Modification of solid propellant deflagration rates due to random nonhomogeneities

Twenty-first Symposium (International) on Combustion/The Combustion Institute, 1986/pp. 2001-2010

MODIFICATION

OF SOLID PROPELLANT DEFLAGRATION RATES DUE TO RANDOM NONHOMOGENEITIES~" ROBERT C. ARMSTRONG AND STEPHEN B. MARGOLIS Combustion Research Facility Sandia National Laboratories Livermore, California 94550

tThis work was supported by the Office of Energy Research, U.S. Department of Energy, and by a Memorandum of Understanding between the Department of the Army and the Department of Energy.

A nonlinear response theory is employed to obtain the modification of the mean burning rate of a nonfiomogeneous solid propellant. Although the methods used here are quite general, attention is focused on the interaction between the three-dimensional structure of the deflagration and the constitutive model. Based on a calculation of the mean burning rate as a function of the correlation length of a random nonhomogeneity, it is shown that the mean burning rate is greater than that of homogeneous burning for small correlation lengths, and is less for larger correlation lengths. Thus, this study indicates that a solid composed of particles of two different propellants would burn more slowly than their mean rate if the size of the particles were large, and more quickly if their size were small. As many of the methods used here are new to this application, some comment is also made on the utility of this approach in computing mean burning rate statistics for random media.

1. Introduction T h e p u r p o s e of this p a p e r is to explore n o n h o m o g e n e o u s solid p r o p e l l a n t deflagration as a f u n c t i o n of its statistics, particularly with r e g a r d to m e a n values o f observables such as the b u r n i n g rate. For example, the solid propellant itself may be c o m p o s e d of two different materials, both of which are capable of supporting deflagration by themselves a n d a r r a n g e d so as to p r o d u c e a r a n d o m isotropic m e d i u m . I n the p r e s e n t work, we will a s s u m e that the difference in physical properties from one region o f the solid to a n o t h e r is sufficiently small such that the n o n l i n e a r e q u a t i o n s describing the deflagration process can be solved using p e r t u r b a t i o n techniques. Previous models, such as those that are related to the petite e n s e m b l e m o d e l 1, usually have also a s s u m e d that n o n h o m o g e n e o u s propellants are r a n d o m media, a l t h o u g h some investigators have claimed that there may be some l o n g - r a n g e o r d e r to a m m o n i u m perchlorate composite propellants, a n d have cited rocket m o t o r stability data 2 as evidence of this. However, these petite e n s e m b l e models generally treat a single particle o f p r o p e l l a n t surr o u n d e d by a n o t h e r p r o p e l l a n t (binder) in a o n e - d i m e n s i o n a l fashion with n o interaction

between adjacent particles. T h e statistics are usually i n t r o d u c e d later w h e n mass and e n e r g y conservation are taken into account. A l t h o u g h there is no restriction o n the differences in physical properties between the two propellants m a k i n g u p the composite m e d i u m , lateral interactions are completely neglected in these models. I n contrast, a l t h o u g h we assume that the variation in physical properties is small, as in nitrocellulose-nitroglycerine propellants, the model a n d analysis e m p l o y e d h e r e are fully three-dimensional. T h e motivation for this work is derived from the fact that most successful (i.e., commercial) propellants are actually a m i x t u r e of two or m o r e substances, at least one of which is usually capable o f s u p p o r t i n g a deflagration by itself. T h e l e n g t h scale over which t h e r m a l conduction interactions occur (defined here as the ratio of the solid t h e r m a l diffusivity to the b u r n i n g velocity) is r o u g h l y o n the o r d e r of 1 pxm ( g u n propellants) to 101xm (rocket propellants). S c a n n i n g electron m i c r o g r a p h studies reveal that propellants often r e f e r r e d to as " h o m o g e n e o u s " , such as the nitrocellulosenitroglycerin variety, are a n y t h i n g b u t homogeneous o n these length scales, a a Recognizing that most models of p r o p e l l a n t c o m b u s t i o n are o n e - d i m e n s i o n a l or steady-state or both, the

2001

2002

COMBUSTION OF PROPELLANTS

object here is to explore the consequences o f such nonhomogeneities on the burning rate and other experimentally observable quantities in a time-dependent, multidimensional format. A previous p a p e r 5 exploited a similar model of propellant deflagration as the one which is considered here. However, that p a p e r focused on the linear response both in a stochastic and deterministic context, and although the application appears to be novel, the mathematics required only s t a n d a r d linear response theory. In this work the nonlinear response o f the propellant as a r a n d o m medium will be explored and, as a consequence, nonlinear perturbation techniques will be necessary. T h e linear response is crucial for studying the fluctuations away from the mean, but cannot provide information about the modification of the mean itself. In this p a p e r it is the modification to the mean observables as a direct consequence of the r a n d o m media that is sought.

or the gas phase. In fact, thermocouple data 7 suggest that this intrusive limit is at least as physical as one where the flame is detached from the gas/solid interface. We also assume, as has been done in the past s, that transverse components of the gas velocity Vg are negligible compared with the longitudinal component, and thus gg = (O,O,vg). Although this assumption is only rigorously valid if the amplitude of the disturbance is small (the weakly n o n p l a n a r limit), the perspective taken here is that hydrodynamic effects are not likely to dominate over thermal-diffusive processes, since self-sustained deflagration is p r o p a g a t e d primarily t h r o u g h the latter. T h e governing equations for energy_ transport in the solid and gas are thus given by 6

a~ -ai ---~,,V2 T=0,

s < @(s s t-)

(2.1)

a~

--=-+

Ot

~3 > ~,(*l,&, t-). (2.2)

2, The Mathematical Model The physical model (Fig. 1) described here will largely follow that of a previous p a p e r (a limiting case of Model I in Margolis and Armstrong6). We assume that the gaseous flame is perfectly intrusive, by which we mean that the chemical reaction occurs extremely close to the gas/solid interface c o m p a r e d to the length scale over which conduction occurs in either the solid

Here, dimensional quantities are denoted by a tilde, the subscripts g and s denote gas and solid, respectively, T is the temperature, ~ is the density, ), is the thermal diffusivity, and denotes the position of the gas/solid interface. The fact that mass flux is continuous across the gas/solid interface is implicit in these'equations, In addition, we have assumed that the Mach number is small so that the pressure can be approximated as constant. T h e r e are two additional conditions at the gas/solid interface. In particular, the pyrolysis law that relates b u r n i n g velocity to conditions at the gas/solid interface is given in the usual manner by

xA exp(-E//~: I ;~,=~),

(2.3)

w h e r e / ] is the frequency factor (usually pressure- and t e m p e r a t u r e - dependent), /~ is the activation energy o f pyrolysis, and/~ is the gas constant. Finally, conservation of heat flux across the gas/solid interface gives •

,

Fro. 1. Definition sketch for the solid propellant model. The three spatial dimensions are represented by ~1, 22 and g3, with the gas/solid interface positioned at ~3 = ~(,2t, ~2, t). To the right of the interface hot burned gas escapes at a velocity l?g,and to the left the gas/solid interface consumes solid at a rate 0t~---~n= 0~, where ~s is the unit vector in the ~rdirection.

a@ = ~':~-~i '

(2.4)

DEFLAGRATION RATES WITH NONHOMOGENEOUS SOLID PROPELLANTS where/3 is the heat of reaction, given in units of t e m p e r a t u r e with respect to the solid phase. [In writing Eq. (2.4), we have assumed that the heat capacity is the same for both the solid and gas, which appears to be a good assumption for most propellants1]. Equations (2.1)-(2.4) are solved subject to the b o u n d a r y conditions 2e = ~ , at 23 = - =

r

i82

o=

~s'

(2.7)

~-L Ta--Tu

where tJ = Os-1 Ah (T~; ~) exp ( -/~//~T~) is the b u r n i n g velocity -O+/Ot for the case of steady p l a n a r burning of the homogeneous solid (i.e. Ah is the average of A). We also introduce the nondimensional parameters

2( r . ~ L(L;~;)' " - ~ ( o ~

A(O)=--

L

; (2.8)

o . - L _ L, o = &/L,

x = ~,#)C.,,

x = xl, y = x2, z = x3 - @(xL, x2, t),(2.10) the nondimensional version o f our model is OO Ot

(2.6)

Note that the form of the surface regression rate has been chosen in the customary Arrhenius form, which can be considered an empirical assumption. We note also that for a truly general r a n d o m media, all material properties will fluctuate as a function of position in the solid. Here it will be assumed, for simplicity, that only the frequency factor.J, is stochastic. In the m o r e general case the correlation functions arising from A would merely be replaced by a linear combination o f similar functions for the other properties, but beyond this little would change in the following derivation. Consequently, it is unlikely that the results generated by one fluctuating quantity will be qualitatively different from that of many. It is not the p u r p o s e of this work to dwell on characteristics that are associated with the material properties of a particular propellant. Here, we focus on illustrating the importance o f including the full three-dimensional structure in a model of this type (i.e., if three-dimensions are not necessary, the problem can be simplified greatly). We now introduce the nondimensional variables

~8

Thus, in terms of the moving coordinate system

(2.5)

= T~ = 7", + /3 at 23 = =.

OcI, OO =V20 Ot Oz

O0

1 OO O0

Ot

o Ot Oz

O = 0 at z = - o o ,

at -

+\-~x/

(2.11)

= XV20

(2.12)

O = 1 at z=oo

(2.13)

\ay./

)

oo

\-~-~/

\T/J\

+(1-ox)

W ~=0+ 0~ ,:0/

+ -~-y

~:o = o-7"

W e observe that a solution that corresponds to steady planar burning o f a homogeneous solid (A = Ah, A = 1) is given by Z<~0

0~

I ez' 1 z>0

aPo=-t.

(2.16)

3. Perturbation Analysis We now consider the case in which the propellant properties are random, and it is particularly useful to consider the purely rand o m element as a deviation from the homogeneous average. Thus,

A = 1 + Ea(xl,

X2, X3);

X3

<-- cI)(Xl, x2, t),

(3.1)

where a is a stationary, ergodic function of position only in the solid propellant matrix, the statistics of which are d e t e r m i n e d solely by the manufacture of the propellant solid. We now assume that the m a g n i t u d e of the scaling p a r a m e t e r ~ is small, and thus seek a solution to the problem (2.11)-(2.15), (3.1) which is a small perturbation of the homogeneous basic solution (2.16). In particular, the perturbation quantities da(x,y,t) and 0 are defined by

9 =(p+~o(t)-bt,

0=~ (2.9)

2003

dO o O = O + O o ( z ) + ( ~ dz '

z>0,z<~

(32)

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COMBUSTION OF PROPELLANTS

where b is the change in the mean burning rate due to the presence of the nonhomogeneity. The second equation in (3.2) implies that

4) = O+(x,y,z=O+,t) - O_(x,y,z=O-, t).

(3.3)

properties of these dependent variables must also be identified. It is required (as a definition) that the deviation from planar burning induced by the nonhomogeneities average to zero with respect to time at any point (x,y). That is, since (6) ~ 0, we require that

We now seek a solution for the perturbation quantities as

~=]~e'4, "), 0_=

e;0_%

i=1

i=l

0+ = :7: o,0s;,,

(3.4)

:7:

i=1

z=l

where e is the (small) scaling coefficient introduced in Eq. (3.1). Substituting these expansions into the nonlinear problem (2.11)-(2.15), expanding the nonlinear terms in Taylor series, and equating coefficients of like powers of e, we obtain a sequence of stochastically forced linear problems for the b(0 and the 0~ as

(0(o9}--(0o(i}_)=0,

where ( ) indicates the usual ensemble average. Moreover, the propellant solid is assumed to be statistically isotropic, and thus the random process a(x,y,z) must reflect this property. In other words, every average composed from the observables in the solid must be isotropic. Consequently,

00ore\ = / 000'2 --g/-/ k--g- / \o,(/

00-")+ 00_(~ +b(, ) 00o _V20(o = ~-o),

Ot

Oz

Oz

-b(i)+'~O(+')l~=o+=w (i) (3.8)

1 O(0+"))

X

Oz

g

(') =


(3.14)

<~-+~')>lzi, (3.15)

~-I~=o-

+ ( .'~'- 1 ) 0 + (,) I~=o+

=,),0),

(3.9)

where

2 /0V

/0V

+t,W +ts)

(3.10)

Here ~), ~), y(i), and w(i) represent nonhomogeneous terms which are themselves functions of the b~/, 0~), and 0~/ for j < i. For brevity, we introduce the simplified notation 0o(~ = o~ - + I::o+

Oz

2(0-0)) = ( ~'_('))[z]

p

00!o z=o_+ao) l

(0)

(313)

and similarly for 0~)+ and q5(i). Applying this information to an ensemble average of Eqs. (3.5)-(3.6) gives

--0(0(-')) " +Ob(i) O0~ Oz z -- ( 0~)

00+('l ~=o+ OOL~)[ Ot Ot ~=o-

Oz

kay/

o -

zO (3.6) Ot IO(+i)l<~atz=+~, 0(/)=0 at z = - ~ (3.7)

X O0+~')[ P ~ z=O+

(3.12)

~

Oz z=o+'

a")--- 0")1:=o-, ~0

-

-

Oz =- Oz ~=o

Since the above system of equations is fundamentally stoichastic in nature, certain statistical

where {z} is a reminder of the z-dependency of the (_i) and ~'(~). The analogous application to eqs. (3.8)-(3.9) yields

lO0(i)\

[O0~

(3.16)

-b(i)+~(O(oi)+)=(wli)).

(3.17)

We emphasize that the mean burning rate deviation b(i) is not a fluctuating quantity. Rather, it is a property of ttae propellant which is to be determined, and which depends on the statistics intrinsic to the solid. An inspection of the form of the nonhomogeneous terms, which are given explicitly in the Appendix, reveals that (~)), (~)), (,~(1)) and (w0)) all vanish. This implies, as found previously 5, that b(1) = 0 and that the first-order mean

DEFLAGRATION RATES WITH NONHOMOGENEOUS SOLID PROPELLANTS quantities in Eqs. (3.14)-(3.17) vanish. Thus, the first contribution to the deviation in the mean burning rate arises from b(2). Since b(2) is generally a function o f 0(-l / a n d 0(+x}, these must be f o u n d in a general form applicable to every realization of the stochastic process. The ability to do this hinges on recasting Eqs. (3.5) through (3.9) into a linear o p e r a t o r formalism (consistent with the b o u n d a r y conditions) for the case i = 1. We now introduce the F o u r i e r transform with respect to x, y and t, and the two-sided Laplace transform with respect to z. U n d e r this transformation Eqs. (3.5)-(3.9) for i = 1 become

( ico-p_-p~_ + k2)O~_l>+ (p_ + l )O~o~

oog_, -

1 ico+ ~p+

..~

• oo~o~ oOoc~ ^ Oz

(3.18)

X k2- X p +2) 0 +^(1) + ( p + X - ~ )10 0 +*(1)

+•

p

o---7 =~

o~ + ~176+ (z-1)0~

ooo,e

=0

(3.19)

=-~

(3.20)

iwtO(1)--O0)~+ ~O0) = --d, '~ 0+ 0-! ~ 0+

(3.21)

L~ = ],

f = (0,0, - d , - d ) T,

(3.27)

~=V o+'-o-' 0z ' g / and

f~ =

~-1

1

pX

-1

ic0+E

-i~0

0

0 J .

rr++ h -- l / p

0

k

0

0

rr_+ +1

0

t

\oz/

+) p+(p+-+) (~-+(2))[p+]/X

(3.22)

(P-+l)<0~

\ 3z /

p_(p_+l) p_(p_+l) +p

1 + x/1 + 402X(Xk 2+ iw) 2 XO

It_• =

--1 + x/1 +4(k2+ ir 2

(3.29)

(3.23)

where 7r+• =

(3.28)

We r e m a r k that the condition det(L) = 0 determines the neutral stability boundary ~ = =c(k) for the homogeneous problem. 6 Thus, in o r d e r for Eq. (3.26) to be meaningthl we require L to be negative definite. We now consider Eqs. (3.14)-(3.17) for i = 2. We observe that although there is no x or y d e p e n d e n c e in Eqs. (3.14)-(3.15), there is a z-dependence. This again motivates applying the two-sided Laplace transform to both sides of these equations to obtain

(0(2)) = Or_+ + 1)00(')00~ Oz = 0 '

(3.26/

where

where, in Fourier space, m is conjugate to t, k~ to x, bv toy, a n d k 2 = k~+/~. In Laplace space,p+ is conjugate to z for z > 0, a n d p_ is conjugate to - z for z < 0. Hats are used to identify variables in the Fourier-Laplace domain. In addition, the boundedness conditions (3.7) require that

l, ^ 1 00( 2 ~++x--~)o'02+• =0

2005

(3.24) (3.25)

T h a t is, the poles for ~0~ atp+ = Iv++ and ~0~ at p_ = -tr_+ are required to vanish. T h e closed subsystem (3.20)-(3.23) may be written in o p e r a t o r form as

( p _ + l ) 2' (3.30)

where, as before, p+ is conjugate to z and p_ is conjugate to - z . As in the previous development for 0(21 and 0(J/, the condition that every realization of 0(2/ be b o u n d e d requires that we eliminate the pole at p+ = I/pX in Eq. (3.29). Similarly, the condition that 0(-2) vanish as

requires that we eliminate the pole at p Eq. (3.30). Thus, we obtain

= 0 in

COMBUSTION OF PROPELLANTS

2006

O

x l OO'o +'\

=~

~

/ 0O(2)\

These two equations plus Eqs. (3.16) and (3.17) are sufficient to calculate b12~(see the A p p e n d i x for details). The final input that is required is the correlation function for a. T h e possibility of measuring the actual correlation function of a particular propellant seems remote. However, for real propellants it is likely that a knowledge of the magnitude of the fluctuation and a characteristic length scale can be determined. If, for example, the solid propellant were composed of a mixture o f two propellants, each having a different pyrolysis rate, then the magnitude IX o f the difference between these two rates would be a measure of the standard deviation of a. Moreover, the average dimension of the particles t h a t make up the solid defines a characteristic length scale C Since we have already assumed that the solid is statistically isotropic, we postulate the correlation function

(a(x 1,x2, x 3)a(x(, x~, x;) ) = C(x~ - x ( , x2 - x ; , ~3 - x ; )

= #2fexp [

(xl--x~)2+(x2-x2)2+(x3-x3)2] 21>2

(3.33) where this particular function was chosen because it has the same form in both Fourier and real space. However, it is expected that functions with the same correlation length and the same standard deviation will produce the same qualitative results. Using Eq. (3.33), we show in Fig. 2 a plot of b(2), normalized by the fluctuation magnitude Ix, as a function of the characteristic length scale (~ (in units of the conduction length scale k,/U). These results were c o m p u t e d by Gaussian q u a d r a t u r e using various values for the density ratio P and the modified activation energy p a r a m e t e r ~. We observe that, in general, the mean burning rate is greater than that c o r r e s p o n d i n g to the case of a homogeneous m e d i u m for small correlation lengths, and is less for larger correlation lengths. 4. D i s c u s s i o n

Although the calculation presented in Fig. 2 is explicit and straightforward, we also wish to provide an intuitive u n d e r s t a n d i n g of the ira-

--= .8

1/15 v

ob,-

.2

tO I

i

I I !!

10 -1

I

10 0

1

'[

101

CORRELATION LENGTH FIG. 2. The results computed with Gaussian quadrature using the correlation function of eq. (2.47) for two different gas/solid density ratios P. The correlation length is a measure of the size of the fundamental particles that compose the solid matrix and is given in units of conduction length scale ~,]/). The change in the mean burning rate b(2) is shown normalized by the standard deviation of the nonhomogeneity, Ix. The parameter -= is a modified nondimensional activation energy which was defined explicitly in Eq. (2.8). portant elements contributing to the modification of the mean b u r n i n g rate. For the purpose of illustration it is useful to divide the previous analysis into two parts: the first reflecting the pyrolysis model dynamics in a one-dimensional sense, and the second incorporating the contribution from the true three-dimensional nature of the solid. For the first part, we a d o p t a one-dimensional version of the pyrolysis rate law used previously, namely

'Ozo Ot =-(a+a(z~

/

-E , To~r(zo) )

(4.1)

where a is a r a n d o m deviation in the mean frequency factor A, 9 is the induced deviation in the surface t e m p e r a t u r e T~, and z~ is the position of the gas/solid interface (one-dimensional analogue to q~). Here, we again have selected the frequency factor as the only propellant material p r o p e r t y to fluctuate in o r d e r to deemphasize the importance of specific propellant material properties in favor of the effects due to including three dimensions. Assuming for simplicity that a and 9 are uncorrelated (such as assumption was not m a d e in the analysis o f the previous section which

DEFLAGRATION RATES WITH NONHOMOGENEOUS SOLID PROPELLANTS p r o d u c e d Fig. 2), then the resulting one-dimensional average b u r n i n g rate to second o r d e r is

d•za• dt

-a exp(-E/T~) [1 +

(a2)/A 2+ (89

2+ E/T o)(r2)/ToZl " (4.2)

We thus note that due to the A r r h e n i u s form of the b u r n i n g rate law the average burning rate is always smaller than the b u r n i n g rate of the average. It is easily shown that the average burning rate is even lower if a and T are perfectly correlated. For the second part o f this argument we speculate that the major effect o f including multidimensionality is a change in surface area available for combustion. On the average, the total such area written as a fractional increase is

-LdxI-L dy l + \ o x / +\Oy/ sg~ = lira L ~

which to second order is ,

s ~ -- l + g

~

~

. (4.4)

Since s ~ = 1 if the disturbance a is absent, this result shows that the b u r n i n g area is always increased. We now speculate that the b u r n i n g rate in the three-dimensional system is the one-dimensional rate given by Eq. (4.2) weighted by a proportionate increase s ~ in the burning front area. This line of reasoning is o f course similar to the "wrinkled flame" rationale in turbulent flame theory 1~ and to that used to explain similar results for pulsating, cellular flames 1l. It suggests that there are two forces at work; namely, a component in the mean burning direction that is d e p e n d e n t on the pyrolysis law and which tends to decrease the burning rate, and a metric c o m p o n e n t that tends to increase the b u r n i n g rate by increasing the burning surface area. F u r t h e r m o r e , differential quantities [such as those a p p e a r i n g in Eq. (4.4)] under the ensemble average become weighted relative to absolute quantities [such as those appearing in Eq. (4.2)] for larger wave numbers. Since the lateral length scale over which z~(x,y) varies cannot be significantly different from that of a(x,y,z), the metric c o m p o n e n t Eq. (4.4) will make a larger contribution when the length scale of the nonhomogeneity is small. This is

2007

related to the fact that the arc length of a sine wave becomes arbitrarily large as it's wavenumber becomes large, even if the amplitude is held constant. This implies, for small length scales, that

) However, the length scale of a(x,y,O) will clearly have no effect on the one-dimensional component (4.2). T h e above heuristic reasoning serves to clarify the qualitative features o f Fig. 2. When the correlation length of the nonhomogeneity is large, the one-dimensional contribution represented by Eq. (4.2) dominates, leading to a decrease in the mean b u r n i n g rate. As the correlation length decreases, a threshold is reached where the metric contribution represented by Eq. (4.4) begins to dominate and the b u r n i n g rate increases. Equation (4.2) also suggests that the decrease in the mean burning rate will be more greater in magnitude tbr larger activation energies, a feature that is verified by Fig. 2. We note that the underlined terms in Eqs. (A.2)-(A.6) are directly related to the three-dimensional nature o f the problem and contain factors similar to Eq. (4.5). It is suggested by Eq. (4.5), and confirmed in Fig. 2, that as the length scale of the nonhomogeneity f approaches zero, the interfacial area, and hence the burning rate, will become arbitrarily large. This anomaly is directly related to our assumption that the intrusive flame is infinitely thin. T h e r e is, in reality, a length scale associated with the flame thickness, below which this analysis ceases to be valid. The interior of the flame is reactive-diffusive in character and it is principally the diffusive character that is likely to reestablish a new regime with a mean b u r n i n g rate that is i n d e p e n d e n t o f r for ~ much less than the flame thickness. It can be said that there is a "cut-off" for r which is on the o r d e r of the flame thickness and which demarks the beginning of this new regime. As a final note, we r e m a r k that although a rather simplistic model of propellant deflagration has been used in this work, the methods used here provide a generalization of linear response theory and are amenable to more detailed models without significant modification. While this nonlinear response theory is enormously powerful, its major limitation is that it is based on a perturbation argument. Although it is difficult to show dramatic effects using a perturbation treatment, important qualitative trends such as those shown in Fig. 2

2008

COMBUSTION OF PROPELLANTS

may nonetheless be deduced. In fact, one indicator of the i m p o r t a n c e of nonhomogeneities is the relative amplification resulting from an input deviation of unit amplitude. By this measure, Fig. 2 demonstrates that nonhomogeneities indeed play a significant role in the deflagration of solid propellants, and that it is necessary to consider multidimensional effects in order to fully u n d e r s t a n d their impact.

4. LENGELLE', G., BIZOT, A., DUTERQUE, J., AND

TRUBERT, J. F.: Fundamentals of Solid Propellant

Combustion (K. K. Kuo and M. Summerfield, Eds.), p. 361, AIAA, 1984. 5. ARMSTRONG, R. C., AND MARGOLIS, S. B.: Nonho-

6.

7. 5. S u m m a r y

From the above analysis a uniform picture of the contribution of nonhomogeneities to propellant deflagration can be drawn. For instance, a solid such as nitrocellulose/nitroglycerin is composed of particles o f nitrocellulose whose size (~ 1 to 10ixm) would be associated with the correlation length. Moreover, the b u r n i n g rate in nitroglycerin rich areas would be different from that in nitroglycerin poor and the magnitude of this difference is identified with the standard deviation Ix (Fig. 2). T h e modification of the mean b u r n i n g rate is depressed (b(2) < 0) for.particle sizes close to the conduction scale (ks~U) because a constitutive relation of the Arrhenius form has been used (Eqs. (2.3), (4.2)). However, for very small particles ( .1}tfl)) the induced roughness in the b u r n i n g surface increases the total area exposed to the flame and thus dramatically increases the burning rate. This, of course, holds true only for particle sizes above the physical thickness of the flame since this entire analysis is predicated on a flat-flame aproximation. T h e flame thickness establishes a "cut-off" length below which the mean burning rate is expected to be independent of particle size. Finally, as the correlation length (particle size) gets large, the solid becomes perfectly correlated and the modification to the mean b u r n i n g rate vanishes (b(2)-->0, see Fig. 2).

8. 9. 10. 11.

mogeneous Propellants as Random Media, Combust. Sci. Technol. 52, 59 (1986). MARGOLIS, S. B., AND ARMSTRONG, R. C,: Cornbust. Sci. Technol. 47, 1 (1985). KUBOTA,N., OHLEMILLER,T. J., CAVENY,L. H., SUMMER~IELD, M.: AMS Technical Report No. 1087, Princeton Univ., March 1973. STRAHLE,W.C.,: AIAAJ. 13, 640 (1975), McQUARRIE, D. A.: Statistical Mechanics, p. 557, Harper k Row, 1976. CLAVIN, P. AND .WILLIAMS, F. A.: J. Fluid Mech. 90, 589 (1979). MARGOLIS,S. B. AND MATKOWSKY,B. J.: SIAM J. AppI. Math. 45, 93 (1985). Appendix

The n o n h o m o g e n e o u s terms in Eqs. (3.5)(3.9) are given by ~'+")=0,

00+(1)'/~-)'00+(1)D~ Oz

- - 2 X ( D~176

(A.1)

(A.2)

0 ~ (1)

~-(_2)=Do0(1).Do(h(1)eZ+ 0(1) _~___~ez -- ~ (l)Do2 q~(l)ez_ 2(D0 r (1).Do O (1))eZ --

//

2~kD~

(1).

00(1) "~

00 (1)

D Do Oz ) + ~ Oz

2"r o~

5,(2) = (X0-1)Do 0(1).Do 0o(~+ 89 0 (1)2'~' 0+ ~

1. RAMOHALLI, K. N. R.: Fundamentals of Solid Propellant Combustion (K. K. Kuo and M. Summerfield, Eds.), p. 409, AIAA, 1984. 2. T'IEN, J. S.: Fundamentals of Solid Propellant Combustion (K. K. Kuo and M. Summerfield, Eds.), p. 791, AIAA, 1984. 3. MANN, D. C., AND PATRICK, M.A.: Scanning Electron Microscope Examination of Cotton Linters and Wood Pulp Fibers Before and After Nitration and Gun Propellant Manufacture, Ballistic Research Laboratory Technical Report ARBRL-TR02476, 1978.

3,(1)=-a

i-+(z)= 1 OO(1) 00+(1) p Ot Oz

+ O,+1 REFERENCES

w~

~'_~

1

,~(1)~2 +

2 %+ -

_~

-[- 0a d~(1)- 0 ( 1 ) ~ Ot "~ v~

q5(1)

4~(1)_ o 0 ) ~

0(1)~ w (2) = _ 89 ~ ~ (1). Do ~ (1) + -0+__ ~ Ou+l

(A.3)

v~ 1 2

(A.4)

0(~2 (A.5)

where

Do=\o/Oy,],

Do2=-~x2+0-~-, (A.6)

and the u n d e r l i n e d terms arise from the multidimensionality o f the problem. It is necessary to

DEFLAGRATION RATES WITH NONHOMOGENEOUS SOLID PROPELLANTS treat these terms, which are ultimately quadratic in

o,,,

o+'

o 1,, oo'o2 Oz

and

W e now c o m p u t e the e n s e m b l e average o f t e r m s o f the f o r m ({(~,)n u(s)}v(s)) as

OO'o1'_ Oz

i

oo

ds'G(s-s')fs(s')

(A.7)

--oo

w h e r e G and H are G r e e n ' s functions. We now assume t h a t f s (s) is a subset o f stochastic process on an infinite domain. T h a t is,

fs(S)=

IJ0(s) Isl ~s Isl >s,

(A.9~

and thus

is ds'G(s-s')fs(s')

(A.10)

is ds'H(s-s')fs(S'),

(A.11)

Us(S)= -s Vs(S)= -s

,1 ,)

Us(S vs(s

= lira

and o n which 0(2), 0(-z) a n d +(9) d e p e n d , with a variation o f the W e i n e r - K h i n t c h i n e t h e o r e m 9. T o d e m o n s t r a t e the c o m p u t a t i o n o f the terms in Eqs. (A.2)-(A.6), it suffices to give a simple o n e - d i m e n s i o n a l e x a m p l e . Let u(s) and v(s) be two i n d e p e n d e n t variables o f interest (e.g., 0(-I), 0(+1), o r +(1)) which are linear responses to a r a n d o m forcing f u n c t i o n f s (s) which is n o n z e r o only o n - S < s < S. T h i s e n s u r e s t h a t f s (s) is F o u r i e r t r a n s f o r m a b l e a n d thus u and v can be written in the f o r m

Us(S) =

2009

S~oo

;:

s ~ = - s ds .,-s ds"

= lim

l(~ ~

a(s-s')

1

X H ( s - s") C ( s ' - s") =

s dq S --o~

dq

q

--no

X H(q")C(q'-q"),

(A. 12)

w h e r e the correlation f u n c t i o n C(s' - s") = (f(s') f(s")). H o w e v e r , f r o m application o f Parceval's and Faltung's t h e o r e m s , we obtain

=

I•

dr'(-ir')"G*(r'

)/:/( r' )d( r'),

(A.13)

w h e r e r' is the F o u r i e r space variable and ~,/2/ and C are the F o u r i e r t r a n s f o r m s o f G, H and C, respectively. T h u s , the t r a n s f o r m a t i o n back to real space is not necessary to c o m p u t e statistics such as those above, a n d h e n c e only q u a d r a tures in F o u r i e r space are r e q u i r e d . T h e formulas for the G r e e n ' s functions n e e d e d to p r o d u c e the desired statistics w e r e given in a p r e v i o u s p a p e r 5.

COMMENTS Cohen-Nir, E.N,S. T.A. The burning rates of composite propellants containing two different oxidizers such as ammonium perchlorate and HMX show two observable mechanisms: 1) at low pressures (p < 100 bars) the burning rate is predominated by A.P.; 2) at higher temperatures, HMX controls the burning rate and the pressure exponent. Therefore it would appear that one cannot assume a mean burning rate as you propose. Does your model predict the burning rates of these propellants? Author's Reply. Our model does not assume a mean burning rate. Rather, it is a quantity which is

presumed to exist (i.e., it is an observable), and which is calculated. The method we employ requires that the spatial mean of the two propellants he used as a starting point; the observed mean is then computed as a deviation from the value. Ultimately the only assumption beyond the simplicity of the test model, is the assumption that the burning rate at two adjacent points on the surface differ by only a fraction of the nondimensional activation energy. If this were not the case large corrugations (greater than several conduction scales) would appear. From an empirical standpoint, particularly in light of SEM data 4, we see no evidence for this in propellants which are thought

201{)

COMBUSTION OF PROPELLANTS

to be applicable to our work. Moreover, evidence concerning HMX-AP propellants that you cite does not contradict our findings. In the discussion section we showed that, if two-dimensional effects are unimportant, the mean burning rate is dominated by the slower burning propellant. In fact any observable of the system is progressively weighted towards that of the slower burning propellant including (we conjecture) the pressure exponent.

F.A. Williams, Princeton University. This is an interesting study, but despite your clear explanation, I still have difficulties with the divergence of the burning rate as the correlation length approaches zero. Five, tall bumps on the propellant surface seem unlikely to me. Could there be a possibility of the existence of multiple solutions, ie. more than one mean burning rate at the same condition, as a consequence of a

leading proportion through one or the other phase? Perhaps there could be another solution without divergence.

Author's Reply. We believe that the divergence in the burning rate in the limit that the correlation length goes to zero is due to the divergence in the surface area (bumpiness) presented to the burning front, rather than due to the presence of another solution. According to linear stability theory, the solution about which we have expanded is stable for the values of the parameters considered here (Margolis and Armstrong, 1986). Consequently, we would not apriori expect a small perturbation to produce a migration to another solution, if indeed one exists. Thus, we believe that a more likely explanation for this divergence is that our model, which assumes an infinitesimally thin reaction front, ceases to be valid in the limit of small correlation length.