Combustion Science and Technology

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Nonlinear Intrinsic Instability of Solid Propellant Combustion Including Gas-Phase Thermal Inertia

K.R. Anil Kumar a; K.N. Lakshmisha a a Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India Online Publication Date: 01 September 2000 To cite this Article: Kumar, K.R. Anil and Lakshmisha, K.N. (2000) 'Nonlinear Intrinsic Instability of Solid Propellant Combustion Including Gas-Phase Thermal Inertia', Combustion Science and Technology, 158:1, 135 — 166 To link to this article: DOI: 10.1080/00102200008947331 URL: http://dx.doi.org/10.1080/00102200008947331

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Nonlinear Intrinsic Instability of Solid Propellant Combustion Including Gas-Phase Thermal Inertia K.R. ANIL KUMAR' and K.N. LAKSHMISHAt Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India (ReceivedMay 01,2000) The problem of homogeneous solid propellant combustion instability is studied with a one-dimensional flame model, including the effects of gas-phase thermal inertia and nonlinearity. Computational results presented in this paper show nonlinear instabilities inherent in the equations, due to which periodic burning is found even under steady ambient conditions such as pressure. The stability boundary is obtained in terms of Denison-Baum parameters. It is found that inclusion of gas-phase thermal inertia stabilizes the combustion. Also, the effect of a distributed heat release in the gas phase. compared to the flame sheet model. is to destabilize the burning. Direct calculations for finite amplitude pressure disturbances show that two distinct resonant modes exist. the first one near the natural frequency as obtained from intrinsic instability analysis and a second mode occurring at a much higher driving frequency. It is found that even in the low frequency region, the response of the propellant is significantly affected by the specific type of gas-phase chemical heat-release model employed. Examination of frequency response function reveals that the role of gas-phase thermal inertia is to stabilize the burning near the first resonant mode. Calculations made for different amplitudes of driving pressure show that the mean burning rate decreases with increasing amplitude. Also, with an increase in the driving amplitude, higher harmonics are generated in the burning rate. Keywords: Combustion instability; Solid propellants; Energetic materials

1 INTRODUCTION Intrinsic, nonacoustic instability of solid propellant combustion is important from the viewpoints of both fundamental combustion theory and practical applications. When subjected to external pressure/radiation flux perturbations of cer-

* Research Student,

E-mail: [email protected]. in

t Assistant Professor, Corresponding Author: E-mail: kn)@aero.iisc.emet.in - Fax: +91-803345 t 34. t35


136

K.R. ANIL KUMAR and K.N. LAKSHMISHA

tain frequencies, a resonant mode of highly amplified oscillatory burning could occur due to intrinsic instabilities (Denison and Baum, 1961; Culick, 1968; T'ien, 1972; Price, 1984). A comprehensive theory explaining its mechanism is not available yet. Its basic complexity is due to an interplay of the thermo-diffusive transport processes in two distinct, separated physical phases, coupled with chemical rate processes occurring in the gas phase, in the gas-solid interface and within the solid phase. Despite early work (Zel'dovich, 1942; Hart and McClure, 1959; Denison and Baum, 1961), the current knowledge on such instability is yet largely based on linear stability theories employing quasi-steady gas phase and activation energy asymptotics (AEA). A summary of previous theoretical research on this problem is presented in Table I. From this, it is evident that a modeling restriction that is common to most works is the Quasi Steady gas-phase and surface, Homogeneous solid One Dimensional (QSHOD) flame. This approximates that the gas-phase and the surface respond instantaneously and adjust to a new steady-state for any perturbations in the external conditions, such as pressure, ambient temperature, or externally incident radiation. Such an instantaneous adjustment is reflected in the form of new values of the thermal gradient on the gas-side and surface temperature (which equal the steady-state value that would exist corresponding to the new value of pressure and/or other conditions). However, the condensed phase response is treated as a truly time-dependent process. Thus, a single time scale tc' of condensed phase thermal relaxation is assumed to govern the regression dynamics and the surface response time t s as well as the gas-phase response time ,s: are taken as zero. Hence the QSHOD model is alternatively known as the "!« approximation" (Novozhilov, 1992). TABLE I A summary of previous theoretical research on intrinsic and pressure-driven combustion

instability Reference

Stability Gas-phase Resonant Modes

Remarks

Zel'dovich (1942)

Linear

QSHOD

One

Denison & Baum (1961)

Linear

QSHOD

One

Variable T;

Novozhilov (1967)

Linear

QSHOD

One

ZN Method

Krier, et al.. (1968)

Nonlinear

QSHOD

One

Distributed flame

Brown and Muzzy (1972)

Nonlinear QSHOD

One

Similar to Z N approach

Two

T'ien (1972)

Linear

Transient

Margolis & Williams (1988)

Linear

Transient

Constant T,

Numerical Asymptotics

Huang & Micci (1991)

Linear

Transient

Two

Numerical; 2-step chemistry

Clavin & Lazimi (1992)

Linear

Transient

Two

Asymptotics

Novozhilov (1991)

Linear

Transient

Two

ZNMethod

Nonlinear QSHOD

One

Theoretical

De Luca et al., (1995)


INSTABILITY OF PROPELLANT COMBUSTION

137

Two different approaches were developed to calculate the transient energy feedback to the solid from the gas phase: the flame modeling (FM) method by solving gas-phase conservation equations and the phenomenological, Zel'dovich-Novozhilov (ZN) method using the steady-state experimental results. The FM approach was evolved and developed in the United States while the ZN method was pioneered mainly within the former Soviet Union. The equivalence of different QSHOD models, both FM and ZN approaches, has been demonstrated by Culick (1968), Son (1993) and De Luca, et al., (1995). Typically, the surface responds 10 times faster, and the gas-phase 100 times faster, than the solid phase to any pressure transient, that is, 102t g = 10\ = t e (Kuo, et al., 1984). Therefore, a fully transient burning model should consider the unsteadiness of the condensed phase, burning surface and the gas phase all together. As discussed before, the QSHOD model considers unsteadiness of the condensed phase only, and hence, its validity is restricted to pressure perturbations in a low-frequency range of below some I kHz (Denison and Baum, 1961; T'ien, 1972; De Luca et al., 1995) and to cases where the indirect effect of adiabatic compression on the gas-phase energy release can be neglected (Clavin and Lazimi, 1992). Further, it is impossible to consider the effect of intrinsic gas-flaine instability on the stability of the deflagrating solid with QSHOD models (Margolis and Williams, 1988). The quasi-steady gas-phase assumption was relaxed by T'ien (1972), who numerically solved the linearized perturbation equations to obtain the pressure frequency response function and observed a second resonant mode. Later, this analysis has been extended to include a two-step reaction in the gas-phase (Huang and Micci, 1991). Margolis and Williams (1988) also relaxed the quasi-steady gas-phase assumption and obtained the linear stability limits for one- and quasi-one-dimensional burning, by employing high activation energy asymptotics (AEA). Their work suggested that when the flame stand-off distance is large, combustion dynamics approaches the limit of a strictly gaseous, one-dimensional premixed flame. The instability observed was strongly thermo-diffusive in nature, and the critical parameters were found to be the Lewis number and the Zel'dovich number of the gas-phase. However, when the flame is very close to the burning surface (for instance, at high pressures), the behavior approaches that of strictly condensed-phase combustion, and the instability becomes independent of the Lewis number. This analysis has been subsequently extended to obtain the linear frequency response under pressure-driven burning (Clavin and Lazimi, 1992). The AEA results of Clavin and Lazimi (1992) confirmed that (I) the QSHOD model fails to properly describe the combustion dynamics under high frequency regimes, and (2) the assumptions in the gas-phase flame affect the response function even at low frequencies. Using the ZN approach, Novozhilov (1990) obtained the linear, intrinsic stability limits and


13M

K.R. ANIL KUMARand K.N. LAKSHMISHA

found that inclusion of gas-phase thermal inertia stabilizes the burning when the derivative of the surface temperature with respect to the initial temperature (dT/dTII ) is low. In a subsequent work, Novozhilov (1991) studied pressure-driven burning, and concluded that the gas-phase unsteadiness is important for burning under pressure frequencies higher than the resonant frequency predicted by the QSHOD model. His analysis showed that inclusion of gas-phase thermal inertia (I) results in a second resonant peak in the real part of the response function, and (2) stabilizes the burning around the resonant frequency in comparison to the QSHOD model. Experimentally, multiple peaks in the response function of dynamic burning of solid energetic materials have been observed for oscillatory pressure (Strand et al., 1980) and for oscillatory external radiation heat flux (Son, 1993). The observed second peak appears to occur at frequencies much below those expected to be caused by the gas-phase unsteadiness. Son (1993) attributed this to the possibility of a finite relaxation time of the condensed-phase reaction zone. Zebrowski and Brewster (1996) examined this by including a zeroth-order, distributed, reaction in the solid phase, but with a quasi-steady gas-phase. However, their results did not show a second peak in the response function for radiation-driven burning. Therefore, it is interesting to examine whether the 'second resonant frequency is significantly altered by the interaction of finite relaxation times of the gas phase and the solid-phase reaction zone. Secondly, the assumption of small perturbations made by many models may give rise to errors when applied to problems where the change in conditions results in considerable nonlinearities. Apart from Krier et al., (1968), Brown and Muzzy (1970) and De Luca et al., (1995), all studies till now, both quasi-steady and transient gas-phase, have used the assumption of small perturbations. It has been experimentally observed (Eisel, 1964) that the finite-amplitude pressure fluctuations reduce the mean burning rate in a wide range of frequencies (up to 10kHz), where the quasi-steady gas-phase assumption is not valid. This emphasizes the need for a model which can take into account both the nonlinearities and transient gas-phase effects, such as the one used in the present study. In their pioneering work on this problem, Denison and Baum (1961) stressed the need for such a model, but interestingly, no workable solution has been found despite several efforts (see De Luca, 1992). In the following section, the basic objectives are set out and the methodology is outlined. Section 4 provides a detailed mathematical formulation, followed by the method of solution in Section 5. The results and discussion are covered in Section 6. The principal conclusions are summarized in Section 7.



139

2 OBJECTIVES AND METHODOLOGY Therefore, the central objective of this paper is to open an avenue for numerical investigation of nonlinear intrinsic instability of combustion including the gas-phase thermal inertia. This also helps to assess the performance of the linear, QSHOD models over a range of parameters by making a comparison with the current results. This is attempted through the formulation and direct solution of the gas-phase and solid-phase equations for the transient combustion of a one-dimensional, adiabatic, homogeneous solid propellant. In the first part of the study, the external conditions of pressure and ambient temperature are kept constant, and a series of calculations are made for different values of chosen parameters such as the activation energy, Lewis number etc, to find out whether a steadily regressing solution changes to nonsteady, oscillatory behavior. The neutral stability boundary is determined by a go/no-go method, where the steady solution changes to an oscillatory one for a small change in the chosen parameter. In the second part, the gas-phase pressure is externally forced to vary sinusoidally with a specified, finite amplitude and frequency, and the resulting variations in the mass burning rate are studied. This exercise is carried out with transient and quasi-steady gas-phase models, and the influence of gas-phase thermal inertia is studied. Finally, the effect of the amplitude of pressure oscillations on the pressure-driven frequency response function is examined. Results presented in this paper refer to a single-step, irreversible, gas-phase chemical reaction because complex chemical modeling (Bizot and Beckstead, 1988; Huang and Micci, 199I; Prasad et al., 1998) is still in a developing stage. However, the formulation has a potential capability for straightforward inclusion of complex chemical models. This paper considers the finite relaxation time of the gas phase and assumes the solid phase chemically inert with the surface regression being modeled according to the classical Arrhenius form. However, it is emphasized that the finite relaxation rate of the condensed-phase reaction layer and the finite rate of transformation of condensed phase to gaseous phase are expected to affect the response function at the same order. Such effects could be easily incorporated into the present model and will form important issues of future research. Currently, the topic of radiation-driven combustion instability is finding increasing interest among various researchers, both experimental and theoretical (Son and Brewster, 1992; Son, 1993; De Luca et al., 1996; Zebrowski and Brewster, 1996; Novozhilov et al., 1998). The present methodology could be straightforwardly extended to such a problem as well.


140


3 DESCRIPTION OF THE PHYSICAL PROBLEM We consider the laminar combustion of a I-dimensional solid propellant (see Figure I) with the following modeling assumptions: (I) the solid propellant is a nonporous, chemically inert mass of homogeneous and isotropic composition, (2) the burning surface is infinitesimally thin, planar, separating the gas phase on the right (extending to +00) from the solid phase on the left (extending to -00), (3) the propellant pyrolysis occurs at the interface via an exothermic sublimation process Solid ~ Gases, with an Arrhenius form of rate, (4) the gas velocities are normal to the burning surface and are very small compared to the local acoustic velocities, (5) the gas-phase combustion takes place via a single-step, overall, irreversible chemical reaction Reactants ~ Products, governed by an Arrhenius rate, and (6) radiative heat transfer is absent.

T

Solid-phase

Gas-phase

x=-oo

x=+oo FIGURE I Schematic of the Problem Domain

Zebrowski and Brewster (1996) showed that the assumption of an infinitesimally thin reaction zone in the condensed phase can lead to errors in linear response magnitude for realistic steady-state properties. So in any further improvement of the present model assumptions (I) and (2) must be relaxed and an appropriate distributed reaction model used so as to include the effect of unsteadiness in the solid-phase reaction zone. A remark about the assumption (3)



141

is in order. Although the Arrhenius law may be satisfactorily applicable for steady burning, it may not be so for unsteady burning. A more rigorous expression derived from AEA (Lengelle, 1970) appears to yield predictions for homogeneous propellants that compare more satisfactorily with experimental data (Brewster and Son, 1995). In view of the assumption (4), the pressure may be considered uniform in space (see p. 609 of Kuo, et ai., 1984). Two kinds of situations are studied in the present work. In the first, the pressure is taken as constant in time, while in the second, pressure is externally forced to vary (e.g, sinusoidally) in time, with a specified amplitude and frequency. The objective of the first study is to ascertain if there exists a parametric range beyond which unstable burning is observed even when the pressure is constant. The second study seeks to investigate the behavior of the nonlinear frequency response function.

4 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS 4.1 Equations in Dimensional Form

The original partial differential equations in a coordinate system fixed to the burning surface governing the transient burning analysis of a solid propellant are given in Kuo, et ai., (1984). In the laboratory-fixedcoordinate system, these take the form. Overall Continuity: (1)

Gas-phase Species:

ay;

ay;

pg[jt +PgUga;; =

a ( aY;) ax t/,1Pg ax

.

+Wi

(i

= 1, ... , N s )

(2)

Gas-phase Energy:

aT

PgCg7it

aT

+ PgUgCg ax

=

a (aT) . dp ax kg ax + qfwf + dt

(3)

Condensed Phase Energy:

(4) Interface Flux Balance: Overall Mass: (5)


142


Species:

Energy:

(7) Initial and Boundary Conditions:

T(x,O)

(aax'1' )

= To(x);

1~(x,0)

= Yi,o(x)

(8)

. = 0; 1~(xs_, t) = 1~,u; T( -00, t) = Tu

(9)

00,
50 atm). In this section the present numerical results are compared with the closed-form expressions of De Luca, et al., (1995) for properties given in Table II, which are typical of high-pressure burning, 03 = 0.044. The lalter consider linear, QSHOD theory for pressure-driven burning, and obtain closed-form expressions for &i?p(llc) and 0(Qc)' De Luca et al., consider three simplistic but different models for the gas-phase heat-release profile. The present work considers the more realistic Arrhenius form. In the present calculations, first a steady-state solution is obtained. Then a sinusoidal pressure variation with an amplitude of I % of the mean pressure (P(t) = I + M sin(Qt), where M= 0.0 I, 0.0004 ::; Q ::; 4) is imposed. The real and imaginary parts of the response function are calculated by taking the Fourier transform of the pressure and burning-rate disturbances. It is found that (Figure 4) a first "resonant mode" burning occurs when the driving frequency is near the natural frequency of the propellant obtained through intrinsic instability analysis, and a second resonant mode (as indicated by a second peak at Q c = 100) occurs at a much higher frequency. The QSHOD theory, as has been argued (Denison and Baum, 1961; T'ien, 1972 and De Luca et al., 1995), performs well at lower driving frequency which is evidenced by the good matching of the first resonant frequency predicted by QSHOD theory with the present results. However, the secondary resonant mode is predicted only by the present non-QSHOD model, and the QSHOD model does not capture this phenomenon. However, this comparison does not provide an exact idea about the effect of introducing gas-phase thermal inertia, because the gas-phase chemistry models used are different. So, numerical QSHOD calculations with an Arrhenius gas-phase chemistry model are performed in order to compare with present results. Such an exercise is carried out as follows. First, steady-state results are obtained at several values of pressure and surface temperature over the range P = 1 ± M. For each steady-state result the gas-side temperature gradient is extracted. For the variable-pressure calculations, these are used.


152


6 5 4

3 2

o

0.1

1

10

100

1000

100

1000

Frequency, .Q. c 40 20

o -20 -40 -60

0.1

1

10

Frequency, .Q. c FIGURE 4 Pressure-driven frequency response. Comparison with QSHOD results. -0- Numerical (Present. Arrhenius); - Numerical (QSHOD, Arrhenius); -- -- Sharp flame (De Luca et al., 1995); --Distributed flame, y 0 (De Luca et al., 1995); ... Distributed flame, y I (De Luca et al., 1995)

=

=



153

The comparison between quasi-steady and transient gas-phase results shows that near the first resonant mode, at moderately low frequencies, introduction of the gas-phase thermal inertia stabilizes the burning, whereas at higher frequencies it is destabilizing. But at low frequencies, the gas-phase chemistry model appears to be more important than unsteadiness of gas phase. Consideration of distributed Arrhenius heat release in the gas phase increases the response function, which is also evident from the comparison of the stability boundaries (Figure 3). Further, all the above results satisfy the theoretical limit of !!lp --t nl for Q c --t 0, where nl is the pressure index (Denison and Baum, 1961). 6.2.2 Comparison with transient gas-phase results

Multiple resonant modes have been also predicted by previous non-QSHOD models (Tien, 1972; Huang and Micci, 1991; Clavin and Lazimi, 1992). All these previous studies assumed infinitesimally small pressure perturbations (linear stability). In Figure 5, the results obtained using the present model are compared with those of T'ien (1972) and Denison and Baum (1961). For this purpose, the present model was modified for a second-order reaction in the gas phase and incorporated the other modeling assumptions used by T'ien (1972). The amplitude of sinusoidal pressure variation used in the present study is I% of the mean value. It is seen from Figure 5 that the resonant frequencies predicted by all models are same. It is also seen that the present results for small values of !:.pIp ref approach the results from linearized equations. It is important to examine the origin and nature of the high-frequency reso-

nance. Calculations with a transient gas phase, but quasi-steady solid phase confinn that (see Figure 5) the second resonant frequency is due to the gas-phase thermal inertia only. The peak occurs at a value of Q~ slightly less than one. This is because the gas-phase thermal relaxation time used to normalize the frequency is calculated using hot boundary conditions. Also, inclusion of the solid-phase thermal inertia is found to stabilize the burning against high-frequency perturbations. In the present case the second resonant frequency is significantly increased with inclusion of the solid thermal inertia. Intrinsic instability at the second resonant frequency has not been studied yet. 6.2.3 Effect of mean pressure

T'ien (1972) showed that the real part of the acoustic admittance at the first resonant frequency decreases with increase in density ratio 03' This is true also for the burning-rate response function at the first resonant frequency if the mean surface temperature does not vary with the density ratio. But Figure 6(a) shows that an increase in mean pressure (which is equivalent to an increase in density ratio)


K.R. ANIL KUMARand K.N. LAKSHMISHA

154

4 3 2

1

o

10

0.001 Frequency, Q ..

40

.. ,

o

.. '

~

.

-40

-80 0.001

0.01

0.1

1

10

Frequency, Q .. FIGURE 5 Pressure-driven frequency response. Comparison between linear stability theory and the Present, present results. --- T'ien (1972); - 0- - Denison and Baum (1961); - Present, nonlinear;

QS-Solid phase

increases the response function at the first resonant frequency if the mean surface temperature is allowed to vary with pressure. The first resonant frequency, 0lJ varies with mean pressure, Pre/" The variation is found to obey the relation



ISS

p;;;

WI ex for Pref< 10 atm, while it is WI ex p~~~" for 50 < Pref< 100 atm. A decrease in the index with mean pressure was observed by T'ien (1972) also. In experiments on Double Base propellant burning in a constant-pressure bomb (Svetlichnyi and Margolin, 1971), the relationship W n ex has been observed over the range of pressures where both the primary and secondary flames merge. A small decrease in the index can be attributed to relaxation of the quasi-steady gas-phase assumption, because for classical QSHOD models, W n ex at all mean pressures. However, in spite of this small decrease in the index, this suggests that the low-frequency resonance is basically due to thermal inertia of the solid phase, although the numerical magnitude of the response function depends on the assumptions made in the gas phase.

p;;;

p;;;

The effect of the mean pressure on the real part of the response function at the second resonant frequency is shown in Figure 6(b). The response function is found to increase with mean pressure, but the rate of increase is small compared to that at the first resonant frequency. The resonant frequency nondimensionalized by the gas thermal relaxation time increases as the mean pressure increases. It is found that at low pressures the effect of the solid-phase thermal inertia on the dynamics of the gas-phase flame is not significant, and the peak frequency approaches that obtained with the quasi-steady solid-phase model. If the instability at high frequency is not affected by the solid phase, the second resonant frequency so nondimensionalized must remain same. 6.2.4 Effect of amplitude of pressure oscillations A major advantage of the present model is its capability to consider finite pressure fluctuations. In order to study the effect of the imposed pressure amplitude on the burning rate, calculations were made at different values of liP/Pref and a frequency equal to the first resonant frequency of the previous case. The results are shown in Figures 7 and 8 for Iip/Pref= 0.05, 0.1 and 0.5 respectively. It is seen that at low amplitudes of pressure fluctuations, Iip/Pref= 0.05, the burning rate variations essentially follow that of the pressure. However, when the pressure amplitude is large, second (and even third) harmonics appear in the burning-rate variations (which was found by the presence of significant values in the higher-order Fourier coefficients of r( T)). Further, a decrease in the mean burning rate is observed at large amplitudes of pressure. In the present case, the mean burning rate is decreased by about 20% when the amplitude is 50% of the mean pressure. This result is in qualitative agreement with experimental observations made by Eisel (1964). These results confirm the importance of nonlinearity in combustion instability. Such nonlinearity could be significant in the behavior of propellant combustion when subjected to large changes in pressures as in L'


156


16

=5 arm Pref =Warm Pref

12

Pref

=50 arm

Pref

= 100 arm

4

a

0.001

0.01 Frequency,

0.1

nco

(a)

2.0 -0- Pref

1.5 ~... 1.0

=5 arm

-a- Pref =10 atm ~ Pref =50 atm -- Pref = 100 atm

0.5

o.a0'-----'-----1..---'-.........-'--'--'----------'-----' .3

1 Frequency,

nco

3

(b) FIGURE 6 Effect of the mean pressure on the real part of the response function. (a) Low-frequency region. (b) High-frequency region. Pg at 10 aIm and 300 K is taken as I g/cm 3

instability or rapid pressurization/depressurization. These effects will be examined in future work.



157

1.6 r

-- p

1.2

..... ~

0.8 0.4

0.0

0

5000

10000

15000

20000

15000

20000

15000

20000

(a)

1.6

..... 1.2 ~

0.8 0.4

0.00

5000

10000 (b)

1.6 1.2

..... ~

0.8 0.4

0.0

0

5000

10000 Time,

't

(c) FIGURE 7 Effect of imposed pressure amplitude on burning rate. (a) M' ~ 0.05. (b) 6P ~ 0.1 and (c) M' ~ 0.5. Note that i becomes non-sinusoidal at large amplitude although the driving pressure remains sinusoidal


158


4 ~

--cr

....

~r::..

--/::-

3

-¢--

C'i

~r::..

r;

2

~~ '"

1

"lo..'"

0.2 0.3 IlP/P . FIGURE 8 Real part of pressure frequency response. Effect of amplitude of pressure oscillations (Q_ = 0.005) on different harmonics

7 CONCLUSIONS

The intrinsic, nonlinear stability limit of solid propellant deflagration has been computed considering finite thermal inertia of the gas phase. This instability limit has been compared with the theoretical, QSHOD and linear non-QSHOD model results. Including the gas-phase thermal inertia stabilizes the deflagration. Distributed gas-phase energy release destabilizes the burning for the same gas-phase and interface activation energies. At large gas-phase activation energies the stability boundary predicted by all models, both QSHOD and transient gas-phase, are the same. For very small gas-phase activation energy, there is a considerable difference in the resonant frequencies predicted by QSHOD models and transient gas-phase models. Results for pressure-driven combustion lead to the following conclusions: I. The response for pressure burning exhibits resonances at two values of the driving frequency. The first resonant mode occurs when the driving frequency is near the natural frequency at the intrinsic stability limit. The second


INSTABILITY OF PROPELLANTCOMBUSTION

159

resonant mode occurs at a higher frequency and can be described only by inclusion of finite gas-phase thermal inertia. 2. Inclusion of the gas-phase thermal inertia decreases the response function near the first resonant mode and increases it at higher frequencies. 3. An increase in the amplitude of the driving pressure results in the appearance of higher harmonics in the burning rate as well as a fall in the mean value of the burning rate. 4. The response function increases with an increase of the mean pressure at the resonant frequencies. The resonant frequency is proportional to p;~j but the exponent decreases as the mean pressure increases.

NOMENCLATURE A

Pre-exponential factor of reaction rate

A DB

Es(Ts.ss - Tu)/'Yi.T;.ss

A

Nondimensional pre-exponential factor, Eq. (22)

c

Specific heat capacity

§1

Mass diffusivity

E

Activation energy

k

Thermal conductivity

Le

Lewis number, kg/Vpgc g Mass burning rate

n

Overall reaction order in the gas-phase Index in the steady-state burning rate law, uc,ss ex: p;: Number of species Pressure Nondimensional pressure, Eq. (22) Nondimensional amplitude of pressure oscillations Heat of combustion of reactant gases per unit mass of fuel (negative) Surface heat release rate (positive for exothermic) Nondimensional surface heat release, Eq. (22) Nondimensional burning rate, pcu/Pg.uuref


160


Rg

Nondimensional gas-phase chemical source term, Eq. (27)

'J\

Universal gas constant

f%p

Real part of pressure coupled frequency response function Dimensional time

Ie

Condensed phase thermal relaxation time, kcPC/CCm;8

18

Gas phase thermal relaxation time, k gpg/C9m ;8

T

Temperature

Tb

t; + qlcg

U

Velocity of the gas normal to the solid-gas interface

Us

Bum rate

ib

Reaction rate

W

Average molecular weight

x

Distance coordinate

Y

Mass fraction

y

Normalized fuel mass fraction

Greek:

a

Nondimensional heat release factor, Eq. (22)

lXDB

Cg T oo,88/((n + 2)/2 + (E g/2'J\L oo ,88))

~

Zel'dovich number

y

Ratio of specific heats of gas-phase, parameter used by De Luca et al., (1995)

e

Nondimensionaltemperature

p

Mass density

't

Nondimensional transformed time coordinate, Eq. (21)

¢

Imaginary part of pressure frequency response function

1;

Transformed distance coordinate, Eq. (14)

'II

Nondirnensional transformed distance coordinate, Eq. (21)

to

Dimensional circular frequency

Qu

Nondimensional circular frequency, wkg,u/ pg,UCgU;,U



n~

161

Nondimensional circular frequency used by T'ien (1972), wkg,oo/ pg,OOCgU~,=

nc

Nondimensional circular frequency, Wl c

nc,DB

Resonant frequency obtained by maximizing Eq. (71) of Denison and Baum (1961).

nc,n

Natural frequency of intrinsic instability

Subscripts:

o

Initial value

C

Condensed phase

f

Fuel

g

Gas-phase

ref

Reference or mean value

s

Burning surface

s+

Gas-side of interface

s-

Solid-side of interface

ss

Steady state

u

Ambient or unburned

00

Hot boundary

_00

Cold end of the propellant

Acknowledgements This research was funded by the ISRO-IISc Space Technology Cell. The authors would like to thank Professor P.1. Paul for technical discussions. References Anil Kumar, K.R., Lakshmisha, K.N. and Paul, PJ. (1997). Computation of Homogeneous Solid Propellant Combustion, Proc. 15th Nat. Con! IC Engines & Comb., Madras, pp. 759-764. Bizet, A. and Beckstead, M.W. (1988). A Model for Double Base Propellant Combustion, Twenty-second Symposium (International) on Combustion, pp. 1827-1834. Brewster, M.Q. and Son, S.E (1995). Quasi-Steady Combustion Modeling of Homogeneous Solid Propellants, Combustion and Flame, 103, pp. 11-26. Brown, R.S. and Muzzy, RJ. (1970). Linear and Nonlinear Pressure Coupled Combustion Instability of Solid Propellants, AIM J., 8(8), pp. 1492-1500. Clavin, P. and Lazimi, D. (1992). Theoretical Analysis of Oscillatory Burning of Homogeneous Solid Propellant including Non-Steady Gas Phase Effects, Combust. Sci. Tech., 83, pp. 1-32. Culick, EE.C. (1968). A Review of Calculations for Unsteady Burning of a Solid Propellant, AIM J., 6( 12), pp. 2241-2254.


162


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164

APPENDIX: NUMERICAL SCHEME TO TRACK THE BURNING INTERFACE In the Figure 9(a), ABCD is the interface cell and XY the interface plane at a time level n. The gaseous part of the interface cell, XYCD and the solid part ABXY have widths Ll1jJ;,. and Ll1jJ~. respectively. At time level (n + I), the interface plane XY has moved to a new location, as shown in Fig. 9(b). The problem of tracking the location of the moving plane XY amounts to calculation I,n+l at the next time level (n + I). This is subject of the widths .6. ol,n+l and .6..'PC,S "Pg,8 to the mass conservation,

(31) From the definition of mass burning rate, the new location of the interface is obtained from

+ (r)n+l.6.r

(32)

.6.1jJn+l = .6.1jJn _ (r)n+l.6.r C,s C,S

(33)

.6.1jJn+l = .6..I,n 9,8 0/9,8

Here the burning rate as given by Eq. (30) requires a knowledge of interface temperature 0~+1, the computation of which is described in the following.

Enthalpy Balance for the Gaseous-part XYCD The enthalpy balance written for the finite-volume XYCD centered at mis,

where prime denotes spatial derivative. Using a fully implicit time difference and approximating 0 m -- 0 •

+ 2'-'-'1-'9,' 1 01, (0') .+, A

this can be written in the discre-

tized form as,

n+1 _ .6..I,n 0 n) ( .6.0I,n+10 '1/9,8 711 0/9,8 m

which is a nonlinear algebraic equation of the form 1 :F1 {0 n+ (0,)n+l) 8 l s+

=0

(36)



~"'~.s

~"'~.s X

.'

165

C 0

0

0

s m

j

y

m+I

D

(a) ~",n+l

~",n+l g.s

c.s

B

X

C

I I

I

0

0

j-I

s

J

y

A

I I I I I I

0

0

m

m+I

I

D

(b) FIGURE 9 Structure of Numerical Grids at the Interface (a) at time level n and (b) at time level n + I

Enthalpy Balance on the Interface plane XV

From Eq. (28), this is written as, (0')~t1

+ (rr+ 1Qs =

"2(0/)~~1

(37)

This can be written in the discretized form as, n+1 F 2 {(0,)n+1 =0 8+ 1 0 s ' (0,)n+1} s-

(38)

Enthalpy Balance for the solid-part ABXV

The enthalpy balance written for the finite-volume ABXY centered at j is,

:T (t::.VJc,s0

j )

= "1(0')8- - "1(0')j-t - r0 8

(39)


166


Again, 0

j

using

= 0s-

a

fully

~.6.¢c,s(0')s-,

(.6.¢,,+10,,+I_.6.¢" 0")

C,s

j

~r

C,s

j

implicit

time

difference

and

approximating

this can be written in the discretized form as,

= Ii (0')"+1_ Ii (0'),:+1 _ 1',,+10,,+1 + 0(.6.T 2 ) 1

8-

J-!

1

S

(40) which is a nonlinear algebraic equation of the form

:F3 {0,,+1 (0,)n+1} 8' s-

=0

(41)

At each time interval, the set of equations (36, 38, 41) are solved by the NewThe gradients ton's method for the three unknowns {0 s' (0')s+> (0')sJ n (0')::;~;\2 and are provided by the solution of the gas-phase and solid 2 phase equations respectively. Typically, at each time level, the equations converge within 4-5 iterations. Then the mass burning rate at the new time level (11 + I) is computed from the pyrolysis law, Eq. (30), corresponding to the new interface temperature 0:,+1. In the actual computations, the interface species condition, Eq. (28) is also solved coupled with the above equations in a similar manner. This is not presented here for the sake of brevity. As a result of surface regression, the gaseous-width ~Ijfg.s continuously increases while the solid-width ~ljfc,scontinuously decreases. When ~ljfc,s = 0, the interface cell ABeD is completely converted to gaseous part, and the next adjacent solid cell is considered as the new interface cell. This process continues.

(0')J':'::i

+'.