Differential-integral equation for dissipation function and derivation of conservation relationship connecting reflection, transmission and dissipation functions of finite sla

Bellman, R. E.

Kagiwada, H. H.

Kalaba, R. E.

Ueno, S.

English

NASA Technical Reports Server

' JULY 1965 
I 
THE INVARIANT IMBEDDING EQUATION 
mAR VUl3 nTCIC!TD A V T A h T  l 3 T T h T T u f l l T / \ h T  nl3 
H HWMUtiHN,MJUS J?'lNlrl'.E SLAB 
R. E. Bellman, H. H. Kagiwada, R. E. Kalaba and S. Ueno 
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~~ 
NASr-21(07) 
MEMORANDUM 
RM-4651-NASA 
JULY 1965 
.- 
THE INVARIANT IMBEDDING EQUATION 
FOR THE DISSIPATION FUNCTION O F  
A HOMOGENEOUS FINITE SLAB 
R. E. Bellman, H. H. Kagiwada, R. E. Kalaba and S. Ueno 
This research is sponsored by the National Aeronautics and Space Administration under 
Contract No. NASr-21. This report does not necessarily represent the views of the 
National Aeronautics and Space Administration. 
1700 H b l N  S l  . S A N l 4  HONIC4 * C 4 L I f O I N I 4  * , 0 4 0 6  
,’ w 
ii 
Published by The RAND Corporation 
iii 
c 
PREFACE 
This Memorandum is part of RAND'S continuing study of Satellite 
Meteorology for the National Aeronautics and Space Administration 
under contract NASr-21(07). The relations which are derived will 
lead to the increasingly effective use of computers in the numerical 
study of radiative transfer in planetary atmospheres. The Memorandum 
will be especially useful to atmospheric physicists, astrophysicists, 
and applied mathematicians. 
V 
SUMMARY 
A commonly occurring situation in planetary and stellar physics 
is that parallel rays of radiation illuminate a slab in which both 
absorption and isotropic scattering take place. The ultimate fate 
of an incident photon is either to be diffusely reflected, trans- 
mitted, or absorbed by the slab. 
in the study of the probability of transmission and reflection, less 
attention has been paid to the probability of absorption as a function 
of the thickness of the slab and the angle of incidence. 
While much has been accomplished 
The authors derive a differential-integral equation for the 
dissipation function. 
tionship connecting the reflection, transmission, and dissipation 
functions. 
tational purposes. 
In addition they derive a conservation rela- 
These relations are useful for both analytic and compu- 
v i i  
CONTENTS 
PREFACE ....................................................... iii 
SUMpfARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v 
Sect  i o n  
I. INTRODUCTION ............................................. 1 
11. DERIVATION OF INVARIANT IMBEDDING EQUATION 
FOR THE ABSORPTION FUNCTION ............................ 1 
111. CHECK FOR THE CASE OF NO REEMISSION ...................... 3 
I V .  CONSERVATION RELATIONSHIP  ................................ 4 
V. D I S C U S S I O N  ............................................... 4 
REFERENCES ...................................................... 6 
1 
I. INTRODUCTION 
The p h y s i c a l  s i t u a t i o n  t o  be cons idered  i n  t h i s  Memorandum i s  
as fo l lows :  p a r a l l e l  r a y s  of  r a d i a t i o n  a r e  i n c i d e n t  on a f i n i t e  
homogeneous s l a b  which abso rbs  and s c a t t e r s  r a d i a t i o n  i s o t r o p i c a l l y .  
These d i f f u s e l y  t r a n s m i t t e d  and r e f l e c t e d  f i e l d s  have been s t u d i e d  
i n t e n s i v e l y .  I n  e a r l i e r  papers  (1-3) we have shown t h e  importance of 
t h e  d i s s i p a t i o n  f u n c t i o n  i n  v a r i o u s  a n a l y t i c a l  s t u d i e s  of t r a n s p o r t  
i n  a rod. I n  t h i s  Memorandum we d e r i v e  a n  e q u a t i o n  f o r  t h e  d i s s i -  
p a t i o n  f u n c t i o n  of  a s l a b  and w r i t e  t h e  c o n s e r v a t i o n  r e l a t i o n  which 
r e l a t e s  t h e  r e f l e c t i o n ,  t r ansmiss ion ,  and d i s s i p a t i o n  f u n c t i o n s .  
11. DERIVATION OF INVARIANT IMBEDDING EQUATION FOR THE ABSORPTION 
FUNCTION 
Consider a p l a n e - p a r a l l e l ,  homogeneous and i s o t r o p i c a l l y  s c a t t e r i n g  
medium of  f i n i t e  o p t i c a l  t h i c k n e s s  T Suppose t h a t  a p a r a l l e l  beam o f  
r a d i a t i o n  o f  cons t an t  n e t  f l u x  v p e r  u n i t  area normal t o  t h e  i n c i d e n t  
d i r e c t i o n  i s  i n c i d e n t  on  t h e  upper s u r f a c e  T = 0 a t  a f i x e d  a n g l e  whose 
c o s i n e  i s  p, (0 < p L 1) w i t h  r e s p e c t  t o  t h e  inward normal. We fo l low 
t h e  s t a n d a r d  nomenclature of  Chandrasekhar. 
1' 
0 0 
( 4 )  
The i n t e n s i t y  o f  r a d i a t i o n  which i s  d i f f u s e l y  r e f l e c t e d  from t h e  
s l a b  w i t h  d i r e c t i o n  c o s i n e  p, i s  S ( ~ ~ ; p , p ~ ) / & ,  and t h e  d i f f u s e l y  t r a n s -  
m i t t e d  i n t e n s i t y  w i t h  d i r e c t i o n  cos ine  p i s  T(Tl;p,,po)/4p,. 
r e c t l y  t r a n s m i t t e d  i n t e n s i t y  i s  TT exp(-.r /p, ) i n  t h e  d i r e c t i o n  o f  
inc idence .  
The d i -  
1 0  
2 
We d e f i n e  t h e  a b s o r p t i o n  f u n c t i o n  L i n  t h e  fo l lowing  fash ion .  Le t  
t h e  ra te  o f  p roduc t ion  o f  t r u l y  absorbed 
p a r t i c l e s  i n  a c y l i n d e r  of  u n i t  base  area 
extending  from T = 0 t o  = T t h e  i n p u t  
having d i r e c t i o n  c o s i n e  p and t h e  n e t  
i n c i d e n t  f l u x  be ing  TT. 
1’ 
0 
(It i s  c l e a r  t h a t  L i s  a l s o  t h e  p r o b a b i l i t y  o f  u l t i m a t e  a b s o r p t i o n  o f  
a p a r t i c l e  w i th  d i r e c t i o n  c o s i n e  p 
t h i ckness  7 ) We add an  i n f i n i t e s i m a l  l a y e r  o f  o p t i c a l  t h i c k n e s s  A 
t o  t h e  lower s u r f a c e  T and w e  c o n s i d e r  i t s  e f f e c t  on t h e  r a t e  o f  
product ion  of  absorbed p a r t i c l e s .  
which i s  i n c i d e n t  on a s l a b  of  
0 
1’ 
1’ 
We o b t a i n  t h e  equa t ion  
TTL ( l+A, lo 1 
where h i s  t h e  a lbedo  f o r  s i n g l e  s c a t t e r i n g .  The f i r s t  term on t h e  
r ight-hand s i d e  of t h e  equa t ion  accounts  f o r  t h e  a b s o r p t i o n  o f  p a r t i c l e s  
which never  e n t e r  t h e  t h i n  s l a b .  The second term accoun t s  f o r  those  
p a r t i c l e s  which i n t e r a c t  i n  t h e  t h i n  s l a b  and then  a r e  absorbed. The 
f i r s t  bracketed expres s ion  r e p r e s e n t s  t h e  r a t e  of p roduc t ion  of  i n t e r -  
a c t i n g  p a r t i c l e s  i n  t h e  c y l i n d e r  of  u n i t  base  a r e a  ex tend ing  from 
T = T t o  T = 7 + A ,  and t h e  second b racke ted  e x p r e s s i o n  i s  t h e  
p r o b a b i l i t y  t h a t  a n  i n t e r a c t i n g  p a r t i c l e  i s  u l t i m a t e l y  absorbed.  
1 1 
A l l  
3 
2 o t h e r  p r o c e s s e s  have p r o b a b i l i t i e s  o f  o r d e r  A o r  g r e a t e r  and a re  ac-  
counted f o r  i n  t h e  term o(A). L e t t i n g  A + 0 ,  w e  o b t a i n  the  p a r t i a l  
d i f f e r e n t i a l  i n t e g r a l  equa t ion  
The i n i t i a l  c o n d i t i o n  i s  
111. CHECK FOR THE CASE OF NO REEMISSION 
For t h e  c a s e  of  no r eemiss ion  of i n t e r a c t i n g  p a r t i c l e s ,  t h e  r a t e  
of produc t ion  of absorbed p a r t i c l e s  i s  
- 7  ILL 
= rr ~ ~ ( 1 - e  O) . 
A s  a check,  w e  p u t  A = 0 i n  Eq. (2),  and t h e  r e s u l t  a g r e e s  wi th  
Eq. ( 4 ) .  
4 
I V .  CONSERVATION RELATIONSHIP 
The p a r t i c l e s  i n c i d e n t  on a u n i t  o f  h o r i z o n t a l  area are  e i t h e r  
d i r e c t l y  t r a n s m i t t e d  , t r u l y  absorbed ,  d i  f f u s e  l y  r e f l e c t e d  , o r  d i f f u s e l y  
t r ansmi t t ed .  This  l eads  t o  t h e  c o n s e r v a t i o n  r e l a t i o n s h i p  
0 
which reduces t o  
1 
V. DISCUSSION 
For  r e f e r e n c e ,  w e  p r e s e n t  t h e  complete  s e t  of i n t e g r o -  
d i f f e r e n t i a l  equa t ions  f o r  t h e  d i f f u s e  r e f l e c t i o n ,  d i f f u s e  t r ansmiss ion  
and abso rp t ion  f u n c t i o n s .  These e q u a t i o n s  a r e  
5 
S(0;P,Po) = 0 ,  (7’) 
T h i s  s e t  of  equa t ions  can be approximated v i a  t h e  use  of Gaussian 
q u a d r a t u r e  formulas  which l e a d s  t o  an e f f e c t i v e  computa t iona l  scheme 
f o r  de t e rmin ing  S ,  T ,  and L. 
The conse rva t ion  equa t ion ,  which can  be d e r i v e d  a n a l y t i c a l l y  as 
w e l l  a s  p h y s i c a l l y ,  connec ts  t h e  s o l u t i o n s  of  E q s .  (7), (8), and ( 9 ) .  
6 
It can s e r v e  as a n  au tomat i c  check on t h e  accuracy o f  t h e  numerical  c a l c u l a -  
t i o n s  o f  S ,  T ,  and L. F o r  t h e  inhomogeneous o r  a n i s o t r o p i c  c a s e s ,  t h e  
a p p r o p r i a t e  equa t ions  f o r  t h e  a b s o r p t i o n  f u n c t i o n s  a re  r e a d i l y  de r ived .  
REFERENCES 
1. Bellman, R. E . ,  K.  L. Cooke, R. E .  Kalaba,  and G. M. Wing, E x i s t e n c e  
and Uniqueness Theorems i n  I n v a r i a n t  Imbedding - I: 
P r i n c i p l e s ,  The RAND Corpora t ion ,  RM-3611-ARPA, May 1963. 
Conservat ion 
2 .  Bellman, R. E . ,  H. H. Kagiwada, R. E. Kalaba,  and M. C. P r e s t r u d ,  
I n v a r i a n t  Imbedding and Time-dependent T r a n s p o r t  P r o c e s s e s ,  
American E l s e v i e r  P u b l i s h i n g  Company, I n c . ,  New York, 1964. 
3. Bellman, R. E . ,  R. E. Kalaba,  and G. M. Wing, " I n v a r i a n t  Imbedding, 
Conservat ion R e l a t i o n s ,  and Non l inea r  Equat ions w i t h  Two-point 
Boundary Values," Proc.  Nat. Acad. Sc i .  USA, Vol. 46, 1960, 
pp. 1258- 1260. 
4 .  Chandrasekhar, S . ,  R a d i a t i v e  T r a n s f e r ,  Oxford U n i v e r s i t y  P r e s s ,  
19 50. 


The invariant imbedding equation for the dissipation function of a homogeneous finite slab

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