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UBC Theses and Dissertations

Natural convection in liquid metals Stewart, Murray John 1970

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NATURAL CONVECTION I N LIQUID METALS by MURRAY JOHN STEWART B.A.Sc. ( H o n o u r s ) , U n i v e r s i t y  of B r i t i s h  Columbia, 1967  A THESIS SUBMITTED I N PARTIAL FULFILMENT  OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of METALLURGY  We a c c e p t t h i s required  thesis  as c o n f o r m i n g t o t h e  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA December,  1970  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree tha  permission for extensive copying of this thesis  for scholarly purposes may by his representatives.  be granted by the Head of my Department or  It is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of  Metallurgy  The University of British Columbia Vancouver 8, Canada  Date  December 7,.  1970  Acknowledgement  The his  a u t h o r would l i k e  research director.  c o n t i n u i n g a d v i c e and course o f t h i s  R.  accorded.  to  Dr. F r e d W e i n b e r g , f o r h i s encouragement throughout  a s s i s t a n c e was  g i v e n by  M a c M l l l a n F a m i l y F e l l o w s h i p and  Fellowship  thanks  the  project.  Financial H.  to express  i n E n g i n e e r i n g and  soecial  the  the S h e l l thanks  are  Canada thus  Abstract  Natural  convection  s t u d i e d by d i r e c t o b s e r v a t i o n  i n liquid  o f the f l u i d  radioactive t r a c e r techniques.  strongly influences  flow,  using  The s t u d y i s o f i m p o r t a n c e  in understanding the s o l i d i f i c a t i o n flow  m e t a l s h a s been  o f metals since  fluid  t h e h e a t and mass t r a n s f e r i n t h e  system which i n t u r n s t r o n g l y i n f l u e n c e s h o m o g e n e i t y , and m e c h a n i c a l p r o p e r t i e s  the s t r u c t u r e ,  of the s o l i d  metal  produced. The a rectangular on  edge.  system examined i n t h i s liquid  A small  imposed across  cell  of variable thickness,  d r i v i n g force  the l i q u i d  i n v e s t i g a t i o n was  cell  positioned  f o r natural convection a n d when s t e a d y s t a t e  was conditions  were r e a c h e d , - a s m a l l  amount o f t h e same m a t e r i a l  a radioactive isotope  was added t o t h e t o p o f t h e c e l l .  The  t r a c e r m a t e r i a l was p i c k e d  given  time i n t e r v a l the l i q u i d  position.  The r e s u l t a n t s o l i d  containing  up by t h e f l o w a n d a f t e r a was q u e n c h e d t o f i x t h e t r a c e r block  was  autoradiographed  t o d e t e r m i n e t h e d i s t r i b u t i o n o f t h e added r a d i o a c t i v e material. Thermal c o n v e c t i o n and  liquid  TI  *  2 0 1  respectively.  increase liquid  lead using  with  cell,  was o b s e r v e d i n l i q u i d t i n  radioactive S n  1 1 3  and r a d i o a c t i v e  The r e s u l t s show t h a t  the flow  rates  i n c r e a s i n g temperature d i f f e r e n c e across the i n c r e a s i n g a v e r a g e t e m p e r a t u r e , and i n c r e a s i n g  liquid  cell  from 10  6  thickness.  t o 10  8  Flow r a t e s w i t h Grashof  were e x p e r i m e n t a l l y  observed.  A f i n i t e difference numerical problem o f thermal  from 2 x 10 liquid  3  t o 2 x 10 .  a r e found  analogies to .metallic Solute  for Prandtl  metals  t o approach t h e t h e o r e t i c a l temperature  o f v a r i o u s types  of fluids  t o show t h a t n o n - m e t a l l i c  f l o w problems have v e r y  limited  convection i sexperimentally  from t h r e e d i f f e r e n t  numbers  results f o r  t h i c k n e s s e s and l a r g e  The f l o w b e h a v i o r  compared w i t h l i q u i d  for the  a n d 0.0127 w i t h G r a s h o f The e x p e r i m e n t a l  7  for large c e l l  differences. is  0.1,  t i n ( P r = 0.0127)  analysis  solution  convection i spresented  numbers o f 1 0 . 0 , 1 . 0 ,  numbers  value.  considered  v i e w p o i n t s ; a) i n d e p e n d e n t s o l u t e  c o n v e c t i o n , b) t h e i n f l u e n c e o f s o l u t e c o n v e c t i o n on t h e r m a l c o n v e c t i o n , and c ) t h e t h e r m a l complete l i q u i d  mixing.  and s o l u t e c o n d i t i o n s f o r  I t was f o u n d  t h a t t h e r e must be a  h o r i z o n t a l d e n s i t y i n v e r s i o n a c r o s s t h e whole l i q u i d for  complete mixing t o occur throughout Interdendritic  natural in  primary  alloys.  liquid  The f l o w p e n e t r a t e d  zone t o a p o i n t o f a p p r o x i m a t e l y dendrite spacings  Several experimental flow.  flow r e s u l t i n g  convection i n the residual  lead-tin  liquid  liquid  the l i q u i d  cell  zone.  from t h e  p o o l was  observed  into the s o l i d 12 - 22 % s o l i d f o r  o f f r o m 700 t o 1000  models a r e p r e s e n t e d  microns.  for interdendritic  A t h r e e - d i m e n s i o n a l w i r e mesh m o d e l p r e d i c t s t h a t  the  finer  the  dendrite  penetration  into  results  the  the  for  the  solid-liquid  lead-tin  castings  was  which  have  castings  3 wt.  Al  distribution found  The  the  flow  experimental  compared f a v o r a b l y  the  with  in  determine  radioactive  flow  stationary,  % A g was  macrosegregation  patterns.  rotated,  determined  silver  considerations,  by  added t o  the  and r o t a t e d  castings.  was case  of  the  long  shown  the to  in  the  range  melt  off is  movement in  the  a direct  of  was  be  the  of  the It  present macroFor  accounted  dendrite  result The  to  c a n be  oscillated  melt.  Extensive  solid-liquid  oscillation. related  and  oscillated casting.  macrosegregation  movement  associated with  transition.  the  and/or  This  is  detected  The  measuring  stationary  break  tration  to  flow  macrosegregation  basis  region.  of  out  fluid  significant  oscillated  which  -  the  t h a t , no  segregation  the  greater  zone.  imposed f l u i d  present  of  of  carried  macrosegregation  in  alloys  an e x t e n s i o n  an i n v e s t i g a t i o n  was  the  model.  As  in  structure,  the  for  on  fragments interface  turbulent  maximum s i l v e r  waves concen-  columnar-to-equiaxed  TABLE OF CONTENTS Page 1.  2.  3.  1  Introduction 1.1.  The i m p o r t a n c e o f l i q u i d  1.2.  The d r i v i n g f o r c e s  1.3.  The d e t e c t i o n o f f l u i d  1.4.  Purpose  1  metal flow  for liquid  metal flow  11 16  flow  o f the present i n v e s t i g a t i o n  G e n e r a l E x p e r i m e n t a l A p p a r a t u s and P r o c e d u r e s  18 20  2.1.  Apparatus  20  2.2.  Experimental procedure  24  2.3-  Validity  27  Thermal 3.1.  o f technique  32  Convection  32  Experimental investigations  3.1.1.  V a r i a b l e a p p l i e d temperature d i f f e r e n c e a n d average, t e m p e r a t u r e  3.1.2. V a r i a b l e :  3.1.3-  liquid  Temperature  39  metal thickness  distribution  i n the  :liquid 3.1.4. • : ; V a r i a b l e h e i g h t o f t h e m o l t e n 3 . 1 . 5 . •••Double c e l l f l o w 3.1.6. .Liquid metal i n v e s t i g a t e d  33  44 zone  49 49 56  3.1.7. 3.2  58  Thermal f l u c t u a t i o n s  The t h e o r e t i c a l p r o b l e m o f t h e r m a l 60  convection 3.2.1.  Problem statement  3.2.2.  Previous  3.3.  60  theoretical  solutions  3.2.2.1.  Solution of Batchelor  3.2.2.2.  S o l u t i o n o f Emery a n d Chu (3*1)  70  Numerical  a n a l y s i s o f thermal  74  3.3.1.  (33)  convection  experimental  85 108  results  3. 4.1.. T h e r m a l p r o f i l e s  108  :  3.4.2.  65  ?4  Technique o f s o l u t i o n  3.3.2. Results o f numerical analysis 3.4. C o m p a r i s o n o f t h e o r e t i c a l and  4.  65  F l o w p a t t e r n s and f l o w r a t e s  109 117  Solute Convection 4.1.  Independent s o l u t e c o n v e c t i o n  118  4.2.  The i n f l u e n c e o f s o l u t e on t h e r m a l c o n v e c t i o n  122  4.3.  convection  Thermal and s o l u t e c o n d i t i o n s f o r complete l i q u i d  128  mixing  5.  Volume Change on F r e e z i n g  136  6.  Interdendritic  111  F l u i d Flow  6.1. 6.2.  Two d i m e n s i o n a l m o d e l T h r e e d i m e n s i o n a l w i r e model  142 145  6.3.  W i r e mesh model  149  6.4.  Interdendritic  flow i n l e a d - t i n  alloys  162  7.  M a c r o - s e g r e g a t i o n i n C a s t i n g s R o t a t e d and O s c i l l a t e d During S o l i d i f i c a t i o n  177  7.1.  Introduction  177  7.2.  Experiment  178  7.3.  Results  182  7.4.  Discussion  190  7.5.  Conclusion  195  7.6.  Appendix t o s e c t i o n 7  196  8.  Summary and C o n c l u s i o n s  198  9.  Suggestions  204  f o r F u t u r e Work  206  Bibliography Appendix I .  ::  Appendix I I .  Appendix I I I .  N o t a t i o n Used  209  The t h e o r e t i c a l s o l u t i o n t o t h e problem o f n a t u r a l c o n v e c t i o n i n l i q u i d metals  212  Computer Program  223  List  of Figures  1.  A closed rectangular f l u i d length i n the z d i r e c t i o n  2.  A c r o s s s e c t i o n view o f the system o f F i g u r e 1 showing (a) the s t e a d y - s t a t e c o n d u c t i o n i s o t h e r m s (b) t h e s t e a d y - s t a t e c o n v e c t i o n i s o t h e r m s , a n d (c) t h e s o l i d - l i q u i d i n t e r f a c e advancing across the system.  5  The l o n g i t u d i n a l m a c r o s e g r e g a t i o n f r o m u n i d i r e c t i o n a l growth w i t h (a) complete c o n v e c t i v e m i x i n g i n t h e l i q u i d , ( b ) m i x i n g by d i f f u s i o n o n l y i n t h e l i q u i d , and ( c ) i n c o m p l e t e •. convective mixing i n the l i q u i d .  8  The e x p e r i m e n t a l a p p a r a t u s f o r o b s e r v i n g n a t u r a l c o n v e c t i o n showing (a) t h e s i d e s e c t i o n view o f t h e o v e r a l l f u r n a c e and ( b ) t h e f r o n t s e c t i o n view o f t h e i n n e r furnace d e t a i l s -  21  E x p l o d e d v i e w o f t h e c o p p e r o r a l u m i n u m mould used i n o b s e r v i n g n a t u r a l c o n v e c t i o n .  22  The t e m p e r a t u r e v e r s u s t i m e c u r v e i n t h e l i q u i d t i n d u r i n g r a p i d water quenching.  28  The t r a c e r movement i n a l i q u i d t i n m e l t (Sn t r a c e r ) w i t h a 0°C t e m p e r a t u r e d i f f e r e n c e l e f t 30 s e c o n d s b e f o r e q u e n c h i n g .  28  The t r a c e r p r o f i l e i n a t i n m e l t w i t h a 3.0°C t e m p e r a t u r e d i f f e r e n c e l e f t 30 s e c o n d s b e f o r e quenching,- ( a ) on t h e as c a s t s u r f a c e , ( b ) 13% i n t o t h e t i n b l o c k , and ( c ) 44$ i n t o t h e b l o c k .  30  The t r a c e r p r o f i l e i n t i n m e l t s w i t h a t e m p e r a t u r e d i f f e r n c e and a t i m e b e f o r e q u e n c h i n g o f ( a ) 0.23°C, 300 s e c o n d s , ( b ) 0.67°C, 90 s e c o n d s , ( c ) 1.11°C, 120 s e c o n d s , ( d ) 1.96°C, 60 s e c o n d s , ( e ) 3.04°C, 30 s e c o n d s , ( f ) 4 . 0 0 ° C , 30 s e c o n d s , (g) 5.05°C, 15 s e c o n d s , and ( h ) 9 . 1 ° C , 15 s e c o n d s .  34  3.  4.  5. 6. 7.  system o f i n f i n i t e  2  1 1 3  8.  9.  10.  The t r a c e r p r o f i l e i n t i n m e l t s w i t h an a v e r a g e t e m p e r a t u r e , t e m p e r a t u r e d i f f e r e n c e , and t i m e b e f o r e q u e n c h i n g o f ( a ) 236°C, 2.00°C, 60 s e c o n d s , (b) 305°C 2.02°C, 30 s e c o n d s , ( c ) 237°C, 3 . 0 0 ° C , 30 s e c o n d s , a n d ( d ) 305°C, 3-02°C, 15 s e c o n d s . 36 i  11.  12.  13.  The t i m e f o r t h e f l o w i n t i n m e l t s t o c o m p l e t e one c y c l e a r o u n d t h e c e l l v e r s u s t h e t e m p e r a t u r e d i f f e r e n c e a c r o s s the c e l l f o r average temoera t u r e s o f 237°C, 260°C, and 305°C. """  38  The t r a c e r p r o f i l e i n t i n m e l t s w i t h a t e m p e r a t u r e d i f f e r e n c e , l i q u i d c e l l t h i c k n e s s , and t i m e b e f o r e q u e n c h i n g o f ( a ) 5 . 6 0 ° C , 0.32 cm., 60 s e c o n d s , ( b ) 19.0°C, 0.32 cm., 15 s e c o n d s , ( c ) 5 . 0 ° C , 0.48 cm., 60 s e c o n d s , and ( d ) 5 . 0 ° C , . 0.95 cm., "90 s e c o n d s .  40  The l i q u i d c e l l t h i c k n e s s v e r s u s t h e t e m p e r a t u r e d i f f e r e n c e a c r o s s the c e l l showing t h e c o n d i t i o n s f o r two d i m e n s i o n a l and t h r e e d i m e n s i o n a l f l o w i n l i q u i d t i n a t 260°C.  42  14.  The t i m e f o r t h e f l o w i n t i n m e l t s t o c o m p l e t e one c y c l e v e r s u s t h e t e m p e r a t u r e d i f f e r e n c e a c r o s s the c e l l f o r l i q u i d c e l l t h i c k n e s s e s o f 0.32 cm., 0.48 cm., and 0.95 cm. -43  15.  The e x p e r i m e n t a l i s o t h e r m a l p l o t s f o r l i q u i d t i n a t 260°C f o r an a p p r o x i m a t e R a y l e i g h number a n d c e l l t h i c k n e s s o f ( a ) 1.4 x 1 0 , 0.32 cm., ( b ) 2.4 x 1 0 , 0.32 cm., ( c ) 8.0 x 1 0 , 0.32 cm., ( d ) 1.4 x 1 0 , 0.95 cm., and ( e ) 2 x 1 0 , 0.95 cm. 45 5  s  s  5  16.  s  The e x p e r i m e n t a l i s o t h e r m a l p l o t f o r l i q u i d t i n a t 260 C i n a 10.8 by 6.4 by 0.32 cm. l i q u i d c e l l .  47  The t r a c e r p r o f i l e i n t i n m e l t s w i t h a l e n g t h t o h e i g h t r a t i o o f ( a ) 1.6 : 1, ( b ) 3.5 : 1, ( c ) 4.9 : 1, and ( d ) 8.3 : 1.  50  A schematic o f t h e heat f l o w i n a s t a n d a r d c a s t i n g , ( a ) , and t h e t e m p e r a t u r e p r o f i l e , ( b ) , in the residual l i q u i d pool.  51  ( a ) The t r a c e r p r o f i l e both ends. ( b ) The t r a c e r p r o f i l e one s i d e o n l y .  53  b  17.  18.  19.  20.  i n a t i n melt c o o l e d from i n a t i n melt c o o l e d from  The t r a c e r p r o f i l e i n a t i n m e l t ( a ) c o o l e d f r o m b o t h ends w i t h e q u a l e n d w a l l t e m p e r a t u r e s and l e f t f o r 300 s e c o n d s b e f o r e q u e n c h i n g and ( b ) c o o l e d f r o m b o t h ends w i t h t h e l e f t hand w a l l c o o l e r t h a n t h e r i g h t hand w a l l , and l e f t 120 seconds b e f o r e quenching.  55  21.  22. 23.  The t r a c e r p r o f i l e i n l i q u i d l e a d w i t h a t e m p e r a t u r e d i f f e r e n c e and t i m e t o q u e n c h o f (a) 2 . 9 6 ° C , 3 0 s e c o n d s , and ( b ) 4 . 9 8 ° C , 1 5 s e c o n d s , and i n l i q u i d t i n w i t h a t e m p e r a t u r e d i f f e r e n c e and t i m e t o q u e n c h o f ( c ) 3.04°C, 30 s e c o n d s , and ( d ) 5 . 0 5 ° C , 1 5 s e c o n d s . The t h e o r e t i c a l f l u i d s y s t e m t o be u s e d analysis of thermal convection.  57  i n the  6 l  The s o l u t i o n o f B a t c h e l o r s h o w i n g ( a ) a n o r m a l i z e d s t r e a m f u n c t i o n p l o t and ( b ) a normalized v e l o c i t y i n the X d i r e c t i o n at a p o s i t i o n o n e - h a l f way down t h e l i q u i d z o n e .  69  24.  The b o u n d a r y l a y e r d e v e l o p m e n t a l o n g t h e w a l l s i n t h e s o l u t i o n o f Emery and Chu.  71  25.  A normalized v e l o c i t y versus the p o s i t i o n w i t h i n t h e b o u n d a r y l a y e r f o r m t h e s o l u t i o n o f Emery and Chu. •  71  26.  The f i n i t e d i f f e r e n c e g r i d s y s t e m f o r t h e f l u i d c e l l used i n the n u m e r i c a l a n a l y s i s .  78  27.  The t h e o r e t i c a l p l o t s o f t h e n o n d i m e n s i o n a l t e m p e r a t u r e f o r a R a y l e i g h number o f ( a ) 2 x 1 0 , (b)'2 x 1 0 , ( c ) 2 x 1 0 , (d) 2 x 10", (e) 2 x 1 0 and ( f ) 2 x 1 0 . 2  vertical  3  6  28. 29.  30.  A p l o t o f the t h e o r e t i c a l average v e r s u s t h e R a y l e i g h number.  N u s s u l t number  86 91  A p l o t o f t h e t h e o r e t i c a l a v e r a g e N u s s e l t number v e r s u s t h e G r a s h o f number f o r P r a n d t l numbers o f 10.0, 1.0, 0.1, and 0.0127.  92  A p l o t o f t h e l o c a l N u s s e l t number v e r s u s t h e p o s i t i o n a l o n g t h e c o l d end w a l l f o r R a y l e i g h numbers o f 2 x 1 0 , 2 x 10 , 2 x 1 0 \ 2 x 1 0 , and 2 x 1 0 .  93  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r l i q u i d t i n a t 260°C w i t h a G r a s h o f number o f ( a ) 2 x 1 0 , ( b ) 2 x 1 0 \ ( c ) 2 x 1 0 , ( d ) 2 x 1 0 , and (e) 2 x 10 -.  94  2  3  s  6  31.  s  3  s  6  7  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r a P r a n d t l number o f 0.1 w i t h a G r a s h o f number o f ( a ) 2 x 10 V, and ( b ) 2 x 1 0 . 5  33.  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r a P r a n d t l number o f 1.0 w i t h a G r a s h o f number o f ( a ) 2.x 10 * and ( b ) 2 x 10 .  98  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r a P r a n d t l number o f 10.0 w i t h a G r a s h o f number o f ( a ) 2 x 10* and ( b ) 2 x 1 0 .  . -99  1  3^..  s  s  35.  The t h e o r e t i c a l f l o w v e l o c i t y ( u and U) a t a p o s i t i o n X = 0.5 f o r v a r i o u s v a l u e s o f t h e G r a s h o f number f o r ( a ) P r = 0.0127, ( b ) P r = 0.1, ( c ) P r = 1 . 0 , and ( d ) P r = 10.0.  101  36.  The f l o w t i m e p e r c y c l e v e r s u s t h e G r a s h o f number and t e m p e r a t u r e d i f f e r e n c e a c r o s s t h e c e l l f o r the s o l u t i o n o f B a t c h e l o r , s o l u t i o n o f Emery a n d Chu, n u m e r i c a l s o l u t i o n , and e x p e r i m e n t a l r e s u l t s . I l l  37.  The f l o w t i m e p e r c y c l e v e r s u s t h e l i q u i d c e l l t h i c k n e s s f o r ^ a t e m p e r a t u r e d i f f e r e n c e o f 1.0°C s h o w i n g t h e e x p e r i m e n t a l and t h e o r e t i c a l c u r v e s .  113  T h e o r e t i c a l flow time per c y c l e f o r l i q u i d t i n versus the average temperature i n the c e l l f o r a t e m p e r a t u r e d i f f e r e n c e o f 0.555°C. .-  114  39.  The e x p e r i m e n t a l i n t i a l c o n d i t i o n s f o r o b s e r v i n g independent s o l u t e c o n v e c t i o n .  118  40.  The t r a c e r p r o f i l e i n t h e s y s t e m o f F i g u r e 39 l e f t m o l t e n 50 m i n u t e s w i t h o u t o p e n i n g t h e g a t e .  120  The t r a c e r p r o f i l e i n t h e s y s t e m o f F i g u r e 39 w i t h o n l y pure t i n p l u s S n i n the l e f t s e c t i o n and q u e n c h e d 3 0 . s e c o n d s a f t e r t h e g a t e i s o p e n e d .  120  The t r a c e r p r o f i l e i n t h e s y s t e m o f F i g u r e 39 w i t h t i n p l u s 0.1 w t . % l e a d p l u s S n i n the l e f t s e c t i o n , q u e n c h e d ( a ) 15 s e c o n d s and ( b ) 30 s e c o n d s a f t e r t h e g a t e i s o p e n e d .  12.1  38.  '41.  113  42.  1 1 3  43.  44.  The tracer d i s t r i b u t i o n i n samples with v a r i o u s  m e l t m a t e r i a l s , a v e r a g e t e m p e r a t u r e s . , and t e m p e r a t u r e d i f f e r e n c e s showing the e f f e c t o f s o l u t e c o n v e c t i o n on t h e r m a l c o n v e c t i o n .  124  Tracer p r o f i l e i n lead melts with a temperature d i f f e r e n c e and t i m e t o q u e n c h o f ( a ) 1.07°C, 60 s e c o n d s , ( b ) 3.07°C. 30 s e c o n d s and ( c ) 5.00°C 15 s e c o n d s w i t h Sn tracer. 127 11 3  45.  46.  47.  The e x p e r i m e n t a l i n t i a l c o n d i t i o n s f o r d e r i v i n g the thermal and s o l u t e c o n d i t i o n s f o r complete l i q u i d mixing/  12 8  The t r a c e r p r o f i l e f o r s t e a d y - s t a t e f l o w s f o r t h e i n i t i a l c o n d i t i o n s as shown i n F i g u r e 45 with various lead contents.  130  The s t e a d y - s t a t e p r o f i l e o f t h e ' t r a c e r r e s u l t i n g f r o m t h e l o w e r s e c t i o n o f t h e c e l l b e i n g comnosed o f t i n , 1.0 w t . % l e a d , and T I " .  132  The e x p e r i m e n t a l i s o t h e r m a l p r o f i l e i n a s y s t e m i n i t i a l l y a s i n F i g u r e 45 w i t h t h e l o w e r s e c t i o n composed o f t i n p l u s 10 w t . % l e a d .  132  The t r a c e r o r o f i l e i n an i s o t h e r m a l m e l t o f ( a ) p u r e t i n a t 2 35°C and ( b ) 4-4.5 w t . % l e a d - 55.5 wt. % b i s m u t h a t 127°C, b o t h l e f t 300 s e c o n d s before quenching.  138  The t r a c e r o r o f i l e i n a m e l t o f ( a ) p u r e t i n and ( b ) 4 4.5 w t . % l e a d - 55.5 w t . % b i s m u t h a l l o w e d t o c o o l , n u c l e a t e , and f r e e z e c o m p l e t e l y w i t h o u t quenching. .  138  2 0  48.  49.  50.  51.  ( a ) The t e m p e r a t u r e v e r s u s t i m e p l o t f o r t h e m e l t t e m p e r a t u r e d u r i n g n u c l e a t i o n and f r e e z i n g o f t h e l e a d - b i s m u t h e u t e c t i c m e l t and ( b ) t h e d i f f e r e n t i a l temperature versus time p l o t b e t w e e n two 0.5 cm. a p a r t p o i n t s i n t h e l e a d b i s m u t h e u t e c t i c d u r i n g n u c l e a t i o n and f r e e z i n g . 139  52.  The two d i m e n s i o n a l e x p e r i m e n t a l observing i n t e r d e n d r i t i c flow.  53.  The e x p e r i m e n t a l r e s u l t s o f t h e m o d e l o f F i g u r e 52 w i t h a. p u r e t i n m e l t w i t h a t e m p e r a t u r e d i f f e r ence a c r o s s t h e p o o l a n d a t i m e b e f o r e Q u e n c h i n g o f ( a ) 3-68°C, 60 s e c o n d s , ( b ) 5.02°C, 60 s e c o n d s , and ( c ) 5.18°C, 600 s e c o n d s . 144  54.  The t h r e e d i m e n s i o n a l w i r e r o d model f o r observing i n t e r d e n d r i t i c flow.  145  The t r a c e r d i s t r i b u t i o n i n t h e s y s t e m o f F i g u r e 54 w i t h a t e m p e r a t u r e d i f f e r e n c e o f 5.73°C a c r o s s t h e p o o l s h o w i n g ( a ) t h e as c a s t s u r f a c e and ( b ) t h e p r o f i l e 0.45 mm. b e l o w t h e s u r f a c e .  147  55.  model f o r  142  x i i. i  56.  The t r a c e r d i s t r i b u t i o n i n t h e s y s t e m o f F i g u r e 54 w i t h a t e m p e r a t u r e d i f f e r e n c e o f 5 . 1 ° C snowing (a) t h e as c a s t s u r f a c e , ( b ) t h e l e f t hand end o f t h e b l o c k w i t h t h e w i r e r e m o v e d , and ( c ) t h e l e f t h a n d end w i t h 3 / 8 i n c h e s o f t h e b l o c k end removed..  147  57.  The t h r e e observing  58.  The q u a l i t a t i v e f l o w l i n e s i n t h e s y s t e m o f F i g u r e 5 7 s h o w i n g t h e two f l o w c e l l s and intermesh flow d i r e c t i o n s .  150  A m i c r o g r a p h o f a c r o s s s e c t i o n o f a w i r e mesh i n an a c t u a l l e a d s a m p l e t h a t has b e e n mounted and p o l i s h e d .  152  The t r a c e r p r o f i l e o f t h e i n t e r m e s h f l o w t h a t occurs i n 1 2 0 seconds a f t e r the t r a c e r i n t r o d u c t i o n w i t h a 6.C)6°C t e m p e r a t u r e d i f f e r e n c e f o r s a m p l e s c o r r e s p o n d i n g t o number ( a ) 5 , ( b ) 4, ( c ) ?, and id) i n s a m p l e o f T a b l e V I I I .  154  59.  60.  d i m e n s i o n a l w i r e mesh model f o r i n t e r d e n d r i t i c flow.  150  61.  The c o u n t i n g a r r a n g e m e n t f o r m o n i t o r i n g t h e a c t i v i t y I n t h e ( a ) l a r g e f l o w c e l l and ( b ) s m a l l flow c e l l i n the intermesh flow samples. 155  62.  The b o u n d a r y l a y e r s a r o u n d t h e mesh w i r e s the n o t a t i o n used i n t h e a n a l y s i s .  showing  155  63.  A p l o t o f the f r a c t i o n flowed versus the r a t i o o f the hole s i z e t o t h e w i r e d i a m e t e r f o r the v a r i o u s mesh s i z e s •• i n v e s t ! g a t e d . 158  64.  A p l o t o f the wire spacing versus the r a t i o o f the h o l e s i z e t o t h e w i r e d i a m e t e r f o r t h e no f l o w c o n d i t i o n from F i g u r e 6 3 . 159  65.  A p e r p e n d i c u l a r view o f the primary d e n d r i t e model u s e d i n - t h e i n t e r p r e t a t i o n o f t h e i n t e r m e s h flow r e s u l t s .  160  A p l o t o f the f r a c t i o n s o l i d i n the s o l i d - l i q u i d i n t e r f a c e versus the primary dendrite spacing s h o w i n g t h e f l o w and no f l o w c o n d i t i o n s o b t a i n e d from the intermesh f l o w r e s u l t s . A l s o shown i s the l e a d - t i n a l l o y i n t e r d e n d r i t i c f l o w r e s u l t s .  1.6l  The e x p e r i m e n t a l c o n d i t i o n s u s e d f o r o b s e r v i n g i n t e r d e n d r i t i c flow i n l e a d - t i n a l l o y s showing t h e t h e r m o c o u p l e and t r a c e r a d d i t i o n p o s i t i o n s .  163  66.  :  67.  i  1  i  e  I  X  68.  69.  70..  The r e l e v a n t p o r t i o n o f t h e l e a d - t i n p h a s e d i a g r a m showing the f o u r a l l o y s used t o o b s e r v e i n t e r d e n d r i t i c f l u i d flow.  163  The t r a c e r d i s t r i b u t i o n i n t h e r e s i d u a l l i q u i d p o o l o f a p u r e t i n c a s t i n g w i t h the e x p e r i m e n t a l c o n d i t i o n s of Figure 67.  165  A a u t o r a d i o g r a p h showing the cast s t r u c t u r e of a t i n - 2 wt. % l e a d a l l o y d i r e c t i o n a l l y c a s t with T l * tracer.  165  The t r a c e r d i s t r i b u t i o n s h o w i n g i n t e r d e n d r i t i c f l o w i n c a s t i n g s o f (a) t i n - 2 wt. % l e a d , (b) t i n - 5 wt. % l e a d , ( c ) t i n - 12.5 wt. % l e a d , and (d) t i n - 20 wt. % l e a d w i t h t h e c a s t i n g conditions of Figure 67.  167  2 0 <  7'i .  72.  73-  1  A p l o t of weight percent s o l i d versus degrees c e n t i g r a d e below the l i q u i d u s t e m p e r a t u r e o b t a i n e d f r o m t h e p h a s e d i a g r a m o f F i g u r e 68 f o r t h e f o u r a l l o y s concerned.  170  A p l o t o f p e r c e n t s o l i d f o r no f l o w v e r s u s weight p e r c e n t l e a d f o r the l e a d - t i n a l l o y i n t e r d e n d r i t i c flow experiments.  171  the  74.  The t r a c e r p r o f i l e i n a t i n - 2 wt. % l e a d c a s t i n g r e s i d u a l l i q u i d p o o l s h o w i n g (a) t h e i n t e r d e n d r i t i c f l o w and t h e o v e r a l l c a s t i n g s i z e and (b) t h e maximum p o s i t i o n r e a c h e d by t h e 0.6 mm. diameter w i r e i n t o the i n t e r f a c e . 173  75-  The t r a c e r p r o f i l e i n a t i n w i t h (a) q u e n c h i n g 30 s e c o n d s i n t r o d u c t i o n and (b) c o m p l e t e s o l i d i f i c a t i o n o f the c a s t i n g  76.  The e x p e r i m e n t a l a p p a r a t u s u s e d f o r p r o d u c i n g s t a t i o n a r y , r o t a t e d , and o s c i l l a t e d c a s t i n g s .  77.  R e p r e s e n t a t i v e i n g o t s c a s t I n (a) (b) r o t a t i n g , and ( c ) o s c i l l a t i n g  78.  E q u i a x e d g r a i n s i n the oscillated casting.  79.  R e p r e s e n t a t i v e i n g o t s c a s t i n (a) (b) r o t a t i n g , and ( c ) o s c i l l a t i n g  2 wt. % l e a d c a s t i n g a f t e r the t r a c e r directional without quenching. 176  stationary, moulds.  central region  of  the  stationary, moulds.  the  179 183 184 184  V  80.  The r a d i a l s i l v e r d i s t r i b u t i o n i n a s t a t i o n a r y casting; ( a ) 1/4 i n c h d r i l l h o l e s i n 1/4 i n c h s t e p s , ( b ) 1/4 i n c h d r i l l h o l e s i n 1/8 i n c h s t e p s , (c) 0.030 i n c h l a t h e t u r n i n g s , and ( d ) 0.050 inch lathe turnings disolved i n acid. 187  81.  The r a d i a l s i l v e r d i s t r i b u t i o n i n ( a ) s t a t i o n a r y , '(b) r o t a t e d , and ( c ) o s c i l l a t e d i n g o t s u s i n g method ( b ) o f F i g u r e 80.  189  An a u t o r a d i o g r a p h o f t h e c r o s s s e c t i o n o f t h e o s c i l l a t e d ingot showing the s i l v e r d i s t r i b u t i o n in the c a s t i n g .  191  The d e v e l o p m e n t o f t h e r a d i a l m a c r o s e g r e g a t i o n i n an o s c i l l a t e d i n g o t , ( a ) p r i o r t o t h e t i m e o f t h e CET, ( b ) a t t h e t i m e o f t h e GET, and ( c ) t h e f i n a l s i l v e r d i s t r i b u t i o n i n the casting.  194  82.  83.  List  I. TI. III. IV. V. VI. VII. VIII. IX.  of Tables  Selected fluid properties  1*1  Properties of liquid  62  l e a d and t i n  Computer r u n c o n d u c t e d  84  E q u i v a l e n t R a y l e i g h and.Grashof numbers c o r r e s p o n d i n g t h e F i g u r e 27  89  The e f f e c t on t h e n a t u r a l c o n v e c t i o n o f a l t e r i n g dimensionless parameters '  107  E x p e r i m e n t a l r e s u l t s f o r combined and s o l u t e c o n v e c t i o n  133  Density of l e a d - t i n of temperature  alloys  thermal  as a f u n c t i o n 134  D a t a on w i r e meshes u s e d i n f l o w and f r a c t i o n f l o w e d r e s u l t s L e a d - t i n -alloy i n t e r d e n d r i t i c  experiments  flow r e s u l t s  151 169  1.  Introduction  The s t r u c t u r e and p r o p e r t i e s o f c a s t i n g s a r e d e t e r m i n e d , i n p a r t , by n a t u r a l c o n v e c t i o n liquid  metal d u r i n g s o l i d i f i c a t i o n .  convection,  as u s e d  motion o f a f l u i d  i n this  thsis,  Natural or free c a n be d e f i n e d as t h e  due t o d e n s i t y c h a n g e s r e s u l t i n g  temperature or compositional This  o f the r e s i d u a l  f r o m any  d i f f e r e n c e s i n the f l u i d .  i n t r o d u c t i o n w i l l d i s c u s s t h e r e a s o n s why t h e s t u d y o f  natural convection solidification  1.1.  i s v i t a l t o a t r u e u n d e r s t a n d i n g o f many  phenomena.  The i m p o r t a n c e o f l i q u i d  metal  The i m p o r t a n c e o f l i q u i d convection liquid  flow  m e t a l f l o w due t o n a t u r a l  has o n l y r e c e n t l y been r e a l i s e d .  metal during s o l i d i f i c a t i o n  The f l o w  of.the  a f f e c t s the heat t r a n s f e r ,  the mass t r a n s f e r , , and t h e f i n a l c a s t , s t r u c t u r e . The h e a t t r a n s f e r r a t e i n a s y s t e m t r a n s f e r o c c u r s by c o n v e c t i o n would r e s u l t  i n which  i s greater than that  from c o n d u c t i o n a l o n e .  With  heat  which  convection'there  are  two modes o f h e a t t r a n s f e r ; by t h e c o n d u c t i o n  of heat  b e t w e e n two p o i n t s a t d i f f e r e n t t e m p e r a t u r e s , ' a n d by mass transport convection  i n the l i q u i d .  The more r a p i d h e a t r e m o v a l by  i n a c l o s e d s y s t e m , s u c h as t h e r e s i d u a l  liquid  m e t a l i n a e a s t i n g , r e s u l t s i n more r a p i d a l o w e r i n g o f temperature gradients  i n the l i q u i d  m e t a l and a more r a p i d  removal o f superheat from the l i q u i d . in a convecting the'thermal alone. fluid  Figure  profile  system w i l l have a d i f f e r e n t shape t h a n  profile  i n a s y s t e m t r a n s f e r r i n g h e a t by  To i l l u s t r a t e , c o n s i d e r s y s t e m shewn i n F i g u r e  1.  The t h e r m a l  the c l o s e d  conduction  rectangular  1.  A c l o s e d r e c t a n g u l a r f l u i d system o f i n f i n i t e length i n the z d i r e c t i o n . '  One  long v e r t i c a l  side i s maintained at a temperature 8 j ,  and t h e o p p o s i t e v e r t i c a l w a l l i s m a i n t a i n e d a t t e m p e r a t u r e 02,  such that  B2 i s g r e a t e r t h a n 8 1 .  lower h o r i z o n t a l the  s u r f a c e s be p e r f e c t l y  c e l l be i n f i n i t e l y  this  L e t the upper  and  i n s u l a t e d and l e t  l o n g i n the z d i r e c t i o n .  Thus, i n  system, a l l the heat t r a n s f e r i s across the c e l l  y = 0 t o y = d.  I f the f l u i d  f l o w , the heat t r a n s f e r  from  system i s not a l l o w e d to  i s g o v e r n e d by t h e g e n e r a l t h r e e  d i m e n s i o n a l heat c o n d u c t i o n e q u a t i o n ( 1 ) : 3_ . 3_8 3x1 3x  azl 3z.  yj  where:  K  +  D C  o3t  (1.1)  k = thermal conductivity q - heat t r a n s f e r p =. f l u i d C  P  = fluid  rate  density specific  heat  t = time 8 = temperature Assuming f r o m - F i g u r e l a t h a t a l l the heat t r a n s f e r i s i n the  y d i r e c t i o n , a l l m a t e r i a l p r o p e r t i e s are constants  and t h e s y s t e m . h a s r e a c h e d s t e a d y - s t a t e equation  conditions,  (1.1) reduces t o : d 8 dv " 2  7  =  0  (1.2)  w i t h the boundary c o n d i t i o n s at  y = 0:0  = 81  y = d: 8 =  8  2  that  Solving equation w i t h t h e use  (1.2)  f o r the r e s u l t i n g  o f c o n d i t i o n s (1.3)  -  (e  2  9  e  x  =  y + e  (1.4).  temperature  The  (i.n)  x  temperature  - 8 i ) / d.  2  i s not In  i n y a c r o s s the  I t s h o u l d be n o t e d  fluid  s i n c e the  the thermal p r o f i l e , the  p r o b l e m has and  be  thermal  are coupled.  and  x  and  is  the direction.  i s allowed  to  fluid  flow  alters  motion i s d e t e r -  f l o w and  the  temperature  They must t h e r e f o r e be  Churchill  (2).  The  solved  convection case.  details  d i s c u s s e d In Chapter  3.  profile  from the  that results The  This  of t h e i r  by  solution  Q u a l i t a t i v e l y , the  isotherms  cell.  iso-  f l o w i n g system i s  are seen t o bend  s i d e a l o n g the bottom o f the  c o l d s i d e a l o n g the top o f the fluid  that  been s o l v e d n u m e r i c a l l y f o r non-metals  shown i n F i g u r e 2b. the hot  cell  the  c o n v e c t i o n t h e s y s t e m becomes much more  t o g e t h e r f o r .the g e n e r a l t h e r m a l  will  and  T h i s i s because the motion o f the f l u i d  i n the c e l l  Wilkes  from  lines  a f u n c t i o n of the v e r t i c a l  t h e t h e r m a l p r o f i l e , and m i n e d by  that result  are v e r t i c a l  a system t h a t the  by n a t u r a l t h e r m a l complex.  isotherms  gradient i s linear  equal to ( 0  profile  results i n :  F i g u r e 2a shows t h e i s o t h e r m a l l i n e s equation  thermal  cell  and  towards  towards  the  T h i s i s c a u s e d by  the  n e a r t h e c o l d w a l l b e i n g c o o l e d , b e c o m i n g more d e n s e ,  thus  f l o w i n g down t h e  of the c e l l  from l e f t  c o l d w a l l and  to r i g h t .  along the  Similarly  the  bottom  fluid  rises  Figure  2.  A c r o s s s e c t i o n view o f the system o f F i g u r e 1 showing (a) the s t e a d y - s t a t e c o n d u c t i o n isotherm's, (b) t h e s t e a d y - s t a t e c o n v e c t i o n i s o t h e r m s , and (c) the s o l i d - l i q u i d i n t e r f a c e advancing across the system.  on t h e h o t s i d e  o f the c e l l  it  the top of the c e l l  moves a c r o s s  left.  This  fluid  as i n F i g u r e  solidification,  from the r i g h t t o the  movement c a u s e s t h e i s o t h e r m s  shape o f t h e s o l i d - l i q u i d  t o bend  interface,  i s a l t e r e d by t h e t h e r m a l  therms i n the r e s i d u a l  liquid  pool.  during  convection  iso-  I f the v e r t i c a l  wall  temperature 8 1 , i s a s o l i d i f i c a t i o n  allowed  and  2b.  The  at  as i t s d e n s i t y d e c r e a s e s  front  and i t i s  t o move i n t o t h e l i q u i d p o o l t h e i n t e r f a c e w i l l n o t  remain i n a v e r t i c a l  plane.  It will  take  up a shape as  shown i n F i g u r e 2 c , i f 81 i s b e l o w t h e m e l t i n g p o i n t o f the  liquid  and  Chhabra (3)  reasonable isotherms  metal.  This  shape h a s b e e n o b s e r v e d  i n a pure lead  system.  by  Szekely  The shape i s  when c o n s i d e r a t i o n i s g i v e n t o t h e shape o f t h e i n Figure 2b.  move more r a p i d l y  A solid-liquid  interface  i n t o a l i q u i d w i t h a lower  will  thermal  gradient  s i n c e t h e r e i s l e s s h e a t t o be c o n d u c t e d away t h r o u g h t h e solid  i n order  regions  f o r the i n t e r f a c e  of the c e l l  near the cold w a l l  than  hence t h e i n t e r f a c e regions  have a l o w e r  temperature  the upper r e g i o n s  will  ;  o f heat removal,  lower  gradient  i n the lower  shown i n F i g u r e l e i .  s y s t e m where t h e r e i s a r a p i d  there i s a corresponding  lowering o f the temperature g r a d i e n t s ; i n a normal c a s t i n g ,  The  o f t h e c e l l , and  a d v a n c e more q u i c k l y  r e s u l t i n g i n the i n t e r f a c e In a c o n v e c t i n g  rate  t o advance.  rapid  the l i q u i d .  In  i n w h i c h no a d d i t i o n a l h e a t i s added t o  the  s y s t e m as  of the As  liquid  solidification  progresses,  superheat w i l l  depend on  constitutional  due  t o the  the  cold wall,  of the  supercooling  d i f f e r e n c e i n the  i n t e r f a c e and  not  constitutional  lower  regions  expected t h a t the  temperature  others.  showed e x p e r i m e n t a l l y  i n d i c a t i n g b r e a k d o w n and  2b  i n d i c a t i n g no  regions  the  lowest  upper  and  be In  these  b e e n o b s e r v e d by  Weinberg  structure  edges o f t h e i n t e r f a c e  a planar region occurred  i n the  breakdown i n u n i d i r e c t i o n a l l y  rods.  and  solidification  i n t e r f a c e , i s very  a f f e c t both the  from u n i d i r e c t i o n a l  In a c a s t i n g .  resulting  from the  s e n s i t i v e t o f l o w i n the The  liquid  with  (b) m i x i n g  The  uni-  liquid  macrosegregation  solidification  i n the  macro-  of a metal rod, with a  advancing i n t e r f a c e .  vective mixing  can  the m i c r o s e g r e g a t i o n  longitudinal.macrosegregation  of the  at the  that a c e l l u l a r  lower  Natural convection  directional  i n some  I t would thus  T h i s phenomenon has  n e a r t h e u p p e r and  segregation  occur  i n t e r f a c e breakdown would occur  (4) who  solidified  gradientalong  From F i g u r e  vertical wall.  first.  c e n t r a l area  Also,  t h u s t h e most f a v o u r a b l e p o s i t i o n  regions  occurred  convection.  of i n t e r f a c e  o c c u r r i n g sooner.  s u p e r c o o l i n g , occurs  of the  removal  s t r u c t u r e , i s enhanced  i n t e r f a c e breakdown can  t e m p e r a t u r e g r a d i e n t s , and for  the r a t e o f  the g r a d i e n t s decrease the p o s s i b i l i t y  breakdown o c c u r r i n g , c h a n g i n g the by  the r a t e o f  planar ahead resulting  (a) c o m p l e t e i n the  liquid  conby  DISTANCE 3-  Figure  The l o n g i t u d i n a l m a c r o s e g r e g a t i o n from u n i d i r e c t i o n a l growth w i t h (a) complete c o n v e c t i v e m i x i n g i n t h e l i q u i d , (b) m i x i n g b y . d i f f u s i o n o n l y i n t h e l i q u i d , and ( c ) i n c o m n i e t e c o n v e c t i v e m i x i n g i n the l i q u i d .  d i f f u s i o n only or (c) incomplete liquid,  i s shown i n F i g u r e  3.  c o n v e c t i v e m i x i n g i n the  This i s a plot  of solute  c o n c e n t r a t i o n "'as a f u n c t i o n o f t h e d i s t a n c e a l o n g solidified of C . o  r o d , w i t h an a v e r a g e c o m p o s i t i o n  The  s e e n t o be  g r e a t l y a l t e r e d by  interface  Due  also occur  example o f t h i s  liquid  rod  i f the The  are  mixing. solidmost  is-inverse segregation  f r e e z i n g o f the  metal,  i s p u l l e d back i n t o the d e n d r i t i c  cause s o l u t e enrichment  casting/  can  t o a volume change on  solute rich to  the e x t e n t o f the  i s of a d e n d r i t i c nature.  commonly o b s e r v e d (5).  i n the  shaoe o f the s o l u t e c o n c e n t r a t i o n c u r v e s  Macrosegregation liquid  the  i n the i n i t i a l  stages  of  network the  T h i s i s i n c o n t r a s t t o the s o l u t e d e p l e t i o n  from F i g u r e . 2 .  w h i c h w o u l d be e x p e c t e d fluid  f l o w , volume  a pressure  considered  w i t h i n the d e f i n i t i o n  here.  cause o f  c o n t r a c t i o n i n the d e n d r i t i c  g r a d i e n t and p u l l i n g new  does n o t f a l l  The  fluid  zone  of natural  convection  type  of flow (6,7,8,9).  has b e e n d e v e l o p e d by F l e m i n g s and h i s c o - w o r k e r s  the  causing  i n t o t h e zone,,  Detailed analysis of this  Their a n a l y s i s includes thermal  this  and s o l u t e c o n v e c t i o n i n  d e n d r i t i c . z o n e but does not c o n s i d e r t h e f l o w i n t h e  residual  liquid  outside the i n t e r f a c e r e g i o n .  Macrosegregation c u t t i n g across channels  c a n a l s o be c a u s e d by t h e f l o w  t h e d e n d r i t e s and t h e f o r m a t i o n  at r i g h t  causes s e g r e g a t i o n  angles  to the dendrites  will and  be l o w e r  the l i q u i d  convective  v e r t i c a l downwards d i r e c t i o n forces w i l l  the elements b e i n g denser than  on  This  solidification.  pool  be i n t h e  i n t h e mushy z o n e .  d e p e n d on t h e r e l a t i v e  segregated.  region  i n the c e n t r a l  forces w i l l  flow  liquid,  i n the i n t e r d e n d r i t i c  -than t h a t o f t h e l i q u i d  hence the t h e r m a l  convective  (10).  by d i s p l a c i n g i n t e r d e n d r i t i c  w h i c h i s e n r i c h e d by t h e s o l u t e r e j e c t i o n The t e m p e r a t u r e . o f  of flow  The  solute  density of  I f the r e j e c t e d s o l u t e . i s  the s o l v e n t the s o l u t e convection  will  be  downwards and h e n c e e n h a n c e t h e n a t u r a l t h e r m a l  convection.  If  convection  the r e j e c t e d s o l u t e i s l e s s  will  be upwards  forces. fluid  and c o u n t e r a c t  Although  the d r i v i n g  dense t h e s o l u t e the thermal  convection  f o r c e s f o r the  interdendritic  f l o w a r e known, t h e a n a l y s i s i s g r e a t l y hampered  by  the  very  complex n a t u r e o f t h e s o l i d - l i q u i d i n t e r f a c e . The phenomena o f s o l u t e b a n d i n g i n a c a s t i n g  has  b e e n shown t o be c a u s e d by c o n v e c t i o n  metal the  ( 1 1 , 12, 1 3 ) .  flow rates  For very  nature.  become t u r b u l e n t  thermal  a t any p a r t i c u l a r p o i n t  and  flow  will  c a n be s u c h  or o s c i l l a t o r y i n flow w i l l  be t h e r m a l  i n the l i q u i d .  occur.  the e f f e c t i v e s o l u t e d i s t r i b u t i o n  This  will  Previous  When interface  result  c o e f f i c i e n t changing  a band o f s o l u t e d i f f e r e n t i n c o m p o s i t i o n  produced.  gradients,  f l u c t u a t i o n s occur near a s o l i d - l i q u i d  a growth r a t e p e r t u r b a t i o n in  convective  Accompanying t h i s t u r b u l e n t  fluctuations the  large temperature  i n a system w i t h  that the system w i l l  i n the l i q u i d  being  w o r k e r s h a v e shown a d i r e c t c o r r e l a t i o n  b e t w e e n t h e r a t e o f t e m o e r a t u r e f l u c t u a t i o n s and t h e r a t e o f band  formation. The manner i n w h i c h n a t u r a l  a cast  structure i s s t i l l  most w i d e l y the  convection  affects  open t o much c o n t r o v e r s y .  The  i n v e s t i g a t e d a s p e c t o f s t r u c t u r e change i s  columnar.to equiaxed t r a n s i t i o n which occurs i n a l l o y  systems.  T h e r e have been a number o f t h e o r i e s  developed  to account f o r the onset o f the equiaxed s t r u c t u r e , p a r t icularly  the source o f the n u c l e i r e s u l t i n g i n t h i s  structure.  These i n c l u d e  leaving solid shearing solid  dendrite  of dendrites  (a) p a r t i a l  fragments with  remelting  of  dendrites  ( 1 4 , 1 5 ) , (b) b r e a k i n g  or without remelting  or  producing,  p a r t i c l e s (1.6), ( c ) s u r v i v a l o f n u c l e i f r o m t h e  chill  of the  c a s t i n g (17),  upper s u r f a c e  o f the  c a s t i n g , the n u c l e i  initial  c a s t i n g (18),  moving i n t o the arms c a u s i n g dendrite  adiabatic local  fragments  (19).  The  e n h a n c e d by  gradients and  breaking  producing  A l s o , due  above  liquid  on  the  to convection,  solid-liquid  solid theories  transition  of dendrite  f l u c t u a t i o n s i n the  ahead o f the  hence t h e  i n t o an also  flow.  dendrite  m e t a l f o r the  f l u c t u a t i n g mechanical forces exerted a turbulent  bending of  m e l t i n g , and  liquid  r e m e l t i n g and  thermal  (e) the  the  subsequently  Almost a l l of the  r e q u i r e some m o t i o n o f t h e to occur.  (d) n u c l e a t i o n a t  arms i s  and  the  dendrites the  by  thermal  i n t e r f a c e are  reduced  p r o b a b i l i t y o f n u c l e i s u r v i v i n g and  growing  equiaxed grain i s increased.  facilitates  t h e movement o f t h e  p o s i t i o n at which they are equiaxed g r a i n s to stop  The  liquid  metal  n u c l e i from  the  f o r m e d t o where t h e y grow  the  f u r t h e r growth of  flow  as  the  columnar g r a i n s . 1.2.  The  d r i v i n g forces The  convection the  for liquid  p r e s e n c e and  i s d e p e n d e n t on  extent the  metal  flow  of f l u i d  f l o w by  d r i v i n g forces present  r e s i s t a n c e of the m e t a l t o f l o w .  There are  of d r i v i n g f o r c e s which produce f l u i d motion i n metals,  i n c l u d i n g thermal  mechanical mixing,  and  imposed magnetic  and  a number liquid  gradients, concentration  Thermal g r a d i e n t s  natural  gradients,  fields.  are perhaps the  most common  driving force In  for liquid  a l lsolidification  motion  i n the casting of metals.  p r o c e s s e s , some t e m p e r a t u r e g r a d i e n t  i s present i n the l i q u i d  a t some t i m e d u r i n g t h e p r o c e s s .  The  temperature d i f f e r e n c e s produce a force r e s u l t i n g  the  t e m p e r a t u r e dependence o f t h e l i q u i d  metal d e n s i t y .  t e m p e r a t u r e g r a d i e n t s c a n r e a d i l y be a l t e r e d many o f t h e e f f e c t s the  o f f l u i d f l o w a r e e x a m i n e d by a l t e r i n g  flow v e l o c i t i e s (20,  gradients  As  experimentally,  g r a d i e n t and c o r r e s p o n d i n g t h e r m a l c o n v e c t i o n .  actual  from  are quite s e n s i t i v e  The  t o the thermal  21).  T h e r e a r e two i m p o r t a n t d i m e n s i o n l e s s p a r a m e t e r s w h i c h a r e u s e d when d i s c u s s i n g t h e r m a l c o n v e c t i o n : t h e P r a n d t l number and t h e G r a s h o f number. is  the r a t i o  diffusivity  P  where:  The P r a n d t l number  o f t h e momentum d i f f u s i v i t y  t o the thermal  o f t h e f l u i d and i s w r i t t e n a s :  =  r  y-  a  =  v/o  k/pC  =  P  Vi  /,  .(1.5)  k  P r = P r a n d t l number v  = kinematic v i s c o s i t y ,  cm.  2  / sec.  2 a  = thermal d i f f u s i v i t y ,  :  cm.  / sec.  u.  = a b s o l u t e v i s c o s i t y ,. p o i s e , gm. / c m . - s e c .  P-  = d e n s i t y , gm. / cm.  k = t h e r m a l c o n d u c t i v i t y , c a l . / cm.-sec.-°C C^ = s p e c i f i c h e a t , c a l . / gm.-°C The  m a g n i t u d e o f t h e P r a n d t l number f o r a f l u i d  the  way t h e f l o w and t h e r m a l p r o f i l e s  relates  are r e l a t e d  i n natural  convection.  Table  I lists  various fluids  r e s p e c t i v e P r a n d t l numbers.  The  are g e n e r a l l y lower than those by  and  values f o r l i q u i d  convection behaviour  metals  f o r other types of  as much as s e v e r a l o r d e r s o f m a g n i t u d e and  thermal  their  will  be v e r y  fluids  hence  the  different. —?  Generally  l i q u i d metals  h a v e v a l u e s o f t h e o r d e r o f 10  g a s e s have v a l u e s c l o s e t o 1, and w a t e r The  Grashof  has  temperature and  a value near  number i s a d i m e n s i o n l e s s  r e l a t i n g the buoyancy f o r c e s i n the f l u i d  ,  due  to  10,  parameter the  d i f f e r e n c e s to the v i s c o u s f o r c e s i n the  fluid  is written: „  _  gBA9d  Or  -  *-JT—  r  where:  Gr = G r a s h o f  3  (1.6)  number 2  g = a c c e l e r a t i o n due 8 = coefficient AG  = 6  2  t o g r a v i t y , cm.  o f volume e x p a n s i o n ,  - 6i= t e m p e r a t u r e the hot  and  the  difference  cold walls,  d = d i s t a n c e between the.hot cold walls, Another is  and  1 /  °C  between  the  cm.  t h e R a y l e i g h number d e f i n e d a s : Ra = Gr • P r S o l u t e g r a d i e n t s cause f l u i d gradients.  sec.  °C  d i m e n s i o n l e s s parameter used i n t h e r m a l  manner t o t h e r m a l  /  convection (1.7)  motion  in a  Solute concentration  similar  TABLE I . Selected Fluid  Fluid  Viscosity  centipoise Tin (37,  38,  39)  Lead (37,  38,  39)  Specific Heat cal/gm-r°C  Thermal Conductivity cal/ cm-sec-°C  1.88  0.054  0.08  2.39  0.038  0.039  0.0094  0.52  0.000052  6.5  0.12  NaCl ( l i q u i d ) (41)  1.27  0  Water  1.38  1.0  NH ( g a s ) 3  Properties  Densi t y  Kinematic Viscosity  Prandtl Number  '. G r a s h o f AT  -.  gm/crrr .  cm / s e c  1/°C  6.95  0.0027  0.013 ,  3.6  x 10  10.62  0 .0022  0.  024  5 . 8  x 10 .  0.0079  0.012 .  0.9  0.07  6.95  0 . 0 0 9 3  0 . 1 1  0.6  x 10  0.03  1.54  o  0.12  1.3  x i d  1.3  x 10  b  6  0 . 3 2 " x  1 0  :  6  (1)  Steel (40)  (1)  .27  0.0014  •1.0  -  .0083  . 0.0138  10.0  6  6  6  ar—' -t  differences liquid the  may  i n t e r f a c e o r any  solute  d e p e n d e n t on the  °C  are  solute  that  the  m i x i n g due  vection. go  the  the  thermal  the  flow  and  suppress the  fluid  discussed  or s u p p r e s s the metals are  (5-30  i s negligible'  rather  than n a t u r a l  develop flows  convection  forced  fields  natural  electrical  can  natural  i s used to i n t e n s i f y  Control  of  i n v e s t i g a t e d by  mould t o enhance 23).  convection  i n Chapter  can  to produce, f o r example,  been e x t e n s i v e l y  16,  flow  con-  which  t h a t would r e s u l t from the  m o t i o n (15,  Magnetic  Winegard  convection.  r o t a t i o n s o f the  e f f e c t s of  the  % lead-tin alloys)  more c o n t r o l l a b l e g r a i n s i z e s .  oscillations  be  wt.  Enhancement o f f l o w  g r a i n s t r u c t u r e has  will  C o l e and  convection  convection  e f f e c t s of natural  i n t o the  a l s o whether  mechanical mixing to induce f l u i d  to forced  forces-.  f i n e r and  and  produced  thermal gradients  M e c h a n i c a l m i x i n g can  against  driving  for large  to s o l u t e  External rise  differences  gradients  d i l u t e a l l o y s (0.01  compared w i t h  gives  i n a p a r t i c u l a r system i s  a d d i t i v e or s u b t r a c t i v e .  and  in  r e l a t i v e Importance between  r e l a t i v e density  have shown t h a t / cm.)  The  convection  t h e r m a l and  effects (22)  the  solid-  other inhomogeneities present  l i q u i d metal a l l o y .  t h e r m a l and  by  a r i s e from s o l u t e r e j e c t i o n at a  An on  the  the various  or  investigation macrosegregation  8. be  convection  used t o e i t h e r enhance in a liquid ;  c o n d u c t o r s t h e y can  be  metal.  induced  Since to  move u n d e r a r o t a t i n g m a g n e t i c f i e l d magnetic f i e l d  i fa direct e l e c t r i c a l  through the l i q u i d field  (25).  metal at r i g h t  A magnetic f i e l d  the n a t u r a l c o n v e c t i o n  of the l i q u i d .  i s a retarding force to magnetic f i e l d .  of natural convection  oscillations  to  solidified  1.3.  a uni-  r o d ( 2 6 ) , and s u p p r e s s t h e c o l u m n a r  e q u i a x e d t r a n s i t i o n by r e d u c i n g  the t h e r m a l  will  ( 1 2 ) , reduce the s o l u t e  b a n d i n g ( 1 3 ) , change t h e m a c r o s e g r e g a t i o n a l o n g directionally  This  as an i n c r e a s e i n t h e v i s c o s i t y  The r e d u c i n g  suppress the thermal  t o the magnetic  c a n a l s o be u s e d t o r e d u c e  as t h e r e  c a n be v i s u a l i z e d  constant  current i s passed  angles  motion o f a conductor In a constant behavior  (24), or a  convection  and i n c r e a s i n g  gradient (27).  The d e t e c t i o n o f f l u i d  flow  The d e t e c t i o n and a n a l y s i s o f n a t u r a l c o n v e c t i o n _ c a n be d i v i d e d i n t o two g e n e r a l d i r e c t and i n d i r e c t . analysis  c l a s s e s o f techniques:,  The i n d i r e c t methods  o f an e f f e c t o f c o n v e c t i o n ,  involve the  t o deduce t h e f l o w  t h a t was o c c u r r i n g a t t h e t i m e t h e e f f e c t was p r o d u c e d . The d i r e c t method  involves the a c t u a l measuring o f the  f l o w o r some p r o p e r t y  associated with the flow, while the  flow i s o c c u r r i n g . The e f f e c t s o f c o n v e c t i o n 1.1., s u c h as s e g r e g a t i o n ,  discussed  i n[section  c a n be u s e d t o d e d u c e t|he l i q u i d I  flow that occurs  during the formation  o f the effecit.  Generally,  this  type  of analysis w i l l  degree o f m i x i n g or a very o f the  flow.  approximate  temperature  gradients  c o n f i g u r a t i o n s to determine  rates  using  Quenched  the  final  shapes ahead o f the i n that  flow  rates  are  not  easily  be  (26)  p o s s i b l e by  - water systems  differences and  inythe  to  This  these  (31),  (16,  basic  The  subject  method t o d e t e r m i n e the f o r a thermal  points (26)  can  and  be  will  determined. temperature  flow  This  comparison. layer  t o deduce  local  the  fluid  28,  the  flow  are  29,  liquids.  30),  sodium  relating  the  between t h e  discussed  be  non-metallic  way  they  f u r t h e r i n Chapter  used  in liquid  as  a  direct  metals.  The  t o t r a v e l between two  near  thermocouples  velocity  between the  procedure, however,.requires  f l u c t u a t i o n s and  of  main drawbacks o f  rates  two  the  transparent  d i f f e r e n c e s i n the be  flow  m o l t e n sodium c h l o r i d e  behavior  oscillation  measured w i t h  hence, the  the  metal s i t u a t i o n s i s  flow the  and  be  methods.  T e m p e r a t u r e measurements c a n  time  on  boundary  in non-metallic  liquid  m e t a l l i c systems o r  solidify.  can  f o r the  used  methods o f o b s e r v i n g  accomplished  observations  direction  However, q u a n t i t a t i v e v a l u e s  have a l l been s t u d i e d .  these  relative  specimen s i z e s  effect  diffusion  Ammonium c h l o r i d e - w a t e r s y s t e m s chloride  and  their  i n t e r f a c e can  region.  Direct very  a  i d e a o f the  solute distribution  i n t e r f a c e shapes and  flow  give  U n i d i r e c t i o n a l growth e x p e r i m e n t s  done a t v a r i o u s and  only  hence has  limited  i n the two  melt  points  large use  at  the  3.  lower  flow rates without  effect  turbulent flow..  Also the l o c a l  o f t h e t h e r m o c o u p l e on t h e f l o w i s n o t known.  The  shape o f t h e i s o t h e r m a l l i n e s  the  fluid  profile  c a n a l s o be u s e d t o d e d u c e  flow d i r e c t i o n s i n the melt.  i n Figure  2b i s o b v i o u s l y  around the l i q u i d  For example, the  f o r a one c e l l  flow  cell.  Direct observation  of flow i n l i q u i d  metals  can  a l s o be made u s i n g r a d i o a c t i v e t r a c e r  The  f l o w c a n be o b s e r v e d u s i n g " i n s i t u " m o n i t o r i n g o f  t h e movement o f t h e t r a c e r s i n t h e m e l t a quenching technique  (20).  1.4  i n the f o l l o w i n g  ( 2 1 ) o r by u s i n g  The q u e n c h i n g t e c h n i q u e  been used e x c l u s i v e l y i n t h e p r e s e n t discussed  techniques.  has  w o r k and i s f u l l y  chapter.  Purpose o f the present i n v e s t i g a t i o n Many f a c t o r s o f s o l i d i f i c a t i o n  are greatly  a l t e r e d by n a t u r a l c o n v e c t i o n , a s d e s c r i b e d section.  For a basic understanding  i n the previous  o f the e f f e c t of l i q u i d  m e t a l f l o w on c a s t i n g s t r u c t u r e s and p r o p e r t i e s , t h e d e t a i l s o f t h e f l o w i t s e l f must be known. e i t h e r experimental the exact  behavior  A great  of l i q u i d  metals  d e a l o f work h a s b e e n done  and e x p e r i m e n t a l l y  convection, but generally t h i s accuracy,  research,  o r a n a l y t i c a l , h a s b e e n done t o d i s c o v e r of natural convection  i n a c l o s e d system. both a n a l y t i c a l l y  Very l i t t l e  on n o n - m e t a l l i c  natural  c a n n o t , w i t h any d e g r e e o f  be a p p l i e d t o l i q u i d m e t a l f l o w .  The r e a s o n s  for  this  will  to  follow.  in  liquid  for  become e v i d e n t  Direct metals  these  observation  has  reasons  a research  experimentally,  thermal  convection  observe were  the  observed.  and the the  volume  Plows  outer  liquid  under was  study  the  This  work  identical  the  was  patterns  achieved.  It  undertaken  to  other  of  from solute  the  the  developed  areas  were  Is  metal.  technique  c a u s e d by  course  of  rotated  and  fluid  to effects  convection  observed  as  convection  was  in  the  of  Mr.  MacAulay.  in  Metallurgical  f o r m as p r e s e n t e d  it  this  work  an  auxiliary  jointly  with  Mr.  L.  in  is  This  in to  work  Weinberg.  during  Chapter appear has  Transactions here.  P.  aluminum -  oscillated  presented  f o r m as  of  Professor  macrosegregation  is  flow  theoretically,  a liquid  freezing  conducted  direction  on t h e  castings  flow  been  sections  pool.  During research  of  several  on  cell  p r o g r a m was  resulting  change  theoretical  the  not  experimetal  flow,  interdendritic  of  and compute  behavior  the  fluid  the  previously  observe  Using  in  been  This  3 wt.  this  the  %  study silver  thesis  in  forthcoming  accepted  and w i l l  MacAulay  solidification.  7 •• o f in  C.  appear  for in  an  thesis  publication the  same  2.  2.1.  General  A p p a r a t u s and  Procedures  Apparatus The  was  Experimental  technique  used i n the present i n v e s t i g a t i o n s  t h a t of u s i n g r a d i o a c t i v e elements to t r a c e but  flow patterns  i n the  restrictions.  The  Figures  4b.  4a and  liquid  m e t a l p o o l under v a r i o u s  experimental The  The  molten metal i s contained  c e n t r a l U shaped p i e c e  a l u m i n u m o r c o p p e r p l a t e and and  cell  sheets  of the  The  mould.  metal.  p l a t e s and m e t a l out  The  thicknesses  m o u l d i s open on 1/16  any  graphite  the U p i e c e  coated  A  with a  side  of the  liquid  o f t h e mould e v e n i f a good c o n t a c t  i n the  mould  solid  metal  obtained.  any  the  leakage  assembly i s not  a l s o stops  flat  the collodial  a t t a c k on t h e m o u l d by l a y e r between the  the  inch thick  the U shaped p i e c e .  A l l s u r f a c e s o f the mould are  an  from  s t e e l b o l t s are used to assemble  g r a p h i t e wash t o p r e v e n t liquid  i s machined  s i d e s o f the mould are  same m a t e r i a l as  number o f s t a i n l e s s  The  in  shown i n  a number o f v a r i o u s  l e n g t h s were p r o d u c e d .  upper s u r f a c e .  boundary  a p p a r a t u s i s shown i n  a l u m i n u m o r c o p p e r m o u l d t h a t i s a s s e m b l e d as F i g u r e 5.  the  A f t e r quenching the  Figure  H.  The e x p e r i m e n t a l a p p a r a t u s f o r o b s e r v i n g natural c o n v e c t i o n s h o w i n g (a) t h e s i d e s e c t i o n v i e w o f t h e o v e r a l l furnace- and -(b) t h e f r o n t s e c t i o n v i e w o f t h e inner furnace d e t a i l s .  Figure  5.  E x p l o d e d view o f t h e copper o r aluminum mould u s e d i n o b s e r v i n g n a t u r a l c o n v e c t i o n .  is  removed  by  d i s a s s e m b l i n g the  mould a h e a t i n g contained passing the  in  heating  the  coil  to  copper  block.  power  up  end of  give  to  to  the  the  block  a variable  bolted  to  mould i s  to  autotransformer  the  a cooling  giving  a  experiment. block  of  heating  block.  A copper  tube  for  passage  of  argon  or  removal.  Both  rate  the  a  power  transformer  80 w a t t s , d u r i n g  the  into  The  voltage  passed through  maintained  hole  lowered  constant  construction  are  heating  the  is  a Sola  opposite  blocks  and t h i s  of  is  from a v a r i a b l e  On t h e  to  tube  end  A chromel winding  supplied  applied  water  attached.  On o n e  is  stable  similar  is  a quartz  through  connected  is  block  mould.  the  of  heat  U shaped p i e c e  and a t i g h t  for  good heat  transfer  by  the  use  The  composite  mould i s  suspended  of  the fit  .002  is  inch  spacers.  stainless steel with  straps. argon  container element for  steel  During  to. reduce is  is  raising  A Honeywell heaters  container  Is  rapidly  placed  an e x p e r i m e n t oxidation,  control  the  stainless  quenching  the  of  steel  within  for  side  the  overall  tank. ± 1/2  is  between This  °C a t  A large steel  the  the  to  furnace heating  controller control  filled  flat  heating  container  furnace  connected  maintain a constant  thermocouple  stainless is  on e i t h e r  to  with  covered  container  water.  controller  a  the  with  temperature used  and  inches  filled  and m a i n t a i n i n g the  The  temperature  6 x 6 x 2  in  temperature. the  plate  temperature. plate  maintains  and a  thermocouple.  The w h o l e a s s e m b l y  Is contained i n a brick insulated outer  aluminum c a s e , c o n s t r u c t e d t o a l l o w d i r e c t v i s u a l  obser-  v a t i o n o f t h e m e l t f r o m above t h e f u r n a c e . Temperature measurements o f t h e m o l t e n b a t h a r e made w i t h l o n g t h e r m o c o u p l e p r o b e s from t h e top o f t h e f u r n a c e . are used.  i n s e r t e d i n t o the melt  Two t y p e s o f t h e r m o c o u p l e s  For simple temperature monitoring o f the bath  i r o n - c o n s t a n t a n t h e r m o c o u p l e s , s h e a t h e d by 1.5  mm.  diameter  q u a r t z t u b i n g , a r e used.  The e n d o f t h e s h e a t h i s l e f t  open a n d t h e t h e r m o c o u p l e  b e a d w h i c h i s 0.5 mm.  i n diameter,  p o s i t i o n e d o u t s i d e t h e t u b e f o r good t e m p e r a t u r e and a c c u r a c y .  For accurate temperature p r o f i l e  response measurements  f o r t h e b a t h , a c o m m e r c i a l l y made i r o n - c o n s t a n t a n t h e r m o c o u p l e c o n t a i n e d i n a 0.5 mm. The  stainless  s t e e l tube i s used.  v e r y s m a l l d i a m e t e r o f t h e t u b e c a u s e s minimum d i s t u r b a n c e  in the melt.  E i t h e r o f t h e s e t h e r m o c o u p l e p r o b e s c a p be  p o s i t i o n e d t o any l o c a t i o n i n t h e m e l t w i t h t h e a i d o f a h o r i z o n t a l and v e r t i c a l top o f t h e f u r n a c e .  c a l i b r a t e d t r a v e r s e m e c h a n i s m on  The t h e r m o c o u p l e s  a r e connected v i a an  ice water c o l d j u n c t i o n t o a Honeywell E l e c t r o n i c temperature r e c o r d e r . 20 m i l l i v o l t s in this 2.2.  Full  scale deflections  (360°C) and 0 . 1  millivolts  194  o f between  ( 1 . 8 ° C ) were  used  work.  Experimental procedure The  l i q u i d metals used i n t h i s  s t u d y were t i n ,  l e a d , l e a d - t i n a l l o y s and l e a d - b i s m u t h a l l o y s . A l l materials  were s p e c i f i e d a s 99.999$ p u r e  impurities. with  The r a d i o a c t i v e t r a c e r s u s e d w e r e t i n ( S n  K and e~ r a d i a t i o n and t h a l l i u m  emitter.  and t r a c e r e l e m e n t of n a t u r a l  1 1 3  )  (TI **) a oure b e t a 2 0  The t r a c e r s were o b t a i n e d f r o m t h e A t o m i c  o f Canada L i m i t e d .  Various combinations o f melt  Energy  material  were u s e d t o i n v e s t i g a t e v a r i o u s a s o e c t s  convection. The  is  formetallic  g e n e r a l p r o c e d u r e f o r an e x p e r i m e n t a l r u n  as f o l l o w s : (a)  The m o u l d i s a s s e m b l e d  Investigated  i s cast  and t h e m a t e r i a l  i n t o t h e mould.  The f i l l e d  t o be  mould i s  p l a c e d i n t h e f u r n a c e and t h e t e m p e r a t u r e o f t h e s y s t e m i s r a i s e d by means o f t h e h e a t i n g p l a t e s . the  With t h e use o f  h e a t i n g and c o o l i n g b l o c k s a p r e s c r i b e d  difference  i s imposed and m a i n t a i n e d a c r o s s t h e molten  or semi-molten r e g i o n .  During this  i s maintained i n the s t e e l (b)  temperature  t i m e an a r g o n  container.  A temperature traverse  i s made o f t h e m o l t e n  zone when t h e s y s t e m h a s r e a c h e d e q u i l i b r i u m The  conditions.  s y s t e m i s assumed t o be i n e q u i l i b r i u m when t h e a v e r a g e  t e m p e r a t u r e and t h e t e m p e r a t u r e d i f f e r e n c e are  atmosphere  both constant f o r a period  hour.  The 0.5 (c)  mm.  of approximately one-half  diameter thermocouple  To o b s e r v e t h e f l o w p a t t e r n ,  of r a d i o a c t i v e m a t e r i a l  across the melt  i s used. a small  i s added t o t h e m e l t .  sphere The  material  0.1  added i s a p p r o x i m a t e l y  gms.  For  the  t i n tracer  113 t h e p a r t i c l e a d d e d i s 25 wt. A t o m i c E n e r g y o f Canada L t d .  % Sn For  as r e c e i v e d f r o m the t h a l l i u m t r a c e r  The the  J  20 4 particle is  a d d e d i s 1.5  added a t  for  the  top  wt. of the  a certain period  cold  (d) • The  The  filling  The  i s allowed to  tracer flow  whole system i s then  the  stainless steel  con-  water.  resultant  active material  as r e c e i v e d .  m e l t and  of time.  q u e n c h e d v e r y r a p i d l y by tainer with  % Tl  solid  block containing  i s removed f r o m t h e  m o u l d and  the  radio-  p l a c e d on  s h e e t o f e i t h e r X-Ray o r C o n t r a s t P r o c e s s O r t h o f i l m . presence of r a d i o a c t i v e effectively  expose the  material  film locally.  developed f i l m then reveals p r i o r t o the  quench.  r a d i a t i o n i n the for  the  flow  the  m a c h i n e d o f f and All  the  The  the  metal c a s t i n g  surface  pattern  i n the  distances  of  short  the (20  microns  internal  progressively  autoradiographed.  tracer w i l l  where a p p l i c a b l e , w i t h are  the  printed  o r i g i n a l autoradiographs. show as  o r i g i n a l autoradiograph.  d i r e c t i o n , and  the  liquid  f o r o b s e r v i n g the be  will  examination of  is relatively  can  The  film  a u t o r a d i o g r a p h s p r e s e n t e d i n t h i s t h e s i s are  p o s i t i o n o f the the  flow  that  layers  f r o m n e g a t i v e s made f r o m t h e  in  An  penetration  t h a l l i u m i n t i n ) , so  pattern  a d j a c e n t to the  a  flow  a darkened region  A l l pictures i n the  a l s o shown a c t u a l  will  be  The as shown,  counterclockwise  size.  2.3.  Validity  of  The this are  most  experimental  fluid  or  the  quench,  will the  quench  be  is  the  fixed  water  in  two  during  curve  is  starting The  solidify  its  be  the  resolved  in  autoradiography  occurs  itself  prior  causes  the  shown  tank  is  The  likelihood prior  quench  rate  A typical  in  Figure of  to  the  significant  solid  for  268°C  one  tracer  quenching.  To  filled  cold  with  estimated left  tin  that  the  second and  in  quench  from the  vs.  melt  point  seconds  rapid  the  temperature  (melting  three  more  that  is  a pure  indicates  after  the  a thermocouple  6,  response  to  completely  quench.  approximately  completely  the  of  temperature  in  is  assume t h a t  response  time  with  a  231.9°C). tin  that  starts  the  from the  to  casting start  of  quench.  A zero pure  tin  i n which there thirty 6.4  that  to  position  seconds.  temperature  the  to  whether  flow  process  greater  steel  temperature  melt  the  is  reasonable  in  stainless  the  of  question  motion.  It the  important  technique  representative  quench,  is  technique  temperature  should produce radioactive  was  no d e t e c t a b l e  seconds - a f t e r  c m . by  tin  6.4;cm.  by  the  no  gradient  fluid  was  in  flow..  added to  temperature addition.  :  a molten bath  A test liquid  gradient, The  0 . 3 2 cm. t h i c k .  was  tin  in  then  sample s i z e Figure  of  conducted which quenched was  7 shows  the  TIME F R O M S T A R T O F Q U E N C H , S E C O N D S  F i g u r e 6.  The t e m p e r a t u r e v e r s u s t i m e c u r v e i n t h e l i q u i d t i n d u r i n g r a p i d water quenching.  F i g u r e 7.  The t r a c e r movement i n a l i q u i d t i n m e l t (Sn t r a c e r ) w i t h a 0°C t e m p e r a t u r e d i f f e r e n c e , l e f t 30 s e c o n d s b e f o r e q u e n c h i n g . ( A c t u a l s i z e ) 1 1 3  resulting The  f l o w as  the  darkened r e g i o n o f the  t r a c e r i s s e e n t o have r e m a i n e d e s s e n t i a l l y  r e g i o n i n w h i c h i t was f l o w due being  t o the  the  the  fluid  liquid  the  cell  should  c a s t i n g can  to  solidify  during this  positions  i n the  s a m p l e had cell,  an  outer  tracer.  cm.  mould.  interval.  I t should  be  temperature gradient.  little  In the  central  region  start  vertical position  8.  This  the  liquid  made i n t h e  i n Figure as  8a,  8b  various  i s pure t i n with  6.4  by  6.4  0.32  by  that a temperature d i f f e r e n c e c o n d i t i o n s r a t h e r than 2b  From F i g u r e  i t Is evident  a constant  cell  might  i s shown i n F i g u r e  thermal  i n the  quenching  at  and  but  and  8c  c a s t s u r f a c e , 13%  a  that  the  a f u n c t i o n of  the  hence i s not  parameter f o r a c e r t a i n . t h e r m a l c o n d i t i o n .  p o s i t i o n at the  of  tracer profile  noted  temperature g r a d i e n t i s not  the  so t h a t  from the  The  c a s t i n g was  used t o d e s c r i b e the  radiographs  surface of  temperature d i f f e r e n c e across  The  determine  rapidly  a v e r a g e t e m p e r a t u r e o f 2 6 0 ° C , and  tin  is  used t o  some e x t r a n e o u s f l o w  s o l i d block  a 3°C  The  extremely  second elapse  different  are r e p r e s e n t a t i v e of  anticipated.  complete s o l i d i f i c a t i o n ;  occur  a l s o be  quenching.  t r a c e r movement w o u l d be there i s a three  downward  radioactive t i n  flow p a t t e r n at  autoradiographs  flow before  i n the  liquid.  change i n the  p o s i t i o n s w i t h i n the what e x t e n t  a d d e d , w i t h some s l i g h t  a d d i t i o n c o n t a i n i n g the  c o l d e r than The  to  autoradiograph.  a unique  The  represent below the  three  the  auto-  tracer  surface,  and  C  Figure  8.  The t r a c e r p r o f i l e i n a t i n m e l t ( S n tracer, 260°C a v e r a g e t e m p e r a t u r e ) w i t h a 3 . 0 ° C t e m p e r a t u r e d i f f e r e n c e l e f t 30 s e c o n d s b e f o r e q u e n c h i n g , (a) on t h e as c a s t s u r f a c e , ( b ) 13% i n t o t h e t i n b l o c k , and ( c ) i n t o the t i n b l o c k . 1 1 3  kH% b e l o w t h e s u r f a c e . flow around the l i q u i d  I t i s apparent that the l i m i t cell  of  ( p o i n t A) i s n e a r l y t h e same  t h r o u g h o u t t h e c a s t i n g w h i c h w o u l d n o t be t h e c a s e i f t h e r e were i n t e r n a l  flow a f t e r  the surface  Prom t h e o b s e r v a t i o n s it  i s very  reasonable  presented  solid.  in this  t o assume t h a t t h e q u e n c h i n g  d o e s n o t c a u s e any s i g n i f i c a n t flow pattern.  l a y e r s were  disturbance  o f the  section operation overall  3.  3.1.  Experimental  Thermal  Convection  investigations  Thermal c o n v e c t i o n i s perhaps  t h e most  form o f c o n v e c t i o n i n s t a n d a r d c a s t i n g p r o c e s s e s  important and  t h e r e f o r e r e c e i v e the g r e a t e s t study i n t h i s work.  will The  e x p e r i m e n t a l r e s u l t s o f the present i n v e s t i g a t i o n w i l l p r e s e n t e d under the f o l l o w i n g (a) average  temperature  divisions:  difference  a c r o s s the l i q u i d  and  temperature  (b)  liquid  metal t h i c k n e s s  (c)  temperature  (d)  liquid  zone h e i g h t  (e)  double  cell  (f)  liquid  metal  (g)  thermal  distribution  liquid  flows investigated  o f pure  t i n , r a d i o a c t i v e t i n was  as t h e t r a c e r and  f o r melts o f pure  t h a l l i u m was  as t h e t r a c e r - .  used  i n the  fluctuations  •For.:a m e l t  will  be .  lead,  Both  radioactive  of these  systems  be r e p r e s e n t a t i v e o f t h e r m a l c o n v e c t i o n o n l y .  used  3.1.1.  V a r i a b l e a p p l i e d t e m p e r a t u r e d i f f e r e n c e and average  The  temperature  thermal convective flow i s very  on t h e t h e r m a l b o u n d a r y The  liquid  cell  dependent  c o n d i t i o n s present i n the melt.  used i n the e x p e r i m e n t s d e s c r i b e d i n t h i s  s e c t i o n i s 6.4 cm. by 6.4 cm. by 0.32 cm. t h i c k . of  changing the l i q u i d  next s e c t i o n . difference  cell  thickness i s discussed i n the  The f l o w p a t t e r n a s a f u n c t i o n o f t e m p e r a t u r e  a c r o s s t h e p u r e t i n m e l t i s shown i n t h e a u t o -  r a d i o g r a p h s i n F i g u r e 9.  A l l t h e s e samples  t e m p e r a t u r e o f 260 ± 1°C.  The e i g h t  have t e m p e r a t u r e d i f f e r e n c e s the  liquid  zone.  Comparing  h a d an a v e r a g e  f l o w p a t t e r n s shown  f r o m 0.23°C t o 9.1°C a c r o s s the samples  shows t h a t t h e  flow rate increases greatly with i n c r e a s i n g  temperature  d i f f e r e n c e , b u t t h e shape o f t h e f l o w p a t t e r n essentially  The e f f e c t  remains  c o n s t a n t over the range o f temperature  considered here.  A l l flow occurs i ns i n g l e  differences  cellular  p a t t e r n s and a p p e a r s t o be l a m i n a r i n n a t u r e a s no s m a l l t u r b u l e n t eddy c u r r e n t s a r e a p p a r e n t . The of  e f f e c t o f changing the average  t h e t i n melt- c a n be s e e n i n F i g u r e 10.  average  temperature  Three  different  t e m p e r a t u r e s have b e e n s t u d i e d , 237°C, 260°C and 305°C.  F i g u r e s 10a and;.10b compare t h e f l o w r a t e s a t 237°C a n d 305°C w i t h a 2°C t e m p e r a t u r e d i f f e r e n c e  a n d F i g u r e s 10c  and l O d compare: t h e f l o w r a t e s f o r a 3°C t e m p e r a t u r e difference.  I t s h o u l d be n o t e d when c o m p a r i n g t h a t t h e  a  b  c  F i g u r e 9.  d  The t r a c e r p r o f i l e i n t i n m e l t s ( S n tracer, ?60°C a v e r a g e t e r m o e r a t u r e , 0.32 cm. t h i c k c e l l ) w i t h a t e m p e r a t u r e d i f f e r e n c e and a t i m e b e f o r e q u e n c h i n g o f ( a ) 0 . 2 3 ° C , 300 s e c o n d s , ( b ) 0.67°C, 90 s e c o n d s , ( c ) 1.11°C, 120 s e c o n d s , ( d ) 1.9fi°C, 60 s e c o n d s . (continued) 1 1 3  e  f  F i g u r e 9 c o n t i n u e d . The t r a c e r p r o f i l e i n t i n m e l t s (Sn t r a c e r , 260°C a v e r a g e t e m p e r a t u r e , 0.32 cm. t h i c k c e l l ) w i t h a t e m p e r a t u r e d i f f e r e n c e and t i m e b e f o r e q u e n c h i n g o f ( e ) 3.04°C, 30 s e c o n d s , ( f ) 4 . 0 0 ° C , 30 s e c o n d s , ( g ) 5 . 0 5 ° C , 15 s e c o n d s , and ( h ) 9.1°C, 15 s e c o n d s . 1 1 3  d  C  F i g u r e 10.  The t r a c e r p r o f i l e i n t i n m e l t s ( S n tracer) w i t h an a v e r a g e t e m p e r a t u r e , t e m n e r a t u r e d i f f e r e n c e , and t i m e b e f o r e q u e n c h i n g o f ( a ) 236°C, 2 . 0 0 ° C , 60 s e c o n d s , ( b ) 305°C, 2.02°C, 30 s e c o n d s , ( c ) 237°C, 3.00°C, 30 s e c o n d s , and ( d ) 305°C, 3.02°C, 15 s e c o n d s . 1 1 3  lower superheat superheat to  f l o w s were l e f t  t w i c e as l o n g as t h e h i g h e r  f l o w s f o r t h e same t e m p e r a t u r e d i f f e r e n c e  t h e quench.  From t h e a u t o r a d i o g r a p h s i t i s e v i d e n t  t h a t an i n c r e a s e i n t h e a v e r a g e increase i n the flow rate For  temperature  c a u s e s an  f o r an e q u a l t e m p e r a t u r e  quantitative analysis  time f o r the flow t o complete been c h o s e n  prior  difference.  of the flow rates the  one c y c l e a r o u n d  as t h e r e l e v a n t p a r a m e t e r .  t h e c e l l has  U s u a l l y two t o f o u r  experiments with d i f f e r e n c e s i n the times before  quenching  are  and average;  done f o r a p a r t i c u l a r t e m p e r a t u r e d i f f e r e n c e  temperature t o obtain the time p e r c y c l e . for  The t i m e p e r c y c l e  a p a r t i c u l a r c a s t i n g i s o b t a i n e d by m e a s u r i n g t h e  a n g u l a r movement o f t h e t r a c e r  f r o n t . a r o u n d t h e c e l l and  then u s i n g t h e t i m e t o quench t o c a l c u l a t e t h e p e r i o d t h e t r a c e r would  take f o r a complete  c y c l e o f 360°.  The s a m p l e  m u s t , t h e r e f o r e , be q u e n c h e d b e f o r e t h e t r a c e r has c o m p l e t e d one  full  it will rate. any  cycle.  This parameter  h a s been c h o s e n  since  allow a s i n g l e value t o d e s c r i b e the o v e r a l l  flow  The a c t u a l a u t o r a d i o g r a p h s c a n be u s e d f o r n o t i n g  changes i n t h e shape o f t h e f l o w p a t t e r n .  per cycle  f o r oure t i n versus the temnerature  across the c e l l is plotted  f o r three d i f f e r e n t  i n F i g u r e 11.  degrees  The t i m e difference  o f superheat  I t i s seen f r o m t h i s  g r a p h , as  with the autoradiographs, that the flow rates increase i n c r e a s i n g temperature d i f f e r e n c e across the c e l l they a l s o i n c r e a s e w i t h i n c r e a s i n g melt  superheat.  with  and t h a t  -A  ,  I  U  I  0  -L-  I  1  1  1  1  i  i  I  i  •  2  3  4  5  6  7  8  9  10  TEMPERATURE  DIFFERENCE,  °C  F i g u r e 11. The t i m e f o r t h e f l o w i n t i n m e l t s t o c o m p l e t e one c y c l e a r o u n d t h e c e l l v e r s u s t h e t e m p e r a t u r e d i f f e r e n c e , across t h e c e l l f o r average temperatures o f 237°C, 260°C, a n d 305°C.  3.1.2.  Variable  l i q u i d metal  Three investigated: a n d 0.95  cm.  zonelength  change size  the  done  flow due  on t h e  flow.  rates  were  All  to  mode o f  metal.  The  to  The  change  dimensional typical  Figure  difference like  all  and  found  that  the  characterized  flow by  of  the  0.32  shows  the  patterns the  single  cell.  By  thickness the  flow  increasing cell  to  to  0.48  changes  the  any  significant  with  a thicker  the  large  flow  These  flat  cell  cell  pattern  increasing cell  and  as  can develop  Figure  12  shows  in  type  shown  9.  Figure flow  circular  temperature  path  by  cm., Figure  12c,  or  0.95  tin.  of  flow  This  c e l l with around  difference  or  two-  liquid  two-dimensional  12b,  the  for  two-  5.60°C t e m p e r a t u r e  and a  laminar  liquid  as  flows  in  the to  surfaces  cm. c e l l  in  different  referred  flow.  cast  thicknesses  a completely  Figure  19°C,  to  molten  if  modes c a n be  forming a continuous  cm.  had a  cm.  differences  tracer  0.32  for  three-dimensional  which  cells  6.4  the  convection  different  is.-for  in.)  (3/16  investigated.  autoradiographs  12a  in  were  of  by  and t h r e e - d i m e n s i o n a l  dimensional  cm.  liquid  occurred  constraints  thermal  two  the  thicknesses  0.48  determine  and i n c r e a s i n g t e m p e r a t u r e stable  in.),  behaviour  both  It:was  zone  cm. and a h e i g h t  6.4  were  molten  (1/8  in.).  (3/8  possibly  faces flow  in  cm.  0.32  of  experiments  different  thickness  flow the  the  liquid  across  i n c r e a s i n g the cm., Figure  t h r e e - d i m e n s i o n a l mode.  is  The  the cell 12d,  three-  C  F i g u r e 12.  d  The t r a c e r p r o f i l e i n t i n m e l t s ( S n tracer, 260°C a v e r a g e t e m p e r a t u r e ) w i t h a t e m p e r a t u r e d i f f e r e n c e , l i q u i d c e l l t h i c k n e s s , and t i m e b e f o r e q u e n c h i n g o f ( a ) 5 . 6 0 ° C , 0.32 cm, 60 s e c o n d s , ( b ) 19.0°C, 0.32 cm., 15 s e c o n d s , ( c ) 5 . 0 ° C , 0 . 4 8 cm., 60 s e c o n d s , and ( d ) 5 . 0 ° C , 0.95 cm., 90 s e c o n d s . 1 1 3  ill  dimensional  mode i s c h a r a c t e r i z e d by a s p i r a l p a t t e r n , w i t h  t r a c e r d e p l e t e d bands coming f r o m t h r e e liquid  corners  zone a n d a t r a c e r r i c h b a n d f r o m t h e t o p l e f t - r h a n d  corner.  The s p i r a l p a t t e r n o b s e r v e d  t i m e , t h e bands r e m a i n i n g  fixed  does n o t c h a n g e w i t h  i n p o s i t i o n with time.  p a t t e r n was f o u n d t o be i d e n t i c a l on b o t h casting. areas  of the  T h e r e a r e no c o n t i n u o u s  where t h e t r a c e r i s p r e s e n t  bands.  sides o f the  flow l i n e s  connecting the  due t o t h e s e  depleted  Thus f o r t h e t r a c e r t o move t h r o u g h o u t t h e c e l l ,  as s e e n i n t h e a u t o r a d i o g r a p h s ,  t h e r e must be an i n t e r n a l  f l o w , i n t h e m o l t e n zone.  T h i s complex  flow p a t t e r n i s developed  f o rhigher  cells.  The  Figure  three-dimensional  g r a d i e n t s and t h i c k e r  13 shows q u a n t i t a t i v e l y  the d i v i d i n g  b e t w e e n t h e two f l o w modes f o r a v a r i a b l e c e l l  line  thickness  and  temperature g r a d i e n t .  A b a n d i s shown s e p a r a t i n g t h e  two  modes o f f l o w a t l o w e r  temperature d i f f e r e n c e s since  at  very  low g r a d i e n t s t h e a u t o r a d i o g r a p h  becomes more d i f f i c u l t . convection for  i s very  two-dimensional  experimental  interpretation  The t h e o r e t i c a l a n a l y s i s o f t h e r m a l  complex and i s u s u a l l y o n l y flow.  Thus t o e n a b l e  attempted  t h e o r e t i c a l and  c o m p a r i s o n s t o be made, most o f t h e p r e s e n t  observations-of thermal two-dimensional The  c o n v e c t i o n h a v e b e e n made f o r t h e  mode. v a r i a t i o n i n the a c t u a l flow rates f o r the  various l i q u i d  cell  the t h r e e  t h i c k n e s s e s u s e d i n F i g u r e 12 e x p e r i m e n t s  cell  t h i c k n e s s e s was a l s o o b s e r v e d .  For  1.2  1.0  - \ _ \  \ \  CO UJ  o  0.8  \ \ \  0.6  3-  \ \  DIMENSIONAL  FLOW  \  \  \  \  \  \ O  X  \  0.4  UJ u  0.2  2 - DIMENSIONAL  FLOW  2  4  TEMPERATURE  Figure  13.  6  DIFFERENCE,  8  °C  The l i q u i d c e l l t h i c k n e s s v e r s u s t h e t e m p e r a t u r e d i f f e r e n c e across t h e c e l l showing t h e c o n d i t i o n s f o r two d i m e n s i o n a l a n d t h r e e d i m e n s i o n a l f l o w i n l i q u i d t i n a t 260°C.  ro  .0.00  I0«  1  0.1  0.2  1  1  0.4  1  1  06  TEMPERATURE Figure  14 .  1 I  I I  10  DIFFERENCE,  1  2  :  1  I  4  »  .  6  °C  The t i m e f o r t h e f l o w i n t i n m e l t s t o c o m p l e t e one cycle versus thetemperature d i f f e r e n c e . a c r o s s t h e c e l l f o r l i q u i d c e l l t h i c k n e s s e s o f 0.32cm., 0.48 cm., a n d 0 . 9 5 cm.  I  I •I  ,  0  were to  conducted  arrive  curve  at  for  with  a time  each  calculated,  pure per  cell.  only  For  to  three-dimensional  to  a false  purposes.  the  three  temperature  different the be  time per  cells  thicker  cannot  difference is  d i s c u s s e d when  3.1.3.  cycle  dimensional  cells  theoretical  time  shown have  Temperature  prior ence To  to  the  across  the  determine  conditions, on e a c h o f total  of  plotted the  quench  is  in  Figure  geometric  15a -  centre  are  are  of  due  the  cell  for  rates for is  fully  versus  the  three  evident  rates.  This  that shall  various  3.4.  in  the  liquid in  of.the  of  readings.  15e n o r m a l i z e d molten  liquid  liquid  for  determined  the  at  a given eight melt  The  results  to  to  a zero The  differ-  thickness.  the  zone.  tin  temperature  the  isotherms  traverses  the  of  compared w i t h  and a l s o  temperature  forty.temperature  It  distribution  zone  vertical  14.  is  tracer  defined  cell  early  comparison  flow  flow  a function  temperatures five  the  Figure  distribution  molten  the  across  be  This  the  parts  to  very  used.  be  The  section  The.temperature  the  reading, for really  difference  value  in  internal  results  in  or  c a n be  more r a p i d  these  solutions  flows  flow.  in  cycle  resulting  the  temperature,  temperature  per  flows  flow  through  A cycle  developed  versus  a time  three-dimensional  circuiting"  give  260°C a v e r a g e  two-dimensional  of  "short  at  cycle  stages the  tin,  set  of  points give in  °C  point  thermal  a are at  F i g u r e 15.  The e x p e r i m e n t a l i s o t h e r m a l p l o t s f o r l i q u i d t i n a t 260°C f o r an a p p r o x i m a t e R a y l e i g h number and c e l l t h i c k n e s s o f ( a ) 1.4 x 10 , 0.32 cm., and (b) 2.4 x 1 0 , 0.32 cm. s  5  (d) F i g u r e 15  continued. The e x p e r i m e n t a l i s o t h e r m a l p l o t s f o r l i q u i d t i n a t 260°C f o r an a p p r o x i m a t e R a y l e i g h number and c e l l t h i c k n e s s o f ( c ) •8.0 x 1 0 , 0.32 cm., (d) 1.4 x 1 0 , 0.95 cm. 5  s  Figure  15 c o n t i n u e d . The e x p e r i m e n t a l i s o t h e r m a l p l o t s f o r l i q u i d t i n a t 260°C f o r an a p p r o x i m a t e R a y l e i g h number and c e l l t h i c k n e s s o f ( e ) 2 x 10* and 0.95 cm.  F i g u r e 16.  The e x p e r i m e n t a l i s o t h e r m a l p l o t f o r l i q u i d t i n a t 260°C i n a 1 0 . 8 by 6.4 by 0.32 cm. ;liquid cell.  profiles 0.95  and that  f o r v a r i o u s temperature cm. t h i c k  vertical,  conductive that  heat  cell  a r e shown.  e q u a l l y spaced transfer.'  the isotherms  The r e s u l t s  i n the l i q u i d  o f convective flow.  a one c e l l  flow p a t t e r n with.the  right  across the c e l l  fluid  moves t o t h e l e f t  itatively,  I t should  isotherms  indicative  f o r the 0.32  differences  would  indicate  i n a manner  The i s o t h e r m cool  fluid  shapes  indicate  moving t o the  the h o t t e r l e s s  dense  across the top o f the c e l l .  f o r a given l i q u i d  noted  represent  clearly  are-bent  bottom w h i l e  a g a i n be  at a constant  Quant-  average  temp-  e r a t u r e , the g r e a t e r the bending  o f the i s o t h e r m s , the  greater  C o m p a r i n g F i g u r e s 15a,  the r a t e o f f l u i d  15b  and 1 5 c , i n c r e a s i n g  the  isotherm bending;  isotherm bending.  thick  cell  similar  with  the c e l l  as s e e n  cell  with  t  a thermal  '  m a t e r i a l :is t i n and t{ie p l o t  manner as t h e -square  indicating  i s very  a 3°C  temperature This  presented previously.  profile  cm. by .0.32 cm. m o l t e n zone c e l l  i s very  cm.  i  )  profile  difference  the flow r a t e r e s u l t s j  ••  The  increases  by c o m p a r i n g F i g u r e s 15c and 15d.  For. c o m p a r i s o n ^ 6.4  increases  The i s o t h e r m a l s h a p e f o r t h e 0 . 3 2  cm. t h i c k  i s - i n agreement w i t h  by  difference  thickness also  a 19°C t e m p e r a t u r e  t o t h e 0.95  difference  the temperature  c o m p a r i n g F i g u r e s 15a and 15d o r  15b. and 15e, i n c r e a s i n g the  flow.  cm.  -  cm.  i s shown i n F i g u r e 16.  ' :• is• normalized  cell ^profiles.  flow p a t t e r n .  I n t h e same  The shape o f t h e  s i m i l a r t q the square  a one c e l l  i n a 10.8  :  cell  profiles,  again  3.1.4.  Variable height The  the  e f f e c t o f the h e i g h t  zone  o f t h e m o l t e n zone o n  f l o w p a t t e r n was i n v e s t i g a t e d i n a 10.8cm. l o n g by  0.32 cm. t h i c k l i q u i d ent  o f the molten  cell.  The m o u l d was f i l l e d t o d i f f e r -  l e v e l s t o o b t a i n the v a r i a b l e h e i g h t  required.  The  r u n s were done w i t h a 9 - 10°C t e m p e r a t u r e d i f f e r e n c e across  the  c e l l , a n a v e r a g e t e m p e r a t u r e o f 260^C a n d were  q u e n c h e d 120 s e c o n d s a f t e r t h e 17 ( a , b , c , d) shows t h e height  tracer introduction.  r e s u l t a n t flows- f o r l e n g t h t o  r a t i o s ; o f 1.6 : 1, 3.5: 1,1.9  respectively. i n each case,  : -1 a n d 8.3 : 1  The t r a c e r was a d d e d t o t h e thecold side being  on the  left-hand  left.  1.6 : 1, 3.5 : 1 a n d 4.9 : 1 l e n g t h t o h e i g h t f l o w i s o f a one f l o w c e l l n a t u r e square mould. the  Figure  However, i n the  corner  In the ratios the  a s was o b s e r v e d i n t h e  8.3 : 1 r a t i o  liquid  cell  f l o w i s s t a r t i n g t o b r e a k i n t o more t h a n one c e l l .  i s i n d i c a t e d by t h e  secondary flow c e l l  l e f t - h a n d end o f the a t i o n s the  sample.  length t o height  forming  at the  Due t o e x p e r i m e n t a l ratio  This  limit-  c o u l d n o t be i n c r e a s e d  further.  3.1.5.  Double c e l l Castings  flow normally  solidify  c o n f i g u r a t i o n , such as i l l u s t r a t e d i s e x t r a c t e d from the and  liquid  In a  i n Figure  three-dimensional 18a.  The h e a t  i n many d i r e c t i o n s a t o n c e  therefore- the r e s u l t i n g flow i n the  l i q u i d p o o l may  d F i g u r e 17.  The t r a c e r p r o f i l e I n t i n m e l t s ( S n tracer, 260°C a v e r a g e t e m o e r a t u r e , 10°C temperature d i f f e r e n c e , 0.32 cm. t h i c k c e l l , t i m e b e f o r e q u e n c h o f 120 s e c o n d s ) w i t h a l e n g t h t o h e i g h t r a t i o o f ( a ) 1.6 : 1. ( b ) 3-5 : 1, ( c ) 4.9 : 1, and 8.3 : 1. 1 1 3  LLI  rr  ZD  5  rr ui o.  POSITION HEAT FLOW  (a) Figure  18.  (b)  A schematic o f the heat flow i n a standard c a s t i n g , ( a ) , and t h e t e m p e r a t u r e p r o f i l e , (b), i n the r e s i d u a l l i q u i d pool.  change f r o m s i m p l e  one c e l l  flow patterns  t o very  complex  forms. Figure with s o l i d - l i q u i d  18a shows a p a r t i a l l y  Figure  across  casting  i n t e r f a c e moving towards the c e n t r e  each s i d e l e a v i n g a c e n t r a l l i q u i d p o o l . gradient  solidified  this pool w i l l  from  The t e m p e r a t u r e  be o f t h e f o r m a s shown i n  18b s u c h t h a t e a c h s i d e i s a t t h e l i q u i d u s t e m p e r -  a t u r e o f t h e m e t a l a n d t h e c e n t r a l r e g i o n i s a t a somewhat higher  temperature.  vertical the  The a c t u a l p r o f i l e  will  d e p e n d on t h e  p o s i t i o n i n t h e m e l t and t h e t i m e e l a p s e d  c a s t i n g was p o u r e d .  since  An e x p e r i m e n t was c a r r i e d o u t  which would approximate  t h i s more g e n e r a l  casting config-  uration. In  t h i s e x p e r i m e n t c o o l i n g b l o c k s were used  b o t h ends o f t h e r e c t a n g u l a r c e l l 0.32  cm.).  The  (10.8 cm.  average temperature o f the l i q u i d  m a i n t a i n e d by r a i s i n g t h e o v e r a l l f u r n a c e sufficiently to  by 6.4  by  t i n was  t o e n a b l e t h e h e a t l o s t t h r o u g h t h e c o o l ends  T h i s c o u l d be done i f t h e h e a t l o s t  c o o l i n g b l o c k s was  kept s m a l l .  f l o w p a t t e r n w i t h two t r a c e r was  cm.  temperature  be b a l a n c e d by h e a t g a i n e d t h r o u g h t h e f l a t  mould.  at  Figure  flow c e l l s .  through the  19a shows a  This  typical  To show b o t h c e l l s  added a t two p o i n t s s i m u l t a n e o u s l y ,  h a l f o f the mould.  faces o f the  c a s t i n g was  one  q u e n c h e d 120  a f t e r t h e t r a c e r i n t r o d u c t i o n , h a d an a v e r a g e  the  i n each seconds  temperature  o f 267°C and a t e m p e r a t u r e d i f f e r e n c e f r o m t h e m o u l d w a l l s to  t h e c e n t r a l r e g i o n o f 1 - 2°C.  r e s u l t s has two  c e l l s of approximately  i n d i v i d u a l flow c e l l s to  t h e one  F i g u r e 19b  cell  The  that result  flow pattern that t h e same s i z e .  are very  The  s i m i l a r i n form  f l o w s d i s c u s s e d i n an e a r l i e r s e c t i o n .  shows a s i n g l e c e l l p a t t e r n f o r l i q u i d  t i n at  260°C w i t h a t e m p e r a t u r e d i f f e r e n c e o f 1.11°C and a t i m e before  t h e q u e n c h o f 120 s e c o n d s .  s i m i l a r t o the double c e l l seen t h a t i n d i v i d u a l experimental  These c o n d i t i o n s a r e  c a s t i n g o f F i g u r e 19a.  flow c e l l s  configurations.  I t can  a r e t h e same f o r b o t h  Although the  temperature  d i f f e r e n c e s c a n n o t r e a l l y be c o m p a r e d , t h e f l o w r a t e s a r e  be  b F i g u r e 19.  ( a ) The t r a c e r p r o f i l e i n a t i n m e l t c o o l e d f r o m b o t h ends a n d l e f t 120 s e c o n d s b e f o r e quenching with a temperature d i f f e r e n c e across o n e - h a l f o f t h e c e l l o f 1.5°C. ( b ) The t r a c e r p r o f i l e i n a t i n m e l t c o o l e d f r o m one s i d e w i t h a t e m p e r a t u r e d i f f e r e n c e o f 1.11°C and l e f t 120 s e c o n d s b e f o r e q u e n c h i n g .  quite  similar  c a n be  indicating  applied  to  The  two  wall  having  temperatures.  while  in  Figure  0.8°C  lower  than  added to  each  case.  cell  is  one-half  case  if  a metal  In  the  on t h e  For  both the  thermal  together. of  the  the  of  be  is  that  gradient  this  region  observed  observed of  referred convection  to  is  of  the  same  of  in  zone the  zone  which  is  sides  since  the  both  the  the  flow  opposite  a region  in  the  in of  gradient the  melt.  thermal the  a "thermal  will  Davis, and  of  in the  point Fryzuk in  They  This the  come  mixing  profile.  rods.  Valve"  on e a c h s i d e  the  from  where  gradients  convective  thermal  liquid  melt  be  cell  cell.  be  two  the  interfaces.  increasing  zero  was  flow  is  the  at  in  gradient  where  the  tracer  the  2 mm. d i a m e t e r  occurs  The  temperatures than  20a  temperatures  molten  for  be  slightly  end w a l l s  f r o m two  can  maintained  wall. the  at  Figure  was  molten  wall  zero  a lack  as  side  cells  along a long  a maximum i n  the. e f f e c t  same  wall  results  gradient  will  point  at  the  temperature  there  of  much l a r g e r  maximum t e m p e r a t u r e  have  since  will  such  This  (32)  c a n be  width  flow  blocks  temperature  non-equal  side  two  the  flow  flow.  sample  side  solidifying  sample w i t h  the  left-hand  equal  the is  directions  held  left-hand  the  system i s  the  cell  of  cooling  right-hand  temperature  If metal  the  the  coldest  were  single  types of  two In  20b t h e  only  liquidus  sizes  the  temperatures  the  complex  relative  m a n i p u l a t e d by different  more  that  phenomenon  liquid  region  but  not  b  F i g u r e 20.  The t r a c e r p r o f i l e i n a t i n m e l t ( a ) c o o l e d f r o m b o t h ends w i t h e q u a l end w a l l t e m p e r a t u r e s and l e f t f o r 300 s e c o n d s b e f o r e q u e n c h i n g and ( b ) c o o l e d f r o m b o t h ends w i t h t h e l e f t hand w a l l c o o l e r t h a n t h e r i g h t h a n d w a l l , and l e f t 120 s e c o n d s b e f o r e q u e n c h i n g .  i n the ally as  region  i n the  the  itself.  the if  quench. any  This  left  cell  flow  implies  and  there  as  must be  two  19a  can  concluded from t h i s  by  20  the  light  by  natur-  Figure  a l s o have  left  there  tracer  flow  this  phenomenon. the  18  double c e l l  they Figure  introduced  f i v e minutes  before  a p p e a r s t o be  little  two  flow  cells.  some s o r t o f q u i e s c e n t  buffer  S u c h a zone i s e v i d e n t  region  between the  result that  two  the  in  cells.  "thermal  i n much l a r g e r s y s t e m s t h a n t h o s e  D a v i s and  3.1.6.  i t was  cells.  Figure  also exists  The  o f m a t e r i a l between the  zone b e t w e e n t h e  be  also occur  thus i t i s p o s s i b l e  t h a t had  I t i s observed that  transport  and  "thermal valve"  shows a d o u b l e c e l l the  19  thermal gradient,  a l s o e x h i b i t the  i n t o only  a maximum.  observed i n Figures  maximum i n t h e  20a  c o n d i t i o n may  casting configuration described  t h e r m a l p r o f i l e has  conditions  will  This  It valve"  investigated  Fryzuk.  L i q u i d metal Several  investigated  e x p e r i m e n t s were c o n d u c t e d w i t h  lead melts f o r comparison with  the  liquid  liquid  t i n results.  These e x p e r i m e n t s were i n t e n d e d t o d e t e r m i n e w h e t h e r  the  results  liquid  metals.  f o r one The  l i q u i d m e t a l can  differences  compared t o t i n are number f o r l e a d 21a  and  21b  not  be  in fluid great  applied  properties  (Tablel)  i s almost double that  show t h e  flow  patterns  to other  f o r l e a d when  a l t h o u g h the  Prandtl  of pure t i n .  Figures  obtained  for  liquid  The t r a c e r p r o f i l e i n l i q u i d l e a d ( T l " t r a c e r , 357°C a v e r a g e t e m p e r a t u r e ) w i t h a t e m p e r a t u r e d i f f e r e n c e and t i m e t o quench o f ( a ) 2.96°C, 30 s e c o n d s and (b) 4.98°C, 15 s e c o n d s , and i n liquid t i n ( S n t r a c e r , average temperature o f 2 6 0 ° C ) w i t h a t e m p e r a t u r e d i f f e r e n c e and t i m e t o quench o f ( c ) 3.04°C, 30 s e c o n d s and (d) 5.05°C, 15 s e c o n d s . 2 0  1 1 3  lead  at  357°C w i t h  difference pattern  a 2.96°C  respectively.  i s developed.  a n d a 4.98°C  Again  temperature  a simple  The t r a c e r  one c e l l  used with  lead  flow is  204 Thallium  , with  addition ence  dropped  between  combined w i t h addition that  1 . 2 $ t h a l l i u m i n l e a d m a k i n g up t h e into  the l i q u i d  the low a l l o y  the difference Thus  only  for  liquid  i.e.  with  a small  t i n under  temperature  3°C  faster flow  3.1.7.  uations or  turbulence  nature  cannot  observed.  the flow  conditions and t h e  are quite  the c e l l  patterns as the  similar  with  rates  are  at a  slightly  a t a 5°C d i f f e r e n c e slightly  lead  superheat  The f l o w  the  higher.  there  based experiments  (13, 22, 26).  Steady  any t h e r m a l  of  laminar  on  thermal  For thermal  m u s t b e some d e g r e e  i n the melt. produce  have  on t h e o b s e r v a t i o n  i n the melt  to occur  is  is  fluctuations  metal convection  oscillations  across  tin is  Many r e s e a r c h e r s liquid  and the a d d i t i o n  rate.  lead while  for the l i q u i d  Thermal  patterns  when  the assumption  convection  difference  difference  f o r the l i q u i d  rate  the melt  i n the flow  (0.7$) w h i c h  justify  t h e same t h e r m a l  The f l o w  differ-  of t h a l l i u m i n the  2 1 c a n d 2 1 d show  the temperature  difference  would  thermal  comparison, Figures  t h e same.  content  between  For  being  The d e n s i t y  l e a d and t h a l l i u m i s s m a l l  made t o t h e l e a d ,  negligible.  lead.  of flow  fluctuations.  fluct-  periodicity by i t s  very  Cole and  W i n e g a r d (22)  d e r i v e d an e x p e r i m e n t a l e q u a t i o n w h i c h  p r e d i c t s when t e m p e r a t u r e found t h a t H G 3  L  where:  fluctuations  f l u c t u a t i o n s would s t a r t  should occur.  They  when  > 3.1  (3.1)  H = height of the l i q u i d G^ = t h e t e m p e r a t u r e  zone,  cm.  g r a d i e n t , °.C / cm.  For the apparatus i n the present i n v e s t i g a t i o n the molten zone h e i g h t i s 6.4 gradient  cm. so t h a t t h e c r i t i c a l t e m p e r a t u r e  f o rturbulent  thermal o s c i l l a t i o n s 0.076°C. the  6.4  f l o w from e q u a t i o n (3.1)  at a temperature  predicts  difference of  A g r e a t d e a l o f t h e r m a l p r o b i n g h a s b e e n done i n cm. by 6.4  cm. c e l l f o r v a r i o u s t e m p e r a t u r e  differ-  e n c e s and v a r i o u s c e l l p o s i t i o n s u s i n g b o t h t h e s h e a t h e d 0 . 5 mm.  diameter thermocouple  bare bead thermocouple.  and t h e 0 . 5 mm.  The maximum t e m p e r a t u r e  o b s e r v e d was 19°C a c r o s s t h e m e l t w h i c h limit  diameter  i s close to the  o f the apparatus u s i n g c o l d water c o o l i n g .  gives a nominal temperature 3°C / cm.  difference  This  g r a d i e n t a c r o s s t h e melt o f about  A t no t i m e were any t e m p e r a t u r e  fluctuations  o b s e r v e d a t any p o s i t i o n i n t h e m e l t o r a t any t e m p e r a t u r e d i f f e r e n c e t h a t c o u l d be a t t r i b u t e d t o f l u i d f l o w i n t h e melt.  R e f e r r i n g t o F i g u r e 12b f o r l i q u i d  temperature  t i n w i t h a 19°C  d i f f e r e n c e , although the flow i s of a three  d i m e n s i o n a l mode, I t i s s t i l l  evident that the flow i s not  composed o f any t u r b u l e n c e .  The f l o w s t i l l  and does n o t p r o d u c e t h e r m a l  fluctuations.  appears  laminar  F l u c t u a t i o n s were o b s e r v e d i n the upper meter o f the l i q u i d  zone r e s u l t i n g  from t h e r m a l  i n t h e a r g o n a t m o s p h e r e above t h e m e l t .  The  fluctuations  fluctuations  w e r e g r e a t e s t n e a r t h e s u r f a c e o f t h e l i q u i d and at  depths  g r e a t e r t h a n one  s c a l e d e f l e c t i o n o f 0.1  centi-  disappeared  c e n t i m e t e r as o b s e r v e d on a  m i l l i v o l t s over the t e n i n c h  full  wide  t e m p e r a t u r e r e c o r d e r c h a r t w h i c h i s e q u i v a l e n t t o 0.18°C / inch of chart.  C o v e r i n g t h e open s u r f a c e o f t h e  r e s u l t e d i n the f l u c t u a t i o n s d i s a p p e a r i n g even  melt  at the  melt  surface.  3.2.  The  3.2.1.  t h e o r e t i c a l problem of thermal convection  Problem The  is  statement theoretical  shown i n F i g u r e 22.  experimental conditions  f l u i d system under  This system  consideration  i s s i m i l a r t o the  f o r observing thermal convection  i n the p r e v i o u s s e c t i o n s .  The  left-hand vertical  i s m a i n t a i n e d a t 61 and t h e r i g h t - h a n d v e r t i c a l m a i n t a i n e d at 62, such t h a t  9 i i s l e s s than 82.  wall  wall is Both  the  u p p e r and  l o w e r s u r f a c e s a r e assumed t o be p e r f e c t l y i n -  sulated.  At t i m e t = 0, t h e f l u i d i s a t t e m p e r a t u r e  such  9o  that 9o = 9  2  + 91  £^  2)  '•2  The  c e l l w i d t h i s d, and t h e c e l l h e i g h t i s 1, and  i t is  '///////////_ y X  LIQUID  9 d  The t h e o r e t i c a l f l u i d s y s t e m t o be u s e d i n the a n a l y s i s o f thermal convection.  22.  Figure  assumed f o r t h e o r e t i c a l c o n s i d e r a t i o n s t h a t t h e f l u i d i s infinite  i n the z d i r e c t i o n .  The c o o r d i n a t e  system i s  s e l e c t e d such t h a t t h e o r i g i n i s I n t h e upper l e f t - h a n d corner  o f the c e l l . The t h e o r e t i c a l s y s t e m o f F i g u r e  solved  f o r l i q u i d t i n a t 237°C, 260°C a n d 305°C f o r com-  parison with the experimental fluid listed  results.  p r o p e r t i e s o f l i q u i d t i n a t these  Table I I l i s t s the temperatures.  i s t h e p a r a m e t e r Gr / AT f o r t h e 6.4  This parameter i s only and  22 i s t o be  c e l l size.  a function o f the f l u i d  cell.  properties  From t h e t a b l e i t i s s e e n t h a t i f t e m p e r a t u r e  d i f f e r e n c e s up t o a p p r o x i m a t e l y be t h e o r e t i c a l l y  cm. w i d e  Also  considered,  5°C a c r o s s  the c e l l are t o  t h e s o l u t i o n must h o l d f o r  _2 P r a n d t l numbers o f t h e o r d e r  7  of the order  o f 10 .  o f 10  Previous  a n d G r a s h o f numbers  analyses  and s o l u t i o n s o f t h e  TABLE I I . Properties  M e t a l . Temper- V i s c o s ature ity  Tin  °C  centipoise  237  2.022  of Liquid  Lead and T i n  S p e c i f i c Thermal ConHeat ductivity c a l/cmcal/gmsec-°C °C  Density  Coefficient o f Volume Expansion  gm/ cm  1/°C  0.05^11 0.0798  6.9698  Kinematic Prandtl V i s c o s i t y Number 2.  1/°C  cm / s e c 0.002901  1.0215  0.01365  1.880  0.05431 0.080.6  6.9538  1.0239 x  305  1.680  0.05463 0.0809  6.9217  x Lead  357  2.39  . 0.0384  0.0386  10.6231  0.002704  0.01270  0.002427  0.01135  1.1503 x 10  _ i <  0.00225  0.02378  6  4.489 x 10  10'^  6  3.601 x 10  10~^  1.0287  3.115 x 10  x 10"^ 260  Grashof AT  6  5.842 x 10  6  present  problem are  low  Prandtl  the  fluid  solved  water. metal  and h i g h  This for  types will  results  so t h a t  the  lead  Table at  fluids  be  II  357°C  as  compared w i t h  difference  in  the  such  also for  Figure  such  with  lists  comparison  22 w i l l gases the  a  also  and  liquid  flow  behaviour  analysed.  analysed  profiles  in  the  governing in  order  cell  equations to  are  as  Momentum e q u a t i o n  3u  systems  system of  of  results  to  number.  liquid  theoretical  other  The be  of  These  c a n be  to  applicable  Grashof  properties  purposes. be  not  +  u  3u +• v|H. 3x  =  in  -g (e 6  obtain  of  thermal  the  flow  convection  rates  and  thermal  follows: the  x  - e ) 0  direction:  - 1 3p p  + v  3x  (3.3)  Momentum e q u a t i o n  3y_ 3t  3v 9y Energy  equation:  in  1 P  the  y  3p* + 3y  v  direction:  f3 v [3lF 2  A +  3 v] W . 2  37  (3.4)  Continuity equation:  3x  All  3y  (3.6)  u  the n o t a t i o n i s f u l l y  t a b u l a t e d i n Appendix  most i m p o r t a n t a s s u m p t i o n s  used  I.  The  i n the d e r i v a t i o n of the  above e q u a t i o n s a r e : (a)  A l l the f l u i d  d e n s i t y , p, w h i c h (b)  The  i s a function of  8.  (c)  The  (d)  Compressibility The  viscous dissipation  t >  0:  0:  effects  i s neglected. are n e g l e c t e d .  b o u n d a r y c o n d i t i o n s t o be u s e d as  F i g u r e 22 a r e as t =  temperature.  temperature d i f f e r e n c e across the melt i s  s m a l l compared w i t h 1 /  in  p r o p e r t i e s are constant, except  follows-:  0  < x < 1  u = v =  0  < y < d  e =  x = 1,  x = 0;  y =  0;  y = d;  of  0  e  0  u = v = 0 39 3x  Three  shown  u = v =  =  (3.7)  0  0,  u = v = 0,  =  9.  8 =  6,  t y p e s o f s o l u t i o n s w i l l be p r e s e n t e d f o r t h i s  thermal convection i n a closed (a)  An i n f i n i t e  problem  system.  s e r i e s s o l u t i o n which  by t h e maximum v a l u e s o f t h e G r a s h o f number.  is limited  (b) theory ratio  An i n t e g r a l  which of  is  the  (c)  be  l i m i t e d by  A finite  in  the  The  transferred  that  would  be  by  number  convection  allowed  to  Therefore,  conduction  numerical  natural  Nusselt  transferred  was  difference. to  to  technique.  by  in  is  the  for.the  a Nusselt The  ratio  in  of  the  the  same  Nu i s  transfer h where  Nusselt  coefficient 6  the  2  -  heat  3.2.2.1.  q  general  the  heat  heat  rate  system i f  number  of  unity  Nusselt  is  defined  number  defined  and h i s  the  local  no  is  equivas:  (3.9)  transfer  rate  is (3.10)  theoretical  Solution  of  solutions  Batchelor  solution  equations  heat  by:  6i  Previous  The  also  temperature  q = -kfae^ UyJ y = o  3.2.2.  of  (3.8)  the  =  also  and w i l l  a system to  conduction  occur  alone.  convection  is  Nu = hd•  where  layer  height  dimensionless parameter  problem of  rate  alent  a minimum l e n g t h  difference  Nusselt  calculated.  convection  boundary  cell.  The interest  method b a s e d on t h e  (33)  developed  described  in  the  by  Batchelor  previous  takes  section  the and  by  t h e u s e o f power s e r i e s , o b t a i n s a s o l u t i o n f o r t h e f l o w rates.  The s o l u t i o n u s e s n o n - d i m e n s i o n a l  governing  equations  forms o f t h e  by u s i n g t h e f o l l o w i n g  dimensionless  parameters: X = x , cT  Y = y_ d  k_ d±  "  u  _  pC^d 3 Y '  -k  "  V  3^  pC d 3X p  (3.11) " 1  6  6  1  R  a  =  9  s- , 2 - 1  7T-  9  f2 9  .  6  -  a  5  6  J - ~ - ^  l ] *  =  k p C  3 d  r  ,  >1  where  p  =  i s d e f i n e d as t h e s t r e a m f u n c t i o n  as t h e v o r t i c i t y .  Substituting  i n t o equations:(3.3), the p r e s s u r e iating  and £ i s d e f i n e d  t h e r e l a t i o n s i n (3.11)  (3.4), (3.5) and (3.6) and e l i m i n a t i n g  t e r m s i n t h e momentum e q u a t i o n  by d i f f e r e n t -  (3.3) w i t h r e s p e c t t o Y and (3.4) w i t h r e s p e c t t o  X resultsi n :  1  Pr  |3£ ^3X  d\l>  3<J> 3C  3Y  3X  3Y)  =  RaH-'+ "3Y  V C 2  (3.12)  99' 3X  3 ^ _ 39J_ 31 3Y 3X 3Y  A l l the time d e r i v a t i v e s  _  ;  ~  v  ,  2  0  u  state  *- H "  X . - I :  Y = l:  (3.14)  * = | f= 0 ,  9' = 0  * - | | - 0 ,,  9' = 0  The s e r i e s s o l u t i o n f o r 9' a n d ^ i s o b t a i n e d by expanding  these  t e r m s i n power s e r i e s o f t h e R a y l e i g h  number a s f o l l o w s :  6 (X,Y)  =  f  iJ/(X,Y)  =  (3.15)  Y + Ra9J(X,Y) + R a 8 ' ( X , Y ) + 2  2  (3.16)  Ra^ (X,Y) + Ra ^ (X,Y) + 2  1  S u b s t i t u t i n g equations  J  0  3X  '  S  form a r e :  iii- - o Y = 0 :  , :  solution i s ofinterest.  The b o u n d a r y c o n d i t i o n s i n d i m e n s i o n l e s s  3  i  ( 3 /3 t ) have b e e n e q u a t e d t o  zero s i n c e only the steady  X = 0  %  2  (3.15) a n d (3.16)  into  equations  (3.12) a n d (3.14) a n d e q u a t i n g l i k e p o w e r s o f R a y l e i g h t h e  equations  8' ,  describing  For values  8'  ,  o f Rayleigh l e s s than  i> , and $ 10  of close to unity the value o f  c a n be o b t a i n e d .  and f o r a 1 / d r a t i o c a n be g i v e n b y :  For t h e geometry under c o n s i d e r a t i o n e x p e r i m e n t a l l y , 1 / d = 1 Therefore  the s o l u t i o n  f o r t h e s t r e a m f u n c t i o n c a n be  written as:  i>  = . \ Ra X  The f l o w v e l o c i t i e s  2  (1 - X )  2  Y  2  may be o b t a i n e d  (1 - Y )  2  (3.18)  directly  from  equation  (3.18) w i t h t h e u s e o f t h e v e l o c i t y r e l a t i o n s h i p s i n e q u a t i o n (3..11).  The s t r e a m l i n e s a r o u n d t h e c e l l  a r e p l o t t e d i n F i g u r e 23a. field  forthis analysis  The shape o f t h e s t r e a m l i n e  i s independent o f R a y l e i g h and t h e r e f o r e t h e f u n c t i o n .  plotted  i s ^/..Ra x 10  .  The s t r e a m l i n e s r e p r e s e n t t h e  paths o f t h e p a r t i c l e s moving i n a l a m i n a r cell. in  F i g u r e 23b shows t h e shape o f t h e v e l o c i t y  profile  t h e X d i r e c t i o n a t a p o s i t i o n h a l f way down t h e c e l l  (X = 0.5).  Again  t h e shape o f t h e p r o f i l e  o f t h e R a y l e i g h number. since this and  fashion i n the  i s independent  The p a r a m e t e r u d i s p l o t t e d . aRa  i s only a f u n c t i o n o f the p o s i t i o n i n the c e l l  i s given by: ud aRa  =' X  2  (1 - X )  2  [ 2Y (1 - Y )  2  - 2Y  2  (1 - Y) ]  (3  (a)  F i g u r e 23.  The s o l u t i o n o f B a t c h e l o r f o r t h e p r o b l e m o f n a t u r a l c o n v e c t i o n showing (a) a n o r m a l i z e d s t r e a m f u n c t i o n ( 4>/ Ra ) p l o t and ( b ) a normalized v e l o c i t y i n the X d i r e c t i o n ( ud / aRa ) a t a p o s i t i o n o n e - h a l f way down the l i q u i d zone.  F i g u r e 23b o f t h e v e l o c i t y p r o f i l e velocity  i s almost  w a l l and t h e c e l l  h a l f way b e t w e e n t h e o u t e r  analysis w i l l  i n a following section.  c a n n o t be compared w i t h t h e e x p e r i m e n t a l tin  vertical  centre.  The r e s u l t s o f t h i s with other results  shows t h a t t h e maximum  s i n c e t h e minimum t e m p e r a t u r e  e x p e r i m e n t a l l y was 0.23°C.  be compared These  results  results f o r liquid  d i f f e r e n c e observed  This temperature  difference  f r o m t h e v a l u e o f P r a n d t l and G r / AT f r o m T a b l e I I 4 corresponds  t o a R a y l e i g h number o f 1.1 x 10  o r d e r o f magnitude g r e a t e r than  w h i c h i s an  t h e maximum v a l u e o f R a y l e i g h  the s o l u t i o n o f B a t c h e l o r i s v a l i d f o r . 3.2.2.2.  S o l u t i o n o f Emery a n d Chu (3*0 The- s o l u t i o n d e v e l o p e d  by Emery a n d Chu i s based,  on a b o u n d a r y l a y e r d e v e l o p m e n t a l o n g e a c h o f t h e v e r t i c a l > w a l l s as shown I n F i g u r e 24. the p r e v i o u s  The g e o m e t r y i s s i m i l a r t o  a n a l y s i s a n d i t i s now assumed t h a t t h e r e i s  a b o u n d a r y l a y e r o f t h i c k n e s s , 5, f o r t h e t h e r m a l a n d velocity profiles  a l o n g each c e l l  side.  The a n a l y s i s w i l l  o n l y a p p l y when t h e b o u n d a r y l a y e r s a r e r e l a t i v e l y t h e y do n o t meet a l o n g t h e c e n t r e o f t h e l i q u i d  t h i n and  cell.  The  l i q u i d between t h e boundary l a y e r s i s i s o t h e r m a l a t temperature B  m  and t h e - l i q u i d  velocity  i s zero i n t h i s  The g e n e r a l b o u n d a r y l a y e r e q u a t i o n s equations  region.  are developed  from  (3-3), (3.4), (3.5) and (3.6) by p u t t i n g a l l t h e  F i g u r e 25.  A n o r m a l i z e d v e l o c i t y [ y/6 (1 - y / 6 ) ] versus the p o s i t i o n w i t h i n the boundary l a y e r f r o m t h e s o l u t i o n o f Emery and Chu. 2  t i m e d e r i v a t i v e s e q u a l t o z e r o and a s s u m i n g v i s n e g l i g i b l e compared w i t h  u.  The t h e r m a l a n d v e l o c i t y b o u n d a r y  a r e assumed t o be e q u a l w h i c h i s v a l i d (22).  Integrating  3_ 3*  u  3_ 3x  u(6 - 9 J  2  for a liquid  dy  (9 - 9 ) dy - v  dy  -a  3u 3.V. w a l l  u  =  plate  (3.21)  shape i n t h e boundary  Equations  i n an i n f i n i t e  u ( x ) jr 1 a o  8 - 6  (  -  8 l  (3.20)  3_6 3.7 w a l l  l a y e r i s assumed t o be s i m i l a r t o t h o s e flat  metal  t h e boundary l a y e r e q u a t i o n s r e s u l t s i n :  The v e l o c i t y a n d t h e r m a l p r o f i l e  vertical  layers  e j  formed f o r a s i n g l e  fluid.  Therefore: (3.22)  I  Ii  -  (3.23)  y-  ( 3 . 2 2 ) a n d (3.'23) a r e s u b s t i t u t e d i n e q u a t i o n s  (3.20) and (3-21) a s s u m i n g u . (6, ct  exponential functions u (x) a 6  =  i  ~  Q  = C  e  co  2  C  o f x such x  1 n  - 8 ) a n d 6 t o be  -L  0 0  that:  n  (3.24)  xm  =  (3.25)  c  3  x  (3.26)  Y  For the boundary c o n d i t i o n s  under c o n s i d e r a t i o n ,  8^ i s a  c o n s t a n t such t h a t y results such  In values  = 0.  Equating l i k e  powers o f x  f o r G^, C , n and m b e i n g  calculated  2  that  240a  „  n  720a 3g3(9! - e j "  =  1/2 ,  Using the f l u i d  m  =  20  (3.27)  v +  1/4  properties  of liquid  t i n a t 260°C t a k e n  f r o m T a b l e I I a n d f o r t h e 6.4 cm. w i d e c e l l  this  results  in  a v e l o c i t y p r o f i l e one h a l f way up t h e c e l l o f :  u  =  o.7 ( e  where:  1  -  e ) 2  6 =  1  /  I [ l  2  0.24 (e»  -  .  e ) 2  " sec.  (3.28)  c m  cm.  (3.29)  1 / 2 j  For  e x a m p l e , w i t h a 1°C t e m p e r a t u r e d i f f e r e n c e  ary  layer  The  shape o f t h e v e l o c i t y p r o f i l e i s shown i n F i g u r e 2.5.  The  plot  i s 0.24 cm. t h i c k  i s o f J yi -f y l] -  differences  will  boundary l a y e r  2  at a point  versus y / 6  t h e bound-  h a l f way up t h e c e l l  since  a l l temperature  g i v e t h e same s h a p e d p r o f i l e i n t h e  according to this  analysis.  3.3-  Numerical  3.3.1.  analysis o f thermal  Technique o f s o l u t i o n The  section w i l l and  n u m e r i c a l a n a l y s i s t o be d e v e l o p e d  show t h e t h e r m a l p r o f i l e  by W i l k e s  (36).  The s o l u t i o n i s b a s e d on  described i n Figure 22.  The r e s u l t s  obtained  a r e n o t a p p l i c a b l e t o t h e p r e s e n t work s i n c e h i s  a n a l y s i s becomes u n s t a b l e a n d b r e a k s  down a t l a r g e v a l u e s  of  the Grashof  to  f l u i d s w i t h a P r a n d t l number o f 0 . 7 3 3 .  number, a n d h i s r e s u l t s w e r e c o n f i n e d  f o r a system  with a v e r t i c a l l y  mainly  The r e s u l t s o f  S a m u e l s and C h u r c h i l l do n o t a p p l y s i n c e t h e i r applied  solution  temperature  d i f f e r e n c e and n o t a h o r i z o n t a l d i f f e r e n c e present  (35)  difference analysis of natural convection i n the  c l o s e d system  is  metals.  i s b a s e d on t h e r e c e n t w o r k s o f W i l k e s  Samuels and C h u r c h i l l  a finite  i n this  and t h e flow p a t t e r n  flow rates f o r thermal convection i n l i q u i d  This analysis and  convection  as i n t h e ;  work. A ^ c o n d e n s e d v e r s i o n o f t h e a n a l y s i s w i l l be  presented  i n -this s e c t i o n so t h a t an u n d e r s t a n d i n g  of the  t e c h n i q u e s may be o b t a i n e d w i t h o u t t h e n e e d t o p r e s e n t all  the equations  required i n the solution.  used i n the computer program a r e a l l p r e s e n t e d :  Equations into a dimensionless  (3-3),  (3.4),  (3.5),  The e q u a t i o n s i n Appendix I I .  and (3.6)  are put  f o r m by t h e u s e o f t h e f o l l o w i n g  dimensionless  X  =  u  =  T  =  parameters.  Y  =  ud  9  AS'  V  -  6  2  0  -  e  =  9  9  and  as  T  V  d" 2  -  P '  D  (3.30)  2  0  e  Q  ,  "  L  J  g3A9'd  In .the s o l u t i o n o f B a t c h e l o r a s t r e a m  a vorticity  function  modified  Grashof  half  temperature  the  Substituting governing results  "  vd  =  Gr'  Also,  -  T  P  -  2  T  d >  the  i s used  introduced.  difference  a c r o s s the  slightly  equations:  on  one  cell.  o f (3-30) i n t o  again e l i m i n a t i n g  following  A  i n t h i s c a l c u l a t i o n based  relationships  equations,  i n the  are  function  the  the p r e s s u r e  terms,  Vorticity  equation:  »S* Energy  V 1/^ 2  =  These  . "  (3.31)  V c  +  -C  (3.33)  function equation:  Velocity  u  ' f t  equation:  Stream  U  +  equation:  = M  v  9Y '  equations  dimensionless  *  v  (3.34)  3X  a r e t o be s o l v e d w i t h  the f o l l o w i n g  boundary c o n d i t i o n s :  1=0:  0 < X  <  L  T = 0,  c = 0  •0 < Y < 1 (3.35) T >  0:  X  =  0, X = L ; 3T  ax  Y = 0;  T > 0 :  ^ = ^.=  0,  T =- l ( 3 . 3 5 )  jji = | | = 0 ,  Y = 1;  The  finite  difference  T =  +1  n u m e r i c a l technique used  f o r t h e s o l u t i o n o f t h e above e q u a t i o n s I s c a l l e d t h e implicit the is  alternating d i r e c t i o n technique.  I n t h i s method,  time step, A T , over which the i t e r a t i o n i s c a r r i e d o u t , split  i n t o two p a r t s o f d u r a t i o n A T / 2 .  For the f i r s t  h a l f time step, a l l the X d i r e c t i o n d e r i v a t i v e s and a l l t h e Y d i r e c t i o n d e r i v a t i v e s  are i m p l i c i t  are e x p l i c i t .  For the  second h a l f time step the X d i r e c t i o n d e r i v a t i v e s a r e explicit  and t h e Y d i r e c t i o n d e r i v a t i v e s  Explicit  derivatives  that  the derivatives  parameters  are i m p l i c i t .  f o r any t i m e s t e p f r o m T I t o T2 mean are c a l c u l a t e d with the values of the  at the time equal t o T I .  Implicit  derivatives  f o r t h i s t i m e s t e p means t h e v a l u e s o f t h e p a r a m e t e r s a r e taken at the time equal t o T . 2  technique i n which parameters  t h e time i s p r o g r e s s e d i n s t e p s and t h e  are calculated  from t h e i r d e r i v a t i v e s  new t i m e s t e p , t h e e x p l i c i t the  implicit  derivatives  The i n F i g u r e 26.  T h u s , f o r an i t e r a t i v e  derivatives  a r e known, w h e r e a s  a r e unknown.  g r i d system used There  f o r each  a r e m-1  i n this analysis  i s shown  g r i d d i v i s i o n s o f l e n g t h AX  _^ Y  n-l  I  n  2 3  I  Ax  m Figure 26.  -I  The f i n i t e fluid cell  i n the X d i r e c t i o n the  Y direction.  AY  k  d i f f e r e n c e g r i d system f o r the used i n the n u m e r i c a l a n a l y s i s ,  and n - l g r i d d i v i s i o n s The  subscripts  o f l e n g t h AY i n  i and j d e n o t e t h e  in  t h e g r i d h a v i n g t h e c o o r d i n a t e s X = i A X and Y =  In  the f o l l o w i n g  analysis  the  star superscript  (*)  position JAY. denotes  v a l u e s o f p a r a m e t e r s a f t e r t h e end o f t h e f i r s t h a l f s t e p and t h e d a s h s u p e r s c r i p t  (')  a f t e r t h e end o f t h e s e c o n d h a l f finite  difference  eauations are:  time  denotes values of parameters or t o t a l time step.  e q u a t i o n s f o r t h e e n e r g y and  The  vorticity  Vorticity  equation  for the f i r s t  LJui_-ILJL +•u AT/2 1,3  V  e  * 1+1,3  rp i  Vorticity r i  ^ 1,3  "  2  ;"*l,3 TAXT2  *1+1,J  _ £*  u  i , j  G r  * 1-1,3  ^  rp  t  1,3-1  *1.3*  C  CAY)*  _  i,3+l  . vi.io;  n  6  )  step  £ *  2AX  _  _  2  ^ 1-1,3 . +  ,  1,3+1  2AY  -L,J  step:  2AY  time  ^ 1+1,3  ^ 1,3-1  1,3 + 1  ,  r *  ^ 1,3 . „  ^ 1,3+1  ,  S  Gr'  f o r the second h a l f  AT/2  V,  +  =  time  * 1-1,3 ,  2AX  £ _ f M J +1 H . j - l 1,3 2AX  *i-l,3  half  1,3-1 .  2AY  1,3  * 1+1,3  TAXP  * i J-l  +  ^ l.J  ?  +  TAYTI  * 1,3+1 (3.37)  Energy e q u a t i o n rp* 1  1,3  _  AT/2  L_ Pr  rn* 1  rp  f o r the f i r s t  1,3  1-1.3  +  rp#  + TT i,3 U  •_• ? T *  1.3 (AX) 2  +  half -  1+1,3 2AX  T *  T *  time  step:  +  T  T  .  1-1,3  x  v  V  i,3  1+1.3 . 1 _ ^ ^ - l Pr  -  i,3+l 2 Y  -  T  i,3-l  _  A  2T 1,3 (AY)*  +  T  1,3+1 (3.38)  Energy  equation  for  AT/2  -  1  T  Pr  u  'i-1..1  •  2  T  the  second  i , j  half  time  2AX  ' l , J  t  T  (AX)  +  V l . . 1  , 1  V  i , j  2AY  •  ^,.1-1  Pr  +  step  "  j  /  ^  TAYP  (3.39) The  velocities  calculated  at  by  points  equations  "3V i , j  U  "  -  13Y  not of  adjacent  the  ^i>J-2  -  the  walls  are  form: 8  *i,.1-l  "  i.J  to  1  2  * A  8  -  » l „ m  *i,j+2  Y  (3.40) A slightly adjacent beside  different  to  the  the cold  converted  applied.  Churchill.  r  n+l  ?  where  +  V  solve  for  the  implicit  This  is  Equation  is  the  the  for  the  points  points  just  to  (3.41)  stream function state  to  direction  method u s e d by then  equation  an u n s t e a d y  alternating  (3.33)  9i|> 3?-  =  refers  integration  for  used  ^  from a steady  r^.n+l * ••  n+1  as  is  wall:  p r o b l e m and the is  such  5AH" To  is  boundary,  =  1,2  approximation  (3.33)  state technique  Samuels  and  becomes:  n+1  the  continued  (3.42) n+1  iteration  until  the  in  T'.  The  stream function  numerical is  H  negligible. half  time  The  finite  difference  forms  are  for  the  first  step  AT'/2  ( A X )  R  (3.43)  (Afr and f o r  the  second h a l f  time  steo  AT'/2  (AX)  2  (3.44)  The grid  points  wall,  for  since  example,  minate  since  of  form  the  equations  they  if  parameters are  outside  the  form  such the  is  apply  to  applied to  interior the  ^ are  indeter-  cell.  Thus,  equations  (3.45)  2  a higher  ^ i , 2 ~  [  cold  as  % 2  Wilkes uses  •i,l  (3.36)  only  " rAYT "  '1,1 used.  vorticity  equation  2  are  of  -  2(AY)  order  »l Z  l 3  approximation  of  (3.46)  but  s e r i o u s problems a r i s e w i t h t h e use o f i t f o r l a r g e  G r a s h o f numbers.  This  i s due t o t h e r a p i d i n c r e a s e i n  stream f u n c t i o n values i c i t y with equation  causing  a s i g n change i n t h e v o r t -  (3.46).  I t i s advantageous t o a l s o s o l v e f o r the l o c a l N u s s e l t number.  Non-dimensionalizing  equation  (3.10)  q  gives:  (3.47)  S u b s t i t u t i n g equation  (3.9) and (3-47) i n t o e q u a t i o n .(3.8)  results i n (3.48)  which i s r e a d i l y Is  solved  f o r a f t e r the temperature  profile  known. The  procedure f o r the s o l u t i o n o f the o v e r a l l  p r o b l e m c a n be s u m m a r i z e d as f o l l o w s : (a) time  The new t e m p e r a t u r e s a r e computed f o r t h e n+1  step. (b)  The new i n t e r i o r v o r t i c i t i e s  t h e n+1 t i m e s t e p (c) is  f r o m t h e new t e m p e r a t u r e s .  The new s t r e a m f u n c t i o n f o r t h e n+1 t i m e  c a l c u l a t e d f r o m t h e new i n t e r i o r :  (d) time  step.  a r e computed f o r  The new v e l o c i t i e s  step  vorticities.  a r e computed f o r t h e n+1  (e) the  n+1  The  time  (f) Nusselt The  new b o u n d a r y  new  local  number  are  calculated  solution  procedure is  vorticity  the  equations For in  is  form:  a  2  S  l  or  from the until  actually  described  and t h e new  for  average  temperatures.  a steady  solving  for  row the  fully  of of  state  for  for  in  writing  the  the  the  of  temper-  individual  appendix.  an e q u a t i o n  immediate i n t e r e s t  various  column a set  parameters,  £,  are  :  for  from  T,  n-l equations  The each the  a n d ty.  obtained  :  "  c  i 2  *  b  2 2  +  of  consists  derived  b^s-^  number  repeated  column or  each row the  Nusselt  and s t r e a m f u n c t i o n s  method e s s e n t i a l l y in  is  method  and columns  point  computed  obtained.  The  rows  are  step.  The  above  ature,  vorticities  s  =  s  +  C  2 3 S  + b^s^  a^S2  =  + c^s^  d  d  i  2  = d^ (3.49)  a  n-2 n-3 s  +  b  n-2 n-2 s  a  for  which  values  of  equations  n-l n-2  a solution the  a,  s will  program used  in  +  b,  s  for  +  s  b  s  =  n-l n^l  =  is  s  available  T,  analysis  or is  shown  n-2  d  d  n - l  knowing  c and d c o e f f i c i e n t s .  represent this  n-2 n-l  C  For  The in  all  the  each set  computer  Appendix  III.  of  TABLE I I I . C o m p u t e r Runs C o n d u c t e d  Run Number  Prandtl Number  Grashof Number  Grid Size  Gr'  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  0 .0127 0.0127 0.0127 0 .0127 0.0127 0.01365 0.01365 0.01135 0.01135 0.733 0.1 0.1 -. 0.1 0.1 1.0 1.0 1.0 10.0 10.0 10.0 :  lojj* 1 0 5 1 0  1 0 1 0  l  7 6  1 0 6 0.865 x 1 0 ° 1 0 6 1.247 x 1 0 ° 1 0 3  5 6 10° 1 0 1 0  ioi 1 0  3  10 i 1 0  5  10°  11 11 11 21 31 21 21 21 21 11 21 21 21 21 21 21 21 21 21 21  x x x x x x x x x x x x x x x x x x x x  11 11 11 21 31 21 21 21 21 11 21 21 21 21 21 21 21 21 21 21  Temperature Average Difference Temperature  0 .000555 0.00555 0.0555 0.555 5.55 0.642 0.555 0.446 0.555  — _  —  _ _  — — — _ —  260 260 260 260 260 237 237 305 305  —  _ _ _ _ _ _ _ _ _  The p r o g r a m  I s w r i t t e n i n F o r t r a n I V l a n g u a g e a n d was r u n  on a n IBM/360 m o d e l 67 c o m p u t e r Columbia.  at the University  Twenty r u n s o f t h e p r o g r a m  of British  were c o n d u c t e d f o r  v a r i o u s v a l u e s o f t h e G r a s h o f a n d P r a n d t l numbers. various r u n parameters for to  are l i s t e d  i n Table I I I .  The r u n s  P r a n d t l e q u a l t o 0 . 0 1 2 7 , 0.01365 a n d 0.01135 l i q u i d t i n at the three d i f f e r e n t  results  correspond  s u p e r h e a t s and t h e  corresponding temperature d i f f e r e n c e s are also e a c h G r a s h o f number.  The  listed for  The p r i n t o u t o f t h e c o m p u t a t i o n  c o n s i s t s o f f i v e m by n m a t r i c e s f o r e a c h o f T,  if), U, V and E, a n d a l s o t h e l o c a l N u s s e l t number f o r e a c h g r i d p o i n t a l o n g the c o l d w a l l p l u s t h e average N u s s e l t number.  3.3.2.  Results of numerical analysis The  t e m p e r a t u r e m a t r i x and t h e s t r e a m  m a t r i x from t h e computer  p r i n t o u t were c o n v e r t e d i n t o  i s o t h e r m a l p l o t s and s t r e a m l i n e p l o t s pretation.  dependent and  for visual  The t h e r m a l r e s u l t s w i l l : b e p r e s e n t e d The  function  thermal p r o f i l e  interfirst.  r e s u l t s were f o u n d t o be  on t h e R a y l e i g h number o n l y , and n o t on t h e G r a s h o f  P r a n d t l numbers i n d e p e n d e n t l y .  The n o n - d i m e n s i o n a l  i s o t h e r m a l p l o t s a r e shown i n F i g u r e 27a - 27f f o r R a y l e i g h numbers o f 2 x 10,  2 x 10 , 2  2 x 10^ r e s p e c t i v e l y . within the l i q u i d  cell.  2 x 10 ,  Each p l o t  3  2 x 10^, 2 x 1 0  5  and .  c o n s i s t s o f seven isotherms  The v e r t i c a l  left  side i s the  -0.75  -OA  -025  0.0  0.25  OS  0.75  (o)  0.75  05 025 0.0 -0.25 -0.i  -0.75  (b) Figure  27.  The t h e o r e t i c a l p l o t s of t h e n o n d i m e n s i o n a l temperature T f o r a R a y l e i g h number o f ( a ) 2 x 1 0 , (b) 2 x 1 0 . 2  (d)  F i g u r e 27  c o n t i n u e d . The t h e o r e t i c a l p l o t s o f t h e n o n d i m e n s i o n a l t e m p e r a t u r e , T, f o r a R a y l e i g h number o f ( c ) 2 x 10 , (d) 2 x 10*. 3  F i g u r e 27  c o n t i n u e d . The t h e o r e t i c a l p l o t s o f t h e n o n d i m e n s i o n a l t e m p e r a t u r e , T, f o r a R a y l e i g h number o f ( e ) 2 x 10 , ( f ) 2 x 10 . 5  6  Equivalent  Rayleigh  and G r a s h o f Numbers  Corresponding t o Figure  Rayleigh Number  2 x 10  2 x 10  2 x 10  2  3  I) 2 x 1(T  2 x 10  5  2 x 10  6  27  Profile in F i g u r e 27  Prandtl Number  27a  0.01 0.1 1.0 10.0  2 2 2 2  0 .01 0.1 • 1.0 10.0  2 2 2 2  0.01 0.1 1.0 10.0  2 2 2 2  0.01 0.1 1.0 10.0  2 2 2 2  27e  0.01 0.1 1.0 10.0  2 X 6 2 X 10° 2 X 10? 2 X lo"  27f  0 .01 0.1 1.0 10.0  2 2 2 2  27b  27c  27d  Grashof Number  X X X  10 10 10  2  10 10 10  2  3  X X X X X X X X X X X X  1 0  3  l  10 ' 10^ 10 2  1 0  5  10? 10* 10 3  1 0  X X X X  10? 6 1 0  10  5  -  1.0  Isotherm  isotherm.  1.0  and t h e v e r t i c a l r i g h t s i d e i s t h e +  Table  IV t a b u l a t e s t h e e q u i v a l e n t R a y l e i g h and  G r a s h o f numbers f o r t h e v a r i o u s P r a n d t l numbers and g i v e s the  corresponding  plot  i n F i g u r e 27.  The p r o g r e s s i v e  increase i n the bending of the isotherms R a y l e i g h number i n c r e a s e s .  i s obvious  The a v e r a g e N u s s e l t  p l o t t e d a g a i n s t t h e R a y l e i g h number i n F i g u r e plot  shows t h a t up t o a v a l u e  with convection p l a y i n g a very  a l l by  small role.  s e e n i n F i g u r e 27 w h e r e b e l o w a R a y l e i g h isotherms  are v e r t i c a l  number i s  28.  This  o f the Rayleigh o f  10 , t h e h e a t t r a n s f e r i s e s s e n t i a l l y  approximately  conduction, This  i s also  o f a b o u t 10  l i n e s as i n c o n d u c t i o n  For comparison of the various types  as t h e  heat  the  transfer.  of f l u i d s , Figure  29  shows t h e a v e r a g e N u s s e l t number p l o t t e d a g a i n s t t h e G r a s h o f number f o r t h e v a r i o u s P r a n d t l n u m b e r s . lower  I t i s seen t h a t the  t h e P r a n d t l number, t h e h i g h e r t h e G r a s h o f number  r e q u i r e d t o cause c o n v e c t i v e  heat t r a n s f e r .  The change  i n t h e l o c a l N u s s e l t number w i t h c h a n g i n g R a y l e i g h is  seen i n F i g u r e  is  fairly  constant  30.  number  F o r l o w R a y l e i g h numbers t h e n u s s e l t  down t h e l e n g t h o f t h e w a l l , w h i l e a t  l a r g e R a y l e i g h numbers a p e a k o c c u r s  i n the l o c a l  Nusselt  number t o w a r d s t h e t o p o f t h e c e l l . The s t r e a m l i n e p l o t s c o r r e s p o n d i n g t i n a t 260°C- a r e shown i n F i g u r e t o G r a s h o f numbers o f 2 x 1 0 , 3  31.  to liquid  The p l o t s  2 x lO**, 2 x 1 0 , 5  correspond 2 x 10  6  7 and  2 x 10  and a r e drawn w i t h t h e p u r p o s e o f s h o w i n g t h e  -o  o I  —J 2 x I0  1  I  L_  2xl0  2  2xl0  RAYLEIGH  3  2xl0  I 4  2xl0  NUMBER  F i g u r e 28. A p l o t o f t h e t h e o r e t i c a l a v e r a g e N u s s e l t number v e r s u s t h e R a y l e i g h number.  I 5  2 x I0  6  2 x I0  3  2xl0  4  GRASHOF F i g u r e 29.  2xl0  5  2xl0  6  2x10  NUMBER  A p l o t o f t h e t h e o r e t i c a l a v e r a g e N u s s e l t number v e r s u s t h e G r a s h o f number f o r P r a n d t l numbers o f 1 0 . 0 , 1.0, 0.1, a n d 0.0127.  ro  0 I 0  1  0.2  I  I  I  I  0.4  0.6  0.8  1.0  POSITION,  F i g u r e 30.  X  A p l o t o f t h e l o c a l N u s s e l t number v e r s u s t h e p o s i t i o n a l o n g t h e c o l d end w a l l , X, f o r R a y l e i g h numbers o f 2 x 1 0 , 2 x 10 , 2 x 10 , 2 x 1 0 , and 2 x 1 0 . 2  k  5  6  (b) Figure  31.  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r l i q u i d t i n a t 260°C ( P r a n d t l = 0.0127) w i t h a G r a s h o f number o f ( a ) 2 x 1 0 , ( b ) 2 x 1 0 . 3  14  Figure  31 c o n t i n u e d . The t h e o r e t i c a l s t r e a m f u n c t i o n f o r l i q u i d t i n a t 260°C ( P r = 0.0127) w i t h a G r a s h o f number o f ( e ) 2 x 1 0 . 7  Figure  32.  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r a P r a n d t l number o f 0.1 w i t h a G r a s h o f number o f ( a ) 2 x 10" and ( b ) 2 x 1 0 . 5  Figure  33-  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r a P r a n d t l number o f 1.0 w i t h a G r a s h o f number o f ( a ) 2 x 10 and ( b ) 2 x 10 . k  s  Figure  34.  The t h e o r e t i c a l s t r e a m f u n c t i o n f o r a P r a n d t l number o f 10.0 w i t h a G r a s h o f number o f ( a ) 2 x 10* and ( b ) 2 x 1 0 . s  shape o f the between the not  f l o w i n the  32,  velocity  33  34  and  to 0.1,  number e q u a l  show t h e 1.0  2 x 10  and  .  through  by  c h a n g i n g t h e P r a n d t l number.  shows t h e  X = 0.5, 35  Figure  s h o u l d be very is  are  noted  Figure  35  flow  shape  down  Each o f the p l o t s  P r a n d t l number and i t  t h a t t h e v e r t i c a l s c a l e on e a c h p l o t i s The  s m a l l e r t h e P r a n d t l number, t h e flow v e l o c i t i e s  f o r a given  cell  o f f l u i d s behave i n the As  i s given  same  the  profile  wall.  A scale of a c t u a l v e l o c i t i e s on t h e p l o t s i n F i g u r e  greater  Grashof  as t h e G r a s h o f number i s i n c r e a s e d .  towards the  velocity  pattern  shows t h e  G r a s h o f number i n c r e a s e s t h e p e a k i n t h e v e l o c i t y shifts  type.  Figures  a t a p o s i t i o n h a l f way  A l l the v a r i o u s types  g e n e r a l way  A comparison of  i n the X d i r e c t i o n .  dimensionless  number.  does  r e s p e c t i v e l y f o r a Grashof  for a different  different.  the  spacing  change i n shape o f t h e  o f t h e v e l o c i t y p r o f i l e , U,  in  the  s t r e a m l i n e s f o r the P r a n d t l  10.0  and  31  cell,  and  increments  5  number o f 2 x 10 34  equal  the  as i s n o r m a l i n p l o t s o f t h i s  4  the  Therefore,  s t r e a m l i n e s are not  represent  Figures  cell.  35.  i s also included  From e q u a t i o n  (3.30)  the a c t u a l  by:  (3.30)  u = Uv •' d  If  a value  of:the kinematic  v i s c o s i t y , v , and  d, i s c h o s e n t h e n  the d i m e n s i o n a l  computed.  o f the k i n e m a t i c  Values  cell  v e l o c i t y , u, may viscosity  size, be  f o r each  value  0  O.l  0.2  0.3  0.4  0.5  DISTANCE,Y Figure  35a.  The t h e o r e t i c a l f l o w v e l o c i t y ( u and U) a t a p o s i t i o n X = 0.5 f o r v a r i o u s v a l u e s o f t h e G r a s h o f number f o r l i q u i d t i n a t 260°C ( P r = 0.0127).  0.6-  0.5  - I  ' \  - 500  \ \ Gr - 2 x 10 \  - 400  0.4 (J UJ CO  5  0.3  - 300  'I  3 200  r-  O O _J  UJ >  100  -50 0.3  0.2 DISTANCE, Figure  35b.  0.4  0.5  Y  T h e t h e o r e t i c a l f l o w v e l o c i t y ( u a n d U) at a p o s i t i o n X = 0 . 5 f o r v a r i o u s v a l u e s o f t h e G r a s h o f number f o r P r = 0 . 1 and assuming NaCl p r o p e r t i e s to c a l c u l a t e u.  0.1  0.2  0.3  0.4  05  DISTANCE, Y Figure  35c  The t h e o r e t i c a l f l o w v e l o c i t y ( u and U) a t a p o s i t i o n X = 0.5 f o r v a r i o u s v a l u e s Of t h e G r a s h o f number f o r P r =1.0 and a s s u m i n g NH^ p r o p e r t i e s t o c a l c u l a t e u.  :  Figure  35d.  The t h e o r e t i c a l f l o w v e l o c i t y ( u and U) a t a p o s i t i o n X = 0.5 f o r v a r i o u s v a l u e s o f t h e G r a s h o f number f o r P r = 10.0 and a s s u m i n g w a t e r p r o p e r t i e s t o c a l c u l a t e u.  o f t h e P r a n d t l number u s e d I n t h e c o m p u t a t i o n c a n be from Table  I f o r the various types  of d equal  to 6.4 centimeters  size  forvelocity  of fluids.  estimated  Also a  value  has been chosen f o r t h e c e l l  comparisons.  A d i s c u s s i o n of the various behaviors  i n flow  for  t h e v a r i o u s P r a n d t l numbers, c a n be shown b e s t by c o m p a r i n g  two  particular  fluids.  A c o m p a r i s o n o f P r a n d t l numbers o f  0 . 0 1 2 7 a n d 10 w i l l be done t o i l l u s t r a t e thermal  convection differences.  0 . 0 1 2 7 can represent  liquid  10 c a n r e p r e s e n t w a t e r .  Comparing F i g u r e  The P r a n d t l number o f  t i n , a n d t h e P r a n d t l number o f  As was p o i n t e d o u t e a r l i e r ,  R a y l e i g h numbers p r o d u c e e q u a l melt.  the Important  thermal  conditions i n the  31e f o r l i q u i d  t i n ( P r = O.OI27)  w i t h R a y l e i g h e q u a l t o 2 . 5 x 10^ and F i g u r e (Pr  = 10.0) w i t h R a y l e i g h e q u a l  flow patterns are not s i m i l a r . still  t o 2 x 10  5  The l i q u i d  quite c i r c u l a r i n nature, especially  regions of the c e l l . i n nature  equal  34a f o r water we s e e t h a t t h e metal  flow i s  i n the c e n t r a l  The w a t e r f l o w i s q u i t e r e c t a n g u l a r  a n d a p p e a r s t o c o n f o r m more t o t h e s h a p e o f t h e  square c e l l  than  the l i q u i d  metal  flow.  The a c t u a l  v e l o c i t i e s o f t h e f l o w s c a n be s e e n by c o m p a r i n g  Figures 5  3 5 a and 3 5 d . liquid  metal  F o r a R a y l e i g h number o f n e a r 2 x 10 f l o w has a maximum v e l o c i t y  at the X = 0 . 5  p o s i t i o n o f a b o u t 1 . 4 cm. / s e c . w h i l e a t t h e same t h e maximum f l o w i n t h e w a t e r c e l l t h e r e i s a f a c t o r o f two o r d e r s  the  position  i s 0.02 cm. / s e c . Thus  o f magnitude d i f f e r e n c e  i n t h e f l o w r a t e s f o r t h e two  m a t e r i a l s w i t h an e q u a l  thermal  profile. The water  and  between the l i q u i d  c a n be made i n a n o t h e r way,  profiles of  comparisons  2.5  f o r equal flow rates. x 10  at a Grashof  r a t e i s a b o u t 0.02  liquid  occurs  cm.  x 10 ,  is still  Nu = 1.0)  2  f o r the water  accomplished  various figures. relates  thermal  t i n at a  Grashof  cm.  are s i m i l a r . two  i n a conductive  /  (Ra = 2 x 10  f l o w s , the nature bending  , Nu  5.7).  =  of the o t h e r v a r i o u s types of comparing  c a n be  simply  i t  summarized.  U s i n g t h e maximum v a l u e o f U a t a p o s i t i o n o f X = 0 . 5 ' comparing  f l o w r a t e s , and  fluids  the  g e n e r a l f l o w b e h a v i o u r and how  t o the v a r i o u s parameters  sec.  However,  while a great deal of  i n t h e same manner by  The  and  t h e maximum f l o w  f o r these  isotherms  A comparison c a n be  o f 2 x 10  / s e c , which  the thermal p r o f i l e s  t i n profile  (Ra = 2.5  For l i q u i d  the  t h e maximum f l o w r a t e i s a b o u t 0.03  f o r the water  comparing  comparing  metal  for  u s i n g t h e N u s s e l t number t o  compare t h e d e g r e e o f i s o t h e r m w a r p i n g  T a b l e V shows t h e  behaviour i n thermal convection. Several important of  comments c a n be made i n t e r m s  the d i f f e r e n c e s between l i q u i d  types of f l u i d the l i q u i d  flow.  metal  f l o w and  other  For equivalent thermal p r o f i l e s  (Nu = c o n s t a n t ) t h e f l o w v e l o c i t i e s  in  liquid  metals  a r e much h i g h e r t h a n i n o t h e r t y p e s o f f l u i d s .  liquid  metals, large  f l o w r a t e s may  be d e v e l o p e d  in  in a  For melt  The e f f e c t on t h e N a t u r a l  Convection  of A l t e r i n g Dimensionless  Parameters  Alteration  A f f e c t on N u s s e l t Number  (U Gr Pr  A f f e c t on Plow Rate  a t X=0.5)  max Decreasing  Constant Increasing  Increasing  Gr - - I n c r e a s i n g P r •*• C o n s t a n t  Increasing  Increasing  Ra •*• I n c r e a s i n g  Increasing  Unknown  while the thermal  profile  conduction  F o r e x a m p l e , w i t h an a v e r a g e  form.  number o f o n l y . 1 . 2 5 n e a r l y 0.2 in  t h e 6.4  is still  (Gr = 2 x 10 ) 5  u n a l t e r e d from the pure Nusselt  flow v e l o c i t i e s o f  cm. / s e c . w e r e c a l c u l a t e d f o r t h e l i q u i d t i n cm. w i d e c e l l .  For equal  flow r a t e s , the l i q u i d  m e t a l f l o w i s l e s s d e p e n d e n t on t h e s h a p e o f t h e c e l l enclosure  as was s e e n by t h e l i q u i d  flows i n t i n  remaining  c i r c u l a r w h i l e other l i q u i d s approach a r e c t a n g u l a r shaped flow c o r r e s p o n d i n g  t o the r e c t a n g u l a r  cell.  Phenomena o f m e t a l l u r g i c a l I n t e r e s t t h a t a r e a f f e c t e d by c o n v e c t i v e in  mentioned  t h e i n t r o d u c t i o n t o t h i s t h e s i s , a r e d e p e n d e n t on  the l i q u i d thermal the  f l o w i n t h e m e l t , as was  both  f l o w r a t e s and f l o w p a t t e r n a n d a l s o t h e  profile  i n the l i q u i d pool.  flow rates;and  effect that occurs.  The n a t u r e  o f both  h e a t f l o w combine t o g i v e t h e r e s u l t a n t I f one o f t h e c o n t r o l l i n g e f f e c t s i s  a l t e r e d , e i t h e r flow rate or thermal  c o n d i t i o n s , the  resulting effect w i l l  be d i f f e r e n t .  By u s i n g f l u i d s  than l i q u i d metals  t o observe  effect i s to alter  the r e l a t i v e nature  thermal  c o n d i t i o n s such  m e t a l l u r g i c a l phenomena, t h e o f t h e f l o w and  t h a t the system w i l l  parable to a l i q u i d metal.  other  n o t be com-  The d i f f e r e n c e s p r o d u c e d a r e  unknown, and h e n c e t h e r e s u l t s are at best only q u a l i t a t i v e  from n o n - m e t a l l i c  analogs  indicators of metallurgical  effects.  3.4.  C o m p a r i s o n o f t h e o r e t i c a l and e x p e r i m e n t a l  3.4.1.  Thermal  results  profiles  The e x p e r i m e n t a l p r o f i l e s  of liquid  t i n .in F i g u r e  15 c a n be compared t o t h e p r o f i l e s o b t a i n e d i n t h e n u m e r i c a l s o l u t i o n i n F i g u r e 27. number n e a r 2 x 10 Although  to  cell  the p r o f i l e  t h e computer s o l u t i o n .  27e.  f i g u r e s are not i n the non-  form t h e shapes w i l l  liquid  a n d 15e w i t h a R a y l e i g h  may be compared w i t h F i g u r e  the experimental  dimensional thick  F i g u r e 15b  be c o m p a r a b l e .  For the  i s very s i m i l a r i n nature  The t h i n  liquid  h o w e v e r , i s i n d i c a t i v e o f a much l o w e r  cell  profile,  R a y l e i g h number.  The good a g r e e m e n t b e t w e e n t h e t h e o r y a n d t h e t h i c k e r liquid  cell  mental c e l l convection.  i n d i c a t e s the large f l a t  faces of the e x p e r i -  a r e n o t c a u s i n g a l a r g e change i n t h e t h e r m a l :  The d i f f e r e n c e s i n t h e t h i n  liquid  cell  c a n be  caused e i t h e r . b y the v i s c o u s drag o f the l a r g e w a l l s o r the thermal  conduction  of the side w a l l s .  The f l o w c o u l d a l s o  be  i n f l u e n c e d by  t h e 0.5  by a s l o w i n g o f t h e to  decrease the  mm.  flow.  f l o w and  diameter  thermocouple probe  These e f f e c t s would a l l t e n d hence the  thermal  profile  A r i g o r o u s comparison o f the t h e r m a l has  n o t b e e n a t t e m p t e d due  of the experimental large  to the  techniques  flow.  the r e s u l t s  favourably.  s e n s i t i v e thermal  F l o w p a t t e r n s and  effects  seem t o compare much more  flow rates  f l o w p a t t e r n s t h a t were o b s e r v e d  i n F i g u r e s 9 and  10 a l l h a v e t h e one  f o r pure  flow c e l l  T h i s i s i n agreement w i t h the a n a l y t i c a l t r e a t m e n t which a l s o generates nature.  The  theoretical the time liquid for  a one  experimental results  f o r the  be  I n a q u a n t i t a t i v e way  various c e l l  curves  full  pattern. presented similar  compared w i t h by  comparing  c y c l e arouhd  velocity  For the  c o m p a r i s o n t o be made, t h e  converted  T h i s i s done by  a l o n g the  the  a r e shown i n F i g u r e  to a time per  cycle  e a c h v a l u e o f t h e G r a s h o f number f o r t h e P r a n d t l 0.0127.  the  t h i c k n e s s e s w i t h p u r e t i n a t 260°C  c o m p u t e r r e s u l t s must be  to  flow pattern of a  results w i l l  experimental  average temperature.  for  cell  f l o w t o c o m p l e t e one  c e l l . • The  the  con-  •  The tin  relatively  I n t h e t h i c k e r c e l l , where t h e s e  are l e s s i m p o r t a n t ,  3.4.2  profiles  limitations  of p u t t i n g a  f o r e i g n body i n t o t h e v e r y  vective  inherent  bending.  equal  a graphical i n t e g r a t i o n of  s t r e a m l i n e s shown i n F i g u r e  31.  A  the  14  number o f i n c r e m e n t s a r e t a k e n a r o u n d knowing the v e l o c i t y printouts  the t o t a l  i n each time  increment  solution  time p e r c y c l e  The Figure  o f magnitude  numbers o f i n t e r e s t . thickness taken  by t h e e x p e r i m e n t a l  on a l o g - l o g  changes  flow rates  from F i g u r e  v e r y poor.-  a r e somewhat  14.  sidewall low  restrictions  flow rates  plates  I t i s evident  the b e t t e r  temperature thick  difference  cell  and G r a s h o f  that  cell  t h e agreement  and t h e o r y a t low G r a s h o f numbers for this  poor  agreement t o the  i n the experimental c e l l .  and t h e h e a t  gradients  At t h e  the hydro-  flow through the side on t h e f l o w .  and t h e o r y i s q u i t e  differences.  Also,  The agreement  good f o r  the t h i c k e r the  t h e agreement becomes a t a p a r t i c u l a r  difference.  liquid  o f the orders  but are probably r e l a t e d  between t h e e x p e r i m e n t s  cell,  a r e shown i n  shown on t h e c u r v e w h i c h a r e  must have a g r e a t e r e f f e c t  higher temperature  which  results.  results  and low t e m p e r a t u r e  dynamic r e s i s t a n c e  cycle  purposes  value  s c a l e because  The e x a c t r e a s o n s  unclear,  one  The t h r e e c u r v e s f o r t h e t h r e e are also  computer  the computer  i n both the time p e r c y c l e  between t h e e x p e r i m e n t s is  Thus,  be a minimum  numerical analysis  36, p l o t t e d  f r o m U and V  F o r comparison  i s taken.  times per c y c l e w i l l  s h o u l d be a p p r o a c h e d  s t r e a m l i n e and  f o r the flow t o complete  on e a c h s t r e a m l i n e I s o b t a i n e d . t h e minimum  each  F o r example, w i t h t h e 0.95:cm.  the experimental point  i s almost  on t h e t h e o r e t i c a l  a t a 3°C line,  temperature  while f o r the  10" T-  10"  -r  4000 EMERY  a  I0  10"  CHU  2000 K>00  EXPERIMENTAL THICK  600  0.32 CM.  CELL  400 200  100 60 40 20  10 6 4 BATCHELOR  0.001  _L 0.01 TEMPERATURE  .Figure-; 36.  _L 0.1  \  1.0  10.0  100.0  DIFFERENCE, °C  The f l o w t i m e p e r c y c l e v e r s u s t h e . G r a s h o f number a n d t e m p e r a t u r e d i f f e r e n c e a c r o s s t h e c e l l ( t i n a t 260°C) f o r t h e s o l u t i o n o f B a t c h e l o r , s o l u t i o n o f Emery a n d Chu, n u m e r i c a l - s o l u t i o n , and e x p e r i m e n t a l r e s u l t s .  A  -  0.32 cm.: c e l l , approximately •  v e r y good a g r e e m e n t i s n o t r e a c h e d 10°C t e m p e r a t u r e d i f f e r e n c e s .  until  The  curve  \  •."  for  the intermediate thickness c e l l  two  as expected.  curves  they a r e approaching expected  infinitely  between.the  other  I t i s e v i d e n t b'y n o t i n g t h e p r o g r e s s i o n  of the experimental  and  lies  as t h e ^ t h i c k n e s s i n c r e a s e s t h a t  t h e computer s o l u t i o n i n a s y s t e m a t i c  manner.  The t h e o r e t i c a l l i n e r e p r e s e n t s t h e  thick cell  and s h o u l d be t h e a s y m p t o t i c  limit.  T h i s c a n be shown g r a p h i c a l l y by p l o t t i n g t h e t i m e p e r cycje f o ra p a r t i c u l a r temperature d i f f e r e n c e against the cell thickness.  F i g u r e 37 shows s u c h a p l o t  temperature d i f f e r e n c e . represents  f o r a 1°C ;  The h o r i z o n t a l d a s h e d  line,  the t h e o r e t i c a l time p e r c y c l e f o r t h i s  temperature d i f f e r e n c e .  This curve  supports  the proposal  that the experimental  r e s u l t s a p p r o a c h t h e t h e o r y and t h a t  for  g r e a t e r than approximately  a very t h i c k c e l l  1.5  Cm.  t h i c k t h e t h e o r y a n d a c t u a l f l o w s s h o u l d be i n c o m p l e t e agreement. C o m p u t e r r u n s were made c o r r e s p o n d i n g the average temperature i n t h e l i q u i d to  305°C and 2 3.7°C.  t i n melt  ical  f r o m 260?C  F o r 305°C t h e P r a n d t l number i s 0.01135  f o r 237°C t h e P r a n d t l number i s 0 . 0 1 3 6 5 .  and  The t h e o r e t -  change i n t h e t i m e p e r c y c l e f o r a c o n s t a n t  d i f f e r e n c e o f 0.55°C f o r t h e v a r i o u s d e g r e e s o f is  to altering  shown i n F i g u r e 38.  temperature superheat  These r e s u l t s a r e i n agreement w i t h  t h e r e s u l t s o f F i g u r e 11 w h i c h a l s o shows t h a t an i n c r e a s e  100  80  EXPERIMENTAL  60  or w  40  -  20  THEORETICAL  0.2  _L  _L  0.4  0.6  CELL  Figure  37.  . ... 0.8  THICKNESS,  ±  1.0  1.2  1.6  CM.  The f l o w t i m e p e r c y c l e v e r s u s t h e l i q u i d c e l l t h i c k n e s s ,. f o r a t e m p e r a t u r e d i f f e r e n c e o f 1 ° C . s h o w i n g t h e e x p e r i m e n t a l and . t h e o r e t i c a l c u r v e s .  M  23  co Q Z 22  o CJ UJ  CO  UJ  o >  20  or  LU Q_  19  UJ 18  17  230  X 250 AVERAGE  Figure  38.  270  290  TEMPERATURE,  °C  The t h e o r e t i c a l f l o w t i m e p e r c y c l e f o r l i q u i d t i n versus t h e average temperature i n the c e l l f o r a temperature d i f f e r e n c e a c r o s s t h e c e l l o f 0.555°C.  310  In t h e a v e r a g e t e m p e r a t u r e per  causes  a decrease  cycle.  The  numerical  compared w i t h stated  a n a l y s i s presented,here  the a n a l y t i c a l  earlier,  solution  the s o l u t i o n  260°C t h i s  corresponds  o r a temperature  manner as f o r t h e n u m e r i c a l solution  c a n be c o n v e r t e d  shows t h e s o l u t i o n numerical  results. line  function  i s directly  10 . J  For l i q u i d  number o f 7.9  o f 0.022°C. results,  into  In a  the numerical  velocity  f o r constant  to Batchelor's solution  ;  solution  theory  o f 7.9  in  this  36  as t h e as a '  The t i m e  profile  layer.-  number, computer  at the  lower  mentioned  T h i s i s expected  i s correct.  p e r c y c l e was t a k e n  a t X = 0.5  i s taken  shown on  assuming the . •  around the c e l l  i n the  approximate but w i l l  discussion.  in a  t o the d i s t a n c e from the  o f t h e maximum v e l o c i t y  This i s very  qualitative  The  x 10  o f Emery and Chu i s a l s o  centre to p o s i t i o n  boundary  Figure  appears  P r a n d t l number.  c i r c u l a r manner a t a r a d i u s e q u a l cell  of Batchelor  p r o p o r t i o n a l to.the Rayleigh  numbers, and below t h e v a l u e  36.  t i n at x 10**  a time p e r c y c l e .  a b o v e , t h e agreement becomes q u i t e c l o s e .  Figure  valid  similar  the r e s u l t s  Batchelor's solution  i s asymptotic  The  As  o f s l o p e equal t o u n i t y s i n c e the stream  andhence G r a s h o f  Grashof  of Batchelor.  o f B a t c h e l o r on t h e same . p l o t  straight  solution  than  t o a Grashof  difference  c a n be  o f B a t c h e l o r s h o u l d be  f o r R a y l e i g h numbers o f l e s s  if  i n the time  The t i m e s  be  used  a r e v e r y much  ionger due to the l a r g e r d i s t a n c e s around the c e l l at the p o s i t i o n o f maximum flow.  The  time•per  c y c l e curve i s  again a s t r a i g h t l i n e with a slope of o n e - h a l f s i n c e u i s p r o p o r t i o n a l t o (A9)  i n t h i s theory.. From the  numerical  r e s u l t s , the h i g h e r the Grashof number, the c l o s e r to the o u t s i d e w a l l i s the peak i n the v e l o c i t y a l s o that the a n a l y s i s o f Emery and Chu  curve.  T h i s means  should become c l o s e r  to the a c t u a l case at very l a r g e Grashof numbers where the flow i s s i m i l a r to a narrow boundary l a y e r on each vertical wall.  The s o l u t i o n o f Emery and Chu w i l l approach  the. numerical s o l u t i o n at very l a r g e iGrashof numbers o f the o in order of lCr to 10  .  T h i s i s out of the range o f e x p e r i -  mental i n t e r e s t , but does q u a l i t a t i v e l y numerical  support  the  results. In summary, s e v e r a l p o i n t s can be made concerning  the' t h e o r e t i c a l numerical s o l u t i o n ;  (a)  The  developed:  s o l u t i o n agrees with the i n f i n i t e  series  a n a l y t i c a l s o l u t i o n at low values of the Grashof number. (b)  The  s o l u t i o n approaches t h a t o f an  boundary layer-model  integral  f o r very l a r g e values o f the  Grashof  number. (c)  The  s o l u t i o n corresponds  t o the  r e s u l t s , when p r o j e c t e d t o a very t h i c k  experimental  cell.  Thus-.it i s very reasonable to conclude that the numerical a n a l y s i s presented here Is a good and s o l u t i o n to the ;problem ;  of thermal  convection.  proper  4.  The  Solute  problem of  frqm other  types  convection  directly  are  set  of  up d u r i n g  system where  the  than unity.  If  the  solute  during  some s o r t  exist  occur.  liquid  in  for  thermal  the  upper  stabilizing to  driving in  the  retarding  liquid  conducted  also  to  are  or  Solute  motion. observe  1  solute  gradients  multicomponent is  a different  other density  gradients  convection  and  a  exists.  m e l t i n g t h e r m a l g r a d i e n t s ,of t h e r m a l and s o l u t e  convection  c a n be r e d u c e d  vertically part the  of  to  the  Several the  the  with  the  there solute  If  in  order  must  first  to  thermal  of  or less  thermal for be  overcome  a the  gradients  experimental  ef-fect  convection  warmer  system.  system then  from the  s y s t e m due  any  be d e n s i t y  be o b s e r v e d ,  force  the  coefficient  of  natural  and hence b o t h  stability  were  distribution  will  convection  and o b s e r v i n g  difficult.  components  force  convection  sufficient  flow  solute  s o l i d i f i c a t i o n of  solidifying  are  separating  very  freezing  The  e l i m i n a t e d by  solute  the  gradients  Generally  gradients  is  the  driving  dense  fluid  solute  potential  will  Convection  solute  investigations convection  alone, and of  t h e combined  o f thermal  a l s o t o observe q u a n t i t a t i v e l y thermal  k.l.  and s o l u t e  Independent An  o f mixing  solute  gradient.  shown i n F i g u r e slots  solute  the r e l a t i v e  convection, strengths.  convection was c o n d u c t e d t-o o b s e r v e t h e  i n a molten pool Schematically 39.  and s o l u t e  convection.  experiment  extent  in  effect  c a u s e d by an  a s y s t e m was s e t up as  A copper sheet  on t h e l a r g e f l a t  faces  two d i s t i n c t  d i v i d e r was p l a c e d  o f t h e mould t o s e p a r a t e  the  liquid  into  regions.  the  liquid  i n e a c h o f t h e two r e g i o n s  the  experiment.  The e x p e r i m e n t  initial  The c o m p o s i t i o n  of  v a r i e d d e p e n d i n g on  c o n s i s t e d o f h o l d i n g the  SYSTEM ISOTHERMAL  Figure  39.  The e x p e r i m e n t a l i n i t i a l c o n d i t i o n s f o r o b s e r v i n g independent s o l u t e c o n v e c t i o n .  entire gate the  system i s o t h e r m a l l y , then  by  pulling  resulting  f l o w was to  not  the  liquid  radioactive  in  cell  the  would  across  cell  and  put  occurred  flow past  o f time  out'with  p u r e t i n on t h e  and'pure t i n p l u s  by  removed and  later.  F i g u r e 41  and  minutes  To  The  the  :  top  o f the  To  wt.  The and  very  the  resultant  f r o m the .motion o f t h e  a l s o t h e movement o f t h e  small  gate liquid  as  left 40  leakage  was  l o n g •. fluid  carried  side l i q u i d  cell  left-hand side,  gate.  The  flow.  flow  gate  bulk  of  position. cell  the A  and  i s t h o u g h t -to  i t i s pulled  t i n to  was  seconds  The  the bottom o f the This  left-  t o what e x t e n t a test  the  plus  Figure  c o n s i d e r i n g the  gate  i n the  liquid  quenched.  system quenched t h i r t y  cell.  put  i n the  slight  did  determine  s y s t e m was  large right-hand  along  gate  % lead  remained i n i t s o r i g i n a l  i s observed  the  liquid  determine  removing the  shows t h e  material has  a l o n g the result  gate.  fluid  necessary  p u r e t i n was  denser  density d i f f e r e n c e across  small-flow  flow.  r a d i o a c t i v e t i n on  rapidly  active  the  involved.  i t was  smaller left-hand  negligible  induced  no  the  casting.  i s considered  gate  observing  of:material occurred  t i n plus.0.1  i n the  f l o w was  i.e.  the  sheet  observed  a c t i o n o f opening  m o l t e n s t a t e f o r 50  observed  t h a t the  fluid  copper  mould and  premature leakage  significant  shows t h e r e s u l t a n t  period  establish  t h a t the  t i n was  I f leakage  o f the  to s o l u t e convection  occurred  right-hand  hand  and  produce  leakage  To  t h a t no  gate  itself  cell.  flow. solely  establish  past  if  due  i t upwards o u t  removing the  fill  the  upwards void  Figure  itf).  The t r a c e r p r o f i l e I n t h e s y s t e m d e s c r i b e d i n F i g u r e 39 l e f t m o l t e n 50 m i n u t e s w i t h o u t opening the gate.  Figure  4l.  The t r a c e r p r o f i l e i n t h e s y s t e m o f F i g u r e 39 w i t h only pure t i n p l u s S n i n the l e f t s e c t i o n and t h e s y s t e m i s q u e n c h e d 30 s e c o n d s a f t e r the gate i s opened. l i 3  b Figure  42.  The t r a c e r p r o f i l e i n t h e s y s t e m o f F i g u r e 39 w i t h t i n p l u s 0.1 w t . % l e a d p l u s S n i n the l e f t s e c t i o n , q u e n c h e d ( a ) 15 s e c o n d s and ( b ) 30 s e c o n d s a f t e r t h e g a t e i s o p e n e d . l l S  left is  by  the  not  g a t e as  i t i s removed.  from a s o l u t e right-hand section  density  section  times.  castings  quenched  both  falls  Figures  right-hand mixing  In  tin-lead  fifteen  side  alloy  of  the  and  in a following  4.2.  influence  the  left-hand  plus  radioactive  show two  thirty  seconds  lower r e g i o n no  cell  respectively  that  of the  the  denser  cell  tracer rising  in  on  i n d i c a t i n g that  s e c t i o n i s not  ,  resultant  the extensive  occurring.  The  the; p u r e t i n Ven > The  stability  below a l e s s d e n s e r e g i o n  discussed  of  with  the  % lead  42b  mixing occurring. liquid  resulting  system quenched at  appears to d i s p l a c e  little  flow  It i s observed  liquid  right-hand  a ;dense l a y e r o f  The  the  and  gate.  wt.  T h e r e a p p e a r s t o be  the  with very  0.1  42a  r a p i d l y t o the  cases.  fluid  pure t i n , a n d  opened and  r e m o v a l .of t h e  liquid  however,  e f f e c t were c a r r i e d out  t i n plus  g a t e was  variable  to determine  containing  containing  The  after  flow,  extensive.  Experiments  tin'.  The  masse" of  such  will  be  section.  solute  convection  on  thermal  convection A s e r i e s of  experiments was.carried  out  in 204  which small  amounts o f  added t o pure- t i n m e l t s thermal convection. difference  t i n c o n t a i n i n g ' 0.1 subject  Initially  between t h e  wt.  % TI  • were  to d i f f e r e n t degrees there  added m a t e r i a l  i s a small and  the  of  density  melt  which  p r o g r e s s i v e l y decreases  as the a d d i t i o n spreads.  shows the r e s u l t s o f these experiments.  The  Figure 4 3  autoradiographs  on the l e f t - h a n d s i d e are pure t i n with t i n t r a c e r f o r a given temperature d i f f e r e n c e and those on the r i g h t - h a n d s i d e are pure t i n with t h a l l i u m t r a c e r f o r the same tempe r a t u r e d i f f e r e n c e across the c e l l . temperature d i f f e r e n c e , F i g u r e s 43a  For the case of a zero and 43b,  fluid  flow  r e s u l t s only from s o l u t e c o n y e c t i o n . ; The h e a v i e r t r a c e r i s observed the c e l l  and  to drop very r a p i d l y to the lower regions o f i n doing so induces a d d i t i o n a l f l u i d  i n the c e l l .  motion  T h i s i s evidenced by the s w i r l i n g a c t i o n  i n the lower r e g i o n s and the r a p i d l y formed band of t r a c e r at  the bottom.  to  1°C,  With an i n c r e a s e i n the temperature g r a d i e n t  F i g u r e s 43c  very d i s s i m i l a r .  and 43d,  The  the two' flow p a t t e r n s are  t h a l l i u m t r a c e r drops down the  much more q u i c k l y than the t r a c e r under thermal only.  There i s s t i l l  an accumulation  lower regions of the c e l l . counterclockwise flow.  still  cell  convection  of t r a c e r i n the  The p a t t e r n , however, shows a  s w i r l s i m i l a r to the thermal  convective  This flow i s i n e f f e c t a composite o f the flows, of  F i g u r e s 43b  and 43c.  For F i g u r e s 43e  and 43f the  temperature  d i f f e r e n c e has-.been i n c r e a s e d to 3 ° C a n d the two. p a t t e r n s ; '  't  are becoming s i m i l a r .  There i s s t i l l  an  accumulation'of  t h a l l i u m i n the lower r e g i o n s of the mould and a s i n g l e cell  flow p a t t e r n i s now  d i f f e r e n c e of.5°C,  r e a d i l y apparent.  F i g u r e s 43g  and 43h,  For a temperature  the flow p a t t e r n s  are very s i m i l a r , apparently dominated by thermal  convection.  d  c  Figure  43.  The e f f e c t o f s o l u t e c o n v e c t i o n on t h e r m a l convection; t h e t r a c e r d i s t r i b u t i o n i n samples with melt m a t e r i a l , average t e m p e r a t u r e , t e m p e r a t u r e d i f f e r e n c e , and t i m e t o quench o f (a) Sn - S n , 2 6 0 ° C , 0 ° C , 60 s e c o n d s , (b) Pb - T I *, 357°C, 0 ° C , 60 s e c o n d s , ( c ) Sn - S n , 2 6 0 ° C , 1.11°C, 30 s e c o n d s , (d) Pb - T I " , 357°C, 0 . 9 8 ° C , 30 s e c o n d s . 1 1 3  201  1 1 3  2 0  g  h  F i g u r e 43 c o n t i n u e d . The e f f e c t o f s o l u t e c o n v e c t i o n on thermal convection; the t r a c e r d i s t r i b u t i o n i n samples w i t h melt m a t e r i a l , average t e m p e r a t u r e , t e m p e r a t u r e d i f f e r e n c e , a n d time t o quench o f ( e ) Sn - S n , 260°C, 3.04°C, 30 s e c o n d s , ( f ) Pb - T l " , 357°C, 3 . 0 0 ° C , 30 s e c o n d s , (g) Sn - S n , 260°C, 4 . 8 0 ° C , 30 s e c o n d s , ( h ) Pb - T l " , 357°C, 5 . 0 0 ° C , 30 s e c o n d s . 1 1 3  2 0  1 1 3  2 0  (  I Any  i  accumulation i n the lower r e g i o n s appears  j  up t h e r i g h t - h a n d s i d e o f t h e [ l i q u i d In was  was  l e s s d e n s e m a t e r i a l was  results.  added t o a m e l t .  This  l e a d m e l t and a d d i n g l e a d p l u s  i n the l i q u i d  lead melt.  For a temperature d i f f e r e n c e  ( F i g u r e s 44a  and  44b  b u l k o f the m e l t .  e n c e o f 5°C  F i g u r e 44 o f 1°C  and  3°C,  spreads  does n o t s i g n i f i c a n t l y  enter  With a l a r g e r temperature  differ-  the f l o w r e s u l t i n g from  thermal  ( F i g u r e 44c)  c o n v e c t i o n c o m p l e t e l y overshadows s o l u t e a normal  shows t h e  and  r e s p e c t i v e l y ) , the t r a c e r  a c r o s s t h e top...of t h e c e l l  convection producing  thermal convection flow pattern. T h e ; r e s u l t s o f t h e a b o v e two  sets of  show t h a t a c e r t a i n t e m p e r a t u r e d i f f e r e n c e for  carried  % r a d i o a c t i v e t i n i n which case the a d d i t i o n s h o u l d  tend to f l o a t  the  material  A l t e r / n a t i v e e x p e r i m e n t s were  done by u s i n g a p u r e  19 w t .  swept  cell.  t h e p r e v i o u s e x p e r i m e n t s more d e n s e  added t o the m e l t .  out i n which  t o be  experiments  i s required  the l e s s dense s o l u t e t o e n t e r the b u l k o f the  liquid  m e l t o r t h e more d e n s e s o l u t e t o be swept f r o m t h e  bottom  of the  the l i q u i d  melt.  The  following section w i l l  c o n d i t i o n s . r e q u i r e d f o r complete  and t h e r m a l  gradients.  Investigate  mixing with both  solute  Figure  44.  T r a c e r p r o f i l e i n lead melts ( S n tracer, 357°C a v e r a g e t e m p e r a t u r e ) w i t h a t e m p e r a t u r e d i f f e r e n c e and t i m e t o quench o f ( a ) 1.07°C, 60 s e c o n d s , (b) 3.07°C, 30 s e c o n d s , and ( c ) 5.00°C, 15 s e c o n d s . 1 1 3  PURE  6  9  4.3.  45.  c  Sn + x wt.%Pb  b  Figure »•  Sn  d  i-.  The e x p e r i m e n t a l i n i t i a l c o n d i t i o n s f o r d e r i v i n g t h e t h e r m a l and s o l u t e c o n d i t i o n s f o r complete l i q u i d m i x i n g .  Thermal  and s o l u t e  c o n d i t i o n s f o r complete :  liquid  mixing The complete  of  a lead-tin  cell  For  the c e l l  across the c e l l  i s observed.  layer  shown i n F i g u r e 4 5 ,  temperature  and t h e f l o w p a t t e r n  A simple  complete  i s composed A  t h e o r y c a n be  when t h e t e m p e r a t u r e  h i g h t o cause  u s i n g the i n i t i a l  l a y e r o f the  o f known c o m p o s i t i o n .  t o determine  be.sufficiently  The u p p e r  t i n and t h e l o w e r  i s applied  the l i q u i d  developed  alloy  conditions required f o r  c a n be o b s e r v e d  shown i n F i g u r e 4 5 .  zone i s p u r e  difference in  and s o l u t e  mixing i n the c e l l  conditions liquid  thermal  gradient w i l l  mixing  i n the c e l l .  e  let and  l e t the  Since  P and  since  and  the  <  p  M  walls  t i n + x wt.  are  c  K  hypothesis  of  this analysis  difference,  p„ > p . a — d.  d i s p l a c i n g the  side.by  Once t h e  lower r e g i o n  the  difference  i s diminished.  solute  difference  stabilizing  f l o w between a d j a c e n t a temperature  i s s i m i l a r t o the Figure  46  hypothesis,  with  with a variable  the  mould.  The  for  lead  the  are  samples w i t h  the  is a  vertical preventing  valve".  This  exists condition  earlier. t e s t the  same t h e r m a l  i n the  the:  a condition  e x p e r i m e n t s done t o  content  the  become com-  s y s t e m by  discussed  on  right-hand  f i x e d and  then  the ;  cell  l a y e r s , even i n  a "solute  the  differ-  cause  liquid  When t h e r e  liquid  approximately  and  will  system w i l l  gradient, as  the  a liquid  thermal valve  shows s e v e r a l  of  temperatures  solute  r e f e r r e d to  This  a l l o y on  m i x i n g b e g i n s the  mixed s i n c e  be  i s that  occur f o r a c e r t a i n temperature  left-hand  w h i c h can  0  (4.3)  d  solute  presence of  % l e a d where x ?  H  the  fluid  ;  Isothermal  pure t i n to e n t e r  pletely  etc.  < p, b  complete m i x i n g to  side.  a t b be  (^.2)  a l l o y i s of  The  ence and  p^,  .  b  vertical P a  a t p o s i t i o n a be  a  < P  d  the  K  (4.i)  x  density  c  p  > e  2  conditions  lower regions  incomplete mixing  above  show a  of small  Figure  46.  The t r a c e r p r o f i l e f o r s t e a d y - s t a t e flows f o r t h e i n i t i a l c o n d i t i o n s as shown i n F i g u r e 45 w i t h l e a d c o n t e n t x , t e m p e r a t u r e d i f f e r e n c e , a n d t i m e t o q u e n c h o f ( a ) 10 w t . %, 3 . 0 2 ° C , 60 s e c o n d s , ( b ) 0 . 2 w t . %, 2 . 6 4 ° C , 60 s e c o n d s , ( c ) 0 . 1 w t . %, 2 . 9 1 ° C , 60 s e c o n d s , a n d ( d ) 0.05 w t . % 2 . 8 9 ° C , 60 s e c o n d s . t  cell  in.the  cell,  upper 46a  Figures  completely  left-hand 46b.  and  The  show a s i n g l e 46c  c o r n e r on t h e c o l d  cell  46d.  pattern  zone,  Figures  shape  o f t h e i n c o m p l e t e l y mixed  was  carried  and  out ±n'which  t i n plus  1% l e a d  and  lower  layer.  under  a 2.82°C t e m p e r a t u r e  reach  complete  The  incomplete line  47  and  due  left  The  The  Some s o r t  o f complex d e n s i t y  as shown'in  two  the  and run. The  used  i n the  odd  upper  thallium  i n the  minutes  resultant  i n nature to the  same c u r v e d  up  and  to  pattern i s  demarcation  the hot  /This cell  s i d e o f the  i s set'up which  cell.  is  a l l o w s no m i x i n g between  regions.  i n T a b l e VI  f o r the  including  the i n i t i a l s o l u t e The  calculated  d e n s i t i e s :of p u r e  full  the temperature  difference  values of p t i n and  s e t o f experiments;, a r e difference  for. each e x p e r i m e n t a l and  tin-lead  p , are also alloys  included.  as a  o f :temperature .were o b t a i n e d i n t h e f o r m ,p = a + bT ;  the  '  liquid  t o a l l o w the system  gradient  figure  The.-results listed  f o r the  to the thermal c o n v e c t i o n i n the lower  s m a l l amount o f s o l u t e  the  the  lower r e g i o n s i s e v i d e n t .  forcing-a  stable  throughout  molten, f o r 60  gradient  mixing patterns. and  have m i x e d  radioactive  i s very s i m i l a r  between t h e u p p e r  c o u l d be  plus  s y s t e m was  o f the  r e g i o n : a n o t h e r experiment  t i n was  steady s t a t e .  shown i n F i g u r e  that  To h e l p a c c o u n t  pure  layer  samples  side  work o f T h r e s h  a and b u s e d .  :  (37).  Table VII l i s t s  l t i s evident  from F i g u r e  function from  the v a l u e s o f 46  and T a b l e VI  Figure  47.  The s t e a d y - s t a t e t r a c e r p r o f i l e r e s u l t i n g f r o m t h e l o w e r s e c t i o n o f t h e c e l l b e i n g comoosed o f t i n , 1.0 w t . % l e a d , and T l " and w i t h a t e m p e r a t u r e d i f f e r e n c e o f 2.82°C. 2  Figure  48.  0  The e x p e r i m e n t a l i s o t h e r m a l p r o f i l e i n a s y s t e m i n i t i a l l y as i n F i g u r e 45 w i t h t h e l o w e r s e c t i o n composed o f t i n p l u s 10 w t . % lead.  TABLE V I . Experimental  Results  Thermal and S o l u t e  Run Number  6i  02  °C  °C  1  275.49  278.51  2  282.43  3  wt. % Pb  S  f o r Combined Convection  p P  a  3  a  > p  d  p  d" a p  Complete Mixing  gm/cm  gm/cm  10  6.9427  7.1978  No  0.2551  ' No  285.58  10  6.9378  7.1925  No  0.2547  No  279-64  282.37  2  6.9398  6.9893  No  0.0495  No  4  273.37  276.64  0.5  6.9442  6.9548  No  0.0106  No "  5  275.5  278.5  0.1  6.9427  6.9431  Equal  0.0004  Yes  6  276.56  279.45  0.05  6.9420  6.9411  Yes  -.0009  Yes  7  274.55  277.46  0.1  6.9434  6.9439  Equal  0.0005  Yes  8  .276.96  274.32  0.2  6.9417  6.9513  No  0.0096  No  9  278.41  281.60  0.5  6.9406  6.9512  No  0.0106  No  10  277.59  280. 41-  1.0  6.9412  6.9650  No  0.0238  No  TABLE V I I . Density  of Lead-Tin Alloys  F u n c t i o n o f Temperature p = a + bT wt. % Pb  that  (37)  (T°C)  a  b  x 10  10  7.4083  7.5583  2  7.1929  7.2117  1  7.1659  7.1683  0.5  7.1525  7.1467  0.2  7.1471  7.1380  0.1  7.1417  7.1293  0.05  7.1403  7.1272  t h e change t o c o m p l e t e  predicted. for  as a  The a c t u a l  the temperature  m i x i n g o c c u r s when p  > p , as  change o c c u r s n e a r 0.1 wt. % l e a d  difference  o f a p p r o x i m a t e l y 3°C.  Corresponding t o the incomplete mixing autoradiographs o f Figures  46 and ;47 i s t h e t e m p e r a t u r e  Figure  48.  across  the c e l l  region.  shown i n  c a s t i n g h a d a 9°C t e m p e r a t u r e and a 10 wt. % l e a d  These ^ c o n d i t i o n s w i l l  throughout a single  This  traverse  the.liquid  flow c e l l  cell.  difference  content i n the a l l o y  n o t a l l o w complete  The t e m p e r a t u r e  i n the upper p a r t  mixing  profile  shows  o f t h e mould as  e v i d e n c e d by t h e b e n d i n g o f t h e i s o t h e r m s i n t h e f i g u r e . The  lower r e g i o n  isotherms  appears  are v e r t i c a l  t o be r e l a t i v e l y  and e v e n l y  spaced.  still  as t h e  The show t h a t taining be  results  obtained  complete mixing  a v e r t i c a l density  obtained  by  of s u f f i c i e n t  will  in this not  clearly  occur i n a l i q u i d  gradient.  imposing a thermal  section  Complete m i x i n g  gradient  on t h e  concan  system  magnitude t o develop a h o r i z o n t a l d e n s i t y  inversion leading to f l u i d  flow.  5.  The may  be  volume change on  a driving  ification. expansion being  o r c o n t r a c t i o n d e p e n d i n g on  isolate  result  The  small  small fluid  was  attempted  in fluid  flow.  % l e a d 55.5  volume  wt.  change on  stabilized tin,  2°C  a t about  pletely.  between the be  liquid  A  i f the  an  metal produce difficult  comparative  volume change d i d  c o n s i s t e d o f comparing 2.6%  on  Both  f r e e z i n g and  s y s t e m s were  above t h e  occurred  d i f f e r e n c e s i n the  p u r e t i n sample and  a t t r i b u t a b l e t o the  solid-  be  w h i c h w o u l d be  melting  a l l o y ) and  i n t o each m e l t .  nucleation  The  the  may  convection.  lead-bismuth  of t r a c e r introduced until  and  metals  during  % bismuth e u t e c t i c a l l o y  freezing.  125°C f o r t h e  cooled  flow  to determine This  flow  volume change would  from normal t h e r m a l  i n pure t i n which c o n t r a c t s wt.  fluid  volume change i s s m a l l  considered.  technique  Freezing  freezing in liquid  force l e a d i n g to  This  a corresponding to  Volume Change on  and  The they  volume change on  44.5  a  w h i c h has  no  Initially  point  (232°C  a small  solidified  sample  freezing.  for  amount  s y s t e m s were  movement o f t h e the' a l l o y  flow  then  com-  tracer should The  p u r e t i n sample had bismuth solute  alloy  had  effects  r a d i o a c t i v e t i n added and  should  not  molten s t a t e at the  occurs.  Figure  49  the  t h a t has  nucleate  and  50b  mixing  i n the  It  must be on  the  a result  must  the  solid  the  moved a r o u n d t h e ation melt  i s that during  liquid  shows  are  t h a n the  recalescence  volume  The  only  system. flow  feathery some  before  the  reasonable  liquid  flow  change  n u c l e a t i o n , the  gradients.are o f the  the  for this  occurred.  very  equivalent.  pattern that  before  The  to  extensive  edges i n d i c a t e t h a t  cell.  the  allowed  patterns  form o f d e n d r i t e s  large thermal  the  flow  volume change i s z e r o  i n the  50  The  f o r p u r e t i n , and  e u t e c t i c flow  tracer profile  state  quenching.  o f m i x i n g a p p e a r s t o be  n u c l e a t i o n has  present  thermal  molten  e u t e c t i c show two  in a  an e u t e c t i c  systems are  50a  of something other  after  o f the was  The  system i s i s o t h e r m a l  occur  nature  degree  f r o m the  f r e e z i n g as  Since  cell.  added  move  nucleation  Figure  two  flow.  melt;  i n the  before  Both F i g u r e  lead-bismuth  liquid  but  the  premature  minutes  r e s u l t e d when t h e  i s evident  as  moved s i g n i f i c a n t l y .  f o r the  dissimilar  five  introduced  freeze.  Figure  no  that  t r a c e r d i d not  shows a t i n c a s t i n g and  t r a c e r was not  the  fluid  t r a c e r a d d i t i o n was  were e l i m i n a t e d and  t r a c e r has flow  that  same t e m p e r a t u r e  c a s t i n g w h i c h were l e f t after  contribute to  to assure  t o n u c l e a t i o n , the  gradients  lead-  r a d i o a c t i v e t h a l l i u m added i n o r d e r  In o r d e r prior  the  tracer  explan-  s e t up  in  the  just  after  the  F i g u r e 49.  The t r a c e r p r o f i l e i n an I s o t h e r m a l m e l t o f (a) p u r e t i n a t 235°C ( S n t r a c e r ) and ( b ) 44.5 wt. % Pb - 55.5 wt. % B i a t 127°C ( T l * t r a c e r ) , b o t h l e f t 300 s e c o n d s b e f o r e quenching. 1 1 3  2 0  u F i g u r e 50.  b  The t r a c e r p r o f i l e i n a m e l t o f (a) p u r e tin (Sn t r a c e r ) and (b) 44.5 wt. % Pb 55.5 w t . % B i ( T l * t r a c e r ) a l l o w e d t o c o o l , n u c l e a t e , and f r e e z e c o m p l e t e l y w i t h o u t quenching. 1 1 3  2 0 1  initial  nucleation. Two  the  system  temperature  d u r i n g the n u c l e a t i o n p r o c e s s .  c o u p l e s were p l a c e d below t h e l i q u i d  thermocouples  Two  one-half centimeter apart,  surface  recorder monitored the  r e c o r d e r s were u s e d t o m o n i t o r  at the middle  the temperature and  bare one  o f the melt  thermocouples.  of  temperature  between t h e two  s y s t e m was  a l l o w e d t o c o o l a t a p p r o x i m a t e l y 1/2°C  minute and  nucleate.  RECALESCENCE  One  f r o m one  difference  51  centimeter  o f the c e l l .  the o t h e r r e c o r d e d the  Figure  thermo-  The  eutectic per  shows t h e r e s u l t i n g  I  temperature  2 MINUTES I  I  Figure  51.  ( a ) The t e m p e r a t u r e v e r s u s t i m e p l o t f o r the m e l t t e m p e r a t u r e d u r i n g n u c l e a t i o n and f r e e z i n g o f t h e l e a d - b i s m u t h e u t e c t i c m e l t and (b) t h e d i f f e r e n t i a l t e m p e r a t u r e v e r s u s t i m e p l o t between two 0.5 cm. a p a r t p o i n t s i n t h e l e a d b i s m u t h e u t e c t i c d u r i n g n u c l e a t i o n and f r e e z i n g .  time curves.  F i g u r e 51a shows t h e o v e r a l l  temperature  c o o l i n g r a t e w i t h t h e s u p e r c o o l i n g and r e c a l e s c e n c e r e a d i l y evident.  The d i f f e r e n t i a l  t h e r m a l c u r v e , F i g u r e 51b, o n  t h e same t i m e s c a l e as t h e o v e r a l l t e m p e r a t u r e a t e n t i m e s e x p a n s i o n on t h e t e m p e r a t u r e  s c a l e , shows t h e  p e r t u r b a t i o n c a u s e d by a l o c a l t e m p e r a t u r e b e t w e e n t h e two t h e r m o c o u p l e s .  curve but  fluctuation  The m a g n i t u d e o f t h e temp-  e r a t u r e p e r t u r b a t i o n i s about o n e - t e n t h o f a degree t h e s m a l l d i s t a n c e and p e r s i s t s  over  f o r a l m o s t two m i n u t e s .  I t i s n o t u n r e a s o n a b l e t o assume t h a t t e m p e r a t u r e g r a d i e n t s i  '  a r e s e t up t h r o u g h o u t t h e m e l t d u r i n g t h e r e c a l e s c e n c e p e r i o d , and t h a t t h e s e cause  thermal c o n v e c t i o n which  i n t h e f l o w o b s e r v e d i n F i g u r e 50.  results  This also accounts,for  some s o l i d b e i n g p r e s e n t p r i o r t o t h e t r a c e r movement. T h e , r e s u l t s o b t a i n e d i n an a t t e m p t t o i s o l a t e t h e f l o w due t p v o l u m e change o n f r e e z i n g h a v e p r o v e n i n conclusive.  The f l o w o b s e r v e d I s d e p e n d e n t more on t h e  degree o f s u p e r c o o l i n g b e f o r e t h e n u c l e a t i o n t h a n t h e volume change. fluid  H o w e v e r , i t s h o u l d be n o t e d , t h a t  significant  movement . r e s u l t e d f r o m v e r y s h a l l o w t e m p e r a t u r e  gradients over r e l a t i v e l y  s h o r t t i m e p e r i o d s , when t h e r e  was a l s o s o l i d p a r t i c l e s p r e s e n t i n t h e m e l t .  6.  Interdendritic Fluid  There has been a g r e a t  Flow  deal of interest  i n the importance o f the i n t e r d e n d r i t i c during s o l i d i f i c a t i o n of t h i s but  flow In metals  (9, 10, 30, 3 D .  Direct  not In l i q u i d metals.  liquid  The d r i v i n g  f l o w a r e t h e volume  change  on f r e e z i n g p u l l i n g  i n t o t h e mushy z o n e , t h e t h e r m a l  convection  l i q u i d pool. ally  and t h e f l u i d  contraction In convection,  motion i n the r e s i d u a l  The p r o b l e m o f i n t e r d e n d r i t i c  a problem o f flow through a very  channels.  (10)  forces f o r i n t e r -  t h e mushy zone p u l l i n g l i q u i d b a c k , t h e r m a l solute  observation  f l o w h a s b e e n made i n ammonium c h l o r i d e c e l l s  dendritic the  recently  flow i s b a s i c -  complex network o f  The n e t w o r k i s a f u n c t i o n o f t h e g r o w t h  conditions  a t t h e i n t e r f a c e s u c h as g r o w t h r a t e , t e m p e r a t u r e g r a d i e n t , composition  and p o s i t i o n i n t h e mushy z o n e .  An  investi-  g a t i o n was c o n d u c t e d t o o b s e r v e t h e i n t e r d e n d r i t i c directly.  Due t o t h e c o m p l e x i t y  experimental  models  flow  o f the problem s e v e r a l  were c o n s t r u c t e d w i t h t h e a i m o f s i m p l i -  f y i n g t h e complex geometry.  This allows  p a r a m e t e r s t o be c o n t r o l l e d a n d a n a l y z e d  the various individually.  A two d i m e n s i o n a l were  used.  for direct  6.1.  Two  model and two t h r e e  Also l e a d - t i n  a l l o y s were  o b s e r v a t i o n o f an a c t u a l  dimensional  52.  This consists  Into the c o l d  interdendritic.spaces, the is  52.  cm.  cm.  interface,  vertical  block The  inserted copper  i t to r e p r e s e n t the  crude  channels  thick  t h i c k mould.  and t h e m e t a l  each o f the i n t e r d e n d r i t i c  Figure  cast  model i s as shown i n  c u t back i n t o  dendrites.;. T h i s i s a very no c o n t i n u o u s  directionally  growing  o f a 0.31  end o f t h e 0.31  b l o c k has l a r g e s l o t s  models  model  The two d i m e n s i o n a l Figure  dimensional  remaining  represents  approximation  as t h e r e  and hence t h e f l o w  regions w i l l  be  into  independent,  The. two d i m e n s i o n a l e x p e r i m e n t a l model f o r observing I n t e r d e n d r i t i c flow.  to  a l a r g e degree, o f the regions  above and below i t .  F i g u r e 53 shows t h r e e r u n s w i t h t h i s m o d e l . was  quenched  60 s e c o n d s  F i g u r e 53a  a f t e r t h e t r a c e r was i n t r o d u c e d  and h a d a t e m p e r a t u r e d i f f e r e n c e o f 3.68°C a c r o s s pool.  F i g u r e 53b was a l s o q u e n c h e d  60 s e c o n d s  t h e molten  a f t e r the  t r a c e r was i n t r o d u c e d b u t h a d a 5.02°C t e m p e r a t u r e ence.  F i g u r e 53c shows a, c a s t i n g q u e n c h e d  differ-  t e n minutes  a f t e r t h e t r a c e r i n t r o d u c t i o n a n d h a d a 5.l8°C t e m p e r a t u r e difference.  A l l t h e c a s t i n g s a r e pure t i n w i t h r a d i o a c t i v e  tin  The a u t o r a d i o g r a p h  tracer.  f i l m was e x p o s e d  to a light  w i t h t h e c a s t i n g on t h e f i l m a n d t h e d e n d r i t e b l o c k This causes t h e i n t e r d e n d r i t i c  regions t o appear  removed.  light  s i n c e t h e y were s h i e l d e d and t h e a r e a s where t h e d e n d r i t e b l o c k w a s , now a p p e a r d a r k i n t h e a u t o r a d i o g r a p h . evident  that there  i s very  interdendritic region.  little  There  It i s  penetration into the  i sa slight  increase i n the  p e n e t r a t i o n by i n c r e a s i n g t h e g r a d i e n t f r o m 3.68°C t o 5 02°C b u t v e r y  little  T  i n c r e a s e i n p e n e t r a t i o n i s shown  by l e a v i n g t h e s y s t e m f o r a n a d d i t i o n a l n i n e m i n u t e s . this  simple model,  i f the assumptions  From  f o r i t s use a r e  c o r r e c t , i t must be c o n c l u d e d t h a t t h e r e i s no s i g n i f i c a n t interdendritic pool  flow r e s u l t i n g from t h e r e s i d u a l  convection.  liquid  F i g u r e 53.  The e x p e r i m e n t a l r e s u l t s o f t h e m o d e l o f F i g u r e 52 w i t h a p u r e t i n m e l t ( S n tracer) with a temperature d i f f e r e n c e across the p o o l and a t i m e b e f o r e q u e n c h i n g o f ( a ) 3.68°C, 60 s e c o n d s , ( b ) 5 . 0 2 ° C , 60 s e c o n d s , and ( c ) 5.18°C, 600 s e c o n d s . 1 1 3  - — M O U L D  Figure  6.2.  54.  Three  The t h r e e observing  dimensional wire The  dimensional the  model i s  wires  the  cold  represent This as  flow. there  end o f  the  model i s  this  two  shown  as t h e  0.035  of  approximately in  experimental  same t h i c k n e s s  steel  dimensional wire rod model Interdendritic flow.  inch  dendrite  in  Figure  cell,  interface than  in  from the  molten pool  into  wires  that  the  c o u l d be  stainless  is  placed  spaced s t e e l  wires  experimental  model.  two  dimensional  channels  for  the  in  this  liquid  the  block,  protruding  block,  this  three  the  main a p p r o x i m a t i o n s  no d e c r e a s e  the  A copper  diameter  evenly in  for  twenty-six  from the The  flow  54.  with  (0.89mm.)  cell.  is  number o f  configuration  more r e a l i s t i c  allows, v e r t i c a l The  model  centimeters the  for  mushy  to  system are solid  zone  equally  liquid  ratio  and t h a t  spaced i n  one metal  that back the  the  block  was  limited.  The  wires  t e n d t o b e n d and  several wires  1  c o u l d come i n t o  contact  w h i c h would change the  significantly.  For the  c a s e s t u d i e d h e r e t h e mushy z o n e  was  17%  s o l i d w i t h a 1.9  model does not w h i c h may  mm.  dendrite  resulting  interdendritic  a p u r e t i n m e l t and  Figure and  55a  Figure  below the  end  of the w i r e d e n d r i t e  the  figure.  all  t h e way  pattern.  f l o w p a t t e r n on  the 0.45  mm...thick s a m p l e . the  The  The  arrows  cell  flow  i n t e r i o r view, Figure  flow i s a l s o s i m i l a r to the from the  surface  f l u i d bulk.  c o m p l e t e d o v e r one  while  f l o w has  the main p a r t of the  progressed three-quarters as  s e e n by  Figure  56a  the  .  i n t e r i o r view t h a t the  zone i s s l o w e r t h a n i n  bulk  55b,  mushy  r a t e through the w i r e d e n d r i t e The  on  t h e mushy zone  flow p e n e t r a t i o n through the  I t i s evident  i n the  flow pattern  zone i s m a r k e d by  fluid.  similar  the p o o l  are  the  f o r a c t u a l growing  shows t h e 3.1  wires  c o p p e r b a s e b l o c k w i t h a one  p a t t e r n i n the b u l k  zone and  branching  flow.  f l o w i s seen t o p e n e t r a t e  to the  shows a v e r y  55b  s u r f a c e of the  The  cell.as  shows t h e  mm.  The  dendrite  a 5.73°C t e m p e r a t u r e d i f f e r e n c e i n  c o o l e r l e f t - h a n d s i d e of the  sample s u r f a c e  This  f l o w t h r o u g h t h i s , model w i t h  m o l t e n p o o l i s shown i n F i g u r e 55.  interfaces.  spacing.  take i n t o account dendrite side  p o s s i b l y a f f e c t the The  geometry '  interdendritic o f t h e way  flow  the  cycle  f l o w has  down t h e mushy  only zone,  t r a c e r f r o n t i n t h e mushy zone i n F i g u r e  shows a n o t h e r r u n t h a t i s e x p e r i m e n t a l l y  55b.  similar  F i g u r e 55.  The t r a c e r d i s t r i b u t i o n i n t h e s y s t e m o f F i g u r e 54 ( t i n m e l t , S n t r a c e r ) w i t h a temperature d i f f e r e n c e o f 5.73°C a c r o s s t h e p o o l s h o w i n g ( a ) t h e as c a s t s u r f a c e and ( b ) t h e p r o f i l e 0.45 mm. b e l o w t h e s u r f a c e . 1 1 3  F i g u r e 56.  The t r a c e r d i s t r i b u t i o n i n t h e s y s t e m o f F i g u r e 54 ( t i n m e l t , S n t r a c e r ) with a temperature d i f f e r e n c e o f 5.1°C across the p o o l showing ( a ) t h e as c a s t s u r f a c e , (b) t h e l e f t h a n d end o f t h e b l o c k w i t h t h e w i r e s r e m o v e d , and ( c ) t h e l e f t hand end w i t h 3/8 i n c h e s o f t h e b l o c k end r e m o v e d . 1 1 3  to  the  s a m p l e o f F i g u r e 55.  dendritic Figure  flow are very  56b  and F i g u r e  l o o k i n g at the  The  s i m i l a r t o the  56c  show t h e  d e n d r i t e end  machined o f f the  simply  block. The  mechanically  No  end  v i e w shows l i t t l e  i n the  by  a p p e a r d a r k due  The  seen i n t h i s p i c t u r e . the  inter55.  The  wire  3/8  inch  dendrites  56b  o f the  sample  56c  and  soft t i n  t r a c e r i n the  wires. casting  o f t r a c e r down f r o m t h e w i r e  upper r e g i o n s .  the w i r e .  view w i t h  from removing the  The  wire  c a s t i n g shows t h a t t h e  h o l e s where t h e  wires  c a s t i n g down t h e  distribution  Figure  dendrite  t o the r a d i a t i o n coming  f r o m t h e more a c t i v e i n t e r i o r o f t h e left  view of the  p u l l i n g them o u t  except f o r small t r a i l s  were p u l l e d out  the  the  flow of Figure  sample f o r F i g u r e s  distortion results  as c a s t end  positions  and  end  f l o w and  sample, r e s p e c t i v e l y .  have been removed from the by  outer  56c  can be  hole  readily  f u r t h e r i n t o the  end  f l o w i s q u i t e complex around  of the  wires. The . r e t a r d i n g f o r c e due dendrites percent  seems s i g n i f i c a n t l y  solid  A l s o the spacing  and  very  the  t o these  artificial  l a r g e c o n s i d e r i n g the  large dendrite spacing  smooth n a t u r e  o f the w i r e s  and  o f 1.9 their  mm. regular  s h o u l d a l l o w g r e a t e r f l o w t h a n an a c t u a l i n t e r f a c e :  of equal branching  average spacing will  and  percent  solid  undoubtedly r e t a r d flow.  since  fluid  side  T h i s model i n -  d i c a t e s t h a t an a c t u a l d e n d r i t i c i n t e r f a c e w i l l retard  low-  very  greatly  m o t i o n as e v e n t h i s m o d e l w i t h i t s many l e s s  r e s t r i c t i v e p r o p e r t i e s reduces the  flow  substantially.  All  the  c o n d i t i o n s o f an a c t u a l d e n d r i t e  a r e i m p o s s i b l e t o meet w i t h an e x p e r i m e n t a l the  two  preceeding  models i t i s e v i d e n t  s i o n a l m o d e l must be  u s e d t o g e t any  A l s o the  model.  From  that a three  correlation.  d e s i r a b l e f e a t u r e s a r e a change i n t h e  solid-liquid  liquid  metal  the d e n d r i t e s p a c i n g .  the  a r t i f i c i a l d e n d r i t e s as i n a r e a l i n t e r f a c e b u t  d i s s o l v e the a r t i f i c i a l d e n d r i t e s s i g n i f i c a n t l y  analysis.  was  chosen f o r e x t e n s i v e  A w i r e mesh i s p l a c e d v e r t i c a l l y  wide mould.  The  w a l l f a c e s and  mesh s l i d e s  into a slot  mould t o p r e v e n t  should  any  not  during  bottom of the  liquid  flow through  cell  a r o u n d the edges o f the  interface  the  and  t o the  A series s i z e and  wire  the  mesh.  cell  a flow develops  as as  f l o w coming a c r o s s  large flow c e l l  there w i l l  be  large flow c e l l  v e r t i c a l mesh r e p r e s e n t s o f an e q u a l  cm.  the  t o w a r d s t h e mesh t h e r e w i l l be  Similarly  small flow c e l l . i n t o  i n t h e 0.31  the  t h e mesh f r o m t h e  small flow c e l l *  The  Due  experimental  i n the U shaped p i e c e o f  shown s c h e m a t i c a l l y i n F i g u r e 58 figure.  model  large  leakage  shown i n t h i s  wet  i n t o s l o t s on t h e two  A temperature gradient i s ' then placed across  wire  percentage  A f t e r - c o n s i d e r i n g a l l t h e s e : c o n d i t i o n s the  shown i n F i g u r e 57  cell.  dimen-  Other  and  a test.  interface  solid-liquid  to  the  a f l o w out o f at the  top o f  a position in a r a t i o and  Table  the the  dendritic  dendrite  o f meshes a r e u s e d w i t h a r a n g e spacing.  a  spacing.  of  V I I I l i s t s the p r o p e r t i e s  WIRE MESH  SIDE PLATE  MOULD  Figure 57.  The three dimensional wire:mesh model f o r o b s e r v i n g i n t e r d e n d r i t i c flow,.  0,  F i g u r e 58.  The q u a l i t a t i v e flow l i n e s In the system o f F i g u r e 57 showing the two flow c e l l s and intermesh flow d i r e c t i o n s .  D a t a on W i r e  Meshes Used i n Plow  and F r a c t i o n F l o w e d  Mesh Sample' •• Number  •MeshNumber wires/inch  , w'  Results  Properties  Wire Diameter d  Experiments  mm.  Experimental  : Wire-' Spacing S  ,  •  mm.  a  Hole Size h  ,  mm.  Counts, large cell  Results  Counts, small cell  Fraction Flowed  a  1  100  0.114  0.254  0.140  9559  1692  0.1073  2  100  0.102  0.254  0.152  7368  662  0.0824  3  80  0.165  0.317  0.152  11374  266  0.0229  4  80  0.127  0.317  0.191  8445  409  0 .0462  5  60  0.241  0.424 .  0.183  7384  259  0.0339  6  60  0.178  0.424  0.246  7348  '720  0.0892  7  40  0.279  0.635  0.356  6934  311  0.0429  8  40  0.216  0.635  0.419  10330  1149  0.1001  9  30  0.330  0.846  0.516  6569  279  0.0407  10  30  0.254  0.846  0.592  7121, -  ...  2410 ....  0.2529  o f t h e s p e c i f i c w i r e meshes u s e d .  A l l t h e meshes  were s q u a r e woven and were e i t h e r o f c o p p e r o r bronze.  For the experiments pure l i q u i d  radioactive thallium tracer.  The  phosphor-  l e a d was  mesh was  studied  used  coated with  with a  s o l d e r i n g t y p e f l u x and t h e n d i p p e d i n a b a t h o f m o l t e n before being i n s e r t e d i n t o the s l o t s  i n the mould.  lead  This  gave a t h i n l a y e r o f l e a d on t h e w i r e mesh s o t h a t t h e r e was  a good w e t t e d s u r f a c e  f o r the experiment.  e x p e r i m e n t s were c o n d u c t e d w i t h a t e m p e r a t u r e across the c e l l  o f 6.05  A l l the difference  ± 0.05°C s o t h e d r i v i n g  p r e s e n t f o r t h e i n t e r m e s h f l o w was  constant f o r a l l t e s t s .  F i g u r e 59 shows a s e c t i o n v i e w o f t h e  F i g u r e 59.  forces  casting  A micrograph of a cross s e c t i o n of a wire mesh i n an a c t u a l l e a d s a m p l e t h a t has b e e n mounted and p o l i s h e d .  s h o w i n g t h e v i e w o f a number 40 mesh embedded i n t h e c a s t i n g . The  mesh i s s e e n t o be p e r p e n d i c u l a r  that  t h e mould s o  t h e l e a d has no p a t h a r o u n d t h e mesh.  nature  o f the wires  dissolution  the various  point  i t would appear t h a t  o f t h e mesh by t h e m o l t e n  Figure for  across  60 shows t h e f l o w  mesh s i z e s .  A on F i g u r e  58.  from t h e : l a r g e  analysis  s u c h as f l o w  past  is little  patterns  that  result  The t r a c e r i s i n t r o d u c e d a t  At l e a s t  two r u n s were done  to the small  to follow.  there  lead.  e a c h p a r t i c u l a r mesh and t h e sample t h a t flow  By t h e r e g u l a r  with  showed t h e l e a s t  flow, c e l l  was u s e d  T h i s method was u s e d as any d e v i a t i o n  t h e s i d e o f t h e mesh, a l a r g e d e g r e e ,  o f d i s s o l u t i o n o f t h e mesh by t h e m o l t e n l e a d o r an woven mesh w o u l d a l l c a u s e an i n c r e a s e flow.  Thus i f t h e minimum f l o w  correspond  t o the o r i g i n a l  10 o f T a b l e  VIII.  was u s e d .  in  61.  Figure  i t would  activity  cell in  Figure  sheilding  6l.  samples a r a d i o a c t i v e  The c o u n t i n g  o f the large  ; :  The  one p a r t  4,.'. 5  F o r q u a n t i t a t i v e a n a l y s i s o f the flow  A two i n c h  o f the s o l i d  The  2,  t o samples  apparatus used  scintillation  flow  block  used,  and s c a l e r .  c e l l ' and o f t h e s m a l l  b l o c k were c o u n t e d s e p a r a t e l y was c o u n t e d . s t i l l  o f t h e sample w i t h  counting  i s shown  t u b e was  c o n n e c t e d t o a Hamner m o d u l a r a m p l i f i e r , t i m e r The  better  d i m e n s i o n s o f t h e mesh.  t h r o u g h t h e w i r e mesh i n t h e s e technique  irregularly  i n the observed  i s used  60 c o r r e s p o n d  four castings o f Figure and  i n the  flow  as shown  intact  by  a lead shield.  A l l  F i g u r e 60.  The t r a c e r p r o f i l e o f t h e i n t e r m e s h f l o w t h a t o c c u r s i n 120 s e c o n d s a f t e r t h e t r a c e r i n t r o d u c t i o n w i t h a 6.05°C t e m p e r a t u r e d i f f e r e n c e f o r s a m p l e s c o r r e s p o n d i n g t o number ( a ) 5, ( b ) «», ( c ) 2, and ( d ) 10 s a m p l e o f T a b l e V I I I .  SHIELDING  (q) 61.  L E A D  S A M P L E — i  SCINTILLATION TUBE  (b)  The. c o u n t i n g a r r a n g e m e n t f o r - m o n i t o r i n g t h e a c t i v i t y i n t h e (a) l a r g e - f l o w c e l l and (b) s m a l l f l o w c e l l i n t h e i n t e r m e s h f l o w samples  BOUNDARY LAYER  62.  The! b o u n d a r y l a y e r s a r o u n d t h e mesh w i r e s showing t h e n o t a t i o n used i n t h e a n a l y s i s .  counts  were c o r r e c t e d f o r b a c k g r o u n d .  mesh f l o w i s t a k e n  as a r a t i o  the s m a l l flow c e l l large  flow c e l l  ially  the  mesh.  over  the  p l u s the  The  sum  o f the  counts  small flow c e l l .  VIII l i s t s this  as f r a c t i o n f l o w e d , The q u i t e complex.  or r a t i o s  i n f o r m a t i o n f o r each  S  a  vertically.  of wire diameter  flow through  proven  some o f  spacing,  to.wire spacing  the percent  give;  any  indication  t h e mesh c a n be t a k e n  a s e r i e s of c y l i n d e r s placed  F i g u r e 62  of. f l o w a r o u n d a p o r t i o n o f s u c h a  ing  up  the  flow i t s e l f w i l l  Due  around the w i r e s r e s u l t i n g be  reduced.  diameter  from the  s i z e and.hence the  shape o f t h e mesh and  boundary  system. and  h„ t h e a  intermesh  A parameter, h , g  c a n be  d i s t a n c e b e t w e e n t h e b o u n d a r y l a y e r s o f two be  taken  adjacent  assumed t h a t t h e  flow, the as  the  wires.  effective  f l o w , i s a f u n c t i o n of the  a l s o the  equiv-  to a boundary l a y e r b u i l d -  f l o w w i d t h between the w i r e s  For t h i s a n a l y s i s i t w i l l  as  horizontally  shows a s k e t c h o f t h e  d i s t a n c e between the w i r e s .  hole  sample  r e s u l t s has  Various p l o t s of flow versus  i s the wire sDacing, d the w i r e -' w  effective  the  f.  i n t e r p r e t a t i o n of these  alent to flow past  l a y e r type  i s essent-  f u n c t i o n a l r e l a t i o n s h i p between the v a r i a b l e s . The  and  the  gone, t h r o u g h  scattered arrays of i n d i v i d u a l points without of a simple  from  This  r e l e v a n t v a r i a b l e s s u c h as w i r e s i z e , w i r e solid  inter-  o f t h e number o f counts- f r o m  f r a c t i o n o f t h e t r a c e r t h a t has  Table  degree o f  geometric  a c t u a l p h y s i c a l size of  the  mesh.  F i g u r e 63 i s a p l o t o f t h e f r a c t i o n o f t r a c e r t h a t  flowed  through  t h e mesh, f , v e r s u s  t h e r a t i o ,of t h e i n t e r -  w i r e s p a c i n g t o t h e w i r e "diameter, non-dimensional  factor.  hVd a w  , a  geometric  The number on e a c h p o i n t i s t h e  n o r m a l mesh s i z e i n - n u m b e r s o f ' w i r e s p e r i n c h . . I t s h o u l d be n o t e d  that a s i n g l e l i n e  impossible. geometric  This  from t h i s  through  t h e mesh.  /d  ;  a  Therefore,  f o r each w i r e s p a c i n g  f o r no f l o w .  The r e s u l t s  are not  completelv .* ••.< . "  w  The z e r o  t h e 80 mesh r e s u l t s , b u t a r e  flow values  against the a c t u a l wire the e x p e r i m e n t a l  (I.e.  i s drawn b a c k t o t h e a b s c i s s a f o r t h e v a l u e  consistant, especially ible.  The r e l e v a n t i n f o r m a t i o n t h a t i s  a n a l y s i s i s t h e c o n d i t i o n s f o r no f l o w  mesh s i z e ) a l i n e of h  a l l the points i s -  shows t h a t t h e f l o w i s n o t r e l a t e d t o t h i s  f a c t o r alone.  desired  through  f r o m F i g u r e 63 a r e now p l o t t e d  spacing  i n F i g u r e 64.  c o n d i t i o n s f o r no f l o w t h r o u g h  straight  l i n e produced i s : /a d.) + 0.0102 w  where S  i s i n centimeters.  This p l o t a;mesh.  S, = 0.06l('h a &  of^equation  (6.1)  The  With a s u i t a b l e i n t e r p r e t a t i o n  I t i s p o s s i b l e t o p r e d i c t f o r an a c t u a l  f o r no i n t e r d e n d r i t i c  65 .shows a m o d e l o f a d e n d r i t i c  and t h e  flow to occur.  stalks.  as t h e r e s u l t s  f r o m t h e mesh e x p e r i m e n t s a r e a t b e s t  For this  Figure  interface perpendicular  to the primary  limited.  gives  (6.1)  i n t e r f a c e t h e f u n c t i o n between t h e f r a c t i o n s o l i d dendrite spacing  reproduc-  Only a very  simple  model i s c h o s e n , very  model:  Fraction solid  = F  s  = Ud  2 w  ) / (4S  2 a  )  (6.2)  Q  0.24  h  0.20  h  0.16  h  0.12  h  0.08  U  0.04  h  LU  o  h-  u  <  h  Figure  63.  a  /  w  d  A p l o t of the f r a c t i o n flowed, f , versus the r a t i o o f the hole s i z e to the w i r e diameter, h / d^, f o r t h e v a r i o u s mesh s i z e s i n v e s t i g a t e d . a  O  I  '  0  0.2  0.4  h  F i g u r e 64.  Q  / d  0.6 w  FOR  I 0.8  I  I  1.0  1.2  NO FLOW  A p l o t of the wire s p a c i n g , S , versus the r a t i o o f the hole s i z e to the wire diameter, h=;. / d , f o r the no flow c o n d i t i o n ( f = 0) from Figure 6 3 . w  I 1.4  @ T i 0  F i g u r e 65.  A p e r p e n d i c u l a r view o f the primary d e n d r i t e model used i n t h e i n t e r p r e t a t i o n o f t h e intermesh flow r e s u l t s .  Also from the  geometry:  S  h  d^=  1  («-3>  dT  +  w  w  Combining equations F  s  0.054 ) + 0.051.  = f  (s  a  where S no  &  S  >  and ( 6 . 3 ) :  (.6.4)  2  Therefore,  t h e r e s h o u l d be  t h e mushy zone when  ( 540 S 510 a  where S  (6.2)  i s i ncentimeters.  flow through P  (6.1),  1  (6.5)  2  +  i s i n microns.  F i g u r e 66 shows e q u a t i o n  (6.5)  CI  plotted  out.  ditions  f o r no i n t e r d e n d r i t e  curve  represents  the p l o t the  The r e g i o n above t h e c u r v e  represents  flow and t h e area below t h e  conditions for interdendritic flow.  i t i s seen t h a t f o r d e c r e a s i n g d e n d r i t e  flow w i l l  penetrate  con-  From  spacings  d e e p e r i n t o a mushy zone s i n c e t h e  LEAD-TIN ALLOY INTERDENDRITIC FLOW RESULTS  200  400 DENDRITE  Figure 66.  SPACING  600 S  Q  ,  800  1000  MICRONS  A p l o t o f the f r a c t i o n s o l i d i n the s o l i d - l i q u i d , i n t e r f a c e , , versus the primary d e n d r i t e s p a c i n g showing the s  fraction solid  f o r no  equation  and  (6.5)  F i g u r e 66  spacing of less than penetrate  flow conditions i s higher.  f o r m a t i o n and  =1,  t o assume t h a t t h e  s o l u t e movement, w i l l  t r a t i o n of the  s t r u c t u r e , due  flow.  6.4.  d e p e n d e n c e on t h e  fraction;solid  Interdendritic To  effects  and  dendrite  flow i n l e a d - t i n  presented  of  channel  that this  theory  temperature  gradient  i s only a function  alloys flow d i r e c t l y  and  flow i n l e a d - t i n a l l o y s .  c o n d i t i o n s a r e s h o w n i n F i g u r e 67.  The  ;  by  0.31  cm.  nucleated  at the  to,progress placed  m o u l d was  across  and  the mould.  t e m p e r a t u r e g r a d i e n t s and When t h e  t h e h a l f way  10.8  cm.  by  6.4  Solid  cm.  :  was  allowed  T h r e e t h e r m o c o u p l e s were cell  monitoring  s o l i d had  the  experimental  s o l i d i f i c a t i o n was  i n the upper r e g i o n s of the  process.  The  used i n the experiments.  c o l d end  to  b a s e d pn a t h e o r e t i c a l mesh  model s e v e r a l e x p e r i m e n t s were c o n d u c t e d t o o b s e r v e interdendritic  the  spacing.  observe i n t e r d e n d r i t i c  t e s t the h y p o t h e s i s  these  t o the depth o f pene-  h e n c e t h e d e p t h o f t h e mushy z o n e , b u t  o f the  From  be more p r o n o u n c e d  I t s h o u l d be n o t e d  i n c l u d e any  dendrite  completely  f l o w , s u c h as d e n d r i t e r e m e l t i n g ,  f i n e r the d e n d r i t i c  and  flow w i l l  t h e mushy z o n e , t h a t i s u n t i l F  interdendritic  does n o t  i t i s seen t h a t f o r a  30 m i c r o n s t h e  r e s u l t s i t i s reasonable  Prom  the  progressed  f o r measuring  the  solidification to  approximately  p o i n t , a r a d i o a c t i v e master a l l o y  of  an  THERMOCOUPLE  x4 HEAT OUT  Figure  67.  — MUSHY ZONE  TRACER J  \ \ \ \ A  HEAT  LIQUID  IN  \  The experimental c o n d i t i o n s used f o r o b s e r v i n g I n t e r d e n d r i t i c flow i n l e a d - t i n a l l o y s showing the thermocouple and t r a c e r a d d i t i o n p o s i t i o n s . :  -  WEIGHT PERCENT  240  LEAD  12.5 ,  o  o  60  70  80  WEIGHT PERCENT  Figure  68.  90  100  TIN  The r e l e v a n t p o r t i o n o f the l e a d - t i n phase diagram showing the four a l l o y s used to observe i n t e r d e n d r i t i c f l u i d flow.  equivalent  density  as the bulk f l u i d was i n t r o d u c e d  and  allowed t o flow f o r a short p e r i o d b e f o r e b e i n g quenched. Pour d i f f e r e n t a l l o y s o f l e a d - t i n were used, 2, 5, 12.5 and  20 wt. % l e a d i n t i n and a l s o pure t i n .  The phase  diagram f o r the l e a d - t i n system f o r the r e l e v a n t i s shown i n F i g u r e  68 with the v a r i o u s  alloys  a l l o y s shown on the  diagram./ The  c a s t i n g produced by u s i n g  i s shown i n F i g u r e  69.  The s o l i d i f i c a t i o n . f r o n t  seen and appears t o be p l a n a r . left The  a pure t i n melt  The s o l i d r e g i o n  and the flow i n the remaining l i q u i d  i s easily i s ori the  i s r e a d i l y seen.  shape o f the i n t e r f a c e i s i n agreement with the shape  described convection  i n the I n t r o d u c t i o n  and i s caused by the thermal  a l t e r i n g the heat flow c o n d i t i o n s .  In t h i s run  as i n a l l others i n t h i s s e c t i o n the s o l i d i f i c a t i o n i s o f the order o f 2 - 5  rate  cm./hour and the t r a c e r i s l e f t  3Q seconds b e f o r e quenching so that very l i t t l e movement occurs while the t r a c e r i s i n the melt.  interface Thus; the  t r a c e r represents  the i n t e r f a c e at an i n s t a n t i n time due  to the r e l a t i v e l y  l a r g e d i f f e r e n c e s i n the r a t e o f the  t r a c e r and i n t e r f a c i a l movements. For: a l l the a l l o y c a s t i n g s growth mode was. observed.  a d e n d r i t i c type  A c a s t i n g o f t i n plus 2 wt. %  l e a d plus r a d i o a c t i v e t h a l l i u m was d i r e c t i o n a l l y s o l i d i f i e d the complete width o f the mould.  The t h a l l i u m seg-  regates with the l e a d t o show the d e n d r i t i c s t r u c t u r e i n  Figure 69.  The t r a c e r d i s t r i b u t i o n i n t h e r e s i d u a l l i q u i d pool o f a pure t i n c a s t i n g ( S n tracer) with the e x p e r i m e n t a l c o n d i t i o n s o f F i g u r e 67. 1 1 3  F i g u r e 70.  An a u t o r a d i o g r a p h s h o w i n g t h e c a s t s t r u c t u r e o f a t i n - 2 wt. % l e a d a l l o y d i r e c t i o n a l l y cast with T I * t r a c e r . 2 0 1  the  autoradiography  The; l e f t - h a n d the a to  side  nucleation  certain  distance  liquid  ditions  Table  gradient  between  in  the  question.  zone  is  graphs.  To  use to  of  versus  ature, plot  of  is  this  the  is  is  where  and  it  requires  dendritic  by  to  the  structure.  area  run  seen  controlled  be  directional  ail  the  is  obtained  an average  at  at  ;  growth,  be  towards  fluid  motion  to  the  front  of of  into  original  of  penetration  is'calculated.  72 t h i s  temperature  difference  the  derived  fraction  solid.  from the solid  at  is  Figure  72,  of  below  the  liquidus  temperature  phase the  mushy alloy the  along  solid, the  ;the  mushy  With  the  converted  weight  percent temper-?  diagram i n Figure  penetration  mushy  autoradio-  a fraction edge  in  #1  the  of  percent  con-  linear'  the  distance  outer  in  temperature,  distance  from the  the  shown  assuming a  temperature  between  are  thermocouple  the  distance  showing  casting  The  penetration  length  difference  various  by  penetration  this  alloys  castings.  liquidus  flow  point  the  temperature  the  castings  lead-tin  temperature  correct  Figure  a weight  solid  the  The  and the  is  the  for  interface  temperature zone  for  solid  t a k e n as  the", e n t i r e  of  directional  IX l i s t s  67 a n d t h e  taken to  as  cast  1  and r e s u l t s in  zone  start  growth d i r e c t i o n  flow  gradient  Figure  the  and t h i s  a typical  irregular  the  resultant  interdendritic 71.  at  70 s h o w s  pool.  The  Figure  very  for  The  upper, r i g h t  i n 'the  is  occurs  stabilize.  the  Figure  distance  68.  A  versus  b  Figure  71.  The t r a c e r d i s t r i b u t i o n s h o w i n g i n t e r d e n d r i t i c f l o w i n c a s t i n g s o f ( a ) t i n - 2 wt. % l e a d and ( b ) t i n - 5 wt. % l e a d w i t h t h e c a s t i n g c o n d i t i o n s of Figure 67.  Figure  71  continued. The t r a c e r d i s t r i b u t i o n s h o w i n g i n t e r d e n d r i t i c flow i n c a s t i n g s o f (c) t i n 12.5 wt. % l e a d and (d) t i n - 20 wt. % l e a d w i t h the c a s t i n g c o n d i t i o n s o f F i g u r e 67.  TABLE I X . Lead-Tin A l l o y wt. % Pb  2  Interdendritic  Temperature L i q u i d u s a t T/C #1 T e m p e r a t u r e °C °C  Plow  Results  Temperature Plow .Gradient Penetration °C/cm mm. (Average)  Percent S o l i d f o r No P l o w -  212.5  229  3.69  2.0  22.5  206  224.5  3.08  3.7  16.0  12.5  192  213.5  5.20  4.0  11.8  20  186  203.5  3.50  20  186.5  203.5  4.07  ' "5  ..•  8.9 . 6.. 1  12.5") V 11.3 .10.oJ  Figure  72.  A p l o t o f w e i g h t p e r c e n t s o l i d v e r s u s °C b e l o w t h e l i q u i d u s temperature o b t a i n e d from the phase diagram o f F i g u r e 68 f o r t h e f o u r a l l o y s c o n s i d e r e d .  o  4  8  WEIGHT  PERCENT  12  16  LEAD  A p l o t o f p e r c e n t s o l i d f o r no f l o w v e r s u s th«* weight p e r c e n t ' l e a d f o r the l e a d - t i n a l l o y i n t e r d e n d r i t i c flow experiments.  20  the percent  l e a d i n the  73.  a l l o y i s shown i n F i g u r e  p e n e t r a t i o n d i s t a n c e appears q u i t e i n s e n s i t i v e  The  t o the  lead  content. All Exact they  c a s t i n g s had  a very  m e a s u r e m e n t s were q u i t e d i f f i c u l t were a l l i n the  lead-tin  alloy  on F i g u r e ions the  66.  interdendritic Due  results  grown due  dendritic  flow  whole  conducted i n which  tracer penetration.  is s t i l l To  do  low  still  past  t h i s an  the  interden-  f l o w e x p e r i m e n t i s done e x a c t l y as d e s c r i b e d  about the  p r i o r t o the quench a  i n diameter,  same as  difficulty  considerable in  not  shown t h a t t h e mushy zone I s  p u s h e d i n t o t h e mushy z o n e .  h a v e no  mesh  compared w i t h a c t u a l i n t e r -  fraction solid  s t e e l w i r e , 0 . 6 mm.  is  approximat-  so t h a t t h e  e x p e r i m e n t can be  except that immediately  and  and  results.  open and t h e  dritic  are p l o t t e d  f i n e r d e n d r i t e s were  be  Figure  depth.  7^a.  but The  c o n s i s t e n t w i t h the  limitations  6 6 c o u l d not  i s experimentally  depth o f the  flow r e s u l t s  are remarkably  A direct  very  to accomplish  t o t h e many l i m i t a t i o n s  to experimental  range of F i g u r e  spacing.  range of 7 0 0 t o 1 0 0 0 microns.  model t h e o r y . . U n f o r t u n a t e l y ,  it  similar dendrite  The  the d e n d r i t e  above  stainless  i s plunged i n t o the  liquid  The  wire  d i a m e t e r o f the  spacing  so i t s h o u l d  i n p e n e t r a t i n g t h e mushy zone t o This experimental c a s t i n g i s o f 2 wt.  a  c a s t i n g i s shown % l e a d i n t i n and  b  Figure  74.  The t r a c e r p r o f i l e i n a t i n - 2 wt. % l e a d c a s t i n g r e s i d u a l l i q u i d p o o l showing the ( a ) i n t e r d e n d r i t i c f l o w and t h e o v e r a l l c a s t i n g s i z e and (b) t h e maximum p o s i t i o n r e a c h e d by t h e 0.6 mm. d i a m e t e r w i r e i n t o t h e i n t e r f a c e .  i s very alloy  s i m i l a r t o the casting of Figure  composition.  71a o f t h e same  The c a s t i n g was m a c h i n e d down f r o m t h e  l e f t - h a n d s i d e a t an a n g l e  t h e same as t h e o b s e r v e d  face u n t i l  the top o f the s t a i n l e s s  Figure  shows t h e o u t l i n e o f t h e c a s t i n g  74b  with the observed i n t e r d e n d r i t i c  s t e e l w i r e was met.  shows t h a t t h e mushy zone i s s t i l l  served  i s low past  for interdendritic  remaining  flow stopping w e l l before .  the p e n e t r a t i o n d i s t a n c e o f t h e w i r e .  fraction solid  This  qualitatively  q u i t e open and t h e  the p e n e t r a t i o n d i s t a n c e s obflow.  Much h a s b e e n s a i d i n t h e l i t e r a t u r e concerning of l i q u i d should  the i n t e r d e n d r i t i c  i n t o t h e mushy zone by v o l u m e c o n t r a c t i o n .  r e s u l t i n the p u l l i n g o f l i q u i d  regions  o f t h e mushy  The solute tin  i n t o the upper  alloys.  case c o n s i d e r e d  h e r e i s when t h e r e j e c t e d  flow .in Figure  71 t h e mushy zone e x t e n d e d b a c k  i s p o s s i b l e that the backflow  from, t h e r m a l  consider-  f r o m the- s h r i n k a g e  t h a n t h e 30 s e c o n d s b e f o r e t h e  quench o f t h e p r e v i o u s  experiments presented,  n o t be o b s e r v e d .  I f this  It  requires  a longer time t o occur  will  lead-  I n t h e e x p e r i m e n t s done t o show t h e i n t e r -  s o t h e w h o l e l e f t - h a n d r e g i o n i s a mushy z o n e .  backflow  regions  zone.  t h e l e f t - h a n d end o f t h e c e l l  ations  This  out o f the  i s d e n s e r t h a n t h e s o l v e n t as i n t h e t i n r i c h  dendritic to  recently  f l o w c a u s e d by t h e p u l l i n g  o f t h e mushy zone a n d t h e r e j e c t i o n o f l i q u i d lower  inter-  such t h a t the  i s s o by l e a v i n g  the t r a c e r a l o n g e r p e r i o d o f time tracer w i l l  be a l l o w e d  before  quenching, the  t o flow i n t o the upper regions o f  t h e mushy zone and t h e n o n - t r a c e r  material w i l l  out  75 shows a c o m p a r i s o n  o f the lower  regions.  Figure  be p u s h e d  b e t w e e n two i d e n t i c a l r u n s o f l e a d - t i n a l l o y e x c e p t t h e c a s t i n g i n F i g u r e 75a was q u e n c h e d a f t e r  30 s e c o n d s a n d t h e  c a s t i n g i n F i g u r e 75b was n o t q u e n c h e d a t a l l b u t was to  solidify  completely.  R a d i o a c t i v e t i n was u s e d so t h a t  t h e r e w o u l d n o t be any o b s e r v e d on  solidification.  segregation of the t r a c e r  No f l o w p a t t e r n i s v i s i b l e  75b s i n c e t h e t r a c e r h a s a l o n g p e r i o d o f t i m e be  evenly  left  mixed throughout t h e melt.  i n Figure i n which t o  It i s easily  t h a t t h e shapes o f t h e i n t e r f a c e s a r e almost  seen  identical.  T h i s shows t h a t i f t h e r e was any f l o w I n t o t h e mushy as s o l i d i f i c a t i o n p r o g r e s s e d  zone  due t o v o l u m e c h a n g e on f r e e z i n g  o r d e n s i t y d i f f e r e n c e i n t h e i n n e r r e g i o n s o f t h e mushy zone i t does n o t c a u s e any l o n g r a n g e movement o f l i q u i d arid t h e f l o w r a t e s i n t h e mushy zone must be small.  extremely  b  Figure 75.  The t r a c e r p r o f i l e i n a t i n - 2 w t . % l e a d c a s t i n g r e s i d u a l l i q u i d p o o l w i t h (a) quenching 30 s e c o n d s a f t e r t h e t r a c e r i n t r o d u c t i o n and (b) complete 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 of the c a s t i n g w i t h o u t quenching.  7.  Macrosegregation i n Castings and  7.1.  O s c i l l a t e d During  Solidification  Introduction Castings  structure  are  w h i c h have a s m a l l  considered  t o be  better mechanical properties a partially the  grain structure  the  mould.  radically  i s by  than equivalent  solidification.  will  an e a r l i e r  CET;  and  The  m i x i n g of the —  mixing). solute  way  can  be  r o t a t i o n and  c a s t i n g may  (a f u n c t i o n of the  kind  and  i n the  and  solvent.  a  cases  mechanical  extent of  l i q u i d i f there  d i f f e r e n c e between s o l u t e  promote  cause macrosegre-  In a d d i t i o n , r o t a t i o n a l f o r c e s  transport  equiaxed  have  oscillation  moving  mould,  mould w i l l  c o n t r o l o f g r a i n s t r u c t u r e by l i q u i d during  with  residual  done by  columnar to  o f the  have  of c o n t r o l l i n g  a s t a t i o n a r y mould w i l l  s t r u c t u r e between the  gation  This  suppress the  (CET); o s c i l l a t i o n  42).  One  to  castings  Constant r o t a t i o n of a c y l i n d r i c a l cooled,  grain  m e c h a n i c a l l y m i x i n g the  transition  }  equiaxed  more homogeneous and  columnar s t r u c t u r e .  l i q u i d during  (l6  Rotated  might  liquid influence  i s a large  density  I f macrosegregation  i s e n h a n c e d by l i q u i d m i x i n g t h i s c o u l d casting  be d e t r i m e n t a l  to  quality. The  purpose o f the present i n v e s t i g a t i o n i s t o  determine the extent o f macrosegregation i n s t a t i o n a r y , rotated,  and o s c i l l a t e d c a s t i n g s ,  to the cast  7.2.  structure.  Experiment The  macrosegregation i n the castings  m i n e d by a r a d i o a c t i v e was  and r e l a t e t h e r e s u l t s  tracer technique.  were c y l i n d r i c a l ,  The a l l o y u s e d  Figure  76  f  The c a s t i n g  apparatus  used,  e n a b l e d t h e c a s t i n g o f A l - A g a l l o y s t o be made  i n s t a t i o n a r y , . r o t a t i n g , o r o s c i l l a t i n g moulds. was  done i n a g r a p h i t e  The  a l l o y was s u p e r h e a t e d t o a p p r o x i m a t e l y  crucible i n a resistance  immediately p r i o r t o c a s t i n g , a small Ag "'" was a d d e d i n t o t h e m e l t . 0  Melting furnace.  800°C a n d  \  amount o f r a d i o a c t i v e  The c a s t i n g s  were a l l p o u r e d  750°C (90°C s u p e r h e a t ) i n t o s t a i n l e s s s t e e l m o u l d s ,  w a t e r c o o l e d b e f o r e and d u r i n g top  The  3 1/4 i n c h e s i n d i a m e t e r and  approximately 6 inches high.  at  deter-  A l - 3 w t . % Ag made up o f 99.99% A l a n d 99.8% A g .  ingots  1  was  casting.  was u s e d t o keep t h e h e a t t r a n s f e r  surface  t o a minimum.  stationary mould.  Three c a s t i n g  mould, r o t a t e d  A graphite hot from the upper  conditions  were u s e d :  m o u l d a t 126 rpm, and a n o s c i l l a t e d  The o s c i l l a t i o n was a r o t a t i o n o f 126 rpm w i t h t h e  d i r e c t i o n o f r o t a t i o n being r e v e r s e d every f i v e seconds.  HOT TOP  S T E E L MOULD  o;-  :  5-  WATER COOLING Al-Ag CASTING  MOTOR-*  Figure  76.  The e x p e r i m e n t a l a p p a r a t u s u s e d f o r p r o d u c i n g the s t a t i o n a r y , r o t a t e d , and o s c i l l a t e d c a s t i n g s o f A l - 3 wt. % A g .  The c a s t i n g m i c r o s t r u c t u r e was d e t e r m i n e d s e c t i o n i n g and e t c h i n g o f t h e c a s t i n g s p a r a l l e l p e n d i c u l a r t o the c y l i n d r i c a l a x i s . a M o d i f i e d Tucker  e t c h ( H C 1 , HNO^,  by  and p e r -  E t c h i n g was done i n HP, and H 0  in a  2  2:2:1:15 r a t i o ) and t h e e t c h i n g p r o d u c t s w e r e w a s h e d o f f immediately with concentrated n i t r i c  acid.  To m e a s u r e t h e m a c r o s e g r e g a t i o n t h e most e x p e d i e n t p r o c e d u r e , measure t h e s o l u t e  i n the ingot,  as commonly u s e d ,  i s to  concentration of cuttings taken at  v a r i o u s p o i n t s - i n t h e i n g o t by d r i l l i n g .  This i s s a t i s -  f a c t o r y i f t h e r e i s no m i c r o s e g r e g a t i o n and no s h o r t variations case.  i n the macrosegregation, which  To i m p r o v e  and-analyses  i s r a r e l y the  t h e a v e r a g i n g p r o c e s s , more  c o u l d be made, o r a l t e r n a t i v e l y  drillings a l a y e r o f the  c a s t i n g c a n be- m a c h i n e d o f f and s a m p l e s t a k e n f r o m F i n a l l y , the e n t i r e  range  this.  c a s t i n g c a n be m a c h i n e d a n d t h e c o n -  c e n t r a t i o n o f a l l t h e c a s t i n g by s e c t i o n s c a n be m e a s u r e d . The t i m e and e f f o r t the f i r s t all  to f i n a l  process l i s t e d  four procedures  accuracy  i n v o l v e d increases very g r e a t l y  were used  and r e p r o d u c i b i l i t y  above.  initially  Accordingly, t o determine  f o r the present  (a)  Holes  the  macrosegregation. o f 1/4  t h e c a s t i n g i n 1/4 f i x e d weight  their  castings.  F o u r methods o f s a m p l i n g w e r e e m p l o y e d t o d e t e r m i n e degree o f r a d i a l  from  I n c h d i a m e t e r were d r i l l e d  through  inch steps i n the r a d i a l d i r e c t i o n .  of solid  t u r n i n g s was p a c k e d  A  into a standard  container. in  The  activity  e a c h s a m p l e was  well  method  Holes  (a)  the  inch  of  except  The  thick  concentric  t a k e n and  (d)  by  a  inch  were  The  silver  present  scintillation  0.050  each  cut  was  acid  and In  Ag^""^  is is  were  samples  then  evaluating  in  drilled  used.  The  the  inch  in  long  as  in  analysis  All  the  in  a small  amount  made u p  to  each  the  to  of (c)  progres-  in  a geometry.  except removed  from  of  in  solution.  500 m l . each  measured. concentration  the  measured  of  activity.  weak  differences  techniques  should  in be  negligible. Autoradiography the  macrosegregation  in  the  was  used to  ingots.  show  Since  qualitatively the  -  sample  solution  and aluminum a  (c)  0.030  fixed  250 o r  geometrical  and  gram  mercury  taken  sample  results  small  of  either  s a m p l e was  the  were  material  a concentrated  of  and  sample measured conditions  thick.  (b)  a lathe  machined as  gamma e m i t t e r  (a),  in  a random f i v e  be p r o p o r t i o n a l  a result,  counted  one  under  were  activity  a strong As  of  A 10 m l .  the  were  mounted  cut  counter  dissolved  taken to  absorber.  were  Inches  adding water.  solution  steps  cylinders  containing  solutions  diameter  (a).  castings  was  silver  inch  activity  well  cut  nitric The  radioactive  measured w i t h  From each  the  scintillation  in  1/8  castings  sivelyv-removed.  the  1/4  same a s m e t h o d  (c)  was  the  counter. (b)  was  then  of  energy  of  the  radiation  the  autoradiograph  distance  is  into  high  the  and t h e  will  represent  ingot.  thin  pendicular  to  the  cylindrical  and p o l i s h i n g  the  discs  sections  ary,  rotating,  shown i n  to  activity  to  axis  of  obtain  required.  low a  a thickness  prepared of  large  reasonable  Thin discs  were  centre, centre large  clearly  rotated  the  The  at  the  applied the  to  higher  have  per-  by  0.020  machining  inches.  and the in  region  a columnar oscillated  the  station-  with  are  The  in  zone  the  to  the  c a s t i n g has previous  oscillated in  a  obser-  ingot  Figure  o b t a i n e d on e t c h e d  79a,  f l u i d - m o t i o n on t h e stationary  cast  to  mould w a l l .  with  shows  growing  respectively.  equiaxed  in  were  solidification  grown d e n d r i t i c a l l y  ingots -.(Figures  79b)  77c  agreement  grains  structures  direction  (Figure  and  which  are 78,  magnification.  radial the  castings  c a s t i n g has  equiaxed  shown  of  a small  casting,  The  grains  77b,  equiaxed:region,  vations.  taken  77a,  c a s t i n g has  the of  sections  and o s c i l l a t i n g d u r i n g  Figures  stationary  of  are  is  Results  Vertical  of  the  Therefore  resolution  7.3  sample a b s o r p t i o n  79b,  The  show  the  effect  columnar  zone.  In  Figure  the  the  columnar region  grains  columnar  a spiral  sections  79c)  ingot all  cross  growing  region  shape w i t h  non-perpendicular  to  the  for the  is  of  in  the  79a a  perpendicular the  rotated  initial  mould  ingot  columnar  walls.  Figure  77.  R e p r e s e n t a t i v e i n g o t s c a s t i n (a) s t a t i o n a r y , ( b ) r o t a t i n g , and ( c ) o s c i l l a t i n g m o u l d s . Approximately 2/3 magnification.  Figure  79.  Representative ingot cast mould, actual s i z e .  in  a  (a)  stationary  Figure  79  continued. R e p r e s e n t a t i v e i n g o t s c a s t i n (b) r o t a t i n g and ( c ) o s c i l l a t i n g m o u l d s , a c t u a l size.  These g r a i n s a r e liquid  pool.  growing towards the  In the  d i r e c t i o n of the  o s c i l l a t e d ingot  In both the  effects.  has  b e e n made by  r o t a t e d and  of the  T h e r e w e r e no  always  c a s t i n g u s i n g the  i s evident  with  80.  The  t h a t they are of the the  from the  experimental  genuine the  1/8  Again the p o i n t s  outer  due  the d r i l l  them.  described  the  is  points  have For  the  various drilled  macrosegregation  mould w a l l t o the c e n t r e l i n e .  hole.  of the  To  concentration  scintillation  t o a change i n t h e test i f this  counter.  Thus  concentration  c y c l i c behavior  is  i n c h s t e p h o l e method i s shown i n F i g u r e show a c y c l i c b e h a v i o u r ,  p e r i o d i s d i f f e r e n t f o r the along  had  in a stationary  four sets of  ( F i g u r e 80a)  a true representation  drill  o f Ag  d i f f e r e n t techniques  sample measured i n t h e  the  curved  accuracy of the p o i n t s p l o t t e d i s such  s c a t t e r must be  along  the  is a  c r y s t a l l o g r a p h i c growth o r i e n t -  Comparing the  inch steps  appears c y c l i c  Schippen  ingot there  four sampling techniques  t h a t the  1/4  flow.  g r a i n s observed which curved or  degrees of s c a t t e r a s s o c i a t e d w i t h holes  the  mould  R o t h and  oscillated  concentration  ;  it  the  maintained.  The solute  shown i n F i g u r e  79c)  columnar g r a i n s to achieve  a k i n k , i n d i c a t i n g t h a t the a t i o n was  in  so t h a t t h e y a l w a y s grow i n t o t h e  A similar observation  ^enucleation  (Figure  c o l u m n a r zone c h a n g e s when t h e  r o t a t i o n i s reversed  (43).  oncoming f l u i d  hole  must be  two the  cases.  but  Thus  cause o f the  the  80b.  cycle  microsegregation cycling.  The  3.2r  (a)  3.0 2.81-  3.2l h o  o  3.0  (b)  2.8 or UJ > _j  <n  \z  UJ  o or  3.2 3.0  0  (c)  6  cP cP  UJ Q. V-  o  CO  2.8  oo o  0  o o o  UJ  5s 2.6 3.2 3.0  n  o oo° e° 0  0  (d)  0  2.8 0  05  1.0  J  1.5  I  DISTANCE FROM MOULD WALL (INCHES) F i g u r e 80.  The r a d i a l s i l v e r d i s t r i b u t i o n i n a s t a t i o n a r y c a s t i n g ; ( a ) 1/4 i n c h d r i l l h o l e s i n 1/4 i n c h s t e p s , ( b ) 1/4 i n c h d r i l l h o l e s i n 1/8 i n c h s t e p s , ( c ) 0.030 i n c h l a t h e t u r n i n g s , a n d ( d ) 0.050 i n c h l a t h e t u r n i n g s d i s s o l v e d i n a c i d .  plot  f o r the s o l i d  l a t h e t u r n i n g s ( F i g u r e 80c)  e x t e n s i v e s c a t t e r between p o i n t s . be  sufficiently of the  lathe turnings.  t u r n i n g s o f the e n t i r e w h i c h shows much l e s s this  case  liquid  s a m p l e g e o m e t r y due The  the c o u n t i n g geometry i s not  nature  a p r o b l e m as  t u b e and  was  the  80d,  In  the  liquid This  used f o r a l l the  measurements.  The. r a d i a l m a c r o s e g r e g a t i o n i n g o t s I s shown i n F i g u r e 8l.  i n the t h r e e types  of  T h r e e s e t s o f d a t a were  f o r the r a d i a l m a c r o s e g r e g a t i o n ,  r e g i o n o f one g r o u p o f c a s t i n g s , and :  the r e s u l t s  the  of Figure  i s a t r u e average o f the sample c o m p o s i t i o n .  n e a r t h e t o p and  a  s c a t t e r than the o t h e r methods.  subsequent macrosegregation  All  having  to the  s a m p l e gave t h e r e s u l t s  method gave r e p r o d u c i b l e r e s u l t s and  obtained  the  f i n a l method o f d i s s o l v i n g  i s contained i n a standard  counted  could  the s e l e c t i o n of  f r o m t h e w h o l e s a m p l e , o r by n o t  constant  an  This large scatter  c a u s e d by t h e m i c r o s e g r e g a t i o n , by  m a t e r i a l taken  shows  one  from the  t h e o t h e r two  bottom o f a second s i m i l a r  central  from  group o f c a s t i n g s .  f o r a p a r t i c u l a r t y p e - o f c a s t i n g were  very  similar. The  s i l v e r c o n c e n t r a t i o n i n t h e s t a t i o n a r y and  r o t a t e d i n g o t s , shown i n F i g u r e s constant  indicating  little  8la  and  8lb,  macrosegregation,  is essentially except  s m a l l drop i n c o n c e n t r a t i o n at the c e n t r e l i n e T h e r e i s no e f f e c t difference  on t h e m a c r o s e g r e g a t i o n  i n . d e n s i t y of the s i l v e r  and  due  for a  of the to  casting.  the  aluminum i n the  (a)  o o  o o o —=—o-o  JX.  3.0  OO  0  o  2.9  ?  o i  0  I I  3.1  or UJ  (b) °  3.0  > _j  o or  „  ©  o  o  "—r-™  o  [ oo  o  3.2  UJ  0-  ? o  3.1 i CD UJ  o  O  o  2.9 Z UJ  O  /  3.0  oo %  I  _£  (c)  °\>  ©_i.  o 2.9  N  U P  2.8 h i  o  2.7I> 0  •1 0.5  1.0  1.5  I  DISTANCE FROM MOULD WALL (INCHES)  Figure  81.  T h e ' r a d i a l s i l v e r d i s t r i b u t i o n i n (a) s t a t i o n a r y , (b) r o t a t e d , a n d ( c ) o s c i l l a t e d i n g o t s u s i n g method (d) o f F i g u r e 8 0 .  igo  rotated casting.  In the  i s p r e s e n t , w i t h an up  to a peak.  The  c e n t r e l i n e o f the the  Co  value.  curve; The  oscillated  initial  i n the  p o s i t i o n of the  r e s u l t s show t h a t t h e  o v e r 0.25%  CET  CET  Ag  i s shown on CET  the below the 77c.  i n Figure  corresponds to the  maximum  concentration. An  (Figure  82)  autoradiograph  of the  oscillated  s i l v e r depleted  centre  of the  areas which correspond  measurements ( F i g u r e  8lc).  Due  s i l v e r d i s t r i b u t i o n i t can  ingot  a n a l y s i s t h a t does not  The  represent  t o the q u a n t i t a t i v e  to t h i s mottled e a s i l y be  effect  in  seen t h a t the  h o l e methods o f a n a l y s i s c o u l d g i v e s p u r i o u s any  Ingot  shows t h e m a c r o s e g r e g a t i o n q u a l i t a t i v e l y .  l i g h t e r areas towards the  can  concentration  concentration then decreases to c a s t i n g to a value  The  macrosegregation silver  t h i s p o s i t i o n corresponds t o the  silver  the  rise  case  i n c l u d e the  results,  total  drill as  radial  sample.  7.4.  Discussion The, c a s t s t r u c t u r e a s s o c i a t e d w i t h s t a t i o n a r y ,  r o t a t e d , and reported be  by  oscillated Cole  and  discussed here. (1)  w i t h the (2)  c a s t i n g s are  Boiling Two  The-; r e l a t i o n  (16)  s i m i l a r to  a n d : o t h e r s and  points w i l l  be  of the observed  shape o f t h e  solute  will  not  considered. macrosegregation  cast s t r u c t u r e . The  those  curves.  Figure  82.  An a u t o r a d i o g r a p h o f t h e c r o s s - s e c t i o n o f the o s c i l l a t e d ingot showing the s i l v e r d i s t r i b u t i o n i n the c a s t i n g , a c t u a l s i z e .  Conditions the o s c i l l a t e d  c a s t i n g than i n e i t h e r the r o t a t e d or s t a t -  ionary casting.  This  temperature gradient causes e x t e n s i v e by  f o r t h e CET a r e met much e a r l i e r i n  i s b e l i e v e d due t o t h e l o w e r i n g o f t h e i n the molten region  (oscillation  mixing) thus a l l o w i n g " n u c l e i " ,  produced  l a r g e s h e a r f o r c e s a t t h e i n t e r f a c e , t o s u r v i v e and grow.  During  r o t a t i o n r e v e r s a l these  large shear forces  cause r e m e l t i n g and/or b r e a k i n g (l4,  15)  o f fof dendrite  w h i c h c a n a c t as " n u c l e i " .  will  fragments  These can e a s i l y  be  swept i n t o t h e c e n t r a l r e g i o n o f t h e l i q u i d p o o l by t h e v i o l e n t t u r b u l e n t f l o w o c c u r r i n g as a r e s u l t Observation  of this  flow i n a rheoscopic  of oscillation.  liquid  shows t h e  e x i s t e n c e o f a t u r b u l e n t wave g e n e r a t e d a t t h e i n t e r f a c e and  r a p i d l y moving t o t h e c e n t r e .  at a constant high face. be  (16)  I f t h e mould i s r o t a t e d  speed t h e temperature g r a d i e n t w i l l  a n d no s h e a r f o r c e s w i l l  Therefore,  be p r e s e n t  natural convection  will  will  For the s t a t i o n a r y ingot .  yield  low shear f o r c e s a t t h e  i n t e r f a c e and t h e t e m p e r a t u r e g r a d i e n t w i l l  be o f some  value.  The stationary  at the i n t e r -  no n u c l e i w i l l be p r o d u c e d a n d t h e CET  s u p p r e s s e d , as o b s e r v e d .  intermediate  remain  l a c k o f m a c r o s e g r e g a t i o n i n t h e r o t a t e d and  i n g o t s c a n be a t t r i b u t e d t o t h e l a c k o f s i g n i f -  icant  fluid  flow i n the i n t e r d e n d r i t i c  fluid  f l o w t h e r e w i l l be no n e t s o l u t e f l u x f r o m t h e i n t e r -  f a c e r e g i o n , and t h e r e f o r e , no  region.  macrosegregation.  Without  For the o s c i l l a t e d in  the v i c i n i t y  c a s t i n g high shear  of the s o l i d - l i q u i d  more e x t e n s i v e i n t e r d e n d r i t i c solute rich  liquid  interface will  flow.  the equation C  s  where C  =  f o r complete  kCoCl-g) "1  produce  This flow w i l l  o u t o f t h e mushy z o n e .  d i s t r i b u t i o n n e a r t h e mould w a l l w i l l to  forces  sweep  The s o l u t e  then tend  t o conform  mixing:  1  i s the s o l i d solute composition,  k the d i s t r i -  s b u t i o n c o e f f i c i e n t , Co t h e a v e r a g e i n i t i a l and  g the f r a c t i o n s o l i d i f i e d .  versed this  columnar growth t h e s o l u t e  t h e r e f o r e be as shown i n F i g u r e  initial  mixing  p a r t o f the curve  incomplete  mixing  corresponds  distribution  When t h e CET i s  t o a complete  B e y o n d t h e p e a k t h e com-  t o w a r d s t h e c e n t r e due t o t h e  of the solute rich  interface.-' Concurrent  83a.  ingot.  be t h a t shown i n F i g u r e . 83b.  s i t u a t i o n up t o t h e p e a k .  p o s i t i o n g r a d u a l l y decreases  the  transport  s o l u t e towards t h e centre o f t h e s o l i d i f y i n g  immenent t h e d i s t r i b u t i o n w i l l The  composition  When t h e r o t a t i o n i s r e -  a t u r b u l e n t wave i s p r o d u c e d w h i c h w i l l  During will  solute  with this  liquid  generated a t  s o l u t e movement i s t h e  r e d u c t i o n o f the temperature gradient which, u n t i l o c c u r s , does n o t a l l o w s u r v i v a l o f n u c l e i . r o t a t i o n r e v e r s a l high shear  dendrite fragments.)  (At every  f o r c e s and t e m p e r a t u r e  u a t i o n s , due t o t u r b u l e n c e , w i l l  t h e CET  fluct-  p r o d u c e a l a r g e number o f  When t h e t e m p e r a t u r e g r a d i e n t i s i  DISTANCE FROM MOULD WALL  F i g u r e 83.  The d e v e l o p m e n t o f t h e r a d i a l m a c r o s e g r e g a t i o n i n an o s c i l l a t e d i n g o t , (a) p r i o r t o t h e time o f t h e CET, ( b ) a t t h e t i m e o f t h e CET, a n d ( c ) the f i n a l s i l v e r d i s t r i b u t i o n i n t h e c a s t i n g .  sufficiently  low these fragments w i l l  be a b l e  and  grow a n d be swept by t h e t u r b u l e n t wave  the  remaining  liquid.  Since  will  i n length.  83b - F i g u r e  (Figure  Appendix, S e c t i o n 7.6. 83c w i l l  Figure  solidification.  83c).  of  be t h a t o f a n i n g o t o s c i l l a t e d  p r o f i l e of during  The m a c r o s e g r e g a t i o n p r e d i c t e d a b o v e was 8lc.  Conclusion  significant  i n v e s t i g a t i o n h a s shown t h a t no  macrosegregation accompanies s o l i d i f i c a t i o n  s t a t i o n a r y o r r o t a t e d moulds f o r t h e system examined. :  This  i m p l i e s t h a t t h e l a r g e d e n s i t y d i f f e r e n c e between t h e  solvent  ( A l ) and s o l u t e  solute distribution. is  zone  reduction In  The s o l u t e d i s t r i b u t i o n  The p r e s e n t  in  t h e mushy  i s c a l c u l a t e d i n the  observed i n the o s c i l l a t e d i n g o t , Figure  7.5  A l s o , due  The mass and c o m p o s i t i o n  fragments n e c e s s a r y t o cause t h i s  composition  i n this  83c.  o f the temperature gradient  be i n c r e a s e d  dendrite  composition  be r e d u c e d as shown i n F i g u r e  to the lowering  throughout  t h e s e f r a g m e n t s a r e o f com-  p o s i t i o n l e s s t h a n Co t h e o v e r a l l region w i l l  to survive  associated with  The i n i t i a l liquid  (Ag) h a s no e f f e c t on t h e r a d i a l  However, a p p r e c i a b l e  macrosegregation  t h e o s c i l l a t i o n mode o f s o l i d i f i c a t i o n .  r i s e i s a t t r i b u t e d to solute mixing  i n t e r f a c i a l r e g i o n due t o t h e t u r b u l e n t  i n the flow.  The  maximum c o n c e n t r a t i o n  i s associated with  the columnar t o  equiaxed t r a n s i t i o n .  The s o l u t e d e p l e t i o n I n t h e c e n t r e  of the  c a s t i n g i s c a u s e d by  concentration  being  mushy zone t o t h e  7.6.  s w e p t , by  ingot  the  t u r b u l e n t waves, from  model p r o p o s e d f o r m a c r o s e g r e g a t i o n i n  swept i n t o t h e  i n the  central liquid  mass  i s the  volume a f t e r t o t a l  distribution profile given  /*  R  /o  where C ( r )  volume o f the  CET.  The  In t h i s  r e g i o n and  2tr r  liquid  assuming a Co  = 3.0  in  linear wt.%  Ag,  rdr  2TT  i s composition  as  a f u n c t i o n o f r a d i u s and  t h e r e f o r e C = 3.02  wt.  estimate CET  wt.  central  C(r)dr  From F i g u r e  3.10  of  average composition  solidification  i s r a d i u s o f CET.  to the  following  by:  C =  an  and  change  required.  r e g i o n j u s t p r i o r to the  83c  the  i s an a p p r o x i m a t e c a l c u l a t i o n f o r t h e  Assume Vg  is  zone, to give 83b  the  fragments  The  fragments which.is  this  dendrite  83c.  d i s t r i b u t i o n between F i g u r e  analysis  the  7.  o s c i l l a t e d i n g o t assumes s u f f i c i e n t are  solute  centre.  Appendix to s e c t i o n The  s m a l l g r a i n s o f low  %  can Ag.  of the be  % Ag.  8lc,  C ( r ) » -^P- r + 83b  Comparing F i g u r e s  average composition  made ( C ( r ) = -^-jjp r + 3.00)  i n V"  E  and  R 2.70, and  just  prior  i s equal  to  Assuming the swept i n t o t h e  central  increase  d e n d r i t e b r a n c h e s as t o be  1.5.  K  Therefore:  a  K  C 0  o  f o r the  central  the  be  in  the  assumed  a l l o y under c o n s i d e r a t i o n .  1  wt.  volume o f d e n d r i t e  molten region  resulting  factor  = < • 5) (0 . 25) ( 3.0) = 1.12  L e t v be  fragments  i n solute concentration  s o l i d i f i c a t i o n progresses,  i s 0.25  Q  of dendrite  r e g i o n i s cxK C , w h e r e a i s a • ° o o  r  to account f o r the  composition  (Vg)  change i n t h e  %  Ag.  f r a g m e n t s swept i n t o  at the  time o f the  CET.  average c o n c e n t r a t i o n  the The  i n Vg  can  then  u s e d t o s o l v e f o r v.  Therefore:  3.10  Therefore:  v = 0.04 This  solid  V  E  + 1.12  v = 3.02  (V  +  E  V"  E  calculation  shows t h a t a s m a l l amount  fragments i s r e q u i r e d r e l a t i v e to the  volume t o c a u s e t h e  l a r g e mushy z o n e , due  to the  the  large interdendritic  f l o w , due  rotation  effect.  low  reversal,  s m a l l volume o f f r a g m e n t s b e i n g the observed  total  decrease i n concentration  The  duced d u r i n g the  v)  thermal  to the should  of  liquid  observed. gradient,  turbulence result  made a v a i l a b l e  to  and  pro-  in this cause  8.  Summary a n d  Natural  convection'.in  Conclusions  liquid  t i n and l i q u i d  l e a d h a s b e e n o b s e r v e d i n a s m a l l c l o s e d s y s t e m by a radioactive t r a c e r technique. that the d i s t r i b u t i o n  I t has been  demonstrated  o f r a d i o a c t i v e m a t e r i a l i n t h e quenched  metal corresponds t o the d i s t r i b u t i o n i n the l i q u i d to  quenching. To e x a m i n e t h e r m a l  c a r r i e d out i n pure l i q u i d conditions. liquid  and For  prior  convection,  t i n and l e a d u n d e r a v a r i e t y o f  I t was o b s e r v e d t h a t t h e f l o w r a t e s i n t h e  increase.:with (a)  i n c r e a s i n g temperature d i f f e r e n c e across  (b)  I n c r e a s i n g average  (c)  increasing thickness  0.23°C.to 19°C ( c o r r e s p o n d i n g g f o r a 6.4 cm. w i d e  c y c l e around the c e l l  the c e l l  temperature o f the l i q u i d  a temperature d i f f e r e n c e across  t o 10  e x p e r i m e n t s were  the c e l l  zone. ranging  from  t o G r a s h o f numbers f r o m 1 0 ^  cell)  the time f o r a complete  v a r i e d b e t w e e n 780 and 12 s e c o n d s .  When e i t h e r t h e t e m p e r a t u r e d i f f e r e n c e o r t h e c e l l  thickness  was i n c r e a s e d  increased  beyond  a c e r t a i n point the flow rate  and  t h e mode o f f l o w c h a n g e d f r o m a s i m p l e  complex t h r e e d i m e n s i o n a l  flow pattern.  o s c i l l a t i o n s were d e t e c t e d  cell  No  i n the l i q u i d  t o a more  thermal  f o r the entire  range o f t e m p e r a t u r e d i f f e r e n c e s and c e l l  thicknesses  examined. The to  f l o w p a t t e r n i n t h e t h i n c e l l was f o u n d  be a f u n c t i o n o f t h e l e n g t h t o h e i g h t  o f 4.9 : 1 o r l e s s while  for a ratio  in nature.  ratio.  the flow p a t t e r n i s a s i n g l e o f 8.3 : 1 t h e f l o w becomes  For ratios cell,  multicellular  The p a t t e r n a l s o c h a n g e s i f t h e t e m p e r a t u r e  d i s t r i b u t i o n i s changed.  I f both  sides o f the c e l l are  cooled w i t h respect t o the c e n t r e , a double flow p a t t e r n i s o b t a i n e d , each c e l l c e l l described oreviously.  corresponding  The w i d t h  cell  t o the simple  o f the c e l l s are  d e t e r m i n e d by t h e p o s i t i o n o f t h e maximum t e m p e r a t u r e i n the l i q u i d .  A b u f f e r zone e x i s t s b e t w e e n t h e two c e l l s  i n h i b i t i n g mass t r a n s f e r f r o m one c e l l  to the other.  In  l a r g e r c a s t i n g s w i t h many s o l i d i f i c a t i o n d i r e c t i o n s a n d complex thermal  c o n d i t i o n s , the present  s u g g e s t t h a t many c e l l s a c t i o n , o r at l e a s t very cells.  results  would  a r e s e t i n m o t i o n w i t h no i n t e r little  i n t e r a c t i o n between  Thus l o n g r a n g e t r a n s p o r t o f s o l u t e o r s o l i d  these parti-  c l e s i n t h e l i q u i d may be s e v e r l y r e s t r i c t e d by t h e m u l t i cellular  behavior. An  a n a l y s i s o f the thermal  convection  problem  h a s b e e n made f o r t h i s numerical  procedure.  P r a n d t l numbers numbers  system u s i n g a f i n i t e d i f f e r e n c e The a n a l y s i s h a s b e e n s o l v e d f o r  o f 1 0 . 0 , 1.0, 0.1 and 0.0127 w i t h  of 2 x 10  3 J  ' 7 t o 2 x 10'.  • The t h e r m a l  Grashof  profile  has  b e e n f o u n d t o be a f u n c t i o n o f o n l y t h e R a y l e i g h number while the flow v e l o c i t i e s and  G r a s h o f numbers  are a f u n c t i o n o f both  independently.  The n u m e r i c a l  a g r e e s w i t h an a n a l y t i c a l s e r i e s s o l u t i o n of the Grashof  values  number. flow i n metals,  have been r e p o r t e d I n t h e l i t e r a t u r e o f f l o w  i n t r a n s p a r e n t - l i q u i d systems w h i c h have been to l i q u i d metals. serious limitations flow behaviour profiles  solution  f o r low  I n an a t t e m p t t o d e t e r m i n e f l u i d observations  the P r a n d t l  The p r e s e n t  show t h a t t h e r e a r e  i n extrapolating non-metallic  to l i q u i d metals.  i n different  simultaneously  results  extended  types  The t h e r m a l  of fluids  and  flow  c a n n o t be m a t c h e d  a n d h e n c e phenomena d e p e n d i n g on  p a r a m e t e r s must be a l t e r e d  fluid  f o r various types  of  both fluids.  Measurements o f t e m p e r a t u r e g r a d i e n t s i n c a s t i n g s have been used t o e s t i m a t e Often  fluid  flow during  no t e m p e r a t u r e d i f f e r e n c e s a r e o b s e r v e d  solidification. i n the centre  o f t h e c a s t i n g ;from w h i c h i t i s c o n c l u d e d  t h a t t h e r e i s no  thermal  erroneous,  convection.  T h i s c a n be e n t i r e l y  shown by t h e t h e o r e t i c a l  solution  as  i n F i g u r e 2 7 f where t h e  c e n t r a l r e g i o n has e s s e n t i a l l y a z e r o h o r i z o n t a l t e m p e r a t u r e  „• •  "  gradient.  '  "  However, t h e r e i s e x t e n s i v e f l o w a s s o c i a t e d  w i t h s i g n i f i c a n t temperature' g r a d i e n t s near the Accordingly to p r e d i c t thermal profile  i n the  between the  g r a d i e n t s i n the The liquid  boundaries  and  not  thermal  relevant  the  para-  difference temperature  cell.  experimental  thermal  convection results  in  t i n tend towards the t h e o r e t i c a l s o l u t i o n f o r l a r g e  temperature  d i f f e r e n c e s and  seen t h a t f o r a l i q u i d the e x p e r i m e n t a l The  The  convection i s the-temperature  flow c e l l  walls.  convection a complete  l i q u i d pool i s required.  meter i n thermal  cell  and  thermal p r o f i l e s  cell  large c e l l  thicknesses.  t h i c k n e s s o f 1.5  i n t h e 0.95  cm.  thick  cell  (b)  t h e . I n f l u e n c e o f s o l u t e c o n v e c t i o n on  mixing.  differ-  experiments:  independent s o l u t e  (c)  are i n  by t h r e e  (a)  convection,  correspond.  results.  Solute convection Is observed of  or greater,  t h e o r e t i c a l r e s u l t s should  c l o s e agreement w i t h the t h e o r e t i c a l  ent s e r i e s  cm.,  convection thermal  and the thermal  and  s o l u t e c o n d i t i o n s f o r complete  I t i s . shown t h a t i f a v e r t i c a l  stable solute  d i f f e r e n c e i s I m p o s e d on a s y s t e m , t h e h o r i z o n t a l c o n d i t i o n r e q u i r e d t o cause complete m i x i n g zone must be inversion.  It is  sufficient  enough, a m u l t i c e l l  i n the  t o cause a h o r i z o n t a l  If. the temperature  liquid  density  d i f f e r e n c e i s not  flow pattern w i l l  thermal  great  result with a buffer  zone between t h e  The  flow  interdendritic  due  to the  The  flow penetrates  in  the  cells.  residual  liquid  fluid  pool  flow i n l e a d - t i n  c o n v e c t i o n has  t o a d i s t a n c e o f from  solid-liquid  zone.  The  alloy  12  been  primary, d e n d r i t e s p a c i n g o f a p p r o x i m a t e l y  examined.  t o 22%  c a s t i n g s have 700  to  alloys  solid a  1000  microns.  Three e x p e r i m e n t a l dritic  The  flow are  presented,  (a)  a two  (b)  a three  (c)  a wire  wire  dimensional  spacings, results  were n o t  The  tensive  and  extent  r o d model,  and  lead-tin  conducted  over  conducted  the wire  carried  the primary  of i n t e r d e n d r i t i c  Although  alloy  rotation  liquid  growth  d i d agree  with  the  mesh m o d e l .  out  in Al-3%Ag  fluid  flow c o n s i d e r a t i o n s ,  to determine  alloys  occurs  the  fluid  macroflow p a t t e r n s .  subjected to  during s o l i d i f i c a t i o n  segregation occurs  dendrite  a wide r a n g e o f d e n d r i t e  i n . c a s t i n g s w h i c h have Imposed  macrosegregation  little  t h a t as  e x t e n s i o n o f the  macrosegregation  lation  but  an  i n v e s t i g a t i o n was  segregation  1  experiments  p r e d i c t e d by As  an  the  increase.  the  wire  model  mesh m o d e l .  spacing decreases,  experiments  channel  dimensional  mesh-model p r e d i c t s  flow w i l l  models t o a n a l y s e i n t e r d e n -  shows t h a t  oscilex-  in oscillated castings,  in stationary  and  rotated  castings. equiaxed  A concentration peak occurs at the columnar to t r a n s i t i o n and a general depletion occurs i n the  central equiaxed  zone.  The segregation i n the o s c i l l a t e d  casting i s accounted for by long range movement of dendrite fragments which break and/or melt o f f i n the s o l i d - l i q u i d interface region.  9.  (a)  Suggestions  Work  Using the t r a c e r techniques developed  work, o b s e r v a t i o n s be  f o r Future  on n a t u r a l  and f o r c e d  in this  convection  made on much more complex s y s t e m s , c o n s i d e r i n g  t h e r m a l and s o l u t e  driving forces.  advancing s o l i d - l i q u i d could  configurations  This  could  both  t h e e f f e c t s o f an  i n t e r f a c e on t h e n a t u r a l  be i n v e s t i g a t e d .  geometric  Also  could  be a p p l i e d  s u c h as o b t a i n e d  convection  to specific  i n continuous  casting.  (b)  The i n t e r d e n d r i t i c f l o w e x p e r i m e n t s  continued ent less  f o r large  a l l o y systems.  ranges o f d e n d r i t e  (c) applied  interface  could  The t h e o r e t i c a l n u m e r i c a l  conditions analysis  liquid  variety  rate  analysis.  differsolute  solutes.  be  s h a p e s and t h e r m a l  situations.  be u s e d i n a more e x a c t  be  to observe  s o l u t i o n could  o f geometric  which o c c u r i n c a s t i n g  could  be a n a l y s e d  between t h i s and more dense  to a great  and  Systems w h i c h have t h e r e j e c t e d  dense than- t h e b u l k  flow d i f f e r e n c e s  spacings  could  This  s o l u t i o n t o a growing  (d)  More e x t e n s i v e e x p e r i m e n t s a r e r e q u i r e d i n o r d e r  to e s t a b l i s h  t h e t h e o r y t h a t a mass o f d e n d r i t e  a r e swept out o f t h e i n t e r f a c e i n o s c i l l a t e d produce the observed macrosegregation.  fragments  castings to  Bibliography  1.  J . P. Holman: Heat New Y o r k , 1 9 6 3 .  2.  J . 0 . W i l k e s and S. W. C h u r c h i l l : v o l . 1 2 , p. 1 6 1 .  3.  J . S z e k e l y and P. S. C h h a b r a : 1 9 7 0 , v o l . 1 , p. 1195.  4.  P. W e i n b e r g : C r y s t a l Growth, P r o c e e d i n g s o f an I n t e r n a t i o n a l C o n f e r e n c e on C r y s t a l G r o w t h , B o s t o n , 20-24 June 1 9 6 6 , p . 6 3 9 .  5.  J . S. K i r k a l d y and W. V. Y o u d e l i s : 1 9 5 8 , v o l . 2 1 2 , p. 8 3 3 .  6.  M. C. F l e m i n g s and G. E . N e r e o : v o l . 2 3 9 , p . 1449.  7.  M. C. F l e m i n g s , R. M e h r a b i a n , and G. E . N e r e o : TMS-AIME, 1 9 6 8 , v o l . 242, p . 4 l .  8.  M. C. F l e m i n g s and G. E . N e r e o : v o l . 242, p . 5 0 .  9.  R. M e h r a b i a n , Transactions,  T r a n s f e r , McGraw-Hill  Book Go.,  A.I.Ch.E. J . , 1 9 6 6 ,  Metallurgical Transactions,  trans  TMS-AIME,  TMS-AIME, 1 9 6 7 ,  trans  trans  TMS-AIME, 1 9 6 8 ,  trans  M. Keane, and M. C. -Flemings: M e t a l l u r g i c a l 1 9 7 0 , v o l . 1, p. 1 2 0 9 .  10.  R. J . McDonald and J . D. Hunt: v o l . 245, p.. 1 9 9 3 .  11.  A. M u l l e r and M. Wiehelm: v o l . 1 9 a ( 2 ) , p. 254 .  trans  TMS-AIME, 1 9 6 9 ,  Z. N a t u r f o r s c h ,  1964,  ;  12.  D. T. J . H u r l e : C r y s t a l Growth, P r o c e e d i n g s o f an I n t e r n a t i o n a l C o n f e r e n c e on C r y s t a l Growth, B o s t o n , 20-24 J u n e , . 1 9 6 6 , p . 6 5 9 .  13.  H. P. U t e c h  14.  K. A J a c k s o n , J . D. H u n t , D. R. Uhlmann,and T. P. Seward, I I I : t r a n s TMS-AIME, 1 9 6 6 , v o l . 2 3 6 , p . 149.  15-  S. W a j c i e c h o w s k i and B. C h a l m e r s : 1 9 6 8 , v o l . 242, p . 6 9 0 .  and M. C. F l e m i n g s :  ibid.,  trans  p . 651.  TMS-AIME,  16.  G. S. C o l e and G. P. B o i l i n g : F o r d M o t o r Company r e p o r t " E n f o r c e d F l u i d M o t i o n and t h e C o n t r o l o f G r a i n S t r u c t u r e s i n M e t a l C a s t i n g s , " March 1 5 , 1 9 6 7 .  17.  B. C h a l m e r s :  18.  R. T. S o u t h i n :  19.  W. A. T i l l e r and S. O'Hara: The S o l i d i f i c a t i o n o f M e t a l s , I S I P u b l i c a t i o n 1 1 0 , December 1 9 6 7 , p . 2 7 .  20.  M. J . S t e w a r t and F. W e i n b e r g : v o l . 245, p . 2108.  21.  L. MacAulay and F. W e i n b e r g : v o l . 245, p . 1 8 3 1 .  22.  G. S. C o l e and W.  J . Aust.  Inst.,  1963, v o l . 8 , p. 255.  t r a n s TMS-AIME, 1 9 6 7 , v o l . 2 3 9 , p . 2 2 0 .  t r a n s TMS-AIME, 1 9 6 9 , t r a n s TMS-AIME, 1 9 6 9 ,  C. W i n e g a r d :  1 9 6 4 - 6 5 , v o l . 9 3 , p. 1 5 3 .  J . Inst,  of Metals,  23.  G. F. B o i l i n g : F o r d M o t o r Company r e p o r t o f S t r u c t u r e and P r o p e r t i e s , " O c t o b e r 3 0 ,  24.  W. C. J o h n s t o n and W. A. T i l l e r : Westinghouse Research R e p o r t , " F l u i d Flow C o n t r o l D u r i n g S o l i d i f i c a t i o n , " December 1 5 , 1 9 5 9 .  25.  G. S. C o l e and G. F. B o i l i n g : v o l . 236 ,.p. 1366.  26.  H. P. U t e c h , W. S. Bower, and J . G. E a r l y : C r y s t a l Growth, P r o c e e d i n g s o f an I n t e r n a t i o n a l C o n f e r e n c e on C r y s t a l Growth, B o s t o n , 20-24 J u n e , 1 9 6 6 , p . 2 0 1 .  27.  D. R. Uhlmann, T. P. Seward, I I I , and B. t r a n s TMS-AIME, 1 9 6 6 , v o l . 2 3 6 , p . 5 2 7 .  28.  A. Ohno: ..The S o l i d i f i c a t i o n o f M e t a l s , I S I P u b l i c a t i o n 110, December 1 9 6 7 , p . 349-  29.  R. J . McDonald and J . D. H u n t : 1970, v o l . 1 , p . 1 7 8 7 .  30.  G. S. C o l e and G. F. B o i l i n g : "Visual Observations o f C r y s t a l l i z a t i o n from Aqueous S o l u t i o n s Under Enforced F l u i d Motion," unpublished.  31.  N. S t a n d i s h and G. L a n g : v o l . 1 5 , p. 120.  32.  K. G. D a v i s and P. F r y z u k : v o l . 233,. p . 1796.  "Manipulation 1969.  t r a n s TMS-AIME, 1 9 6 6 ,  Chalmers:  Metallurgical Transactions,  J . Aust.  Inst.,  1970,  t r a n s TMS-AIME, 1 9 6 5 ,  33.  G. K. B a t c h e l o r : v o l . 12, p. 209.  34.  A. Emery a n d N. C. Chu:  35.  J . 0 . W i l k e s : Ph.D. t h e s i s , U n i v e r s i t y Ann A r b o r , M i c h i g a n , 1963.  36.  M. R. S a m u e l s and W. v o l . 13, p . 77.  37.  H. R. T h r e s h , A. F. C r a w l e y , and D. W. G. t r a n s TMS-AIME, 1968, v o l . 242, p . 819.  38.  H. T h r e s h : "The V i s c o s i t y o f L i q u i d T i n , L e a d a n d Lead-Tin A l l o y s , " t o be p u b l i s h e d .  39.  R. N. L y o n : L i q u i d M e t a l s H a n d b o o k , The C o m m i t t e e on the B a s i c P r o p e r t i e s o f L i q u i d M e t a l s , O f f i c e o f Naval R e s e a r c h , D e p a r t m e n t o f t h e N a v y , 1954.  40.  H. E. McGannon, e d . : The M a k i n g , S h a p i n g a n d T r e a t i n g o f S t e e l , U n i t e d S t a t e s S t e e l , 1964. "  Q u a r t , o f A p p l i e d M a t h , 1954, T r a n s , o f ASME, F e b . 1965,  Churchill:  p.  110.  of Michigan,  A . I . C h . E . J . , 1967, White:  :  !  41.  Molten S a l t s : Volume 1, E l e c t r i c a l and V i s c o s i t y D a t a , N a t i o n a l B u r e a u O c t o b e r 1968.  Conductance, D e n s i t y , o f S t a n d a r d s 15,  42.  G. S. C o l e and G. F. B o i l i n g : v o l . 245, p. 725.  43.  W. R o t h and M. S c h i p p e n : p. 78.  44.  M. H a n s e n : . C o n s t i t u t i o n o f B i n a r y A l l o y s , 2nd. e d . , M c G r a w - H i l l Book Co., New Y o r k , 1958.  trans.  TMS-AIME,  1969,  Z. M e t a l l i c , 1956, v o l . 47,  Appendix I Notation  Used  heat  Cp  =  Specific  d  =  Width o f l i q u i d  =  D i a m e t e r o f t h e w i r e mesh w i r e s  f  =  F r a c t i o n o f t h e t r a c e r t h a t has f l o w e d t h r o u g h t h e mesh  F„' °  =  Fraction s o l i d i n the interface region  g  =  A c c e l e r a t i o n due t o g r a v i t y  L  =  Temperature g r a d i e n t  Gr  =  G r a s h o f number, b a s e d on temperature d i f f e r e n c e  Gr'  =  G r a s h o f number, b a s e d on o n e - h a l f the temperature d i f f e r e n c e  h  =  Convective heat  d  w  G  H h  =  ;  =  a  h  .= e  Height  zone  metal  The e f f e c t i v e h o l e s i z e mesh  =  Thermal  L  =  Height zone  to length ratio  1  =  Height  o f the l i q u i d  =  N u s s e l t number  P  =  Dimensionless  p»  =  P r e s s u r e d e v i a t i o n from s t a t i c value  Pr  =  P r a n d t l number  :  coefficient pool  i n t h e w i r e mesh  k  Nu  total  transfer  o f a molten  Actual hole size  solid-liquid  i n the wire  conductivity of the l i q u i d cell  pressure initial  q  =  Heat f l u x a t t h e c o l d  Ra  =  R a y l e i g h number  S  =  S p a c i n g d i s t a n c e o f t h e w i r e mesh wires >  T  =  Dimensionless  t  =  Time  u  =  Velocity  U  =  Dimensionless direction  v  =  Velocity  V  =  Dimensionless  a  wall  temperature  i n the x d i r e c t i o n v e l o c i t y i n the X  i n the y d i r e c t i o n v e l o c i t y i n the Y  direction x'  =  V e r t i c a l coordinate  X y  = =  V e r t i c a l dimensionless Horizontal coordinate  =  Horizontal  =  S u p e r s c r i p t denoting values o f the parameters a f t e r a o n e - h a l f time step  =  S u p e r s c r i p t denoting values o f the parameters after, a f u l l time step  a  =  Thermal  8  =  Coefficient  6  =  Boundary l a y e r  C  =  Vorticity  0  *  Temperature  =  Temperature o f the c o l d  =  Temperature o f the hot  :  Y  6i  :  6. 2  coordinate  dimensionless  coordinate  diffusivity o f volume  expansion  thickness  function  wall wall  v  =  Kinematic  viscosity  p  =  Density  T  =  Dimensionless  T'  =  Dimensionless time f o r the function iteration  =  Stream  time  function  stream  Appendix I I The T h e o r e t i c a l S o l u t i o n t o t h e P r o b l e m of N a t u r a l Convection i n L i q u i d Metals  The  g o v e r n i n g e q u a t i o n s and t h e boundary  c o n d i t i o n s t o be s o l v e d a r e g i v e n by e q u a t i o n s (3-32), (3.33), (3.34), and (3.35).  Also  (3.31),  equations  (3.36), (3.37), (3.38), and (3.39) g i v e t h e v o r t i c i t y and e n e r g y equations  finite  difference equations  f o rthe f i r s t  general f i n i t e  _  the f i r s t  a* 6  for  u x i,j  approximation  U  i+l,J  " i-l,J U  m Y  ±  )  d e r i v a t i v e s and  -  ily 3X "" ~  U  1-1,J  "  2  i s : used  -  8T  "  U'  2 U  i,J * i+U A X ^ U  2  derivatives.  iH  The  are the  2AX  U  t h e second  used  The  forms such as  3X  for  h a l f time s t e p .  difference approximations  central difference M  and s e c o n d  f o r the governing  The f o r w a r d  ( ? ) ( 2 )  difference  f o r t h e t i m e d e r i v a t i v e s such as - u  i,,1 AT  i , J  f  general procedure  of the s o l u t i o n i s  given i n Section 3 o f the t h e s i s , s t a t e a l l the equations used  ^ ° '  and t h i s a p p e n d i x  i n t h e computer program.  will  1.  C o m p u t a t i o n o f t h e new For  the boundary  temperature v a l u e s o f the second  derivative  a s e r i e s e x p a n s i o n i s used such as:  (4)  1,J From t h e b o u n d a r y u p p e r and  condition  o f no h e a t t r a n s f e r  lower w a l l the f i n i t e  3 T  difference  through the  becomes:  Z  AX " 7  US For  the f i r s t  ( T  2,J " 1,J T  (5)  )  h a l f time step the f o l l o w i n g  equations are solved  simultaneous  f o r c o l u m n s j = 2, 3,  .... n-1  i n turn  with  1*1: 2_ *  .AT  r  2  Pr(AX)'  T  *1,J  " Pr(AX)  T  AT 1,J  *2,j  T  +  Pr  fi2  y l,J T  (6) 1 = 2,3>  2AX U  1,J _ 2AX  ...  m-1:  T  Pr(AX)*  1 Pr(AX)*  Vi,j  •  [h* P r ( A X )  T*  T 1  i+l,j  AT i j j i , j V  i,J  +  i„j+1 ~ 1,J-1 T  2AY  1 = m:  Pr(AXr  T  *m-l,j  (AT  +  Pr(AX)<) *mJ  +  T  AT m,,1  =  T  Fr y m,  +  6 2  T  <  (8) For the second h a l f time equations  step the f o l l o w i n g  simultaneous  a r e s o l v e d f o r row i = 1 w i t h  j - 2: 2_ ,  2  AT •  Pr(AY)*  PHAIT-  ( T  T  'l,2  ~ Pr(AY)  *2,2 " * l , 2 T  )  2 T  5  P1TATP-  +  — T * AT  'l 3  T  1,2  +  (9)  l , l  j = 3,4, ... n-2:  1 , Pr(AY)' 'l,j-1 T  AT  l , j  1  +  Pr(AX)<  Pr(AY)* 'l,n-2 T  Pr(AX)1  Solve  ( T  (2 [AT  +  *2,n-l  {  i  2 Pr(AY)*  +  2,J  1  1,J  1,J  Pr(AY)'  1  1,J+1  (10)  ;  2_ +  Pr(AY)*  AT  ~ ^ . n - l  5  +  Pr(AY)  l,n-l  1 4  f o r rows i = 2,3, ... n - l w i t h  T  l ,n  AT  1  l,n-l  J = 2:  2 AT  2 PFTAYT "  +  1  2  1,2 1 2AY " P r ( A Y ) '  1,2  1,3  1,2 ^  AT  V.  w  1,2 , 1 2AY P r (AY) ' T  - " i / * * ' 2AX " * - ' 1  l  t  l  1  2  T  1  1  2  + >P r " xr» 1,2 (12)  n-2:  j = 3,4,  V  2AY  Pr(AY)  Pr(AY)  Pr  T  ,  i , j - i  +  PTTIYT-.  [h+  T 1  1  i,J  +  1,J 2AY  i,j+l  (13)  x i,J  0  1  j = n-1:  1 ,n-l 1 ~~ 2AY ~ P r ( A Y ) *  i ;  i,n-2  i—  k  AT  +  ,  2  Pr(AY)'_  i,h-l  • i,n-1 . 1 T. - U. / 1+1,n-1 l , n - l " 2AY Pr(AY)" l,n V  AT  +  i,n-l  Pr  6  Solve J = 2:  1-1,11-1  k  x i,n-l 1  f o r row i = m w i t h  (14)  (  2_ +  AT  2  Pr(AY)*  +  Pr(AX)*  ( T  in,2  Pr(AY)*  * m - l , 2 " *m,2 T  ) +  m,3  1  "  AT  1  m,2  (15)  Pr(AY)* m , l T  j = 3, 4, ... n-2  1 Pr(AY)  i T  'm,j-l  AT *m,J  +  T  +  f2 (A?  2 ) 1 Pr(AY) J 'm,j " P r ( A Y )  +  2  T  PHfxT- < V l , J - m,J T  T#  2  T  'm,J+l  (16)  )  J - n-1: 1 , Pr(AY)' 'm,n-2 T  ¥Fmr  +  2.  ( T  +  (2 [A?  2 Pr(AY)<  +  V l , n - l - *m,n-1> T  m,n-l  simultaneous  equations  m,n-l  P T T W m,n T  +  C o m p u t a t i o n o f t h e new i n t e r i o r For the f i r s t  AT  h a l f time  ( 1 7 )  vorticity  step the f o l l o w i n g  a r e s o l v e d f o r columns j = 2,3, .  n-  i n turn with i = 2:  2 AT  +  2 AX'  r*  +  rp I 2 X  AX  7  3,j  u  r  rn t 2AY (over)  2^1 2AY  U  +  AY"  AIT  1_AT  5  2„t  AY'  2AY  +  '2,3+1  1  2,3  AX'  2AX  (18)  '1,3  i = 3 , 4 , ... m-2 ( U  Ix*  2AX  i-l,J  5  rp t  V  (AT  V..  2AY  1,3 • 1  AY "  2AY  C  2  _ rp | '  U  2 AX "  +  I l l  2AY  +  AY'  AX  2AX  i>3  ?  i , j - l  +  1+1,3  7  |AT  AY'  (19)  '1,3+1  7  '1,3  1 = m-1 f U  n-l,3  JL, AX'  2AX  mt  u  1  C  V 2 , j  +  if? +  rp t n - l , J-1  n-l,j+l  V  2  n-l,3  n-l,3  AY " 7  ' V. AY"  ?  n-l,3  2AY  +  v AY*.  C  n-l,j-l  n-l,J 2AX  'n-l,3+1  1  2_  . 1  2AY  2AY  r  AT"  +  (  AT  _1.  ~ AX  4  'n,3 (20)  F o r t h e second, h a l f t i m e s t e p t h e f o l l o w i n g e q u a t i o n s a r e s o l v e d f o r rows i = 2,3,  simultaneous  ... n - l i n t u r n w i t h  3 = 2: (V i , 2  2 . 2 AT  A T  7  C  'l,2  +  2AY  1 AY " 7  V  1,3  =  Gr  rp » i,3  2AY  _ rp l 1,1  (continued  over)  +  2AX  1 AX " 2  2AX  2AY  G r r  AT  AY  AT  +  +  1,2  Q  1  2AY  AY " 7  (21)  '1,1  n-2  i,j-l  7  +  +  2  [AT  i,3  i  AY " 7  U  +  1,3  2AX  5  'i,J  AX'  1 AT "  2AY  +  c*  +  { 2AX  +  (V  2  f -  rp » _ rp i — i - i i l i i,J-l 2AY  AX'  c*  AX"  1+1,2  AX'  J = 3,4,...  +  ^ 1-1,2  1,3+1  2  +  (22)  1+1,3  AX^  3 = n-l: V  i,n-l 2AY  T' .  1 ~  -  AY " 7  C»  T'  AX " 7  c* Q  i,n-l  +  +  2  IAT  i,n-l 2AX  2AY  2  f—  +  ^ i,n-2  +  2  AY "  1 AX " 7  7  1 ,n-l  ^ 1-1,n-l  fv  U 2AX  AX  7  1+1,n-l  r  IAT  i,n-l 2AY  1 AY " 7  (23) 3.  Computation  o f t h e new  For the f i r s t  stream  function  h a l f time step the  following  s i m u l t a n e o u s e q u a t i o n s are s o l v e d f o r columns j = n-l  in turn  with  2,3,  2 2  AT " * 2 , 4 - 1  +  2  (  i =  2,j  [AT'  2,4  AY'  (24)  )  +  m-2  2  AJF^'i-l.j  AT'  ^2,4 l  +  i = 3,4,  '  AX^ * 3 , 4  2,4  AX "  IAT  '1,4  AY'  +  UT'  +  +  T  2  AX  i,4  AY "  +  2  AX"  T  1+1,4  C  i,4  +  (25)  n . j - ^  m-1:  AT^*m-2,4  2 AY " 2  ,AT'  (sfr  +  m-1,4  +  AT  7  AT " V l , 4 - l  +  2  (  " m-l,j  m-1,4  ?  +  ^m-l,4+l  For the second h a l f time step t h e f o l l o w i n g equations  are solved  f o r rows i = 2 , 3 ,  +  (26)  )  simultaneous  ... m-1  i n turn  with  4=2: 2 . 2 AT " 2  AT'  +  1,2  AT " < * V l , 2 7  +  AY^  1,3  *Vl,2  )  h,2  +  [AT  !  AX'  1,2  (27)  j = 3,4, ... n-2  AY  IAT'  R  AT'  ^ i , j  AX'  i,J  AY  AX ^ 2  i - l , j  k V  AY^'ij+i  +  =  ?  +  (28)  i+i,y  v  i j  j = n-1:.  2 , 2  AY**'i,n-2  AT'  -2  2 **i,n-l  AT "  AT'  2  i,n-l  AY "  AY "  +  1-1,n-1  2  4.  Computation o f the v e l o c i t i e s  For  c o l u m n j = 2, row i = 2,3, ...  U  3  1,2  For  U  +  6  - *l,m-l  i,j  +  6  8  *i„1-l  (30)  m-1  (3D  c o l u m n j = 3,4, ...  *i,J+l 12AY +  8  (29)  m-1  * i , m - 2 - »l,m-3 6TY  row i = 2,3, ... m-1,  »l,j_2 -  +  ^ 1+1,n-1 )  +  column j = n - 1 , row i = 2,3, ...  i,n-l  i,n-l  *i,3 ~ *i,4 ~6"AY  3  For  U  ~ *i,2  5  - »l ,1+2 t  m-2  (32)  For  v  row I = 2, c o l u m n j = 2,3, ... n-1  -  2J  For  3  *2„1 - *3,.1 6AX 6  row 1 = m-1, c o l u m n j = 2,3, ... n - 1 :  - . Vl,J 3  v  m-lj  For  v V  i,j  5.  +  "  "  6  V 2 6AX  l  + V-3.J  3  row i = 3,4, ... m-2, c o l u m n j = 2,3, ... n - 1 :  "  1-2,1  +  8  ^i-l„1 " ^ 1 , J + * i + 2 , J 12AX 8  C o m p u t a t i o n o f t h e new b o u n d a r y  vorticities  Expanding the stream f u n c t i o n i n t o a Taylor series results i n :  From t h e b o u n d a r y c o n d i t i o n s o f :  d ^> , 2  94,  e q u a t i o n ( 3 6 ) c a n be r e d u c e d t o : 9 <K 2  ,  2<|,  Thus t h e e q u a t i o n  used t o s o l v e f o r t h e w a l l  vorticity  a r e as f o l l o w s f o r 1 = 1:  J = 1 2i> AX  •l,j  2*  (38)  i = m:  .1 = n: 2^  ~ l ?J- ( i j n ) AX<  9T? 8Y  7.  4  U  •i,n  ;  Computation o f the Nusselt  n  1,1  The s o l u t i o n  2  s  ~  Y  =  ±  g i v e n by e q u a t i o n  y  ±  1  =  b. - "* i -  »•  ±  =  i  -  a  l'  b  1  1p  d  Y  matrix (3-49)  1  =  n-2,n-3,...  1  ^ h  =  j  (43)  s $  where:  1  coefficient  2  n-1 c  s  ^  0  y  o f the t r i d i a g o n a l  n-l  (in)  number  -11T, + 18T, - 9T, + 2T, 1 >1 i , 2 i , 3 ij.4 "r5AY  of the s e t of l i n e a r equations  6,  2<J> 1*11=1 AY  m  V  6.  (39)  AY  '1,1  " l  i i - l  1  ,  y  g  ,  Y  l-  6^  I = 2 , 3 , ... n - l • o , i = 2 , 3 , ... n - l  (44)  Appendix I I I Computer  1.  Program  Flow c h a r t o f t h e computer  C  program  Start  )  Read F l u i d Parameters I  7  Set I n i t i a l Conditions Compute Temperature Compute Interior Vorticity  <  Compute Stream Function Test Stream F u n c t i o n Convergence  \  No  Yes Compute Velocities  I  Compute Boundary V o r t i c i t y  I  Compute N u s s e l t Number Test O v e r a l l Convergence  No  Yes Print T,t|/,U,V,C,Nu Subroutine TRIMA  C  stop  ~y  7  2.  Notation  A,B,C,D  used i n t h e computer program  Arrays used i n the t r i d i a g o n a l coefficient  matrix  AN  Local  Nusselt  number  AVNU  Average N u s s e l t  DT  Time i n c r e m e n t , A T  DTI  Time i n c r e m e n t , A T  DX, DY  Grid spacing,  GR  G r a s h o f number, G r '  I,J  Grid positions, i , j  K  Number o f i t e r a t i o n s t h e p r o g r a m  number  1  AX, AY  i s t o complete M,N  Maximum v a l u e o f I a n d J r e s p e c t i v e l y  PR  Prandtle  S  Stream  SS  S t r e a m f u n c t i o n , \p*  SPC  Convergence  T  Temperature, T  TS  T e m p e r a t u r e , T*  TD  Temperature, T  TRIMA  Tridiagonal  U,V  V e l o c i t i e s , u,v  VI  Solution of tridiagonal  Z  Vorticity, C  ZS  Vorticity,  C*  ZD  Vorticity,  V  number, P r function,  limit  on s t r e a m  function  1  matrix  solution  subroutine  matrix  H1«TCAN  IV  G C l « I M t ER  VAIN  !  OC 1  <•.'"••? _ _  .  <  N  0007 ooos  nn or  T(I,JI  =  :''!'' : 11 . "12 ; on r>H ^^^r? •'.•"if. 17  S( I,J I n i , j) U( I , J ) v 11, j i SSI I , J) viiii 0 0 5 1 1 TCI,1) T ( I , N I  = 0.<~. = •:•.<'• = '.0 = o. c = O. ', c = 1,21 = -1.0 = 1 . C  r  r--.-"'J  50  SI  19  vri"  5f. &f  = =  >:? > ..'•:•?6 r  >i*  !  '<• < "•*'> • ''-1 1  ' ^ 7  E  r  1  a  =  si  r  >.<-•  = 0.02  S2  r  = i,2i - 1,21  SFC r?2  002 3 "C?4 n?5 OC?*T V ? '•'"26 "'/?<3 0V3C •'•'U ','•"»•<2 :< ?3  i j  OTI  •:• : ? i  '-"'.2  u.-t, 2.o 2.C  / /  nTi DTI  • -  2 . 0 2.0  / /  c C  DX**2 0X**2  I »  S3 = 1 . 0 / ( nx**2 I Nl = N - 1 N ? = N - 2 Ml = M — 1 M2 = M - 2 K5 = C 170 CONTINUE K5 - K5 • I READ (5 , 2 0 0 ) GR, P R , OT 20C FORMAT ( F 2 C . 6 / F 2 C . 6 / F 2 0 . 6 I IF ( GR . L T . 1 0 . 0 I GO TO 603 E l = 2 . 0 / F-T + 2 . 0 / 1 P R * DX**2 I F2 = 2 . 0 / (. P 3 * 0 X * * 2 ) • E? = 2 . 0 / DT f» - E 2 / 2 . 0 E 6 = 2 . 0 / (IT * 2 . 0 / 0 X * * 2 E7 = l . f . / ( D X « » 2 I 8 = 1.0 / ( 2 . 0 * 0 X I E" = 2 . 0 / DT 2.0 / DX*»2 n o = 0X**2 E 11 = 1 . 0 / 1 6.0*0Y I E12 = 1 . 0 / C 1 2 . 0 * OY I _ ' =13 = 2 . 0 / OY**? £5 = GR / ( 2 . C * 0 Y I WRITE ( 6 , 7 0 0 ) M,N,NUM,GR,PR,OX,OY.DT 7 0 c. FORMAT ( 1 H 0 , ' v; = • , I 5 , • N - • • . I 5 , » N U M - ' , 1 5 , • GR = • PR = ' , 1 P F 1 2 . 1 , ' OX = • . I P E 1 2 . 3 , ' OY = • l P E 1 2 . 3 t 2 • OT = ' , 1 P E 1 2 . 3 ) < = 0.0 " 1 K = K. «• 1 C C A L C U L A T E NFw TEMPERATURE S FtlP F I R S T HALE T I M F S T F P f  ;  ."/.--•  PAGE 0 0 0 1  ?  NUM = 50C OX = . 0 5  '.•""< 0?o ' .'•1  12:56:05  O i l ENS ION T ( 2 1 , 2 1 ) , TS ( 2 1 , 211 , T D ( 2 1 , 2 1 1 , 2 1 2 1 , 2 1 1 , Z S C 2 1 , 2 1 1 . 1 2 -K21 , 2 1 I , S ( ? 1 , 2 1 I , S S 1 2 1 , 2 1 t , U ( 2 1 , 2 1 ) , V I 2 1 , 2 1 1 , A N C 2 1 I . B l 2 1 ) , , 2 A l 21 ) , r ( 21) , D ( 21) , VII 211 ;"•'•>'.'?« -1=21 j :  0• • '<••<, 0005  ,  CB-12-7C  '.1PE12.3,  (UKTi'JM OC-51  0053 f ;5<. Cf5f> v'0S7 -•058 V~5<5  0  'I ! 2  r.• -• '. •(• J  r-Zb<> ;;• >•.'  -:.e6 .  :^if•^ CO*.?  : C-.7.!• • 1 7  ' VTZ  1,07 3 1-74 '••07s •)'-7fr "',77 •;^7M "••"> ("OHr OCR 1 "H-H2  IV G COMPILER  08-12-70  12:56:05  PAGE  r » n IOC J = 2 . N 1 8(1) = El C(l) = -E2 A(M) = - C Z 81M ) = E1 3111 = E 3 * T ( 1 , J | • E 4 * ( T 1 1 , J - 1 I - 2 . 0 * T I 1 , J I • /T(l,J»n) niMI = F.3*T(M,J1 • E 4 * < T ( M , J - 1 1 - 2 . 0 * T « H , J > • T ( M , J - H > ) DP H;l I = 2 , M l 5(1) = - U ( I , J I * E 8 - E 4 HID - El CI I I = U( l , J ) * E 8 - E 4 Dill = F 3 - T I I i J l - V I I, J I * E 8 * ( T I 11 J * l l - T ( I . J - l I I * E 4 * ( T ( I , J - l l 1 2 . ' . * T ( I , J I • T( I , J * 1 I ) l'.l CONTINUE C A L L TRIMA ( A , 3 . C 0 , V 1 , M , 11 DO 10? I = 1 , M ISll.jl = v i l l i 10? CONTINUE I Of . CUNT INUE C C A L C U L A T E NEW T E M P E R A T U R E S F O R S E C O N D H A L F T I M E S T E P C FOR ROW 1 = 1 ' ' n(21 = E l C ( 2 I = -E<V 0(2) = E3*TS<1,21 + E 2 * ( T S ( 2 , 2 » - T S ( l , 2 » » • E 4 * T » l . l l A (N-l I = - E 4 ' . • . .'. ' F(N-l) = El r V-: ~ 01 N - l » = E3*TS(l.N-ll*E2*(TS(2,N-ll-tS(l,N-l»l*£**Tll,NI DO 1 0 3 J = 3.N2 A( J I -E<» » U ) = El  0002  -  t  1  V  cui  =  1  rni I , J I CONTINUE  C.  ROWS  FOR  »»r>  I  1CS7 ^:PH OCB'-i C'C<K "o I  =  l.JII  .  = 2,.  HO 105 I 0(2)  -TSI  = V K JI  1?'<  '".'.'B*  - E *  n(JI = E3«TS(1,JJ • E2»(TS<2,JI CONTINUE CALL TRIMA 1 A , B . C , D, V U N - 1 , 21 on 104 J = 2 i N l  1C-3  f-i»•<  .M-1  =2,Ml  El  C(2 ) = V( 1,2 )*E8 - E 4 0(21 = E 3 » T S ( I . 2 I • (V(I,2l*E8*E41*T«l,ll-U(l,2l*E8*<TSIl*lt2»1 TS(I-1,2))+E4*(TS(I-1,2> - 2.0*TS(I.2I* TS(I*1,2I1 M N - 1 ) = (-VI 1 N - l l * E 8 - E 4 I B ( N - 1 I = FI DIN-11 = F 3 » T S ( I , N - 1 ) * I - V ( 1 , N - 1 ) » E 8 * E « I « T ( I . N I - U I 1 , N - 1 ) » £ 8 * ( 1 TSI I+ 1 , N - l I - T S I t - l , N - l ) l + E 4 * < T S < I - l i N - l l - 2 . 0 * T S U , N - l ) 2 • TSI1*1,N-l11 HO 106 J = 3 t N 2 BUI = El ~ ." " ' " C U I = V ( I , J I * E 8 - EA A l JI = -VI I J ) » E 8 - E « • f)(Ti = E3*TS( I t J I - U ( I t J ) * E 8 * < T S « l * l . J I - T S C I - l , J I I » E 4 * « T S C I - l , 1 JI - 2 . 0 * T S I I , J I * T S U + l t J l l CONTINUE :  :  '.  1  1  p  C092 •• ' •""> 3 It "! '•^°5 ''""a  MAIN  0  '.  W  t  ' 1C6  '  "  °^  r.npPI I E K  ! 1" Tf- A ",' 1 V C 1:  "!' 2 0 1! 5  .0 1 0 4 CI'. 5  : l : t>  'ii ; r m r  C I 16 01 1 7  •3IN-1)  =  = E 3 « TS( M , N - D + E 2 " ( TS( t»-1 , N - 1 I - T S I M , N - l I > + E 4 * T I M , N )  012', C 1 ' 1  '•12?  '/! 2 3 *l?4 •M2 5  ''126 " 1 27 ~12H :i21  0 1 •* .-  013 1 0 1 3 2 v" 1 H  '.: 1 3 4 •'. 135 •". 1 36  11 37 'ISP. 013° 014C 0141 0142 0 143 0144 0145  PAGE  0003  J  El =  3,N?  AIJI  =  b( J ) C(J)  = El = -ft.  - E 4  "^TJl = E3*TSO, J ) E 2 * ( T S ( M - 1 , J | CONTINUE C A L L TRIM* ( A , S , C , 0 , V I , N - 1 , 2 1 00 1 0 9 J = 2 . N 1 10(M,J) = VKJ) CONTINUE  1 JO  CHJCK •• '• 1 1 <=  12:56:05  O(N-l) )•: l u P  +  1 - *  '115  '" 1 1 8  C8-12-7C  C A L L T ° IMA ( A , 8 , C O . V I , N - 1 , 2 1 00 107 J = 2 . N 1 TD( I , J l - VI ( J ) CONTINUE 107 ~TT5 CONT INUF C FOR ROW I = M B(21 = E l C(2) = - F 4 0(2) = E 3 » T 5 ( « . 2 ) + E 2 « « T S ( M - 1 , 2 | - TS(M,2)> + E 4 » T ( M , l ) A(N-1I = -E4  '107" ' loi ci . -n  < 112 ? "•1 1 •••114  MAIN  FOR  TEMP.  CONVERGENCE—(OVERALL  -  TSCM.J))  CONVERGENCE!  AMAX = 0 . 0 00 30C I = l , H 00 30C J = 2 , N 1 OIFF = ABSl T I 1,J ) T0(I,J)t IE ( O I F F , L T . AMAX) G O T O 300 AM A X = O I F F CONTINUE IF ( A M 4 X . L T . O . C O O l I GO TO 400 IF ( K . E Q . 81 . A N D . K5 . E O . 1 i GO TO 4 0 0 IF 1 K . E O . 50 . A N D . K5 . G T . 1 ) GO -TO 400 IF ( K . E U . 100) GO TO 4 0 0 C C A L C U L A T E NEW INTERIOR V O R T I C I T I E S c r if ST HALF TIME S T E P OP 1 U J = 2.N1 H(?) = E6 C( 2) = U ( 2 , J ) * E 8 - E 7 E5*( TD( 2 , J +1)• T D I 2 , J - l > ) • ! V I 2 , J ) * E 8 * E 7 ) * Z ( 2 , J - l I • E 9 * Z I 2 , J I 'Jl 21 1 • I - V I 2 , J I * E 8 + F 7 ) * Z ( 2 . J + 1 l - l - U ( 2 , J l * E 8 - E 7 ) * 2 ( 1,J) A I M - 1 ) = - U « M - 1 , J ) * E 8 - E7 6 I M - 1 ) = E6 O I M - 1 ) = E 5 * l T O ( M - 1 , J * l I - T 0 ( M - 1 , J - 1 ) I •< V I M - 1 , J »•£ 8+E7 I « 2 ( N - 1 , J - 1 ) 1 + F9*Z< M - 1 , J l + I - V I M - 1 , J I * E 8 * £ 7 ) * Z I M - 1 , J + 1 I - 1 U l M-1 , J I » . E B - E 7 ) *Z I M , J I 00 111 I = 3.M2 5TT! = f - U l ! , J ) * E 8 - E7) B< I > = E6 C( I ) = U l I , J ) * E 8 - E7 DI I I . « E 5 * ( T D ( I , J * l l - T D I I , J - 1 I ) * t V l I , J 1*E8 + E7 I *Z I I . J - l ) 1 •EQ'ZI I, J ) • ! - V ( l , J ) * E 8 + E 7 ) * 2 I I , J * l > CONTINUE 111 CALL TRIMA I A , B , C 0 , V 1 , N-1, 2 » DO 112 I * 2 , M l Z SI I , JI » VII I )  '  FORTRAN -1*6 1*7  0  IV G COMPILER  112 CONTINUE 11C CONTINUE C SECOND HALF TIME  0155 C-156 0 157 01 59 oi5<»  r  01 74 0 175 "176 0 177 0 17? CM 7-1 0183 f I'M c 1 :> CM 3 o 1 p<. 01 3* "IP'. 01 6 7 \uH , 1 ^ ' 1' " 1 >\  114  c .  000*  H A l1NN--1l 11 = E6 -VII , N - l )*E8 - E7 DIN-1) = E5*ITD<1,N)-TDII,N-2)l + IUII,N-11«E8+E7l*ZSII-l,N-ll*E9» 1 Z S I I , N - 1 l + l - U I I , N - 1 ) * E 8 « - E 7 ) * Z S < I + l . N - l l - ( VI I , N - l 1 * E 8 - E 7 I * l 1 1 , N ) DO 114 J = 3 . N 2 AlJI = - V I I , J ) * E 8 - E7 d(J) =56 C(JI = V I l , J I * E 8 - E7 OIJ) = E5*(T0(I,J+1)-TD(I,J-1)»+1UII,J)»E8*E7)*ZS(1-1,JI 1 • F 9 « Z S ( I « J I + ( - U ( I» J)»E8*E7I*ZS(I*-1.JI CONTINUE CALL TRIMA ( A , B , C , D , V 1 . N - 1 , 2 I 00 115 J = 2 . N 1  TO STANDARD  NOTATION  I=  DO N1 00 116 J » 2 ,1,M TII.J) = TDII.JI 1 16 CONT INUE 00 117 I = 2 , M l 00 117 J = 2 . N 1 ZII.JI = ZD(l.J) 117 CONTINUE C COMPUTE NEW STREAM FUNCTION Kl » 0 - K lO » 1TIME C12CF I R S TK lHALF F 3C5  IF  STEP  FOR STM FCN  00 3C? = T2 N 1 I GO TO 4 0 3 ( K lJ . G . .NUM B I 2 I = SI CI2I = - S 3 A(M-l) = - S 3 H I M - 1 I = SI D(2I = S 2 * S 1 2 . J I • S 3 * ( S ( 2 , J - 1 I :i(M-l| = Z ( M - l , j | • S2*SIM-1,J)  • S12.J*1)> • Z I 2 . J I • S3*(S(M-1 , J - 1 ) • S I M - l . J + l ) )  I= I ) = si  A = - S 3 3,M? 00l l )3C3 C( 1) • = -S3 B( 303  CALL 3 . SC 2D».SVIII, ,M , 2 )S3*IS(I,J*1I O i l ) TRIMA = Z I I . J I) A , «• J I- 1 •  3C4 312  DD 304 2,Ml CONTINUE SSI J I = VII I CONTINUE  J  : 1  PAGE  STEP  115 CONTINUE ZD I I,J) = VIIJI 113 CONTINUE C CONVERT NEW T AND VORT  0  .  12: 5 6 : 0 5  3 ( 2 )113 = E6 00 2,Ml C ( 2 I = V I 1 , 2 >*E8 - E7 0 1 ? ) = E5*(TDlI , 3 1 - T D I I , 1 ) ) • 1 U 1 1 . 2 ) * E 8 * E 7 ) * Z S 1 1 - 1 , 2 ) * E 9 * Z S 1 I,2 ) • 1 l-UIIt 2 I * E 8 + E7 > « Z S ( 1 * 1 , 2 1 - 1 - V I I , 2 ) * F 8 - E 7 ) * Z I I,1)  0152 f 153 C154  C 166 0167 0168 L'16Q 01 7C 171 0 172 0173  08-12-7.'  I=  0148 0140 015C 0151  JM6C '.161 0162 0163 r 164 '"165  MAIN  I,  I=  I  • S(I.J-ll)  PIL P  "r~;.-T"n*'""|v'"'-» 0  SfC'-iH  r ;  HtlF  TIME  O R - 1 2 - 70  STEP  12:56:05  PAGE  0005  O  l->. •:•>•• •V 1  I = 2 , « i SI -S3 . * -S3 * SI 7 1 1 , 2 1 • S 2 * S S ( I , 2 I • S3* (SS ( 1 - 1 » 2 I • S S t l * l , 2 ) t = M I . S - l l • S2*SS(I,N-1) • S 3 « l S S I 1-1,N-1) • S S I I + l . N - l ) ) J = 3,N?  0JV7  M J ) =  -S3  '.'*'  P  •!'-7  = si f 1" '• ('. I J I = - S 3 >?..•> , 3.7 !.i(JI = M I , J ) • S ? » S S I I , J ( • S3*( S S I I - l t J I >2'6" ••" ' ' ' C A L L ' T3 1 A ( A , < < , C D , V I , N - l , 2 1 - •>7 on 3c>• J = 2 , N I 'j:. KSI 1 , J ) = V I I J I <• p••• o T-";; c ;,Tl vJF C C H E C K F L ' R ST3M F C N CONVERGENCE ?1 722 = S 3 M 4 . 0 * S ( 2 , ? I - S I 3 . 2 I S<2,3)l VI! CHFCK - ABSI IZ22 - Z < 2 , 2 ) ) / Z < 2 , 2 ) ) IF ( K . O T . 4 ) GO TO 311 02 n -/RITE ( 6 . 3 1 2 ) K l , CHECK nrz  3T2—FORMAT 311  '. ??1 CZ22 * ??3 c 2 ?  T  J  r  ,77  " ? '-' '>?"« 023/ •'?U V V  C?33  COMPUTE  ;  •  '  IIH ,  '  K I = • , i4,  •—CHECK  • SSII*1,J)I ^  ', < ' • • '  ' :  ;  ' '  = * ', I P E H . 6 )  '  , :  ;  I ! j I \  •  ;  ;  j |  :  :  v  on 125 J = 2 . N 1 V I 2 . J ) = ( 3 . 0 * S 12. J ) - 6 . 0 * S I 3 , J I + S I 4 . J I » * E 1 1 VIM-l.JI =-(3.0*S(M-l J)-6.0*SIM-2.J)+S(M-3,JI)*Ell 00 126 I = 3 . M 2 V(I,J) = (-S(I-2,JI+8.0*(S(I-1,J)-S(I*1,JII*S( I*2,J»»*E12 126 CONTINUE , 125 CONTINUE C C O " P U T ! NF W BOUNDARY V O R T I C I T I E S " OC 127 I = 2 . M l Z( 1,1) = - E 1 3 « S 1 I . 2 )  i  f  -7TJ-T4  :  ~?<5 0-736 02 37 <">.'>*• J0230 0 24O  127  C24I  " '  ;  CONTINUE IF ( CHECK . G T . S F C I GO TO 1 2 0 CONTINUE " C COMPUTE NEW V E L O C I T I E S on 123 I = 2 , M l C COMPUTE U 0(1,2) = l-3.C;*S(I,2)*6.0*S(I,3)-SI I » 4))*E11 ~ JII.N-l) = - l - 3 . 0 * S ( I , N - l l » 6 . 0 » S I I , N - 2 ) - S I I,N-3I)*E11 OH 124 J = 3 , N 2 J 1 I . J ) = ISI I, J - 2 ) - 8 . 0 * ( S ( I , J - 1 ) - S ( I , J * l l » - S ( I , J * 2 ) ) * E 1 2 124 CONTINUE . 3 CDNT K.iiF ' : ~ : c  0225 26  .  (JI  0 215 :-?16 L 7! 7 "21-3 :..'!'<  ~ "  FnR STM FCN  oii 3 r * "(21 = CI 2> = U K - l l r i ' M I 0(^1 = C!IN-1 > 0 0 3? 7  - 1 "> 5  \ f  '•'AIM  R  zii,NI  = -Ei3 *  .  -  i  ; -  . . . .. '  :  '  : j j j  ~-  :  so.N-n  CONTINUE 00 12B J = 2 . N 1 Z.ll.J) = -E13 * SI2.J) ZIM.J) = -E13 * SIM-l.JI 129 CONTINUE AVNU = 0 . 0 C COMPUTE NEW NUSSELT NUMBER on 129 I = 1 , M  .  , i j. ! 1  CO r\J VO  f  r'T<T = A »  IV  0  C i v i l •r  F J  ?45 r - Uh c  0 ?41  0 ' 25 02 5'  '  2'>1  F  5 01  AVNU  129  CONTINUE  6  FORMAT (1H , 0 0 TO 4 0 1  ;-"i5 ^257 r 253 r ? 5o '?»•)  266  .  GO  To  wPITF  (6,6^2 )  ' , 1 4 , '  AV&.NU  =  ',E16.fe,«  AMAX  =  ' , E 1 6 . 6 »  U U l i  606  FORMAT  00  7  FORMAT WRITE WRITE  (lHl,  "J  FORMAT  • 60 3  1  =  ' . 1 4 , '  HAS NOT C O N V E R G E D  CHECIC=  ' , E 1 6 . 6 , '  STM FCN UNCONVERGEO'  1  1  )  J = l , N ) , I = l , M I  FUNCTION  S I I . J ) ,  MATRIX')  J « 1 , N ) . I - 1 , M I  VELOCITY U U . J ) , VELOCITY  (( V ( I , J ) , VORTICITY (( Z ( I , J ) , (  -  MATRIX' »  T(I.J),  1H0,IP11E11.3  (6,609~)  /  MATRIX'! J= l,N),1=1,M) MATRIX')  J=1,N),I=1,NI  i.  MATRIX') J=l,N),1=1,M1 1P10E11.3  )  AN(I),I =1,MI  (IHO,'LOCAL  NUSSFLT  NUMBER'/  l P U E l l . 3  * f•  -  17u  G O N TTION U E C ST0»  0 2«0  END ME»0RV  ((  ( 1 H 1 , 'V  FORMAT  K  ((  (6,604) (6.6C8)  WRITE (( 66, .66014C) I FORMAT (1H1,' WRITE (6,604)  .. 6 0 . 9  • . E 1 6 . 6 , '  CHI-CK.  ( 1 H 1 , 'STREAM  610  WRITF  ((  (b,604) (6,607)  FORMAT  -  =  'TEMPERATURE  60 8  614  AMAX  (6,605) (6,6041 (6,606)  WHITE WRITE  'A,M1A4X , '  Kl ,  (lHOi'.Kl  0 279  T01 A L  A V N U , AMAX  60 3  WRITE Wkt TF  r. 2 7 6 0277 0 2TII  K, 'K. =  FORMAT  0 22 77 43  0275  (6.<>0CI  605 4~C  0272  r  * OX  rriRMAl  0 2 7,-. 0 2 71  2 .0  = A.VMJ . • F  602  0 26 7 '•: 2 6 ? 0 ?69  A M I 1  nU I T E  - 265 R  =  403  02 6 3  "2e>4  /  t-r.PYUT" M0t1. ' )K K = w<MTh ( 6( ,l 6 , r,n TO 6 ( , 3  0 2 M  •2 '6?  nx  6'. 1  •"2 5 4  .: ? < = -s  *  4 02  •-•253  .  0006  TO 5 0 1  500  .0  PAGE  r  = ANI I )  <>f  12:56:05  = ( - 1 1 . 0 * T ( 1 , 1 ) + 1 8 . 0 * T C 1 , 2 ) - 9 . 0 * T ( I, 3 1 * 2 . 0 * T ( I . 4 1 ) * E 1 2 . N . l.AMO.i . N E . M) G O TO 5 0 0  WP I T E  2 40  03-12-70  ->A1N  AMI) IF (I  02<.2 0 2* ? ,->;>4 4  REUUIPEMENTS  /  1P10E11.3  )  .  *  O06ED4  BYTES  \  •  *•  • -,-• -  —  —  -  -  -  .  /  f-r-^TWAi'J 00 -l •y.-r? CCC3 r.r-o.i, r  OCOf. 0 1C7 cqc« -JCO ~ \/_ \\ .1? "1 ? ' : 14 '.IS '••:!'. T P ' •'.! • -• y  r  ICTAl  :  iv  (i C C M P I L E P TP I MA 08-12-71, 12:56:58 S I W T U T IME T P I MA ( A , B , C , D , V 1, L , K I 01 " F N S I C N B ( 2 1 ) , A I? 11 , C ( 2 1 ) , 0 ( 2 1 1 , V I ( 21 ). B A ( 2 1 ) ,GA( 2 1 1 Kl = K • 1 LK = L - K . LI - I - 1 HA(K) = 3 ( K ) i i M K I » OIKI / BAIKI or* i i ~K i , L B M I I = BI I 1 - ( A( I l*C( I — 1 » ) / B A d - l l 5 M I M (PI I ) - A ( I I*GAI 1 - 1 I) / B A ( I I _ i CONTINUE V11L) = GA(LI on 2 I = 1»LK J = L I V K J I = GA1JI - ( C ( J ) * V l ( J + l » l / BA(JI ? CONTINUE HE TUPN END  «FMnnY  PFUUIFEMENTS .0C(.3AF  3YTES  PAGE  OcOl  Position-. 25  Figure Line  1  Table Line  IV h  5 lines bottom  from  2 lines bottom  from  Line  11  Corrected  Original form  from  intial  initial  the  to  thsis  thesis  Id  2c  I  Caption  ( b ) Pb ( d ) Pb  -  (b) Sn (d) Sn  -  Caption  ( f ) Pb ( h ) Pb  -  ( f ) Sn (h) Sn  -  

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