RADIATIVE-THROUGH

CONDUCTIVE OPTICALLY

Yu.

P.

TRANSMISSION DENSE

OF

HEAT

MEDIA.

Konakov

UDC 536.3

The p r o b l e m of heat t r a n s m i s s i o n by conduction and radiation through a s e m i i n f i n i t e optically dense m e d i u m is analyzed, with incident external radiation and with convective heat t r a n s f e r taken into account. An e x p r e s s i o n for the radiant t h e r m a l flux is d e r i v e d f r o m the solution to the equation of radiation flux propagation by the method of a s s o c i a t i v e a s y m p t o t i c expansions. The effect of the t e m p e r a t u r e gradient at the s u r f a c e on the e m i s s i v i t y of the body is e s t a b l i s h e d for the m e d i u m r a n g e of a b s o r p t i v i t y v a l u e s . In an a n a l y s i s of the r a d i a t i v e - - c o n d u c t i v e heat t r a n s m i s s i o n through optically dense media, the R o s seland a p p r o x i m a t i o n usually s e r v e s as the e x p r e s s i o n for the radiant component of the t h e r m a l flux. This a p p r o x i m a t i o n has been d e r i v e d f r o m a p a r a m e t r i c expansion of the solution to the equation of radiation p r o p a g a t i o n [1]. An expansion in t e r m s of the p a r a m e t e r r e s u l t s in the elimination of the a r b i t r a r y constant f r o m the solution to the equation of r a d i a t i v e heat t r a n s m i s s i o n and r e n d e r s it unsuitable for the inhomogeneous r e gion adjoining the boundary, w h e r e a boundary condition m u s t be s a t i s f i e d . Obviously, such an expansion cannot be c o n s i d e r e d equally useful o v e r the e n t i r e r a n g e of the p r o b l e m . The existence of an inhomogeneous region is, as a rule, r e l a t e d to the expansion of the solution in t e r m s of the p a r a m e t e r by which the f i r s t d e r i v a t i v e is multiplied [2], and in the equation of radiation p r o p agation through optically dense m e d i a such an expansion p a r a m e t e r is e = 1 / k . To a r r i v e at a u n i v e r s a l l y useful p a r a m e t r i c expansion in this situation is the object in the p r o b l e m of p a r t i c u l a r (singular) p e r t u r b a t i o n s . A u n i v e r s a l solution is m o s t often obtained by c o n s t r u c t i n g an app r o x i m a t i o n which is u n i f o r m l y c l o s e within the inhomogeneous region and then a s s o c i a t i n g it with a s t r a i g h t p a r a m e t r i c expansionby the method of a s s o c i a t i v e a s y m p t o t i c expansions. Several i n t e r e s t i n g facts about r a d i a t i v e - - c o n d u c t i v e heat t r a n s m i s s i o n through optically dense m e d i a can b e r e v e a l e d , if the method of a s s o c i a t i v e expansions is applied to the solution of such p r o b l e m s . We will c o n s i d e r the following p r o b l e m : It is d e s i r e d to d e t e r m i n e the s t e a d y - s t a t e t e m p e r a t u r e field and the t h e r m a l fluxes in a semiinfinite solid body on whose s u r f a c e i m p i n g e s external radiation u n i f o r m l y f r o m all d i r e c t i o n s . Through the s a m e s u r f a c e heat is t r a n s f e r r e d f r o m that body to the ambient m e d i u m by convection. The p r o b l e m is solved u n d e r the following a s s u m p t i o n s : 1) the boundary between the solid body and the adjoining m e d i u m is t r a n s p a r e n t to the external radiation and is also diffusive, 2) the a b s o r p t i v i t y of the solid m a t e r i a l does not v a r y with the radiation frequency, 3) the hypothesis of local dynamic equilibrium applies to the radiation, 4) the p h y s i c a l p a r a m e t e r s of the m a t e r i a l a r e not t e m p e r a t u r e - d e p e n d e n t , 5) the t e m p e r a t u r e field is u n i f o r m , and 6) the r e f r a c t i v e indices of the solid m a t e r i a l and the adjoining m e dium a r e both equal to unity. We ~rill r e s o l v e the radiation intensity in the solid into two components in opposite directions: and I - (see Fig. 1).

I+

T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s l d i Zhurnal, Vol. 23, No. 3, pp. 459,464, S e p t e m b e r l 1972. Original a r t i c l e s u b m i t t e d D e c e m b e r 20, 1971. 9 1974 Consultaz~ts Bureau, a division of Plenum Publishing Corporation, 227 g'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.

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Both intensity fields a r e d e s c r i b e d by the equation of

]+

t-

dI ~ B (x) --I. dx

em - -

In o r d e r to find the I T intensity distribution in the solid, we l e t I = I + a n d m > 0 i n Eq. (1).

O m

To r(o) h

r(x}

X

The conditions of radiation at the boundary will be stated as

I+~=I+(0) when x = 0 .

-

Fig.

I.

(1)

Schematic d i a g r a m of the p r o b l e m .

(2)

The quantity I+(0) will be d e t e r m i n e d l a t e r . The outer expansion of the solution will be e x p r e s s e d in t e r m s of the following a s y m p t o t i c sequence

i+= l+o+el++ ~2I+ +...

(3)

The unknown functions a r e d e t e r m i n e d by i n s e r t i n g this expansion into Eq. (1) and subsequently equating the r e s p e c t i v e e q u a l - p o w e r t e r m s in e. As a r e s u l t , we obtain

I+----B ( x ) - emB' (x) + 0 (e~).

(4)

The p r i m e sign denotes a d e r i v a t i v e with r e s p e c t to x. E x p r e s s i o n (4) is not a u n i f o r m approximation to the solution to Eq. (1), b e c a u s e it does not m a t c h the boundary condition (2). This expansion is, t h e r e f o r e , unsuitable for the inhomogeneous region adjoining the boundary. We will now e x a m i n e the solution in the inhomogeneous region. new v a r i a b l e and a new function defined by the equalities: X-

For this p u r p o s e we introduce a

x ; i +(x, e) -----J+ (e, X). 8

With the aid of t h e s e e x p r e s s i o n s we t r a n s f o r m the original equation (1):

dd+

m -----B (eX) - - J+ dX and the boundary condition (2).

(4')

The l a t t e r is now written as J+=:l+ (0) when X=0.

(5)

The t r a n s f o r m a t i o n s have r e s u l t e d in the elimination of the s m a l l p a r a m e t e r by which the f i r s t d e r i vative is multiplied. An expansion of the solution to Eq. (4) in t e r m s of e will m a k e it p o s s i b l e to r e t a i n s e v e r a l e s s e n t i a l p r o p e r t i e s of the solution which a r e lost in the outer expansion (3). We will now r e p r e s e n t the inner expansion by such an a s y m p t o t i c sequence

j+ =g~ + ~j+ ~_~2~ +...

(6)

The unknown functions in expansion (6) a r e d e t e r m i n e d by i n s e r t i n g (6) into (4) and (5). It m u s t be c o n s i d e r e d h e r e that B (x) = B (eX) = B (0) + eB' (0) X + . . .

(7)

It is e a s y t o s e e that the unknown functions in expansion (6) a r e the s o l u t i o n t o f i r s t - o r d e r differential equations with boundary conditions defined a c c o r d i n g to (5). The binomial inner expansion is

J+:=~B (0)~[1+ (0)-- B (0)] exp ( - X - )

-!- emB' (o) [ X -- l + exp (-- X ) ] + O (e2). This expansion d e s c r i b e s the solution to Eq. (1) w h e r e expansion (3) is u s e l e s s .

1128

(8)

,

r(x)

By combining expansions (3) and (6), one can construct a uniformly approximate solution to Eq. (1) e v e r y w h e r e within the r a n g e of the p r o b l e m . The sought composite expansion obtained by the method of addition [2] is

kX =l#O

,oo! 00!

2

l~ = B (x) & [I+ (O) -- B (O)] exp ( - - ~m )

Rx

Fig. 2. T e m p e r a t u r e field in a semiinfinite body with (r = 5.67 910 -8 W / m 2- (OK)4,~ = I0 W / m 2.~ R = 0.5, In = 1300 W/m2, T O= 300~

[

( x)]

-- em B' (x) -- B' (0) exp - - -~m

+ 0 (~2).

(9)

A uniform approximation for I- is constructed in an analogous m a n n e r . The inhomogeneous region of intensity I- lies at infinity and, t h e r e f o r e , the outer expansion alone will suffice for I-, namely

I- = B ( x ) - emB' (x) + 0 (~).

(10).

The radiant flux in the solid is 1

--1

qr = 2a j'. ml+dm-- 2~ ~ ml-dm. o

(il)

*/

0

With the aid of (8) and (10), this e x p r e s s i o n can be expanded as follows qr .... 2n [P (0) -- B (0)] E a (kx) -- 4~ 3~- B' (x) + -2~ 7 B'

(12)

(O)E, (kx).

H e r e E3(kx) and E4(kx) a r e exponential integrals whose p r o p e r t i e s and values a r e given in [3]. We note that the use of the outer expansions (3) and (10) in (11) yields, after integration, the well known Rosseland approximation for the radiation flux through optically dense media: qr ......

4u B' (x). 3k

(13)

The relation derived h e r e will be inoperative where the outer expansions (3) and (10) a r e unsuitable, i . e . , within the region adjoining the boundary. Expression (12) yields an estimate of the e r r o r i n c u r r e d by the often used Rosseland approximation. Through media with a high absorptivity (metals, for example), t h e r e f o r e , the radiative t r a n s m i s s i o n of heat is appreciable only within small regions n e a r the boundary. According to e x p r e s s i o n (13), on the other hand, radiative heat t r a n s m i s s i o n o c c u r s almost nowhere. The magnitude of I+(0) in (12) i s found f r o m the balance of radiant energy at the solid surface: a/+ (0) = n (1 - - R) I~ + Rq- (0)

(14)

Expanding the respective terms here yields 2~

nl § (0) -- ~ (I - - R) It, -~- ~RB (0) + - ~ RB' (0).

(15)

With the aid of this equality, we obtain f r o m (12) an e x p r e s s i o n for the radiation flux at the solid surface:

qr (0) --= a (l -- R) [ I ' ~ - B ( 0 ) -

23k B'(0)].

(16)

From here it is easy to determine the intrinsic radiation from the body into the adjoining space, if one assumes that In = 0. In media with a high absorptivity the intrinsic radiation is proportional to the transparency of the boundary between the body and the adjoining medium, defined by 1 - - R , and to the Planck function. At moderate values of k, an appreciable effect on the intrinsic radiation has the t e m p e r ature field gradient at the boundary. The Fourier hypothesis and expression (12) for the radiation flux has yielded an equation for one-dimensional radiative heat transmission and conduction

1129

ZT" + 4n3kB" (x) = - - 2~k {(1 - - R) [Is - - B (0)]

i 2 B' (0)} E~ (kx) --

2~B' (0) E~ (kx).

(17)

U s u a l l y equations d e s c r i b i n g s u c h p r o c e s s e s do not contain exponential i n t e g r a l s . The total t h e r m a l flux at a c o n s i d e r a b l e d i s t a n c e f r o m the b o u n d a r y s u r f a c e we set equal to z e r o . Then T'-+0 a s x-+co.

(18)

The m o l e c u l a r h e a t t r a n s f e r t h r o u g h the b o u n d a r y is g o v e r n e d by the following b o u n d a r y condition of the t h i r d kind (19)

~T' = a (T - - To) when x = 0. The solution to Eq. (17) with the b o u n d a r y conditions (18) and (19) is 2 (1 - - R) [~I~ - - o'T' (0)] E, (kx) -4- C, k

ZT (x) + 4 a T* 3k (x) --

(20)

wh~re Z C~

(l - - R) [nI~ - - ~T* (0)]

-T- 9

8(y T 3 (0) 1 + 0 - - R") --~-~ + 2~(1--R)I,~+ 2a(I+R)T'(0) 3k 3k

k_~,ro"

The t e m p e r a t u r e g r a d i e n t is

T' (x) (

/

6/e

6~

+ ~ 16tr T3 (x)

(21)

where

r' (0) . . . . . 0_--~)__[~(-_ -- or' (0)] Z + ( 1 - - )R" - ~ -8o' T3 (0) The a c c u r a c y of t h e s e e x p r e s s i o n s is of the o r d e r O(1/kk2). The unknox~aa s u r f a c e t e m p e r a t u r e T(0) is found as the solution to the a l g e b r a i c equation 1 + ...8_~__/ aT 4 (0) - -

3k~ !

-~- 1--Ra T (0) - - [_ [-~ a

8 ~ aToT 3 (0)

3kz,

Z o -~- zl~]

= 0.

(22)

In Fig. 2 a r e shown t e m p e r a t u r e fields in a solid m e d i u m c a l c u l a t e d on a c o m p u t e r f o r the following v a l u e s of the g o v e r n i n g p a r a m e t e r s : = 5.67.10 -s w/m L-K; a :.: 10 w/m 2-~

' R == 0.5;

I~ = 1300 W/m2; T O=: 300 ~ A d e c r e a s e in the p r o d u c t kk r e s u l t s in a l o w e r s u r f a c e t e m p e r a t u r e of the body, which in t u r n r e d u c e s the c o n v e c t i v e t h e r m a l flux t r a n s f e r r e d to the adjoining m e d i u m . T h e r a d i a t i v e c o m p o n e n t of the t h e r m a l flux at t h e body s u r f a c e , which has been stipulated in the p r o b l e m to be equal to the c o n v e c t i v e component, behaves analogously.

1130

E x p r e s s i o n (16) indicates that a d e c r e a s e in the radiation flux is r e l a t e d to an i n c r e a s e in B'(0), which is proportional to the t e m p e r a t u r e gradient at the boundary and, t h e r e f o r e , the other component B(0) d e c r e a s e s with d e c r e a s i n g s u r f a c e t e m p e r a t u r e . This is confirmed in Fig. 2. We note, in conclusion, that, for analyzing the e m i s s i v i t y of bodies, t h e i r boundary with the s u r r o u n ding medium must be c h a r a c t e r i z e d only in t e r m s of its reflectivity and t r a n s m i t t i v i t y with r e s p e c t to radiation. Radiation is g e n e r a t e d in the body within the boundary l a y e r , where the radiati on intensity is d e t e r m i n e d according to e x p r e s s i o n s (8) and (9). In optically v e r y dense media, where heat is t r a n s m i t t e d onJy by conduction, the radiative component of the t h e r m a l flux is appreciable within the boundary l a y e r . The heat t r a n s m i s s i o n c h a r a c t e r i s t i c s h e r e a r e strongly affected by the boundary conditions with r e s p e c t to radiation and the m o l e c u l a r flux. NOTATION

T(x) To

In I +, lm = cos e; B(x) = ( ~ / ~ ) r ' k k O~

R (Y

is the t e m p e r a t u r e ; is the t e m p e r a t u r e of adjoining medium; is the intensity of external radiation; a r e the radiation intensities along the positive and the negative x-axis; is is is is is is

the the the the the the

e m i s s i v i t y of black body; radiation absorptivity; t h e r m a l conductivity; heat t r a n s f e r coefficient; s u r f a c e reflectivity; Stefan--Boltzmann constant. LITERATURE

1o

2. 3.

CITED

Bai-Shi-i, Dynamics of a Radiating Gas [Russian translation], Izd. Mir, Moscow (1968). M. Van-Dyke, Perturbation Methods in Fluid Mechanics [Russian translation], Izd. Mir, Moscow(1967). E. M. Sparrow and R. D. Sess, Radiative Heat Transfer [Russian translation], Izd'. ~nergiya, Leningrad (1971).

1131