Iterative solutions for one-dimensional diffusion with time varying surface composition and composition-dependent diffus
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AND
C. R. HOUSKA
Solutions are given for one-dimensional diffusion problems with a time varying surface composition and also a composition dependent diffusion coefficient. The most general solution does not require special mathematical functions to fit the variation in surface composition or D(C). In another solution, a series expansion may be used to fit the time dependent surface concentration. These solutions make use of iterative calculations that converge rapidly and are highly stable. Computer times are much shorter than that required for finite difference calculations and can efficiently make use of interactive graphics terminals. Existing gas carburization data were used to provide an illustration of an iterative approach with a time varying carbon composition at the free surface.
T H E c h e m i c a l , m e c h a n i c a l and e l e c t r i c a l p r o p e r t i e s of a solid s u r f a c e can be a l t e r e d g r e a t l y by i n t r o d u c ing a l o cal c o m p o s i t i o n c h a n g e . T h i s m a y be b r o u g h t about by a d i f f u s i o n flux o r i g i n a t i n g f r o m a v a p o r , liquid, or s o l i d s o u r c e in c o n t a c t with a f r e e s u r f a c e . T he c l a s s i c a l e x a m p l e of t h e s e p r o c e s s e s is the use of a c a r b o n s o u r c e to p r o v i d e a h a r d w e a r r e s i s t a n t s u r f a c e l a y e r . In a l l of t h e s e d if f u s io n p r o b l e m s , the s u r f a c e c o n c e n t r a t i o n m u s t change f r o m s o m e i n i t i a l v a l u e to a new v a l u e which m a y or m a y not a p p r o a c h a constant v a l u e with i n c r e a s i n g t i m e . T h i s p a p e r d e a l s with s o l u t i o n s to the d i f f u s i o n e q u a t i o n when the s u r f a c e c o m p o s i t i o n c h a n g e s c o n t i n u o u s l y with t i m e . U s u a l l y , the d i f f u s io n c o e f f i c i e n t is c o n c e n t r a t i o n dependent, and t h i s must be c o n s i d e r e d if the d e t a i l e d shape of c o m p o s i t i o n p r o f i l e is to be d e t e r m i n e d . F i nite d i f f e r e n c e c a l c u l a t i o n s have b e e n the only m e a n s of obtaining p r o f i l e s f o r this m o r e g e n e r a l c a s e of both a t i m e d e p e n d e n t s u r f a c e c o m p o s i t i o n and a d i f fusion c o e f f i c i e n t that v a r i e s with c o n c e n t r a t i o n . A l though this a p p r o a c h is c a p a b le of g i v i n g n u m e r i c a l r e s u l t s , the c o m p u t e r t i m e can a p p r o a c h a c t u a l d i f fusion t i m e s . T h e s e e x c e s s i v e c o m p u t e r t i m e s a r e r e q u i r e d to a s s u r e that the c a l c u ] a t i o n s c o n v e r g e to the p r o p e r v a l u e s and a r e s t a b l e . If the f i n i t e d i f f e r ence n e t w o r k is i m p r o p e r l y c h o s e n , d i f f i c u l t i e s will be e n c o u n t e r e d . Confidence in the n u m e r i c a l r e s u l t s is d e v e l o p e d only a f t e r an i n i t i a l e x p l o r a t i o n of the e f
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