P R E S S U R E V A R I A T I O N D U R I N G T H E I N I T I A L

P E R I O D O F C O M B U S T I O N O F A S O L I D P R O P E L L A N T

B. T . E r o k h i n a n d B. A. R a i z b e r g UDC 536.46+662.215.2

Pobedonos t sev and Zel 'dovich es tab l i shed that a d e c r e a s e in duct a r e a causes a sha rp i nc rea se in p r e s s u r e and leads to unstable c h a m b e r p r o c e s s e s . In ce r ta in c i r c u m s t a n c e s "chuffing" may develop; in this case the p r e s s u r e d i a g r a m has a discontinuous c h a r a c t e r . This c rea ted a need for a c r i t e r i on that would make i t poss ib le to s e l ec t initial conditions such that the instabi l i ty is suppres sed .

The c r i t e r i on ~ , i .e. , the ra t io of the burning sur face of the duct a rea , p roposed by Pobedonos tsev has p roved to be the m o s t viable. At the s a m e t ime, it has been noted that this c r i t e r i o n by no means c o m - p le te ly c h a r a c t e r i z e s the p r o c e s s and in this sense lacks genera l i ty . The p r e s s u r e fluctuations a re a lso affected by turbulent combust ion, hydrodynamic l o s se s , flow velocity, the geome t r i c c h a r a c t e r i s t i c s , and other fac tors .

Natural ly, these effects should be re f l ec ted in the c r i t e r ion . In order to obtain a c r i t e r ion c h a r a c t e r - izing the p r e s s u r e r i se , as our s t a r t ing s y s t e m we shall take the s y s t e m of equations of motion of the c o m - bustion products in a duct with a gas supply d is t r ibuted along its length. Using the quas i s ta t ionar i ty p r inc i - ple (which e s sen t i a l ly cons is t s in neglecting the t e r m s ref lec t ing the nonsta t ionar i ty of the p r o c e s s in the genera l s y s t e m of equations) for the one-d imens iona l flow model , a s suming that the gas is an ideal nonheat- conducting fluid, we obtain the following sys t em,cons i s t i ng of the flux, momen tum, and ene rgy continuity equations :

0 -~x 9Fw = p s hit,

oF&" - - F o f Ox '

-~x 9Fw C.~, T 4- A = 9s huQs A -~x p r ~ . (1)

Here , h = 7r d is the wetted p e r i m e t e r of the duct; p is the gas density; p is p r e s s u r e ; T is t e m p e r a t u r e ; F is the duct a rea ; Ps is the propel lan t densi ty; u is the burning ra te ; Qs is the heat r e l ea sed in the combus - t ion of unit m a s s of propel lan t ; /~ is the t h e r m a l equivalent of work.

In o rde r to c lose s y s t e m of equat ions (1) it is n e c e s s a r y to add the equation of s ta te

p/o=g•r (2) and the burning law

u=u(gw). (3)

Other things being equal, the burning ra t e in a turbulent flow is bas i ca l ly de te rmined by the flow veloci ty and degree of turbulence. The deg ree of turbulence at the head of the duct depends chief ly on the development of the burning sur face , i .e. , on the p r e s e n c e or absence of a compensa to r . The express ion for the burning ra t e in a turbulent flow has the fo rm

u = u l p " [ l + K ~ V w - - w t 8]; (4) 0, if w < W t

6 = t, if ~ > / w t ,

Moscow. T rans l a t ed f rom Fizika Goreniya i Vzryva, No. 4, pp. 488-492, O c t o b e r - D e c e m b e r , 1971. Original a r t i c l e submi t ted June 3, 1971.

�9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 [Fest 17th Street, New York, N. Y. 10011. 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. ,1 copy of this article is available from the publisher for $15.00.

419

where u I is the unit burning rate ; v is the p re s su re exponent in the combustion power law; K w is the tu r - bulent combustion coefficient; w t is the threshold flow velocity corresponding to the cr i t ical nondirnensional number

Yet = r~Yll (5)

(The cr i t ica l number Ycr is understood as the number at which :turbulent eddies penetrate into the fizz zone.)

Here, Ylt is the width of the flame zone and v is the kinematic viscosi ty of the combustion products. The quantitative value of the threshold velocity is p r imar i l y determined by the degree of turbulence and the flow velocity in the entrance pa r t of the duct. In this sense to speak of the value of the threshold velocity for a specific type of condensed sys tem independently of the combustion conditions (presence or absence of a compensator) is meaningless , since it has a par t i cu la r value in each individual case. We make the addi- tional assumption that in sys t em of equations (1) the a rea of the duct is constant along its length, i. e., OF/~x e 0. This assumption cannot have much effect on the quantitative and qualitative variat ion of the

p r e s s u r e r ise , since the inc rease in a rea along the length of the duct is small as compared with the a rea itself, i.e., A F / F << 1.

F rom the second and third equations of sys tem (1) we find the p re s su re and density in t e rms of the veloci ty

p=Ho

~)0 ~--- l-[o

W ~

2 (6) k + l w~ '

i~ 2(k-- 1)

k k -- i (7)

io + k + 1 w 2(k--O

Here,

FIo= Po + pow~;

�9 k po Wo 2 t ~ Po +- '2-"

F rom the joint solution of the equation of continuity for the flux and equation (7) we obtain

or af ter differentiation

d H k ----:--F w o ps hu

i + k - a l 2(k-- I) w2 F

(8)

(9)

i o - - k + 1 w 2 k 2 (k-- I) dw _ Ps hu (10)

[ k+, ]~ dx P Hok--I Io4 2(k--l) w~

Combining relations (4) or (5), depending on the shape of the duct, (6), and (10), we obtain

~o 2 ( k - g 2 a~ _ psh.,k--~ ~ - ~ " ( 1 - K ~ ~ a ) . (ii) =- /~ +_____~I w~ k + l ax pn~-~ k i o + 2 ( k _ i )

i0 + "2 ( k - 1)

Equation (11) is the s ta r t ing equation for determining the law of variat ion of flow velocity along the length of the due t on the turbulent combustion interval. In deriving the lat ter equation it was assumed that the burning rate depends on the static p r e s s u r e in the gas flow, i.e., that it inc reases along the length of the duct owing to the turbulent combustion effect and dec reases as a resul t of the fall in static p res su re . The la t ter effect is especia l ly wel l -expressed at gas flow velocit ies commensurable with the speed of sound.

We go over in Eq. (11) to the relat ive fl0w velocity

] • - - I) ' ~ T r lo

420

obtaining

1 - - ~ d ~

l + ~ dx k--1 psh u ( k + !

- - k Fl'1~ -v 2 ~ = 1) l~

'1 - - k - - 1 )v �9 _ L + t ~" ~:Vf t0 (L- -~ t )~ , (12)

where

/~,t = wt

~#-TT io (13)

After separat ing the var iables and integrating (12), we obtain

I/: (1 --s k - -1 p~ hullo h~- 1 (14)

,0 (1-}-L z) I - - Z2 14-Kw - ~ _ f lo ()~--~t)5

We note that, if X < Xt, inevaluating the quadrature on the Ieft side of Eq. (14) it is necessa ry to set Kw=0. To simplify the calculations we denote

k - - I \v V 2 i k _ l ) (I+~2)2-v I - - 7 , I ~ ) i-}- Kw k + l

where �9 is the pr imit ive.

It is easy to see that for known values of X 0 and II 0 Eq. (14) determines the dependence X =X(x), i.e., the law of variat ion of relat ive velocity along the length of the duct. F rom (14), (15), and the relat ion for the s teady-s ta te p r e s s u r e it is possible to determine the p r e s s u r e r i se : a) for a condensed sys tem with a compensator

l ( p~0 = ~k--4-T] s F t (16)

P0 = P s t r (~l) - - r (~)

b) for a ducted sys tem with ze ro flow veloci ty at the duct entrance

L po ~ ~-~ -- = - - = S F l

Pst ' �9 ( ~ t ) '

where l is the duct length; S is the combustion surface. In determining the polynomial ~ (X) for the outlet par t of the duet it is f i rs t necessa ry to find the relat ive velocity coefficient k z . With allowance for the hydrodynamic losses this can be determined f rom the t ranscendental equation

where

fn q (Z,) = (~t ~) -- n (xt, ~)-&[ F~ [ ' ~,:a

IJ(Lt ,~)= P0cr= 1 -- ~k , ~ , ) ~ . Pot k- - l ~(M) '

is the hydrodynamic loss coefficient in the duct outlet - n o z z l e throat zone; S e is the leading-end combus- tion surface; S Z is the total combustion surface; q(X), v(X), r (x) are gas -dynamic functions; P0l and P0cr are the total p r e s s u r e s at the duct outlet and in the nozzle throat . From the known value of the relat ive veloci ty coefficient it is possible to de termine the quantitative value of the polynomials �9 (X/) and �9 (X0). The distinguishing feature of the function ~ (X) for ducts with and without a compensator [in accordance with the express ions for the turbulent burning ra te (4)] is the difference in the threshold velocit ies.

From relat ions (16) and (17) it is possible to determine the p r e s s u r e r i se due to turbulent combustion and hydrodynamic losses and thus select the optimal duct dimensions.

421