I n th e microwave d is c h a r g e a p p a r a tu s a s m a ll b o re tu b e , 2 mm i n t e r n a l d ia m e te r , was u sed t o c o n ta i n th e d is c h a r g e volume. S in ce a s tr o n g r e a c t i o n i s to be e x p e c te d betw een a f r e e h a lo g e n atom and s i l i c o n from th e q u a r tz tu b e , i t i s im p o rta n t t o c o n s id e r th e l o s s o f h a lo g e n r e s u l t i n g from d i f f u s i o n t o th e w a l l. T his d i f f u s i o n i s d e s c r ib e d i n th e f o llo w in g w ith th e d i f f u s a n t produced i n r e p e t i t i v e s h o r t p u ls e s accom panied by h e a t p ro d u c tio n . The th erm al b e h a v io u r i s t r e a t e d f i r s t to allo w c a l c u l a t i o n o f te m p e ra tu re d ep en d en t d i f f u s i o n .
3.1 D e r iv a t io n o f th e C a l c u l a t i o n Method
The prim ary s i m p l i f i c a t i o n s a r e t h a t th e p u ls e d u r a t i o n i s assumed t o be n e g l i g i b l e compared w ith th e p e rio d betw een th e p u l s e s , and t h a t th e th erm al c a p a c i ty o f th e c o n ta in in g tu b e i s much g r e a t e r th a n t h a t o f th e g a s . The l a t t e r s i m p l i f i c a t i o n p e r m its t h e tu b e h e a ti n g to be t r e a t e d s e p a r a t e l y from th e gas h e a t i n g .
Two h e a t l o s s p r o c e s s e s a r e d e s c r ib e d ; f r e e c o n v e c tio n and f o rc e d c o n v e c tio n . Under f r e e c o n v e c tio n , h e a t i s l o s t from th e o u t e r
5/ii
s u r f a c e o f th e tu b e a t a r a t e p r o p o r t io n a l t o T , w h ile f o rc e d c o n v e c tio n i s re g a rd e d as h o ld in g th e o u t e r s u r f a c e o f th e tu b e a t room te m p e ra tu re .
Chapter 3 . 5 3
3 .1 .1 B asic E q u a tio n s
The fun d am en tal e q u a tio n s g o v ern in g d i f f u s i o n i n i s o t r o p i c m edia a r e
f = -D g rad C ' (1)
and
a c / a t = div(D g rad C ), (2)
where C i s th e p a r t i c l e c o n c e n t r a t io n , D th e d i f f u s i o n c o e f f i c i e n t and f th e normal p a r t i c l e flow i n number p e r u n i t tim e p e r unj.t a r e a . For h e a t t r a n s p o r t by c o n d u c tio n i n b o th th e gas and th e tu b e w a ll we can w r i t e C = phT , and D = K/ph , where T i s th e te m p e ra tu re , p th e d e n s i t y , h t h e s p e c i f i c h e a t c a p a c i ty , and K th e th erm al c o n d u c ti v it y o f t h e g as ( o r th e w a l l ) . I t i s u s u a l to w r i t e k = K/ph
where k i s known as th e d i f f u s i v i t y . The s o l u t i o n o f t h e d i f f u s i o n e q u a tio n (2) i s d is c u s s e d i n d ep th by Crank (1956) and, f o r h e a t c o n d u c tio n , by C arslaw and J a e g e r (1 9 5 9 ).
The p a r t i c u l a r s o l u t i o n o f (2) r e q u i r e d h e r e i s f o r c y l i n d r i c a l geom etry w ith v a r i a t i o n f o r th e r a d i a l c o o r d i n a te o n ly . The te m p e ra tu re dependence o f th e d i f f u s i o n and d i f f u s i v i t y w i l l be t r e a t e d by s e p a r a t i n g th e tim e b eh av io u r i n t o sm a ll elem en ts w ith c o n s ta n t d i f f u s i o n and d i f f u s i v i t y . The n e c e s s a ry changes due to te m p e ra tu re w i l l be i n s t e p s p r o v id in g an av erag e v a lu e to a s s i g n to each tim e e le m en t. T his means t h a t a s o l u t i o n o f (2) f o r c o n s ta n t d i f f u s i o n c o e f f i c i e n t can be u sed , th u s a v o id in g co n sid erab le- m ath em atical c o m p le x ity .
For th e c a se o f a c o n s ta n t d i f f u s i o n c o e f f i c i e n t and by u s in g c y l i n d r i c a l c o o r d in a te s w ith r a d i a l v a r i a t i o n o n ly , e q u a tio n (2) becomes a c / a t = D(ô^C/ôr^ ( 1 / r ) ô C / a r ) , (3) I t w i l l be u s e f u l to u se th e fo llo w in g d im e n s io n le s s v a r i a b l e s i n t h i s e q u a tio n ; r = r / a t = D t/a ^ , C =,C/Cq f o r d i f f u s i o n , and 2 t = k t / a and T = T/T^ f o r h e a t t r a n s p o r t ,
Chapter 3 . 5 5
where a i s th e i n n e r tube w a ll r a d i u s , 0^ i s th e c o n c e n t r a t io n of* d i f f u s a n t r e s u l t i n g from a s i n g l e e x c i t a t i o n p u ls e , and i s th e room te m p e ra tu re (293K ). 3 .1 .2 Heat T ra n sp o rt The a p p r o p r i a t e d im e n s io n le s s form o f (3) i s ô f / ô t = Ô^T/ôr^ + ( 1 /r ) Ô T /ô r . (4) For t y p i c a l boundary c o n d it io n s th e a n a l y t i c s o l u t i o n o f t h i s e q u a tio n in v o lv e s le n g th y sum m ations. S o l u ti o n s a r e t h e r e f o r e o b ta in e d u s in g th e f i n i t e d i f f e r e n c e method (C rank 1956) w ith r d iv id e d i n t o a number o f i n t e r v a l s . I n th e f o llo w in g , th e i n t e g e r v a r i a b l e s i and j g iv e th e number o f s p a t i a l in c r e m e n ts , 5 r , and tem p o ral in c re m e n ts, 5 t , r e s p e c t i v e l y . The f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n s o f th e d e r i v a t i v e s app ro x im ate t o th e f o llo w in g ,
(a^T/apZ) = (T - 2T, . + T .)/(5r)^ , (5)
1 + 1 , J 1 - I , j
= ( V l . j -
.
(G)
( a î / a ? ) , _ j = - T , ^ j ) / 5 t . (7)
r = i 5 r , (8)
S u b s t i t u t i n g th e s e i n t o (5) g iv e s
T, . , = T. . + 5tt(2i+l)T. , . - ItiT. , + (2i-1)T. , .]/2i(5r)^ .1 ; J + 1 I j J 1 + 1 , 0 1 , 0 1™1, 0
. . . . ( 9 )
T his e q u a tio n g iv e s a fo rm u la whereby th e r a d i a l d i s t r i b u t i o n o f th e d im e n s io n le ss te m p e ra tu re can be c a l c u l a t e d f o r s u c c e s s iv e tim e i n t e r v a l s s t a r t i n g from an i n i t i a l d i s t r i b u t i o n . I t can be s e e n t h a t th e v a lu e o f th e m ain v a r i a b l e a t a g iv e n p o i n t i s c a l c u l a t e d from t h r e e s p a t i a l v a lu e s one tim e in c re m e n t p r e v i o u s ly . These t h r e e s p a t i a l p o i n ts a r e th e c u r r e n t p o i n t and i t s n eig h b o u rs a t s m a lle r and l a r g e r r a d i u s . T his fo rm u la can n o t be u se d , t h e r e f o r e , f o r th e two end p o i n ts o f th e r a d i u s v a r i a b l e . I n a d d i t i o n , th e s e end p o i n t s may be i n c o n ta c t w ith h e a t s o u r c e s ,o r s in k s . S in c e , i n g en eral., a g r e a t v a r i e t y o f c o n d it io n s may o c c u r, i t w i l l be n e c e s s a ry to e l a b o r a t e th e model used h e r e t o p e rm it c a l c u l a t i o n o f t h e s e te r m in a l v a lu e s .
For th e tu b e , th e l a r g e th erm al c a p a c i ty means t h a t c o n d it io n s i n th e gas w i l l have s t a b i l i s e d b e f o r e th e tube te m p e ra tu re s t a r t s to r i s e . As a consequence, th e h e a t flo w in g i n t o th e in n e r tube w a ll can be re g a rd e d a s c o n s ta n t , sm ooth, and eq u a l t o th e av erag e r a t e a t which th e gas i s h e a te d . For th e p r e s e n t c irc u m s ta n c e s e q u a tio n (1) can be w r i t t e n
Chapter 3 57
£ = -.DôC/ôr ;
which i n d im e n s io n le ss v a r i a b l e s o f h e a t t r a n s p o r t i s
q = (-KT / a ) a T / 6 r , (10)
where q i s energy p e r u n i t a r e a p e r u n i t tim e . The f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n o f (10) i s
= -aq/KTo • (11)
For th e i n n e r tu b e w a ll, th e term T. , i n (11) and (9) w i l l be3.“ I f i c t i t i o u s and must be e lim in a te d from (9) and (11) t o g iv e
T, . , = T. , + (25t/(5r)^)CI. , . - T. . + (2i-1)5raq/2iKT.] .1 , J+1 J 1 + ‘ , J ^I J u
(12)
For th e o u t e r tube w a l l, q i s th e h e a t l o s s r a t e and T^^^ i s e lim in a te d t o g iv e
T. 1 ,J + | = T. . + (25t/(5r)^)[T ;J - T. , - (21+1)6raq/2iK?n] ■1 ,J U
..(13)
Under f o rc e d c o n v e c tio n , th e o u t e r w a ll te m p e ra tu re i s c o n s ta n t a t T^