CRITICAL IN

THE

CONDITIONS PRESENCE

MECHANICAL A.

OF OF

HEAT M.

Stolin

THERMAL

CHEMICAL

EXPLOSION AND

SOURCES and

A.

G.

UDC 536.46

Merzhanov

As yet little attention has been given to t h e r m a l explosion (TE) r e g i m e s a s s o c i a t e d with heat r e l e a s e due to f o r c e s of internal friction (mechanical heat source), which is important in connection with the motion of highly viscous liquids. It is possible to point to only one study [1] involving the numerical calculation of the c r i t i c a l conditions of t h e r m a l explosion of a reactive liquid with allowance for this heat source. With r e f e r e n c e to the c a s e of a Newtonian liquid we have investigated Poiseuille flow in an infinite c i r c u l a r tube for a given p r e s s u r e drop and Couette flow between two infinite cylinders at a given rate of rotation of one of the latter. It is shown that the c r i t i c a l value of the F r a n k - K a m e n e t s k i i p a r a m e t e r 5 d e c r e a s e s as the strength of the m e c h a n i c a l heat s o u r c e s i n c r e a s e s . The possibility of using the nonstationary methods of t h e r m a l explosion t h e o r y to solve t h e r m o h y d r o dynamic p r o b l e m s f o r a f a i r l y general rheological law and an a r b i t r a r y dependence of v i s c o s i t y on t e m p e r a ture was d e m o n s t r a t e d in [2] with r e f e r e n c e to the Couette-type flow of an inert liquid. In the p r e s e n t paper, on the basis of a simplified q u a s i s t a t i o n a r y approach to the solution of t h e r m o hydrodynamic p r o b l e m s [2], we analytically derive the t h e r m a l explosion conditions for the Couette flow of a viscous r e a c t i v e liquid. Different t e m p e r a t u r e dependences of the v i s c o s i t y and different methods of a s signing the boundary conditions on the moving boundary are considered. Statement

of the

Problem

We will consider the one-dimensional Couette flow of a reacting liquid enclosed between cylindrical surfaces. We start by formulating the assumptions on which the model is based. I. The reaction taking place in the liquid is zero-order, expressed by the Arrhenius law

two plane or

and the heat release due to the reaction is

Here, Q is the r e a c t i o n energy, k0is the p r e - e x p o n e n t i a l coefficient; E is the activation energy; T is t e m p e r a ture; R is the gas constant. 2. There is no t e m p e r a t u r e distribution in the Iiquid, which, in a c c o r d a n c e with heat t r a n s f e r t h e o r y [3], is the ease when Bi = ar/~<< 1, where Bi is the Biot number c h a r a c t e r i z i n g the relation between external and internal heat t r a n s f e r ; r is a c h a r a c t e r i s t i c dimension; X is the t h e r m a l conductivity of the liquid; oz is the coefficient of heat t r a n s f e r between the liquid and the ambient medium. 3. The rheological equation d e t e r m i n i n g the relation between the s h e a r rate D and the s h e a r s t r e s s cr has the form (2)

D=:(,~)/~(T).

Moscow. T r a n s l a t e d from Fizika Goreniya i Vzryva, No. 4, pp. 502-510, O c t o b e r - D e c e m b e r , 1971. Original a r t i c l e submitted September 30, 1971.

9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~'est 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. A copy of this article is available from the publisher for $15.00.

431

H e r e , f (o-) is the r h e o l o g i c a l function (particular f o r m s of this function a r e given in [2]); p (T) .is the t e m p e r a t u r e - d e p e n d e n t v i s c o s i t y coefficient. The t e m p e r a t u r e dependence of the v i s c o s i t y is usually e x p r e s s e d in the f o r m ~t (T)=Aexp ( ~ T ) '

(3)

where U is the flow a c t i v a t i o n energy; A is a coefficient that depends on the nature of the liquid. A s s u m i n g that ( T - T 0 ) / T 0 <<1, w h e r e T Ois the a m b i e n t t e m p e r a t u r e , and expanding the exponential in a c c o r d a n c e with the F r a n k - K a m e n e t s k i i method, we obtain for the v i s c o s i t y the well-known Reynolds equation ~ (7") = a exp [-- g (7` -- 7`0)/(~7`~)1,

a = A exp g ~

.

If we a s s u m e that U ( T - T 0 ) / ( H T ~) is small, then f r o m (4) we obtain the hyperbolic law u

Ix (T) : a/[1-k ~ (T -- T~ RTo

(4)

(5)

In what follows we shall m a k e u s e of Eqs. (3), (4), and (5). 4. The p r o c e s s is q u a s i s t a t i o n a r y f r o m the h y d r o d y n a m i c standpoint (the nonstationary t e r m in the equations of h y d r o d y n a m i c s Ov/Ot = 0, while in the heat conduction equation OT/Ot~ 0). A s calculations have shown [1], q u a s i s t a t i o n a r y conditions a r e m a i n t a i n e d up to the s h a r p t e m p e r a t u r e r i s e at P r = ~/ (9a) <<1, w h e r e P r is the P r a n d t l n u m b e r ; O is density; a is t h e r m a l diffusivity. F o r p o l y m e r m e l t s and solutions, viscous oils, etc. at low T 0 the Prandtl number is usually of high o r d e r (greater than 1000). A m a t h e m a t i c a l f o r m u l a t i o n of the p r o b l e m satisfying t h e s e a s s u m p t i o n s r e d u c e s to the e l e m e n t a r y heat balance equation

S ( T - - To), cp dT --~-= qchem(T) +qmech(T) -- ~ "-9-

(6)

where qmech(T) is the d i s s i p a t i v e heat r e l e a s e function; c is the specific heat; S/V is the liquid s u r f a c e volume ratio. In [2] it was shown t h a t the d i s s i p a t i v e function qmech(T) has the f o r m

(7)

k%x~

Here, e b is the s t r e s s at the moving boundary; x is the coordinate along the n o r m a l to the boundary s u r f a c e ; the quantities k, l , and m c h a r a c t e r i z e the shape of the g e o m e t r i c region and the type of flow (Fig. 1); x 0 is the coordinate of the inner boundary; x 1 is the coordinate of the outer boundary. E ~ p r e s s i o n (7) gives qmech(T) if o-b = ~ - const is given on the moving boundary. If, however, v b = v I - const is given, then the value of o-b should be found f r o m the e x p r e s s i o n Xl

=

m 1

I• b x0

It is i m p o r t a n t to note that, in investigating Couette flows with a given s t r e s s on the moving boundary, the h e a t r e l e a s e function a s s o c i a t e d with internal friction f o r c e s i n c r e a s e s with t e m p e r a t u r e , and under c e r t a i n conditions t h e r e is no s t a t i o n a r y flow r e g i m e (hydrodynamic " t h e r m a l explosion ~) [21. On the other hand, if flows with a given boundary velocity a r e c o n s i d e r e d [4], then t h e r e is always a s t a t i o n a r y t e m p e r a t u r e and v e l o c i t y distriN~tion, since in this e a s e the heat r e l e a s e function d e c r e a s e s with i n c r e a s e in ternperature. The subsequent investigation extends to both methods of specifying the boundary condition on the moving boundary. We note that the e a s e when qmeeh_(T) is an i n c r e a s i n g exponential function of t e m p e r a t u r e is s o m e w h a t anaIogous to the investigation of the t h e r m a l explosion r e g i m e s of a t w o - c o m p o n e n t m e d i u m by Bowes [5]. For given kinetic, m e c h a n i c a l , and t h e r m o p h y s i e a l p a r a m e t e r s , f r o m the conditions of equality of the heat supply and heat r e m o v a l functions and equality of the d e r i v a t i v e s of t h e s e functions, it is p o s s i b l e to

432

find the c r i t i c a l t e m p e r a t u r e of t h e r m a l e x p l o s i o n T o and the t e m p e r a t u r e Of the liquid in the c r i t i c a l r e g i m e T*

Type of flow iFlow between ! two planes

0

i

0

fqz ( T * ) - - rz (S~V) (T* - - To) = O, t q'~ (T*) - - a (S., V) = 0,

Vb

[A xial flow between : two cylinders

2

r

(9)

where

0

qz (T)=qchem (T)+qmech (T)iTa ngential flow between two cylinder~

2

2

r

I

.i

We s h a l l not e m p l o y s p e c i a l notation f o r the p a r a m e t e r s r e l a t i n g to the c r i t i c a l r e g i m e .

Thermal explosion for a constant distributed h e a t s o u r c e was i n v e s t i g a t e d b y M a r g o l i n [6]. On the b a s i s of his r e s u l t s we m a y c o n c l u d e that at q m e c h ( T ) - c o n s t the p r e - e x p t o s i o n t e m p e r a t u r e r i s e A T = T* - T Oc a n be divided into a c h e m i c a l c o m p o n e n t (AT)x = T ~ - T ~ , found f r o m the r e l a t i o n s F i g . 1. G e o m e t r y and types of flow.

q chem (T;) - - a (S/V) (V~ - - To::) --= 0] q'chem (T~) -- a S / V -- 0

/

(10)

and a d i s s i p a t i v e c o m p o n e n t ( h T ) m e c h = q m e c h / ( ~ S / V ) , i.e., AT = (AT)~ + (AT)met h,

(11)

while T* does not depend o~ the c o n s t a n t heat s o u r c e . In the g e n e r a l e a s e Eq. (11) f o r A T does not hold, and it is p o s s i b l e to e s t a b l i s h the following: 1) T o d e c r e a s e s when an additional h e a t s o u r c e is added; 2) T* depends on the s t e e p n e s s (derivative) of the total h e a t r e l e a s e function. H q~ (T) > q~hem (T) T (qmech(T) is an i n c r e a s i n g function of t e m p e r a t u r e ) , then T * < T ~ (Fig. 2b); if qE (T) < q c! h e m (T) (qmech(T) is a d e c r e a s i n g function of t e m p e r a t u r e ) , then T* > T ~ (Fig. 2a). Derivation

of the

Critical

Conditions

for

Given

Stress

crb = (r 1 = c o n s t

Using (1), (3), and (7), we w r i t e Eq. (6) in the f o r m CO ~dT

=

(12)

Qkoe_el(sr) + (G/A) e-U~(~r) -- a (S/V) (T - - To),

where

~o

I n t r o d u c i n g the n o n d i m e n s i o n a I v a r i a b l e s and p a r a m e t e r s E 0 = -~o

(T--To),

aS x = [~-~ t,

[7]

RTo ~ = --E-- ,

•

QkoVE

E

and, m o r e o v e r , U

GVE --x.~ = AzSRT~~ e RTo,

(14)

dO d--~ = •176176 -[- x~ ee~iO+~~ -- 0.

(15)

B = U/E,

we r e d u c e Eq. (9) to the f o r m

Here, ~ is the S e m e n o v n u m b e r of n o n s t a t i o n a r y t h e r m a l e x p l o s i o n t h e o r y ; ~r the i n t e n s i t y of h e a t r e l e a s e due to i n t e r n a l f r i c t i o n f o r c e s .

is a c r i t e r i o r , c h a r a c t e r i z i n g

433

a

b

The functional relation between x and x a corresponding to the c r i t i c a l r e g i m e is determined f r o m the e x p r e s s i o n s • ~176 ,,

+ • e~~

, z . . . . - e O/(t+fi~

(ltpv)-

%%= C=T"

rO ~= ~'T7 .r

~ -- 0 = 0 .7-:Bff---~, e B~

-~ ( l t O~#

The dependence ~ = f 0r p r e s e n t e d in the p a r a m e t r i c f o r m

Fig. 2. Semenov d i a g r a m : a) for qb = const; b) for vb = const.

•

/

(16)

1 = OJ"

d e t e r m i n e d from (16) can be r e -

(I+~O)'--BO e_O/(i+~o) ----f~(O,B) [ 1-

.

8

(17)

• _ 0 . - ( 1 + I~0)~ e - ~o/(~+ ~ 0) = f~ (O, B ) 1--B Usually B = U / E < 1. F r o m this condition and the r e q u i r e m e n t that ~ and ~ possible to establish the interval of variation of 0

2~

2~'

be positive quantities it is

(18)

An investigation of the behavior of the functions f l ( 0 , B ) and f 2(0 ,B) f r o m (17) shows that assigning B and 0 f r o m the interval defined by inequalities (18) uniquely d e t e r m i n e s the relation between ~r and ~r in the c r i t i c a l r e g i m e , x = f ( x ~ ) d e c r e a s i n g as B i n c r e a s e s . We make the usual a s s u m p t i o n that fl0 = A T / T 0 << 1. As fl --~ 0 Eq. (3)for the viscosity goes over into the Reynolds equation (40, and the relation between ~ and ~r is r e p r e s e n t e d in the f o r m

n= ~

~

O-- I -Be

~ = T-~-B e

I

(19)

'

in which ease at B -< 1 the following interval of variation is defined for

1 <~ 0.<~ 1/B,

1 ~< B ~

1

(20)

A c o m p a r i s o n of Eqs. (17) and (19) shows that when the intensity of the heat r e l e a s e due to internal friction forces n q is specified, taking into account the c o r r e c t i o n for/3 gives values of the critical x that a r e somewhat too low. F o r a hyperbolic t e m p e r a t u r e dependence of the v i s c o s i t y [Eq. (5)] valid for s m a l l B, the c r i t i c a l c o n ditions a r e given by the equation (I--

Bxo ) e_.;(i_B,o),

(21)

and when B = 0, which c o r r e s p o n d s to the case of constant viscosity and constant dissipative function, we have 1

= - T e-Xo .

(22)

When ~cr = 0 (absence of mechanieal heat sourees) f r o m Eqs. (17), (21), and (22) for Reynolds and hyperbolic t e m p e r a t u r e dependences of the v i s c o s i t y and for constant viscosity, respeetively, we obtain ~ = l / e , 0 = 1 the Semenov c r i t i c a l condition. As ~ i n c r e a s e s , the critical ~'~ d e c r e a s e s , and 0 - the s t a t i o n a r y f r e e explosion t e m p e r a t u r e r i s e - i n c r e a s e s . The r a t e of these changes depends on the f o r m of the t e m p e r a t u r e dependence of the v i s c o s i t y : the s t r o n g e r this dependence, the s m a l l e r for a given ~(r and B the critical vt and the g r e a t e r 0. We note that, as 0 i n c r e a s e s , the e r r o r a s s o c i a t e d with the F r a n k - K a m e n e t s k i i expansion of the A r r h e n i u s multiplier i n c r e a s e s and the c r i t i c a l condition should be calculated from Eqs. (17). In the absence of c h e m i c a l heat s o u r c e s (n = 0) for a Reynolds dependence of the v i s c o s i t y on t e m p e r a t u r e x ~ = 1/(Be), 0 = l / B , for a h y p e r b o l i c dependence ~ = l / B , 0 - ~ ~; for the c a s e of constant p r o p e r t i e s t o- --> ~. in Fig. 3 we have plotted z = f (B~r and B0 = f { B ~ ), calculated from (16). In the calculations we a s s u m e d that B = 1, 0.5, and 0.2 and that fl = 0.02. When B = 1, for points (~,~r below the straight line + ~r = 0.375 a s t a t i o n a r y flow r e g i m e is always realized. Along this s t r a i g h t line 0 = const = 1.042. For ~4 + Zcr > 0.375 the flow is essentially nonstationary. It is also c l e a r f r o m Fig. 3 that the weaker the t e m p e r a t u r e dependence of the v i s c o s i t y (the s m a l l e r B), the s m a l l e r the region of steady development of the process. 434

7 c):~:::-:--~-::Z ....

.o.

o,~ o,2 o,5 nzo.

-o,5

~''"~-,'1

o

o,5 ~,o r

Fig. 3

,

,o

0

7,gz~

Fig. 4

4

~.o,;5,1 /era z

Fig. 5

F i g . 3. V a r i a t i o n o f ~ = f (Bxo-) a n d B0 = f (B~cr) a t B = 1 (1), 0.5 (2), a n d 0.2 (3). F i g . 4.

V a r i a t i o n of ~ = f(log ~v) and log 0 = f(log ~ v ) at B = 1(1), 0.5 (2), and 0.2 (3).

F i g . 5. T e m p e r a t u r e of t h e a m b i e n t m e d i u m To, l i q u i d t e m p e r a t u r e T ~ , a n d t e m p e r a t u r e r i s e A T a s f u n c t i o n s o f t h e s t r e s s ~ in the c r i t i c a l r e g i m e . Derivation

of the

Critical

Conditions

for

a Given

Velocity

v b =v 1 =const

In t h i s c a s e in o r d e r to find q m e c h ( T ) it is n e c e s s a r y t o e x p r e s s cr (T) in t e r m s of v v T h i s c a n b e done e x p l i c i t l y o n l y f o r c e r t a i n f o r m s of f (or). H e n c e f o r t h we a s s u m e t h a t f (e) - ~ , w h i c h c o r r e s p o n d s to a Newtonian liquid. A s f o l l o w s f r o m [2],

G=

qmech(T) -----O~ (T),

~. X I --

-

(23)

if /z - - 21 ~ 0 if k - - 2 1 = 0 ,

tin (XdXo), I(x~-.,-,

~ F~ Xo

x~-.,-*)i(l

m -

-

O,

F = lln (xl/xo),

if if

1--m--l ~O I--m--l=0.

U s i n g (23) a n d (3), w e o b t a i n t h e h e a t b a l a n c e e q u a t i o n i n the f o r m cp ~-~ = Qk6e -el(Rr) + GAe u}(~r) - - c, (S/V)

(T

To).

(24)

I n t r o d u c i n g n o n d i m e n s i o n a l r e l a t i o n (13) and, m o r e o v e r , GAVE

• = ;~Tozex p (UI(RTo)),

(25)

we o b t a i n dO -_ ~

xeo/(l+~o)

+ x~ e-~01(I+~0) --0.

(26)

For determining the critical parameters we obtain the following system of equations: ze ~176

+ x ~ e-a~

(1 4- ~0)'

~

~o) - - 0 = 0

}

Bxu

I

H e n c e ~ a n d X v c a n b e r e p r e s e n t e d in the p a r a m e t r i c :~

e-~~ 1+~~ - - 1 = 0

form

B0 + ( t + 30)~ e_O/(l+~o) - - - -

t+e

(27)

/ = ~L (0,B,{~)

~--(I § ,~ ~0)2 e::O/(t+,~0) m l+e =%(O,B,~)

(28)

I

If it i s a s s u m e d t h a t B -< 1, t h e n f r o m t h e p o s i t i v e n e s s of x and x v t h e r e f o l l o w s 2> AS s h o w n b y a n i n v e s t i g a t i o n of t h e b e h a v i o r of ~o1(0 , B, fi) a n d ~02(0 , B, fl), f o r f i x e d B a n d fl t h e c r i t i c a l condition curve is unique.

435

/

F o r a R e y n o l d s d e p e n d e n c e of the v i s c o s i t y on temperature

BO+I

G,]

•

~ e- [ 0 - - 1 ~o [ ' z~ ----T W - g e j 0

1

2

,.~5.1o-2,rps

$

~0

~,z:

Fig. 6

(29)

~o5 for a h y p e r b o l i c d e p e n d e n c e

Fig. 7

F i g . 6. To, T * , a n d A T a s f u n c t i o n s of the r a t e of r o t a t i o n n in t h e c r i t i c a l r e g i m e .

• ~ e~---(1+ B ~ ) e-x~/(a+B,.~,

(30)

a n d in t h e c a s e of c o n s t a n t v i s c o s i t y

F i g . 7. D i s s i p a t i v e r a t e s of h e a t r e l e a s e q~ a n d qv f o r t h e r e g i m e s ~rb = c o n s t and v b = c o n s t , respectively, as functions of the critical temp e r a t u r e of t h e r m a l e x p l o s i o n T O.

1 e

(31)

AS in the p r e c e d i n g c a s e , when ~ v = 0, f r o m (29)(31) we o b t a i n ~ = l / e , 0 = 1, and a s ~
T h e d e p e n d e n c e s ~ = f flg~v) and l g 0 = f (lg~v) f o r v a r i o u s B c a l c u l a t e d f r o m (28) a r e p r e s e n t e d in F i g . 4. T h e c u r v e s a s y m p t o t i c a l l y a p p r o a c h the a x i s of a b s c i s s a s . S i n c e the t e m p e r a t u r e r i s e 0 i n c r e a s e s m o n o t o n i c a l l y w i t h i n c r e a s e in ~ v , for s u f f i c i e n t l y l a r g e ~ v ( s m a l l ~ ) the c r i t i c a l c o n d i t i o n s s h o u l d b e c a l c u l a t e d f r o m E q s . (28). Example

of

Calculation

of the

Critical

Parameters

A s a n i l l u s t r a t i o n w e p r e s e n t a t y p i c a l e x a m p l e of t h e c a l c u l a t i o n o f t h e c r i t i c a l t h e r m a l e x p l o s i o n p a r a m e t e r s f o r a C o u e t t e flow of t h e e x p l o s i v e DINA b e t w e e n two c y l i n d e r s . A c c o r d i n g t o [8], DINA h a s the following kinetic constants. E = 3 5 . 1 0 3 cal/mole, K0=2,5 9 103 1/sec, Q =950 cal/cm 3. In [9] t h e t e m p e r a t u r e

d e p e n d e n c e of the v i s c o s i t y of DINA i s g i v e n in t h e f o r m ,

~(T) = ~oexp ~

,

w h e r e ~0 -:- 9.238 g / e r a - s e e , U = 6.8 9 103 e a l / m e I e , R = 1.99 c a l / d e g 9 The c a l c u l a t i o n s w e r e macle f o r t h e v a l u e s ~ ; 10 .4 c a l / c m 2 9 s e e 9 d e g , R 0 = 2 era, R~ = 2.5 era, V / S = 0.25 era. T h e c a l c u l a t i o n s c h e m e w a s a s f o l l o w s : ~ w a s d e t e r m i n e d f r o m . t h e s e l e c t e d v a l u e of the a m b i e n t t e m p e r a t u r e T 0. F r o m r e l a t i o n (17) (or F i g . 3) for the r e g i m e o"b = e o n s t a n d (28) (or F i g . 4) f o r t h e r e g i m e Vb = c o n s t , u s i n g the ~t o b t a i n e d , w e d e t e r m i n e d t h e c o r r e s p o n d i n g x o ' , ~ v , a n d 0. A f t e r t h i s we c a l c u l a t e d the d i m e n s i o n a l p a r a m e t e r s in t h e c r i t i c a l r e g i m e . In F i g . 6 t h e c r i t i c a l t e m p e r a t u r e of t h e r m a l e x p l o s i o n To, the l i q u i d t e m p e r a t u r e in t h e c r i t i c a l r e g i m e T * , and the p r e - e x p l o s i o n t e m p e r a t u r e r i s e A T a r e shown a s f u n c t i o n s of the r a t e of r o t a t i o n of the u p p e r c y l i n d e r n = V b / ( 2 7r R2) r p s f o r the r e g i m e v b = c o n s t . A s n i n c r e a s e s , T o d e c r e a s e s f r o m 396.5~ a t n = 0 (liquid a t r e s t ) to 380~ a t n = 200 r p s , the p r e - e x p l o s i o n t e m p e r a t u r e A T i n c r e a s e s f r o m 9~ (n = 0) to 26~ (n = 200 r p s ) , w h i l e T* r e m a i n s p r a c t i c a l l y u n c h a n g e d . T h e r e s u l t o b t a i n e d f o r T* m a y be a t t r i b u t a b l e to the f a c t t h a t d q m e e h / d T d e c r e a s e s with t e m p e r a t u r e (see F i g . 2a), a n d a t s u f f i c i e n t l y l a r g e T the s t e e p n e s s of t h e i n t e g r a l h e a t r e l e a s e f u n c t i o n a l m o s t c o i n c i d e s with t h a t of t h e c h e m i c a l h e a t r e l e a s e function, w h i c h a l s o l e a d s to T * b e i n g i n d e p e n d e n t of n. In F i g . 5, To, T * , a n d A T a r e s h o w n a s f u n c t i o n s of t h e s t r e s s on the m o v i n g o u t e r c y l i n d e r in the r e g i m e ~ b = c o n s t . T* a n d T Od e c r e a s e a s ~ b i n c r e a s e s , the f a l l in T* b e i n g s o m e w h a t l e s s p r o n o u n c e d . A s ~ v a r i e s f r o m 0 t o 100 g / c m 2, the p a r a m e t e r s in q u e s t i o n v a r y on the i n t e r v a l s T o = 396.5~ - 375~ T ~ = 405.5 ~ , 397*, A T = 9* -- 23 ~.

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The dependence of the c r i t i c a l r a t e of heat r e l e a s e on the a m b i e n t t e m p e r a t u r e T O is shown in Fig. 7 for the r e g i m e s ~b = const and v b = coast. We note that qv (T0)>qa (To). H e n c e w e m a y conclude that, other things being equal, the explosion l i m i t is m o r e e a s i l y r e a c h e d in the r e g i m e ~b = eonst in the s e n s e that reaching that l i m i t at a given T o r e q u i r e s l e s s of an i n c r e a s e in the d i s s i p a t i v e r a t e of heat r e l e a s e . This r e s u l t is an obvious co~sequence of the different b e h a v i o r of the d i s s i p a t i v e heat r e l e a s e function in the two r e g i m e s (see Fig. 2). LITERATURE 1o

2. 3. 4. 5. 6. 7. 8. 9.

CITED

S. A. Bostandzhiyan, A. G. Merzhanov, and N. M. Pruchkina, Zh. P r i M . Mekhan. i Tekh. Fiz., No. 5 (1968). A. G. Merzhanov and A. M. Stolin, Dokl. Akad. Nauk SSSR, 198, No. 6 (1971). V. P. Isachenko, V. A. Osipova, and A. S. Sukomel, Heat T r a n s f e r [in Russian], Energiya, M o s c o w Leningrad (1965). S. A. Bostandzhiyan, A. G. Merzhanov, and S. I. Khudyaev, Zh. P r i M . Mekhan. i Tekh. Fiz., No. 5, 45 {1965). 1). S. Bowes, Comb. and F l a m e , ~ No. 5, 521 (1969). A. D. Margolin, Zh. Fiz. Khim., 4_, 137 (1963). D. A. F r a n k - K a m e n e t s k i i , Diffusion and Heat T r a n s f e r in Chemical Kinetics [in Russian], Nauka, Moscow (1967). A. S. Shteinberg, B. M. Slutsker, and A. G. Merzhanov, Fiz. Goreniya i Vzryva, 6__,No. 4 (1970). V. V. Barzykin, }~. A. S h t e s s e l ' , e t a l . , F i z . Goreniya i Vzryva, 7, No. 2 (1971).

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