Heat transfer and fluid flow phenomena in electroslag refining
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a p r e v i o u s p a p e r z t h r o u g h the s t a t e m e n t of M a x w e l l ' s equations and the t u r b u l e n t N a v i e r - S t o k e s e q u a tions a f o r m u l a t i o n was p r e s e n t e d , d e s c r i b i n g the v e l o c i t y f i e l d s in both the s l a g and the m o l t e n m e t a l p o o l in ESR s y s t e m s . T h i s work, in which only DC o p e r a t i o n was c o n s i d e r e d , was s u b s e q u e n t l y g e n e r a l i z e d 2 to allow for s y s t e m s with AC, which c o m p r i s e the m a j o r i t y of i n d u s t r i a l i n s t a l l a t i o n s . The p r e d i c tions b a s e d on t h e s e m o d e l s w e r e found to be c o n s i s t e n t with both p r a c t i c a l o b s e r v a t i o n s 3 and with d a t a that could be deduced f r o m p l a n t s c a l e m e a s u r e m e n t s .4 It is noted, h o w e v e r , that a n u m b e r of s i m p l i f y i n g a s s u m p t i o n s have b e e n m a d e in the d e v e l o p m e n t of this p r e v i o u s model; the m o s t i m p o r t a n t of t h e s e a r e l i s t e d in the following: 1) I s o t h e r m a l b e h a v i o r was a s s u m e d ; thus, h e a t t r a n s f e r p h e n o m e n a w e r e not taken into c o n s i d e r a t i o n . 2) No a l l o w a n c e was m a d e for the effect of d r o p l e t formation. 3) The s h a p e and s i z e of the liquid p o o l was a s s u m e d to be given. 4) In addition, c y l i n d r i c a l s y m m e t r y was a s s u m e d , and the d a m p i n g of t u r b u l e n c e by the e l e c t r o m a g n e t i c f o r c e f i e l d was n e g l e c t e d . In the following we s h a l l c o n s i d e r the r a t h e r m o r e g e n e r a l c a s e by r e l a x i n g a s s u m p t i o n s (1) and (2). T h u s , in the t r e a t m e n t to be outlined s u b s e q u e n t l y , due a l l o w a n c e will be m a d e for h e a t t r a n s f e r p h e n o m e n a and f o r the effect of t h e r m a l l y d r i v e n n a t u r a l c o n v e c tion. The m o d e l l i n g e q u a t i o n s d e v e l o p e d on t h i s b a s i s w i l l a l l o w a much i m p r o v e d insight into the b e h a v i o r of the s y s t e m . We note f u r t h e r m o r e that, u n d e r t h e s e conditions, it is p o s s i b l e to m a k e a d i r e c t c o m p a r i s o n b e t w e e n the t h e o r e t i c a l l y p r e d i c t e d and the e x p e r i mentally measured melting rates. A. H. DILAWARI and J. SZEKELY are Visiting Engineer and Professor of Materials Engineering, respectively, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. A. H. DILAWARI is presently on leave from Institute of Chemical Engineering and Technology, Punjab University, Lahore-I, Pakistan. Manuscript submitted May 3, 1977. METALLURGICAL TRANSACTIONS B
In the following Section 2, we s h a l l p r e s e n t a m a t h e m a t i c a l s t a t e m e n t of the p r o b l e m ; the c o m p u t e d r e s u l t s w i l l be given in S e c t i o n 3, while the concluding r e m a
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