Macrosegregation during solidification resulting from density differences in the liquid
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M A C R O S E G R E G A T I O N caused by i n t e r d e n d r i t i c fluid flow has b e e n t r e a t e d a n a l y t i c a l l y in a n u m b e r of r e c e n t p a p e r s . Most of t h e s e models a r e for i n v e r s e s e g r e g a tion, where an a n a l y t i c a l solution can be obtained by c o n s i d e r i n g backflow through a v o l u m e e l e m e n t as the solid and liquid c o n t r a c t d u r i n g s o l i d i f i c a t i o n . Chill face s e g r e g a t i o n u n d e r t h e s e conditions was f i r s t p r e dicted b y Scheil ~ and his model was l a t e r extended by K i r k a l d y and Youdelis to p r e d i c t the solute d i s t r i b u t i o n along the whole c a s t i n g . 2'3 A m o r e g e n e r a l s o l u t i o n for m a c r o s e g r e g a t i o n cons i d e r i n g fluid flow in t h r e e d i m e n s i o n s for a b i n a r y a l loy with a c o n s t a n t p a r t i t i o n r a t i o was f i r s t published b y F l e m i n g s and N e r e o 4 and used to p r e d i c t solute d i s t r i b u t i o n s for i n v e r s e s e g r e g a t i o n in A1-Cu alloys. The model was b a s e d on the lafann equation, which gives the c o m p o s i t i o n of solid at the s o l i d - l i q u i d i n t e r f a c e ( C s ) when t h e r e is c o m p l e t e m i x i n g in the adjoining liquid: C s = kCo(1 - fs)/~l
where k = equilibrium partition ratio = weight f r a c t i o n solid ( i . e . , c o m p o s i t i o n s a r e in wt pet) C O = i n i t i a l alloy c o m p o s i t i o n
fs
The model has tended s'6 and has and F l e m i n g s 7 to a c o m b i n a t i o n of
s u b s e q u e n t l y b e e n r e f i n e d and e x b e e n applied by M e h r a b i a n , Keane p r e d i c t m a c r o s e g r e g a t i o n c a u s e d by s o l i d i f i c a t i o n c o n t r a c t i o n and solute
N. STREAT, formerly Research Associate, Department of Metallurgy, University of British Columbia, is now MetallurgicalResearch Engineer, British Columbia Hydro and Power Authority, Vancouver, Canada. F. WEINBERGis Professor, Department of Metallurgy, University of British Columbia, Vancouver, Canada. Manuscript submitted October 22, 1973. METALLURGICALTRANSACTIONS
convection. They c o n s i d e r the fluid d y n a m i c s through a v o l u m e e l e m e n t where the f o r c e s acting a r e solid c o n t r a c t i o n , liquid c o n t r a c t i o n and g r a v i t y . The liquid is of v a r i a b l e d e n s i t y , and the s o l i d - l i q u i d r e g i o n is t r e a t e d as a porous m e d i u m of v a r i a b l e p o r o s i t y . E q u a t i o n s a r e d e r i v e d r e l a t i n g i n t e r d e n d r i t i c fluid p r e s s u r e , i n t e r d e n d r i t i c flow velocity, f r a c t i o n liquid and liquid c o m p o s i t i o n which can, in t h e o r y , be solved to give t h e s e v a r i a b l e s as a function of position. In p r a c t i c e , solutions for the g e n e r a l c a s e a r e difficult to obtain, s i n c e this would involve the solution of s i m u l t a n e o u s p a r t i a l d i f f e r
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