Pearlite growth rate of Fe-C-X eutectoid alloys
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wa~ = ( - ~ )
v
[1I]
w h e r e V is an a v e r a g e m o l a r v o l u m e ( a s s u m e d constant), and MAI i s the a t o m i c weight of a l u m i n u m . Fig, 4 a l s o shows the 5 l a y e r t h i c k n e s s and s u r f a c e d i s p l a c e m e n t a s f u n c t i o n s of s u r f a c e c o n c e n t r a t i o n c a l c u l a t e d f r o m the s t a n d a r d e r r o r f u n c t i o n solution, u s i n g a s e r i e s of a v e r a g e diffusion coefficients for the 5 p h a s e given by
1
; y ~ DdN) ~v.
[12]
I t i s r e m a r k a b l e that the 5 l a y e r t h i c k n e s s e s s o c a l c u l a t e d a r e w i t h i n 1 pct of t h o s e obtained by the n u m e r i c a l method. The s u r f a c e d i s p l a c e m e n t (or weight gain) i s p r e d i c t e d l e s s a c c u r a t e l y and, a s a l r e a d y m e n t i o n e d , t r u e c o n c e n t r a t i o n p r o f i l e s cannot be obtained by the u s e of a n a v e r a g e d i f f u s i v i t y . T h e n u m e r i c a l method i s e a s i l y g e n e r a l i z e d to the c a s e i n which diffusivity i s c o m p o s i t i o n - d e p e n d e n t i n a l l p h a s e s . It is only n e c e s s a r y to s e l e c t as a s t a r t i n g point for the solution, a v a l u e of c o m p o s i t i o n , N~t , i n the t e r m i n a l p h a s e K, s u c h that the v a r i a t i o n of D~ i s n e g l i g i b l e b e t w e e n N~t and N~o , with a n a v e r a g e v a l u e D} i n this r a n g e . P r o c e e d i n g as b e f o r e , the diffusion equation i s t r a n s f o r m e d to a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n by the s u b s t i t u t i o n , z = x/2c~ t ~';Di (choose D = D} f o r c o n v e n i e n c e ) :
I n this equation a t i s r e l a t e d to the p o s i t i o n xt of the c o m p o s i t i o n , N~t , by x t = 2 a t ~ : , F u r t h e r t r a n s f o r m a t i o n of Eq. [13] i s p o s s i b l e if D(N) c a n be e x p r e s s e d a n a l y t i c a l l y a s i n Eq. [3]. M o r e g e n e r a l l y , however, l e t u s s u p p o s e that the c o m p o s i t i o n v a r i a t i o n of D(N) i s gPten n u m e r i c a l l y . T h e p r o b l e m m a y t h e n b e solved b y s e l e c t i n g a v a l u e of cq and c a l c u l a t i n g (&~/dZ)z ; ~ a n d 4.r ~ zdN a s s m n i n g a n e r r o r - f u n c t i o n s o l u t i o n f o r the c o n c e n t r a t i o n p r o file in ~ for z > 1 (x > xt):
N(Z)= N ; o + (Net- N~;o)~~)-~-} ( d N / d Z ) z -'~ =
~ ( 1 - e r r at)
[14a] '
T h i s i s the s t a r t i n g p o i n t for the s t e p - w i s e c a l c u l a t i o n u s i n g the i n t e g r a t e d f o r m o~ Eq. [t3] to o b t a i n the c o n c e n t r a t i o n - g r a d i e n t a t the n e x t point, (N~t + ~ V , 1 - Zaz): (dN/dz)~.~ z
=
o N~t*AN r ..... 2 , ~ .~ zdN. D(N~t + ~uY) N~o
[15]
F o r the p a r t i c u l a r c a s e of the a l u m i n i z a t i o n of Ni, d u r i n g which A1 is gained but Ni is not lost, the c a l c u l a t i o n is continued, a s b e f o r e , u n t i l A r e a A = A r e a B i n Fig. 1. M o
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