AVERAGE FOR
ABSORPTION OPTICALLY S.
P.
COEFFICIENT THIN
MEDIA
Detkov
M o d i f i c a t i o n s a r e c o n s i d e r e d of the mean P l a n c k a b s o r p t i o n coefficient f o r a s e c t i o n of an a b s o r b i n g m e d i u m a d j a c e n t to a s o u r c e s e c t i o n , t a k i n g into account t e m p e r a t u r e i n e q u a l i t i e s of b l a c k b o d y r a d i a t i o n and the a b s o r p t i o n c r o s s s e c t i o n of the m e d i u m , a s well a s the effect of the length of the s e c t i o n , which is s m a l l . E q u a t i o n s a r e p r e s e n t e d for an analogous c o e f f i c i e n t d e t e r m i n i n g s e l f - a b s o r p t i o n of r a d i a t i o n by the gas.
S p e c t r a l mean a b s o r p t i o n c o e f f i c i e n t s a r e i n t r o d u c e d in o r d e r to u s e g r a y - g a s r a d i a t i v e equations in r a d i a t i v e t r a n s f e r c a l c u l a t i o n s . In p a r t i c u l a r , the t r a n s m i s s i o n of a m e d i u m is d e s c r i b e d by an exponential law. However, the need to u s e this law r e s u l t s in the a v e r a g e a b s o r p t i o n c o e f f i c i e n t s depending on the r a y length x. An e v a l u a t i o n of t h i s dependence a l l o w s the s o - c a l l e d o p t i c a l l y thin gas a p p r o x i m a t i o n to be i m p r o v e d a n d extended. A n o t h e r m a t t e r which must be c o n s i d e r e d is the effect of d i f f e r e n c e s between the r a d i a t i o n s o u r c e t e m p e r a t u r e T 1 and that of the a b s o r b i n g m e d i u m T k. An a v e r a g e P l a n c k i a n a b s o r p t i o n coefficient a c is u s e d in the l i t e r a t u r e . It gives the a b s o r p t i o n of a b l a c k b o d y flux by a thin l a y e r of gas. A t e m p e r a t u r e d i f f e r e n c e between the b l a c k b o d y s o u r c e and the m e d i u m i s a l l o w e d f o r in [1,2]. The quantity a c , with a c o r r e c t i o n f a c t o r , is c a l l e d the m o d i f i e d m e a n P l a n c k i a n a b s o r p t i o n coefficient. In the well-known g r a y gas a p p r o x i m a t i o n [2] and e l s e w h e r e , the s e l f - a b s o r p t i o n of the gas is d e s c r i b e d by the s a m e coefficient. It i s a l s o well known that in t h i s c a s e the a c t u a l a b s o r p t i o n c o e f f i c i e n t a . is m o r e than an o r d e r of m a g nitude l a r g e r than a c. The p r o b l e m is to d e t e r m i n e a c, a , , and to c a l c u l a t e to what extent they a r e affected, in the f o r m of c o r r e c t i o n s , b y t h e q u a n t i t i e s T i, Tk, Xk, xi. We c o n s i d e r the b a s i c combustion p r o d u c t s of h y d r o c a r b o n f u e l s , CO 2 and H20. T h e i r s p e c t r a a r e r e p r e s e n t e d in a s i m p l i f i e d way in the f o r m of a group of n o n - o v e r lapping bands. R e s t r i c t i n g o u r s e l v e s to o p t i c a l l y thin m e d i a , we r e p r e s e n t any d i r e c t i o n along a r a y path in the f o r m of two a d j a c e n t i s o t h e r m a l s e c t i o n s , i and k. The s o u r c e i can be a b l a c k b o d y point s o u r c e . The s e c t i o n k is a s e g m e n t in the gas. The total p r e s s u r e is a s s u m e d c o n s t a n t , and the f i e l d of p a r t i a l p r e s s u r e of the r a d i a t i v e c o m p o n e n t s is unchanging. A s an a r g u m e n t we take the r a y path x = Spdl, m- arm, r e f e r r e d to the p a r t i a l p r e s s u r e p. T h e r e f o r e , a l l the c o e f f i c i e n t s denoted by a have d i m e n s i o n (m.atrn) -1. L o c a l t h e r m o d y n a m i c e q u i l i b r i u m is a s s u m e d . The m o r e c o r r e c t heat t r a n s f e r e q u a t i o n s use the d i r e c t a b s o r p t a n c e of s e c t i o n k f o r r a d i a t i o n f r o m s e c t i o n i: aik ---- ~I~k / ei(~Ti 4,
Ilk = e~aTi4/~ - - Ii~"'
H e r e e i is the e m i t t a n c e of s e c t i o n i; oT 4 is the d e n s i t y of b l a c k b o d y r a d i a t i o n ; Ilk, W / m 2 . s t e r is the d i f f e r e n c e in i n t e n s i t y b e t w e e n the ends of s e c t i o n k; I l k . is the r a d i a t i v e i n t e n s i t y of the s e c t i o n (point) i, t r a n s m i t t e d by s e c t i o n k. S v e r d l o v s k . T r a n s l a t e d f r o m Z h u r n a l P r i k l a d n o i Mekhaniki i T e k h n i c h e s k o i F i z i k i , Vol. 11, No. 1, pp. 12-15, J a n u a r y - F e b r u a r y , 1970. O r i g i n a l a r t i c l e s u b m i t t e d May 13, 1969.
9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g/est 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available [tom the publisher for $15.00.
10
If a i k is not d e f i n e d b y the d i r e c t e q u a t i o n , a n e x a m p l e of which is g i v e n below, but the a b s o r p t i o n c o e f f i c i e n t s a a r e e m p l o y e d , then ark :
t - - exp(--ai~ xh),
ai~* ----- I - - exp(--ai~*xk).
(!)
The a s t e r i s k d e n o t e s the v a l u e f o r s e l f - r a d i a t i o n of the m e d i u m . It can be shown that f o r b l a c k b o d y r a d i a tion a c o n s i d e r a b l y m o r e a c c u r a t e e q u a t i o n is ai~ aik ~ aik-~ [i -- exp (-- ~i~:*x~)], The e q u a t i o n s g i v e n a r e a p p l i c a b l e only f o r f a i r l y thin l a y e r s . T h e e x p a n s i o n exp (--u) ~ 1 - u i s a p p l i c a b l e within known l i m i t s . T h e n the q u a n t i t i e s Cqk c a n be e v a l u a t e d by the e q u a t i o n s
ai~aik]x
k,
a~:* ~---~a~* / x ~ .
(2)
T h e p l a n f o r the r e s t of t h i s p a p e r i s : 1) the e q u a t i o n s f o r a i k , is d e r i v e d , b a s e d on a v e r y s i m p l e s p e c t r a l m o d e l ; 2) the a p p r o x i m a t e e q u a t i o n s (2) a r e u s e d to d e t e r m i n e the c o e f f i c i e n t s a i k a n d a i k * a s a f u n c t i o n of the p a r a m e t e r s ; 3) the c o e f f i c i e n t s o b t a i n e d a r e r e c o m m e n d e d f o r u s e in the v e r y s i m p l e e q u a t i o n s (1). F o r a g r o u p of n o n - o v e r l a p p i n g b a n d s the e q u a t i o n for a i k , t a k e s the f o r m
j
(3)
Z~co,j e~o: l exp (-- % x). -
-
H e r e w (cm -1) is the wave n u m b e r ; e w a n d a w a r e the s p e c t r a l v a l u e s of the e m i t t a n c e a n d the a b s o r p t i o n c o e f f i c i e n t ; A w , j is the width of the n a r r o w e s t b a n d , c h o s e n s o that t h e r e is no a b s o r p t i o n within it. If i i s a b l a c k b o d y s o u r c e point, then e i = ewi = 1. The p a r t i c u l a r e q u a t i o n for aik is o b t a i n e d f r o m Eq. (3). The only s o u r c e of e r r o r in Eq. (3) is a s s o c i a t e d with c h o i c e of the d i s c r e t e v a l u e of the P l a n c k f u n c t i o n (I0j). A d i s c r e t e v a l u e of the P l a n c k f u n c t i o n can be u s e d to a d j u s t the b a n d so that the c o e f f i c i e n t ~w v a r i e s m o n o t o n i c a l l y . T h e r e a f t e r , a b a n d p r o f i l e m u s t be c h o s e n . A n a l y s i s shows that f o r f a i r l y t h i c k l a y e r s the a b s o r p t i o n is a c o m p a r a t i v e l y weak f u n c t i o n of the b a n d p r o f i l e , but the s i t u a t i o n c a n be quite d i f f e r e n t in the c a s e c o n s i d e r e d h e r e , x i ~ Xk ~ 0. We u s e t h r e e d i f f e r e n t p r o f i l e s : r e c t a n g u l a r , t r i a n g u l a r , and symmetrical exponential c~ = a0,
a~ = a0(l - - v/Q),
cr = %0exp (--2v/~),
Here and below ~ is either the profile width or a quantity proportional to it; w 0 is the position profile center where the spectral absorption coefficient has the maximum value s 0.
of the
The c o r r e s p o n d i n g i n t e g r a l b a n d i n t e n s i t i e s have the f o r m S =aoQ,
t
S =-~-a0Q ,
S=
ao~.
(4)
E q u a t i o n s (4) a l l o w u s to d e t e r m i n e the f u n c t i o n a0(T ) in t e r m s of two i n d e p e n d e n t f u n c t i o n s S(T) a n d ~(T). A c c o r d i n g to r e c e n t e x p e r i m e n t a l data in [3-5] a n d e l s e w h e r e , we can u s e a p o w e r r e l a t i o n S ~ T • w h e r e ~ > 0 is a n e x p o n e n t g i v i n g the d e p e n d e n c e of i n t e n s i t y on t e m p e r a t u r e i n a c a l c u l a t i o n f o r a s i n g l e p a r t i c l e ; the u n i t in the e x p o n e n t t a k e s into a c c o u n t the change in the n u m b e r of p a r t i c l e s a s s o c i a t e d with a c o n s t a n t e x p a n s i o n . In thin l a y e r s , b a n d s with f u n d a m e n t a l f r e q u e n c i e s p l a y a n e x c e p t i o n a l r o l e . F o r t h e s e ~ 0. The r e l a t i o n ~2 ~ T m was a s s u m e d f o r the h a l f - w i d t h o r the width of a l l the b a n d s . F r o m s p e c t r o s c o p i c data m ~ 0.5.
il
The result is a unique t e m p e r a t u r e dependence for the values of a 0. Independently of the profile we have a o ~ T ~,
u=x--m--J.
(nmo)
(5)
Here and below we use the relations 7~
Sj = j
u ~ i o j ~ a2do), Here ~ ,
~ ~. do)
Ao~j
SJ~ ~ ~d0). ~ AJ
(6)
(in contrast with ~2j) is the total band with.
We consider the absorption of a blackbody ray. The quantity aik was obtained f r o m Eq. (3) under the conditions 8 i = 8io~ = t ,
X;~ ,~, O,
~cok ~
O~o)k X k - -
1/20~a~k 2 X k 2 o
Then
ai~,~Ioi(T~) i [~,~Xk--+O~,k~Xk~Jdo)". j
A~j
Use of Eqs. (2) and (4)-(6) leads to the result
ac (Ti)
- ~ 0%
.
(7)
It is noteworthy that the quantity ~ik does not depend on the profile shape nor on the inequality T i -~ Tk 9 Next we calculate the s e l f - a b s o r p t i o n by the gas. It is not difficult to show that if the band profiles of the hotter section a r e r e c t a n g u l a r , then the band profiles of the other section a r e not important. With p r o files identical for the two sections, the r e s u l t s a r e different. In the derivation of Cqk, f r o m Eq. (3), we r e s t r i c t ourselves to two t e r m s of the s e r i e s in the e x pansions of ewi and ec0k 8o~iSo~k ~--~ O~o~iO;cokX i X k .
C o r r e s p o n d i n g l y we put ei ~ c~cxi. Equation (3), taking into account Eq. (2),becomes s i m p l e r :
j
A~.j
The limit of integration for Aw.j has the peculiarity that, for a limited band (rectangular or t r i a n g u lar), the integration is c a r r i e d out over the n a r r o w e s t band a f t e r the maxima have been made to coincide. The following r e s u l t s a r e obtained:
a, (Td a~:*(T~, Tk) ~ __ ~u/s (~)
:~, (TOj
12
for ~ > I.
We p r e s e n t e x p r e s s i o n s f o r t h e f u n c t i o n s f l and f 2 o f E q s . (8) f o r d i f f e r e n t b a n d p r o f i l e s r e c t a n g u l a r f l = 1, f2 = 1 t r i a n g u l a r f l = 1/2(3 - ~m), f2 = 1/2 (3 - ~ - m ) e x p o n e n t i a l f l = 2(1 + ~ m ) - l , f2 = 2~rn( 1 + ~ m ) - i F o r ~ = 1 and a n y p r o f i l e , t h e c o r r e c t r e s u l t ~ i k * = o~, is o b t a i n e d . F o r ~m << I o r ~m >> 1 t h e t r i a n g u l a r a n d the e x p o n e n t i a l p r o f i l e s give r e s u l t s w h i c h a r e g r e a t e r b y f a c t o r s of 1.5 a n d 2 than f o r t h e r e c t a n g u l a r p r o f i l e . In the c a s e s m >> 1 t h i s c a n n o t have m u c h i m p o r t a n c e b e c a u s e of the w e a k a b s o r p t i o n (u ~ - 1 , 5 , Su << 1, Oqk, << ~ , ) . In c o n t r a s t to t h e r e c t a n g u l a r a n d t r i a n g u l a r p r o f i l e s , t h e e x p o n e n t i a l p r o f i l e has d i s t a n t w i n g s , s i m i l a r to an a c t u a l p r o f i l e . But it a l s o h a s the d e f e c t t h a t the d e p e n d e n c e of ~ i k * / ~ , on $ d o e s not c h a n g e f o r m in p a s s i n g t h r o u g h the point ~ = 1. T h i s c h a n g e s h o u l d b e in a g r e e m e n t with the r e p r e s e n t a t i o n of the s p e c t r a l "windows," which a p p r o x i m a t e l y r e f l e c t s the r e a l s i t u a t i o n . P o s s i b l y the t r i a n g u l a r p r o f i l e is t h e m o s t r e a l i s t i c of t h o s e c o n s i d e r e d in o u r w o r k . It is c o n s i d e r a b l y m o r e c o m p l e x to e v a l u a t e t h e d e p e n d e n c e of ~ik,(Xk), s i n c e we m u s t t a k e a m i n i m u m of t h r e e t e r m s in the s e r i e s f o r ewk. The c o e f f i c i e n t s ~c a n d oz, a r e b a s e d on the e q u a t i o n s f o r ~ i k a n d Oqk*. T h e y can b e d e t e r m i n e d f r o m Eq. (6) with s p e c t r o s c o p i c d a t a i n c l u d e d . A n o t h e r m e t h o d is to u s e the i n t e g r a t e d e m i t t a n c e s e(x,T) p u b l i s h e d f o r CO 2 and H20 in the w e l l - k n o w n m o n o g r a p h s [6]. T h e e q u a t i o n s ~ = (08 / az)~=0,
czo~, = -
( O28 / ax~)~=o
are used. Data on ~c are given in [6-8] and elsewhere. Values of ~, are given in [6] in implicit form. The accuracy of ~c, and particularly, of ~,, is not sufficient for CO 2 and H20. In conclusion we note that the relation for ~ik (Ti, Tk), represented in Eq. (7) with Xk = 0, was obtained in complete agreement with [1,2,9]. We note also that the exponent m does not come in, since u + m = ~4 - i. Estimates of the effect of the path length according to Eq. (7), and of the dependence of O~ik* (Ti, Tk), represented by Eqs. (8), have been obtained for the first time. LITERATURE 1.
2. 3. 4. 5.
6~
7. 8. 9.
CITED
R. D. C e s s a n d P. M i g h d o l l , " M o d i f i e d P l a n c k m e a n c o e f f i c i e n t s f o r o p t i c a l l y thin g a s e o u s r a d i a t i o n , " I n t e r n a t . J. H e a t M a s s T r a n s f e r , 10, no. 9, p. 1291, 1967. R. D. C e s s , P. M i g h d o l l , a n d S. N. T i w a r i , " I n f r a r e d r a d i a t i v e h e a t t r a n s f e r in n o n g r a y g a s e s , " I n t e r n a t . J. H e a t M a s s T r a n s f e r , 1.00, no. 11, p. 1521, 1967. J. C. B r e e z e , C. C. F e r r i s o , C. B. L u d w i g , a n d W. M a l k m u s , " T e m p e r a t u r e d e p e n d e n c e of the t o t a l i n t e g r a t e d i n t e n s i t y of v i b r a t i o n a l - r o t a t i o n a l b a n d s y s t e m s , " J. C h e m . P h y s . , 42, no. 1, p. 402, 1965. J. C. B r e e z e a n d C. C. F e r r i s o , "Shock w a v e i n t e g r a t e d i n t e n s i t y m e a s u r e m e n t s of t h e 2,7 m i c r o n CO 2 b a n d b e t w e e n 1200 ~ a n d 3000 ~ K," J. C h e m . P h y s . , 139, no. 10, p. 2619, 1963. C. C. F e r r i s o a n d C. B. L u d w i g , " S p e c t r a l e m i s s i v i t i e s a n d i n t e g r a t e d i n t e n s i t i e s of the 1 . 8 7 - , 1 . 3 8 - , and 1 . 1 4 - m i c r o n H20 b a n d s b e t w e e n 1000 ~ a n d 2400 ~ K," J. C h e m . P h y s . , 41, no. 6, p. 1668, 1964. A. S. N e v s k i i , H e a t T r a n s f e r in O p e n - H e a r t h F u r n a c e s [in R u s s i a n ] , M e t a l l u r g i z d a t , M o s c o w , 1963. S. P. D e t k o v a n d V. N. G i r s , " M e c h a n i z a t i o n of r a d i a t i o n c a l c u l a t i o n s in c a r b o n d i o x i d e , " I z v e s t i y a VUZ, Chernaya metallurgiya, no. 2, p. 162, 1967. M. M. Abu-Romia and C. L. Tien, "Appropriate mean absorption coefficients for infrared radiation of gases," Trans. ASME, ser. C. J., Heat Transfer, 89, no. 4, 1967. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities [Russian translation], Izd-vo inostr, lit., Moscow, 1963.