Convective combustion of porous compressible propellants

COMBUSTION A N D F L A M E 89:260-270 (1992)

260

Convective Combustion of Porous Compressible Propellants N. N. S M I R N O V and 1. D. D I M I T R I E N K O Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia, C.1.S. This abstract describes the combustion of solid porous propellants, focusing on burning regimes characterized by a convective mechanism of propagation. A new mathematical model of the dynamics of multiphase chemically reacting media, including interphase heat and mass transfer, is developed in the course of the investigation. Mathematical simulations of the process prove theoretically that there exists a self-sustained combustion regime, driven by convection. This regime is supersonic with respect to the gas phase and subsonic with respect to the condensed phase.

NOMENCLATURE p

density projections of velocity on 0 X and 0 Y axes accordingly e internal energy E total energy T temperature Lamet's parameters ~ 2 , /~2 P x ~ , P x y , P y y components of the stress tensor of the solid phase h thickness of the phase layer burning rate constant Ufo n burning rate index pressure ( = - p l 6 i j + 7ij ) Po crack length L ot volume concentration of phase H,

I)

Subscripts 1 2 0 s w e

gas phase solid phase initial phase interface ( y = h j) on the boundary ( y = H ) in the chamber ( x < 0)

INTRODUCTION This article deals with the process of flamespreading and combustion of solid propellants that contain longitudinal cracks and pores. The physical problem may be described as a solid-propellant combustion process that is sustained by convection of heat into the unignited sections of the propellant crack. This convection is achieved by the hot gaseous products that flow 0010-2180/92/$5.00

into the crack and ignite the surface of the solid propellant, generating additional hot gaseous products. Since solid propellants have high burning rates, the rate of gas generation within the crack is high. Therefore, the pressure increases rapidly during this flame-spreading process. The pressure rise enhances the pressure-dependent burning rate of the propellant, the result of which is a rapidly accelerating combustion process, with generally steep gradients along the length of the crack. Thus, a high-velocity gas flow towards the crack end is achieved. The rate of combustion due to this convective mechanism of heat transfer surpasses that resulting from thermal conduction by several orders. Further, this convective mechanism of flame spreading may evolve into a regime in which the crack surface is ignited by means of the shockcompressed gas flow, in which case the flamespreading velocity is determined by the velocity of the shock wave. The high pressures that arise within the crack because of the gas generation on the propellant surface cause solid-propellant deformations. Stress waves spread through the solid phase ahead of the flame front, causing partial narrowing or even closing of the channel or pore. This, in turn, exerts an influence on the flame-spreading regime, especially in the case of small pore diameters. Therefore, it is necessary to examine the combined wave field of both gas and solid phases in the flame-spreading process, taking into account the deformation tensor for the solid phase, the phase interactions, and the energy generation due to burning. References 2 - 9 deal with different models of Copyright © 1992 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc.

COMBUSTION OF POROUS PROPELLANTS flame spreading in pores and cracks of an incompressible solid propellant. Reference 3 contains a mathematical model of heterogeneous two-phase reacting media (obtained as a result of averaging), in which the condensed phase is considered deformable but incompressible, and pressure discontinuities on the phase interface are neglected. In Ref. 4, an empirical law for the compressibility of the solid phase is assumed. Experimental results for the flame-spreading in an individual pore or crack are presented in Ref. 10 and 12. A description of the process of crack propagation, without consideration of the dynamic processes in the solid phase, is given in Refs. 13 and 14. Numerical and experimental results of flame spreading in the crack are produced in Ref. 15: the solid phase is assumed to be incompressible, and the solid-phase stresses and deformations are determined by the static balance equation after determination of the gas pressure. Hence the spreading wave of stresses in the solid phase is tied to the gas pressure wave, and the dynamic processes in the solid phase are not considered. Solid-propellant compressibility is taken into account in Ref. 16, where the mass, momentum, and energy equations for the gas and solid phases are solved numerically, and the solid phase is assumed to be linear-elastic. The present article deals with averaging of the two-dimensional differential equations, the consequence of which is that a one-dimensional model of the flame-spreading process is constructed and solved numerically, as described later.

261

I4

t"it0 8

0

~/

T 2 = 2 T o. For the solid propellant at the sides (x = O) the following conditions are assumed: Pxx = - P c ; Pxy = Pxz = 0; T2 = Te.

F ( x , y , t) - y - h , ( x , t) = 0.

Vj = O,

rxy = O;

y = H:

u2

Pxy = 0.

=

O,

+ 1

(4) '

with the initial condition Yo = h~(x, to). If the angle between the phase interface and Ox axis is small, then the projections of the normal ~ ( n x, ny), towards the phase interface, for each point of the surface are determined by Oh l OX

nx=

(1)

(3)

If D = D n , the interface velocity in the direction normal to the surface, then the interface form is determined from

)

y = O:

(2)

At the side x = 0, Pe and Te are distributed uniformly; thus the plane deformed state is realized in the solid phase, and all functions are independent of coordinate z. The form of the phase interface is assumed to be

- D

We consider a system of periodically repeating (along the axis 0 y ) cracks: on the Oxy plane the crack has a rectangular section. Because of crack symmetry and the equivalence of all stresses for each crack, we consider half the crack (Fig. 1) with the boundary conditions

5e

assumed that the phase interface temperature must he greater than some critical temperature, say

0h, (t0h, 12 at \ ax

MATHEMATICAL MODEL

/,

Fig. I. Geometricalmodelof the crack.

ny

1,

(5)

and from Eq. 4 Ohi

The section x = L is the solid wall. At the side ( x = 0) the gas pressure rapidly rises at the rate d P e / d t , and hot gaseous products reach the pores or cracks. For ignition, it is

D -

Ot

(6)

The velocity of the phase interface D is the sum of the burning rate and the solid-phase veloc-

262

N . N . SMIRNOV AND I. D. DIMITRIENKO

ity in the direction normal to the interface: D = u r + u2n x + U2ny Oht u 2 OX + % .

= uf-

(7)

The mass, momentum, and energy conservation equations in unsteady, two-dimensional forms are considered for gas volume (i = 1) and solidpropellant volume (i = 2):

OPi

3PiUi

Ot

~

--+

OpiJdi - +

-

Ot

-

3PiVi

+

3y

OPiUi 2 - + 3x

i = 1,2,

-0,

(8)

O(rxx - Pi) + -Orxy Ox

Oy

,

3Pxy

+

OpildiUi

at

Ox

i=1;

i=2;

o--7'

Ot

-~=x +

OPxy

3y

Opyy

Tx + 77'

OPxy at

OPiUi2

O(ryy -- Pl)

Pxy = 21Z2exy,

ap.~

+ - Oy

O'rxy

It is necessary to consider the equations of state for the solid phase. In terms of stresses

(exx, exy, ezy are components of the deformation tensor). These were differentiated and the dependence between mechanical deformations and stresses invoked:

(9) -OPiVi - + - -

[ [JU2

3x

-

(14)

' 3u z

+ x2 0-7

(15)

i=1,

The internal energy of the solid phase in this case is determined as follows: i = 2; e 2 = Q + c 2 7"2

1 + --(p2xx+2/x2P2

3t

(13)

OIJz )

- " 2 t aTy +

0,

,

a%

au z

- (K2 + 2#2)-ffx--x + ~ O--y-'

3 p . y = (X2 + 2/~2) 0 %

(10) -3piEi - + -3piEiu - +i

(12)

e, = c v i T 1.

pyy = ()k2 "4- 2]~2)6yy -'[- ~2exx

Oy

3Pxx

Pl = p I R T I ,

Pxx = (X2 + 2~2)exx + X2%y,

OPildiO i -

-

For the gaseous state the perfect gas law is assumed:

OpiEil)i -

2PxY+PZx)

X

3y

4/z202(~k2 + /A2) (Pxx "~ pyy)2,,

0

~XX((--Pl + rxx)U 1 --qxl + "rxyVl)

+0-70 (,rxyUl+(fyy_pl)vl_qyl) i=1 O

where Q is the specific chemical energy of the condensed fuel. The boundary conditions on the phase interface are o , ( D - u l n x - Vlny) = O2uf;

(16)

(rxx - pt)nx + rxyny

--0 (PxyU2 + PyyV2 -- qy2)

+ O,u1(D - u,n~ - v,ny)

+ 3y

= P x x n x + Pxyny +

i=2 (11)

r x y n x "4- ( T y y -- p l ) n y

O2U2/gf;

(17)

COMBUSTION OF POROUS PROPELLANTS + Olvl(D-

ulrt x - Vlny )

= PxyHx + p y y H y + fl2U21df;

263 to decrease because of gas generation:

(18)

in (1 + B)

Zxy~f

u2)

B

'

Txys. f

(20b)

+ [~x.,,~+ (-p, + ~.),,.] o,- qo, + p , E , ( D - uln x - 1)lrly )

Heat flows are determined by q., = aet(Tt - Ts)'

q~2 = a ~ 2 ( T s -- T2)' (21)

= (Pxxnx + p x y n y ) u 2 + (p~yn~ + pyyny)V 2 + P 2 " f E2 -- q.2

-

B=

Txy s

[ ( - p , + rxx),,~ + ~ . . ~ ] u ,

p2uf(u,

-

(19)

Thus the complete system of 12 differential equations is formed for the following functions:

;,(x, y, t), . , ( x , y, t), ~,(x, y, t), el(x, y, t) (i = 1,2), Pxx(X, y, t),

I where ael = ~-CfSRP~CplIUl -- U21. SR = ~r CR~ /XI and X1 is the coefficient of heat conduction; ae2 = ~ / h 2 when Ts < T*. ae2 = CzpeU f when T~ > T*. The phase interface temperature Ts may be obtained from the boundary condition (Eq. 19). The burning rate law is a pressure-dependent law allowing for erosion

u

[P')"

(22)

Uf = K e • fo~ Po

pxy( X, y, t), pyy( X, y, t), hi(x, t). No account is taken of the gas viscous stresses which are small in comparison with the component of the spherical component P t of the stress tensor, and the axial heat conduction Xi(aTi/Ox), which is small compared with the convective heat transfer. The drag force between the gas and the solid propellant 7"xys, and the heat transfer from the gas to the propellant, q, through the interface were considered. These interactions must be represented by empirical correlations. When the crack surface is not burning the drag force Txy s is determined by Txx , Tyy,

Txys

"o,(u,

-

.~)~

= cf = k Re-1/4,

The averagings for the gas and the solid propellant were introduced in the following way:

(¢),

1 f0h~(x,t) ( x,

- ~

(¢,)~ -

y , t)

ay,

~( x, y , t) ay. H

-

h I

~{x, t)

The differential equations for the gas phase (i = 1) (Eqs. 8-11) were integrated from 0 to ht(x, t) by means of the rules for integration with variable upper limit: oht(x, t) 0

a---~(x, y,t) ay

(20a)

3 Oh 1 = -d-dx(h,l~l,) - ~ x ~ ( X .

h , ( x . t) , t), (23)

where

pl(Ul - u2) 4Fl Re-

/zl

Pl

/xt is the coefficient of dynamic viscosity, F~ is the cross-sectional area, P~ is the cross-sectional perimeter of the crack, and k is the friction coefficient, which may be explained as a coefficient of the rough surface. Theoretical and experimental investigations show the drag force Txysf of the burning surface

hj( x, t)

fo

OtCe(x, y, t) dy a (h~(¢4,) =at

3h~ ¢~(x, h i ( x , t) , t), ~7 (24)

ohm(x,t) (9

~y~(X,y,t)

dy

= ¢,(x, h , ( x , t), t) - ¢/(x,O, t).

(25)

264

N.N. SMIRNOV AND I. D. DIMITRIENKO

Equations 8-11 and 13-15 for the solid propellant were integrated from h i ( x , t) to H , using the relations 5-7 and the boundary conditions 16-19. For example, the mass balance equation was obtained as O

3

3

3

t)2s

3---t( h,(o,),) + -#-£x( hl(p,u,),)

3

I)2s

=

-Jo2s

where subscript s corresponds to the value of the function on the phase interface. From these averaging equations new unknown functions appeared, such as (plU~)l, (pl//12)l, (plvl)~, etc., and the interface values of the functions &s, Uls, Vms, etc. Hence the number of the equations became less than the number of variables. To solve the complete system we introduce an additional hypothesis concerning the distributions of the main functions along the 0 y axis; these distributions are assumed to be constant or linear. It is possible to postulate more complicated distributions, for example,

(31) 3 3 3 3 t u , PlE, + f f - x 0 / l P l E l U l + -ff-x0/lPlUl 30/1 30/! + TxysV2S-~x + PyysO2s OX

= JE2~ + (rxysU 2 + pyysV2 - q ) / H , (32) 3

3

3

3 % 332 -~xPxYV2S y + rxysV2s 3 x 30/2 + PYysO2s 3 x

~b°( X, y , t) = ~b,°( X, t) y + ~b2°(x, t) y 2. However, in this case the number of functions to search for increases. By averaging the two-dimensional differential equations we obtain the quasi-one-dimensional system of equations:

= - J E 2 s - (r~ysU 2 + pyysV2 - q ) / H ,

(33) 3

3

- ~ o t 2 P x x -- P x x - ~ O t 2

P212f

0/1 : hl / H ,

3

~xOqP, u, :

+

J,

(26)

3

3

3

3 0/I -

P'

ox

-g0/2o .2 +

Txys

-

3

JU,s

0/2m"2

3 0/2 + Pxx-= - Ju2 3x

2

3

2pxx

Pxys --

-

-

H

00/2 [d'2O2s3 x '

(35)

30/2

3u 2 = 2~0/2- 2(;~ + 2t~2)V2s/H. 3X (36)

3

-

3a 2

-~0/2(Pyyw + Pyys) - 2pyys 3t

(28)

-h-'

+

(34)

-- Pxys 3 t

O O2s = ~2~X0/2"-2-

(27)

~0/,p,u, + O---£0/,p,u12+ 3--x%p,

3

3 ~ot2Pxy

0

-~-7Od2P2 "~ 0-~0/~02U 2 = - - J ,

~k2I)2s

= (k2 + 2t~2)C%~xU2 - - - H - - '

H

3

a

3

ol 2 : 1 - Oil, d : - -

-~elP,

3or 2 - Pxys 3 X

-- ( P y y s -- P y y w ) / H ,

p2Uf,

=

Jols ,

3t0/202 ~ - + -~xx0/202u2 2 3 OX0/2Pxy

P l s ( D - U|sn x - V|s )

- -

(30)

3

=

30/1

3~0/lPlO I + 3 x 0 / l P l U I U 1 -Jr- 7"xys 3X

,

With the boundary conditions 16-19 and the condition of sticking on the phase interface (29) L/I -- UIF/x ~- U 2 -- 0 2 n x ,

(37)

C O M B U S T I O N OF POROUS P R O P E L L A N T S the boundary values of the functions are U,s = / / 2 -

(p2/p,

-

265

7/3,4 : //1 --F al ,

1)Ulnx, /

l) l s =

U2S-

(02

/Pl

7]5,6 : /12 --F

--

1)/,/f,

77,8 = /'12 H- ]/

X2 +

V Pyy, : - P , + PzUf( 1 - P2/Pl),

(Pl +

Pxys = r x y s - -

PZ"~(P2/P!

+pxx).x.

(38)

In this way, the complete system of 12 differential equations (Eqs. 7, 2 6 - 3 6 ) using the empirical relations for the phase interactions is obtained. In the present calculations a modified form of the two-step method of the L a x - W e n d r o f f scheme [ll was used. If the subsonic regime of gas penetration into the crack is achieved, then two functions of the gas phase may be set at the crack entrance ( x = 0), for example, Pe and T~. It is enough to set only one boundary condition (for example, Pe) in the case of subsonic flow out of the crack at x = 0; for supersonic flow out of the crack it is unnecessary to set the boundary conditions for the gas phase at x = 0. The values of the other parameters can be determined from the conditions on the characteristic lines approaching the section at x = 0 . To obtain the conditions on the characteristic lines, the systems of Eqs. 2 6 - 3 6 and 7 was written as

OW --+C 3t

OW =~,

3x

where W = {p~, P2, ul, u2, vl, V2s, el, e2, Pxx, Pxy, Pyyw, ~r}. With the characteristic equation ("2

-

-

x((.= - .)2_ ((//2 -- ")2

-

.)2

_

P2

~k2 -I-O 2//'2) ' p 2:

we obtain characteristic directions

, , ( x , t, W ) = d x / d t ,

"1

=

//1'

"2

=

P2

2#2

P2

Then, obtaining the self-vectors ]k (see Ref. 14), we find the characteristics conditions:

I)

-

-- ,

//2'

]kdW= ]*gdt : f k ( x , t, W ) dt. COMPUTED RESULTS The calculations were for the values Q = 3.58 × 10 6 J/kg, Po = 1 atm, P2o = 1700 k g / m 3, n = 0.41, Ufo = 2.5 × 10 -3 m/s, T O = 295 K, T* = 2To, c 2 = 1400 J / k g K, cpl = 1 0 0 0 J / k g • K, col = 713 J / k g • K, ~2 = 0.92 × 10 9 N / m 2, #2 = 0.96 × 10 9 N / m 2. Figure 2 presents the characteristic development of the combustion process within a crack of dimensions: L = 10-~, m, hi0 = 2.5 × 10 - 4 m, H = 2.5 × 10 -3 m. The distributions of velocity, pressure, and gap width are presented for various times. The boundary condition of a pressurization rate at the crack entrance of 2.5 × 10 4 M P a / s was assumed. Figure 2c shows that the calculated pressure distributions for various times have characteristic shapes. In the initial period, the gas flow from the high-pressure chamber forms the compression wave, which moves to the tip of the crack. The pressure behind the front rapidly rises, due to the high rate of gas generation at the solid propellant surface. The flame front reaches the compression wave and the maximum pressure in the crack cavity becomes higher than in the chamber ( x < 0). The gas compression wave and the wave of solid-phase stress component pyy m o v e with nearly the same speeds, but the wave of stress component Pxx essentially outstrips the gas compression wave. Ahead of the flame front, the local minimum of the gap width is formed. It moves with the wave to the crack tip, and the minimum width decreases in time. The gap at the location of this minimum is partially closed (Fig. 2a). In these figures the initial form of the crack is indicated by the dashed line. The velocity distributions for various times are shown in Fig. 2b. As time progresses, the veloc-

266

N . N . SMIRNOV A N D I. D. D I M I T R I E N K O

:i: t.03 .OZ

.00 fL'~

t

0

120 ca.

\

4O F i g . 2. C a l c u l a t e d g a p w i d t h (a), v e l o c i t y (b), a n d p r e s s u r e (c) d i s t r i b u t i o n s for t i m e s tj = 6 . 0 7 ×

0 &0

02

0.4

0.6

0.8

92/L ity and gas pressure profiles change. During the initial period when the crack surface does not burn the curve has a " p l a t e a n " form. With the generation of burned gases as the flame front spreads through the channel, the pressure p~ increases in the combustion zone. As p~ increases, the velocity u 1 at the left edge decreases, and after the rate of the pressure increase within the channel has become higher than that in the chamber, the gas begins to flow out of the channel. The increase of the pressure in the combustion zone leads to the increase of the velocity u~. The secondary wave forms (the u I velocity peak. Fig. 2b) and the front of convective combustion reaches it. The secondary wave with the combustion front reaches the leading shock wave (at time

1.0

10

.5 s, t 2 = 1.05 × 10 - 4 s, t 3 = 1.24 × 10

s,

t 4 = 1.45 × 10

t 6 = 1.82 × 10 4 t 8 =2.38

dPe/dt =

× 10 - 4

4 s, s,

s,

t 5 = 1.64 × 10

t 7 = 1.99 × 10 - 4 and

s,

pressurization

4

4 s,

and rate

2.5 × 104 M P a / s at c r a c k entrance.

moment t s, Fig. 2b). Here the self-propagating wave structure forms. It contains the leading shock wave and the combustion zone just behind it. The ignition of the propellant surface is caused here by a flow of shock-compressed gas, so that a shock-wave (not convective) regime of reaction zone propagation takes place. When the right solid wall at x = L is reached, the reflected shock wave forms in the channel. The calculations for different boundary conditions show that with the development of the combustion spreading process along the channel a speed of the flame front propagation D , , reaches a constant value. Values of these speeds are nearly equal for the cracks with the same geometrical dimensions and different boundary condi-

COMBUSTION OF POROUS PROPELLANTS

267

'1600 £.3 'u3

IzO0 ,~00

4-00 0

D.O

02

0.4

0.6

t. 0

O.g

~/L Fig. 3. Speed of the flame front propagation D , for the following boundary conditions: (1) ignition of phase interface near crack entrance (0 < x < O. 1 L ) T s = T*; (2) pressurization rate d P e / d t = 2.5 × 104 MPa/s in chamber ( x < 0); and (3) Pe = 30 MPa in chamber ( x < 0).

tions, but during the initial period the speeds are essentially different. Near the solid wall, x = L, the speed of the flame front propagation decreases. In Fig. 3 are presented the distributions of the speed of the flame front propagation along the channel for the following boundary conditions: (1) a constant rate of pressure increase in the chamber d p e / d t = 2.5 x 10 4 MPa/s at x --- O, (2) the forced ignition of the solid surface near the crack entrance (0 < x < O. l ' L ) , where the phase interface temperature is determined as equal to the ignition temperature ( Ts = T*) and the pressure in the chamber is constant (pe = O. 1 MPa) and equal to the initial pressure in the crack and (3) the chamber pressure is constant, Pe = 30 MPa at x = 0, and the initial pressure in the crack is essentially less ( p o = 0.1 MPa). In the numerical experiments of the convective combustion in cracks, the following parameters were varied: the gap width hlo, the channel length L, the pressurization rate d p e / d t in the chamber, the friction coefficient of the phase interface k, the burning rate constant Ufo.

6O \

40

tZ2t.

20

60 40

_

-

-

(b)

-

pt

2

\

20 0 O.

i

O.6

i

0,8

Fig. 4. For time t = 1.41 x 10 -4 see the calculated pressure and velocity distributions with the coefficient k in friction law Eq. (20a): (a) k = 0.033; (b) k = 1.00.

268

N . N . SMIRNOV AND I. D. DIMITRIENKO

The calculations show that in the case of narrow channels (h < 1 0 - 4 m) gases hardly flow into the channel, and the compression wave hardly penetrates into it, when the friction is high. For wider channels (h > 10 -3 m) the flame front propagation is slower, because the incoming gasphase reaction products from the phase interface

have less influence on the gas pressure increase within the channel, due to the larger volume occupied by the gas phase. Consequently, an optimal dimension of the gap width hlo = 0.25 - 0.5 × 10 -3 m exists, when the used speed of flame front propagation is a maximum. Comparisons in Fig. 4 of the results of numeri-

Z3Z

.oo 2

-i

6O

t5 4O \

,tz

20 m

Cd]

£3

6OO 400 ¢q 20o 0

•

I

0.0

0.1

~/L

O.2

Fig. 5. Calculated gap width (a), velocity (b), and pressure (c) distributions within short crack ( L = 2.5 × 10 _2 m) for times t I = 3.3 × 10 - 5 s, t 2 = 7.4 × 10 5 s, t 3 = 9.6 × 10 _5 s, t 4 = 1.06 × 10 4 s, t 5 = 1.16 × 10 4 s, 16 = 4.08 × 10 4 s, and speed of flame front propagation D . (d) for pressurization rate dPe/dt = 2.5 × 104 M P a / s in c h a m b e r ( x < 0).

COMBUSTION OF POROUS PROPELLANTS

269

cal calculations for the channels with h~0 = 0.25 X 1 0 - 3 m, L = 10-~ m, and various values of friction coefficient in Eq. 20a show that with the lower value of friction coefficient the compression wave in the gas spreads about twice as fast, but with the higher value it is more intense, and the gas flows from the channel earlier. Analysis of the numerical results show that for different channel sizes (h~0, L) there can be different regimes of flame propagation. When the pressure rapidly decreases in the chamber, ~ombustion may be extinguished. Investigations of the propagation of combustion were made for different rates of rise of chamber pressure Pe. At first

it rises with d p e / d t = 2.5 x 1 0 4 MPa/s, until it reaches Pe*, then it falls with d p e / d t = 1 x 105 MPa/s. The calculations for the crack with the same dimensions as are presented in Fig. 2 and with the last described boundary conditions show that, after attaining the self-sustaining propagation regime, the speed of flame front propagation D . no longer depends on the variation of the boundary conditions as the entrance x = 0. Figure 5 presents values of gap width (a), velocity (b), pressure (c) and the speed of flame front propagation D . (d) for the short crack (L~ = 2.5 × 10 -2 m). All these are for the following boundary conditions at the entrance x =

\ o

,

•

80

4o 20 0

~D ca3 ¢.o

3

.+a

2 1 0

, i.O

t

0.2

I

I

l

0.4

a:/L

i

0,6

I

I

O.g

:0

Fig. 6. Calculated velocity (a), and pressure (b) distributions for times t I = 1.22 x 10 - 4 s, t 2 = 2.06 X 10 - 4 s, t 3 = 2.50 x 10 - 4 s, t 4 = 2.72 × 10 - 4 S, t 5 = 2.92 x 10 - 4 s, t 6 = 3 . 3 1 x 10 - 4 s, t 7 = 3.50 X 10 - 4 s, and trajectory o f flame front (c) for pressurization rate d P e / d t = 2.5 X 104 M P a / s in c h a m b e r ( x < 0) and with coefficient Ufo = 1 x 10 - 3 m / s in burning rate law Eq. 22.

270

N . N . S M I R N O V A N D I. D. D I M I T R I E N K O

0: pressure Pe rises with d p e / d t = 2.5 x 10 4 M P a / s , until it reaches p *e = 30 atm, and then it falls with d P e / d t = 1 X 10 5 M P a / s . Formation o f the reflected compression wave occurs prior to the onset o f the convective propagation regime. As a consequence, the m a x i m u m speed D . o f the flame front propagation decreases. The region 0.9L~ < x < L~ of the phase interface near the solid wall ( x = L~) generally does not burn, since it is heated extremely slowly. The process o f combustion propagation for the smaller value o f the coefficient Ulo in Eq. 22 is shown in Fig. 6 (Ufo = 1 x 10 - 3 m / s ) . Because o f the lower burning rate of solid propellant a smaller volume o f gaseous products is generated and a lower gas pressure p j occurs within the channel. As a result the speed o f the flame front propagation D . in this case is less than D . in Fig. 2. The gaseous products o f reaction move ahead o f the combustion wave and they heat up the phase interface. Hence, with less intensive burning the propagation mechanism is by convection, not shock-wave. CONCLUSIONS 1.

2.

3.

4.

A new quasi-one-dimensional mathematical model o f the dynamics o f a multiphase chemically reacting media, including interphase heat and mass transfer, is presented. The calculations were for different conditions at the crack entrance, and values o f the speed o f flame front propagation compared. The dependence o f the speed o f flame front propagation on the characteristic parameters (the gap width hi0, the channel length L , the friction coefficient o f the phase interface k, the burning rate constant Ufo ) has been determined numerically. The calculations show that with sufficient time the non-stationary wave process might

enter the self-sustaining propagation regime, independent of the conditions at the crack entrance. This regime is supersonic with respect to the gas phase, but subsonic with respect to the condensed phase.

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