Studies in vacuum degassing: Mass and momentum transfer to gas bubbles rising in melts, the freeboard of which is evacua
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ttons it was a s s u m e d that the bubble was in dynamic e q u i l i b r i u m with its surroundings, m o r e s p e c i f i c a l l y : a) I n e r t i a l effects were neglected, i.e., the p r e s s u r e within the bubble was a s s u m e d to be the s a m e as the p r e s s u r e in the liquid at the s u r f a c e of the bubble, f u r thermore, b) It was also a s s u m e d that the bubbles would move with t h e i r t e r m i n a l r i s i n g v e l o c i t y c o r r e s p o n d i n g to t h e i r instantaneous size. With r e g a r d to a s s u m p t i o n a), it has been shown, both through t h e o r e t i c a l a r g u m e n t s and by d i r e c t e x p e r i m e n t a t i o n , that at r a p i d growth r a t e s i n e r t i a l effects may not be neglected b e c a u s e the need to a c c e l e r a t e the fluid surrounding the bubble will r e q u i r e a p r e s s u r e gradient and hence, a p r e s s u r e d i f f e r e n t i a l between the gas and the liquid. M o r e o v e r , r e c e n t work a'9'le has shown that for r a p i d l y growing bubbles, the r i s i n g v e l o c i t y may differ a p p r e c i a b l y f r o m the t e r m i n a l value. It would a p p e a r , t h e r e f o r e , that p r e v i o u s t r e a t m e n t s of m a s s t r a n s f e r to r i s i n g bubbles for vacuum d e g a s sing conditions w e r e somewhat o v e r s i m p l i f i e d and that a m o r e a c c u r a t e r e p r e s e n t a t i o n of these phenomena would be d e s i r a b l e . In the following, we shall p r e s e n t a m o r e p r e c i s e f o r m u l a t i o n d e s c r i b i n g the behavior of r i s i n g bubbles which grow due to both m a s s t r a n s f e r and a p r o g r e s s i v e l y d e c r e a s i n g environmental p r e s s u r e ( e x p e r i e n c e d by the bubbles during t h e i r r i s e ) . The p r e d i c t i o n s b a s e d on this m o d e l will then be c o m p a r e d to m e a s u r e m e n t s on a p h y s i c a l model s y s t e m ; having thus v e r i f i e d the model the computed r e s u l t s will then be g e n e r a l i z e d to s y s t e m s of m e t a l l u r g i c a l i n t e r e s t .
1. FORMULATION Let us c o n s i d e r a gas bubble r i s i n g in a melt, the f r e e b o a r d of which is evacuated, as shown in Fig. 1. The liquid (melt) in which the bubble is contained is s u p e r s a t u r a t e d with r e s p e c t to a solute (hydrogen, VOLUME 5, JUNE 1974-1429
resent a reasonable compromise between the faithful representation of the growth phenomena and the need to deal with a mathematical problem of manageable complexity. Thus a mass balance on the bubble may be written
Co
as: C~
dNi dt
Co
p=
1 A
sat co
0
t
R(t)
Fig. 1--Schematic representation of the physical system.
nitrogen, CO), which may then be t r a n s f e r r e d from the melt into the gas bubble. It follows that during its ascent the bubble will grow due to two mechanisms, namely: a5 growth due to the p r o g r e s s i v e l y decreasing environmental p r e s s u r e , b) growth due to mass transfer
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