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Oscillatory burning of solid composite propellants
OSCILLATORY BURNING OF SOLID COMPOSITE PROPELLANTS W. A. WOOD Analytical consideration of the interaction of an oscillatory pressure disturbance with the burning zone of a composite propellant leads to the conclusion that acoustic waves induce periodic variations in the ratio of the mass release rates of composite propellant components that give rise to harmonic composition and adiabatic flame temperature fluctuations in the gases above the propellant. These mass-release-rate, composition, and flame-temperature fluctuations may make large contributions to the amplification of acoustic waves by burning composite propellants. The magnitudes of these contributions to the admittance are particle-size-dependent and decrease as the ratio of particle burning time to oscillation period decreases. In contrast to homogeneous propellants, the amplification tendency of composite propellants is not expected to be a strong function of the burning rate pressure index of the over-all propellant. On the other hand, composition and flame-temperature contributions to the admittance may obtain for double-base propellants at high frequencies. Experimental work shows that bands of varying luminosity are released from the burning surfaces of propellants that are burned under the influence of an oscillatory pressure. The release of these bands can be related to composition and temperature fluctuations in the gases, but as yet not unambiguously.
Introduction Oscillatory combustion and combustion instability problems associated with solid rocket propellants have existed since the early days of modern rocket technology. The fact that the subject has warranted discussion at two consecutive Combustion Symposia is in itself witness that the problems persist. At the last Symposium 1,~ a number of hypotheses concerned with the interaction of solid propellant combustion zones with acoustic disturbances were reviewed and compared. For the most part these hypotheses were based on response properties expected of homogeneous propellants and did not deal specifically with problems that stem from the heterogeneity of composite-type propellants. Composite propellants consist of oxidizer and additives embedded in a continuous binder phase, usually a polymeric fuel. Owing to the nature of preparation of these propellants, the oxidizer particles are distributed randomly throughout the binder. Since at least two discrete phases having different regression characteristics are present, these propellants may not be expected to respond to acoustic disturbances as do homogeneous propellants of the type described by H a r t and McClure. 3 Approaches to the composite problem have been made by Cheng, ~ who considers a two~omponent system with each component assigned an 335
arbitrary response function, and Barr~re, 4 who applies distributed reaction time lags to a Greentype theory. 5 The present discussion advances the multi-component picture to include the effects of time-dependent variations in the composition of the flame zone brought about by differences in the regression rates of the individual components of the propellant and the implications of their subsequent effects on the acoustic admittance of the burning zone.
Response at Low Frequencies The surface and flame-zone properties of a composite propellant burned under the influence of a very-low-frequency acoustic disturbance can be represented adequately by a series of steadystate (constant-pressure) models that span the pressure range of the acoustic disturbance. During steady-state burning of one face of a semi-infinite slab of composite propellant the ratio of the mass release rate of oxidizer to that of the binder, ~bo/ff~,, is equal to the ratio of the weight fractions of these components, W,,/Wb, incorporated in the propellant. The mass release rate of a component may be expressed in terms of its condensed phase density, p, its surface area, S, and its instantaneous regression rate, r, to give r
w-; =
S~Ooro =
Wo (1)
336
COMBUSTION INSTABILITY
If the propellant is burned under the influence of a very-low-frequency acoustic disturbance, both the surface area and regression rate will adjust so that Eq. (1) will be satisfied at all times.
Response at High Frequencies The response of a composite propellant to high-frequency disturbances cannot be represented by a series of steady states. However, some insight into the problem may be obtained through examination of the general stoichiometry relationship expressed by Eq. (1). At high frequencies only the burning rates of the constituents will respond to the pressure fluctuations3; their surface areas will be invariant. The Sample surface may exhibit irregularities comparable in size to the larger oxidizer particles incorporated in the propellant, or roughly 20-200 microns. The burning rates of oxidizer and binder are generally less than 1 cm per second at 5001000 psi pressure; hence, during the period of a half-cycle of an imposed acoustic oscillation, the surfaces of each regress by approximately (5 X 10a/frequency in cps) microns (or 0.5 # at 104 cps) or less. At high frequencies these regression distances are small compared to the surface irregularities and the surface areas of both oxidizer and binder should remain essentially constant throughout a cycle. If a small change in pressure, dP, due to the acoustic disturbance,~effects changes dro and drb in the regression rates, then the following equation results for stoichiometric flow:
Sopo(ro ~- dro) _, Soporo SbPb( rb + drb) Sbpbrb
(2)
Elimination of terms leads to d In ro = d In rb
(3)
From Eq. (3) it is readily seen that stoiehiometric flow will occur only in the unique case in which the relative changes in the regression rates of the oxidizer and binder are equal, i.e., their pressure responses are identical. In general this similarity of burning rate response of the components is not expected since they may differ considerably in their chemical and physical natures, and their surfaces will be subjected to different temperature and environmental regions of the propellant combustion zone. As a result of these differences, nonstoichiometric flow should obtain and give rise to time-dependent fluctuations in composition and adiabatic flame temperature of the gases above the propellant surface. Of importance in most rocket work are the conditions under which burning goes from stoiehiometric to nonstoichiometric and the
magnitude of surface and composition effects on the admittance of a propellant. In line with this, a quasi-steady-state analysis of the interaction of a composite propellant burning zone with an acoustic disturbance is presented in the following section. Although this analysis will break down at moderate frequencies, it points out many essential features of the problem.
Response of a Composite Propellant Combustion Zone to a Acoustic Disturbance The reaction of a boundary to an acoustic disturbance may be presented in terms of the specific acoustic admittance at the boundary. An acoustic disturbance incident from the gas side (positive x side) on a boundary will be reflected with increased intensity if the real part of the admittance at the boundary is negative and attenuated if positive. 3,6 The admittance is given by Y = - - d V J d P where P is the pressure at the boundary and V~ the normal velocity of the gas into the boundary. The problem, therefore, lies in deriving an expression to relate the gas velocity at the combustion zone boundary to the acoustic pressure at the boundary. A model representative of the burning of an ammonium perchlorate-containing plastisol nitrocellulose propellant has been set up whose principal assumptions are: 1. The combustion of the propellant is considered a three-step process. The first two steps are independent volatilization and primary combustion of (i) the binder and (ii) the oxidizer. The third step is comprised of secondary combustion reactions between oxidizer and binder primary combustion products. 2. Linear regression rates of the surfaces of both the oxidizer and the binder are considered independent of the secondary combustion reactions. These rates are each represented by a burning rate law of the form r = L P q where L and q are constants and P the pressure. 3. The binder surface remains planar; therefore, its average surface area is constant. The regression rate of the binder is set equal to that of the propellant: It will be shown later that these restrictions may be relaxed without seriously affecting the results. 4. Oxidizer particles are spherical and deflagrate in a narrow zone adjacent to the solid propellant surface. The rate of release of these particles into this deflagration zone is governed by the regression rate of the solid propellant surface. 5. Combustion reactions between oxidizer and binder decomposition products are rapid compared to the period of the pressure oscillation. 6. Only surface-average properties are con-
337
O S C I L L A T O R Y B U R N I N G OF C O M P O S I T E P R O P E L L A N T S
sidered. Compositional fluctuations within the propellant are small and random. At any time during the deflagration of a composite propellant the total mass flow rate, w, may be expressed as r = Wb + wo
(4b) So.,)Bpmpo
f'o
BPmdt
t~ -- tf < r
(8)
f
=0
L:--U>r
Equations (7) and (8) may be solved analytically, omitting higher-order terms of e, in the following manner. Integration of Eq. (8) gives
(4c)
where ~bo., and So.~ are the mass flow rate and oxidizer surface al~a required to give stoichiometric burning of the binder, CP ~ is the regression rate of the binder (and propellant surface), and B P m is that of the oxidizer. The first two terms of Eq. (4c) represent the steady-state mass flow rate of the propellant and may be replaced by SpCP"pv where Sp is the cross-sectional area of the propellant over which So, So, and So.~ are determined and pp is the density of the composite propellant. If the combustion products follow ideal gas behavior then the gas velocity may be written as V
R = R, --
(4a)
= Wb qt_ tbo,. + 'tbo -- Wo,, = SbCP"pb + So,.BP"po + (So-
R may be determined from the following equation
R=
R~(t]_
tc--~)
T
me [-exp (--~ty) -- exp ( - - ~ t ~ ) ]
(9)
where ~ -- R~/B/5 m = average burning time of a particle. The variable limit t~ -- r of Eq. (7) may be replaced by a constant limit t~ -- ~ since Eq. (7) has the form
fJ ( t g - f ) f(t/) _to
dtf =
f:;_j(,,)
dtf
KT S ~ - P [SpCP"pv + (So -- So,.)Bpmpo]
tc" ~
(~) where M and T are the molecular weight and temperature of the gases and K is the gas constant. In order to determine the admittance, all variables of Eq. (5) must be cxpresse~t as functions of time. This is facilitated by imposition of a sinusoidal pressure fluctuation, of small amplitude e, normal to the propellant surface. The pressure fluctuation will be written in the form P = t5(1 -[- e cxp --/cot)
(6)
Surface-Area Relationships. The surface area, So, of all deflagrating oxidizer particles at some time Q may be obtained by integration over the interval r (where v is the burning time of a particle that terminates combustion at G) of the product of the surface area at tc of particles released from the propellant surface at prior time t f and the rate of release of particles at t]: So =
f(~
c--r
and, by the mean-value theorem, this may be expressed as
c
Consequently Eq. (7) reduces to So =
"[
47rR 2
>( {NSpC/5"['I -~- eexp (-/r
dt/
(10)
since [4~R~J{NSpC/5"Et + e exp (--/~tf)]"}D -- ~] = 0
attf=
t~-- ~.
Integration of Eq. (10) yields So -- So f t ~- 3e exp (--~.ot) (m -- u) [ i
[ 4r R 2-]
X [-YSvC/sn(1 -~ e exp --i~tf) n-] dO
f
(1
2i
2i
2 exp (/oJ~))]
(11)
(7)
R is the radius at t~ of a particle released at tf, N is the number of oxidizer particles per unit volume of composite propellant, and C/5"(1 -[- e exp --ixotf) n is the burning rate of the propellant at time tj.
where ~o =
47rN SpCD" R~3 3B/5m
Here the subscript has been dropped from t, since tc may have any preassigned value.
338
COMBUSTION INSTABILITY
The oxidizer surface area So,, required for stoichiometric burning at time t is calculated from the oxidizer surface area which would exist under a constant pressure equal in magnitude to the instantaneous pressure; i.e., So,, is So evaluated at P rather than at P . Therefore
S o . , - 47rNS~CP"R'3 3B P "
(12)
and for small values of e may be written
So,, = So[1 -- (m -- n)eexp ( - - / w t ) ]
(13)
The relationship between So and So,, is made clear by consideration of the variation of So with time at various values of w+. So is given by the real part of Eq. (11) which is So
=
~o[I Jr 3e(m -- n)fl]
where #=--eos~t
~
1
w+ / ]
proaches the surface area required for stoichiometric burning, S .... since # --* - - { cos ~t = Re [---{ exp (--/wt)]. In this limiting case the composite propellant behaves as a homogeneous propellant. As ~+ increases the oxidizer surface area cannot adjust fast enough to maintain stoichiometric burning. In the limit, as o0+ --+ oo the oxidizer surface area remains constant. Here #--* 0 and the instantaneous surface area, So, approaches the surface area, So, that obtains under constant pressure conditions at the pressure /5. How the oxidizer surface area fluctuations vary between these limits is shown in Fig. 1. The difference between the instantaneous oxidizer surface area and that required for stoichiometric burning is of importance since fluctuation of this difference (So -- So,,) gives rise to corresponding fluctuations in composition and temperature of the burning zone above the propellant surface. This difference function, given by
So
-
So.,
-
2 (1--eosoJ+)]
3(l
603~.3
At low values of oJ+, i.e., at low frequencies or short particle burning times (small particle sizes), the cosine term predominates. As o~ --~ 0 the instantaneous oxidizer surface area, So, ap-
-
n)~ exp (--#.~t)~/,,
(14)
2i
2
2exp
vanishes as o~r --* 0 and approaches a maximum value as w+ --* oo. The nature of this difference
LINE OF I
03
l
o
-
I
0
~ !A
So(m
where
--sin,.,t[~-
o3t
-=
'
LINE OF MAXI~
o_
,, /
+E
0"5 u ;IIU')"
-0.1
-E
=
T
~Oml
-E
-0.3
0
0.2
Q4
Q6
0.8
1.0
OOt 217 FIG. 1. Form of the deviation of the instantaneous oxidizer surface area from its mean over one pressure cycle.
o 'o:2 '0.4
0'.6
0'.8
,.o
~t
27/ Fro. 2. Form of the deviation of the instantaneous oxidizer surface area from the stoichiometric requirement over one pressure cycle.
339
OSCILLATORY B U R N I N G OF COMPOSITE P R O P E L L A N T S
function is shown in Fig. 2, in which Re [(So -- So,.)/3e(m -- n)So] is plotted versus r
Molecular Weight and Temperature Fluctuations. Compositional fluctuations that arise when So # So,. give rise to variations in M and T that may be expressed as M
(0 In M.~dW~ ] = M. L1 + k0-gi~o /
(15)
and
The pressure oscillation will also produce variations in M and T; however, changes of M will in many cases be small and here will be neglected. Following Hart and Cantrell, ~ the temperature fluctuation due to the acoustic wave may be written as T~O exp (--/wt), where T,O is the temperature fluctuation due to a pressure variation P+ when burning is stoichiometrie. Since only small fluctuations are considered the effects of composition and pressure changes will be considered independent of each other and Eq. (16) expands to
[0 In T,\ T = T, 1 + 0 exp (--/~t) + ~ - ~ - ) d W o T = T, 1 + \
OWo ]dW~
(16)
where M, and T, are the molecular weight and temperature for stoiehiometric burning at P = / 5 .
dWo=
(17) Consideration of only small fluctuations permits
dWo to be written as
So(m -- n)e exp (--ioot)~bBPmpo SpCP~p~ + So(m -- n)~ exp (--io~t)~bBpmpo
_ ~bo-- ~2o,.
Wo -t- Wb
-]
I
-
Setting 0 In MJO Wo and 0 In TJO Wo as constants ~band K, respectively, and substituting the foregoing expressions in Eqs. (15) and (17) gives
(is)
M = M , [ 1 + 4~Wb S p C P ~ ( ~ --~o~mn)e~bBP' n-~e~po--i~t~xp p~ --~i] So(m -- n)e~bBP~po exp , / w t
Determination of the Admittance. Since all variables of Eq. (5) are now expressible as functions of time, the admittance may be determined. Substitution for (So -- So.,), M, and T from Eqs. (14), (18), and (19), respectively, followed by expansion of the resultant expression and retension only of terms to first order in ~ gives the gas velocity
V-
KT, { SvCP,,ppE 1 ._[_0 exp (--/cot)] SpM,P + [-1 + Wb(K -- ~b{1 + 0exp --/cot})
+ o exp -io, tX~o(m - n~r
exp - i , oq}
(2o) The admittance,
Y = --dV/dP = -- (dV/dt) (dt/dP), may be determined from Eq. (20). For those cases in which the thickness of the combustion zone is small compared with the wavelength of
]
(19)
the oscillation
Y
KT.
S p M f ~ { SpCi'nppE(O/e -- 1) + n + Zx3 + So(m -- n)poBD~b X E l + Wb(K--~b) + Z , - ] }
(21)
where E1 and 2~2 are terms that contain 0 and in their numerators only. The term 0/~ -- 1 will generally lie between --1 (the value proposed by Summerfield et al.s for the limiting low-frequency case where dT/dP = 0) and --1/~, (the value obtained by Hart and McClure 3 for the isentropic limiting case where
dT/dP = ['1 -- I/7-]T/P, with ~/ = %/c~). Hence 0 < e(1 -- 1/~,) ~ 0.2~. As e approaches zero both 2h and Z2 approach zero and Eq. (21) becomes Y = (?/P)E(1
- 0/e) - n - ~]
(22)
340
COMBUSTION INSTABILITY
1.0
.8
.2
0.I
o.oi
caY 2/r
I
I0
FIG. 3. Real part of r as a function of the acoustic frequency and average particle burning time. where
M,P and --
WoEm
-
n 3 E l 4-
-
The sign of ~, t h e contributions to the admittance by fluctuations in oxidizer surface area, gas composition, and flame temperature is of importance, since an acoustic disturbance will be amplified if Re [ Y] is negative and attenuated if Re [ Y ] is positive. The term [1 + Wb(~ -- ~b)] will generally be positive for high~nergy oxidizers currently employed in solid propellants. Re [-~b] varies from + 1 (large values of w?) to zero (small values of w?) as shown in Fig. 3. Consequently the sign of Re [~] is determined by the difference between the pressure exponents of the
regression rate relations of Che constituents. If m > n the amplification tendency of the propellant will be increased and if m < n it will be decreased. I t is possible to make an order-of-magnitude estimation of the effect of ~ on the admittance. Sibbett 9 has found values of B = 0.007 era/seearm. and m = 1.2 for ammonium perchlorate. Values of other constants for a typical ammonium perchlorate-containing plastisol nitrocellulose propellant are n = 0.6, T = 1.2, Wo = 0.35, = 1.03, and r = 0.53. Using these values along with values of Re [~b] from Fig. 3, one obtains the results shown in Table I for a sample case at a mean pressure of 1000 psig fluctuating at 1000 cycles per second. These calculations show that ~ may make a significant contribution to the admittance and that a decrease in w? may effect a change in sign of the real part of the admittance. These results are especially important since perturbations in the burning rates of the solid components due to composition-induced temperature fluctuations and to acoustic oscillationinduced temperature and radiation fluctuations are not included and should in many cases cause an increase in Re [$]. I t is pertinent to point out that a decrease in Ri and, consequently, in w? (where w is held constant) will not necessarily lead to a reduction in the amplification tendency of a burning propellant. The real part of ~ is relatively insensitive to changes of w? if w? > 2r, whereas V may increase considerably with an oxidizer particle size decrease and thus lead to an increased value of I R e [ Y ] l .
Effect of Variable Binder Surface Area. Up to this point, the binder has been assumed to exhibit a flat surface and to have the same regression rate as the propellant. This restricts the interpretation of the results since individual effects of binder and propellant burning proper-
TABLE 1 Example of effect of oxidizer particle size or acoustic frequency on the dimensionless admittance Dimensionless admittance
~/2~-
Ri (microns)
Re (~)
Low-frequency limit [1-n-Re ( ~ ) ]
Isentropic limit [(1/~.)-n-Re (~)]
2.7 1.8 0.9 0.45
30 20 10 5
0.28 0.27 0.23 0.13
+0.12 -~0.13 -{-0.17 -k0.27
--0.05 --0.03 0.0 -{-0.10
OSCILLATORY BURNING
OF COMPOSITE PROPELLANTS
ties on the admittance are indistinguishable. In order to estimate the potential contribution of the binder, the admittance function has been rederived for the case in which the binder and the propellant have different regression rates and in which the binder is assigned a fluctuatingsurface-area function. I n this case both binder and oxidizer are assumed to burn as particles and the binder, oxidizer, and propellant regression rates are given by D P ~, B P " , and CP ~, respectively. Analysis of this example by the procedures already demonstrated leads to the expression
341
from steady-state behavior. The theoretical treatment of H a r t and McClure ~ indicates that, at low frequencies and low amplitudes, heat and mass transfer processes in homogeneous propellants approach their steady-state values. F o r composite propellants, we expect a similar behavior; however, deviations from steady-state behavior may appear at lower frequencies than those expected of homogeneous propellants. The effects which may ensue from the acoustic oscillation on the instantaneous flame temperature and on the radiant heat exchange between the gases and the propellant have been neglected. The former should increase in significance with Y = (?//5)[-(1 -- O/~) -- n -- ~o.bJ (23) increase in d P / d t ( dP/dt = --i~Pe exp --/cot); hence, it should increase with increase of either where frequency or oscillation amplitude. Radiation ? = ( K T s / M , P ) CPnpp effects are geometry dependent; however, in motors with large cavities they should increase a,b = (y - n) (Wb,h + Wo~o) with increase in amplitude and decrease in frequency. At high frequencies and short wave-k- Wo[-1 -k- Wb(K -- 4a)3 lengths the propellant surface sees only an averX [-(m -- n)~o -- (y -- n)~bb7 age gas temperature, whereas, at low frequencies, and ~boand ~bbare values of ~b for the oxidizer and the surface sees a gas temperature that varies in phase with the pressure. While the influence of binder, respectively. As W~o and ~ b approach zero, ~bo and tb ap- instantaneous variations in gas composition on proach zero and the admittance equation reduces the flame temperature~ has been considered in the model, the potential effects of such changes to that for a homogeneous propellant. in composition and temperature on the regresf = (I?//5)[-(1 -- O/e) -- n-] (24) sion rates of binder and oxidizer have not been included, and may alter the calculated results in At high values of both r and w~b, ~bo and ~bb either a positive or negative direction. both approach 1 and the admittance is given b y At high amplitudes and/or high frequencies there is a possibility that the compression-inr = ( ? / P ) [ - ( 1 -- Ole) -- W~y -- Worn duced temperature variations will couple with -- WoWt,(~ -- 4))(m -- y)] (25) those arising from changes in gas composition, resulting in appreciable fluctuations in gas-phase This relationship shows that the sign of the ad- reactions and in the regression rates of the conmittance depends strongly on regression rate stituent solids. Since dP/dt maxima precede those pressure exponents y and m of the binder and of P by 90 ~ and the composition-induced flame the oxidizer and that the burning rate pressure temperature fluctuations for the case calculated exponent of the propellant, n, enters only through are either in phase with or lead the pressure the term V. This result is noteworthy in view of waves, the coupling of the two thermal effects the fact that heretofore no correlation between would result in increased reaction and regression n and the extent of instability shown by pro- rates as pressure increased, followed b y depellants has been reported, although theoretical creased rates as pressure decreased. treatments of homogeneous propellants indicate In omitting the interaction of the gas composithere should be such a correlation. tion and flame temperature fluctuation with the regression rates of the propellant constituents, Discussion. The model used here neglects several the indicated method of calculating stoichiofactors that can become important during oscil- metric effects on the admittance becomes indel a t o r y burning of composite propellants. pendent of the burning zone thickness as long as Time-dependent fluctuations in heat and mass this is small relative to the acoustic wavelength. transfer and in the rates of chemical reaction and The acoustic perturbations in turn are congaseous diffusion in the propellant combustion sidered sufficiently small that the thermodynamic zone are not included. At low frequencies, chemi- properties of the combustion zone vary linearly cal reaction and gaseous diffusion times should with respect to composition. Although highbe small compared with the oscillation period speed motion pictures indicate particle burning and consequently cause only small deviations as far as 1 em from the propellant surface, the
342
COMBUSTION INSTABILITY
wavelength of the acoustic oscillation at, say, 1000 cps is 120 cm. On the other hand, calculation of the amplitudes of thermal fluctuations that arise from compositional variations in the gases may be considerably in error because the thickness of the particle-burning zone is, in general, larger than the "wavelengths" associated with these thermal fluctuations. These wavelengths are strongly dependent on frequency, propellant regression rate, and chamber pressure. As an example, a propellant burning at 189 cm/sec at 1000 psi exhibits a thermal "wavelength" of 0.3 to 0.4 cm at 1000 cps. Since most deviations from the constant-pressure regression rate relationships discussed above become more important at high frequencies, the model should be more representative of composite propellant burning at low rather than at high frequencies. Even at high frequencies, however, composition and temperature effects are expected. Although the present treatment deals with true composite (i.e., polyphase) propellants the same arguments may be applied, at least hypothetically, to so-called homogeneous propellants, in which during deflagration a liquid zone is predicated, 10 from which the constituents may volatilize at rates that have different pressure indices. If each of N components is assigned a mass vaporization rate of the form L P q gm sec-~ cm-~, where q is a constant and L is determined from stoichiometric relationships that must exist under constant-pressure deflagration, then the admittance is given by the equation Y = (V//5){(1 -- 0/e) -
~ ['Wjqr
-- (K~"-- ~bj)(1 -- W i ) ( W j ) ( q i -- q~)']}
(26)
where 0 In T~ ,,j -
4,~ =
owj
0 In M,
~
owj
Consider J P N propellant, a double-base formulation u containing about 51% nitrocellulose (NC), 43% nitroglycerin (NG), 3% diethyl phthalate (DEP), and 1% ethyl centralite (EC). Values of Kj and r may be determined graphically from theoretical curves of T, and M, as functions of Wj.. The ~ -- ~b differences for EC, DEP, NG, and NC, are 0.7, 0.6, 0.4 and 0.1, respectively; these are the same order of magnitude as the value of 0.5 estimated for the composite propellant example.
Experimental Experiments have indicated the existence of thermal waves in the combustion zone above a
propellant surface. Streak photographs taken of the gaseous region above propellant samples burning under the influence of imposed acoustic oscillations show the periodic release of luminous bands of gases parallel to the propellant surface varying in intensity in regular fashion. Apparatus. A schematic diagram of the experimental apparatus is shown in Fig. 4. A modified T-motor 1~ was used as an acoustic driver. This motor generates acoustic oscillations primarily in the fundamental longitudinal mode across the arms of the T, the frequency of which may be varied by changing this length. The sample propellant, an ammonium perchlorate-containing plastisol nitrocellulose type, was a 1.8-inchdiameter cylindrical waver approximately 88inch thick. The windows of the sample section were formed from ~-ineh-thick Herculite or ~--inchthick colorless synthetic sapphire. The photographs were taken with a Beckman and Whitley Dynafax high-speed framing camera, modified for operation as a streak camera. The acoustic pressure was measured by means of a straingauge transducer located near the propellant surface and oriented parallel to the direction of the gas flow. Correspondence between the pressure traces and the streak photographs was obtained by means of an exploding wire that simultaneously put a spot on the film and effected a discontinuity in the pressure trace. Results. Streak photographs (Fig. 5) show two distinct patterns of radiant energy emission. The first is an oscillation in the over-all luminosity of the gases, of frequency equal to that of the driver pressure. From correlation of the pressure and streak traces, it has been found that this radiant oscillation is in phase with the acoustic oscillation within 4-20 ~ at 500 cps. The second pattern is evidenced by the release of bands of varying luminosity, parallel to the propellant surface. Since this release occurs at the same frequency as the acoustic oscillation the separation of bands of similar intensity can be related directly to the deflagration rate of the propellant, the average pressure, and the frequency and amplitude characteristics of the imposed acoustic oscillation. The phase relationship between the emergence of the luminous bands and acoustic pressure appears to change with experimental conditions. Data at 500 and 200 cps, at average chamber pressures of 400 and 800 psi, for an ammonium perchlorate-containing plastisol nitrocellulose composite propellant, indicate that the release of luminous bands from the propellant surface generally precedes the pressure maximum, by approximately 180 ~, and that the phase difference decreases with increase in frequency and decrease in pressure.
OSCILLATORYBURNINGOF COMPOSITEPROPELLANTS
A N P R T O L P E L~
~ WINDOW/---
SAMPLE 7Y
II
~
343
/:-
STREAK
C_~M~ERA
~-PRESSURE GAUGE I SHUTTER CAMERAt ACTIVATOR ~"'--DRIVER MOTOR GRAIN .
t
DELAYUNIT I I
FIRINGCIRCUITI
/
L T O MOTOR IGNI'T'KRS
tC~CILLOSCO~ (SINGLESWEiP)]~OSCILLOSCOPE ~.,CAMEP,A
f
FIG. 4. Schematicof apparatus for observingpropellant combustion in a T-motor.
3
cm{Z
P~V I~1t""
~9 IO
TIME
D
I~I I'~I~" Y
-"
I
I
iO-a SEC.
O n," 13. - PROPELLANT
SURFACE
FIG. 5. Typical streak photograph of the gaseous region above a propellant sample burned under oscillatory burning conditions.
344
COMBUSTION INSTABILITY
Discussion. There are at least three explanations for bands of varying luminosity in the gases above a propellant that is burned under the influence of an oscillatory pressure fluctuation. The first involves the release of strata of gases of varying composition and temperature discussed earlier. Either an increase in average pressure or a decrease in frequency reduces the burning time of ammonium perchlorate particles relative to the period of an imposed oscillation and results in increased phase lags between the thermal and pressure waves. The second involves particle release from the surface. Coupling of the propellant burning rate to the acoustic pressure can lead to an oscillatory rate of particle release and consequently give luminous bands in the gases above the surface corresponding to particle concentration gradients. The third, postulated by Summerfield et al., s is based upon the difference in pressure-volume work done by the acoustic oscillation on propellant combustion products having flame temperatures independent of pressure. This mechanism gives rise to thermal waves that are 180 ~ out of phase with the pressure at low frequencies. At the present time it is not possible to separate the individual contributions of the three mechanisms of luminous band formation described above. At 200 cps the phase relationship of luminous band release to pressure is close to that predicted from the pressure-volume work mechanism. The decrease in lead of luminous band release to pressure with increase in frequency and decrease of average pressure, however, could result from either composition-temperature or particle-release mechanisms or from a combination of the two. ACKNOWLEDGMENTS This program was conducted under Contract DA-01~321-ORD-12024 with the Office of the Chief of Ordnance, administered by the Army
Ordnance Missile Command. The author wishes to extend special acknowledgment to Dr. R. H. Cantrell of the Applied Physics Laboratory, The Johns Hopkins University, for suggesting the method of analytical integration of the surfacearea equations. REFERENCES 1. Panel Discussion: Solid Propellant Combustion Instability, Eighth Symposium (International) on Combustion, p. 904. Williams and Wilkins, 1962. 2. CHENG, S. I.: ibid, p. 81. 3. HART, R. W. and McCLURE, F. T.: J. Chem. Phys. 30, 1501 (1959). 4. BARR~RE~M. and BERNARD,J. J.: see reference 1, p. 886. 5. GREEN~L., JR.: Jet Propulsion 28, 386 (1958). 6. MORSE, P. M.: Vibration and Sound, p. 367. McGraw-Hill, 1948. 7. HART, R. W. and CANTRELL,R. H.: The Johns
Hopkins University Applied Physics Laboratory Report TG-335-11, April 1962. 8. SUMMERFIELD, M., WAESCHE, R. I~. W., and WENOGRAD,J. : Aeronautical Engineering Report No. 56~-b. Princeton Univ., Dec. 20, 1961. 9. SmSETT, D. J. and LOBATO,J. M. : Investigation
of the Mechanism of Combustion of Composite Solid Propellants, Aerojet General Corporation, AFOSR-TR-60--6$, 1960. 10. LEWIS, B., PEASE, R. N., and TAYLOR, H. S.: Combustion Processes, Vol. II of High Speed Aerodynamics and Jet Propulsion, p. 532. Princeton Univ. Press, 1962. 11. WIMPRESS, a . N.: Internal Ballistics of SolidFuel Rockets, p. 4. McGraw-Hill, 1950. 12. WooD, W. A. : Rohm & Haas Company Quarterly Progress Report No. P-61-7, Oct. 3, 1961, (Conf.). For original description see PRICE, E. W. and SOFFERIS,J.: Jet Propulsion 28, 109 (1958).
Discussion PROF. H. EMMONS (Harvard University): Dr. Wood's theory supposes oxidizer particles which burn as spheres. If both oxidizer and fuel particles were present would the r functions show two positions of rapid rise and the resultant burning show two regions of rapid admittance change?
DR. W. A. WOOD (Rohm & Haas): The case in which both oxidizer and binder burn as particles has been included in the written text. The frequencyaverage particle burning time r is a function of ( ~ ) only and may be used for either binder or oxidizer. If ~b differs sufficiently from ~o then the admittance will show two regions of rapid change.
Introduction Oscillatory combustion and combustion instability problems associated with solid rocket propellants have existed since the early days of modern rocket technology. The fact that the subject has warranted discussion at two consecutive Combustion Symposia is in itself witness that the problems persist. At the last Symposium 1,~ a number of hypotheses concerned with the interaction of solid propellant combustion zones with acoustic disturbances were reviewed and compared. For the most part these hypotheses were based on response properties expected of homogeneous propellants and did not deal specifically with problems that stem from the heterogeneity of composite-type propellants. Composite propellants consist of oxidizer and additives embedded in a continuous binder phase, usually a polymeric fuel. Owing to the nature of preparation of these propellants, the oxidizer particles are distributed randomly throughout the binder. Since at least two discrete phases having different regression characteristics are present, these propellants may not be expected to respond to acoustic disturbances as do homogeneous propellants of the type described by H a r t and McClure. 3 Approaches to the composite problem have been made by Cheng, ~ who considers a two~omponent system with each component assigned an 335
arbitrary response function, and Barr~re, 4 who applies distributed reaction time lags to a Greentype theory. 5 The present discussion advances the multi-component picture to include the effects of time-dependent variations in the composition of the flame zone brought about by differences in the regression rates of the individual components of the propellant and the implications of their subsequent effects on the acoustic admittance of the burning zone.
Response at Low Frequencies The surface and flame-zone properties of a composite propellant burned under the influence of a very-low-frequency acoustic disturbance can be represented adequately by a series of steadystate (constant-pressure) models that span the pressure range of the acoustic disturbance. During steady-state burning of one face of a semi-infinite slab of composite propellant the ratio of the mass release rate of oxidizer to that of the binder, ~bo/ff~,, is equal to the ratio of the weight fractions of these components, W,,/Wb, incorporated in the propellant. The mass release rate of a component may be expressed in terms of its condensed phase density, p, its surface area, S, and its instantaneous regression rate, r, to give r
w-; =
S~Ooro =
Wo (1)
336
COMBUSTION INSTABILITY
If the propellant is burned under the influence of a very-low-frequency acoustic disturbance, both the surface area and regression rate will adjust so that Eq. (1) will be satisfied at all times.
Response at High Frequencies The response of a composite propellant to high-frequency disturbances cannot be represented by a series of steady states. However, some insight into the problem may be obtained through examination of the general stoichiometry relationship expressed by Eq. (1). At high frequencies only the burning rates of the constituents will respond to the pressure fluctuations3; their surface areas will be invariant. The Sample surface may exhibit irregularities comparable in size to the larger oxidizer particles incorporated in the propellant, or roughly 20-200 microns. The burning rates of oxidizer and binder are generally less than 1 cm per second at 5001000 psi pressure; hence, during the period of a half-cycle of an imposed acoustic oscillation, the surfaces of each regress by approximately (5 X 10a/frequency in cps) microns (or 0.5 # at 104 cps) or less. At high frequencies these regression distances are small compared to the surface irregularities and the surface areas of both oxidizer and binder should remain essentially constant throughout a cycle. If a small change in pressure, dP, due to the acoustic disturbance,~effects changes dro and drb in the regression rates, then the following equation results for stoichiometric flow:
Sopo(ro ~- dro) _, Soporo SbPb( rb + drb) Sbpbrb
(2)
Elimination of terms leads to d In ro = d In rb
(3)
From Eq. (3) it is readily seen that stoiehiometric flow will occur only in the unique case in which the relative changes in the regression rates of the oxidizer and binder are equal, i.e., their pressure responses are identical. In general this similarity of burning rate response of the components is not expected since they may differ considerably in their chemical and physical natures, and their surfaces will be subjected to different temperature and environmental regions of the propellant combustion zone. As a result of these differences, nonstoichiometric flow should obtain and give rise to time-dependent fluctuations in composition and adiabatic flame temperature of the gases above the propellant surface. Of importance in most rocket work are the conditions under which burning goes from stoiehiometric to nonstoichiometric and the
magnitude of surface and composition effects on the admittance of a propellant. In line with this, a quasi-steady-state analysis of the interaction of a composite propellant burning zone with an acoustic disturbance is presented in the following section. Although this analysis will break down at moderate frequencies, it points out many essential features of the problem.
Response of a Composite Propellant Combustion Zone to a Acoustic Disturbance The reaction of a boundary to an acoustic disturbance may be presented in terms of the specific acoustic admittance at the boundary. An acoustic disturbance incident from the gas side (positive x side) on a boundary will be reflected with increased intensity if the real part of the admittance at the boundary is negative and attenuated if positive. 3,6 The admittance is given by Y = - - d V J d P where P is the pressure at the boundary and V~ the normal velocity of the gas into the boundary. The problem, therefore, lies in deriving an expression to relate the gas velocity at the combustion zone boundary to the acoustic pressure at the boundary. A model representative of the burning of an ammonium perchlorate-containing plastisol nitrocellulose propellant has been set up whose principal assumptions are: 1. The combustion of the propellant is considered a three-step process. The first two steps are independent volatilization and primary combustion of (i) the binder and (ii) the oxidizer. The third step is comprised of secondary combustion reactions between oxidizer and binder primary combustion products. 2. Linear regression rates of the surfaces of both the oxidizer and the binder are considered independent of the secondary combustion reactions. These rates are each represented by a burning rate law of the form r = L P q where L and q are constants and P the pressure. 3. The binder surface remains planar; therefore, its average surface area is constant. The regression rate of the binder is set equal to that of the propellant: It will be shown later that these restrictions may be relaxed without seriously affecting the results. 4. Oxidizer particles are spherical and deflagrate in a narrow zone adjacent to the solid propellant surface. The rate of release of these particles into this deflagration zone is governed by the regression rate of the solid propellant surface. 5. Combustion reactions between oxidizer and binder decomposition products are rapid compared to the period of the pressure oscillation. 6. Only surface-average properties are con-
337
O S C I L L A T O R Y B U R N I N G OF C O M P O S I T E P R O P E L L A N T S
sidered. Compositional fluctuations within the propellant are small and random. At any time during the deflagration of a composite propellant the total mass flow rate, w, may be expressed as r = Wb + wo
(4b) So.,)Bpmpo
f'o
BPmdt
t~ -- tf < r
(8)
f
=0
L:--U>r
Equations (7) and (8) may be solved analytically, omitting higher-order terms of e, in the following manner. Integration of Eq. (8) gives
(4c)
where ~bo., and So.~ are the mass flow rate and oxidizer surface al~a required to give stoichiometric burning of the binder, CP ~ is the regression rate of the binder (and propellant surface), and B P m is that of the oxidizer. The first two terms of Eq. (4c) represent the steady-state mass flow rate of the propellant and may be replaced by SpCP"pv where Sp is the cross-sectional area of the propellant over which So, So, and So.~ are determined and pp is the density of the composite propellant. If the combustion products follow ideal gas behavior then the gas velocity may be written as V
R = R, --
(4a)
= Wb qt_ tbo,. + 'tbo -- Wo,, = SbCP"pb + So,.BP"po + (So-
R may be determined from the following equation
R=
R~(t]_
tc--~)
T
me [-exp (--~ty) -- exp ( - - ~ t ~ ) ]
(9)
where ~ -- R~/B/5 m = average burning time of a particle. The variable limit t~ -- r of Eq. (7) may be replaced by a constant limit t~ -- ~ since Eq. (7) has the form
fJ ( t g - f ) f(t/) _to
dtf =
f:;_j(,,)
dtf
KT S ~ - P [SpCP"pv + (So -- So,.)Bpmpo]
tc" ~
(~) where M and T are the molecular weight and temperature of the gases and K is the gas constant. In order to determine the admittance, all variables of Eq. (5) must be cxpresse~t as functions of time. This is facilitated by imposition of a sinusoidal pressure fluctuation, of small amplitude e, normal to the propellant surface. The pressure fluctuation will be written in the form P = t5(1 -[- e cxp --/cot)
(6)
Surface-Area Relationships. The surface area, So, of all deflagrating oxidizer particles at some time Q may be obtained by integration over the interval r (where v is the burning time of a particle that terminates combustion at G) of the product of the surface area at tc of particles released from the propellant surface at prior time t f and the rate of release of particles at t]: So =
f(~
c--r
and, by the mean-value theorem, this may be expressed as
c
Consequently Eq. (7) reduces to So =
"[
47rR 2
>( {NSpC/5"['I -~- eexp (-/r
dt/
(10)
since [4~R~J{NSpC/5"Et + e exp (--/~tf)]"}D -- ~] = 0
attf=
t~-- ~.
Integration of Eq. (10) yields So -- So f t ~- 3e exp (--~.ot) (m -- u) [ i
[ 4r R 2-]
X [-YSvC/sn(1 -~ e exp --i~tf) n-] dO
f
(1
2i
2i
2 exp (/oJ~))]
(11)
(7)
R is the radius at t~ of a particle released at tf, N is the number of oxidizer particles per unit volume of composite propellant, and C/5"(1 -[- e exp --ixotf) n is the burning rate of the propellant at time tj.
where ~o =
47rN SpCD" R~3 3B/5m
Here the subscript has been dropped from t, since tc may have any preassigned value.
338
COMBUSTION INSTABILITY
The oxidizer surface area So,, required for stoichiometric burning at time t is calculated from the oxidizer surface area which would exist under a constant pressure equal in magnitude to the instantaneous pressure; i.e., So,, is So evaluated at P rather than at P . Therefore
S o . , - 47rNS~CP"R'3 3B P "
(12)
and for small values of e may be written
So,, = So[1 -- (m -- n)eexp ( - - / w t ) ]
(13)
The relationship between So and So,, is made clear by consideration of the variation of So with time at various values of w+. So is given by the real part of Eq. (11) which is So
=
~o[I Jr 3e(m -- n)fl]
where #=--eos~t
~
1
w+ / ]
proaches the surface area required for stoichiometric burning, S .... since # --* - - { cos ~t = Re [---{ exp (--/wt)]. In this limiting case the composite propellant behaves as a homogeneous propellant. As ~+ increases the oxidizer surface area cannot adjust fast enough to maintain stoichiometric burning. In the limit, as o0+ --+ oo the oxidizer surface area remains constant. Here #--* 0 and the instantaneous surface area, So, approaches the surface area, So, that obtains under constant pressure conditions at the pressure /5. How the oxidizer surface area fluctuations vary between these limits is shown in Fig. 1. The difference between the instantaneous oxidizer surface area and that required for stoichiometric burning is of importance since fluctuation of this difference (So -- So,,) gives rise to corresponding fluctuations in composition and temperature of the burning zone above the propellant surface. This difference function, given by
So
-
So.,
-
2 (1--eosoJ+)]
3(l
603~.3
At low values of oJ+, i.e., at low frequencies or short particle burning times (small particle sizes), the cosine term predominates. As o~ --~ 0 the instantaneous oxidizer surface area, So, ap-
-
n)~ exp (--#.~t)~/,,
(14)
2i
2
2exp
vanishes as o~r --* 0 and approaches a maximum value as w+ --* oo. The nature of this difference
LINE OF I
03
l
o
-
I
0
~ !A
So(m
where
--sin,.,t[~-
o3t
-=
'
LINE OF MAXI~
o_
,, /
+E
0"5 u ;IIU')"
-0.1
-E
=
T
~Oml
-E
-0.3
0
0.2
Q4
Q6
0.8
1.0
OOt 217 FIG. 1. Form of the deviation of the instantaneous oxidizer surface area from its mean over one pressure cycle.
o 'o:2 '0.4
0'.6
0'.8
,.o
~t
27/ Fro. 2. Form of the deviation of the instantaneous oxidizer surface area from the stoichiometric requirement over one pressure cycle.
339
OSCILLATORY B U R N I N G OF COMPOSITE P R O P E L L A N T S
function is shown in Fig. 2, in which Re [(So -- So,.)/3e(m -- n)So] is plotted versus r
Molecular Weight and Temperature Fluctuations. Compositional fluctuations that arise when So # So,. give rise to variations in M and T that may be expressed as M
(0 In M.~dW~ ] = M. L1 + k0-gi~o /
(15)
and
The pressure oscillation will also produce variations in M and T; however, changes of M will in many cases be small and here will be neglected. Following Hart and Cantrell, ~ the temperature fluctuation due to the acoustic wave may be written as T~O exp (--/wt), where T,O is the temperature fluctuation due to a pressure variation P+ when burning is stoichiometrie. Since only small fluctuations are considered the effects of composition and pressure changes will be considered independent of each other and Eq. (16) expands to
[0 In T,\ T = T, 1 + 0 exp (--/~t) + ~ - ~ - ) d W o T = T, 1 + \
OWo ]dW~
(16)
where M, and T, are the molecular weight and temperature for stoiehiometric burning at P = / 5 .
dWo=
(17) Consideration of only small fluctuations permits
dWo to be written as
So(m -- n)e exp (--ioot)~bBPmpo SpCP~p~ + So(m -- n)~ exp (--io~t)~bBpmpo
_ ~bo-- ~2o,.
Wo -t- Wb
-]
I
-
Setting 0 In MJO Wo and 0 In TJO Wo as constants ~band K, respectively, and substituting the foregoing expressions in Eqs. (15) and (17) gives
(is)
M = M , [ 1 + 4~Wb S p C P ~ ( ~ --~o~mn)e~bBP' n-~e~po--i~t~xp p~ --~i] So(m -- n)e~bBP~po exp , / w t
Determination of the Admittance. Since all variables of Eq. (5) are now expressible as functions of time, the admittance may be determined. Substitution for (So -- So.,), M, and T from Eqs. (14), (18), and (19), respectively, followed by expansion of the resultant expression and retension only of terms to first order in ~ gives the gas velocity
V-
KT, { SvCP,,ppE 1 ._[_0 exp (--/cot)] SpM,P + [-1 + Wb(K -- ~b{1 + 0exp --/cot})
+ o exp -io, tX~o(m - n~r
exp - i , oq}
(2o) The admittance,
Y = --dV/dP = -- (dV/dt) (dt/dP), may be determined from Eq. (20). For those cases in which the thickness of the combustion zone is small compared with the wavelength of
]
(19)
the oscillation
Y
KT.
S p M f ~ { SpCi'nppE(O/e -- 1) + n + Zx3 + So(m -- n)poBD~b X E l + Wb(K--~b) + Z , - ] }
(21)
where E1 and 2~2 are terms that contain 0 and in their numerators only. The term 0/~ -- 1 will generally lie between --1 (the value proposed by Summerfield et al.s for the limiting low-frequency case where dT/dP = 0) and --1/~, (the value obtained by Hart and McClure 3 for the isentropic limiting case where
dT/dP = ['1 -- I/7-]T/P, with ~/ = %/c~). Hence 0 < e(1 -- 1/~,) ~ 0.2~. As e approaches zero both 2h and Z2 approach zero and Eq. (21) becomes Y = (?/P)E(1
- 0/e) - n - ~]
(22)
340
COMBUSTION INSTABILITY
1.0
.8
.2
0.I
o.oi
caY 2/r
I
I0
FIG. 3. Real part of r as a function of the acoustic frequency and average particle burning time. where
M,P and --
WoEm
-
n 3 E l 4-
-
The sign of ~, t h e contributions to the admittance by fluctuations in oxidizer surface area, gas composition, and flame temperature is of importance, since an acoustic disturbance will be amplified if Re [ Y] is negative and attenuated if Re [ Y ] is positive. The term [1 + Wb(~ -- ~b)] will generally be positive for high~nergy oxidizers currently employed in solid propellants. Re [-~b] varies from + 1 (large values of w?) to zero (small values of w?) as shown in Fig. 3. Consequently the sign of Re [~] is determined by the difference between the pressure exponents of the
regression rate relations of Che constituents. If m > n the amplification tendency of the propellant will be increased and if m < n it will be decreased. I t is possible to make an order-of-magnitude estimation of the effect of ~ on the admittance. Sibbett 9 has found values of B = 0.007 era/seearm. and m = 1.2 for ammonium perchlorate. Values of other constants for a typical ammonium perchlorate-containing plastisol nitrocellulose propellant are n = 0.6, T = 1.2, Wo = 0.35, = 1.03, and r = 0.53. Using these values along with values of Re [~b] from Fig. 3, one obtains the results shown in Table I for a sample case at a mean pressure of 1000 psig fluctuating at 1000 cycles per second. These calculations show that ~ may make a significant contribution to the admittance and that a decrease in w? may effect a change in sign of the real part of the admittance. These results are especially important since perturbations in the burning rates of the solid components due to composition-induced temperature fluctuations and to acoustic oscillationinduced temperature and radiation fluctuations are not included and should in many cases cause an increase in Re [$]. I t is pertinent to point out that a decrease in Ri and, consequently, in w? (where w is held constant) will not necessarily lead to a reduction in the amplification tendency of a burning propellant. The real part of ~ is relatively insensitive to changes of w? if w? > 2r, whereas V may increase considerably with an oxidizer particle size decrease and thus lead to an increased value of I R e [ Y ] l .
Effect of Variable Binder Surface Area. Up to this point, the binder has been assumed to exhibit a flat surface and to have the same regression rate as the propellant. This restricts the interpretation of the results since individual effects of binder and propellant burning proper-
TABLE 1 Example of effect of oxidizer particle size or acoustic frequency on the dimensionless admittance Dimensionless admittance
~/2~-
Ri (microns)
Re (~)
Low-frequency limit [1-n-Re ( ~ ) ]
Isentropic limit [(1/~.)-n-Re (~)]
2.7 1.8 0.9 0.45
30 20 10 5
0.28 0.27 0.23 0.13
+0.12 -~0.13 -{-0.17 -k0.27
--0.05 --0.03 0.0 -{-0.10
OSCILLATORY BURNING
OF COMPOSITE PROPELLANTS
ties on the admittance are indistinguishable. In order to estimate the potential contribution of the binder, the admittance function has been rederived for the case in which the binder and the propellant have different regression rates and in which the binder is assigned a fluctuatingsurface-area function. I n this case both binder and oxidizer are assumed to burn as particles and the binder, oxidizer, and propellant regression rates are given by D P ~, B P " , and CP ~, respectively. Analysis of this example by the procedures already demonstrated leads to the expression
341
from steady-state behavior. The theoretical treatment of H a r t and McClure ~ indicates that, at low frequencies and low amplitudes, heat and mass transfer processes in homogeneous propellants approach their steady-state values. F o r composite propellants, we expect a similar behavior; however, deviations from steady-state behavior may appear at lower frequencies than those expected of homogeneous propellants. The effects which may ensue from the acoustic oscillation on the instantaneous flame temperature and on the radiant heat exchange between the gases and the propellant have been neglected. The former should increase in significance with Y = (?//5)[-(1 -- O/~) -- n -- ~o.bJ (23) increase in d P / d t ( dP/dt = --i~Pe exp --/cot); hence, it should increase with increase of either where frequency or oscillation amplitude. Radiation ? = ( K T s / M , P ) CPnpp effects are geometry dependent; however, in motors with large cavities they should increase a,b = (y - n) (Wb,h + Wo~o) with increase in amplitude and decrease in frequency. At high frequencies and short wave-k- Wo[-1 -k- Wb(K -- 4a)3 lengths the propellant surface sees only an averX [-(m -- n)~o -- (y -- n)~bb7 age gas temperature, whereas, at low frequencies, and ~boand ~bbare values of ~b for the oxidizer and the surface sees a gas temperature that varies in phase with the pressure. While the influence of binder, respectively. As W~o and ~ b approach zero, ~bo and tb ap- instantaneous variations in gas composition on proach zero and the admittance equation reduces the flame temperature~ has been considered in the model, the potential effects of such changes to that for a homogeneous propellant. in composition and temperature on the regresf = (I?//5)[-(1 -- O/e) -- n-] (24) sion rates of binder and oxidizer have not been included, and may alter the calculated results in At high values of both r and w~b, ~bo and ~bb either a positive or negative direction. both approach 1 and the admittance is given b y At high amplitudes and/or high frequencies there is a possibility that the compression-inr = ( ? / P ) [ - ( 1 -- Ole) -- W~y -- Worn duced temperature variations will couple with -- WoWt,(~ -- 4))(m -- y)] (25) those arising from changes in gas composition, resulting in appreciable fluctuations in gas-phase This relationship shows that the sign of the ad- reactions and in the regression rates of the conmittance depends strongly on regression rate stituent solids. Since dP/dt maxima precede those pressure exponents y and m of the binder and of P by 90 ~ and the composition-induced flame the oxidizer and that the burning rate pressure temperature fluctuations for the case calculated exponent of the propellant, n, enters only through are either in phase with or lead the pressure the term V. This result is noteworthy in view of waves, the coupling of the two thermal effects the fact that heretofore no correlation between would result in increased reaction and regression n and the extent of instability shown by pro- rates as pressure increased, followed b y depellants has been reported, although theoretical creased rates as pressure decreased. treatments of homogeneous propellants indicate In omitting the interaction of the gas composithere should be such a correlation. tion and flame temperature fluctuation with the regression rates of the propellant constituents, Discussion. The model used here neglects several the indicated method of calculating stoichiofactors that can become important during oscil- metric effects on the admittance becomes indel a t o r y burning of composite propellants. pendent of the burning zone thickness as long as Time-dependent fluctuations in heat and mass this is small relative to the acoustic wavelength. transfer and in the rates of chemical reaction and The acoustic perturbations in turn are congaseous diffusion in the propellant combustion sidered sufficiently small that the thermodynamic zone are not included. At low frequencies, chemi- properties of the combustion zone vary linearly cal reaction and gaseous diffusion times should with respect to composition. Although highbe small compared with the oscillation period speed motion pictures indicate particle burning and consequently cause only small deviations as far as 1 em from the propellant surface, the
342
COMBUSTION INSTABILITY
wavelength of the acoustic oscillation at, say, 1000 cps is 120 cm. On the other hand, calculation of the amplitudes of thermal fluctuations that arise from compositional variations in the gases may be considerably in error because the thickness of the particle-burning zone is, in general, larger than the "wavelengths" associated with these thermal fluctuations. These wavelengths are strongly dependent on frequency, propellant regression rate, and chamber pressure. As an example, a propellant burning at 189 cm/sec at 1000 psi exhibits a thermal "wavelength" of 0.3 to 0.4 cm at 1000 cps. Since most deviations from the constant-pressure regression rate relationships discussed above become more important at high frequencies, the model should be more representative of composite propellant burning at low rather than at high frequencies. Even at high frequencies, however, composition and temperature effects are expected. Although the present treatment deals with true composite (i.e., polyphase) propellants the same arguments may be applied, at least hypothetically, to so-called homogeneous propellants, in which during deflagration a liquid zone is predicated, 10 from which the constituents may volatilize at rates that have different pressure indices. If each of N components is assigned a mass vaporization rate of the form L P q gm sec-~ cm-~, where q is a constant and L is determined from stoichiometric relationships that must exist under constant-pressure deflagration, then the admittance is given by the equation Y = (V//5){(1 -- 0/e) -
~ ['Wjqr
-- (K~"-- ~bj)(1 -- W i ) ( W j ) ( q i -- q~)']}
(26)
where 0 In T~ ,,j -
4,~ =
owj
0 In M,
~
owj
Consider J P N propellant, a double-base formulation u containing about 51% nitrocellulose (NC), 43% nitroglycerin (NG), 3% diethyl phthalate (DEP), and 1% ethyl centralite (EC). Values of Kj and r may be determined graphically from theoretical curves of T, and M, as functions of Wj.. The ~ -- ~b differences for EC, DEP, NG, and NC, are 0.7, 0.6, 0.4 and 0.1, respectively; these are the same order of magnitude as the value of 0.5 estimated for the composite propellant example.
Experimental Experiments have indicated the existence of thermal waves in the combustion zone above a
propellant surface. Streak photographs taken of the gaseous region above propellant samples burning under the influence of imposed acoustic oscillations show the periodic release of luminous bands of gases parallel to the propellant surface varying in intensity in regular fashion. Apparatus. A schematic diagram of the experimental apparatus is shown in Fig. 4. A modified T-motor 1~ was used as an acoustic driver. This motor generates acoustic oscillations primarily in the fundamental longitudinal mode across the arms of the T, the frequency of which may be varied by changing this length. The sample propellant, an ammonium perchlorate-containing plastisol nitrocellulose type, was a 1.8-inchdiameter cylindrical waver approximately 88inch thick. The windows of the sample section were formed from ~-ineh-thick Herculite or ~--inchthick colorless synthetic sapphire. The photographs were taken with a Beckman and Whitley Dynafax high-speed framing camera, modified for operation as a streak camera. The acoustic pressure was measured by means of a straingauge transducer located near the propellant surface and oriented parallel to the direction of the gas flow. Correspondence between the pressure traces and the streak photographs was obtained by means of an exploding wire that simultaneously put a spot on the film and effected a discontinuity in the pressure trace. Results. Streak photographs (Fig. 5) show two distinct patterns of radiant energy emission. The first is an oscillation in the over-all luminosity of the gases, of frequency equal to that of the driver pressure. From correlation of the pressure and streak traces, it has been found that this radiant oscillation is in phase with the acoustic oscillation within 4-20 ~ at 500 cps. The second pattern is evidenced by the release of bands of varying luminosity, parallel to the propellant surface. Since this release occurs at the same frequency as the acoustic oscillation the separation of bands of similar intensity can be related directly to the deflagration rate of the propellant, the average pressure, and the frequency and amplitude characteristics of the imposed acoustic oscillation. The phase relationship between the emergence of the luminous bands and acoustic pressure appears to change with experimental conditions. Data at 500 and 200 cps, at average chamber pressures of 400 and 800 psi, for an ammonium perchlorate-containing plastisol nitrocellulose composite propellant, indicate that the release of luminous bands from the propellant surface generally precedes the pressure maximum, by approximately 180 ~, and that the phase difference decreases with increase in frequency and decrease in pressure.
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344
COMBUSTION INSTABILITY
Discussion. There are at least three explanations for bands of varying luminosity in the gases above a propellant that is burned under the influence of an oscillatory pressure fluctuation. The first involves the release of strata of gases of varying composition and temperature discussed earlier. Either an increase in average pressure or a decrease in frequency reduces the burning time of ammonium perchlorate particles relative to the period of an imposed oscillation and results in increased phase lags between the thermal and pressure waves. The second involves particle release from the surface. Coupling of the propellant burning rate to the acoustic pressure can lead to an oscillatory rate of particle release and consequently give luminous bands in the gases above the surface corresponding to particle concentration gradients. The third, postulated by Summerfield et al., s is based upon the difference in pressure-volume work done by the acoustic oscillation on propellant combustion products having flame temperatures independent of pressure. This mechanism gives rise to thermal waves that are 180 ~ out of phase with the pressure at low frequencies. At the present time it is not possible to separate the individual contributions of the three mechanisms of luminous band formation described above. At 200 cps the phase relationship of luminous band release to pressure is close to that predicted from the pressure-volume work mechanism. The decrease in lead of luminous band release to pressure with increase in frequency and decrease of average pressure, however, could result from either composition-temperature or particle-release mechanisms or from a combination of the two. ACKNOWLEDGMENTS This program was conducted under Contract DA-01~321-ORD-12024 with the Office of the Chief of Ordnance, administered by the Army
Ordnance Missile Command. The author wishes to extend special acknowledgment to Dr. R. H. Cantrell of the Applied Physics Laboratory, The Johns Hopkins University, for suggesting the method of analytical integration of the surfacearea equations. REFERENCES 1. Panel Discussion: Solid Propellant Combustion Instability, Eighth Symposium (International) on Combustion, p. 904. Williams and Wilkins, 1962. 2. CHENG, S. I.: ibid, p. 81. 3. HART, R. W. and McCLURE, F. T.: J. Chem. Phys. 30, 1501 (1959). 4. BARR~RE~M. and BERNARD,J. J.: see reference 1, p. 886. 5. GREEN~L., JR.: Jet Propulsion 28, 386 (1958). 6. MORSE, P. M.: Vibration and Sound, p. 367. McGraw-Hill, 1948. 7. HART, R. W. and CANTRELL,R. H.: The Johns
Hopkins University Applied Physics Laboratory Report TG-335-11, April 1962. 8. SUMMERFIELD, M., WAESCHE, R. I~. W., and WENOGRAD,J. : Aeronautical Engineering Report No. 56~-b. Princeton Univ., Dec. 20, 1961. 9. SmSETT, D. J. and LOBATO,J. M. : Investigation
of the Mechanism of Combustion of Composite Solid Propellants, Aerojet General Corporation, AFOSR-TR-60--6$, 1960. 10. LEWIS, B., PEASE, R. N., and TAYLOR, H. S.: Combustion Processes, Vol. II of High Speed Aerodynamics and Jet Propulsion, p. 532. Princeton Univ. Press, 1962. 11. WIMPRESS, a . N.: Internal Ballistics of SolidFuel Rockets, p. 4. McGraw-Hill, 1950. 12. WooD, W. A. : Rohm & Haas Company Quarterly Progress Report No. P-61-7, Oct. 3, 1961, (Conf.). For original description see PRICE, E. W. and SOFFERIS,J.: Jet Propulsion 28, 109 (1958).
Discussion PROF. H. EMMONS (Harvard University): Dr. Wood's theory supposes oxidizer particles which burn as spheres. If both oxidizer and fuel particles were present would the r functions show two positions of rapid rise and the resultant burning show two regions of rapid admittance change?
DR. W. A. WOOD (Rohm & Haas): The case in which both oxidizer and binder burn as particles has been included in the written text. The frequencyaverage particle burning time r is a function of ( ~ ) only and may be used for either binder or oxidizer. If ~b differs sufficiently from ~o then the admittance will show two regions of rapid change.