Mathematical model for the unidirectional solidification of metals: II. Massive molds
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T. W. CLYNE,
AND
M. P R A T E S
In this p a p e r , a new m a t h e m a t i c a l m o d e l r e c e n t l y outlined by the p r e s e n t a u t h o r s x is applied to the case of u n i d i r e c t i o n a l s o l i d i f i c a t i o n v i a heat e x t r a c t i o n t h r o u g h m a s s i v e uncooled molds of effectively s e m i - i n f i n i t e t h i c k n e s s . The m o d e l p e r m i t s m e a s u r e m e n t of the Newtonian heat t r a n s f e r coefficient at the metaL/mold i n t e r f a c e and a complete d e s c r i p tion of the k i n e t i c s and t h e r m a l c h a r a c t e r i s t i c s of s o l i d i f i c a t i o n is s u b s e q u e n t l y p o s s i b l e . E x p e r i m e n t a l r e s u l t s a r e c o m p a r e d with p r e d i c t i o n s for the case of lead and the effect of mold t h i c k n e s s in the effectively finite r e g i m e is a l s o i n v e s t i g a t e d .
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HE m a t h e m a t i c a l t r e a t m e n t of the g e n e r a l i z e d u n i d i r e c t i o n a l s o l i d i f i c a t i o n p r o b l e m p r e s e n t s cons i d e r a b l e c o m p l e x i t y and exact s o l u t i o n s a r e a v a i l able for only a few r e s t r i c t e d c a s e s . The b a s i c m a t h e m a t i c a l o b s t a c l e to a n a l y s i s is the s i m u l t a n e o u s t r e a t m e n t of heat flow through m e t a l a n d / o r mold by t h e r m a l conduction and a c r o s s the m e t a L / m o l d i n t e r face by Newtonian heat t r a n s f e r . The only g e n e r a l ized s o l u t i o n s which place no r e s t r i c t i o n on the i n t e r facial heat t r a n s f e r coefficient a r e those u t i l i z i n g m a t h e m a t i c a l a p p r o x i m a t i o n s 2-6 The p r o p o s e d model u t i l i z e s a n o v e l a p p r o a c h to the heat flow p r o b l e m . A b a s i c a s s u m p t i o n is that the heat t r a n s f e r coefficient r e m a i n s c o n s t a n t d u r i n g the s o l i d i f i c a t i o n p r o c e s s . * The t h e r m a l r e s i s t a n c e *Physically, this may be approximately valid in some situations but not in others. The formation of an 'air gap' between metal and mold as a result of contraction has frequently been postulated 7"9 as a mechanism for a time dependence of h i (or a position dependence in a steady state process such as continuous casting). In practice, the importance of this probably depends on metal, mold surface, geometry and so forth; in the experimental set-up used in this study, changes of h i with time would be expected to be small and this supported by the h~ternal conconsistency of experimental measurements both in this study and in the previous one. Mathematically, the case of variable h i is certainly not treatable by exact analytical methods.
p r e s e n t e d by the i n t e r f a c e is then m o d e l e d by a " p r e e x i s t i n g " t h i c k n e s s of solid, which, for the p u r p o s e s of heat flow c a l c u l a t i o n s , is additive to the r e a l p h y s i cal t h i c k n e s s . Heat flow may then be c o m
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