A generalized expression for lag-time in the gas-phase permeation of hollow tubes
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n u m b e r of y e a r s ago J a e g e r t and B a r r e r z developed an e x p r e s s i o n r e l a t i n g a n o n s t e a d y state p a r a m e t e r , obtained in a p e r m e a t i o n e x p e r i m e n t and known as l a g t i m e , to the bulk coefficient of diffusion for a p e r m e a t ing s p e c i e s through a m e m b r a n e . T h e i r d e v e l o p m e n t was b a s e d on the a s s u m p t i o n that phase b o u n d a r y r e a c t i o n s can be neglected, and that the b o u n d a r y c o n d i tions to F i c k ' s second law solution a r e c o n s t a n t s c o r r e s p o n d i n g to i n s t a n t a n e o u s e q u i l i b r i u m b e t w e e n the s p e c i e s in the e n v i r o n m e n t s above the e n t r a n c e and exit s u r f a c e s and the s p e c i e s in the l a t t i c e j u s t below the e n t r a n c e and exit s u r f a c e s . R e c e n t s t u d i e s involving the g a s - p h a s e p e r m e a t i o n of hydrogen through a - p h a s e t i t a n i u m s'4 have indicated that phase b o u n d a r y r e a c t i o n s m a y be p a r t i a l l y , if not c o m p l e t e l y , c o n t r o l l i n g s t e a d y state h y d r o g e n p e r m e a tion. Although l a g - t i m e m e a s u r e m e n t s yield an a c t i v a tion e n e r g y which c o r r e s p o n d s c l o s e l y to p r e v i o u s l y p u b l i s h e d data for h y d r o g e n diffusion in a - p h a s e t i t a n i u m , the use of J a e g e r ' s and B a r r e r ' s equation yields a p r e e x p o n e n t i a l c o n s t a n t for diffusion, Do, which is low b y a f a c t o r of about t h r e e . The p u r p o s e of this p a p e r is to p r e s e n t a m o r e g e n e r a l i z e d e x p r e s s i o n for l a g - t i m e obtained f r o m F i c k ' s second law when phase b o u n d a r y r e a c t i o n s need not be n e g l e c t e d . This p r o b l e m is s i m i l a r to heat conduction p r o b l e m s d e s c r i b e d by C a r s l a w and J a e g e r . s'8 The g e n e r a l solution will be s i m p l i f i e d for t h r e e l i m i t i n g c a s e s and applied in each c a s e to the e x p e r i m e n t a l l y o b t a i n e d l a g - t i m e data for a - p h a s e t i t a n i u m and a phase iron.
For a hollow cylinder Fick's second law is given by
D
82c
1 8c
at - ar--r -~- + r
Dr
ac - D ~ - ( a , t) = ha [cg
[2]
and at r = b b e c o m e s -D~7In Eq. [3] no t e r m c o r r e s p o n d i n g to cg in Eq. [2] app e a r s b e c a u s e the exit s u r f a c e is a s s u m e d to be exposed to v a c u u m . Using Laplace t r a n s f o r m s , Eq. [1] b e c o m e s d2~-
1 d~ _ q ~
[4]
dr--~ + r dr where q2 = p/D and the b o u n d a r y conditions, Eqs. [2] and [3] b e c o m e
-D
ha
d~~(a,p)
ce = p
~-(a, p)
[~]
s
and -D d~ - hb -~-(b,p)=
~(b, p) s "
[6]
Eq. [4] is a modified B e s s e l equation of o r d e r z e r o and its s o l u t i o n is given b y
= AIo(qr ) + BKo(qr )
[7]
[I]
U-- Cg [ 1 K Dq Kl(qb)] I o(qr) pA Ls ~ ) -ffb-b Cg Dq pA [ l i o ( q b ) + W I~(qb)] Ko(qr)
KIRITKUMARK. S
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