Nonlinear analysis of a pulse combustor model with exhaust decoupler and vent pipe

Combustion and Flame xxx (2013) xxx–xxx

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Combustion and Flame

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Nonlinear analysis of a pulse combustor model with exhaust decouplerand vent pipe

0010-2180/$ - see front matter � 2013 The Combustion Institute. Published by Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.combustflame.2013.07.003

⇑ Corresponding author. Address: School of Astronautics, Beihang University, 37Xueyuan Road, Beijing 100191, China. Fax: +86 010 82339571.

E-mail address: [email protected] (L.-j. Yang).

Please cite this article in press as: L.-j. Yang et al., Combust. Flame (2013), http://dx.doi.org/10.1016/j.combustflame.2013.07.003

Li-jun Yang ⇑, Zhi-ren Yin, Ming He, Run-ze Duan, Fang-yan LiSchool of Astronautics, Beihang University, Beijing 100191, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 July 2012Received in revised form 1 June 2013Accepted 2 July 2013Available online xxxx

Keywords:Pulse combustorCritical intervalExhaust decouplerAsymptotic solutionLump parameter model

A nonlinear pulse combustor model with an exhaust decoupler and vent pipe was solved using the Poin-caré–Lindstedt perturbation analysis method. The solutions included expressions for the pressure in thecombustion chamber and the exhaust decoupler respectively. Experiments were made to validate thetheoretical analysis; results showed that an exhaust decoupler and a vent pipe can affect the frequencyof the pulse combustor and the pressure in the exhaust decoupler (both amplitude and phase). There arefive main dimensionless parameters: dimensionless exhaust decoupler volume v, dimensionless ventpipe length l, dimensionless vent pipe area s and dimensionless enthalpy ht0, hd0. Following the valuesof these five parameters, the working domain of the pulse combustor was divided into three parts: theinphase zone, critical interval and antiphase zone. In the critical interval domain, the pulse combustorcannot work stable. To make a pulse combustor work at the antiphase zone (recommended in engineer-ing applications), a large exhaust decoupler volume, and fairly long vent pipe with a proper cross-sectionarea are required.

� 2013 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

Pulse combustion is a process in which pressure, velocity andtemperature vary periodically with time [1]. Periodical combustionis caused by the interaction between the combustion process andthe acoustic characteristic of the combustor, which is also the mainreason for combustion instability in modern combustion devices[2]. Since the fluctuating flow in the combustor can improve theheat and mass transfer process [3–7], the pulse combustor is char-acterized by high efficiency, low cost and low emission.

The pulse combustor consists of intake valves of fuel and air, acombustion chamber, tailpipe, exhaust decoupler and vent pipe, asdepicted in Fig. 1. After gas and air enter the combustion chamber,the pressure and temperature in the chamber rise due to combus-tion. When the pressure rises above the atmospheric pressure, in-take valves begin to close and combustion products are pushed outthrough the tailpipe, exhaust decoupler and vent pipe. Because ofthe inertia of the product flow, pressure in the combustion cham-ber becomes lower than atmospheric pressure; then the valvesopen and a new work cycle begins [1].

For a better understanding of the mechanism of the pulse com-bustion process, researchers have developed several theoreticalmodels, such as the approximate analysis models [8–13] and

computational fluid dynamic (CFD) models [14,15]. Because pulsecombustion is a particular combustion instability problem, involv-ing physical and chemical matters that are still not well under-stood, it is difficult to offer a suitable formulation in a CFD model[2]. Therefore, the approximate analysis method which fulfillsbasic requirements is a good choice for better understanding ofthe pulse combustion.

One approximate analysis model which describes pulse com-bustion was developed by Ahrens et al. [8]. This lump parametermodel was based on a standard conservation of energy analysisfor the combustion chamber and a standard conservation ofmomentum analysis for the tail pipe. Both heat transfer and fric-tion were neglected and a simplified burning rate model basedon a plane flame front moving at a constant velocity was used.The model was described by a second-order nonlinear equationand was capable of predicting the pressure oscillation in thecombustion chamber. Ref. [16] gave an asymptotic solution ofthe model equation based on Poincaré–Lindstedt perturbationanalysis. To avoid complications caused by the discontinuity ofthe mass flow rate model, a simplified sub-model was used. Theform of the analytical solution was the sum of a series of trigono-metric function, which seemed reasonable. But the solved pressureamplitude in the combustion chamber was too small comparedwith the real situation. In Ref. [17], the tailpipe friction was addedto the model. The analysis process was similar to Ref. [16]. Resultsshowed that the velocity had a phase shift and the amplitudedecreased compared to the no-friction situation. The amplitude

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Fig. 1. Typical Helmholtz type pulse combustor with exhaust decoupler and vent pipe.

Fig. 2. Time sequence of pressure in combustion chamber and exhaust decoupler(inphase mode).

Fig. 3. Time sequence of pressure in combustion chamber and exhaust decoupler(antiphase mode).

2 L.-j. Yang et al. / Combustion and Flame xxx (2013) xxx–xxx

problem still existed. Ref. [18] used an improved mass flow ratemodel which is closer to the real situation. The results were moreconvincible compared with the earlier work and more numericalresults were given.

The model of Richards et al. [10] is also widely used in analysisof the pulse combustor. Their model was based on the thermalpulse combustion process in which fuel and air were supplied con-tinuously. This is a well-stirred reaction (WSR) model. By applyingconservation of mass, energy and species to the combustion cham-ber, and coupling this with the momentum conservation in a tail-pipe, Richards et al. derived a system of four ordinary differentialequations. Numerical analysis was required to solve these equa-tions. Their model was used and extended by other researchers.Heravi et al. [19] investigated the characteristics of primary pollu-tants of the pulse combustor using an integrated chemical kineticmodel. Datta et al. [20] considered the effects of optically thin radi-ation from the combustor gases to the wall and the qualitative re-sults suggest a transition to chaos through a period-doubling routeprior to extinction. Xu et al. [21] considered many factors might af-fect the combustion stability and the results showed qualitativeagreement with the experimental data.

It should be noted that in the literature mentioned above, theexhaust decoupler and vent pipe were not included in the model.Though the combustor can work without them (which is called abasic pulse combustor), these two components are mostly usedin commercial pulse combustion heating systems [22,23]. Themain reason for their popularity is that the pulse combustor ishighly sensitive to the geometric parameter; but there are safetystandards relating to the height of the exhaust vent [24], whichthe tailpipe cannot satisfy. An exhaust decoupler can act as an iso-lator, separating the basic pulse combustor with the vent pipe. Thevent pipe can then be used to satisfy the standards without infect-ing the basic pulse combustor. In addition, the exhaust decouplercan increase the heat transfer surface and reduce the noise levelof the pulse combustor [25].

The geometric parameters of the exhaust decoupler and ventpipe are also important. A proper design can stabilize the pulsecombustor, while a bad design can cause the entire system to breakdown. At present, exhaust decoupler design is based largely ontrial-and-error. One empirical rule is that larger is the better.

In the process of developing an 180 kW pulse combustor for acommercial furnace, the importance of the exhaust decouplerand vent pipe was noted. When the volume of the exhaust decou-pler and the length of vent pipe were both small, the pressure inthe combustion chamber was inphase with the pressure in the ex-haust decoupler and the operating frequency of the combustor waslow. At the same time, the pulse combustor tended to be unstableand the emission of carbon monoxide was higher. Figure 2 showsthe time sequence of pressure in the combustion chamber andthe exhaust decoupler in this situation. It can be seen that the fre-quency is about 30 Hz. When the volume of the exhaust decouplerand the length of vent pipe were large, the two pressures were inantiphase and the frequency was larger (36.5 Hz), as shown inFig. 3. The pulse combustor was also more stable.

Please cite this article in press as: L.-j. Yang et al., Combust. Flame (2013), htt

Similar results were reported in Ref. [25]. The experimental re-sults showed that a critical value existed for the decoupler volumeand vent pipe length respectively. When the volume or the lengthwas smaller than the critical value, the pulse combustor worked ina relatively low frequency mode and the pressures in the decouplerand combustion chamber were in phase. But when the volume orthe length was larger than the critical value, the pulse combustorworked in a higher frequency mode and the pressure in the decou-pler and the combustion chamber had a phase difference of 180�. Ifthe volume or the length is close to the critical value, the pulsecombustor failed to start. This phenomenon presented a question:What is the relation between critical value and the geometricparameters? Unfortunately, there has been little effort to offer atheoretical explanation in this field.

p://dx.doi.org/10.1016/j.combustflame.2013.07.003


L.-j. Yang et al. / Combustion and Flame xxx (2013) xxx–xxx 3

In this paper, the critical value problem will be treated, basedon a lump parameter model in which the exhaust decoupler andthe vent pipe were considered. The model equation was solvedusing the asymptotic analysis method and experimental validationwas made.

2. Model setup

The model of Ahrens et al. is one of the successful modelsdescribing the pulse combustor system. This model is started byintroducing instantaneous energy balance on the combustionchamber and it is based on the following assumptions:

(1) The combustion chamber contains two uniform regions(reactants and products) separated by a flame front. Thegases in the combustion chamber obey the ideal gas law.The pressure in the chamber is considered to be uniform atany instant of time.

(2) To simplify the energy balance, the heat loss term is omitted.The effect of heat transfer can be approximated by choosinga reduced enthalpy (h0), in the term representing outflow ofcombustion products from the chamber.

(3) The flow in the tailpipe is approximated as incompressibleand frictionless.

In the model of Ahrens et al., the model equation of the basicpulse combustor is a second order nonlinear equation of chamberpressure [8]:

d2pc

dt2 �R

cvVc

� �hR

ddpc

_mRðpcÞ þ CDH

1þ r

� �dpc

dtþ Rh0St

cvVcLt

� �pc ¼ 0 ð1Þ

where pc is the gauge pressure in the combustion chamber, Vc thevolume of the combustion chamber, R the gas constant, cv the spe-cific heat at constant volume, St the flow area of the tailpipe, Lt thelength of the tailpipe, hR the enthalpy per unit mass of the reactants,h0 the enthalpy per unit mass of the combustion products leavingthe combustion chamber, and DH is the heat of combustion per unitmass of gas.

_mRðpcÞ is the mass flow rate of reactants through a flapper valve.The valve is open only when the supply pressure is higher than thepressure in the combustion chamber. The flow through the valve istreated as orifice flow when the valve is open.

_mR ¼ ð1þ rÞ _mg ¼ð1þ rÞCg

ffiffiffiffiffiffiffiffiffi�pcp

; pc < 00; pc P 0

�; Cg ¼

ffiffiffiffiffiffiffiffi2qg

qCDgSg

ð2Þ

where _mg is the mass flow rate of gas, r the air–fuel ratio, qg thedensity of gas, Sg the effective flow area of the gas valve, CDg the dis-charge coefficient of the gas valve, and C is a constant which is de-fined as

C ¼ ScUf

RTcð3Þ

where Sc is the flow area of combustion chamber, Tc the gas temper-ature in combustion chamber, and Uf is the plane flame velocity inthe combustion chamber defined as

Uf ¼ 0:433CgRTa

Ac

ffiffiffiffiffiffiffiffiffiffiffiffihR

PaDH

sð1þ rÞ3=2 ð4Þ

where Ta is the temperature of the reactants entering combustionchamber and Pa is the absolute pressure of the atmosphere.

Similarly, the pulse combustor with an exhaust decoupler andvent pipe (Fig. 1) can be treated as two subsystems in a series:one is the combustion chamber and tailpipe system, another is


the exhaust decoupler and vent pipe system. Then, the modelequations for each system can be established respectively usingthe same process developed by Ahrens et al.

A. Combustion chamber and tailpipe systemThe model equation of this system is almost the same as that of

a basic pulse combustor except that the pressure of the tailpipeoutlet is the pressure in the exhaust decoupler pd instead of thegauge pressure of the atmosphere pa(=0).

d2pc

dt2 �R

cvVchR

ddpc

_mRðpcÞ þ CDH

1þ r

� �dpc

dtþ R

cvVc

h0St

Ltðpc � pdÞ ¼ 0

ð5Þ

B. Exhaust decoupler and vent pipe systemTwo different conditions are presented compared with the basic

pulse combustor: (1) no reaction takes place in the exhaust decou-pler and (2) the flow into the exhaust decoupler is the productleaving the tailpipe which is originally from the combustion cham-ber. Following the method of Ahrens et al., the model equation ofthis subsystem can be derived:

d2pd

dt2 þR

cvVd

Stht

Ltþ Szhd

Lz

� �pd �

RcvVd

Stht

Ltpc ¼ 0 ð6Þ

where Vd is the volume of exhaust decoupler, Sz the flow area ofvent pipe, Lz the length of vent pipe, ht the enthalpy of flow enteringthe exhaust decoupler, and hd is the enthalpy of flow leaving the ex-haust decoupler.

Eqs. (5) and (6) describe the entire pulse combustor system. Bydefining the following parameter,

ku ¼ RCcvVc

� DH1þ r

� DH1þ r

� ScUf

cvTaVcð7Þ

kclðpcÞ ¼

RhR

cvVc

ddpc

_mRðpcÞ ð8Þ

x2c0 ¼

Rh0St

cvVcLtð9Þ

x2t0 ¼

RhtSt

cvVdLt¼ Vcht

Vdh0x2

c0 ð10Þ

x2d0 ¼

RhdSz

cvVdLz¼ hdSzVcLt

h0StVdLzx2

c0 ð11Þ

these two equations become

d2pc

dt2 � kclðpcÞ þ kl

� dpc

dtþx2

c0pc �x2c0pd ¼ 0 ð12Þ

d2pd

dt2 þ x2t0 þx2

d0

�pd �x2

t0pc ¼ 0 ð13Þ

Because the mass flow rate described by Eq. (2) is a piecewisefunction which may complicate the analysis process, a simplifiedmass flow rate model using exponential function was used hereto replace the original one.

1ð1þ rÞCg

_mRðpcÞ ¼1B

e�bpc ; B > 0; b > 0 ð14Þ

where b and B are constants. In Refs. [16,17], b was set to 1 for con-venience. However, the results of the pressure amplitude aredependent on b; b = 1 will lead to very small amplitude. Ref. [18]used b to control the rate of change of the mass flow rate at pc = 0to obtain a reasonable amplitude.

Now Eq. (8) becomes

kclðpcÞ ¼ �

bBð1þ rÞC RhR

cvVce�bpc ð15Þ




If we set e = ku/xc0, Eqs. (12) and (13) become

d2pc

dt2 � exc0ð1� ae�bpc Þ dpc

dtþx2

c0pc �x2c0pd ¼ 0 ð16Þ

d2pd

dt2 þx2c0

Vcht

Vdh0þ hdSzVBLt

h0StVdLz

� �pd �

Vcht

Vdh0x2

c0pc ¼ 0 ð17Þ

where

a ¼ bð1þ rÞCg

BkuRhR

cvVc

� �ð18Þ

Set

g ¼ xc0t ð19Þ�pðgÞ ¼ pðg=xc0Þ ð20Þfað�pÞ ¼ 1� ae�bpc ð21Þ

r ¼ Vcht

Vdh0; d ¼ Vcht

Vdh0þ hdSzVcLt

h0StVdLzð22Þ

Eqs. (16) and (17) become

d2�pc

dg2 � efað�pcÞd�pc

dgþ �pc � �pd ¼ 0 ð23Þ

d2�pd

dg2 þ d�pd � r�pc ¼ 0 ð24Þ

Furthermore, if we set

h ¼ �xg ð25Þp̂ðhÞ ¼ �pðh= �xÞ ð26Þ

the final nonlinear model equations of the entire pulse combustorsystem can be obtained

�x2 d2p̂c

dh2 � e �xfaðp̂cÞdp̂c

dhþ p̂c � p̂d ¼ 0 ð27Þ

�x2 d2p̂d

dh2 þ dp̂d � rp̂c ¼ 0 ð28Þ

3. Nonlinear analysis of the model equations

The nonlinear model equations are analyzed by applying thePoincaré–Lindstedt procedure. The Poincaré–Lindstedt method(or Lindstedt–Poincaré method), is a technique for uniformlyapproximating periodic solutions to ordinary differential equa-tions, when regular perturbation approaches fail. The method re-moves secular terms (terms growing without bounds), arising inthe straightforward application of the perturbation theory toweakly nonlinear problems with finite oscillatory solutions [26–28].

Assume the solutions have the following form:

p̂cðhÞ ¼ pc0ðhÞ þ pc1ðhÞeþ pc2ðhÞe2 þ � � � ð29Þp̂dðhÞ ¼ pd0ðhÞ þ pd1ðhÞeþ pd2ðhÞe2 þ � � � ð30Þ�xðeÞ ¼ 1þ �x1eþ �x2e2 þ � � � ð31Þ

Substitute these equations into Eqs. (27) and (28):

1þ 2 �x1eþ 2 �x2 þ �x21

�e2 þ � � �

�p00c0 þ p00c1eþ p00c2e

2 þ � � � �

� ð1� ae�bp̂c Þðeþ �x1e2 þ �x2e3 þ � � �Þ p0c0 þ p0c1eþ p0c2e2 þ � � �

�þ ðpc0 þ pc1eþ pc2e2 þ � � �Þ � ðpd0 þ pd1eþ pd2e2 þ � � �Þ ¼ 0 ð32Þ

1þ2 �x1eþ 2 �x2þ �x21

�e2þ��

�p00d0þp00d1eþp00d2e

2þ��

þdðpd0þpd1eþpd2e2þ��Þ�rðpc0þpc1eþpc2e2þ��Þ¼0 ð33Þ

Define n = 1/e and retain terms up to order O(e2), we get


p00c0þ 2 �x1p00c0þp00c1

�eþ 2 �x2þ �x2

1

�p00c0þp00c2þ2 �x1p00c1

� e2

þðpc0þpc1eþpc2e2Þ�neðpd0þpd1eþpd2e2Þ� p0c0eþ �x1p0c0þp0c1

�e2� þae�bp̂c p0c0eþ �x1p0c0þp0c1

�e2� ¼0 ð34Þ

p00d0 þ 2 �x1p00d0 þ p00d1

�eþ 2 �x2 þ �x2

1

�p00d0 þ p00d2 þ 2 �x1p00d1

� e2

þ d pd0 þ pd1eþ pd2e2 �� rðpc0 þ pc1eþ pc2e2Þ ¼ 0 ð35Þ

In Eq. (34),

e�bp̂c ¼ e�bðpc0þpc1eþpc2e2þ��Þ ¼ e�bpc0 e�ek

¼ e�bpc0 1� ekþ 12!

e2k2 � � � ��

ð36Þ

where k = b (pc1 + pc2e + pc3e2 + � � �).By ignoring the terms higher than order O(e2), Eq. (36) become

e�bp̂c ¼ e�bpc0 1� bpc1eþ b12

bp2c1 � pc2

� �e2

� �ð37Þ

Substitute Eq. (37) into Eq. (34), and group the terms in Eqs.(34) and (35) by the order of e (to second order). A sequence ofequations is now obtained

p00c0 þ pc0 ¼ 0 ð38Þp00c1 þ pc1 ¼ npd0 � 2 �x1p00c0 þ p0c0ð1� ae�bpc0 Þ ð39Þ

p00c2 þ pc2 ¼ npd1 � 2 �x1p00c1 � 2 �x2 þ �x21

�p00c0 þ abe�bpc0 pc1p0c0

þ �x1p0c0 þ p0c1

�ð1� ae�bpc0Þ ð40Þ

p00d0 þ dpd0 � rpc0 ¼ 0 ð41Þp00d1 þ dpd1 � rpc1 ¼ �2 �x1p00d0 ð42Þp00d2 þ dpd2 � rpc2 ¼ �2 �x1p00d1 � 2 �x2 þ �x2

1

�p00d0 ð43Þ

The solving process of these equations is given in the Appendix.The solutions of the first two components of Eq. (29) to Eq. (31) aresummarized below:

pc0 ¼ A0ðeih þ e�ihÞ ð44Þ

pc1 ¼ �iac2

3bðe2ih � e�2ihÞ � iac3


15bðe4ih

� e�4ihÞ ð45Þ

pd0 ¼r

d� 1A0ðeih þ e�ihÞ ð46Þ

pd1 ¼2 �x1A0rðd� 1Þ2

ðeih þ e�ihÞ � rd� 4

iac2

3bðe2ih � e�2ihÞ � r

d� 9

� iac3


d� 16iac4

15bðe4ih � e�4ihÞ ð47Þ

�x1 ¼ �rn

2ðd� 1Þ ð48Þ

�x2 ¼ �1

2A0

2 �x1A0rn

ðd� 1Þ2þ �x2

1A0 þa2

3bb1c2 þ

a2

8bb2c3 þ

a2

15bb3c4

!

ð49Þ

where

A00 ¼ b � A0 ð50Þ

X1k¼0

A02k0

k!

1k!� A020ðkþ 2Þ!

" #¼ 1

að51Þ




cn ¼ ð�1ÞnX1k¼0

A02kþn0

k!

1ðkþ n� 1Þ!�

A020ðkþ nþ 1Þ!

!ð52Þ

b1 ¼X1k¼0

A02kþ10

k!

2ðkþ 1Þ!�

1k!

� �þ 2

X1k¼0

A02kþ30

k!ðkþ 3Þ!þX1k¼0

A02kþ50

k!ðkþ 4Þ!

b2 ¼X1k¼0

A02kþ20

k!

1ðkþ 1Þ!�

3ðkþ 2Þ!

� �� 3

X1k¼0

A02kþ40

k!ðkþ 4Þ!�X1k¼0

A02kþ60

k!ðkþ 5Þ!

b3 ¼X1k¼0

A02kþ30

k!

4ðkþ 3Þ!�

1ðkþ 2Þ!

� �þ 4

X1k¼0

A02kþ50

k!ðkþ 5Þ!þX1k¼0

A02kþ70

k!ðkþ 6Þ!ð53Þ

Then, by recalling the variable transformation

hðtÞ ¼ �xgðtÞ ¼ �xx0t ð54Þ

the asymptotic solution of the nonlinear pulse combustor modelcan be written as

pcðtÞ ¼ 2A0 cos hðtÞ

� 2aec2

3bsin 2hðtÞ þ c3

8bsin 3hðtÞ þ c4

15bsin 4hðtÞ

� �ð55Þ

pdðtÞ¼r

d�11� rðd�1Þ2

!2A0 cos hðtÞ

�2are1

d�4c2

3bsin 2hðtÞþ 1

d�9c3

8bsin 3hðtÞþ 1

d�16c4

15bsin 4hðtÞ

� �ð56Þ

�x ¼ 1� r2ðd� 1Þ �

r2

8ðd� 1Þ2þ r2

2ðd� 1Þ3

� a2e2

2A00

13

b1c2 þ18

b2c3 þ1

15b3c4

� �ð57Þ

4. Experimental setup

A 12 kW Helmholtz type pulse combustor was designed to per-form the experiments. The liquefied petroleum gas (LPG) was se-lected as the fuel. Figure 4 presents a schematic of thecombustor system. One-way flapper valves are located just beforethe reactants’ inlet. Fuel and air are injected into the head of thecombustion chamber perpendicularly. The exhaust decoupler hasa fission structure. The front plate and the back plate can movealong the cylinder so the volume of the exhaust decoupler can becontinuously changed; the vent pipe is connected to the back plateand is also replaceable. Since the cross-sectional area of the ventpipe is difficult to constantly change, only the volume of the ex-haust decoupler and the length of the vent pipe are changed duringthe experiments.

Measurements of the pressure and temperature were obtainedfor this study. The pressure was measured using a dynamic pres-sure transducer with a natural frequency of 100 kHz and rise timeresponse lower than 1 ls. The temperature was obtained using athermocouple. The arrangement of the transducer is also shownin Fig. 4. The data acquisition rate was maintained at 10 kHz, a ratesufficiently fast to accurately resolve the combustor operating fre-quency of 50–150 Hz. The tests on each working condition werecarried out three times repeatedly; two items of data were re-corded each time. The experimental results were based on the sta-tistics of these six items of data. The Fast Fourier Transform (FFT)method was used to analyze the time sequence of the pressure.


5. Results and discussion

The validation of the model results were carried out first. A typ-ical work condition of the experimental pulse combustor was cho-sen to perform the validation. The parameters of the combustor arelisted in Table 1. According to the experimental results, the pres-sure oscillation in the combustion chamber was 87.4 Hz in fre-quency and 9060 Pa in amplitude. Therefore, the constant b inthe mass flow rate model was set to 0.001 due to the order ofthe pressure amplitude, while the constant B was set to1.55 � 10�4 to fit the pressure amplitude. By using these parame-ters, the variables used in the models were calculated, and the re-sults are shown in Table 2.

The pressure oscillation can now be written in the followingform:

pcðtÞ ¼ 9060 cos �xx0t � 395 sin 2 �xx0t þ 166 sin 3 �xx0t

� 80 sin 4 �xx0t ð58Þ

pdðtÞ ¼ �0:043� 9060 cos �xx0t þ 0:026� 395 sin 2 �xx0t

� 0:011� 166 sin 3 �xx0t þ 0:006� 80 sin 4 �xx0t ð59Þ

�x ¼ 1þ 0:0672� 0:0023� 0:0118� 0:0048 ¼ 1:0483 ð60Þ

Eqs. (58) and (59) indicate that the amplitudes of the multiplefrequencies are relatively smaller than the base frequency. Thisconclusion is more obvious in view of the time series of the pres-sure, as shown in Fig. 5. In the following analysis, only the informa-tion of the base frequency will be considered.

In Eq. (60), the last three terms are also small but only the lastterm is neglected because it is not directly concerned with the ex-haust decoupler and vent pipe system. Next, Eq. (55) to Eq. (56) aresimplified to

pcðtÞ ¼ 2A0 cos hðtÞ ð61Þ

pdðtÞ ¼ l2A0 cos hðtÞ where l ¼ rd� 1

1� rðd� 1Þ2

!ð62Þ

�x ¼ 1� r2ðd� 1Þ �

r2

8ðd� 1Þ2þ r2

2ðd� 1Þ3ð63Þ

From these expressions, some information can be obtainedregarding the pulse combustor system:

(1) The pressure oscillation in the exhaust decoupler has thesame form as that in the combustion chamber, except for acoefficient l. The modulus of l represents the amplitudeof the pressure, while the sign of this coefficient indicatesthe phase relation between the pressure in the exhaustdecoupler and the combustion chamber: positive for thesame phase; negative for the reversed phase. The two pres-sure oscillation has the same frequency:

p://dx.d

f ¼ �xx0=2p ð64Þ

(2) There is a singular point (d = 1) for Eqs. (62) and (63). At thispoint, the two expressions fail, which may also mean thatthe pulse combustor is unstable.

(3) The main factors influencing pressure oscillation are somedimensionless parameters in the expressions of r and d(Eq. (22)); three of these are concerned with the geometricparameters:Dimensionless exhaust decoupler volume:

v ¼ Vd=Vc ð65Þ

Dimensionless vent pipe length:

oi.org/10.1016/j.combustflame.2013.07.003


Fig. 4. Schematic of the experimental pulse combustor system.

Table 1Parameters of the experimental pulse combustor.

Sc 5.809 � 10�3 m2 Ta 300 KVc 1.046 � 10�3 m3 c 1.4St 8.042 � 10�4 m2 hR 301,560 J/kgLt 1.3 m h0 1,395,970 J/kgSz 2.827 � 10�3 m2 ht 821,950 J/kgLz 2.2 m hd 533,980 J/kgVd 6.011 � 10�3 m3 DH 45,254,000 J/kgr 17.7 qg 2.12 kg/m 3

R 287 J/(kg K) Sg 4.71 � 10�6 m2

CDg 0.6 cv 829 J/(kg K)

Table 2Variables used in the nonlinear model.

Cg 5.819 � 10�6 x c0 534.62 radUf 0.774 m/s f0 85.1 Hze 0.078 kl 41.768A0 4530 Pa a 4.15 � 10�3

d 0.241 r 0.102c2 1832.64 c3 �2060.45c4 1845.89 b1 404.556b2 �454.846 b3 407.481

Fig. 5. Model results of pressure in combustion chamber and exhaust decoupler.


Please

l ¼ Lz=Lt ð66Þ

Dimensionless vent pipe area:

cite this article in press as: L.-j. Yang et al., Combust. Flame (2013), http://dx.d

s ¼ Sz=St ð67Þ

The other two are dimensionless enthalpy,

ht0 ¼ ht=h0 ð68Þ

hd0 ¼ hd=h0 ð69Þ
The influence of these parameters on the pulse combustor will
be discussed in the following section. The parameters not men-tioned use the values given in Table 1. Unlike the conventionalcombustors, the physical processes in pulse combustor are peri-odical at the standard running conditions. Before analysis of theexperimental data, the stability of the pulse combustor shouldbe defined. The perfect situation is that the pressure (and otherparameters) oscillation is sustained with constant frequency andamplitude. But in practice, these two parameters are usually notinvariant, especially the amplitude. In engineering, a pulse com-bustor is stable when it can work continuously as long as needed.Considering the lack of cooling in the experiments, 5 min waschosen as the standard time here. Besides, the oscillation of thepressure amplitude was also restricted. A stable pulse combustorshould satisfy two qualifications: (1) it can work 5 min withoutblowing out in all three tests and (2) the oscillation of the pres-sure amplitude in the combustion chamber is less than 40% ofthe average amplitude. Otherwise, the pulse combustor isunstable.

The dependence of the coefficient �x and l on the dimensionlessvolume v is shown in Figs. 6 and 7, respectively. The critical valueof v is 3.5 when l = 0.46, s = 3.52. It can be seen that the theoreticalvalues of the two coefficients diverge when v is close to the criticalvalue. In the experiments, there is also a region in which the pulsecombustor cannot work stably, as marked by the dashed lines.When v is smaller than this critical value, �x is smaller than 1,which means that the frequency of pressure oscillation (f) is lowerthan the natural frequency of the basic pulse combustor f0 (=xc0/2p). Also in this situation, the phase difference between the pc

and pd (/cd) is zero (inphase). When v is larger than the critical va-lue, f is higher than f0 and gets closer to f0 as v increases and thephase difference /cd is 180� (antiphase). The theoretical resultagrees well with the experimental result both qualitatively andquantitatively. There are some differences between them on thelocation of the unstable region; the reason is that in the theoreticalanalysis, the enthalpy of the gas was assumed to be constant. In theexperiments, however, enthalpy was related to heat release of thecombustion processes, which differed with the working conditions,especially when the combustor was unstable.

oi.org/10.1016/j.combustflame.2013.07.003


Fig. 6. Dependence of the coefficient �x on the dimensionless volume v.

Fig. 7. Dependence of the coefficient l on the dimensionless volume v.


Figures 8 and 9 show the dependence of the coefficient �x and lon the dimensionless vent pipe length l. The result is similar to thatcaused by v. The theoretical critical value of l is about 0.26 whenv = 5.75, s = 3.52. Both theoretical and experimental results containa ‘‘bad’’ region. f is lower than f0 and /cd = 0� if l is under this region,while f is higher than f0 and /cd = 180� if l is above it. The differ-ences still exist between the locations of the unstable region.

The dependence of the coefficient �x and l on the dimensionlessvent pipe area s is shown in Fig. 10. The critical value is abouts = 6.23 when v = 5.75, l = 0.46. When s is smaller than 6.23, f is

Fig. 8. Dependence of the coefficient �x on the dimensionless vent pipe length l.


higher than f0 and /cd = 180�. When s is larger than 6.22, f is lowerthan f0 and /cd = 0�.

Basically, these results agree with the experimental data in Ref.[25]: a critical value for frequency mode and phase difference. Thecritical value corresponds to the singular point mentioned above inthe theoretical model. Though expressions are not tenable only atthe singular point, the results near this point are obviously not rea-sonable in the real word, such as the frequency f being much higheror lower than the natural frequency f0. The experimental resultsalso prove this. It is possible to conclude that there is not only acritical value, but a critical interval in which the pulse combustorcannot work stable.

If the combustion process is neglected, the pulse combustorwith an exhaust decoupler and vent pipe becomes a pure acousticsystem. It can be treated as two Helmholtz resonators installed in aseries. In acoustics, this system has two resonant frequencies [29].Considering the temperature difference in the two resonators andusing the parameters above, the result can be expressed as

�x1;2¼1ffiffiffi2p

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þz2 � s

v � lþz2 �1v�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þz2 � s

v � lþz2 �1v

� �2

�4z2 � sv � l

svuutð70Þ

where z is the ratio of the speed sound in the two Helmholtzresonators:

z ¼ Zd=Zc ð71Þ

Zc is the average speed sound in the combustion chamber and tail-pipe system and Zd is the average speed sound in the Exhaustdecoupler and vent pipe system.

Though the two resonant frequencies are in equal position inacoustics, the combustion process and other important processestake place in the combustion chamber. So the frequency closer tothe natural frequency of the basic pulse combustor was chosenas the operating frequency. As an example, Fig. 11 shows thedependence of �x on the dimensionless vent pipe length l whenv = 5.75, s = 3.52. The frequency f1ð¼ �x1xc0=2pÞ is always largerthan f0, while the frequency f2ð¼ �x2xc0=2pÞ is always smaller thanf0. At l = 0.32, these two frequencies have equal deviation from f0.When l < 0.32, f2 is closer to f0. So the pulse combustor works atthe low frequency mode (f2). For the same reason, the combustorworks at the high frequency mode (f1) when l > 0.32. The criticalinterval is denoted as the transition region of the two modes andlocates at the region near l = 0.32. In this region, it’s not sure which

Fig. 9. Dependence of the coefficient l on the dimensionless vent pipe length l.



Fig. 10. Dependence of the coefficients �x and l on the dimensionless vent pipearea s.


frequency predominates over the other. And this may be the rea-son why pulse combustor cannot work stably in the critical inter-val. Because Eq. (70) was the result of purely acoustic analysis, itonly agreed with the theoretical and experimental results abovequalitatively.

It can be seen that the low frequency region is small and thepulse combustor working in this region would easily move to thecritical interval by a small change in the system parameters. Ref.[25] also suggested that pulse combustor working at high fre-quency mode was more stable and clean. If the boundary of thecritical interval could be determined, the design of the exhaustdecoupler and vent pipe would be more effective. Based on theabove results, the phase difference was chosen as the criterionhere. It was assumed that the points at which l = 0 were theboundary of the critical interval. Then the working domain of thepulse combustor was separated into three parts: inphase zone,antiphase zone and the critical interval between them.

Figure 12 shows a contour map of l in v � l plane when s = 2.6.The lines of l = 0 were marked by a thick black line. The inphasezone is at the lower left, while the antiphase zone is at the upperright. The critical interval of v for a certain l becomes smaller as lincreases. At the same time, the critical interval moves towardv = 0, which means that the inphase zone decreases. The criticalinterval of l for a certain v shows a similar trend. Larger v and l isneeded to allow the pulse combustor to work at the antiphasezone.

Fig. 11. Dependence of the coefficient �x on the dimensionless vent pipe length lbased on acoustic method.


The contour map of l in v � s plane when l = 1.33 is shown inFig. 13. The inphase zone is at the upper left, while the antiphasezone is at the lower right. The critical interval of v becomes largeras s increases, and it moves toward v = +1. The critical interval of sis more complex. There are not always three zones. The criticalinterval appears when v > 0.2 and the antiphase zone appearswhen v > 1.35. The critical interval of s becomes larger as v in-creases and moves toward s = +1. To allow the pulse combustorto work at the antiphase zone, smaller s and larger v are better.

The contour map of l in s–l plane when v = 3 is shown in Fig. 14.The critical interval of s(l) becomes larger as l(s) increases. Larger land smaller s is preferred for a stable pulse combustor.

To more intuitively depict the influence of the geometricparameter on the working status of the pulse combustor, Fig. 15shows the zone partitioning in a 3D space of v–l–s. The two sur-faces are the boundaries of the critical interval. Some conclusionscan be drawn from this graph and the results above:

(1) v is the most important parameter affecting the working sta-tus of the pulse combustor. As v grows, the critical intervaland the inphase zone quickly diminish and tend to vanish.Since the geometry of the basic pulse combustor cannot sig-nificantly change, the volume of the exhaust decouplershould be as large as possible to make the pulse combustorwork at the antiphase zone.

(2) The critical interval is sensitive to small l; it becomes stableas l increases. So, when the length of the vent pipe is largeenough, changing the length will not improve the perfor-mance of the pulse combustor further.

(3) Smaller s can make the pulse combustor more stable; butsince the friction of the vent pipe is not included in themodel, the results are only valid if the diameter of vent pipeis not too small. Therefore, a proper s is required. It is sug-gested that the value of s should between 1.0 and 3.0.

The influence of the dimensionless enthalpy will be studiednext. The enthalpy is directly concerned with the temperature ofthe exhaust; therefore the values of these two dimensionlessenthalpies reflect the heat transfer process in the pulse combustorsystem. Under normal condition, the exhaust decoupler and thetailpipe are set in the same cooling environment. It is assumed thatthe heat transfer at the exhaust decoupler is 30% of that at the tail-pipe. Then hd is set to

Fig. 12. Contour map of l in v–l plane (s = 2.6).



Fig. 13. Contour map of l in v–s plane (l = 1.33).

Fig. 14. Contour map of l in s–l plane (v = 3).

Fig. 15. Zone partitioning in a 3D v–l–s space.

Fig. 16. Contour map of l in v–ht0 plane (l = 1.33, s = 2.6).



hd ¼ ht � 0:3ðh0 � htÞ ð72Þ

to simulate real conditions.Figure 16 shows the contour map of l in v � ht0 plane. It can be

seen that there is only an antiphase zone when v is large and ht0 issmall. When ht0 is greater than 0.25, inphase zone and criticalinterval begin to appear, and larger v is needed to stabilize thepulse combustor. The results also indicate that, in real world appli-cation, increasing the efficiency of the cooling system is also amethod to improve the stability of the pulse combustor.

6. Conclusion

A nonlinear model of a pulse combustor with exhaust decouplerand vent pipe was set up based on Ahrens’ basic pulse combustormodel. The model equations were solved using the Poincaré–Lind-stedt perturbation analysis method. The solutions included expres-sions for the pressure in the combustion chamber and the exhaustdecoupler respectively. Experimental validation was made andagreed well with the theoretical results. On further analysis ofthe results, several conclusions were drawn:

1. The pressure in the exhaust decoupler was uniform in fre-quency with that in the combustion chamber. This frequencywas �x times the frequency of the corresponding basic pulsecombustor. The pressure amplitude damped while spreadingfrom the combustion chamber to the exhaust decoupler, witha damping coefficient l. Both �x and l were decided using fivedimensionless parameters: dimensionless exhaust decouplervolume v, dimensionless vent pipe length l, dimensionless ventpipe area s and dimensionless enthalpy ht0, hd0.

2. For these five dimensionless parameters, there was a criticalinterval in which the pulse combustor cannot work stable.The critical interval divided the entire working domain intothree parts. At one side of the critical interval, there was aninphase zone. A pulse combustor working in this zone had a rel-atively low frequency mode and the pressures in the combus-tion chamber and the exhaust decoupler were inphase; at theother side, it was the antiphase zone. A pulse combustor work-ing in this zone had a relatively high frequency mode and thepressures in the combustion chamber and the exhaust decou-pler have a phase difference of 180�. The antiphase zone isthe recommended stable working domain for a pulsecombustor.




3. In order for the pulse combustor to work in the antiphase zone,a large volume exhaust decoupler and a long – but slender –vent pipe was required. Also, a better cooling system wasproved to be more effective for a stable pulse combustor.

Acknowledgments

This work was supported by the Beijing Natural Science Foun-dation (Support Number: 8102026). The authors thank Prof. XiJiang from Lancaster University for providing language help.

Appendix A. Solving process of model equations

The equations to be solved are repeated here again forconvenience.

p00c0 þ pc0 ¼ 0 ðA1Þp00c1 þ pc1 ¼ npd0 � 2 �x1p00c0 þ p0c0ð1� ae�bpc0Þ ðA2Þ

p00c2 þ pc2 ¼ npd1 � 2 �x1p00c1 � 2 �x2 þ �x21

�p00c0 þ abe�bpc0 pc1p0c0

þ �x1p0c0 þ p0c1

�ð1� ae�bpc0 Þ ðA3Þ

p00d0 þ dpd0 � rpc0 ¼ 0 ðA4Þp00d1 þ dpd1 � rpc1 ¼ �2 �x1p00d0 ðA5Þp00d2 þ dpd2 � rpc2 ¼ �2 �x1p00d1 � 2 �x2 þ �x2

1

�p00d0 ðA6Þ

The solution of Eq. (A1) has a general form

pc0 ¼ A0ðeih þ e�ihÞ ðA7Þ

with A0 is a constant and will be determined later.Substitute Eq. (A7) to Eq. (A4)

p00d0 þ dpd0 ¼ rA0ðeih þ e�ihÞ ðA8Þ

Eq. (A8) has a solution as

pd0 ¼ B0 eiffiffidp

h þ e�iffiffidp

h�

þ rd� 1

A0ðeih þ e�ihÞ ðA9Þ

with B0 is a constant.Considering Eq. (A2), the exponent term can be treated as

following

e�bpc0 ¼X1n¼0

ð�1Þn

n!bnpn

c0 ¼X1n¼0

ð�1Þn

n!A0n0 ðeih þ e�ihÞn

¼X1n¼0

ð�1Þn

n!A0n0Xn

m¼0

nm

� �eimhe�iðn�mÞh

¼X1n¼0

ð�1Þn

n!A0n0 e�inh

Xn

m¼0

nm

� �e2imh

!ðA10Þ

where A00 ¼ b � A0.Expanding Eq. (A10) and regrouping the terms, we get

e�bpc0 ¼X1k¼0

1ð2kÞ!

2kk

� �A02k

0

þX1q¼1

ð�1ÞqX1k¼0

1ð2kþ qÞ!

2kþ qk

� �A02kþq

0 ðeiqh

þ e�iqhÞ ðA11Þ

Substituting Eqs. (A7), (A9) and (A11) to Eq. (A2), the first termright of Eq. (A2) becomes

npd0 ¼ nB0 eiffiffidp

h þ e�iffiffidp

h�

þ rd� 1

nA0ðeih þ e�ihÞ ðA12Þ

The second term becomes


�2 �x1p00c0 ¼ 2 �x1A0ðeih þ e�ihÞ ðA13Þ

And the third term becomes

p0c0ð1� ae�bpc0 Þ ¼ iA0 1� aX1k¼0

1ð2kÞ!

2k

k

� �A02k

0

"

þ aX1k¼0

1ð2kþ 2Þ!

2kþ 2k

� �A02kþ2

0

#ðeih � e�ihÞ

þ aA0

X1q¼1

ð�1Þqþ1X1k¼0

1ð2kþ qÞ!

2kþ q

k

� �A02kþq

0

"(

þ ð�1ÞqX1k¼0

1ð2kþ qþ 2Þ!

2kþ qþ 2k

� �A02kþqþ2

0

#

�ðeiðqþ1Þh � e�iðqþ1ÞhÞ�

ðA14Þ

To avoid the secular term in the solution of Eq. (A2), the coeffi-cients of exp(ih) + exp(�ih) and exp(ih) � exp(�ih) must be zero.Sum Eq. (A12) to Eq. (A14) up, which equals

2 �x1 þr

d� 1n ¼ 0 ðA15Þ

X1k¼0

A02k0

k!

1k!� A020ðkþ 2Þ!

" #¼ 1

aðA16Þ

From Eq. (A15), �x1 is easily solved.

�x1 ¼ �rn

2ðd� 1Þ ðA17Þ

Using the results above, Eq. (A5) can be written as

p00c1 þ pc1 ¼ iaA0

X1q¼1

ð�1Þqþ1X1k¼0

12kþ qð Þ!

�2kþ q

k

� �A02kþq

0 ðeiðqþ1Þh � e�iðqþ1ÞhÞ

þ iaA0

X1q¼1

ð�1ÞqX1k¼0

1ð2kþ qþ 2Þ!

2kþ qþ 2k

� �A02kþqþ2

0

� eiðqþ1Þh � e�iðqþ1Þh �þ nB0 ei

ffiffidp

h þ e�iffiffidp

h�

ðA18Þ

Reform Eq. (A18),

p00c1þpc1¼ iaA0

X1n¼2

ð�1ÞnX1k¼0

A02kþn�10

k!

1ðkþn�1Þ!�

A020ðkþnþ1Þ

!( )

� einh�e�inh �

þnB0 eiffiffidp

hþe�iffiffidp

h�

ðA19Þ

Set

cn ¼ ð�1ÞnX1k¼0

A02kþn0

k!

1ðkþ n� 1Þ!�

A020ðkþ nþ 1Þ!

!ðA20Þ

Then Eq. (A19) can be written as

p00c1 þ pc1 ¼iab

X1n¼2

cn einh � e�inh �

þ nB0 eiffiffidp

h þ e�iffiffidp

h�

ðA21Þ

A particular solution for Eq. (A21) is found

p�c1 ¼iab

X1n¼2

cn

1� n2 ðeinh � e�inhÞ þ 1

1� dnB0 ei

ffiffidp

h þ e�iffiffidp

h�

ðA22Þ

So the general solution has the form

pc1 ¼ A1ðeih þ e�ihÞ þ iab

X1n¼2

cn

1� n2 ðeinh � e�inhÞ

þ 11� d

nB0 eiffiffidp

h þ e�iffiffidp

h�

ðA23Þ




where A1 is a constant to be determined subsequently. By limitingthe value of n to 2, 3, 4,

apc1p0c0e�bpc0 ¼ iaA1

X1k¼0

A02kþ20

k!

A020ðkþ3Þ!�

1ðkþ1Þ!

!" #ðeih�e�ihÞ

þ a2 A0 c23

X1k¼0

A02kþ20k!

A20

ðkþ4Þ!þ 1ðkþ2Þ!

� �A02k

0k!

A020ðkþ2Þ!þ 1

k!

� " #þa2A0 c3

8

X1k¼0

A02kþ10k!

A020ðkþ3Þ!þ 1

ðkþ1Þ!

� �A02kþ3

0k!

A020ðkþ5Þ!þ 1

ðkþ3Þ!

� " #þa2A0 c4

15

X1k¼0

A02kþ40k!

A020kþ6ð Þ!þ 1

ðkþ4Þ!

� �A02kþ2

0k!

A020ðkþ4Þ!þ 1

ðkþ2Þ!

� " #( )

� ðeihþe�ihÞþKðhÞðA31Þ

pc1 ¼ A1ðeih þ e�ihÞ � iac2


8bðe3ih � e�3ihÞ

� iac4

15bðe4ih � e�4ihÞ þ 1

1� dnB0 ei

ffiffidp

h þ e�iffiffidp

h�

ðA24Þ

Using the solutions of pc0, pc1 and pd0, Eq. (A5) becomes

p00d1þdpd1¼rnB0

1�dþ2 �x1dB0

� �eiffiffidp

hþe�iffiffidp

h�

þr A1þ2 �x1A0

d�1

� �ðeihþe�ihÞ

�r iac2

3bðe2ih�e�2ihÞþ iac3


15bðe4ih�e�4ihÞ

� �ðA25Þ

For the solution of Eq. (A25), secular terms disappear when thecoefficient of the term is exp(id1/2 h) + exp(�id1/2h) zero, which meansrnB0

1� dþ 2 �x1dB0 ¼ 0 ðA26Þ

Eq. (A26) is satisfied only when B0 = 0.Therefore, Eq. (A25) becomes

p00d1þdpd1¼r A1þ2 �x1A0

d�1

� �ðeihþe�ihÞ

�r iac2




� �ðA27Þ

One of the particular solutions of Eq. (A27) is

p�d1 ¼r

d� 1A1 þ

2 �x1A0

d� 1

� �ðeih þ e�ihÞ � r

d� 4iac2

3bðe2ih

� e�2ihÞ � rd� 9

iac3


d� 16iac4

15bðe4ih

� e�4ihÞ ðA28Þ

Then the general solution of Eq. (A27) is

pd1 ¼ B1 eiffiffidp

h þ e�iffiffidp

h�

þ p�d1 ðA29Þ

where B1 is a constant to be determined.There is now sufficient information to solve Eq. (A3). In the

fourth term at the right side,

abpc1p0c0¼ab½iA0ðeih�e�ihÞ� A1ðeihþe�ihÞ� iac2


�

� iac3

8bðe3ih�e�3ihÞ� iac4


�

¼�a2A0c2

3ðe�ihþeihÞþ iaA00A1ðe2ih�e�2ihÞ

�a2A0c3

8ðe2ihþe�2ihÞþ a2A0c2

3�a2A0c4

15

� �ðe3ihþe�3ihÞ

þa2A0c3

8ðe4ihþe�4ihÞþa2A0c4

15ðe5ihþe�5ihÞ ðA30Þ


Multiply Eqs. (A11) and (A30), and the forth term can be writtenas

where K(h) is the sum of terms without exp(ih) + exp(�ih) andexp(ih) � exp(�ih).

The fifth term at the right side of Eq. (A3) has the followingexpression

�x1p0c0 þ p0c1

�ð1� ae�bpc0 Þ ¼

i �x1A0 þ A1ð Þðeih � e�ihÞ þ 2ac2

3bðe2ih þ e�2ihÞ þ 3ac3

8bðe3ih þ e�3ihÞ

�

þ 4ac4

15bðe4ih þ e�4ihÞ

�� 1� a

X1k¼0

1ð2kÞ!

2k

k

� �A02k

0

!"

þ aX1q¼1

ð�1Þqþ1X1k¼0

1ð2kþ qÞ!

2kþ qk

� �A02kþq

0 ðeiqh þ e�iqhÞ#ðA32Þ

Reform Eq. (A32) and the result is

�x1p0c0 þ p0c1

�ð1� ae�bpc0 Þ ¼ ð �x1A0 þ A1Þi

� 1þ aX1k¼0

1ð2kþ 2Þ!

2kþ 2k

� �A02kþ2

0 � 1ð2kÞ!

2kk

� �A02k

0

!" #

� ðeih � e�ihÞ þ 2a2c2

3b

X1k¼0

1ð2kþ 1Þ!

2kþ 1k

� �A02kþ1

0 þ 1ð2kþ 3Þ!

� 2kþ 3k

� �A02kþ3

0

�ðeih þ e�ihÞ � 3a2c3

8b

�X1k¼0

1ð2kþ 2Þ!

2kþ 2k

� �A02kþ2

0 þ 1ð2kþ 4Þ!

2kþ 4k

� �A02kþ4

0

!

� ðeih þ e�ihÞ þ 4a2c4

15b

X1k¼0

1ð2kþ 3Þ!

2kþ 3k

� �A02kþ3

0

þ 1ð2kþ 5Þ!

2kþ 5k

� �A02kþ5

0

�ðeih þ e�ihÞ þCðhÞ ðA33Þ

where C(h) is the sum of terms without exp(ih) + exp(�ih) andexp(ih) � exp(�ih).

Eq. (A33) can be written to a simpler form.

�x1p0c0 þ p0c1

�ð1� ae�bpc0 Þ ¼ ð �x1A0 þ A1Þi

� 1þ aX1k¼0

A02k0

k!

A020ðkþ 2Þ!�

1k!

!" #ðeih � e�ihÞ

þX4

n¼2

ð�1Þnna2cn

Aðn2 � 1ÞX1k¼0

A02kþn�10

k!

A020ðkþ nþ 1Þ!þ

1ðkþ n� 1Þ!

!" #( )

� ðeih þ e�ihÞ þ CðhÞ

Substituting Eqs. (A31) and (A34) into Eq. (A3) and reformingthe equation, it is found thatwhere P(h) is the sum of terms without exp(ih) + exp(�ih) andexp(ih) � exp(�ih).

Eq. (A35) can be reformed more concisely.



p00c2 þ pc2 ¼ nr

d� 1A1 þ

2 �x1A0

d� 1

� �ðeih þ e�ihÞ � 2 �x1 �A1ðeih þ e�ihÞ

� � 2 �x2 þ �x2

1

�½�A0ðeih þ e�ihÞ�

þ iaA1

X1k¼0

A02kþ20k!

A020ðkþ3Þ!� 1

ðkþ1Þ!

h i !ðeih � e�ihÞ þ

a2A0c23

X1k¼0

A02kþ20k!

A020ðkþ4Þ!þ 1

ðkþ2Þ!

� � A02k

0k!

A020ðkþ2Þ!þ 1

k!

� " #

þ a2A0c38

X1k¼0

A02kþ10k!

A020ðkþ3Þ!þ 1

ðkþ1Þ!

� � A02kþ3

0k!

A020ðkþ5Þ!þ 1

ðkþ3Þ!

� " #

þ a2A0c415

X1k¼0

A02kþ40k!

A020kþ6ð Þ!þ 1

ðkþ4Þ!

� � A02kþ2

0k!

A020ðkþ4Þ!þ 1

ðkþ2Þ!

� " #

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

�ðeih þ e�ihÞ

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

þð �x1A0 þ A1Þi 1þ a

X1k¼0

A02k0k!

A020ðkþ2Þ!� 1

k!

� " #ðeih � e�ihÞ

þX4

n¼2

�1ð Þnna2cnbðn2�1Þ

X1k¼0

A02kþn�10

k!

A020ðkþnþ1Þ!þ 1

ðkþn�1Þ!

� " #( )ðeih þ e�ihÞ

8>>>><>>>>:

9>>>>=>>>>;þPðhÞ ðA35Þ


p00c2 þ pc2 ¼ cðeih � e�ihÞ þ vðeih þ e�ihÞ þPðhÞ ðA36Þ

with

c ¼ iaA1

X1k¼0

A02kþ20

k!

A020ðkþ 3Þ!�

1ðkþ 1Þ!

" #( )

þ �x1A0 þ A1ð Þi 1þ aX1k¼0

A02k0

k!

A020ðkþ 2Þ!�

1k!

!( )ðA37Þ

v ¼ nr

d� 1A1 þ

2 �x1A0

d� 1

� �þ 2 �x2 þ �x2

1

�A0 þ 2 �x1A1

þ a2

3bb1c2 þ

a2

8bb2c3 þ

a2

15bb3c4 ðA38Þ

In Eq. (A38), ci (i = 2, 3, 4) is defined by Eq. (A20), and bi (i = 1, 2, 3)are defined as

b1¼X1k¼0

A02kþ10

k!

2ðkþ1Þ!�

1k!

� �þ2X1k¼0

A02kþ30

k!ðkþ3Þ!þX1k¼0

A02kþ50

k!ðkþ4Þ!

b2¼X1k¼0

A02kþ20

k!

1ðkþ1Þ!�

3ðkþ2Þ!

� ��3X1k¼0

A02kþ40

k!ðkþ4Þ!�X1k¼0

A02kþ60

k!ðkþ5Þ!

b3¼X1k¼0

A02kþ30

k!

4ðkþ3Þ!�

1ðkþ2Þ!

� �þ4X1k¼0

A02kþ50

k!ðkþ5Þ!þX1k¼0

A02kþ70

k!ðkþ6Þ!ðA39Þ

For Eq. (A36), secular terms in the solution disappear when c = 0and v = 0. The second term of c is zero using Eq. (A16). The expres-sion of two variables can be derived:

A1¼0 ðA40Þ

�x2¼�1

2A0

2 �x1A0rn

d�1ð Þ2þ �x2

1A0þa2

3bb1c2þ

a2

8bb2c3þ

a2

15bb3c4

!ðA41Þ

Substituting Eqs. (A40) and (A41), Eq. (36) can be written in theform

p00c2 þ pc2 ¼ PðhÞ ¼ nB1 eiffiffidp

h þ e�iffiffidp

h�

þXðhÞ ðA42Þ

where X(h) represents terms in P(h) without exp(id1/2h)+exp(-id1/

2h). The general solution of Eq. (42) has the following form:

pc2 ¼ A2ðeih þ e�ihÞ þ 11� d

nB1 eiffiffidp

h þ e�iffiffidp

h�

þHðhÞ ðA43Þ


with H(h) is the particular solution corresponding to X(h).Now the last equation Eq. (A6) can be resolved. Reform the

equation by moving terms:

p00d2 þ dpd2 ¼ rpc2 � 2 �x1p00d1 � 2 �x2 þ �x21

�p00d0 ðA44Þ

For Eq. (A44), only the first and second term include factors thatcan result in secular term. The second term is

2 �x1p00d1 ¼ �2 �x1B1d eiffiffidp

h þ e�iffiffidp

h�

þ 2 �x1 p�d1

�00 ðA45Þ

Substitute Eqs. (A43) and (A45) into Eq. (A44), and find

p00d2 þ dpd2 ¼ B1r

1� dnþ 2 �x1d

h ieiffiffidp

h þ e�iffiffidp

h�

þUðhÞ ðA46Þ

where U(h) is the sum of terms without exp (id1/2h)+exp(-id1/2h).To avoid the secular terms, the coefficient of exp(id1/2h)

+exp(-id1/2h) must be zero, which is only satisfied when

B1 ¼ 0 ðA47Þ

The first two components of Eq. (29) to Eq. (31) have beensolved in the main section, and summarized below:

pc0 ¼ A0ðeih þ e�ihÞ ðA48Þ

pc1 ¼ �iac2



15bðe4ih � e�4ihÞ

ðA49Þ

pd0 ¼r

d� 1A0ðeih þ e�ihÞ ðA50Þ

pd1 ¼2 �x1A0rðd� 1Þ2

ðeih þ e�ihÞ � rd� 4

iac2


d� 9

� iac3


d� 16iac4

15bðe4ih � e�4ihÞ ðA51Þ

�x1 ¼ �rn

2ðd� 1Þ ðA52Þ

�x2 ¼ �1

2A0

2 �x1A0rn

ðd� 1Þ2þ �x2

1A0 þa2

3bb1c2 þ

a2

8bb2c3 þ

a2

15bb3c4

!

ðA53Þ




References

[1] B.T. Zinn, in: F.J. Weinberg (Ed.), Advanced Combustion Methods, AcademicPress, London, 1986.

[2] F.E.C. Culick, Unsteady Motions in Combustion Chambers for PropulsionSystems, RTO AGARDograph AG-AVT-039, 2006.

[3] J.E. Dec, J.O. Keller, Combust. Flame 77 (1989) 359–374.[4] J.E. Dec, J.O. Keller, V.S. Arpaci, Int. J. Heat Mass Transfer 35 (1992) 2311–

2325.[5] V.S. Arpaci, J.E. Dec, J.O. Keller, Combust. Sci. Technol. 94 (1993) 131–146.[6] E.A.M. Elshafei, M.S. Mohamed, H. Mansour, M. Sakr, Int. J. Heat Fluid Flow 29

(2008) 1029–1038.[7] B.T. Zinn, Proc. Combust. Inst. 24 (1992) 1297–1305.[8] F.W. Ahrens, C. Kim, S.W. Tam, An analysis of the pulse combustion burner,

ASHRAE Trans. 84 (1978) 488–507.[9] P.K. Barr, J.O. Keller, T.T. Bramlette, C.K. Westbrook, J.E. Dec, Combust. Flame 82

(1990) 252–269.[10] G.A. Richards, G.J. Morris, D.W. Shaw, S.A. Keeley, M.J. Welter, Combust. Sci.

Technol. 94 (1993) 57–85.[11] J.H. Lee, B. Dhar, W. Soedel, J. Sound Vib. 98 (1985) 379–401.[12] L. Narayanaswamy, G.A. Richards, J. Eng. Gas Turb. Power 118 (1996) 461–468.[13] S.B. Margolis, Combust. Flame 99 (1994) 311–322.[14] T. Lieuwen, J. Propul. Power 19 (2003) 765–781.[15] P. Chatterjee, U. Vandsburger, W.R. Saunders, V.K. Khanna, W.T. Baumann, J.

Sound Vib. 283 (2005) 573–588.


[16] F. Bloom, P. Timothy, Nonlinear Stud. 15 (2008) 243–270.[17] F. Bloom, T. Patterson, Nonlinear Anal.-Real. 10 (2009) 3002–3017.[18] F. Bloom, O. Terlyga, T. Patterson, Nonlinear Anal.-Real. 11 (2010) 2314–2334.[19] H.M. Heravi, J.R. Dawson, P.J. Bowen, N. Syred, J. Propul. Power 21 (2005)

1092–1097.[20] S. Datta, S. Mondal, A. Mukhopadhyay, D. Sanyal, S. Sen, Combust. Sci. Technol.

13 (2009) 17–38.[21] Y. Xu, M. Zhai, P. Dong, F. Wang, Q. Zhu, Combust. Theor. Model. 15 (2011)

623–643.[22] D.L. Brenchley, H.J. Bomelburg, Pulse Combustion – An Assessment of

Opportunities for Increased Efficiency, Report No. DE-AC06-76RLO 1830, U.S.Department of Energy, 1984.

[23] Pulse HW (PHW) Hydronic Boiler, Fulton Heating Solutions, Inc. <http://www.fulton.com/products/productprofile.php?ptc=hw&uid=12>.

[24] ASHRAE Handbook, Fundamentals, Duct Design, American Society of Heating,Refrigerating and Air-conditioning Engineers, Inc., Atlanta, GA, 2005 (Chapter35).

[25] P.S. Vishwanath, Decoupler design guidelines for improving sound levels andstability of pulse burners, in: Proceedings of the 1984 International GasResearch Conference, Washington, DC, 1984, pp. 912–921.

[26] Poincaré–Lindstedt Method. <http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Lindstedt_method> (07.03.13)

[27] P.G. Drazin, Nonlinear Systems, Cambridge University Press, 1992.[28] J.F. Navano, Appl. Math. Comput. 195 (2008) 183–189.[29] M.B. Xu, A. Selamet, H. Kim, Appl. Acoust. 71 (2010) 822–829.


http://refhub.elsevier.com/S0010-2180(13)00261-7/h0005
































http://www.fulton.com/products/productprofile.php?ptc=hw&uid=12

http://www.fulton.com/products/productprofile.php?ptc=hw&uid=12

http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Lindstedt_method

http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Lindstedt_method






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Nonlinear analysis of a pulse combustor model with exhaust decoupler and vent pipe

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