Solution of binary multiphase diffusion problems allowing for variable diffusivity, with application to the aluminizatio
- PDF / 435,195 Bytes
- 4 Pages / 612 x 792 pts (letter) Page_size
- 98 Downloads / 182 Views
dxs_ 1 { ON6} dt 1-N6~ 0+D6~ - Xs
[la]
s o l u t i o n p h a s e s , r e s p e c t i v e l y , and the s i g n i f i c a n c e of the s u b s c r i p t s i s c l a r i f i e d in F i g . 1. A v a i l a b l e d a t a 6"~'12 i n d i c a t e t h a t D v a r i e s by l e s s than a f a c t o r of two o v e r the E and ~ p h a s e s and a cons t a n t a v e r a g e v a l u e of D m a y be u s e d for e a c h of t h e s e p h a s e s . The c o n c e n t r a t i o n p r o f i l e s in the E and ~ p h a s e s a r e , t h e r e f o r e , t a k e n to be of the u s u a l e r r o r - f u n c t i o n type: ,~ N~ = N e h - ( N c 5 - : ~ c ~
~ { e r f 0 " c a ~ c ) - e r f ( x / 2 D4~et)} {erf(rc~Se )_erf(rcac~) }
[2a] ~{1 - e r r (x/2 D~D~t)} ,\r~ = N~ ~ _ ( N ~ o _ l ~c) {1 - erf(r~c~c~) }
[2b]
where )-r = x/-D-~; ~'~ = fD~D-~; D = 0 5 (N = Nhr As shown in F i g . 2, the i n t e r d i f f u s i o n c o e f f i c i e n t in the 5 p h a s e v a r i e s s t r o n g l y with c o m p o s i t i o n . * The *Measurements of D in ~ were carried out by analysis of the concentration profiles of Nl specimens pack-aluminized under conditions of constant surface composition 4`s using Wagner's equation 9
c o n c e n t r a t i o n p r o f i l e w i l l not c o n f o r m to an equation of the type [2], and is u n l i k e l y to obey any s i m p l e a n a l y t i c a l e x p r e s s i o n . In this c a s e , it is e x p e d i e n t to s e e k a n u m e r i c a l s o l u t i o n of the diffusion equation f o r the 5 p h a s e and this i s s i m p l i f i e d if D 5 can be e x p r e s s e d a s a function of c o m p o s i t i o n . A r e a s o n a b l e fit of the diff u s i v i t y d a t a i s o b t a i n e d by an e x p r e s s i o n of the type D 6 = D0 exp (pN 6) [3] for e a c h b r a n c h of the d i f f u s i v i t y c u r v e , with the cons t a n t s Do and p o b t a i n e d f r o m a l e a s t - s q u a r e s fit of e a c h b r a n c h of the log D vs N 5 plot shown in F i g . 2. The d i f f u s i o n equation in the 5 p h a s e
ON5 = a [ D
a~__fi~
ot o x k 6 ox ) is then t r a n s f o r m e d to an o r d i n a r y d i f f e r e n t i a l equation ~
d2y
2 D
dy
- + 2~-A~ ~d~ =0 Do
[4]
IS]
by s u b s t i t u t i o n of the new v a r i a b l e s 1~ y = exp(pN5) and z = x / 2 a 6 c 4 - ~ (z = 1, at x = x6e ).
[6]
To s o l v e Eq. [5], the following s t e p s a r e c a r r i e d out: 1) U s i n g the e r r o r - f u n c t i o n s o l u t i o n s [2a] and [2b] f o r the ~ and ~ p h a s e s , Eq. [ l c ] m a y be w r i t t e n r e { e r f (reote~) - e r f ( t e a S e ) }
dt
N6c--Ne51-
dxcc _ 1 { dt N~- N~e.-Dr
60x ONe
+Dc-~-Jx6c
[lb]
ON~ ] + D~-~-Ixr
[lc]
In t h e s e equations N r e p r e s e n t s a t o m f r a c t i o n of A1; 5, r and ~ r e p r e s e n t the NiA1, Ni3A1 and N i - r i c h s o l i d A. K. SARKHEL and L. L. SEIGLE are Research Assistant and Professor, respectively, Department of Materials Science, State Umversity of New York, Stony Brook, NY 11790. Manuscript submitted July 14, 1975. METALLURGICAL
Data Loading...
