unstable combustion in solid-propellant rocket motors

UNSTABLE COMBUSTION IN SOLID-PROPELLENT ROCKET MOTORS 81

where df/dt = 0). Because the reaction rate Q is, at low temperatures, the term most sensitive to temperature we will assume that the ignition and blow-off limit for the case of an optically thin gas is given by the condition that, at the point where ~ is a maximum, ~ = R. In the case in which the enthalpy is maximum on the exhaust boundary the mixing ratio at the point of maximum ~ is given by

d f)e.-An~} ~f {/(1 -- -- 0 (68)

or, approximately, for small f

h2 kT2 (69) f

The maximum value of ~ is approximately given by

nQ

1 - - ~ ~ + 1 (70)

e x p [ - - ( l + ~ ) 1"

In the case in which the maximum enthalpy oc- curs within the mixing region the maximum value of ~ will obtain roughly at the point of maximum enthalpy. The results of an exempletive calcula- tion are shown in Figure 4. The vibration-rota- tion bands of CO, H20 and CO2 were assumed to be the emitters. In Figure 4 is shown the reaction source and radiation loss terms as a function of temperature and density.

REFERENCES

1. KIVEL, B., MAYER, H., AND BETHE, H.: Ann. Phys., 2, 57 (1957).

2. KECK, J. C., CAMM, J. C., KIVEL, B., AND WENTINK, T., JR.: Ann. Phys., 7, 1 (1959).

3. GOODY, R. M.: The Physics of the Stratosphere, Cambridge University Press, New York, 1954.

4. PLASS, G. N.: J. Opt. Soc. Am., 50, 868 (1960) 5. PENNER, S. S.: "Quantitative Molecular Spec-

troscopy and Gas Emissivities," Addison Wesley Publishing Company, Inc., Reading, Mass., 1959.

6. MONTROLL, E. W., AND SHULER, K. E.: J. Chem. Phys., 26, 454 (1957).

7. PENNER, S. S.: J. Chem. Phys., 32, 617 (1960).

VIII

UNSTABLE COMBUSTION IN SOLID-PROPELLANT ROCKET MOTORS

By SIN-I CHENG

Introduction

The combustion of a solid propellant in a rocket motor is often irregular and unpredictable. The problem becomes serious as the scaling of a successful motor to larger units often runs into unpredicted high pressure, which is detrimental to the successful operation of the motor. First reported by Smith and Sprenger, 1 it is now generally accepted that the abnormal phenomena is characterized by the presence of high-frequency, large-amplitude pressure oscillations of the gas inside the cavity. Two questions are of fundamental interest. (i) The origin of these large amplitude oscillations, and (ii) the characteristics of the large amplitude oscillations which produce the damage.

In many cases, large amplitude oscillations

have been shown to develop from small disturbances in smoothly operating motors through amplification. 2 Because smM1 disturbances cannot be avoided, the control of the stability of small disturbances becomes essential but it is not always sufficient because large amplitude oscillations may also be triggered by initially large disturbances. The nonlinear problem, in this Symposium, is dealt with in the section on Ero- sive Burning. The present review will be concerned only with the linear stability of small disturbances.

Mathematical methods of analyzing linear systems are well established, although their application to practical systems may be very tedious. Any experimental findings in disagree- ment with the correctly predicted analytic trends

82 SURVEY PAPERS

can be at tr ibuted to the deficiency of the physical model• However, this fortunate situation has not been helpful in obtaining a solution to the problem of linear dynamic stability in the solid- propellant rocket motor. The incomplete understanding of the combustion of solid propellants is mostly responsible. Lack of pertinent experimental data and lack of precise control of the experiments also contribute to the unsatisfactory state of the art. Under these circumstances, it is fair to evaluate the theoretical developments purely from the self-consistency of the mathematical procedure and of the physical model.

I t is generally agreed that the temporal variation of the instantaneous burning rate, either inherently or in response to the variations of the environmental conditions, must play a crucial role. The importance of acoustic oscillations in the cavity was presupposed. 5' 6 This view was shared by some investigators 7's and contested by others2, ~0, ~, 27 Green 9, ~0 predicted the presence of a critical frequency with a set of discrete values of the time lags, determined solely by the physicochemical behavior of the propellant. The effect of different physical and chemical properties of the propellant on the stability of the system is examined in terms of their role in changing the critical frequency• Although the resonance frequency is an inherent characteristic of the propellant, the "oscillations in the parallel flow conditions" outside the boundary layer parallel to the burning surface of the propellant are considered as the primary forcing agent2,10 Other authors either ignore the oscillations of the parallel gas velocity 5 or indicate that the effect of the oscillation of the burned gas velocity is negligible, 6 except possibly for some degenerate cases32 As such, the work of Nachbar and Green ~ , ~0 stands in the forefront of all the con- troversies and will be discussed first. The other three theories will be given a unified presentation as all three are based upon a closed feed-back circuit involving the interaction of the acoustic oscillations on the burning surface with the mass rate of generation of the final combustion products. (In nonsteady-state this rate should be carefully distinguished from the mass rate of disintegration of the solid propellant or the rate of regression of the "burning surface.")

T h e o r i e s W i t h o u t A c o u s t i c I n t e r a c t i o n - - I n h e r e n t I n s t a b i l i t y

In the work of Green and Nachbar, 9, ~0 the oscillatory burning of a solid propellant slab is

analyzed within the framework of one-dimensional heat conduction in solid phase, with an oscillatory convective heat transfer from the gas which flows over the solid, and with an oscillatory rate of chemical reaction at the propellant surface governed by the quasisteady Arrhenius law.

The heat transfer across the boundary layer to the propellant solid is expressed through a film coefficient F that oscillates with a small amplitude

F = Fo + eR[F~ exp (io~t)] (2.1)

where e << 1. Exp (ioJt) is the complex form of sinusoidal oscillation of angular frequency o~. R indicates the real part of the following complex quantity. No expression for F~ is given but it is implied to be a nonzero constant, uncoupled from the oscillations in the solid. The instantaneous temperature in the solid is expressed in a similar form

T(x, t) = To(x) + ~R[f(x) exp (i~ot)] (2.2)

with origin x = 0 located at the burning surface• The instantaneous burning rate r(x, t) is assumed, according to the quasisteady argument to be

r(x, t) = ro( Tso)

{ E }(2.3) • 1 + ~ ~ R[f(O)e i~(t-T)]

where

ro(Tso) = B exp ( - E / R T s o )

is the steady-state burning rate at a temperature Ts0 with activation energy E. The time delay T, is for the combustion processes to respond to macroscopic disturbances, a Referring to the mov- ing coordinate x, fixed on the burning surface x = 0, the governing heat diffusion equation stands as

OT 02T OT - a + (2•4) ot ~ r -g~ .

The last term arises because heat is required to bring the propellant at a distance r from the burning surface to the surface temperature.

This is the chemical part of the time lag of the burning of the solid propellant, which is only a part of the overall time lag, including the diffusion lags as was adopted by Grad 5 and Cheng. 6 See next section for further explanation.

UNSTABLE COMBUSTION IN SOLID-PROPELLANT ROCKET MOTORS 83

The differential equation for f(x) to the first order of ~ stands as

~z~ _t_ r0 d a d x - f ( x ) (2.5)

= Af(O)e -ro~/a

with the boundary conditions

l i m / ( x ) = 0 (2.6) x ~ oo

expressing the asymptotic decay of the temperature disturbance far from the burning surface to the undisturbed, preignition temperature Ti ;

- ( ~ x d f ) z = o + C f ( O ) = D (2.7)

expressing the heat balance at the burning surface. Here

A = (ro/a)2[E(Tso -- Ts)/RT2so] exp (-icor)

C = (ro E/RT~o){[(Tso - T~)/al ~- 2p~Ls/k}

• exp (--i¢or) -t- Fo/kro (2.8)

D = ( F f f k r o ) ( T f - Tso).

Attention will be called to the boundary condition at the burning surface (2.7) that D represents the variation of the convective heat transfer rate due to the variation of the film coefficient, and that the last term in C, (i.e., Fo/kro), the same due to the variation of the surface temperature Ts . (df/dx)x=o represents the variation of the rate of heat conduction out of the burning zone into the solid; and the rest of the terms, the variation of the latent heat needed in the burning z o n e .

The functionf(x) satisfying Equation (2.5) and the condition (2.6) is

f (x) = f(O)[(iaA/co)exp(--ro x /a)

-t- (1 -- iaA/co) exp (r0 X2 x /2a ) ]

where

X2 = --1 -- (1 + 4icoa/r~) u~. (2.9)

The condition (2.7) on the burning surface then gives

D = f (0){C + iro A/co (2.10)

-- (X2/2)(ro/a -- iroA/@}.

When the complex quantity in the { } of (2.10) vanishes for a discrete set of values (co*, r*), a

critical condition is reached. This is interpreted by Nachbar and Green as the critical condition of resonance, that is, the amplitude of the oscillation of the propellant surface temperature (I Tz - Ts01 ~ If(0)]) attains inordinately large values of the order of e -1. The effect of individual factor on the stability of the system is investigated by evaluating the partial derivatives of the critical frequency co* with other parameters fixed. Rather detailed discussion of the effect of many physical factors are numerically exempli- fied.

Nachbar and Green should be complimented for showing that an inordinately large burning rate may result under a given small oscillation of the heat transfer rate to the burning surface without referring to the acoustic situations outside the solid propellant.

Much of the discussion of Green and Nachbar is concerned with the significant increase of the propellant erosion rate which is, undoubtedly, a phenomenon at large amplitudes. For periodic small perturbations, the mean value of any instantaneous quantity over a cycle differs from the steady state value only to the square magni- rude of the small perturbation, negligible by assumption of the linearized perturbation analysis (Appendix 2,8). The burning rate of a solid propellant is no exception. The erosion rate will be influenced heavily by the nonlinear characteristics and should not be discussed from the stability boundary of a linearized analysis. Accordingly, no discussion of their conclusions will be made in this connection except referring to the arguments between Price 'a and Green and Nachbar. '4

The mathematical and the physical formulation of the linearized problem in (9) and (10) is not suitable for determining the stability boundary of a coupled system. I t serves to indicate possible instability when the "condition of resonance" appears. This is because the problem is formulated in (9) or (10) as a forced oscillation problem with D in the boundary condition (2.7) nonzero, and yet the system is required to respond at the same frequency in a non-decaying and nonamplifying form. If, in Equation (2.2), co is replaced by a complex constant co - ia with representing the amplification coefficient, Equa- tion (2.10) becomes

D = f ~ ( x = 0, t = 0) (2.11)

× G(co, ~; r, a, E . . . )

84 SURVEY PAPERS

in which G will reduce to the form given in (2.10) if o = O. ef,~(x = 0, t = 0) is the initial value of the amplitude of the component oscillation of the surface temperature of frequency co. The amplitude will change with time according as exp (at) with a determined from (2.11). If

a(co, 0; r " ' ) ~ 0

then

f ,o(x = O) = D I G ( c o , O; r - . . ) = 0(2)

will represent the amplitude of the standing periodic oscillation that the system will eventually achieve, when the external forcing function of this frequency is maintained. If G(co, 0; r - - • ) turns out to be small say 0(e), Equation (2.11) is for the determination of the initial amplification constant a for a given initial amplitude f,o(x = 0, t = 0), not of the amplitude of the limiting standing oscillation.

Physically, an oscillation of the film coefficient F1, uncoupled with the burning rate fluctuation ~f(0) is dubious. The evolution of the burned gas possesses a significant transpiration effect of reducing the heat transfer rate to the surface. With F0 depending on r0, one should take F1 = /~f(0) or D = 8f(0) even when the free stream condition is frozen (i.e., no oscillations in outside parallel flow). A homogeneous boundary value problem will result with the characteristic equation

G(~o, ~; r, a, E . . . ) = 8. (2.12)

The stability boundary, given by the equation,

G(co, 0; ~, ~, E . . . ) = ~ (2.13)

will define a curve (or curves) on (co; v) plane separating the stable and the unstable ranges of co and r when all but one of the other physical parameters are fixed. When (2.12) admits real solutions of a with co = 0, the system will diverge (¢ > 0) or converge (a < 0) exponen- tially in an aperiodic (nonoscillating) manner. When 5 = 0, (nonoscillating film coefficient), Equation (2.13) is precisely the "resonance condition.,,9, 10 But this relation

G(co, 0; ~ , . . . ) = 0

stands as the stability boundary with "frozen" free stream, and has nothing to do with the oscillatory conditions in the parallel flow. The unstable region, if any, revealed by Equation (2.13), is inherent in the propellant burning mechanism

which involves retarded response, either postulated in the solid phase D' ,0 or expressed through a diffusion process in the gas phase. 11

An experimental demonstration of such an "inherent instabili ty" of a solid propellant in open atmosphere (or a very large acoustic chamber) is then of interest. Kumagai and Sakai 15 reported that the burning of some propellant under oscillatory electric heating indicates "resonance" of the burning rate a t a specific frequency, but not so under oscillatory heating through radiation. The observed "resonance frequency" varies to some extent with tile geometry of the specimen. Whether or not this "resonance" belongs to the experimental apparatus or to the propellant is not certain. (Private communication from M. Summerfield.) Actually, there are inherent instabilities, well demonstrated experimentally; for example, those leading to turbulent flow conditions in the burning zone or to cellular flames/6 Inherent instabilities deal with the behavior of an isolated system, or at most a system very loosely coupled with the sur- roundings, not a coupled system.

If the solid propellant burning should be loosely coupled with the surrounding gas system in the rocket chamber, the gas system would not oscillate appreciably in response to the "resonance burning" of the solid propellant. This contradicts the important manifestation of the "combustion instability" in solid-propellant rocket motors; i.e., the severity of the gas oscillations. Conse- quently, the presence of inherent instability serves to indicate only that serious instability may result when the propellant is coupled with the gas system in an appropriate manner. To draw inferences beyond the above qualitative statement from the analysis of inherent instability, and to apply these inferences to the combustion instability without reference to the coupling mechanism, can hardly be justified. In the theories with acoustic interaction, the importance of the coupling mechanism between the propellant burning and the gas oscillations in the chamber is presumed. The mathematical formulation of this coupling or interaction becomes the crucial question.

T h e o r i e s W i t h A c o u s t i c I n t e r a c t i o n

In all the theories with acoustic interaction, the burned gas of the solid propellant inside the cavity of the rocket motor is visualized as an acoustic system, bounded by (i) burning surface


of the propellant, (ii) nonburning solid surfaces of the propellant and motor wall casing and (iii) the entrance section of the deLavae nozzle with supercritical gas outflow. In the steady-state, the mass of the gas leaving the system through the nozzle is balanced by the mass of gas generated by the combustion processes on the burning surface of the propellant (allowing for some mass to fill the volume previously occupied by the solid propellant). Both the mass outflow rate at the nozzle and the mass generation rate on the burning surface will vary with their local gas-dynamic conditions. Suppose a slight pressure increase is accidently produced on the burning surface, the rate of mass generation will increase, (for example) not instantaneously, but at some later instant. At the same time, the pressure disturbance will propagate into the gas flow field, reflect from the boundaries, and eventually reach the propellant surface again. By then, the magnitude of the pressure disturbance wil] have been reduced by the dissipative agents, like viscosity of the gas and by the nondissipative damping through the reflection from nonrigid boundaries. If the timing of the wave propagation process in the acoustic cavity is such, that the arrival of the weakened disturbance on the burning surface almost coincides with the increased gas generation rate in response to the initial incident disturbance, the weakened disturbance will be strengthened by the mass addition possibly to a magnitude larger than the magnitude of the incident disturbance. This process repeats and the magnitude of the disturbance grows with time. The system is then unstable to small disturbances.

The variation of the rate of burned gas generation in response to gas oscillations on the propellant burning surface is the excitation mechanism, because i t will tend to excite unstable oscillations when the time condition is met. Whether or not unstable oscillations will result in the chamber depends on whether or not the sum of the excitation contributions over the entire burning surface is able to counterbalance the sum of all the damping contributions. Within the linearized framework, the analysis may be con- eerned only with periodic disturbances with a fixed but arbitrary frequency, because any arbitrary disturbance may be considered as a synthesis of all its Fourier components.

Any theory of combustion instability in a solid propellant rocket motor with acoustic interaction will, hence, consist of three main parts, dealing with the determination of:

(i) The ratio of the fractional variation of the rate of generation of the burned gas to the fractional variation of the gas oscillations on the burning surface of the propellant. This is to establish quantitatively the coupling between the combustion processes of the propellant and the gas oscillations, i.e., the excitation mechanism.

(ii) The conditions of wave reflection on different parts of the boundary of the acoustic system other than the burning surface of the propellant.

(iii) The acoustic wave forms in the cavity under the prescribed boundary conditions and the volumetric damping (and/or excitation, if present) of dissipative nature. The wave form serves to correlate the excitation contributions and the damping contributions.

Different theories, 5-7 advanced, differ in details in formulating the excitation and the damping contributions. The acoustic wave forms and frequencies in the cavity are approximately determined in all theories by assuming the conditions of no gas flow. This is adequate only for rocket motors with nozzles of large contraction ratio (entrance area to throat area) when the Mach number (the ratio of gas flow velocity to local speed of sound) of the flow is negligibly small, e In estimating the acoustic frequency, the entrance to the nozzle of large contraction ratio is approximately a closed end, (rather than an open end) slightly displaced to the downstream direction. The compressibility of the unburned solid propellant 7 and the flexibility of other acoustic boundary will also modify the acoustic frequency.

The thickness of the burning zone is usually assumed to be negligibly small compared with the wave length of the acoustic oscillation. Under this assumption, all of the excitation contributions are present only on the bounding surface of the acoustic system and enter into the acoustic analysis only as a boundary condition. There may be circumstances that the evolution of gas or of heat continues beyond the thin burning zone over the propellant surface and is distributed in some arbitrary manner over the volume of the acoustic cavity or even over the volume in the nozzle. The sensitivity of the gas evolution (mass sources) and the heat evolution (heat sources) to the oscillations in the gas will tend to excite or drive the gas oscillations when properly timed. The possibility of periodic heat addition to excite unstable oscillations is noted by Rayleigh who also established the time and the space

86 SURVEY PAPERS

TABLE 1

Authors Excitation Damping Acoustic Modes Analyzed

Grad 5 Phenomenological con- Nondissipative nozzle Composite stant time lag damping ~, open end

Cheng 6 Phenomenological vari- Nondissipative nozzle Composite and purely able time lag damping .~ close end longitudinal

McClureT. s Hart, Bird Comprehensive conduc-Dissipative damping Purely transversal tion model

Green, Nachbar ~°, Shin- Comprehensive convec- nar, Dishon 1~, Smith ~7 tion model - - - -

EQUIVALENT CLOSED COUPLED MASS AND/OR END OF SUPERCRITICAL

HEAT SOURCES-EXCITATION NOZZLE FOR FREQUENCY DISTRIBUTED OVER V O ? / E V A L U A T I O N

COUPLED MASS SOURCE 1 I, J ON PROPELLANT

ACOUSTIC VOLUME BURNING SURFACE ~EXCITATION

f v t MASS SINK COUPLED HEAT ~ AND PRESSURE WORK SINK, DISSIPATIVE

SURFACE DAMPING AT NOZZLE ENTRANCE - NONDISSI PATIVE

DAMPING

US DAMPING ETC DISTRIBUTED OVER VOLUME

FIG. 1

criteria. 24 The use of reactive additives, such as aluminum powder, will lead to such a situation. The stability analysis of an acoustic system with arbitrarily distributed mass and heat sources is tedious even when the simplest possible coupling between the mass and the heat sources to gas oscillations is assumed.

The major differences between the different theories will be outlined in Table 1. A schematic diagram is shown in Figure 1 to indicate the different aspects of the stability analysis of the system. The theoretical investigations without acoustic interaction, as was discussed in the previous section are also included because of their possible connection with the aspect of excitation in the acoustic theories.

Acoustic Interaction on Burning Propellant Surface--Excitation

The mechanism of burning of a solid propellant in the steady-state is not yet understood to

general satisfaction2 Many thermal theories have been developed but this need not mean that diffusion of active species has no place in the solid propellant burning. Under a rapidly varying gas-dynamic environment, any burning mechanism is necessarily more speculative and contro- versial. Nevertheless, it may be safe to assume that the rate of burning of the solid propellant will inevitably vary in response to the rate of heat transfer to the solid propellant. Hence, the variation of any state (pressure p, density p and

temperature T) and dynamic variables, (the velocity v of the gas relative to the propellant surface) that influence the heat transfer rate will

produce a corresponding variation of the burning rate. If the instantaneous rate of burned gas generation rh(t) is designated functionally as F(t, p, p, T, v, etc.) then the burning rate response

function


~m ~= F p + F T o l n p

Olnp . Fv Olnvt (3.1) + F p ~ + ~Olnp

+ F~ 0 in Vn "01~p + ""1

The acoustic admittance defined as the fractional variation of the gas velocity component v. normal and relative to the propellant burning surface to the fractional variation of gas pressure is

OlnT F Olnp = F , W F r 0 1 ~ n p + ° O l n p

(3.2) + F'~Olnvt + " ' l

Olnp

0 In v.-]

in which subscripts to F indicate partial loga- rithmic differentiation F~ = O in F/O in z. If the gas oscillation is isentropic then

0 1 n T _ ~ ' - 1 0 1 n p _ 1 (3.3) 0 In p ~, 0 in p ~,

where the specific heat ratio -/ of the gas is assumed constant. I t is worthwhile to note that the fractional variation of the tangential gas velocity vt and of the normal velocity component v,~ are not so simply related to the fractional variation of the gas pressure. In addition to % 0 in vt/O in p depends on other fluid dynamical and geometrical factors. Because the convective heat-transfer rate to the solid surface will depend upon vt, Equations (3.1) and (3.2) indicate that the response function and the acoustic admittance depend, to some extent, on the fluid mechanical situation in the acoustic chamber rather than on the physical and the chemical properties of the propellant alone. To adopt a response function (or its equivalent) as an inherent characteristic of the propellant 5' 6.8 is conceptually somewhat in error. The error may become serious if convective heat transfer should play a prominent role in the burning mechanism.

The terms including the normal velocity component v~ represent the transpiration effect of the mass injection into the thermal boundary layer. Its importance is magnified in the acoustic admittance ratio since it appears in the denominator

of (3.2) although its contribution is merely additive to others in the response function (3.1). The energy equation for the temperature oscillations in the gas film, (the thermal boundary layer) is

0 [c~ pv,~ T'] [cv(~T' + p'~)] + (3.4)

0 (cpT) = 0 FxOT'~ + (pv,~)'~ E~L ~ J

written in the single spatial coordinate n normal to the propellant surface (not stationary) with a macron indicating steady-state values and a prime the oscillatory part. The viscous dissipation and the radiative heat transfer are net- leered. The mean mass flow pv,~ is small and may be neglected in the convective derivative (this depends on the frequency of the oscillation). Thus, the heat transfer model is formally conductive. However, the term with (pv,~)', representing the transpiration effect in the unsteady state, cannot be dropped. Equation (3.4) has to be solved simultaneously, at least, with the mass continuity Equation (3.5) when the oscillating pressure is taken as uniform across the flame zone at any instant. If the convective tangential mo- tion of the gas is not consideIed, this transpiration effect brings in, only, the coupling between the unsteady heat and mass transfer, not the fluid mechanical or geometrical parameters like Reynolds number, etc.

o(p~,~)' Op-- + _ _ _ O. (3.5) Ot On

In principle, the acoustic admittance (or the burning rate response function) can be calculated if (i) the rates of the processes in a given model of chemical reaction in generating the burned gas are assumed and (ii) the functional dependence of the rates of the different modes of heat transfer (conduction, connection and radiation) on the state and the dynamic variables of the burned gas is assumed. Both the real and the imaginary part of the ratio

o r

dvo/ p

will depend on the frequency of oscillation and a great many physical and chemical characteristics

88 SURVEY PAPERS

of the propellant. An example was carried out by Hart and McClure s (Smirov ~ reported a similar analysis. Details are not available.) based on the following model. (1) An endothermic reaction on the surface of the solid propellant generates intermediate gaseous products. The rate of this surface reaction is visualized as determined by the quasisteady-state Arrhenius factor in terms of the instantaneous temperature in a thin "surface reaction zone." The detailed formulation involves the assumption of an ignition temperature on the cold side of the surface reaction zone and the assumption that the frequency factor in the kinetic rate expression is inversely proportional to the square of the local temperature. Although some of the argument is not clear, it may be visualized at least as a reasonable phenomenological expression of the reaction rate in terms of the local temperature and the local heat transfer rate.

(2) The gaseous intermediates react to form final combustion products upon reaching an ignition temperature almost instantaneously.

(3) The solid and the gaseous intermediates are energized only by the conductive heat transfer process under the oscillatory temperature field. The transpiration effect in the gaseous zone is included.

(4) Physical and chemical constants required for the determination of the time dependent temperature field are adopted from the steady state constants through the quasi-steady argument.

The burning-rate response function

i

(in their notation) is found to involve at least eleven parametric groups of the constants of the physicochemical system, in addition to the frequency co of oscillation. I t is, hence, very difficult to draw practical conclusions or even to discuss the results. Extensive numerical work has been performed and many curves of the real part of (t~i/~l) as a function of frequency w, are illustrated in Reference 8. The general shape of these curves indicates that R(ftl/~) increases from low frequencies, reaches a maximum at a frequency ~0(10 4) cycles per second (or higher) and tends to oscillate. The details are irregular. The magnitude of the ratio is of the order of unity. The phase lag (or lead) also varies with frequency.

Bird, Haar, Har t and McClure 17 considered the effect of propellant compressibility and thermal expansion on tile burning rate response function. The effect of the thermal expansion is found to be small. The effect of propellant compressibility is to decrease the R(f*l/~O. The qualitative dependence of the response function on frequency is unchanged.

If one wishes to introduce the convective heat transfer or the radiative heat transfer into the model, several physical variables will appear in the burning rate response function. If one wishes to have a more sophisticated combustion model, many more chemical parameters would enter. The problem of extracting useful practical information concerning the effect of each group of parameters on the response function (responsible for excitation) is a formidable one. How much the high speed computing machine can help without first consolidating the model is open to individual opinion. The need of meticulously planned experiments under the simplest possible acoustic conditions in determining the response function of a propellant cannot be over-empha- sized.

Earlier investigators 5, 6 sought to bypass the aforementioned difficulty, basically by postu- lating this response function through intuition, visualized as a property of a given propellant with appropriate frequency dependence. I t is mathematically convenient and physically sen- sible to introduce a time lag to approximate the exponential variation by almost discontinuous variation. The variation is exponential because of the Arrhenius factor in chemical kinetics and of the diffusive nature of the processes of heat and mass transfer. For oscillatory variations, the lags of different processes may differ in sign. The significance and the convenience of introducing an over-all time lag as a gross parameter in analyzing the combustion instability in rocket motors was first suggested by yon K~rm~n} 8 This is no more and no less arbitrary than the use of the ignition lag and the ignition temperature in many older flame theories, or the use of the film coefficient and the thermal boundary layer thickness in heat transfer. I ts use is not desirable if we know better but is not fundamentally objection- able if such an over-all parameter could be demonstrated to be almost an entirety in a certain range of applications. The task is laid on the experimentalists to establish the validity and usefulness of such a gross parameter or to suggest that the theorists make better guesses.


Grad s adopted a constant time lag as a physical property of a propellant much like the ignition lag of the ignition process. The amplitude of the interaction was derived from the quasisteady- state dependence of the burning rate on the state and the dynamic variables of the gases. Cheng 6 adopted a variable time lag 19' 20 and in- troduced an over-all index of interaction to represent the strength of the coupling. The index of interaction and the mean time lag characterize a given propellant. The Grads' scheme would also be a two-parameter description if the pressure exponent in the quasisteady burning law is permitted to take values different from the steady-state value. Grad scheme, so modified, the Cheng scheme might appear to be very much the same. There is, however, an important distinction in the form of the frequency dependence of the response function;

b exp ( - io~r) and

(, respectively, in which b and S are the interaction indices and r the time lag in the two theories. n was taken as a steady state exponent for con- venienee 6 but was basically visualized as the interaction index for the solid phase processes. The qualitative validity of the inherent frequency dependence of the response function should be a good criterion of the usefulness of the model.

The calculated results of Hart and McClure s offer some interesting comparisons. The modified Grad model implies

R(¢1/~1) "~ b cos ~r

i.e., an initial fall-off from the maximum value b as the frequency increases from small values. This is an intolerable situation despite the two- parameter description of the burning rate variation (Figs. 2-4). The Cheng model implies

which increases from the initial value n to a maximum value S when ~T = 7r and then oscillates for larger w. In this sense, n may be con- sistently interpreted as the steady-state exponent (in the limit of ~0 ~ 0). The parameter S may be visualized as an interaction index for the unsteady process and r an over-all time lag. They

may be selected to approximate the real and the imaginary parts of the response function in the appropriate frequency range. If we should adopt a representative set of results from reference 8, a reasonable approximation can be achieved for the range of frequencies w up to the order of (say) 10 4 cycles/see. For higher frequencies, the R(fil/~l) as postulated in Cheng's model falls off and oscillates much too fast (Fig. 2). This might be interpreted to mean, if we strive to match the calculated results of reference 8, that the time lag r must decrease with frequency. This alterna- tive is physically acceptable, although not desirable, from the phenomenologieal point of view. In fact, several of the assumptions in the calcu- lation 8 will break down for oscillations of frequencies in the range of 10 5 cycles/see. Hence, the behavior of the response function in such ultrahigh frequency range may not be taken too seriously.

One must not lose sight of the approximate nature of the analysis, 8 complicated as it is. There are questions in the details of their analysis. But the crucial question is to what extent the calculated response function will be changed if a different combustion model is adopted. If we limit ourselves to the purely thermal theory, should we distinguish the combustion models of a double based and of a composite propellant and others? Even for a single based propellant is the gaseous reaction much faster, much slower or comparable to the solid phase reaction? Will the unsteadiness affect the gas layer more than the solid phase? If the qualitative dependence of the response function on frequency ~o should turn out to be essentially the same, there may be some hope of constructing convenient phenomenological description of this burning rate response function. The phenomenologieal model of Cheng might be quite adequate. Such is not assured, however, because we are not confident that the response function will, in carefully designed experiments, show the behavior as predicted 8 and postulated in Cheng's model!

The accomplishment of Hart and MeClure represents a step toward a better understanding of the complicated excitation mechanism on the burning surface. The problem is still open. The effect of different combustion models, of different heat transfer mechanisms should be examined in a consistent manner. I t would be fortunate if the results of such comprehensive analyses can be analyzed to isolate the effect of each individual

90 • SURVEY PAPERS

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physical and chemical parameter. Even if not, we may be in a better position to construct phenomenological theories to facilitate the study of the stabili ty of the system. The eventual justification of the phenomenological as well as the comprehensive theories will have to depend on reliable measurements of the response function.

D a m p i n g s a n d A c o u s t i c s i n R o c k e t C h a m b e r

Let us assume that the gas in the rocket chamber is inert and acts merely as a wave carrier. The acoustic chamber is bounded part ly by the burn-

ing surface of the propellant, partly by the wall of the motor casing, (including the nonburning surface of the propellant) and closed by a con- verging-diverging nozzle through which gas is leaving the chamber with sonic or supersonic velocity. No disturbances in the ambient atmosphere can influence the oscillations inside the motor through the supercritical flow.

If all of the solid boundaries, including the solid propellant, are rigid, the gas undergoing oscillations, will not do mechanical work on the surrounding solid boundary. If the viscosities and the heat conductivity of the gas are further


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ignored, the gas oscillation wilt be isentropie and neutral. Otherwise, the viscous dissipation and the heat conduction within the acoustic volume and over the bounding surface tend to damp the acoustic oscillations. These are the dissipative dampings of the acoustic oscillations which are of the order of square magnitude of the oscillation, and are proportional to the viscosity and the heat conductivity coefficients. They will be negligibly small, in the limit of very small viscosities and heat conductivity, when compared with any other quantity of the square magnitude of the oscillation, but independent of these dissipative coefficients. Any damping mechanism independent of the viscosity and the heat conductivity of the gas will hence be the predominating damping agent in gases with small diffusivities.

Such a nondissipative damping is provided in the solid propellant rocket motor by the nozzle with supereritical outflow. The mass which flows out of the nozzle carries with it the energy of acoustic oscillations (convective loss). The pressure and the velocity oscillations of the gas at the sonic throat (or at the entrance to the nozzle) are not completely out of phase so that work is done by the oscillating gas. The acoustic condition imposed by a supercritical nozzle on the acoustic oscillations in the cavity is very much different from the conventional condition of zero pressure oscillation at an open end where there can be no pressure work done by the gas oscillation and where there is little net mass outflow (only to the square magnitude of the oscillation). Hence, the

conventional open or closed end condition should not be applied even to a subsonic nozzle (or just an opening) if there is a significant mass outflux.

The determination of the acoustic admittance i.e., the ratio of the local fractional variation of velocity to pressure, at the entrance of the nozzle (super or suberitical) is a very tedious linearized problem in gas dynamics. The damping action as expressed through the acoustic admittance ratio will depend on many factors including the frequency of oscillation, and the geometry of the nozzle. A supereritical nozzle with a constant mean velocity gradient was first treated by Tsien ~1 and completed by Crocco (Appendix 2 5°) on one-dimensional basis. Analysis of three- dimensional oscillations in a quasi-one-dimensional nozzle has been carried out by Crocco. The stability of the acoustic oscillations in the gas system is determined primarily by the relative magnitude of the total excitation over the entire burning propellant surface and the total damping contributions of dissipative as well as nondissipative origin. In reference 6, the nondissipative damping contribution is included in a crude manner with dissipative dampings ignored. In reference 7, only the purely radial modes in the absence of any nozzle are treated and therefore, only the dissipative damping over the acoustic media is estimated (in gas and solid).

The acoustic admittance is the ratio of the fractional variation of the amplitude of the local velocity oscillation to that of the local pressure (or density) oscillation. If the nozzle is located in

92 SURVEY PAPERS

the pressure antinodal region, (or nodal region) the nozzle damping action is the most (or the least) pronounced. The excitation contribution (fractional variation of the mass burning rate) of a given element of the propellant burning surface depends, likewise, on its location relative to the standing wave forms. If the propellant is sensitive only to the pressure like oscillations (density and temperature in phase with pressure), the propellant surfaces in the pressure antinodal region contribute the most to the excitation while those in the pressure nodal region will remain almost inactive. If on the other hand, the propellant is sensitive to the velocity oscillations only (90 ° out of phase with pressure oscillation) the propellant burning surface must be located in the velocity antinodal region (pressure node) to be effective. The acoustic analysis is to identify the phase relation of the damping and the excitation with the standing acoustic wave form so that the stability of the system, determined by the balance of the damping and the excitation contributions, may be correctly evaluated.

An intelligent use of this principle is made by Price and Sofferis ~ to evaluate the relative importance of the pressure-like and the velocity oscillations in producing the burning rate variation of certain propellants. Experiments were conducted in a motor with the nozzle located in the midsection of a cylindrical tube and with the propellant as the radial boundary. This is a "low damping" system for the fundamental longitudinal mode and its odd harmonics since the nozzle is located in their pressure nodal region. Unstable longitudinal oscillations are observed when propellant segments are located in the end portions of the tube (the pressure antinode) but barely observable when located in the midportion of tile tube (the velocity antinode). The pressure like oscillations are, hence, more important than the velocity oscillations of the gas in producing burning rate variation. The analysis summarized in Appendix 2, 6 concludes that the excitation contribution from the velocity oscillation alone is not able to counter-balance the damping of the nozzle located in the pressure antinodal region. The velocity mechanism was, hence, neglected when compared with the pressure like mechanism in exciting instability. The experiment of Price and Sofferis 2 indicates that a stronger statement may be possible because the nozzle damping was essentially eliminated. This justifies to some extent the formulation of the

excitation mechanism without considering the velocity oscillation in the phenomenological 5, 6 and the comprehensive treatments. 8 The oscillation of the drifting velocity parallel to the burning surface modifies the burning rate primarily through the variation of the convective heat transfer rate. With convective heat transfer of lesser importance, the burning rate response function, discussed in the previous section, may actually be visualized as the characteristic property of a propellant in exciting unstable oscillations in a rocket motor.

The apparatus of Price and Sofferis ~ may be modified to reveal much information of fundamental importance, because of the possibility of varying the excitation contribution by using propellant segments of different length and located at different parts of the cylinder. The nozzle damping can be minimized as well as maximized by relocating the nozzle. I t appears that the condition of nearly neutral self-sustained oscillations can be achieved. This apparatus will, then, promise a good opportunity to experimentally determine the burning rate response function

P of a given propellant. This will help immeasur- ably, the formulation of the excitation mechanism and the eventual understanding and control of the phenomenon.

The importance of determining the standing acoustic wave form in the chamber is by now evident. I t is fortunate that, in spite of the important damping action of the nozzle, the characteristic frequency of the acoustic cavity may be determined approximately by replacing the nozzle entrance with a slightly displaced closed end condition, provided that the area ratio of the nozzle contraction is large. The acoustic wave form in the acoustic cavity may, however, be seriously distorted by the lack of sufficient rigidity of the solid boundary of the acoustic cavity, i.e., the motor wall casing and the propellant. In extreme cases, the different ways of mounting a thin walled motor to other structures or the different ways of attaching the solid propellant to the outer casing may significantly alter the acoustic standing wave form, the balance of the excitation and the damping contributions, and consequently the stability of combustion in the rocket motor. The most significant factors are the changes of the positions of the burning sur-


face and the nozzle relative to the standing wave form of the acoustic oscillation.

The over-all effect of the lack of rigidity of the motor structure and the propellant on the stability of the system cannot be concluded in general without referring to the acoustic condition of the system. To illustrate this point, it may be noted that in the experimental cylindrical motor of Price 2 the longitudinal oscillation will be little affected by the compressibility of the propellant, being a radial boundary. The stability will be improved by the propellant compressibility due to the decrease of the excitation. 17 Suppose the nozzle is located at one end rather than at the midlength of the tube, and suppose the length to diameter ratio of the tube is such that the fundamental spiral mode for the rigid propellant is in the frequency range beyond the frequency corresponding to maximum

the compressibility of the propellant may decrease the frequency of the spiral mode to such an extent as to be accompanied by an increase in the

and a decrease in nozzle damping. The propellant compressibility may then bring about instability rather than improve the stability in the system.

These simple examples clearly point out the necessity of considering the acoustic situation in a system as a whole, in evaluating the system stability. Of course, the understanding of the role played by each individual component is a pre- requisite. The stability of the system is the one of practical concern and is the one that has been investigated experimentally. Unless the experiment is specially designed with over-all acoustics in mind to test the role of an individual factor, it will be very difficult to interpret the test results of the system stability to obtain the effect of the variation of the single parameter.

Propellant Additives and Acoustics

Additives of metals or metallic oxides or other types of additives are currently considered as an important cure to combustion instability in solid rocket motors and as a source of additional energy. In view of the previous discussions, the roles of the additives should be studied by con-

sidering their effects on the excitation and the damping contributions, and the over-all acoustic situation.

Some apparently nonreactive additives have suppressed instability. The increase in the viscous damping by the fine solid (or liquid) particles in the oscillating gas has been suggested as a primary stabilizing mechanism? ~, 23 The oscillation of the suspended particles (or droplets) relative to the surrounding gas gives rise to increased dissipative dampings. The effect of the increased dissipative damping may, however, be easily overshadowed by any change in the nozzle damping when present. The change in the nozzle admittance and the dispersion effect of the particles will also modify the acoustic wave form with complicated consequences.

Both the inert and the reactive, additives can affect the rate of burned gas generation under oscillatory or steady-state conditions because of the difference in the physical properties, for example, the heat conductivity, of the additives and of the basic solid propellant. Some additives, although inert, may serve as a chain inhibitor at some stage of the gaseous reaction if not as a catalyzer. The reactive additives may affect the burning rate of the base propellant perhaps in a more adverse manner. Furthermore, the reactive additives may or may not burn with the base propellant near the propellant burning surface. The combustion of the additives will respond to the gas oscillations in a manner different from the basic solid propellant. In principle, a basic propellant with different amounts of additives, inert or reactive, should possess different response functions

We do not know in which way the response function will be changed when additives are intro- duced into a basic propellant. Consequently, a basic propellant with a given amount of a certain additive should be considered as a distinct propellant.

There is an interesting aspect of the stability problem presented by the reactive additives, for example, the aluminum powder. The reactive powders, if not completely burned in the burning zone over the propellant surfaces, will drift into the acoustic cavity. They may burn in the rocket chamber or in the nozzle or they may escape the nozzle and never have a chance to burn. When

94 SURVEY PAPERS

the powders burn, they give off heat to the surrounding gas (if the mass of gas generated is neglected). Thus, the steady-state temperature (stagnation value) of the burned gas may increase significantly from the burning surface toward the nozzle. This will distort the standing acoustic wave form and the characteristic frequency of the chamber. The balance of the excitation on the burning surface and the damping of the system will, hence, be disturbed with consequent change in the stability behavior even if the acoustic admittances on the burning surface and that at the nozzle entrance remain unchanged.

When the burned gas oscillates, the rate of combustion of the powders, and, therefore, the rate of heat release to the gas will oscillate also, and in general, not in phase with the gas oscillations. Based upon the well-known Rayleigh principle, ~4 this retarded oscillatory heat addition to the burned gas (in contrast to the oscillatory mass addition on the burning surface) from the reactive powders may excite instability of the gas oscillation if properly timed. The response function,

q

of oscillatory heat release to the local gas pressure oscillation will have to be determined based upon some combustion nmdel for the reactive additives. Such a function would have to be postulated in the phenomenological approach by introducing some index of interaction and time lag, different from those constants characterizing the basic propellant. This is a novel feature presented by the reactive additives (absent in inert additives) that there may be excitation contributions distributed all over the volume of the acoustic cavity. Because the distribution of the unburned powders in the cavity may be controlled to some extent by controlling the size distribution of the reactive powders, could we take advantage of the situation?

The effect of the steady-state temperature (or the acoustic wave speed) variation in the burned gas is not a simple analytic problem. A prelimi- nary investigation was reported by Cheng, 2~ based upon his phenomenological model. A small temperature increase of the burned gas away from the propellant surface serving as a radial boundary was considered. The qualitative effect of the reactive additives through the steady-state temperature variation depends on the geometry

of the acoustic chamber and the location of the propellant surface (naturally of the nozzle as well).

The effect of the distributed oscillatory heat sources was studied through a one-dimensional model with the heat source response function

/dp represented through an interaction index m' and a time lag r ' different from the corresponding values m and r for the basic propellant (which need not be the same as their values without the additive). An arbitrary one-dimensional distribution of the steady state heat source ( ~ arbitrary variation of the powder sizes) is treated. The following conclusions are obtained. 25 (1) The resultant excitation of the system is the algebraic sum of the excitation due to the mass sources on the burning surface of the propellant and the excitation due to the distributed heat sources. (2) The excitation of each individual heat source (considered as a ~-funetion) is proportional to the square of the local amplitude of the standing acoustic pressure oscillation. (3) The excitation of the heat sources due to the additives and that of the mass sources of the basic propellant are out of phase in general. The phase difference is the product of the frequency and the difference in time lags, (i.e., the difference in the phase lags of the response functions to the local pressure oscillations).

The last conclusion is particularly significant, because it implies the possibility for the escilla- tory heat release of the reactive additives either to counteract or to augment the excitation contribution of the basic propellant in stimulating unstable escillations in the rocket motor. The second conclusion indicates: where the reactive powders should burn relative to the standing acoustic wave form in order to achieve the maximum effect which may be stabilizing or destabilizing. These observations indicate strongly the necessity of discriminating the acoustic situation in interpreting the over-all stabilizing or destabilizing effects of the reactive additives. Whether or not the burning characteristics of the basic propellant is-essentially modified by the presence of the additives is a separate matter.

The observation from the acoustic analyses that the reactive additives distributed over the volume can be destabilizing as well as stabilizing depending on circumstances, should serve some


warning against the complacency that aluminum powder is a universal cure. To illustrate this point, consider the ideal condition that the addition of aluminum powders renders the basic propellant completely insensitive to the gas oscillations in giving off the burned gas. If the heat release of the aluminum powder should be coupled to the gas oscillations in the chamber, this coupled oscillatory heat addition alone may excite an unstable acoustic mode of its own choice according to the Rayleigh criteria. If the over-all time lag of the heat release response function of the reactive additives should be comparable with the residence time of burned gas in the rocket chamber (the mean rate of burned gas generation divided by the mean mass-flow rate out of the chamber), very low frequency oscillations like the chugging in the liquid-propellant rocket motors, 2° may be encountered. Acoustic oscillations of high frequency may also be excited thermally if the time condition is met.

The residual or incompletely burned propellant performs the function of oscillatory mass addition distributed over the chamber volume, and may contribute to the excitation if properly timed like the oscillatory heat sources from additives. In a composite propellant with very coarse grains, the over-all time lag of tile residual propellant might be large enough to fall into the gas residence time in a large rocket chamber. A chugging-type instability might also be excited in solid-propellant rocket motors with conventional propellants. Chugging is a limiting ease of acoustic interaction. The intermediate frequency oscillations in between the chugging frequencies and the acoustic frequencies as a result of the acoustic reflection from the entropy waves 2° might also be a remote possibility. The acoustic situation in a rocket motor is just too complex. One must be very cautious to infer the stability behavior of such a complex system from a single element, fundamental as is the burning-rate response function.

Conclusions

Research activities during the past five years have made definite progress in the understanding of the problem of combustion instability in a solid-propellant rocket motor.

The analyses of propellant resonance, and of inherent stability lend substantiation to the con- viction that the variation of the burning rate in response to the oscillations of the heat transfer

rates to the solid propellant is the important excitation mechanism. The analytic determination of tile burning rate response function based on a rather comprehensive model has demonstrated tile possibility of correlating this excitation contribution with the familiar, physical and chemical parameters. The demonstration of the lesser importance of the oscillation of the drifting velocity compared with the oscillations of the pressure-like state variables, gives more confi- dence in presuming the burning rate response function as the important propellant characteristics. The analyses of the effect of propellant compressibility, tile estimate of the dissipative effect of the acoustic oscillation without and with additives, the effects of the reactive additives in distorting the acoustic wave form and in provid- ing additional excitation, also contribute in one way or another, to the fundamental understanding of tile problem.

In spite of all of these achievements, our understanding of the complete problem is still rather meager. The necessity of considering the stabil- i ty behavior of the system as a whole imposes a rather formidable problem to put all of the pieces of information together in a tractable form. Carefully designed experiments to determine the burning-rate response function of a given propel- ant is vitally needed. If it should turn out to be at all possible, to represent it in phenomenologieal terms, we would prefer to do so in order to facilitate the analysis of tile over-all problem. No matter how comprehensive the combustion model may be, there will always be phenomenologieal quantities involved in the burning-rate response function. The crucial question is how accurately we can determine those phenomenologieal quantities and how conveniently they may be used in the system analysis. This must be resolved by the concerted efforts of both the theoretieians and the experimentalists.

Acknowledgment

The author wishes to express his appreciation of the support of the Office of Ordnance Research, U. S. Army under Contract USA-OOR-DA-36 034 0RD-2183.

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1. SMITH, R. P., AND SPRENGER, D. F.: Fourth Symposium (International) on Combustion, p. 893. The Williams & Wilkins Company, BMtimore, 1953.

96 SURVEY PAPERS

2. PRICE, E. W., AND SOFFERIS, J. W.: Jet Pro- pulsion, 28, 190 (1958).

3. GECKLER, R. D.: Fifth Symposium (Interna- tional) on Combustion, p. 29. Reinhold Pub- lishing Corporation, New York, 1955.

4. VON KXRM.~N, T., WITH MILLAN, G., AND PENNER, S. S.: Sixth Symposi~tm (Interna- tional) on Combustion, p. 1. Reinhold Pub- lishing Corporation, New York, 1957.

5. GRAD, H.: Comm. Pure and Applied Math., 2, p. 79-102, 1949.

6. CHENG, S. I.: Jet Propulsion, 24, Part I, p. 27; Part II, p. 102, (1954).

7. McCLURE, F. T., HART, R. W., AND BIRD, J. F.: Progress Series in Astronautics and Rocketry: Solid Propellant Rocket Research, Vol. I. Academic Press, Inc., New York, 1960.

8. HART, R. W., AND McCLURE, F. T. : J. Chem. Phys., 30, No. 6, 1801 (1959).

9. GREEN, L. J.: Jet Propulsion, 28, 386 (1958). 10. NACHBAR, W., AND GREEN, L., JR.: J. I. A. S.,

26, 518 (1959). 11. SHINNAR, R., AND DISHON, M.: ARS Progress

Series in Astronautics and Rocketry: Solid Propellant Rocket Research, Vol. I. Academic Press, Inc., 1960.

12. HART, R. W., BIRD, J. F., AND McCLuRE, F. T.: ARS Progress Series in Astronautics and Rocketry: Solid Propellant Rocket Re- search, Vol. I. Academic Press, Inc., 1960.

13. PmCE, E. W.: Jet Propulsion, 30, 574 (1960). 14. GREEN, L. J., AND NACHBAR, W.: Jet Propul-

sion, 28, 769 (1958). 15. KUMAGAI, S., AND SAKAI, T.: Eighth Sympo-

sium (International) on Combustion, p. 873. The Williams & Wilkins Company, Balti- more, 1962.

16. MARKSTEIN, G. H.: Fourth Symposium (International) on Combustion, p. 44. The Williams & Wilkins Company, Baltimore, 1953.

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18. SUMMERFIELD, ~/L: J. Am. Rocket Soc., 21, 108 (1951).

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No. 8, Theory of Combustion Instability in Liquid Propellant Rocket Motors, p. 8, 160, 168, Butterworth's Scientific Publications, London, 1956.

21. TSIEN, H. S.: J. Am. Rocket Soc., 22,139 (1952). 22. EPSTEIN, P., AND CARHART, R.: J. Am. Acous-

tic Soc., 25, No. 3 (1953). 23. DELSASSO, L. P. Attenuation and dispersion of

sound by solid particles suspended in a gas. ASTIA Report No. AD-144 065, Department of Physics, University of California, Los Angeles. June, 1957.

24. RAYLEIGH, J. W. S.: Theory of Sound, Vol. II. Dover Publications, Inc., 1945.

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unstable combustion in solid-propellant rocket motors

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