combustion parameters of gas-permeable fuels

COMBUSTION PARAMETERS OF GAS-PERMEABLE FUELS

V. G. Korostelev and Yu. V. Frolov UDC 536.463:662.612.3

In [i], an analysis was undertaken of the dynamics of perturbed combustion of gas- permeable fuels. In the present work, on the basis of the results of this analysis, the dependence of the basic combustion parameters on various parameters is determined.

The broad range of variation of the combustion rate of gas-permeable fuels [2] indicates a variety of forms of combustion of heat transfer determining the combustion conditions of such systems. In the general case, the following mechanisms may be noted:

i) conductive, the basis for layer-by-layer combustion;

2) convective, which predominates in convective conditions of combustion and acts to- gether with conduction in perturbed combustion;

3) radiant heat transfer, which is evidently of definite importance only in the combustion of systems of loose packing and gas suspensions;

4) heat transfer as a result of phase transitions: heat is transported by the combustion products into the depth of the fuel, and is then liberated on phase transition. For example, the propagation of combustion in a porous charge of dusty powder occurs as a result of the motion of jets of molten salts [3, 4].

The possibility of heat transfer is larger in a gas--liquid phase transition, since, as a rule, the latent heat of vaporization exceeds the latent heat of fusion, and the gas phase is of higher mobility than the liquid phase.

The possibility of the condensation of gaseous combustion products in pores was noted in [2], but its influence on the heat transfer was not considered. At the same time, esti- mates show that the heat liberation on condensation may be considerable even in the case when only some of the combustion products are involved. Thus, if the combustion products include 30% water (compositions based on PCA), then 18% of the total heat supply in the gas is liberated when this water condenses. This heat is not scattered over the whole of the heated region but is liberated in a relatively narrow region, where the gas is cooled and, therefore, convective heat transfer is relatively ineffective. It should be emphasized that the combustion products may also include compounds with a larger latent heat of vaporization than water.

Since the condensation temperature and the heat of phase transition depend on the pressure, the contribution and even the possibility of the condensation mechanism of heat transfer will be determined by the pressure in the heated region.

Phase transitions are associated with a sharp change in heat-carrier properties and, hence~ also in heat-transfer laws. This inevitably influences the filtration conditions in the heated region (as a result of change in gas-permeability of the fuel). If the given temperature is large, condensation is evidently able to give rise to ignition of the condensed phase.

Heat liberation in the heated region may be associated with chemical reactions between the condensed phase and the combustion products. Both the liberation and the absorption of heat are possible here. The total heat liberation in the heated region will depend on the combustion of the fuel and combustion products. This offers the possibility, in principle, of regulating the combustion rate.

In reactions between the combustion products and the condensed phase, associated with decrease in the number of moles of gas, and also on gas--liquid phase transition, there is a decrease in pressure in the heated region. This intensifies the gas filtration from the combustion region and the heat-transfer processes. If the reaction proceeds with the libera-

Translated from Fizika Goreniya i Vzryva, Vol. 18, No. 3, pp. 25-32, May-June, 1982. Original article submitted May 27, 1981

280 0010-5082/82/1803-0280507.50 �9 1982 Plenum Publishing Corporation

tion of gas, on the other hand, the total heat transfer is reduced. Thus, there must be a definite correlation between the various heat-transfer mechanisms, and also between the heat transfer and the chemical reactions in the heated region.

With perturbed combustion, the above-noted forms of heat transfer may occur simul- taneously. The contribution of each will be determined by the combustion conditions and the properties of the system. Therefore, the calculation of combustion processes requires data on the dynamics of the real heat transfer in the pores of the fuel. However, such integral characteristics as the combustion rate may, in some cases, be calculated even without knowing the heat-transfer dynamics. Such a calculation must be based on the heat balance of the system, without the use of heat-transfer coefficient, whose value is not known in advance.

An example of the calculation of the parameters of perturbed combustion will now be considered, using the ideas regarding the peculiarities of such combustion developed in [i]. The following assumptions are made.

I. In the direction of propagation of combustion, only some of the heat required for ignition of the fuel is transported by the gas. The rise in rate of combustion u C is proportional to the increase in the convective component of the heat transfer.

2. With steady combustion in the pores preceding the change-over region [5] by a length equal to the width of the heated layer of condensed phase, there is the same amount of gas there as participates in convective heat transfer in the lag time of condensed-phase ignition t L [i].

3. The ignition temperature of the condensed phase T I is equal to the temperature of the combustion surface in normal combustion.

4. The time t L is equal to the time for the creation of the heated layer in normal combustion. This condition is satisfied if the pore diameter does not exceed 40-100 pm [i].

5. When convective combustion occurs, the amount of heat transferred in the condensed phase by convection is the same as the amount of heat present in the condensed phase for normal combustion (heat liberation due to chemical reactions in the heated region is in- significant).

6. Combustion does not influence the gas-permeability of the fuel.

7. There is no passage of combustion through individual pores. Accumulation of gas in the filtration region is eliminated by the mechanism of discrete dispersion [5].

8. The whole mass of the condensed phase participates in the heat transfer in the heated region.

9. Filtration occurs under the action of the initial pressure difference Apo at the inlet to the pore. The filtration region may be modified by changing the external pressure po (quasisteady filtration conditions).

The heat-balance equation is written in the form

Here VG = STmoL, V C = ST(I -- m)l C are the volumes of the gas and condensed phases participating in the heat transfer; cC, CG, specific heats of the condensed phase and the combustion products; ST, cross-sectional area of the charge; IC, width of the heated region; L, penetration depth of gas participating in the creation of the heated layer; TG, maximum (adiabatic) combustion temperature; mo, m, open and total porosity of the fuel; AT---C, mean temperature rise of the condensed phase in the heated region as a result of convective heat transfer from the gas; T, mean temperature of the gas which has penetrated into the pores; 0G~ PC, densities of the gas and condensed phase.

Taking into account that cCPC = ~C/aC and CGOG = p/(k -- I)T, Eq. (i) yields

L (i -- ~) ~ C(k-- I) AT---6 (2)

zc ~c ~oT(r~/~-- ~)

where aC, X C are the thermal diffusivity and thermal conductivity of the condensed phase; p, mean pressure of the gas which has penetrated into the pores; k, adiabatic index.

281

It is expedient to introduce the coefficient n, taking the contribution of convection in the overall heat transfer into account. Knowing the heat supply in the condensed phase in normal combustion (e.g., [6]), the following relation may now be written

%c ---~(~ - - TN) n = CCOC.Ic (J - - m) A-Tc=-~N u: C (l -- m) A'-IC

where u N is the rate of normal combustion; T N is the initial temperature of the condensed phase. Taking Eq. (3) into account, Eq. (2) takes the form

(3)

L ( k - - t ) Xc (TI --TN) n UN ( 4 )

A c c o r d i n g to t h e a s s u m p t i o n s which have b e e n made, L = l C and p = po -- Ap/2 (hp i s t h e p r e s s u r e d i f f e r e n c e o v e r t h e l e n g t h , e q u a l to l C p r e c e d i n g t h e c h a n g e - o v e r r e g i o n ) . Us ing Eq. ( 4 ) , i t i s found t h a t

y_c= 2(k--1)Xc(~--zN) =

m0 (2p 0 - - Ap) (T,G,/Y -- Q"

The f o l l o w i n g n o t a t i o n i s now i n t r o d u c e d

',. c ' o ( r t) ,,c a = 2 ( k _ l ) ~ c ( T i _ r N ) , b = ~ , z - - UN,

-

p = PN(T-/TN) • T = T N + ' ( T o - - T . ) / l n T._----~N ],

where ~N i s t h e v i s c o s i t y o f t h e c o o l e d g a s e o u s c o m b u s t i o n p r o d u c t s ; V, mean gas v i s c o s i t y of t he h e a t e d r e g i o n ; T , , c u t o f f t e m p e r a t u r e o f t h e t e m p e r a t u r e p r o f i l e o f t h e g a s i n t h e f u e l p o r e s ; K, g a s - p e r m e a b i l i t y o f t h e f u e l .

S u b s t i t u t i n g t h e v a l u e o f hp f o r po ~ PB (PB i s t h e e x t e r n a l p r e s s u r e a t which to = t L) [1] g i v e s

2n Z = a ( 4 p o - - b z 2 ) ' Z > t , n > O . ( 5 )

The constraint on the values of z and n arises because the case when n = 0 and z = 1 corre- sponds to normal combustion~ which is not described by this equation; in fact, substituting n = 0 into Eq. (5) yields an indeterminacy of 0/0 type, since -T = T G when n = 0, and so

a=0.

At the onset of convective combustion (po = P,) [I], the relation p = (p, + pN)/2 applies, and n = i. This allows the dependence z,(p,) to be determined from Eq. (4): z, = i/a(p, + PN)" The pressure p, is determined from the condition t L = 4to(hp,) [i]:

P* = PN+ bz2./4. (6)

S u b s t i t u t i n g t h i s v a l u e of p , i n t o t h e p r e c e d i n g e x p r e s s i o n l e a d s to t h e e q u a t i o n

z3, -}- 8PN/b. z , -- 4/ab ---- O; (7) 8

when p, >>PN' z, ~ ]/'6/ab. Taking account of the foregoing, normalization of n is intro-

duced: when z = i, n = 0; when z = z,, n = I. At other points of the curve z(po), it is

assumed that n is proportional to the combustion rate

n = (z - - t ) / ( z , - - t ) = x (z - - t ) , ( 8 )

and then Eq. (5) is written in the form

2 ( x ) 2x (9) za-t- --b- -7--- 2Po z - - - ~ - = 0 , z > l , Po~PB"

Knowing Ap [i] and taking Eq. (8) into account, Eq. (4) leads to an equation for the combustion rate in the region PB < Po < P,

Z2 + Cb-'-~O ~ a ] aCbAPo

282

rUK7 a

- era/me A ~ f

I I I I 1 I t I

O. 1 10 100 /

100

10

L

/

+ o..9 + I

A / /

uCN >zX / -

Z ; /~u 'L

071 !0

+1 02 ",3 A4 05

100 '~ atrn

I __J______a

100 POI atm

Fig. i. Pressure dependence of the combustion rate for 85% PCA + 15% PMMA, mo = 0.14, K = 7"10 -*2 cm ~ (a), 85% PCA + 15% PMMA, mo = 0.118, K = 4"10 -12 cm = (b), and PCA, m = 0.056,

mo = 0.005, K = 1.265"10 -12 cm=(c): 1-4) combustion rate at the first (from the igniter), second, third, and first two bases; the curves correspond to calculation.

Using Eqs. (9) and (i0), the rate of perturbed combustion may be determined in the already- formed filtration region.

The initial ("start") rate of perturbed combustion is determined by the heat supply in the gas which has penetrated into the pores in a time t L after the onset of filtration at a depth L B [i]. The gas pressure in the pores p = (Po + pN)/2.

The mean gas viscosity in the filtration region varies as a function of the relation between the initial width of the heated region ZCN and L B . If the notation ~z = ~ -- ~N is introduced, then

283

TABLE I

... arm

4 10 30 50 8O

/ C - i 0 s , a i m

7,7 5,8 6,0

t3,9 102,0

s e c

2,94 3,t2 t,37 0,t5

0,002

ti~ toL s e c

35 t5,4 5,8 3,6 2,35

n . t O 3

0,6 i,7 8,3

4i,5 443

g

t,07 1,2 2,0 6,0

54,4

~C era/ s e c

0,22 0,37 t,0 3,9

43,4

TABLE 2

Par ameters

m 0

K. 10 ~, cm ~ u ~P0), crdsec TG, ~ TI, ~ g $

p,, atm Zbr Pbr ' arm

Z n

Pn' atm Pl' atm

~C

Composition

A B

0,14 0,1t8 7 4

0,ti t .poO,45 3000 3000 723 723 130,4 i2i,4 92,1 117,5 t,53 t,52

t6,2 20,5 20,6 t9,6 46,7 59,5 t4,5 t8,5 t,643 t,642

C

0,005 t,265

0,0i "Po t300 630 888,7 553,8

1,5 93,t5 73,5 277,5 t38,7

Note. PN = i arm, T N = 300~ ~N = 1.8"10-~n, k = 1.25,~ =i,

T, = 310~ ~C = 8"i0-~ cal/cm' sec'deg, a N = 1.5"10 -2 cm2/sec.

= [/CN~ + (LB--IoN )~N]/LB= ~N + ~'IcN/LB Substituting ZCN/L B from here into Eq. (4), it is found that

On the other hand

Hence

"~= ~tNq- ~t~ . a(po + p N ) /n �9 z N

r

lCN /L B ~ 2 N V 0,656Kacm~ = a (p~ -}- PN:) ZN /n"

(ii)

= [2a(p0 + pN)/n]2"~Apffb. (12)

Equating Eqs. (ii) and (12) and substituting for n from Eq. (8), an equation is obtained for determining the initial combustion rate

Z~ 2X~N+ . . . . . . a~l (Po + PN ) . T " X~N-- 4a2~AP~ (Po + PN)2/b = O. (13) X~N@a~l (P0 + p N ) x[x~N+a~l (Po + PN) ]

I n t h e c o u r s e o f f o r m a t i o n o f t h e f i l t r a t i o n r e g i o n , t he c o m b u s t i o n r a t e w i l l d e c r e a s e f r o m t h e i n i t i a l v a l u e i n Eq. (13) to a f i n i t e v a l u e t h a t may be d e t e r m i n e d f r o m Eqs . (9) and ( 1 0 ) .

In F i g . l a - c , e x p e r i m e n t a l [1] and t h e o r e t i c a l d e p e n d e n c e s u c ( p o ) a r e p l o t t e d f o r c o m p o s i t i o n s A-C, r e s p e c t i v e l y , t o g e t h e r w i t h t h e r e g i o n o f c l o s u r e of t h e f l u x o f o u t f l o w - ing c o m b u s t i o n p r o d u c t s [7]

W ~ ~ ,Po _ f ( k @ i ) M vC~'--m) V 2kBTG (14)

284

where M is the mean molecular mass of the combustion products. Additional data on the calculation are given in Tables I (see Fig. la) and 2.

Overall, the calculation agrees with experiment. The existing differences are associated mainly with the influence of passage of combustion through individual pores of the fuel. In any case, the greatest discrepancy is observed precisely in the region where the development of through passage is most probable [i].

As follows from the experimental data, the dependence uc(Po) (in logarithmic coordinates) in the region of convective combustion (Po ~ P,) may be represented as a straight line. If it is assumed that the slope of this straight line coincides with the slope of the function uCN(Po) at the point Po = P,, then the pressure dependence of the rate of convective combustion may be plotted.

Differentiating Eq. (13), and taking into account that z = bz'p~ z and dz/dpo = ~z" z/po, it is found that

4a2~ [2a~lp~ + (3x~ N + 4a~lp ~) p~ + 2p Np, (x~N-- ~ 1 ; N) -- b v C ~ VN@ z~.*" x[2xPN(z,--t)@abl(po+pN)(2z,--t)][X~N@ "-+

- x ,Np~] + x2a, N,I(~ . -- t) --+ + ~ (P* + PN )]

The theoretical values of p,, z,, and ~C are shown in Table 2, and from these data the dependence uc(Po) is plotted in the region of convective combustion in Fig. la, b.

With decrease in porosity of the sample, the discrepancy between calculation and experiment increases (Fig. la, b). This may be explained by the influence of the sample strength on the pressure in the combustion region. If, in the calculation, po is increased by the amount of the pressure difference fixed in the sample pores on combustion (for composition B with po ~ p,, it is 15-20 arm), the agreement between calculation and experiment is improved.

In [I], it was noted that the form of the dependence uc(po ) is not always as in Fig. la, b. In fact, whereas closure of the flux of outflowing gas occurs in combustion under a pressure p,, beyond the breakdown pressure (Pbr) even a relatively small increase in po causes transition of the combustion to explosion. Hence, calculating u, and p, from Eqs. (6) and (7) and the region of closure in Eq. (14), the character of the combustion when po > Pbr may be determined. For example, it is evident from Fig. la, b that u, < w; therefore, transition of the combustion to explosion does not occur. Calculation for hexogen shows that its combustion, in those conditions where it has been observed [i], must pass to explosion. Experiment confirms this conclusion.

On the curve of uc(Po) , several characteristic points may be isolated -- above all, the point corresponding to the breakdown pressure of perturbed combustion, at which the following condition is satisfied [I]

d/dp.(to/t L) = O. Using t h i s c o n d i t i o n and Eq. ( 9 ) , t he breakdown p a r a m e t e r s may be d e t e r m i n e d

4PN--x/a Zbr= ]/2(6Pbr--4PN--x/a)/3b' Pbr ~ 6

Values o f Zbr and Pbr a r e g i v e n i n Table 2 and F i g . 1.

I t f o l l o w s f rom the c h a r a c t e r o f the dependence uc (Po) when po < p , t h a t t h e r e i s a p r e s s u r e Pn a t which the c o m b u s t i o n r a t e i s most s e n s i t i v e to change in po . At t h i s p o i n t ( t h e p o i n t o f i n f l e c t i o n on the c u r v e ) , d2z /dp~ = 0. For t h i s p o i n t , Eq. (9) g i v e s

~ f ~ x z ~ = ] / ~ , p ~ = ~ .

Values of z n and Pn are given in Table 2 and Fig. la, b.

The parameters of points corresponding to the condition t L = to(Apo) are determined graphically -- by the intersection of the curves described by Eqs. (9) and (i0). The points obtained (with coordinates PB and u B) are shown in Fig. la, b.

285

With normal combustion, the width of the heated region l N decreases continuously with rise in pressure (l N ~ p~ VN ). With perturbed combustion, the width of this region l C changes nonmonotonically. At small Apo, convective heat transfer is only able to restrain the decrease in l C. But at a definite pressure Pl, the decrease in size of the heated region ceases and this region then begins to grow. At the point of the minimum dz/dpo = 0. This condition allows the combustion parameters at which 1 C is minimum to be determined

X

zz : ~ 2 [2p~ (i + i/~N) - - x/a]/3b, pz ~ ~a (l + I/~N)"

The pressure Pl may be both larger (Fig. le) and smaller (Fig. la, b) than Pbr. Dis- crepancy between Pl and Pbr indicates that expansion of the heated zone in itself does not create the conditions for rapid rise in combustion rate.

In conclusion, note that the present work has compared calculation and experiment only when PN = i arm and p, >> PN" The applicability of the given equations with other relations between the pressures p, and PN requires additional verification.

l,

2.

.

4. 5. 6.

.

LITERATURE CITED

V. G. Korostelev and Yu. V. Frolov, Fiz. Goreniya Vzr~za, 18, No. 2 (1982). A. F. Belyaev, V. K. Bobolev, et al., Transition of Combustion in Condensed Systems to Explosion [in Russian], Nauka, Moscow (1973). J. D. Blackwood and F. P. Bowden, Proc. R. S.c. London, A213, 285 (1952). F. A. Williams, AIAA J., 14, No. 5, 637 (1976). V. G. Korostelev and Yu. V. Frolov, Fiz. Goreniya Vzryva, 15, No. 2, 88 (1979). B. V. Orlov and G. Yu. Mazing, Principles of Rocket-Motor Design [in Russian], Mashinostroenie, Moscow (1964). B. S. Ermolaev, Candidate's Dissertation, Inst. Fiz. Khim., Akad. Nauk SSSR, Moscow (1978).

ANALYSIS OF THE THERMAL COMBUSTION REGIME OF LIQUIDS

A. V. Guzhiev, G. S. Sukhov, and L. P. Yarin

UDC 536.46

Diffusion combustion of liquids whose surface is exposed to the flow of a gaseous oxidizer is described in [i]. Similar solutions of dynamic, thermal, and diffusion problems were obtained on the assumption of an infinitely large reaction rate, and the temperatures of the flame and free surface and mass burning rate of the liquid were found. Owing to the known limitation of the model used the results obtained do not reflect a number of important effects caused by the occurrence of the reaction in bulk- the phenomena of flameout, flashback of the reagents through the combustion zone, etc. In connection with this it is expedient to examine the process of combustion of a liquid in a more general formulation reflecting the finite rate of transformation of the components.

Henceforth we will confine ourselves to an examination of the problem of combustion of a semiinfinite liquid layer exposed to a homogeneous flow of a gaseous oxidizer (Fig. i). As before [I], for the analysis we will use an approximation of boundary-layer theory and the assumption of constancy of pressure in the entire flow region. In this case the initial system of equations has the form

p~z~ �9 8 ~ 1 8 x + p~v~ . Ou~lOy = OISy �9 (~t~ . OuJSy) , (1)

Op~uJSx + 8p~v~/Oy = O, (2)

OT i OT~ 0 ~,~ _~ (2__ i) q tW ' (3) cp~piu~ ~ -1- cp~piv~ Oy coy Oy /

Leningrad. Translated from Fizika Goreniya i Vzryva, Vol. 18, No. 3, pp. 32-39, May- June, 1982. Original article submitted August 6, 1981.

286 0010-5082/82/1803-0286507.50 �9 1982 Plenum Publishing Corporation

combustion parameters of gas-permeable fuels

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