T R A N S I E N T C O M B U S T I O N P R O C E S S E S IN

S O L I D - P R O P E L L A N T R O C K E T C H A M B E R S

Yu . A. G o s t i n t s e v , L. A. S u k h a n o v , I . M. K o f m a n , a n d A. V. S h m e l e v

UDC 536.46+662.311

A solution is obtained for a previously formulated [1, 2] sys tem of nonlinear equations for the volume- averaged in ternal -bal l i s t ics p a r a m e t e r s with nozzle opening. As distinct f rom previous t rea tments [3-8], the effects of the nonadiabaticity of the nonstat ionary flame and the incompleteness of the chemical react ions in it on the chamber p rocesses are taken jointly into account.

Previous theoret ical studies of the nonstat ionary p roces se s associa ted with propellant combustion in a rocket chamber can be conventionally divided into two ca tegor ies . Those in the f i rs t ca tegory [3, 4] are chiefly concerned with the inert ia proper t ies of the sol id-phase induction zone and the associated combus- tion nonstationarity. In this case the flame above the combust ion surface is assumed to be isothermal , and the energy equation for the gas is d isregarded.

In the second ca tegory [5-7] the energy equation is used for determining the gas t empera tu re in the chamber, and somet imes [8] the p r e s s u r e dependence of the net heat r e lease is taken into account. However, the actual combustion of the propellant is assumed to be quasis ta t ionary (burning rate and flame t empera - ture dependent only on the p ressure ) . At the same time, in a number of cases it is neces sa ry to give s imul- taneous considerat ion to the incompleteness of combustion and the recons t ruc t ion of the sol id-phase induc- tion zone. In fact, the charac te r i s t i c t ime for the induction zone t s ~ yc/K-2, and the charac te r i s t i c p r e s s u r e equalization t ime t k ~ W/(Pc) (% is the therm~il diffusivity, u is the s teady-s ta te burning rate , W and K are the free volume and the nozzle throat area , c is the speed of sound.) Accordingly, if the rat io X = tk / t s is close to unity, then the combust ion nonstat ionari ty must be taken into account, and only if • >> 1.0 is it possible to employ s teady-s ta te relat ions for the burning rate. In [I, 2] a complete sys tem of equations for the vo lume-averaged pa rame te r s was obtained on the basis of a combustion model with a quasis ta t ionary flame and a quas is ta t ionary sol id-phase react ion zone with allowance for both the above-mentioned effects.

(1) 1 0 9 , 0 9 0 %

~?(0,'0 = ~ (s~,cp), "e,( - ~,~) = ',~o, 9 (~ ,0 )=-0o+ ~ , -- ~~ e~ , (2) v = v (a,(p), -~'~ : e~ (a,q~),

% = % ( ~ , ~ ) , (3)

dn ~ , (4)

d}} g ,~,

. . . . . . , ' ~ ) - a = , (5) dw - - - : r 0 ~ . d, (6)

Here, we have introduced the nondimensional pa r ame te r s and var iables

To, ( ~ )-t+~ ~ T.~ T~ - - , "-- - - , - - : - - , - " 7 . , ~ d g : ~------, ' ~ s 7 : ~ ; ,

TF W ~]'1/2 ]// R'FF u p ,.,:.;,~- ~.{ "; Tg

TF t ~'x W { Ov '. I

Moscow. Transla ted f rom Fizika Goreniya i Vzryva, No. 4, pp. 476-482, Oc tober -December , 197i. Original ar t ic le submitted April 21, 1971.

�9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~'est 17th Street, New York, N, Y. lO011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. /t copy of this article is available from the publisher ]s,r $15.00.

409

where T0, Ts, TF, and Tg a re the initial t empera ture of the propellant , the surface t empera ture , the flame �9 t empera tu re , and the t empe ra tu r e of the gas in the chamber , respect ive ly ; S is the combustion surface; u is the burning ra te ; p is p r e s s u r e ; g is the nozzle throat a rea ; R is the gas constant; ~ is the t empera tu re gradient at the sur face in the direct ion of the solid phase. Steady-state values a re denoted by a bar .

Equation (1) de te rmines the recons t ruc t ion of the t empera tu re prof i le in the solid phase during the t rans ien t p r o c e s s e s . In this case for a complete formal descr ip t ion of nonstat ionary combustion within the f r amework of the phenomenological theory it is n e c e s s a r y to know th ree re la t ions [9] v(v, ~o), Os(~,r ~ F(V,~), which can be de te rmined e i ther exper imenta l ly f rom the s teady-s ta te re la t ions ~(p, T0) , Ts(P, T0) , TF(P,T~) , or theore t i ca l ly a f te r par t icular iz ing the s t eady-s ta te combust ion mechanism.

Equations (4) and (5) a re the volume-averaged continuity and energy equations. Averaging in the gas - dynamic equat ions is c a r r i e d out in the assumption that the tempera ture , gas velocity, and p r e s s u r e g ra - dients in the chamber a re small . This assumption is justified for re la t ive ly small engines with a complex geomet ry in the p re sence of intense turbulent mixing of the products escaping f rom the combustion surface . Averaging the energy equation leads to the appearance of a t empera tu re discontinuity at the flame front under nonsta t ionary conditions. In the case of an exact solution of the unaveraged equations of gas dynamics (this is possible only for a one-dimensional product flow), the re is no such discontinuity, and the f lame genera tes p r e s s u r e and ene rgy waves in the products . System of equations (1)-(6) is nonlinear, and in the genera l case an analytic solut ion is impossible. A l inear approximation of (1)-(6) was cons idered [2]; the same a r t i c le gives the r e su l t of a numerical calculat ion of the nonlinear sys tem for the case when the pa r am- e t e r s in the combustion chamber undergo changes owing to nozzle opening. The initial conditions for (1)- (6) a re thei r s t eady-s ta te solutions

~g (o) = ~f(o) = if(o) = u (o) = ~(o) = 2 ( o ) = 1,o.

The law of var ia t ion of the nozzle throat a r ea is taken in the form

0 ~ " ~ In ~

Z I e x p ~ ' a In Y.~

(7)

In solving (1)-(7) the pr inc ipa l difficulty is the nonstat ionary heat conduction equation (1) de termined in a region with a moving boundary. In o rde r to s implify (1), we employ the method of integral re la t ions [10]. The t empera tu re dis t r ibut ion in the solid phase is found among the functions

~ (~,~) = ~0 + (0~ - - ~0) exp [ (~) ~, (S)

where &s and f a re functions of t ime subject to de terminat ion that sa t is fy the initial values

#s (0) = b~, [ ( 0 ) ---- 1 ,0 . (9)

Then, af ter integrating (1) with r e spec t to the ~ coordinate between -~o and 0 and af ter ce r ta in t r an s fo r - mations, we obtain

- : X ( ~ - - ~o) ( [ - - v ) , (10)

In o rde r to de te rmine the nonstat ionary re la t ions ~s(V,~), v(~r, ~0), it is possible to employ the exper imen- ta l ly substantiated equations [11]

= u ~ p ~ e x p [ ( f ~ p - I - ~o) To], u = Dexp (-- . ._~ ) , E s

which af te r the usual convers ion to the gradient dependence give

Os q~ Oo ],

\

bs (~,~) ---- 1--elnv ' (12)

where k0r) = (/3 lzrp + fi 0) @s - To), ~ = (8 lP + fl 0) @s ~ To), ~ = RTs/Es" Now, using (l l) and (12) and the fact that f = ~o(~ s - $ 0)/(~s - $0), we el iminate the function b( r ) f rom {10) and obtain

410

- - g s - - ~ ( 1 - - e l--n v) ~ @ - k(~) - - d , fiiY) + A ) - s +

0s - - 0o 7~ - - ~o "

Here, A - In v - - v ln~ k 8.0

The d e t e r m i n a t i o n of the f l ame t e m p e r a t u r e dependence ~F(~r,~) is based on the fol lowing c o n s i d e r a t i o n s . At high p r e s s u r e s in the c o m b u s t i o n p r o c e s s the fuel e n e r g y Q is c o m p l e t e l y r e l e a s e d ; acco rd ing ly , the p r e s - s u r e dependence of T F is a s s o c i a t e d only with the ad iabat ic c o m p r e s s i o n or e xpa ns i on of the gas in the f l ame ,

i .e . ,

r e ( p , T o ) = - ~ To +'~-;~ ~ W ] "< ' ( P ~ P ~ ) '

where p~ is the p r e s s u r e above which c o m b u s t i o n is comple t e . At p r e s s u r e s p < P3 the c h e m i c a l c o m p o s i t i o n of the p r o d u c t s e s c a p i n g f rom the f l ame a l so beg ins to depend on p r e s s u r e and the f lame t e m p e r a t u r e is a p p r o p r i a t e l y d e s c r i b e d by the funct ion

Here ,

r ~ ( p , r d = ( 1 - - ~ . ) T ~ ( p , T d + ~ ~ o~-i-2~]\-~; ] .

~={10 at o>~m at p ~ p ~ ,

and T~(p, To) i s t aken f r o m the e x p e r i m e n t a l data . We a p p r o x i m a t e T~(p, To) with a p o l y n o m i a l of t h i rd d e - g r ee in the p r e s s u r e with coe f f i c i en t s tha t depend on the i n i t i a l t e m p e r a t u r e

Cs Q T~ (I3,To) -- -c;- To + ~ + b (To) ( p - - p~) + c (To) (p - - p~)~ + d (To) (p - - p~):'.

If for a c e r t a i n p a i r of spec i f i c p r e s s u r e va lues pl and p~ we know the e x p e r i m e n t a l dependences

T~ (p~,r0) = "Fe~ (To), TF (p,,To) = TF2 (To),

which can be r e p r e s e n t e d by the l i n e a r func t ions

T'--F~ (To) = a~To + b~, ~ ' m (To) = a~To + b.~ ;

then , a f t e r f inding the expans ion coef f ic ien t s and going through the u sua l p r o c e d u r e of the T O ~ ~ t r a n s i t i o n , for the n o n d i m e n s i o n a l f l ame t e m p e r a t u r e we have

t~F = (1 --- ~) g~A + O, 1 'v ~ - - ~ ~a [ ~_~_v_~ ,~ ~ ' + n (~ - - ~)"- + m (~ - - ~ ) ~ . + ~g~A + ~ , ~ ~ ) ~ , (14)

where

B 1

[ % \ ~ n~ + A l p ~- (~'l - n3) ~ -I- A.,.p ~ (.~ - - ~3) '~ ,

c s r .~ \~-__t _ BI-~ B . ~ - - - - Y ~ ] T / - - - -

O, - 0 A, = (m-- p3) c~ __ (m -- p~) c.. c~ T F ' (Pz--P~)~(P~--P 1) (P2--P3)2(P~--P 1) '

(Pl -- P3) s (P2 -- pl) (p*. -- p3) ~ (p~ -- pl) ' (P'- -- P3) ~ (P~ -- P0 (pl -- pa) ~ (p~. -- pl) '

d.. dx T- - I pl--p:~ , B2 ---- (p~__p3)~(p~__pl) - - (pl__p3)~(pe__px) , cl = aj. - - 1 - - ~ p3

ca = a.a - - I T - - I p2--p~ d 1 = b~ - - + T P3 ' Cp T

Q ( I + T - I P , - - P 3 ) d2 -~ b2 - - c--f T p~ "

411

We note tha t the ob ta in ing of the d e p e n d e n c e SF0r,~0) f r o m the e x p e r i m e n t a l da t a can be r e p l a c e d by a t h e r - m o d y n a m i c c a l c u l a t i o n .

S y s t e m of e q u a t i o n s (4)- (7), (11)-(14) was s o l v e d n u m e r i c a l l y on a c o m p u t e r fo r v a r i o u s nozz le opening r a t e s and v a l u e s of the p a r a m e t e r X (the change of f r e e v o l u m e d u r i n g the t r a n s i e n t p r o c e s s was not t aken into accoun t , i . e . , i t was a s s u m e d tha t 6 = 0.) As the i n v e s t i g a t e d p r o p e l l a n t we s e l e c t e d a m o d e l m i x t u r e c o n s i s t i n g of 75% NH4C104 wi th a 4 5 - p p a r t i c l e s i z e and 25% p o l y b u t a d i e n e b i n d e r . F o r th i s m i x t u r e m a n y of the p a r a m e t e r s have been e x p e r i m e n t a l l y d e t e r m i n e d [11], in p a r t i c u l a r ,

v=0,45, ~o= 1,2.10 -3 I/~ ~z=3" 19 -6 l/arm. ~

tl I = 0.092 c m / s e c (atm) u , D = 2241 c m / s e c , E s = 16,000 c a l / m o l e , R = 1.98 e a l / m o l e �9 deg , ~ = 1.87 �9 10 -~ c m 2 / s e c , c s = 0.03 c a l / ~ �9 g, p s = 1.5 g / c m ~, Q = 800 c a l / g , T O = 300~

U n f o r t u n a t e l y , no d a t a a r e a v a i l a b l e on the d e p e n d e n c e of the f l a m e t e m p e r a t u r e T F on the i n i t i a l t e m - p e r a t u r e for t h i s fuel . A c c o r d i n g l y , a s an a p p r o x i m a t i o n we e m p l o y e d the on ly known p u b l i s h e d func t ion T---F(T0) ob t a ined e x p e r i m e n t a l l y fo r a b a U i s t i t e [12]. Then the c o e f f i c i e n t s in (14) a r e equa l to a 1 = 0 .27 /~ b 1 = 1079~ a 2 = 11/~ b2 = 1650~ Pl = 1 a tm , P2 = 10 a t m , p~ = 30 a tm .

T A B L E 1

Variant - ~ / ~ - ~',cm 2 ~ ~

9,5 9,5

19,0 28,5

0,068 0,068 0,068 0,068

1,0

1,5

1,0

1,0

6,0

6,0

6,0

6,0

3 0 ,042

3 0,042 6 0,084 9 0,126

r i

/

- - Osl

~, v,

o,e ' \ [ e r ' N s

O,J " 3

0>4 0 ,8 2 6 rE

Fig. 1. - ,Yg, ~F, ~s' v, ~r, a as functions of the nondimen-

sional time T = t}t k for variants 1 and 2.

~6

o,o" 1,2

Fig . 2

"i " ~ J

,",a 2,4 ~

In the c o m b u s t i o n p r o d u c t s i t was a s s u m e d tha t 3, = 1.22, c 2 = 0.40, c s = 0.30, # = 29. The c a l c u l a t e d v a r i a n t s a r e p r e s e n t e d in the t ab l e .

The r e s u l t s of the c a l c u l a t i o n s fo r the d e p e n d e n c e s of ~_ , ,Y~, ~o , v, 7r, ~ on non- JL" O d i m e n s i o n a l t i m e a r e shown g r a p h i c a l l y in F ig . 1 for v a r i a n t s 1 and 2. F i g u r e s 2 and 3 i l l u s t r a t e the c o r r e s p o n d i n g d e p e n d e n c e s fo r v a r i a n t s 3 and 4. He re , in add i t i on to the a b o v e - m e n t i o n e d quan t i t i e s , the n o n d i m e n - s i ona l func t ion ~ fo r the net hea t r e l e a s e in the f l a m e i s a l s o inc luded . The e x p r e s s i o n for ~ can be ob ta ined f rom the e n e r g y c o n s e r - va t ion equa t ion for the i n e r t i a l e s s r e a c t i o n zones in the s o l i d p h a s e and the f l a m e [9]

_ - - _ _ ( , - - eo) - - e,, + .~ ,~. ( 1 5 ) Cp TF Cp

F r o m (15) and the g r a p h s i t fo l lows tha t l o s s of p r e s s u r e i s a s s o c i a t e d with the i n c o m - p l e t e r e l e a s e of c h e m i c a l e n e r g y in the c o m - bus t i on c h a m b e r , the u n d e r c o m b u s t i o n b e i n g d e t e r m i n e d not on ly b y the q(p) d e p e n d e n c e

I

o o,# ~,2 7,8 ,c

Fig . 3

v(~}

0,8

\\\ I ~

0,2 V~

0 1,o o,8 0,8 o,4 0,2 o

Fig . 4

~sCg') O,,~2a

O,&8a

Fig . 2. I n t e r n a l - b a l l i s t i c s p a r a m e t e r s a s funct ions of n o n d i m e n s i o n a l t i m e for v a r i a n t 3.

F ig . 3. I n t e r n a l - b a l l i s t i c s p a r a m e t e r s a s func t ions of n o n d i m e n s i o n a l t i m e for v a r i a n t 4.

F ig . 4. Burn ing r a t e v and s u r f a c e t e m p e r a t u r e ~s a s func t ions of the v a r i a b l e va lue of the t e m p e r a t u r e g r a d i e n t r in the s o l i d p h a s e for v a r i a n t s 1, 2, and 4 (the d e v i a t i o n of the v ( r c u r v e f r o m a s t r a i g h t l i ne c h a r a c t e r i z e s the d e g r e e of n o n s t a t i o n a r i t y of the p r o c e s s . )

412

but a lso by the ra te of r econs t ruc t ion of the so l id -phase induction zone. (Under nonsta t ionary conditions at l a rge values of the gradient ~ the combust ion p r o c e s s is qual i ta t ively the s a m e as at low initial t e m p e r a - tu res under s t a t iona ry conditions.)

Noteworthy is the osc i l l a to ry nature of the heat r e l e a s e and f lame t e m p e r a t u r e functions with per iod of the order of the chamber re laxat ion t ime t k. In Fig. 4 the su r face t e m p e r a t u r e and burning ra t e v a re plotted as functions of ~ for va r i an t s 1, 2, and 4. Clear ly, the combust ion of the propel lant in the chamber can be cons idered quas i s t a t iona ry only at X -> 10.

We have intentionally r e f r a ined f r o m invest igat ing the question of quenching of the propel lan t with loss of chamber p r e s s u r e . This is because of the l ack of a ma thema t i ca l quenching c r i t e r ion for a p r o p e l - lant with a va r i ab le su r face t e m p e r a t u r e and because , if the p r e s s u r e falls s teeply, the f lame t e m p e r a t u r e is sha rp ly reduced, and the c h a r a c t e r i s t i c reac t ion t ime m a y become comparab l e with the res idence t ime of the gas in the chamber . Clear ly , in this case our phenomenological model of nonsta t ionary p rope l lan t combust ion with a thin f lame is inapplicable, and the engine is desc r ibed by the s y s t e m of equations for a homogeneous chemica l r eac to r , whose solution in the genera l case should a l so provide the answer to the question whether sha rp p r e s s u r e drops r e su l t in the quenching of the propel lant .

LITERATURE CITED

1. Yu. A. Gost intsev, L. A. Sukhanov, and P. F. Pokhil, Dokl~ Akad. Nauk SSSR, 195, No. 1 (1970). 2. Yu. A. Gost intsev, L. A. Sukhanov, and P. F. Poldlil, Zh. Pr iM. Mekhan. i Tekh. Fiz. , No. 4 (1971). 3. Ya. B. Zel 'dovich, Zh. Pr iM. Mekhan. i Tekh. Fiz. , No. 1 (1963). 4. B . V . Novozhilov, Zh. Pr ik l . Mekhan. i Tekh. Fiz., No. 5 (1962). 5. A . A . Shishkov, Gas Dynamics of Solid Prope l lan t Rocket Motors [in Russian], Moscow (1968). 6. N. Cr ie r , J. Tjen, et al. ,AIAA J. , No. 2 (1968). 7. J. Tjen, W. Sirignano, and M. Summerf ie ld , AIAA J. , No. 1 (1970). 8. R . E . Sorkin, Gas The rm odynam i cs of Solid Prope l lan t Rocket Engines [in Russian], Moscow (1967). 9. Yu. A. Gost intsev, L. A. Sukhanov, and P. F. Pokhil, Zh. Pr iM. Mekhan. i Tekh. Fiz., No. 3 (1971).

10. Yu. A. Gost intsev, Fiz. Goreniya i Vzryva, 3, No. 3 (1967). 11. M. Summerf ie ld , L. Kaveny, e t a l . ,AIAA Pape r 70-667 (1970). 12. A . A . Zenin, O. I. Leipunskii , e t al. , Dokl. Akad. Nauk SSSR, 169, No. 3 (1966).

413