LAWS OF COMBUSTION OF A SOLID-PROPELLANT

B. S. Ermolaev, A. I. Korotkov, and Yu. V. Frolov

S A N D W I C H

UDC 536.468

In studying the laws of combust ion of he te rogeneous condensed mix tu r e s it is impor tant to know the re la t ionsh ip between the burning r a t e on the one hand and p r e s s u r e , pa r t i c l e s ize, and the nature of the components , on the other~

However~ the complexi ty of the mix tu re combust ion p r o c e s s makes the theore t ica l genera l iza t ion of the avai lable expe r imen ta l da ta ve ry difficult. Accordingly, it is unders tandable why inves t iga tors pay such attention to the study of s implif ied models more amenable to analysis .

Such a model is a s y s t e m composed of many a l ternat ing thin l a y e r s of fuel and oxidizer(sandwich s y s t e m (Fig. 1)), in which the th ickness of the l a y e r s is equal to the par t i c le s ize of the cor responding c o m - ponent. The o rde red nature of the s y s t e m leads to the e l iminat ion of va r ious t r ans ien t p r o c e s s e s that in- va r i ab ly accompany the genera l ly s t eady - s t a t e r e g i m e of mix ture combustion. The fixed location of the components makes the sandwich model convenient for the expe r imen ta l study of f l ames , m e a s u r e m e n t of the t e m p e r a t u r e f ie lds in the gas and condensed phases , the invest igat ion of the action of combust ion ca t a - lys t s [1], etc.

A number of expe r imen ta l and theore t i ca l s tudies of the combust ion of o rde red condensed s y s t e m s have been published.

Most at tention has been paid to the p r o c e s s of f l ame propaga t ion along the contact sur face between oxidizer and fuel s labs [2-5]. This r e g i m e c o r r e s p o n d s to the l imit ing case of mix tu re s with an infinitely la rge par t i c le size (for a sandwich s y s t e m infinitely thick layers ) . These inves t iga tors have studied the shape of the pit dug by the f lame during the combust ion p r o c e s s and have shown that the burning ra t e is de te rmined by kinetic fac to rs .

In [6] a mul t i l aye r sandwich s y s t e m was used to analyze the effect of the mic ronons ta t ionar i ty of the mix tu re combust ion p r o c e s s on the burning ra te . In [7-9] the theory of combust ion of unmixed gases [10] f o r m s the bas i s of a theore t ica l t r e a t m e n t of the combust ion of mul t i l aye r s y s t e m s s ta r t ing f r o m the a s - sumption that the p r o c e s s is diffusional in c h a r a c t e r .

We have now made an expe r imen ta l invest igat ion of the f l ame s t ruc tu re and the laws of combust ion of mul t i l aye r sandwich sys t ems . We have measu red the dependence of burning ra te on p r e s s u r e , l aye r th ickness , and the nature and p ropor t ions of the components . The r e su l t s obtained are analyzed within the f r a m e w o r k of the t h e r m a l theory of combust ion.

E X P E R I M E N T A L

The sandwich spec imens were p r e p a r e d by batch p r e s s i n g in a r ec tangu la r mold. Weighed por t ions of the powdered components were a l te rna te ly poured into the mold, each l aye r being carefu l ly leveled and compacted to obtain a smooth and even sur face . After the n e c e s s a r y number of l a y e r s (15-30 or more , de - pending on the i r thickness) had been formed, the spec imen was c o m p r e s s e d to n e a r - m a x i m u m density. The finished spec imens had a r ec t angu la r c r o s s sect ion m e a s u r i n g 5x8 m m and were 10-20 m m long. The fo l -

Moscow. Trans la ted f rom Fiz ika Goreniya i Vzryva , Vol. 6, No. 3, pp. 277-2 85, Ju ly -Sep t embe r , 1970. Original a r t i c le submit ted D e c e m b e r 12, 1968.

�9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

251

a b

Fig. 1. Combust ion sur face of a mul t i l aye r A P - P F sandwich s y s t e m in re f lec ted light, p = 20 a tm abs. , h 0 = 1 m m (a) and d i a g r a m of an individual f l ame (b). 1) Diffusion flame; 2) f l ame "root"; 3) p remix ing zone (B is the width of the f l ame "root") .

lowing components were employed: ox id ize rs - ammonium p e r c h l o r a t e (AP) and po t a s s ium p e r c b l o r a t e (PP); f u e l s - po lyformaldehyde (PF), po lys ty rene (PS), naphthalene, uro t ropin , polymethyl me thae ry la t e , and s accha rose .

The c rys ta l l ine subs tances were ground in a v ibra t ing mil l (mean pa r t i c l e size 5-10#); the po lymet r i c m a t e r i a l s were used in the f o r m of f inely d i spe r sed powders with a pa r t i c l e s ize of l e s s than 50 p.

The spec imens , whose l a t e r a l su r f aces were coated with epoxy res in , were burned in a c o n s t a n t - p r e s - sure bomb in a n i t rogen a tmosphere . The burning ra t e was de te rmined by means of a p i ezoe lec t r i c p r e s s u r e probe. The appara tus was designed to p e r m i t h igh-speed f i lming of the combust ing p roce s s . The t e m p e r a - tu re d is t r ibut ion in the spec imen and the f l ame was r eco rded by means of 15 mic ron tungs ten - rhen ium 5/ 20 the rmocoup les using the method desc r ibed in [11].

R E S U L T S O F T H E E X P E R I M E N T S

F lame Structure . A typica l photograph of pa r t of the combust ion su r face of a mul t i l aye r A P - P F s y s - t em, obtained in re f l ec ted light, is p resen ted in Fig. 1. These components a re quite volat i le and decompose without a condensed res idue . Clear ly , the r e l i e f of the combust ion su r face is v e r y jagged; pi ts along the contact p lanes pene t ra t e deep into the condensed phase . The depth of the p i t s at a p r e s s u r e of 20-40 a im abs. is usual ly 0.3-1.0 ram. However , in individual c a se s cons iderab le devia t ions a re observed. The f lame front is also d i s to r ted to confo rm with the su r face re l ief . Essen t ia l ly aplanar combust ion is observed on the en- t i r e th ickness in te rva l 0.1-2.0 m m at p r e s s u r e s above 10 a tm abs.

There is an advantage in studying the t e m p e r a t u r e d is t r ibut ion in sandwich as compared with mixed s y s t e m s , since the locat ion of the thermocouple is s t r i c t ly fixed and the component in which it is inser ted is accura te ly known. An osc f l l og ram giving the t e m p e r a t u r e d is t r ibut ion in the middly sect ion of a l aye r of AP is p resen ted in Fig. 2. Close to the combust ion sur face is the ammonium pe rch lo ra t e decomposi t ion product in te rac t ion zone (T ~1200~ above l ies the m a x i m u m t e m p e r a t u r e (about 2600~ zone, the region of the diffusion f lame. The la t t e r t e m p e r a t u r e is c lose to the calculated adiabatic combust ion t e m p e r a t u r e of the s to ich iomet r i c mix tu re ( P F - A P ) . The two zones a re sepa ra ted by a plateau. The length of the pla teau, reckoned f r o m the combust ion sur face with the depth of the pit taken into account, is approximate ly p r o p o r - t ional to the burning ra t e and the square of the th ickness of the layer . This is typica l of a l amina r diffusion flame.

However, at a layer thickness of 2 mm and pressures higher than 40 atm abs. the burning rate de- pendence is d is turbed (at the s ame t ime the reproduc ib i l i ty of the expe r imen t s sha rp ly de te r io ra te s ) . As f i lming has shown, th is is a consequence of the f l ame becoming turbulent . The scale of the f luctuat ions is 0.4-0.5 m m .

When the the rmocoup les were inser ted in a l aye r of fuel, only the sur face t e m p e r a t u r e could be r e l i - ably es tabl ished; this was found to be 460~ fo r po lys ty rene and 230~ for polyformaldehyde. The t e m p e r a - tu re d is t r ibut ion is evidently se r ious ly d is tor ted when the thermocouple e m e r g e s f r o m the condensed phase owing to the act ion of the me l t or condensed res idue fo rmed at the surface .

252

Fig. 2. Typical osc i l - logram of the t e m p e r - ature dis tr ibut ion in a layer of AP (system A P - P F , ho=0.5 , (~n = 1).

Dependence of the Burning Rate on P r e s s u r e and the Nature of the Components. As p a r a m e t e r s charac te r iz ing the sandwich sys tem we used the thickness of the oxidizer layer h 0 and the mean component rat io a m.

The resul ts of a study of the effect of the nature of the components on the burning rate of a layered sys tem on the p re s su re interval 10-100 atm abs. are presented in Fig. 3. Clearly, on t ransi t ion f rom one component to another the burning rate may va ry by a factor of three. An increase in p r e s - sure also has a s t rong influence on the burning rate. For different pa i r s of components the exponent v in the combust ion law u= bpU lies on the interval 0.0-0.9. An exception is the composit ion based on naphthalene, for which at p r e s s u r e s above 50 atm abs. the u(p) dependence grows weaker.

The curve for the sys tem A P - s a c c h r o s e (Fig. 4) was recorded over the b roader p r e s s u r e interval 1-500 arm abs. Starting f rom 13-15 arm abs. the p r e s s u r e dependence of the burning rate remains very s trong with a

constant value of the exponent ~ = 0.8. At lower p r e s s u r e s the dependence is dist inctly weaker (v = 0.4), which is associated with the approach of the diffusion flame to the surface (Table 1).

Effect of the Thickness of the Layers on Burning Rate. These exper iments were conducted at the constant value a m = 1 for h 0 = 0.1-2.0 ram. The resul t s are presented in Table 2 and Fig. 5. As was to be expected, s tar t ing f rom a cer ta in thickness, which is the smal le r the higher the p r e s su re , the burning ra te increases . If we represen t the function u(h0) on this interval in the fo rm ah~ n , then for the sys tems P P - P F , A P - P F , and A P - P S at low p r e s s u r e s we obtain a value n = 0.3-0.4. (This may be compared with the value n = 0.9 obtained numer ica l ly in [9] for the sys t em ammonium p e r c h l o r a t e - p o l y s t y r e n e . The h{gh value of n, close to 1, is associated with the assumption that the p rocess is diffusional,) It is worth noting the constancy (and even a cer ta in tendency to fall, for the sys tem AP-naph tha lene , for example) of the burning rate of sandwich sys tems based on AP at elevated p r e s s u r e s and the presence of a plateau on the u(h0) curve for the sys tem P P - P S , which is observed at all p r e s s u r e s for h 0 <0.2-0.5 mm. The lat ter are too large for this effect to be attributable to t rans i t ion to the region of the kinetic regime, when complete mixing is possible before the chemical reac t ion s ta r t s [3].

/S

~ :

21

fj arm abs,

Fig. 3

2

2 5 to 20 50t00200500 ~, atm abs.

Fig. 4

Fig. 3. Dependence of the burning rate of mul t i layer sys - t ems on the nature of the components and p r e s s u r e (h 0 = 4.0 mm, cam= I). i) P P - P S ; 2) P P - P F ; 3) A P - P F ; 4) AP:-PS; 5) A P - fiaphthalene.

Fig. 4. P r e s s u r e dependence of the burning rate of A P - saccharose sandwich sys tems (h 0 = 0.5 ram, ~ = 0.12 ram).

253

20

i 7- "~3 A S s . 2 ,

o,, 42 4~ 45 7,~--~,o bOX' m m

Fig. 5. Dependence of the burn- ing rate of a sandwich sys tem on the thickness of the oxidizer layers, i) PP-PS; 2) AP-PF; 3) Corresponding curves for mixtures of the same compo- nents.

TABLE 1. Height of Diffusion Flame L above the Surface of a Layer of Ammonium Pe rch lo ra t e ( A P - P F mult i layer sys tem, am= 1)

Thiclmess of oxidizer layers, mm

0,5 1.0 2,0

L, mm, at a burning rate, mm/sec, of

2 P 4 I ~ to

0:i 10.% l 0.3, ~ ~176 o. o ,,o 1,2 1 , 3 0 1 , 9 0 [ - - - -

0,45 1,30

Other exper iments were per formed to determine the effect of the spec imen-averaged component rat io am on the burning rate of a sandwich system. The value of a m was varied by varying ~ (0.6-0.9 mm). According to the resul ts of the exper iments , the burning rate does not depend on a m (Table 3).

A N A L Y S I S O F T H E R E S U L T S

We assume that the components are gasified and then reac t chemically. If the layers are sufficiently thin, the gaseous components are able to mix completely within the heating zone before the chemical r e a c - tion begins. As the thickness of the l ayers increases , the mixing p roce s s begins to l imit the rate of con- sumption in the flame.

However, the resu l t s of a theoret ical analysis of the combustion of sandwich sys tems, based on the assumption of pure diffusion [7-9] (the burning rate depends strongly on h 0 and a m and does not vary with p ressu re ) , proved not to cor respond with the experimental data. This, apparently, is a consequence of ne- glecting the actual s t ruc ture of the flame.

By filming the p r o c e s s of combustion of sandwich sys tems it has been shown that the f lame front is not plane but highly distorted. In fact, the f lame is broken down into a number of tongues, which penetrate deeply into the condensed phase forming ver t ica l pits along the contact surfaces. If the layers are only f rac t ions of a mi l l imete r thick, the individual tongues may interact intensely with one another. If the l ayers are thick, there is little interact ion and the burning ra te of the sys tem will be s imi lar to the rate of propa- gation of the f lame front along the contact surface between infinitely thick slabs of fuel and oxidizer. We will consider this l imiting case in more detail, denoting the corresponding burning rate by moo.

TABLE 2. Effect of the Thickness of the Layers on the Burning Rate of Sandwich Systems (am= 1)

AP_ PF

AP- naphtha- lene

Pressure, atm abs.

10 40

100

Burning rate, mm/sec, at an oxidizer layer thickness ram of

2,3 5.7 8,2

3,0 6,8

8 ,6

Components 0,12 0,2 0,4 0,8 1,6

i0 4,4 3,8 3,2 -- 2.2 PP-PF 40 13,5 ' 10,3 8,0 6,3 5,8

100 26,5 21,2 16,0 ll,O 10,3

10 3,8 [ 3,3 2,5 2.2 9.2 40 9,2 i 7.8 6.8 6,2 6,2

lO0 14,3 13,2 13,3 12.0 12,0

2,6 6,1 9,0

254

o, o5'[

~/u

Fig . 6. R e l a t i v e b u r n i n g r a t e as a func t ion of the d i m e n - s i o n l e s s r a d i u s of c u r v a t u r e of the f l a m e f ron t . The f o l - lowing c o n s t a n t s w e r e u sed in the i n t e g r a t i o n : T c = 2800 ~ K; T f = 1500~ T o= 300~ @(T) ~= A(T- Tf) 2 exp. { - 2 0 0 0 / T } .

T A B L E 3. R e l a t i o n b e t w e e n the Burn ing Ra te of Sandwich S y s t e m s and a m (h 0 = 0.4 r am, p = 70 a i m abs.)

I Burning rate, mm/sec, at a ratio a m or" Components

1 0,2 0,35 0,5 0,7 1~0 J 6

PP-PS . . . . 16,5 14,7 16,5 15 ,5 i6,0 15,3 AP-PF . . . . 9,7 - - -- i0,3 10,0 9.5

In [5] the shape of the p i t was c a l c u l a t e d , and it was shown tha t t a k i n g into accoun t the c ond i t i ons of c o m b u s t i o n of the g a s m i x t u r e at the " r o o t s " of the f l a m e l e a d s to the e m e r g e n c e of a d e p e n d e n c e of the b u r n i n g r a t e m~ on k ine t i c f a c t o r s (as is a l so o b s e r v e d e x p e r i m e n t a l l y ) .

The r o l e of the f l a m e " r o o t " m a y be e x p l a i n e d a s fo l lows . The b u r n i n g r a t e moo is the r a t e of p r o p a g a t i o n t h r o u g h the s t a r t i n g m a t e r i a l of the t ip of a w e d g e - s h a p e d p i t and i s d e t e r m i n e d by the hea t f low r e a c h i n g the t ip f r o m the c h e m i c a l r e a c t i o n zone. In the g a s p h a s e r e - mo te f r o m the f l a m e " r o o t " the t h e o r y of c o m b u s t i o n of unmixe d g a s e s

i s a p p l i c a b l e [10]. Th is r e g i o n is c h a r a c t e r i z e d by d i f fu s ion b u r n i n g and the hea t f low f r o m the c h e m i c a l r e a c t i o n zone i s e n t i r e l y expended on hea t ing the m a t e r i a l d i f fu s ing into the r e a c t i o n zone to the c o m b u s t i o n t e m p e r a t u r e . The r a t e of p r o p a g a t i o n of the t ip of the p i t i s c h i e f l y s u s t a i n e d by the hea t r e l e a s e d in the " r o o t " of the f l a m e . It i s an i m p o r t a n t c h a r a c t e r i s t i c of the c o m b u s t i o n p r o c e s s tha t the r e a c t i o n at the " r o o t " of the f l a m e invo lves a p r e p a r e d m i x t u r e of g a s e o u s p r o d u c t s of fuel and o x i d i z e r p y r o l y s i s f r o m the h e a t i n g zone n e a r the t ip of the pi t . The wid th of the f l a m e " r o o t " is a p p r o x i m a t e l y equM to the width of the p r e m i x i n g zone , which i s of the o r d e r of the c h a r a c t e r i s t i c d i m e n s i o n of the p r o c e s s D / w ~ 100p (D is the m e a n d i f f u s i o n c o e f f i c i e n t , w is the g a s f low r a t e ) . At such s m a l l v a l u e s of D / w the f l a m e f ron t in the " r o o t " should be s h a r p l y c u r v e d , which l e a d s to the d i s s i p a t i o n of h e a t f ro rn the r e a c t i o n zone in the t r a n s - v e r s e d i r e c t i o n . T h e s e hea t l o s s e s , h o w e v e r , a r e c o m p e n s a t e d b y the p r e s e n c e of a d i f fu s ion f l a m e , which maintains the temperature in the chemical reaction zone at the level of the adiabatic combustion tempera- ture of the mixture.

This m o d e l of the f l a m e " r o o t " m a k e s i t p o s s i b l e to ob t a in a s i m p I e and conven ien t e x p r e s s i o n f o r the b u r n i n g r a t e m ~ . The p r o c e d u r e f o r c o m p u t i n g m ~ r e d u c e s to the s o l u t i o n of the p r o b l e m of the s t e a d y - s t a t e c o m b u s t i o n of a h o m o g e n e o u s m i x t u r e in a f l a m e of g iven c o n s t a n t c u r v a t u r e . The p r o p o s e d m o d e l d i f f e r s f r o m tha t adop ted in [5] b a s e d on b l o w - o f f of the d i f f u s i o n f l a m e owing to the coo l ing e f fec t of the g a s e s d i f fu s ing t o w a r d the f l a m e " r o o t " .

We a s s u m e tha t the t h e r m o p h y s i c a l p r o p e r t i e s of the c o m p o n e n t s a r e s i m i l a r . In t h i s c a s e n e a r the end of the p i t the p a r t i a l d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g the t e m p e r a t u r e and c o n c e n t r a t i o n d i s t r i b u t i o n m a y be a s s u m e d to p o s s e s s ax i a l s y m m e t r y . A c c o r d i n g l y , we rosy n e g l e c t the a n g u l a r c o m p o n e n t of the unknown func t ions . If we m a k e the u s u a l a s s u m p t i o n tha t the d i f fu s ion c o e f f i c i e n t s and t h e r m a l d i f f u s i v i t i e s a r e equa l , we c a n ob t a in the m a s s b u r n i n g r a t e f r o m the s o l u t i o n of the fo l lowing b o u n d a r y v a l u e p r o b l e m f o r a s e c o n d - o r d e r o r d i n a r y d i f f e r e n t i a l equa t ion :

) ] d ( r d r ) dT+Qr -7- d--T ~ + m~ c,, ~t r

(1)

The b o u n d a r y c o n d i t i o n s a r e : r --~ 0, T- - , Tc; r = r f , T = Tf; r = ~o, T = 0.

H e r e , r i s the d i s t a n c e r e c k o n e d f r o m a po in t wi th in the f l a m e wi th t e m p e r a t u r e equa l to the a d i a b a t i c c o m b u s t i o n t e m p e r a t u r e of the m i x t u r e Tc; r f i s the r a d i u s of c u r v a t u r e of the f l a m e f ront ; TS i s the t e m p e r - a t u r e at the f l a m e f ront ; and ~ ( T ) i s the hea t r e l e a s e func t ion d e t e r m i n e d b y the r e a c t i o n k i n e t i c s .

255

Equation (I) was reduced to the dimensionless form: d~O AC(O) d ~2 g ~f)

~ --~ co 0 ~ 0 ;

: ~ f , 0 : Of;

a~ I = ~ + f T-- Te r

0= - - ; R = r c - - Ta ~.

cpm~

g (~) : { exp [-- R (~)] ,}2;

; } (R) = -- e~ ( - R);

A = x ( 7 7 - To)(~-~.)'~

(2)

and integrated by a difference method in accordance with the formula

Q,,+ , = 20 . - 0n_, A �9 r (A ~,,)* (3) g (~,)

f r o m the limit ~ = ~f.

The burning rate moo was computed by a t r i a l - a n d - e r r o r method with an accuracy of • [13]. The resu l t s of the calculat ion can be represented in the form:

i n .

where rn 0 ~ pn/2 is the ra te of propagat ion of the f lame for plane symmet ry : a is the overall o rder of the chemical reaction; and F is a coefficient that depends on the dimensionless radius of curvature of the f lame front and is less than unity. A typical calculation is presented in Fig. 6.

We assume that the radius of curvature rf is equal to half the width of the premixing zone; then its value is determined by the distance H f r o m the" combust ion surface to the f lame front and by the rat io of the mixing rate to the gas flow rate. The quantity H is a function of m~ and is computed together with the m~ (rf) curve, as soon as the t empera tu re at the combustion surface is given.

If we assume that mixing is by molecular diffusion (width of mixing zone much less than the width of the pit), then the quantity

rf __ = 0.5 to 0.7 (5) ~.l cp r a

and in explicit f o r m does not depend on p re s su re . The corresponding value of the coefficient F is equal to 1/4-1/3.

The coefficient F does not depend on pressure. This means that the burning law moo (p) is determined by the order of the reaction; the obvious assumption of a bimoleeular chemical reaction leads to a burning rate proportional to the pressure (moo = bp).

The proposed flame model makes it possible to distinguish certain characteristics of the combustion of ordered systems.

The composition of the gas mixture that reacts in the flame "root" is one of the principal parameters determining the burning rate rn~. In this mixture the fuel-oxidizer ratio (a) depends on the nature of the components and the mixing mechanism and may differ appreciably from stoichiometry. The assumption of free molecular mixing implies that the quantity ~ is determined chiefly by the chemical composition of the component decomposition products (their mean molecular weight). An increase in the thermal stability of the components leads only to a change in the geometry of the pit, but does not affect mixture formation or the burning rate moo.

Observat ions have shown that the assumption of f ree mixing is valid for such volatile fuels as naph- thalene and PF.

Since we lack accurate data on the composit ion of the products of fuel pyro lys is , we can make only a rough es t imate of the value a for the components investigated. This shows, in par t icu lar , that in the case of composi t ions based on AP the gas mixture is highly overenr iched in fuel (a < 0.5). Dilution of the react ion

256

zone with fuel means that the combust ion t e m p e r a t u r e in the "root" T c may be comparab le with the t e m - p e r a t u r e of the p r i m a r y AP decompos i t ion f l ame (about 1200~ This enables us to offer some explanation of such fac ts (observed at e levated p r e s s u r e s ) as the absence of a dependence of the burning ra t e u of sandwich s y s t e m s based on AP on the th ickness of the l a y e r s (see Fig. 5) and the s imi l a r i t y of the burning ra te u to the burning ra te of pure AP.

The expe r imen ta l data on the effect of the nature of the components on the burning ra te of sandwich s y s t e m s (see Fig. 3) can also be explained if they are r ega rded f r o m the standpoint of the value of ~ for the gas mix tu re reac t ing in the f l ame "root" . There is a good co r r e l a t i on between u and calculated value of a .

In the combust ion of sandwich s y s t e m s the in te rac t ion of neighboring tongues of f lame, supe r imposed on the i r independent propagat ion, should lead to a weakening of the u(p) dependence (a reduced value of v in the combust ion law) at low and med ium p r e s s u r e s as compa red with moo (p). The jog in the u(p) curve for the s y s t e m A P - s a c c h a r o s e (see Fig. 4), obse rved at p = 13-15 arm, co r r e sponds to the onset of this effect. Obviously, a d e c r e a s e in the th ickness of the l aye r s should lead to an i nc rea se in the p r e s s u r e at which this effect begins [12].

Equation (4) makes it poss ib le to e s t ima te the m a x i m u m var ia t ion of the burning ra t e of a he t e ro - geneous s y s t e m due to reducing the pa r t i c l e s ize of the mix tu re components . In fact , m 0 is the burning ra t e of the homogeneous mix tu re (plane f l ame front) obtained when the component pa r t i c l e s ize is reduced to a min imum, and moo is the burning ra t e for infinitely l a rge pa r t i c l e s . In this case

mo 1 2,5 to 3,0 . (6) m F

This value is ingood ag reemen t with expe r imen t for mix tu re s based on volat i le components [3, 12]. It should be kept in mind, however , that the s ta r t ing a m of the f inely d i spe r sed mix tu re should be equal to the a in the " root" of the f l ame of the coa r s e sys tem.

We will briefly compare the laws of combustion of sandwich systems and ordinary mixtures. Curves 3 in Fig. 5 represent the variation of the burning rate of PP-PS and AP-PF mixtures with oxidizer par- ticle size. On the given interval of component particle sizes, the burning rates of the mixtures are every- where lower than those of sandwich systems similar with respect to the thickness of the layers and com- position. The difference does not exceed 30% and is observed for ail the pairs of components studied. As the thickness of the oxidizer layers is reduced (to 100-200#), the curves approach each other. This sug- gests that for near-stoichiometric mixtures the transient processes inherent in steady-state combustion do not play a very important role. At large oxidizer particle sizes an important factor is the transmission of combustion through the intermediate layers of fuel, which reduces the mean burning rate as compared with a sandwich system.

LITERATURE CITED

1. B . S . E r m o l a e v , A. L Korotkov, and Yu. V. Fro lov , FGV, 5, No. 2 (1969). 2. N .N . Bakhman and D. P. Pol ikarpov , Izv. AN SSSR, OTN, Energe t ika i Avtomat ika , No. 4, 37 (1961). 3. iN. N. Bakhman and A. F. Belyaev, Combust ion of Heterogeneous Condensed Sys tems [in Russian] ,

Nanka, Moscow (1967). 4. V . B . Librovich, PMTF, No. 4, 33 (1962). 5. J. Powling, Eleventh Syrup. (Internat.) on Combust ion (1967), p. 447. 6. N .N . Novikov, P. F. Pokhil , et al., Dokl. Akad. Nank SSSR, 174, No. 5, 1129 (1967). 7. B . V . Novozhilov, ZhFKh, 36, No. 11, 2508 (1962). 8. W. Nachbar , in. So l id-Prope l lan t Rocket R e s e a r c h , M. Summerf ie ld (editor), Academic P r e s s , New

York-London (1960), p. 146. 9. W. Nachbar and G. B. Cline, Fifth AGARD Colloquium, New York (1963), p. 317.

10. Ya. B. Zel 'dovich , Zhurn. Tekhn. F iz . , No. 10, 1199-1210 (1949). 11. A . A . Zenin, Candidate ' s Disse r ta t ion , Insti tute of Chemical Phys ics ASUSSR, Moscow (1962). 12. A . F . Belyaev, Yu. V. Fro lov , and V. F. Dubovitskii , FGV, 4, No. 1 (1968). 13. D . B . Spalding, Comb. and F lame , 4, No. 1 (1960).

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