are on a per-unit-mass (i.e., specific) basis, time variation becomes irrelevant. All that is needed to 

complete a classical Atkinson cycle analysis are the heat added, and the system volume change during 

compression and expansion. For the PGC Atkinson cycle, the static states at stations 0, 3, and 9 are fixed 

in time, as in the Brayton cycle. However, the state at station 4 in the PGC Atkinson cycle combustion 

chamber varies with time through the range of values shown by the vertical red arrow in Figure 2. This 

occurs over the course of the blowdown and refill stages as fluid crosses the fixed volume boundaries. It 

results in non-uniformity in the exhaust flow which is the primary manifestation of fundamental 

unsteadiness. This combination of steady and unsteady phases of the cycle makes the PGC Atkinson

cycle analysis approach different from both classical Atkinson and Brayton cycle analyses. Assessing the 

impact of this unsteadiness is the main subject of this paper. 

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

0.0 0.5 1.0 1.5 2.0

T/T0

s/cp

Brayton

Atkinson/Humphrey 3

4

0

9

4

9

γ=1.4

M0=1.0

T0=400 R

q0=5.0

Rg=53.57 ft-lbf

/lbm/R

ˆ

ˆ ˆ
NASA/TM—2013-217831 4

Figure 3.—Illustration showing the dependence of the PGC Atkinson cycle states on both 

time and physical location. Colors represent qualitative temperature.

For reference and clarity, a graphical rendering of the issues just discussed is shown in Figure 3. For 

the Brayton cycle, in this illustration a ramjet, the thermodynamic properties are a function only of 

physical location. In contrast, for the classical Atkinson cycle, the thermodynamic properties are a 

function only of time, or cycle period. The PGC Atkinson cycle is essentially a hybrid of these two 

cycles, in that the inlet properties are a function only of physical location, while the combustor and nozzle 

properties are a function of both physical location and time due to its unsteadiness. 

The remainder of this paper will provide an appropriate means of analyzing the unsteady PGC 

Atkinson/Humphrey cycle, and some potential consequences of inappropriate analyses. Mass and energy 

conservation equations will be developed first. From these it will be shown that the mass-averaged 

temperature at station 4 has significance in that it represents the total specific enthalpy available for 

expansion. Next, the specific thrust equation will be developed from the unsteady momentum 

conservation equation. It will be shown that analyses which assume a uniform state 4 for the PGC 

Atkinson/Humphrey cycle, though convenient in that they typically have closed form algebraic solutions, 

will generally over-predict ideal specific thrust, and often under-predict specific thrust when component 

losses (e.g., nozzles) are included. The significance of the under and over-predictions will be discussed.

Fundamental Relationships

All of the analysis to follow will utilize normalized (i.e., non-dimensional) variables, as this 

simplifies the equations. Temperature, pressure, and density are normalized by the freestream static 

values at station 0. Velocity is normalized by the freestream speed of sound. The brief subsections below 

will develop some relationships that will be useful in the analysis to follow 

+Q

-Q

In a ramjet (Brayton cycle) 

engine, properties vary only 

by physical location

In a classical Atkinson 

cycle, properties vary only 

with time, i.e. cycle period

In a PGC Atkinson cycle, 

properties can vary with 

physical location and time

0 3 4 9 0 3

4

9

0

3

4

9

Time

Low High

Temperature
NASA/TM—2013-217831 5

Conservation of Mass and Energy

Consider a control volume through which periodically flows an inviscid, reacting fluid. For 

simplicity, the control volume is assumed quasi-one-dimensional. The equation of continuity is 

( ) = 0

∂

ρ∂ +

τ∂

ρ∂

x

uAA (2) 

Here, τ is the time normalized by L/a0. The spatial dimension is normalized by L. Integration of 

Equation (2) over the length and period respectively, yields 

( ) ( ) ∫∫∫∫ ρ−τρ=τρ τ

Τ T T

Adxd dAudAu

0

outoutout

0

ininin

1

00

(3) 

The subscripts “in” and “out” refer to the entrance and exit planes of the volume, respectively. By the 

definition of periodicity (i.e., nothing can be stored within the volume over the course of one cycle), the 

left hand side of Equation (3) is zero. Thus, whatever mass flows in must flow out in one limit cycle. 

An analogous process can be followed for the energy equation. The result, in terms of a single species, 

premixed, reacting flow (again, for simplicity) is

( ) ( ) ∫ ∫ ρ ρ=τ+ τ+

T

t

T

t dzqTAudzqTAu

0

out0out_outoutout

0

ininin _ in 0 in (4) 

Here, z is the reactant mass fraction which is 1 before the reaction starts and zero when it is finished. Like 

Equation (3), Equation (4) simply verifies that no energy is stored in the volume over one cycle. In the 

simple control volume cycle that is under consideration, it is stipulated that the reaction must take place 

within the vessel. Thus zout=0 at all times. It is also stipulated that no partial fueling of the volume occur. 

Thus, zin = 1.0, at all times. Note that nothing has been stipulated about the nature of the reaction 

(detonative or deflagrative).

Mass averaging, in this case of the total temperature, is defined using the overbar notation as

∫

∫∫

ρ

ρ

== T

T

t

M

t

t

uAdt

dtuAT

M

dmT

T

0

0 0 (5) 

Combining Equations (3), (4), and (5) and using the station numbering shown in Figure 2 yields the 

following important relationship

tt += qTT 034 (6) 

For steady flow control volumes such as the Brayton cycle combustor, it is easily shown that 

Equation (6) is valid with the mass averages replaced by the steady values. For the PGC Atkinson cycle 

under consideration, it has been assumed that Tt3 is steady, hence = TT tt 33 . 
NASA/TM—2013-217831 6

Constant Volume Combustion

In presenting Equations (2) to (6), no assumptions were made about the uniformity or velocity of the 

fluid within the control volume. For the remainder of this paper, discussion will return to the originally 

stated simplifying assumptions that the fluid is uniform and at very low (often zero) velocity. Constant 

volume combustion implies both a fixed volume and mass during the reaction. Thus, the density of the 

fluid is constant. From the first law of thermodynamics, it is found that the release of chemical energy 

(combustion) can only increase the internal energy of the fluid. Using the heat addition parameter of 

Equation (1), the following relationship may therefore be obtained for the instantaneous, peak value of Tt4

in the PGC Atkinson cycle.

t4_peak t γ+= qTT 03 (7) 

Two important observations are brought to light with Equations (6) and (7). The first is simply that 

while Tt4_peak is always greater than the peak temperature of the constant pressure Brayton combustor (for 

the same specific heat addition), the specific enthalpies (essentially the mass averaged total temperatures) 

of the flows exiting the two combustors are the same.

The second, related observation is that while all of the fluid in the PGC combustor momentarily

reaches Tt4_peak, only an infinitesimal amount actually leaves the combustor at this temperature. In the 

notional PGC cycle, the blowdown phase commences immediately following the constant volume 

combustion event. An infinitesimal mass of fluid, exits the combustor, accelerates through the nozzle, and 

leaves the system. The mass has the energy associated with Tt4_peak. If an infinitesimal amount of mass and 

energy leave the combustor, and nothing is being added to the combustor, it follows that the energy 

(temperature) and mass remaining in the combustor must decrease. Therefore, the next infinitesimal mass 

of fluid to leave the combustor does so at a lower temperature than Tt4_peak. This incremental decrease of 

the combustor temperature continues until the blowdown process is complete. The mass average of all 

these temperatures isT 4t , from Equations (6). Another interpretation of this process is that each 

infinitesimal mass of exiting fluid is forced from the combustor by infinitesimal expansion of all the mass

that remains. In other words, there is work done by the remaining static fluid on the mass that is leaving

(Ref. 17). The important implications of this observation are that: a) Tt4_peak does not fully represent the 

energy available for thrust or useful work extraction, b) Tt4_peak does not represent a temperature to which 

any downstream component (e.g., a nozzle) would be continually exposed or able to fully utilize, and c) 

the difference between Tt4_peak and Tt4 is work done internally to accelerate the combustion products from 

the combustor. 

With reference to Figure 2, these observations illustrate that simply because the PGC Atkinson Tt4_peak

is greater than the Brayton Tt4 does not, by itself, indicate greater work available from the PGC Atkinson 

cycle. Conversely, when efficiencies are applied to the PGC Atkinson cycle (discussed later), the cycle 

cannot be debited solely based on Tt4_peak, since it does not fully represent the actual energy of the flow 

exiting the combustor. In fact, a more representative expansion ‘state’ from which to compare the PGC 

Atkinson cycle to the Brayton cycle is Tt4 at s4. 

Entropy Change 

From the definition of isobaric and isochoric processes, and the ideal specification that the entropy 

change in the PGC Atkinson cycle from point 3 to point 4 in Figure 2 is the same as from point 9 to point 

0, it may be shown that 

γ















 =

1

3

_4 peak

9

t

t

T

T

T (8) 
NASA/TM—2013-217831 7

In the ideal PGC Atkinson cycle, the exit static pressure and entropy are fixed. For a perfect gas, this 

means that T9 is also constant, even though Tt4 varies.

Velocity

The instantaneous velocity may be found from total and static temperatures as follows. 

( ) u ( ) t −TT

−γ = 1

2 2 (9) 

Ideal Efficiency and Kinetic Energy

The efficiency of any thermodynamic cycle may be defined in terms of rejected and supplied heat. In 

the nomenclature of the present work, this takes the following form.

( )

0

9 1 1

q

T

th

− −=η (10) 

As noted above for the ideal PGC Atkinson cycle, T9 is constant even though the cycle is periodic and 

unsteady. As such, Equation (10) does not require a steady flow assumption and correctly represents 

thermal efficiency. Equations (7), (8) and (10) can be combined to yield the familiar cycle thermal 

efficiency which requires only fixed parameters, but which is valid for the unsteady cycle (Ref. 2). 





















− 













 γ +−=η

γ

1 1 1 1

1

3

0

0 t

th T

q

q

(11) 

In the context of the pure thrust device under investigation, efficiency can also be defined as the ratio 

of net kinetic energy produced to heat added. Since the PGC Atkinson cycle is unsteady however, it must 

be formally stated in terms of mass averages where they are relevant.

( )( ) 2

0

2

9 2 0

1

uu

q

th − −γ =η (12) 

The fact that Equations (10) and (12) yield the same result may be verified by substituting 

Equation (9) as the mass averaged variable in Equation (5), and using Equation (6) with the result.

Equation (12) can be rearranged and solved for the mass averaged square of the exit velocity.

( )

2

0

2 0

9 1

2

u q u th +

−γ

η = (13) 

Evidently, evaluating the mass averaged square of the exit velocity, which involves integrating the 

non-uniform square of the exit velocity (see Eq. (5)), has a closed form, algebraic solution which depends 

only on known flow parameters. This was also found to be the case for the evaluating the mass average of 

the non-uniform Tt4 (see Eq. (6)). As will be seen in the next section, this is not true of other mass 

averaged quantities.
NASA/TM—2013-217831 8

Specific Thrust

Specific thrust is typically defined as the thrust per unit of mass flow rate and is often associated with 

steady flow devices. In the context of an unsteady PGC device, it is defined as the time averaged thrust 

per unit of time averaged flow rate. However, specific thrust is dimensionally the same, and may also be 

interpreted as specific impulse. Specific impulse is the change in momentum per unit mass of fluid, or, in 

the case of PGC, the change in momentum per cyclic mass of fluid processed. For the ideal notional PGC 

device under consideration the exiting momentum per cyclic mass of fluid is simply the mass averaged 

velocity at station 9 (see Eq. (5)). The net specific thrust is therefore found by subtracting the (assumed 

steady) inlet velocity from u9 . 

09

0

uu

a

sp gT c −= (14) 

The mass averaged exit velocity cannot be evaluated in closed form. However, it can be evaluated 

using a straightforward numerical integration as follows. Because the flow is isentropic after the 

instantaneous constant volume combustion process, and due to simplifying assumptions presented earlier, 

the temperature and pressure in the combustion chamber and nozzle exit (stations 4 and 9) are known 

functions only of the chamber density, as is the exit velocity. During the blowdown period, spatial 

integration of Equation (2) shows that 

t 9994 dAud τρ=ρ− (15) 

Blowdown ends when the chamber pressure, pt4 reaches pt3. At this point, refill begins. Using the 

constant volume combustion relationship pt4_peak/pt3= Tt4_peak/Tt3, the chamber density is found as follows. 

γ γ

















ρ= 















ρ=ρ

1

_4 peak

3

3

1

_4 peak

3

refill_4 3

t

t t

t

t t t T

T

p

p (16) 

During the refill period, the assumption of an exit throat maintaining negligible fill Mach number, 

insures that no further expansion takes place in the chamber. Thus, the conditions in the chamber and at 

the nozzle exit remain constant. 

Equations (15) and (16) can be used in the general form of Equation (5) to show that the mass average 

of any function of the chamber density, f(ρt4) may be evaluated as

( ) ( )

3

44 refill_4refill_4

_4 refill

3

t

ttt t fdf

f

t

t

ρ

ρρ+ρρ−

=

∫

ρ

ρ (17) 

Thus, from Equations (9) and (17), and standard isentropic relationships, the mass average exit 

velocity is written as

( ) ( ) ( ) ( )

3

refill_4 2

1

9

1

_44 peak refill_4 2

1

9

1

_4 peak 4

9

_4 refill

3

1

2

t

Tt t TdT tt t T t

u

t

t

ρ

















ρ−ρ−ρ−ρ

−γ −

=

−γ

ρ

ρ

−γ

∫

 (18) 
NASA/TM—2013-217831 9

As mentioned, this integral must be evaluated numerically. As such, Equation (18) will be referred to 

as the integrated analysis for the remainder of the paper. Figure 4 shows the distribution of pressure and 

temperature in the chamber for the particular cycle illustrated in Figure 2. Also shown for reference are 

the exit static and mass average chamber temperatures. These distributions are used to evaluate u9, the 

integrand of Equation (18), which is shown in Figure 4. Also shown are the square of the exit velocity and 

the numerically obtained, mass averages. It is noted that the numeric 20-point, trapezoidally integrated 

value of 26711 2

9 = .u . The analytical value from Equation (13) is 11.270, which indicates that the 

numerical integration is done correctly.

Figure 4.—Distribution of pressure and temperature in the combustion 

chamber for the idealized Atkinson cycle under the parametric conditions 

shown.

Figure 5.—Distribution of nozzle exit velocity for the idealized Atkinson cycle 

under the parametric conditions shown.

0

2

4

6

8

10

12

14

0.0 0.5 1.0 1.5

T, p

ρt4

Pt4

Tt4

T9

Tt4_bar

γ=1.4

M0=1.0

T0=400 R

p0=14.7 psia

Rg=53.57 ft-lbf/lbm/R

q0=5.0

pt4

Tt4

T9

Tt4

ˆ

ˆ

0

5

10

15

20

25

0.0 0.5 1.0 1.5

u9, u9

2

ρt4

u

u_bar

u^2

u^2_bar

u_bar^2

γ=1.4

M0=1.0

T0=400 R

q0=5.0

u9

u9

2

u9

u9

2

u9

2

ˆ
NASA/TM—2013-217831 10

Figure 5 illustrates a central tenet of this paper which is that in fundamentally unsteady, thrust 

producing PGC systems, the momentum (specific thrust) is not algebraically related to the kinetic energy 

(essentially Eq. (13)). In equation form, this means 

2

9 9 ≠ uu (19) 

In steady systems such as the ram Brayton cycle, 2

9 9 = uu . As such, ideal thermal efficiency can be 

obtained algebraically from known cycle parameters, and the steady forms of Equations (13) and (14) still 

hold. Thus, for steady thrust producing systems, the following may be written. 

( ) 0

2

0

0

0 1

2

uu q

a

gTsp c th −+

−γ

η = (20) 

Failure to recognize Equation (19), which is quite subtle, has led researchers to apply the steady 

Equation (20) to unsteady PGC Atkinson cycle analyses (Ref. 2). These will be referred to as algebraic 

analyses for the remainder of the paper. Their results, even if the appropriate thermal efficiency is used 

(Eq. (11)), are incorrect.

A more general statement than Equation (19) can be made on the relationship between u9 and 2

9 u . 

Consider an arbitrary periodic exit velocity profile arising from detonative or deflagrative PGC. At any 

point in the cycle, this velocity may be written as uuu 999 += ′ , where u9

′ represent a deviation from the 

mass averaged value obtained via Equation (17). Squaring this quantity yields 99

2

9

2

9

2

9 += ′ + 2 uuuuu ′ . 

Taking the mass average of 2

9 u and noting that u9

′ = 0 by definition yields 

2

9

2

9

2

9 += uuu ′ . (21) 

Thus, 2

9 9 = uu only if 0 2 u9

′ = , that is, only if the flow is uniform or steady. Furthermore, since the 

mass averaged square of the deviation 2

9 u′ is greater than zero for all periodic unsteady flows, it follows 

that

2

9 9 ≤ uu . (22) 

As such, the correct ideal net specific thrust from Equations (18) (combined with Eq. (14)) is always 

less than that computed from Equation (20). 

Results and Discussion

Equation (22), describing the relationship between mass averaged momentum and energy fluxes 

represents the unavoidable impact of unsteadiness on PGC performance as measured by specific thrust. In 

fact, Reference 18 describes the ratio 2

9

2

9 uu as the ‘efficiency of non-uniformity’. It is interesting to note 

as a relevant aside that if the PGC Atkinson cycle considered here were implemented as a classical 

control mass Atkinson cycle (e.g., with pistons), the fact that it is unsteady would have no impact on 

performance. In this case, Equation (11) would still hold and the performance metric, net specific piston 

work, would be directly calculable from it. In other words, the control volume nature of the PGC 
NASA/TM—2013-217831 11

Atkinson cycle, and the fact that momentum is the relevant performance metric (not piston work) give 

rise to a sort of ‘loss’ described by Equation (22). 

Figure 6 compares the specific thrust of the algebraic (Eq. (20)) and integrated (Eqs. (18) and (14)) 

analyses over a range of flight Mach numbers, at a constant freestream temperature. For reference, the 

specific thrust for the ideal Brayton cycle is also shown. As expected, the algebraic analysis over-predicts 

performance compared to the integrated state-based model, and the disparity is greatest where the nonuniformity of 2

9

2

9

′ uu is greatest over the course of the cycle.

Performance With Losses 

Thus far the analysis has focused on ideal performance analysis. In this section the impact of 

component efficiency and fundamental unsteadiness will be examined. Specifically, the nozzle efficiency 

will be varied, since it is in the expansion phase of the PGC Atkinson cycle that the effects of 

unsteadiness are manifested. The nozzle efficiency is defined as follows

















γ

γ−

−

−

= −

− =η 1

4

4

9

94

94

1

1

t

t

it

t e

P

T

T

TT

TT (23) 

Equation (23) complicates the PGC Atkinson T-s diagram of Figure 2 since each infinitesimal unit of 

mass not only has a different Pt4, and Tt4, but also has a different T9. In other words, each unit of mass 

follows a different path to the specified exit static pressure line. Mathematically however, it is a simple 

matter to solve Equation (23) for T9, isentropically relate Pt4, and Tt4 to Pt4, and ρt4, insert it into the 

integral of Equation (18) and subsequently obtain specific thrust. It is also noted (though results will not 

be shown), that cycle thermal efficiency is no longer algebraically obtained. It too requires numerical 

integration. 

Figure 6.—Ideal specific thrust as a function of Mach number for the PGC 

Atkinson cycle using the algebraic and integrated analyses.

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5

Tsp,lbf

-s/lb

m

M0

Brayton

PGC Atkinson-Algebraic (Eq. 20)

PGC Atkinson-Integrated (Eq. 18)

γ=1.4

T0=400 R

q0=5.0

ˆ
NASA/TM—2013-217831 12

Algebraic analyses, which do not recognize Equation (19), and which rely on steady relationships 

such as Equation (20) often utilize a single value for T9 and it is typically found by using Tt4_peak and 

Pt4_peak in Equation (23) (i.e., using condition that generate maximum entropy) (Ref. 2). This is equivalent

to treating the entire PGC Atkinson cycle as a steady process.

1 This results in a larger performance 

penalty being incurred by the algebraic model than by the integrated state model for the same specified 

expansion efficiency.

Figure 7 shows this effect using the same format as Figure 6. The expansion efficiency chosen is 

ηe=0.90. It is seen that the algebraic analysis incorrectly penalizes the PGC Atkinson cycle so severely 

that its performance drops below that of the integrated analysis, and even below the performance of the 

Brayton cycle.

It is noted in passing that more realistic performance results could be obtained from the algebraic 

analysis by calculating T9 by using Tt4 from Equation (6) and 

1

t4_peak

4

_44 peak

−γ

γ















 = T

T

pp t tt in Equation 

(23). Doing so would yield algebraic performance results which were everywhere better than the 

integrated results, as would be expected from neglecting the effects of non-uniformity. Such results are 

not shown however, as it is the proposition of this paper that algebraic analyses are not appropriate for 

PGC Atkinson cycle performance analysis.

Figure 7.—Specific thrust with expansion loss as a function of Mach 

number for the PGC Atkinson cycle using the algebraic and integrated 

analyses.

 

1

It is noted that if a classical (control mass, piston-based) Atkinson cycle were under consideration, and Equation (23) 

was a piston work efficiency rather than a nozzle expansion efficiency, the use of a single T9, along with Pt4_peak and 

Tt4_peak would be perfectly appropriate since all the mass in this case would follow the same path.

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5

Tsp,lbf

-s/lb

m

M0

Brayton

PGC Atkinson-Algebraic (Eq. 20)

PGC Atkinson-Integrated (Eq. 18)

γ=1.4

T0=400 R

q0=5.0

ηe=0.90

ˆ
NASA/TM—2013-217831 13

Energy Conservation in Equation (17) 

This paper will close with a brief discussion that does not directly follow the previous sections. It is 

included because the authors believe that Equation (17), with its state-based integration (as opposed to 

time-based), is a somewhat unconventional manner by which to describe mass averaging and therefore 

requires some additional proof as to its validity. To do so, Equation (6), which is a mass average 

demonstrating the cyclic conservation of energy in a control volume is re-derived from Equation (17) as 

follows. 

3

refill_4refill_444

4

_4 refill

3

t

tttt

t

TdT

T

t

t

ρ

ρ+ρ−

=

∫

ρ

ρ

(24) 

Using isentropic relations, Tt4 can be made a function of ρt4 and other known quantities. 

( ) ( )

[( ) _43 peak ] 03

3

4 refil_4

1

1 4

3

_4 peak

4 1 1

_4 refill

3 qTTT

d

T

T tt t

t

t t lt

t

t

t

t

t +=+−γ

γ = ρ

















ρ−ρρ

ρ −

=

γ

ρ

ρ

−γ

−γ ∫

(25) 

Here, Equation (7) has also been employed to produce the final result, which is Equation (6).

Conclusions

This paper has described and quantified the influence of inherent unsteadiness on fixed volume 

pressure gain combustion (PGC) system performance and the analysis thereof. Using a simple notional 

PGC device executing an Atkinson cycle, it has been shown that unsteadiness impacts the fundamental 

cycle diagram interpretation. Specifically, it was shown that a direct comparison of unsteady cycle and 

steady cycle states cannot be used to compare performance. A consistent means of addressing mass, 

momentum and energy conservation has been presented which provides the ability to properly compare 

unsteady and steady cycle performance. Finally, it has been shown that only through the proper 

accounting for unsteadiness can component losses be properly applied to cycle performance.

NASA/TM—2013-217831 15

Appendix—Symbols List

a0 freestream speed of sound

cp specific heat at constant pressure

cv specific heat at constant volume

f/a fuel to air ratio (by mass)

gc Newton constant 

m mass

p nondimensional pressure

pˆ pressure 

q0 nondimensional heat addition parameter

s entropy 

u nondimensional velocity 

x nondimensional axial distance

z reactant concentration

A area

L length

T nondimensional temperature

Tˆ temperature

Tsp specific thrust

Τ nondimensional period

γ ratio of specific heats

η efficiency

ρ nondimensional density

τ nondimensional time

∆hv fuel heating value 

Subscripts

0 freestream

3 ram diffuser outlet 

4 combustion chamber

9 nozzle exit plane

e expansion 

i isentropic

in entering quantity

out exiting quantity 

peak maximum in the cycle

t total

th thermal

Superscripts

- overbar signifying a mass average

′ deviation from the mean 
NASA/TM—2013-217831 16

References

1. Schauer, F.R., et al., “Detonation Initiation Studies and Performance Results for Pulsed Detonation 

Engine Applications,” AIAA 2001-1129, Jan. 2001. 

2. Heiser, W. H., Pratt, D. T., “Thermodynamic Cycle Analysis of Pulse Detonation Engines,” Journal 

of Propulsion and Power, Vol. 18, No. 1, pp. 68-76, 2002. 

3. Kentfield, J.A.C., “Thermodynamic of Airbreathing Pulse-Detonation Engines,” Journal of 

Propulsion and Power, Vol. 18, No. 6, pp. 1170-1175, 2002. 

4. Nalim, M.R., “Thermodynamic Limits of Work and Pressure-Gain in Combustion and Evaporation 

Processes,” Journal of Propulsion and Power, Vol. 18, No. 6, pp. 1176-1182, 2002. 

5. Brophy, C., Sinabaldi, J., “Performance Characterization of a Valveless Pulse Detonation Engine,” 

AIAA-2003-1344, Jan. 2003.

6. Kaemming, T., Dyer, R., “The Thermodynamic and Fluid Dynamic Functions of a Pulsed 

Detonation Engine Nozzle,” AIAA-2004-3916, Jul. 2004. 

7. Paxson, D.E., “Performance Evaluation Method for Ideal Airbreathing Pulse Detonation Engines,” 

Journal of Propulsion and Power, Vol. 20, No. 5, pp. 945-950, 2004.

8. Perkins, H.D., et al., “An Assessment of Pulse Detonation Engine Performance Estimation Methods 

Based On Experimental Results,” AIAA-2005-3831, Jul. 2005. 

9. Rasheed, Furman, and Dean, “Experimental Investigations of an Axial Turbine Driven by a Multitube Pulsed Detonation Combustor System”, AIAA 2005-4209. 

10. Kojima, et al., “Design Study of Turbine for Pulse Detonation Combustor,” AIAA 2007-5081, Jul. 

2007. 

11. Van Zante, Envia, and Turner, “The Attenuation of a Detonation Wave by an Aircraft Engine Axial 

Turbine Stage,” ISABE–2007–1260, Sep. 2007. 

12. Caldwell, et al., “Experimental Analysis of a Hybrid Pulse Detonation Combustor / Gas Turbine 

Engine,” AIAA-2008-121.

13. Suresh, A., Hofer, D.C., Tangirala, V.E. “Turbine Efficiency for Unsteady, Periodic Flows,” AIAA 

2009-0504, Jan. 2009. 

14. Paxson, D.E., “A Simplified Model for Detonation Based Pressure-Gain Combustors,” AIAA 2010-

6717, Jul. 2010. 

15. Rouser, K.P., et al., “Unsteady Performance of a Turbine Driven by a Pulse Detonation Engine,” 

AIAA-2010-1116, Jan. 2010. 

16. Paxson, D. E., Brophy, C., Bruening, G., “Performance Evaluation of a Pulse Detonation 

Combustion Based Propulsion System Using Multiple Methods,” JANNAF Journal of Propulsion 

and Energetics, Vol. 3, No. 1, May, 2010, pp. 44-55. 

17. Sun, G., Akbari, P., Gower, B., Muller, N., “Thermodynamics of the Wave Disk Engine,” AIAA 

2012-3704, Aug. 2012. 

18. Foa, J. V., Elements of Flight Propulsion, Wiley, New York, 1959, pp. 274-283. 

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The Influence of Unsteadiness on the Analysis of Pressure Gain Combustion Devices 

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14. ABSTRACT

Pressure gain combustion (PGC) has been the object of scientific study for over a century due to its promise of improved thermodynamic 

efficiency. In many recent application concepts PGC is utilized as a component in an otherwise continuous, normally steady flow system, 

such as a gas turbine or ram jet engine. However, PGC is inherently unsteady. Failure to account for the effects of this periodic unsteadiness 

can lead to misunderstanding and errors in performance calculations. This paper seeks to provide some clarity by presenting a consistent 

method of thermodynamic cycle analysis for a device utilizing PGC technology. The incorporation of the unsteady PGC process into the 

conservation equations for a continuous flow device is presented. Most importantly, the appropriate method for computing the conservation 

of momentum is presented. It will be shown that proper, consistent analysis of cyclic conservation principles produces representative 

performance predictions. 

15. SUBJECT TERMS

Propulsion; Combustion 

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