Combustion and Flame 159 (2012) 854–858

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Combustion and Flame

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Characterization of solid propellants by damped combustion oscillations

Eric Baum ⇑, Mathew Richard Denison 1

29528 Oceamport Rd., Rancho Palos Verdes, CA, United States

a r t i c l e i n f o

Article history:Received 27 April 2011Received in revised form 23 July 2011Accepted 27 July 2011Available online 23 August 2011

Keywords:CombustionModelSolidPropellant

0010-2180/$ - see front matter 2011 The Combustdoi:10.1016/j.combustflame.2011.07.020

⇑ Corresponding author.E-mail address: eljbaum@verizon.net (E. Baum).

1 Died 7/27/2010. A student of H. Emmons, his giftproblems to their most basic elements was displayemodeling of compressible flows in boundary layers andto the engineering of hypersonic vehicle heat-shieldsubsequent choice of ablation as a protective mecmodeling contains most of the relevant physics, he mo[1] and the current paper revisits that study.

a b s t r a c t

The linearized Denison–Baum solid propellant combustion model is characterized by two parameters (a,A). Over a large portion of the a, A space for which stable combustion is predicted, the return of a per-turbed burning rate to the steady state takes the path of damped oscillations. This paper provides explicitanalytical expressions for a and A in terms of the frequency and attenuation rate of these oscillations,which can be measured directly. This provides a previously unexploited rout toward the evaluation ofthe acoustic admittance, which the model had previously expressed in terms of a and A.

2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

The combustion rate of solid propellants may experienceinherent instabilities evidenced by an exponential rate of changeor an exponentially-bounded periodicity. Denison and Baum [1]proposed a simple model that exhibits these features, and which,in its linearized form, has been used for many years as a baselinefor examining effects extending beyond the simplifying assump-tions of the model. It was clear in the results presented in [1] thatthe D–B propellant model can also exhibit ‘‘ringing’’ even whenburning stably, i.e. the return of the burning rate to the steady statewhen disturbed can take the path of damped oscillations. The D–Bmodel determines the frequency and attenuation rate of theseoscillations by solving a mathematical problem that is character-ized by two parameters; one that is related to the ablation of thesolid propellant at the surface (A) and another that is related tothe structure and kinetics of the flame (a). These parameters aredifficult to evaluate for any given real propellant because the asso-ciated processes are not well enough understood and because thesimplifications introduced to make the model tractable introducecorresponding uncertainties. It might prove useful if we could

ion Institute. Published by Elsevier

for simplification of complexd in his contributions to thewakes. His early contributionss played a major role in thehanism. While his ablation

deled ‘‘combustion’’ only once

provide a direct relationship between measured properties of theringing phenomenon (its frequency and attenuation rate) and anassociated A, a pair.

2. Model equations

Our starting point is the non-linear D–B model, which we sim-ply restate here without reviewing the associated approximationsand simplifications. The propellant occupies the negative y half-plane. A time-varying mass ablation rate m keeps the propellantinterface at the origin. The propellant temperature T is given by

qscs@T=@t ¼ mcs@T=@yþ Ks@2T=@y2 y 6 0 ð1Þ

where m is assumed to follow an Arrhenius law

m ¼ I expðEw=RTð0ÞÞ ð2Þ

The (steady state) von Karman flame model [2] relates the flametemperature Tf to the ablation rate

m ¼ CPn=2Tn=2þ1f expðEf =2RTf Þ ð3Þ

In a comprehensive1968 review paper [3], Culick examines the useof various proposed improvements to this model, but a 1973 paper[4] by Williams uses a generalization of the theory of Bush andFendell [5] to show that Eq. (3) is indeed the appropriate modelin the limit of large activation energy.

Finally, an energy balance between the propellant interface andthe flame gives a boundary condition on Eq. (1)

Ks@T=@yð0Þ ¼ m½CsTð0Þ CpTf þ Qrer L Q radðtÞ ð4Þ

Inc. All rights reserved.

Nomenclature

cs specific heat of solidC constant in von Karman flame modelI constant in Arrhenius law for propellant mass ablation

rateKs thermal conductivity of the propellantL heat of vaporization and decomposition at the propel-

lant surfacem propellant mass ablation rateP pressure

Qrer specific heat of reactionQrad external incident radiant heat flux at interface (assum-

ing transparent gas phase), negative for incoming radia-tion

t timeT(y) propellant temperatureTf flame temperaturey (negative) distance from the propellant/gas interfaceqs propellant density

E. Baum, M. Richard Denison / Combustion and Flame 159 (2012) 854–858 855

2.1. Steady state

The steady state temperature distribution, from Eq. (1), is

T0 ¼ ½T0ð0Þ Ti expðm0Csy=KsÞ þ Ti ð5Þ

where m0 is the propellant ablation rate associated with the steadystate interface temperature and Ti is the initial (cold) propellanttemperature. From Eqs. (4) and (5), the steady state flame temper-ature is

Tf 0 ¼ ½CsTi þ Q rer L Q radð0Þ=m0=Cp ð6Þ

2.2. Dimensionless form

Define new independent variables

g ¼ ðm0Cs=KsÞy ð7Þs ¼ m2

0Cst=qsKs ð8Þ

along with dependent variables

h ¼ T=T0ð0Þ ð9Þ# ¼ Tf =Tf 0 ð10Þl ¼ m=m0 ð11Þ

and parameters

Ef ¼ Ef =2RTf 0 ð12ÞEw ¼ Ew=RT0ð0Þ ð13Þf ¼ CpTf 0=CsT0ð0Þ ð14ÞH ¼ Ti=T0ð0Þ ð15Þ

The dimensionless form of Eq. (1) is then

@h=@s ¼ @2h=@g2 þ l@h=@g ð16Þ

and the corresponding dimensionless forms of Eqs. (2)–(4) are

l ¼ expðEw=hð0ÞÞ= expðEwÞ ð17Þ

l ¼ ðp=p0Þn=2#n=2þ1 expðEf=#Þ= expðEfÞ ð18Þ

@h=@gð0Þ ¼ l½hð0Þ fð# 1Þ H Q ðsÞ

where

Q ¼ ðQ radðtÞ Q radð0ÞÞ=ðm0CsT0ð0ÞÞ ð19Þ

The non-linear D–B model consists of the differential Eq. (16)describing the solid propellant temperature, with boundary condi-tions (19) at the propellant surface. Eqs. (17), (18) determine land # as functions of h0(0), while p/p0(s) and Q(s) are imposed forc-ing functions. The parameters are Ef, Ew, f, H and the pressureexponent n/2. The initial conditions are given by the dimensionlessform of Eq. (5)

h0 ¼ ½1Heg þH ð20Þ

and we wish to evaluate h(0)(s).

3. Perturbation expansion

We will use the notation

h ¼ h0 þ h01 ¼ h0 þ h1 þ h02 ¼ . . . ð21Þ

When a small perturbation in pressure or radiation is introduced,the corresponding (non-linear) response in h is h01. When thedescribing equations are linearized, the result is a linearizedapproximation (h1) of this response. Because both h0 and h1 are thenknown, the bootstrap process can be repeated to get the non-linearequations describing h02 and the linearized approximation h2 that isthe next level of the perturbation expansion.

Series expansions assuming small h01 (0) yield, from Eq. (17),

l ¼ 1þ Ewh1ð0Þ þ ð22Þ

The 1st order approximation to Eq. (16) is then described by

@h1=@s ¼ @2h1=@g2 þ @h1=@g Ah1ð0Þeg ð23Þ

where

A ¼ ð1HÞEw ð24Þ

Eq. (18) can be approximated by

l ¼ 1þ f½n=2P1=P0 þ ½Ef þ 1þ n=2#1g þ ð25Þ

Comparing equally ordered terms from Eqs. (22) & (25) leads to arelation for #1

#1 ¼ ½ðn=2Þ=ðEf þ 1þ n=2ÞP1=P0 þ ½Ew=ðEf þ 1þ n=2Þh1ð0Þð26Þ

The 1st order linearized approximation to Eq. (19) can now be eval-uated using Eqs. (22), (25), and (26)

@h1=@gð0Þ ¼ ½1þ Að1 aÞh1ð0Þ aBP1=P0 Q 1 ð27Þ

where

a ¼ f=½ð1 hÞðEf þ 1þ n=2Þ ð28ÞB ¼ ð1HÞn=2 ð29Þ

The parameters A, B and a are defined to be consistent withReference [1].

4. Approach to (or departure from) the steady state: 1st order

In Ref. [1], it was demonstrated that externally imposed pressuretransients excite a burning rate response that asymptotically grows(or decays) exponentially with time or as an exponentially boundedperiodic function of time. The initial behavior of the first order

DECAYING OSCILLATORY

GROWING OSCILLATORY

GROWING MONOTONIC

DECAYING MONOTONIC

NO

SOLUTIO

NS

5

A

00 1/α 5

Fig. 1. Stability diagram.

5

856 E. Baum, M. Richard Denison / Combustion and Flame 159 (2012) 854–858

temperature disturbance contains information about the corre-sponding details of the pressure transient, but the asymptotic form

h1 ¼ f1ðgÞ expðiXsÞ where X ¼ xqsKs=m20Cs ð30Þ

loses this detail and retains only a measure of the amplitude andphase of the initiating disturbance. The evaluation of f1 proceedsby inserting this into Eq. (23)

d2f1=dg2 þ df1=dg iX f 1 ¼ Af1ð0Þeg ð31Þ

and into the boundary condition (Eq. (27) without contributionsfrom forcing terms)

df1=dgð0Þ ¼ ½1þ Að1 aÞf1ð0Þ ð32Þ

The characteristic solutions of the homogeneous equation have roots

r1 ¼12½1 ð1þ 4iXÞ1=2 ð33Þ

r2 ¼12½1þ ð1þ 4iXÞ1=2 ð34Þ

Since the possibility of exponential growth has not been explicitlyexcluded, X is complex (and, as yet, undetermined). The form ofthe solution to Eq. (31) becomes

f 1 ¼ C1 expðr1gÞ þ C2 expðr2gÞ þ iðAf1ð0Þ=XÞeg ð35Þ

which, evaluated at the interface, gives

f1ð0Þ ¼ ðC1 þ C2Þ=ð1 iA=XÞ ð36Þ

The interfacial boundary condition becomes

f1ð0Þ ¼ ðC1r1 þ C2r2Þ=ðiA=X 1 Að1 aÞÞ ð37Þ

Eliminating f1(0), this requires that

C1fr1ð1 iA=XÞ iA=Xþ 1þ Að1 aÞg¼ C2fr2ð1 iA=XÞ iA=Xþ 1þ Að1 aÞg ¼ 0 ð38Þ

The left and right sides of the equation must individually be zero inorder that the equations apply for arbitrary values of the complexconstants C1 and C2 that scale the magnitude and phase of the ini-tiating disturbance. Since

2r1 þ 1 ¼ ð1þ i4XÞ1=2 ð39Þ

and

2r2 þ 1 ¼ þð1þ i4XÞ1=2 ð40Þ

the two eigenvalue equations in (38) can be written

ð1þ i4XÞ1=2 ¼ ½iA ð1þ 2Að1 aÞÞX=½X iA ð41Þ

where the plus sign corresponds to Eq. (40) and the minus sign toEq. (39).

When this is squared and solved for X we get a quadratic withsolutions

X ¼ 12

A½Að1 aÞ2 ð1þ aÞi

12

Aj1 aj½4A ðAð1 aÞ þ 1Þ21=2 ¼WiiWr ð42Þ

Ω r =

3

0.50.30

3

0 5

1/αα

0

A

1

Fig. 2. Eigenvalue Xr in the parameter plane (1/a, A).

4.1. Stability diagram

Fig. 1 shows the stability diagram derived from these eigen-values. Since a perturbation from the steady state is proportionalto exp([Wi ± iWr]s), the boundaries between monotonic andoscillatory regions occur when Wr changes from real to imaginary.Thus, the perturbation is monotonic for

4A < ðAð1 aÞ þ 1Þ2 ð43Þ

which can be written in the form

1=a > A=ð1 A1=2Þ2 ð44Þ

The boundary between growing and decaying oscillatory regionsoccurs where Wi changes sign. Thus, the perturbation is stable for

A < ð1þ aÞ=ð1 aÞ2 ð45Þ

The region labled ‘‘no solutions’’ is defined by

Refr2g > 0 ð46Þ

or

1=a < A=ð1þ AÞ ð47Þ

We can show, by substituting the eigenvalues into Eq. (41), that theappropriate eigenfunction is of the second family within this region.Since the spatial dependence is then proportional to exp(Rer2g),the disturbance ‘‘solutions’’ within this region increase without lim-it with depth. This is inconsistent with the implied condition thatthe propellant is initially at a constant temperature.

4.1.1. Decaying oscillatory region: Direct solutions Xr(1/a, A) andXi(1/a, A)

The dimensionless frequency is Xr = Wr and the decay rate isXi = Wi when the eigenfunction is periodic. Substituting thesolution back into Eq. (41) shows that 1/a = 1 divides this stable-periodic region into two sub-domains; the solution is the 1steigenfunction for 1/a > 1 and the 2nd eigenfunction for 1/a < 1.The boundary between the decaying oscillatory region and theregion labled ‘‘no solutions’’ is where Re(r2)) 0, and correspond-ingly, the characteristic thickness of the temperature profile inthe propellant (1/r2) becomes infinite. Beyond this boundary, the‘‘solutions’’ have a temperature disturbance that increases without

5

00 5

Ω

Ω

i

r

A=

5

4

321

0.5

Fig. 4. Parameter A in the eigenvalue plane (Xr, Xi).

5

0

Ωi

0 5Ωr

53

2

1/α=

1.5

1.21.1 .85

.75

.60

.45

.25

Fig. 5. Parameter 1/a in the eigenvalue plane (Xr, Xi).

E. Baum, M. Richard Denison / Combustion and Flame 159 (2012) 854–858 857

limit with distance from the gas–solid interface, and therefore can-not satisfy the initial condition that there is a uniform (cold) tem-perature Ti in the propellant. Figure 2 shows Xr(1/a, A). It isobvious that Xr = 0 on the boundary between the decaying oscilla-tory and the decaying monotonic regions, but the figure also showsthat the boundary between the 1st and 2nd eigensolution domains(1/a = 1) is also an Xr = 0 contour. Locally, there is a symmetryXr(1 + e, A))Xr(1 e, A) as e) 0. The boundary between thedecaying and growing oscillatory regions is not evident in Fig. 2since the frequency is continuous across this boundary. Figure 3shows Xi(1/a, A) and this is, of course, zero on that neutrally-stable boundary, but is continuous across the sub-domainboundary (1/a = 1).

4.1.2. Decaying oscillatory region: Inverse solution 1/a(Xr, Xi) andA(Xr, Xi)

In the decaying oscillatory region, one can measure the fre-quency and attenuation rate of a disturbance as it relaxes back tothe steady state, and thereby construct a good approximation tothe dimensionless frequency Xr and relaxation rate Xi. It wouldthen be convenient if the inverse relationships 1/a(Xr, Xi) andA(Xr, Xi) were available in explicit form. We start by defining anauxiliary function

X ¼ Að1 aÞ ð48Þ

and solving each of the eigenvalue equations in (42) for A:

A ¼ X2r=X2 þ 1

4ðX þ 1Þ2 ð49Þ

A ¼ Xi þ12ðX2 þ XÞ ð50Þ

Eliminating A results in a quadratic equation in X2, with only one ofthe solutions being positive (which is required for X to be real). Theapplicable solution of the quadratic is solved for X:

X ¼ f2Xi þ12þ 2½X2

r þ ðXi 14Þ21=2g1=2 ¼ jXj ð51Þ

Substituting back into the eigenvalue equation gives

A ¼ Xi þ12ðX2 þ jXjÞ 1=a ¼ 1=ð1 jXj=AÞ ð52Þ

A ¼ Xi þ12ðX2 jXjÞ 1=a ¼ 1=ð1þ jXj=AÞ ð53Þ

There are just these two candidate (1/a, A) pairs that could make agiven (Xr, Xi) pair consistent with the model. The candidate corre-sponding to negative X (53) has 1/a ( 1 and therefore lies outside ofthe stable-and-periodic region of the stability diagram whenever1/a < A/(1 + A). In that case the remaining (positive X) candidateprovides a unique (1/a, A) pair. When 1/a > A/(1 + A), there aretwo candidate (1/a, A) pairs that must be compared on physicalgrounds, one having 1/a > 1 and the other having 1/a < 1.

0.30.5

1

3

Ωi=

5

0

A

0 51/α

Fig. 3. Eigenvalue Xi in the parameter plane (1/a, A).

Figures 4 and 5 present A(Xr, Xi) and 1/a(Xr, Xi) with bothEqs. (52) and (53) shown. The condition for which Eq. (53) yieldsa non-physical solution (1/a < A/(1 + A)) transforms into X2

r > Xi

so the corresponding curves stop when that limit is reached. Somealgebraic manipulation shows that 1/a from (52) is equal to a from(53), as can be seen in Fig. 5.

5. Discussion

The acoustic instability that is the most likely source of unstableburning in solid propellant rockets has not been discussed directlyhere. The key to modeling that instability is the evaluation of theacoustic admittance at the propellant surface, and the D–B model[1] evaluates that in terms of the parameters (1/a, A). The majordifficulty in evaluating the parameters for a specific propellant isthat the very complex mechanisms have been simplified to anArrhenius-type vaporization at the surface and a von Karman lawflame. The appropriate activation energies for the two processesare difficult to evaluate from independent measurements andthere will always be large associated uncertainties.

Our examination of the behavior of the D–B model whenslightly perturbed from the steady state has shown that, after theperturbing forcing terms are removed, the return to the steadystate has a very simple asymptotic behavior. Over most of the(1/a, A) domain, the surface temperature disturbance approachesan exponentially damped sinusoid. The frequency and dampingrate of this ‘‘ringing’’ phenomenon are potentially measurable sothat the dimensionless forms Xr and Xi can then be evaluatedwith some confidence. Eqs. (51–53) provide two candidate solutionpairs (1/a, A) for each point in the eigenvalue plane (Xr, Xi), onepair with 1/a < 1 and the other with 1/a > 1. The von Karman flamestructure model [2] is a limiting case that requires Ef 2. This

858 E. Baum, M. Richard Denison / Combustion and Flame 159 (2012) 854–858

cannot be satisfied when 1/a < 1 (i.e. for the solution given by Eq.(53)) for any realistic values of the parameters H and f. Theremaining solution given by Eq. (52) provides an estimate of 1/aand A that becomes more consistent with the flame model withincreasing 1/a.

The measurement of the frequency and damping rate of pertur-bations from the steady state does not require a rocket engine andis, indeed, most conveniently performed on propellant strands in alaboratory setting. This therefore allows a propellant to be charac-terized by Xr and Xi using a novel small scale and hence poten-tially quick and low cost approach. The D–B model relates theseparameters to 1/a and A, and hence to the acoustic admittance,but modeling uncertainties dictate using appropriate caution inextrapolating these relationships to real propellants. Applicationsas a predictive tool are not anticipated, but if the retrieved value

of 1/a is sufficiently large, some qualitative inferences can beconsidered. The connection between ringing and the acousticadmittance in the model is not so surprising, since both are inti-mately connected to the phase difference between temperatureand heat flux at the propellant surface. These mechanisms shouldextrapolate to the more complex structure at the surface of realpropellants, so it is not unreasonable to expect that some qualita-tive trends can inferred.

References

[1] M.R. Denison, E. Baum, ARS J. 31 (1961) 1112–1122.[2] T. von Karman, Proc. Combust. Inst. 6 (1956) 1–11.[3] F. Culick, AIAA J. 6 (1968) 2241–2255.[4] F. Williams, AIAA J. 11 (1973) 1328–1330.[5] W. Bush, F. Fendell, Combust. Sci. Technol. 1 (1970) 421–428.