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UBC Theses and Dissertations

Inverse segregation and centreline shrinkage Minakawa, Sakae 1984

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INVERSE SEGREGATION AND CENTRELINE SHRINKAGE by SAKAE MINAKAWA B.E., Tohoku U n i v e r s i t y , J a p a n , 1977. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES D e p a r t m e n t o f M e t a l l u r g i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s a s c o n f o r m i n g tp_ t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J u n e 1984 © Sakae Minakawa, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of M e t a l l u r g y The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: June 8, 1984 i i A b s t r a c t One of the major f a c t o r s c o n t r o l l i n g macrosegregation and shrinkage p o r o s i t y i n c a s t i n g s i s the extent of i n t e r d e n d r i t i c f l u i d flow which occurs d u r i n g s o l i d i f i c a t i o n . I n t e r d e n d r i t i c flow r e s u l t s from movement of l i q u i d to f i l l the v o i d s l e f t by s o l i d i f i c a t i o n and thermal shrinkage, which in the case of many a l l o y s c a s t a g a i n s t a c o l d c h i l l , r e s u l t s i n i n v e r s e s e g r e g a t i o n . In the present i n v e s t i g a t i o n , Inverse s e g r e g a t i o n has been examined t h e o r e t i c a l l y . Q u a n t i t a t i v e v a l u e s f o r the s e g r e g a t i o n both at and adjacent to the c h i l l face have been determined using a mathematical model and computer c a l c u l a t i o n s . The a l l o y s c o n s i d e r e d were Al-Cu, Al-Zn and Sb-Bi, the l a t t e r being of p a r t i c u l a r s i g n i f i c a n c e s i n c e i t expands d u r i n g s o l i d i f i c a t i o n . The model r e s u l t s show reasonable agreement with p u b l i s h e d data of c h i l l face i n v e r s e s e g r e g a t i o n . In the p r e v i o u s models, the volume shrinkage used to c a l c u l a t e i n v e r s e s e g r e g a t i o n d i d not i n c l u d e the thermal c o n t r a c t i o n which, i n the p r e s e n t model p r e d i c t i o n s , was found to have more s i g n i f i c a n t r o l e i n the s e g r e g a t i o n than expected. The r e s u l t s from the present model d i f f e r s s i g n i f i c a n t l y from those of the e a r l i e r models, p a r t i c u l a r l y i n the Al-Zn a l l o y s . C e n t r e l i n e shrinkage i n a s t e e l p l a t e c a s t i n g , a s s o c i a t e d with back flow of i n t e r d e n d r i t i c l i q u i d , was examined u s i n g the i n t e r d e n d r i t i c f l u i d flow model combined with a heat t r a n s f e r model of the system. Regions of the p l a t e Where c e n t r e l i n e p o r o s i t y should occur were p r e d i c t e d with the new model and compared to p u b l i s h e d experimental r e s u l t s of p o r o s i t y i n s t e e l p l a t e s . E x c e l l e n t a g r e e m e n t was o b t a i n e d b e t w e e n t h e p r e d i c t e d a n d e x p e r i m e n t a l r e s u l t s . I n t h e f l u i d f l o w m o d e l , D a r c y ' s Law was u s e d t o d e t e r m i n e t h e e x t e n t o f f l o w i n t h e i n t e r d e n d r i t i c c h a n n e l s . The p r e s e n t r e s u l t s i n d i c a t e D a r c y ' s Law i s v a l i d i n t h i s a p p l i c a t i o n . Table of Contents A b s t r a c t i i L i s t of Tables .... v i L i s t of Figures v i i L i s t of Nomenclature x Acknowledgement x i v PART-A INVERSE SEGREGATION IN BINARY ALLOYS 1 Chapter A-I INTRODUCTION AND LITERATURE REVIEW 2 1.1 I n t r o d u c t i o n 2 1.2 Previous Models f o r C h i l l Face Inverse Segregation 3 1.3 Inverse Segregation away from the C h i l l Face ( p o s i t i o n a l Segregation) 7 1.4 Present Objectives ,...11 Chapter A-II MODELING PROCEDURE 13 2.1 Mathematical Formulation 13 2.1.1 A l l o y D e n s i t i e s 16 2.1.2 Shrinkage Along The C h i l l Face 17 2.1.3 Segregation i n the F i r s t Column Adjacent To The C h i l l Face 19 2.1.4 P o s i t i o n a l Segregation 20 2.1.5 The Length of Inverse Segretation Zone 23 2.2 Computer Programming 24 Chapter A - I I I RESULTS AND DISCUSSION 25 3.1 The Aluminium Copper System 25 3.1.1 Volume Change during S o l i d i f i c a t i o n 25 3.1.2 Segregation at the C h i l l Face 27 3.1.3 P o s i t i o n a l Segregation 28 3.2 The Aluminium Zinc System 30 3.3 The Antimony Bismuth System 33 3.4 Examination Of Previous Model P r e d i c t i o n s 36 Chapter A-IV CONCLUSIONS 37 PART-B CENTERLINE SHRINKAGE IN STEEL PLATE CASTINGS 68 Chapter B-I INTRODUCTION .69 1.1 Gross Shrinkage ........69 1.2 C e n t e r l i n e Shrinkage • ..70 1.3 Previous Models 72 1.4 Pe r m e a b i l i t y ...76 V 1.5 Present O b j e c t i v e s 79 Chapter B-II MODELLING PROCEDURE 80 2.1 Mathematical Formulation 80 2.1.1 Temperature C a l c u l a t i o n s 80 2.1.2 Pressure Required To Feed Shrinkage 82 2.2 Computer Programming 84 2.3 V a l i d a t i o n Of The Heat Transfer Model 84 Chapter B-III RESULTS AND DISCUSSION 86 3.1 S o l i d i f i c a t i o n Sequence .' 86 3.2 P r e d i c t i o n Of C e n t e r l i n e Shrinkage 87 9.3 The E f f e c t of an End C h i l l on the Length of the P o r o s i t y Free Region 89 3.4 Proposed S o l i d i f i c a t i o n Parameters D e f i n i n g the T r a n s i t i o n from Porous to Nonporous Castings 90 Chapter B-IV CONCLUSIONS 92 BIBLIOGRAPHY 107 APPENDIX A - RECALCULATION OF THE SCHEIL-YOUDELIS MODEL PREDICTIONS 110 APPENDIX B - DERIVATION OF NODAL EQUATIONS 111 APPENDIX C - FORTRAN PROGRAM FOR THE PREDICTION OF CENTRELINE SHRINKAGE 113 v i L i s t of T a b l e s I. The comparison of a l l o y d e n s i t i e s with those c a l c u l a t e d from data f o r pure metals 39 I I . The thermal p r o p e r t i e s employed f o r the c a l c u l a t i o n of temperature d i s t r i b u t i o n 40 I I I . P h y s i c a l data employed i n c a l c u l a t i o n s 93 IV. Dimension of the s t e e l p l a t e c a s t i n g examined 94 V. Comparison of the c r i t i c a l v a l u e s of s o l i d f i c a t i o n parameters f o r c e n t e r l i n e shrinkage 94 v i i L i s t of Fig u r e s 1. E q u i l i b r i u m phase diagram and s p e c i f i c volume f o r the Al-Cu a l l o y s 41 2. Schematic d e s c r i p t i o n of s o l i d i f y i n g zone; (a) dendrite morphology, (b) s o l i d and l i q u i d compositions along c h i l l face, and (c) composition of i n t e r d e n d r i t i c l i q u i d 42 3. D e n s i t i e s of copper and aluminium vs temperature 43 4. D e n s i t i e s of Zinc, Aluminium and Bismuth vs temperature 44 5. Schematic c o n f i g u r a t i o n of the model i n v e s t i g a t e d and the s u b d i v i s i o n for the numerical s i m u l a t i o n 45 6. Schematic c o n f i g u r a t i o n of the feeding sequence; (a) F l u i d flow, and (b) p r o f i l e of the l i q u i d composition 46 7. Summary of the f l u i d flow induced by the shrinkage during the s o l i d i f i c a t i o n of the model with 10X10 s u b d i v i s i o n s 47 8. Schematic d e s c r i p t i o n of the d i l u t i o n e f f e c t of the i n t e r d e n d r i t i c f l u i d flow during s o l i d i f i c a t i o n 48 9. Flow chart for the c a l c u l a t i o n of inverse segregation 49 10. Flow chart for the c a l c u l a t i o n of the length of s o l i d / l i q u i d zone adjacent to the c h i l l face 50 11. C a l c u l a t e d volume change during s o l i d i f i c a t i o n of A l -10%C U. The number on the curve denotes each element s o l i d i f i e d 51 12. C a l c u l a t e d volume change and l i q u i d composition i n the r e p r e s e n t a t i v e volume f o r Al-5%Cu. The number on the curve denotes each element s o l i d f i e d 52 13. S o l i d i f i c a t i o n shrinkage r a t i o vs composition of the A l -Cu a l l o y s . The dotted l i n e was deri v e d from F i g 1. ...53 14. P o s i t i o n a l segregation p r o f i l e s f o r various compositions of the Al-Cu a l l o y s 54 15. Comparison of the inverse segregation at the c h i l l face f o r the Al-Cu a l l o y s 55 v i i i 16. E f f e c t of a i r gap at the metal/mold i n t e r f a c e on the length of s o l i d / l i q u i d zone adjacent to the c h i l l face 56 17. Comparison of p o s i t i o n a l segregation f o r A 1 - 1 0 % C U ....57 18. E q u i l i b r i u m phase diagram f o r Al-Zn a l l o y s 58 19. T o t a l shrinkage r a t i o vs composition f o r the Al-Zn a l l o y s 59 20. C a l c u l a t e d volume change and temperature i n the r e p r e s e n t a t i v e volume for Al-lO%Zn. The number on the curve denotes each element s o l i f i e d 60 21. C a l c u l a t e d p r o f i l e s of temperature i n the r e p r e s e n t a t i v e volume f o r var i o u s compositions of the Al-Zn a l l o y s i n d i c a t e d 61 22. Comparison of the inverse segregation at the c h i l l face for the Al-Zn a l l o y s 62 23. E q u i l i b r i u m phase diagram f o r Sb-Bi a l l o y s 63 24. T o t a l shrinkage r a t i o vs composition f o r the Sb-Bi a l l o y s 64 25. C a l c u l a t e d volume change and temperature i n the re p r e s e n t a t i v e volume for Sb-lO%Bi. The number on the curve denotes each element s o l i d i f i e d 65 26. Comparison of the inverse segregation at the c h i l l face for the Sb-Bi a l l o y s 66 27. C a l c u l a t e d volume change and temperature i n the re p r e s e n t a t i v e volume for Sb-20%Bi. The number on the curve denotes each element s o l i d i f i e d 67 28. Feeding r e l a t i o n s h i p determined ex p e r i m e n t a l l y i n the s t e e l c a s t i n g s ; (a) P l a t e s 3 0 , and (b) Square b a r s 3 1 ..95 29. Comparison of measured p e r m e a b i l i t i e s vs volume f r a c t i o n l i q u i d ..96 30. C o n f i g u r a t i o n of the system i n v e s t i g a t e d 97 31. Flow chart of the computer program f o r the p r e d i c t i o n of c e n t e r l i n e shrinkage 98 32. Temperature d i s t r i b u t i o n along the c e n t e r l i n e of the p l a t e c a s t i n g 99 33. S o l i d u s movement along the c e n t e r l i n e of p l a t e c a s t i n g 100 i x 34. The d i s t r i b u t i o n of s o l i d i f i c a t i o n contour l i n e s at the ( a ) i n i t i a l , (b) middle and (c) l a s t stages of s o l i d i f i c a t i o n 101 35. D i s t r i b u t i o n of the pressure required to feed shrinkage at the end of s o l i d i f i c a t i o n (s=5cm) 102 36. D i s t r i b u t i o n of the pressure r e q u i r e d to feed shrinkage at the end of s o l i d i f i c a t i o n (s=2.5cm) 103 37. D i s t r i b u t i o n of the pressure required to feed shrinkage at the end of s o l i d i f i c a t i o n (s=0.5cm) 104 38. Soundness of s t e e l p l a t e c a s t i n g s 105 39. D i s t r i b u t i o n of the pressure required to feed shrinkage at the end of s o l i d i f i c a t i o n -end c h i l l e d c a s t i n g ...106 40. Comparison of the inverse segregation at the c h i l l face fo r the Al-Cu a l l o y s 126 41. Comparison of the inverse segregation at the c h i l l face for the Al-Zn a l l o y s 127 42. Comparison of the inverse segregation at the c h i l l face fo r the Sb-Bi a l l o y s 128 43. D i f f e r e n t types of nodes i n the model i n v e s t i g a t e d ..129 L i s t o f N o m e n c l a t u r e a r e a , ( c m 2 ) e u t e c t i c c o n t r a c t i o n c o e f f i c i e n t , " B i o t number,* mean c o m p o s i t i o n o f volume e l e m e n t , ( w t % ) amount o f s e g r e g a t i o n , ( w t % ) s e g r e g a t i o n f o r m e d d u r i n g t h e s o l i d i f i c a t i o n of e l e m e n t ( i , j ) , ( w t % ) s e g r e g a t i o n i n j - t h c o l u m n , ( w t % ) c o m p o s i t i o n s o f l i q u i d and s o l i d , r e s p e c t i v e l y , ( w t % ) c o m p o s i t i o n o f l i q u i d a t e u t e c t i c , ( w t % ) sum of t h e c o m p o s i t i o n d i l u t i o n e f f e c t s i n j - t h column when i - t h e l e m e n t s o l i d i f i e s , ( w t % ) t o t a l d i l u t i o n o f c o m p o s i t i o n i n j - t h column,(wt%) i n i t i a l c o m p o s i t i o n , ( w t % ) s p e c i f i c h e a t , ( c a l / g ° C ) l o c a l a v e r a g e c o m p o s i t i o n o f s o l i d , ( w t % ) mean c o m p o s i t i o n o f c o r e d s o l i d a t e u t e c t i c , ( w t % ) l e n g t h of p l a t e c a s t i n g , ( c m ) p r i m a r y d e n d r i t e arm s p a c i n g , ( c m ) g r e y body shape f a c t o r , * F o u r i e r number, + w e i g h t f r a c t i o n s o l i d and l i q u i d , r e s p e c t i v e l y , + t e m p e r a t u r e g r a d i e n t , ( ° C / c m ) x i volume f r a c t i o n of e u t e c t i c , * volume f r a c t i o n of s o l i d and l i q u i d , r e s p e c t i v e l y , * a c c e l e r a t i o n due t o g r a v i t y , 9 8 1 ( c m / s e c 2 ) l a t e n t h e a t o f s o l i d i f i c a t i o n , ( c a l / g ) h e a t t r a n s f e r c o e f f i c i e n t , ( c a l / c m 2 s e c ° C ) h e a t t r a n s f e r c o e f f i c i e n t f r o m s a n d mold and m e t a l t o a m b i e n t a i r , r e s p e c t i v e l y , ( c a l / c m 2 s e c ° C ) p e r m e a b i l i t y , ( c m 2 ) t h e r m a l c o n d u c t i v i t y , ( c a l / c m . s e c ° C ) t h e r m a l c o n d u c t i v i t y o f a i r , ( c a l / c m . s e c ° C ) t h e r m a l c o n d u c t i v i t y of s a n d mold and m e t a l , r e s p e c t i v e l y , ( c a l / c m . s e c ° C ) e q u i l i b r i u m p a r t i t i o n r a t i o , * l e n g t h o f s o l i d / l i q u i d zone,(cm) l e n g t h o f s o l i d / l i q u i d z o ne a t s t e a d y s t a t e , ( c m ) l e n g t h o f f l u i d f l o w , ( c m ) c a p i l l a r y f e e d i n g d i s t a n c e , ( c m ) s o l i d and l i q u i d masses, r e s p e c t i v e l y , i n a r e p r e s e n t a t i v e v o l u m e , ( g ) mg and , r e s p e c t i v e l y , a t e u t e c t i c t e m p e r a t u r e , ( g ) r a d i u s o f c a p i l l a r y , ( c m ) p r e s s u r e , ( d y n / c m 2 ) ambient p r e s s u r e , ( d y n / c m 2 ) p r e s s u r e d r o p due t o f l u i d f l o w , ( d y n / c m 2 ) p r e s s u r e a c t i n g due t o c a p i l l a r y f l o w , ( d y n / c m 2 ) X I 1 g r a d i e n t o f f r a c t i o n s o l i d , ( 1 / c m ) h e a t f l u x , ( c a l / c m 2 s e c ) c o o l i n g r a t e , ( ° C / s e c ) t h i c k n e s s o f p l a t e c a s t i n g o r s i z e of s q u a r e b a r c a s t i n g , ( c m ) t e m p e r a t u r e , ( ° C ) t e m p e r a t u r e a f t e r A t , ( ° C ) i n t e r f a c e t e m p e r a t u r e o f i n g o t and m o l d , r e s p e c t i v e l y , ( ° C ) a m b i e n t t e m p e r a t u r e , ( ° C ) t e m p e r a t u r e i n m e t a l and sand mold, r e s p e c t i v e l y , ( ° C ) i n i t i a l t e m p e r a t u r e , ( ° C ) p o u r i n g t e m p e r a t u r e , ( ° C ) t i m e , ( s e c ) r e p r e s e n t a t i v e volume o r f l u i d volume t o f l o w , ( c m 3 ) v o l u m e - t o - a r e a r a t i o o f c a s t i n g , ( c m ) f l u i d f l o w v e l o c i t y i n a c a p i l l a r y , ( c m / s e c ) l o c a l f l o w v e l o c i t y o f i n t e r d e n d r i t i c l i q u i d , ( c m / s e c ) s u p e r f i c i a l f l o w v e l o c i t y o f i n t e r d e n d r i t i c f l u i d , ( c m / s e c ) v e l o c i t y o f s o l i d u s , m o v e m e n t a l o n g t h e c e n t e r l i n e of a p l a t e c a s t i n g , ( c m / s e c ) l o c a l f l o w v e l o c i t y o f i n t e r d e n d r i t i c l i q u i d i n x - d i r e c t i o n , ( c m / s e c ) X I 1 1 e l e m e n t s i z e , ( c m ) m e t a l l o s t a t i c head,(cm) t h e r m a l d i f f u s i v i t y , ( c m 2 / s e c ) s o l i d i f i c a t i o n s h r i n k a g e r a t i o , * t h i c k n e s s o f a i r gap,(cm) e m i s s i v i t y , + v i s c o s i t y , ( p o i s e ) s p e c i f i c volume o f l i q u i d and s o l i d , r e s p e c t i v e l y , ( c m 3 / g ) d e n s i t y , ( g / c m 3 ) l o c a l mean d e n s i t y o f s o l i d and l i q u i d , ( g / c m 3 ) i n i t i a l d e n s i t y of l i q u i d , ( g / c m 3 ) d e n s i t y o f l i q u i d and s o l i d , r e s p e c t i v e l y , ( g / c m 3 s o l i d d e n s i t y o f e u t e c t i c c o m p o s i t i o n , ( g / c m 3 ) 1 . 3 7 x 1 0 - 1 2 ( c a l / c m 2 s e c ° K f t ) t o r t u o s i t y f a c t o r , * d i m e n s i o n l e s s x i v Acknowledgement I wish to express my s i n c e r e g r a t i t u d e to P r o f . Fred Weinberg and Dr. I n d i r a V. Samarasekera f o r t h e i r i n v a l u a b l e a d v i c e , guidance and immense encouragement throughout the course of t h i s i n v e s t i g a t i o n . Thanks are a l s o extended to my f e l l o w graduate students f o r t h e i r voluntary a s s i s t a n c e and co-operation. The a s s i s t a n c e of f a c u l t y members i n the Department of M e t a l l u r g i c a l Engineering, U n i v e r s i t y of B r i t i s h Columbia i s g r e a t l y a p p r e c i a t e d . I am g r a t e f u l to RIKEN Corporation, Tokyo, Japan, f o r p r o v i d i n g me with t h i s opportunity and f i n a n c i a l support. I w i l l not be doing j u s t i c e to myself without acknowledging hard time and s a c r i f i c e s which my w i f e , Yuko, and my daughter, Momoko, had to stand during the course of the present study. 1 PART-A INVERSE SEGREGATION IN BINARY ALLOYS 2 A-I. INTRODUCTION AND LITERATURE REVIEW 1 .1 I n t r o d u c t i o n During the long i n d u s t r i a l h i s t o r y of foundry technology, the c a s t i n g process has been considered from an e m p i r i c a l point of view rather than t h e o r e t i c a l l y . This can be a t t r i b u t e d to the complexity of the c a s t i n g process which has discouraged b a s i c i n v e s t i g a t i o n s of the process. For example, the s o l i d i f i c a t i o n of multiphase systems i s heterogeneous, and occurs under unsteady s t a t e and n on-equilibrium c o n d i t i o n s . During s o l i d i f i c a t i o n of simple binary a l l o y s , s o l i d of one composition forms from l i q u i d of another composition. This occurrence makes i t impossible to cast homogeneous a l l o y s with uniform s t r u c t u r e and p r o p e r t i e s . Recently mathematical modelling of the s o l i d i f i c a t i o n process has enabled c a l c u l a t i o n s to be made which give the temperature d i s t r i b u t i o n i n the s o l i d and l i q u i d during non-equilibrium s o l i d i f i c a t i o n . With computers, these c a l c u l a t i o n s can be done reasonably, q u i c k l y and cheaply. The r e s u l t s of the c a l c u l a t i o n have been shown to agree w e l l w i t h the corresponding temperature measurements i n l a b o r a t o r y experiments and i n i n d u s t r i a l s i t u a t i o n s . The major defects i n c a s t i n g s are macrosegregation and shrinkage. Both have a very strong i n f l u e n c e on the q u a l i t y of the c a s t i n g s . I t i s g e n e r a l l y recognized that these defects are 3 l a r g e l y related to f l u i d flow during s o l i d i f i c a t i o n , p a r t i c u l a r l y i n t e r d e n d r i t i c flow re s u l t i n g from s o l i d i f i c a t i o n and thermal shrinkage. The occurence of inverse segregation, in which the solute gradient d i s t r i b u t i o n i s opposite to that predicted by the equilibrium phase diagram during s o l i d i f i c a t i o n - , has long interested metallurgists. Inverse segregation has been examined by a number of researchers. It i s generally accepted that t h i s type of segregation i s due to the back flow of solute r i c h i n t e r d e n d r i t i c l i q u i d to f i l l the void created by s o l i d i f i c a t i o n and thermal contractions during s o l i d i f i c a t i o n . 1.2 Previous Models for C h i l l Face Inverse Segregation Scheil was the f i r s t person to derive precise a n a l y t i c a l expressions for the amount of inverse segregation which occurs at the c h i l l face during s o l i d i f i c a t i o n . For a t y p i c a l binary eutectic a l l o y system, the segregation (AC) i s given by: AC = c" - Co (A-l) with " W i s - — — + m s E c S E c - - E Kirkaldy and Youdelis 2 extended the Scheil equation (A-2) 4 t o d e t e r m i n e t h e s o l u t e c o n c e n t r a t i o n p r o f i l e b o t h a t t h e c h i l l f a c e and i n t h e body of t h e m e t a l a s a f u n c t i o n o f d i s t a n c e f r o m t h e c h i l l f a c e f o r a u n i d i r e c t i o n a l l y s o l i d i f i e d i n g o t . A r e p r e s e n t a t i v e volume, V, i s r e l a t e d t o mass and s p e c i f i c v o l u m e s o f l i q u i d and s o l i d a s : V = V S m S + V j A (A-3) On t h e a s s u m p t i o n t h a t no c o n t r a c t i o n v o i d f o r m s , and t h a t V i s c o n s t a n t , e q u a t i o n (A-3) r e d u c e s t o , dnL. = - a dm„ X s (A-4) where a = l s + ^ V L V L d m S (A-5) The s o l u t e mass b a l a n c e f o r i n c r e m e n t a l s o l i d i f i c a t i o n i n t h e volume i s g i v e n by: f V T _ V Q d v T x d(m LC L) = - C s d m s + | c L d m s ( - ^ j - ( A _ 6 ) where t h e change of s o l u t e mass i n t h e l i q u i d i s e q u a t e d t o t h e change of s o l u t e mass i n t h e s o l i d p l u s t h e s o l u t e mass t r a n s p o r t e d i n t o t h e volume by c o n t r a c t i o n . The c o n t r a c t i o n c o n t r i b u t i o n i s made up o f two t e r m s : c o n t r a c t i o n due t o t h e s o l i d / l i q u i d p hase change, a n d c o n t r a c t i o n due t o t h e s p e c i f i c volume change o f t h e l i q u i d . N o te t h a t t h e r m a l c o n t r a c t i o n i s n e g l e c t e d i n t h i s m odel. C o m b i n i n g e q u a t i o n s (A-4) t o (A-6) g i v e s t h e b a s i c d i f f e r e n t i a l e q u a t i o n ; L d m L dc ^ " " X (A-7) 5 where r A.I.-SL LL (A-8) The i n t e g r a t i o n of equation (A - 7 ) gives m L which when s u b s t i t u t e d i n t o equation (A - 4 ) gives a value f o r ms. The change of the mass i n a cored c r y s t a l i s r e l a t e d to that i n the remaining l i q u i d as: d ( m S C S ) = CS d mS (A-9) Combining equations (A - 4 ) and (A - 7 ) with equation (A - 9 ) y i e l d s : d ( m S C S ) = ^ V ^ L (A-10) A c c o r d i n g l y , the s u b s t i t u t i o n of m C , c a l c u l a t e d by the i n t e g r a t i o n of equation ( A - 1 0 ) , together w i t h mT and m i n t o IJ S equation (A-2) w i l l give the maximum inverse segregation at the c h i l l f ace. For the Al-Cu system, the temperature dependence of the s p e c i f i c volumes of the s o l i d and l i q u i d under e q u i l i b r i u m c o n d i t i o n s were compiled by Sauerwald 3. These values are shown in F i g 1 along with c o n s t i t u t i o n a l phase diagram of the a l l o y . Using t h i s data, S c h e i l 1 c a l c u l a t e d the maximum segregation at the c h i l l face f o r a u n i d i r e c t i o n a l l y s o l i d i f i e d i ngot. The c a l c u l a t e d values were found to be i n e x c e l l e n t agreement with measured values of the s o l u t e c o n c e n t r a t i o n at the c h i l l face over the e n t i r e composition range examined. The above theory was a l s o a p p l i e d to the Al-Zn and Sb-Bi a l l o y s by Youdelis et a l " j 5 using the s p e c i f i c volume data reported by P r e z e l and Schne i d e r 6 . Again the t h e o r e t i c a l values 6 of the c h i l l face c o n c e n t r a t i o n agreed w e l l with the experimental measurements. In the Al-Zn a l l o y the s c a t t e r of the experimental data i s l a r g e , making the comparison between theory and experiment l e s s defined than i n other systems. Of p a r t i c u l a r i n t e r e s t i n comparing theory and experiment are the r e s u l t s for the Sb-Bi a l l o y . In t h i s system s o l i d i f i c a t i o n i s accompanied by volume expansion at the high B i c o n c e n t r a t i o n s and c o n t r a c t i o n at low Bi values. The change from p o s i t i v e to negative inverse segregation, corresponding to the t r a n s i t i o n from expansion to c o n t r a c t i o n , occurs at approximately 30%Bi. In the S c h e i l and Youdelis development of the mathematical formulation of inverse segregation, the f o l l o w i n g assumptions were made. 1) The s p e c i f i c volume and composition of the primary s o l i d i f i e d metal are constant. 2) The thermal c o n t r a c t i o n during s o l i d i f i c a t i o n i s n e g l i g i b l e . These assumptions are i n v a l i d or only approximately v a l i d f o r the f o l l o w i n g reasons. The model i s only a p p l i c a b l e to systems wi t h n e g l i g i b l e s o l i d s o l u b i l i t y to comply with assumption (1) and f o r a l l o y s with short f r e e z i n g ranges to comply with assumption ( 2 ) . Note t h a t , with complete c o r i n g i n the s o l i d , the f i n a l s o l i d i f i c a t i o n occurs at the e u t e c t i c temperature, not at the s o l i d u s temperature. 7 1.3 Inverse Segregation away from the C h i l l Face ( p o s i t i o n a l  Segregation) Assuming l i n e a r mass d i s t r i b u t i o n of l i q u i d i n the s o l i d / l i q u i d mushy zone, and a s p e c i f i c gradient of l i q u i d composition i n the d i r e c t i o n of s o l i d i f i c a t i o n , K i r k a l d y and Y o u d e l i s 2 have extended S c h e i l ' s a n a l y s i s to the c a l c u l a t i o n of the p o s i t i o n a l v a r i a t i o n of inverse segregation. At the c h i l l face, during s o l i d i f i c a t i o n , i n t e r d e n d r i t i c l i q u i d flows i n t o the elemental volume with ho outward flow. Away from the c h i l l face i n t e r d e n d r i t i c l i q u i d flows both i n t o and out of the elemental volume to take up volume change. To account for the outward f l u i d flow the equation f o r the s o l u t e mass balance, equation (A-6), must be r e w r i t t e n by adding a term due to l i q u i d flow out at the p o s i t i o n L: d ( V t ) - - C sd. s + j C L d ^ ( ) - _ i 1 - ^  a s ^ where a C A'L i s the mean l i q u i d c o n c e n t r a t i o n gradient and S i s the inward flow distance past point L due to c o n t r a c t i o n i n inner regions. Equation (A-6) was replaced with equation (A-11) and the p o s i t i o n a l segregation p r o f i l e s for A 1 - 1 0 % C U and A l -15%Cu were c a l c u l a t e d i n the s i m i l a r manner to the c a l c u l a t i o n of the maximum segregation at the c h i l l face as described i n the previous s e c t i o n . With t h i s theory, c a l c u l a t i o n s of the p o s i t i o n a l segregation gave r e s u l t s which agreed w e l l with 8 expe r i m e n t a l l y measured values of the segregation as a function, of p o s i t i o n from the c h i l l face. Prabhakar and Weinberg 7 questioned the co n c l u s i o n s a r r i v e d at by K i r k a l d y and Y o u d e l i s 2 on the b a s i s of the accuracy of t h e i r measurements. Prabhakar and Weinberg 7 c a r r i e d out some c a r e f u l inverse segregation measurements using r a d i o a c t i v e t r a c e r techniques and showed that the maximum inverse segregation at the c h i l l face i s h i g h l y dependent of c a s t i n g c o n d i t i o n s , i n c l u d i n g the superheat of the melt when c a s t . With superheat temperature greater than 40°C, the co n c e n t r a t i o n at the c h i l l face drops r a p i d l y with i n c r e a s i n g superheat f o r a copper c h i l l . With a s t a i n l e s s s t e e l c h i l l which has a lower thermal c o n d u c t i v i t y than copper, the inverse segregation i s higher at higher superheats when compared to the copper c h i l l , but a l s o drops with increased superheat. Moreover, there was no i n d i c a t i o n of inverse segregation i n the region w e l l away from the c h i l l face as reported by K i r k a d l y and Y o u d e l i s 2 . These r e s u l t s i n d i c a t e that v a r i o u s c a s t i n g parameters, not included i n the t h e o r e t i c a l model, can s t r o n g l y i n f l u e n c e the extent of i n t e r d e n d r i t i c f l u i d flow during s o l i d i f i c a t i o n and a c c o r d i n g l y , the inverse segregation. Based on the concept that a l l s o l u t e macrosegregation r e s u l t s from i n t e r d e n d r i t i c f l u i d flow of s o l u t e r i c h l i q u i d to feed s o l i d i f i c a t i o n shrinkage and thermal c o n t r a c t i o n , the S c h e i l - Y o u d e l i s model was extended by Flemings et a l 8 ' 1 1 to account f o r a wide range of segregation e f f e c t s . This included 9 c e n t e r l i n e s e g r e t a t i o n , banding, u n d e r - r i s e r p o s i t i v e s egregation, and the negative cone of segregation as w e l l as inverse segregation. Conservation of so l u t e mass i n the volume element during s o l i d i f i c a t i o n r e q u i r e s t h a t : £CpC) = - n^LCLv) (A-12) Conservation of t o t a l mass i n the volume element g i v e s , Combining equations (A-12) and (A-13), The change i n s o l u t e mass of the volume element i s the sum of the changes i n the l i q u i d and s o l i d phases; £ ( P C ) = ^ ( C s P s g s + C L P L g L ) (A-15) S i m i l a r l y , the change i n t o t a l mass of the volume element i s , TT = TT (% 8s + P L 8 L ) (A-16) Assuming l o c a l e q u i l i b r i u m at the s o l i d / l i q u i d i n t e r f a c e , no d i f f u s i o n i n the s o l i d , and a constant s o l i d d e n s i t y , g i v e s : at ( V s ^ = k o CL p s T T (A-17) and dg = - dg . L (A-18) S u b s t i t u t i n g equations (A-17) and (A-18) i n equation (A-15), and combining i t with equations (A-16) and (A-14) gives the f i n a l p a r t i a l d i f f e r e n t i a l equation; l^k = _ / lzg_i n , vVT . 8 L 3C T \l-k ) \ 1 + C~ (A-19) L o L For one dimensional problem, that i s u n i d i r e c t i o n a l 10 s o l i d i f i c a t i o n , e q u a t i o n (A-19) i s r e w r i t t e n t o : 9 e L 1 - e V X G g L 3 C L = (A-20) M a c r o s e g r e g a t i o n i s d e f i n e d f o r a b i n a r y e u t e c t i c s y s t e m a s , - m C S M S + p S E % C L E P s ( l - g E ) + p g E g E • (A-21) AC = C - C o w h i c h i s a n a l o g o u s t o S c h e i l ' s e x p r e s s i o n , e q u a t i o n (A-1) and ( A - 2 ) . M a c r o s e g r e g a t i o n c a n now be c a l c u l a t e d w i t h e q u a t i o n (A-21) and e q u a t i o n (A-19) o r ( A - 2 0 ) . Two l i m i t i n g c a s e s c a n be e a s i l y s t u d i e d . F o r s t e a d y s t a t e , _ R ~ T-p" CA-22) S u b s t i t u t i n g e q u a t i o n (A-22) i n t o e q u a t i o n (A-20) and i n t e g r a t i n g y i e l d s , C L l - k Q 8 L = ( c ; ) ° (A-23) The o t h e r l i m i t i n g c a s e c a n be d e f i n e d a t c h i l l f a c e where, and s i m i l a r l y , 1- e C L . l - k Q 8 L = IC 7"/ (A-24) o A s s u m i n g a c o n s t a n t p a r t i t i o n r a t i o , k =0.172, f o r A l - 4 . 5 % C u , o and t a k i n g t h e v a l u e f o r s o l i d i f i c a t i o n s h r i n k a g e a s /3=0.055 F l e m i n g s e l a l 8 c a l c u l a t e d t h e amount o f s e g r e g a t i o n a s : AC = 0.43% a t c h i l l f a c e 11 AC = 0 at steady s t a t e The c a l c u l a t e d inverse segregation at the c h i l l face agrees w i t h the experimental values reported by S c h e i l 1 and the c a l c u l a t e d segregation at steady s t a t e with the r e s u l t s of Prabhakar and Weinberg 7. Note t h a t , i f the s o l i d i f i c a t i o n shrinkage i s defined by: 3 = —k = (A_25) PS V L as Flemings et a l 8 d i d , the values of 0 obtained from F i g 1 are negative i n the a l l o y s c o n t a i n i n g more than 9%Cu. When a negative value of 0 i s adopted to above theory, the r e s u l t i n g segregation at the c h i l l face i s a l s o negative, which does not agree with p o s i t i v e inverse segregation reported by S c h e i l 1 and K i r k a l d y et a l 2 . To v a l i d a t e the model of Flemings et a l 8 ' 9 , /3 must always be p o s i t i v e , t h e r e f o r e , the d e f i n i t i o n of /3 i s not v a l i d f o r the Al-Cu system. 1.4 Present Objectives In the segregation models thus f a r reviewed,the s o l i d i f i c a t i o n shrinkage i s a c r i t i c a l v a r i a b l e . I t i s b e l i e v e d that t h i s i s not c l e a r l y defined i n the models, and the values adopted to c a l c u l a t e the inverse segregation are i n c o r r e c t . In a d d i t i o n other assumptions made in the theory are not e n t i r e l y s a t i s f a c t o r y . In order to c l a r i f y these problems and to p r e d i c t inverse segregation i n a u n i d i r e c t i o n a l l y s o l i d i f i e d ingot 12 p r e c i s e l y , a new model, with minimum assumptions, was developed. 13 A - I I . MODELING PROCEDURE 2.1 Mathematical Formulation In a u n i d i r e c t i o n a l l y s o l i d i f i e d ingot three dimensional c r y s t a l growth occurs since a de n d r i t e grows three d i m e n s i o n a l l y because of microsegregation between den d r i t e s and the d i f f e r e n c e of thermal d i f f u s i v i t i e s i n s o l i d and l i q u i d . To s i m p l i f y t h i s complex problem, i t i s assumed that the primary d e n d r i t e has a pyramid shape without secondary or higher order d e n d r i t e branches. The f o l l o w i n g assumptions are a l s o made i n the mathematical f o r m u l a t i o n ; 1) No surface exudation occurs at the c h i l l face. 2) There i s no d i f f u s i o n i n the s o l i d during s o l i d i f i c a t i o n . 3) Residual i n t e r d e n d r i t i c l i q u i d i s completely homogeneous i n the plane perpendicular to the growth d i r e c t i o n . 4) Local e q u i l i b r i u m c o n d i t i o n s e x i s t at the s o l i d / l i q u i d i n t e r f a c e . 5) The dendrite shape does not change during s o l i d i f i c a t i o n . 6) The microsegregation i s not a f f e c t e d by i n t e r d e n d r i t i c f l u i d flow. 7) No shrinkage p o r o s i t y occurs during s o l i d i f i c a t i o n . 14 This r e q u i r e s low gas l e v e l s and u n r e s t r i c t e d flow of l i q u i d through the d e n d r i t e channels. Under these c o n d i t i o n s , the f i r s t small amount of s o l i d to form i s of composition \}CQ at the l i q u i d u s temperature. During subsequent s o l i d i f i c a t i o n , s o l u t e i s enriched i n the r e s i d u a l l i q u i d which leads to higher s o l u t e c o n c e n t r a t i o n s i n the s o l i d as s o l i d i f i c a t i o n progresses. Since d i f f u s i o n i n the s o l i d i s assumed n e g l i g i b l e , the s o l u t e d i s t r i b u t i o n i n the s o l i d does not change a f t e r s o l i d i f i c a t i o n occurs. A q u a n t i t a t i v e s o l u t e mass balance can be expressed by equating the amount of s o l u t e r e j e c t e d from the s o l i d / l i q u i d i n t e r f a c e to the s o l u t e increase i n the l i q u i d . This balance i s (C - C ) df_ = f T d C T L s S L L (A-26a) since C = k C and f = 1-f : s o L L s dC df„ C s u V l - f s (A-26b) I n t e g r a t i n g equation (A-26b) from C g = k Q C o at f g=0 y i e l d s the s o l i d composition as a f u n c t i o n of f r a c t i o n s o l i d CS " k o C o ( 1 - f S ) V 1 , CA-27a) or i n terms of l i q u i d composition and f r a c t i o n l i q u i d , C L = C o f L k ° 1 (A-27b) Equations (A-27) have been deri v e d by S c h e i l 1 2 and P f a n n 1 3 , are c a l l e d the S c h e i l equation or Pfann equation. The S c h e i l equation has been used to p r e d i c t microsegregation f o r normal 15 c o n d i t i o n s of c a s t i n g and ingot s o l i d i f i c a t i o n i n binary a l l o y s by Brody et a l 1 * and Bower et a l 1 5 . They reported a good c o r r e l a t i o n between the equation and experimental r e s u l t s . At the i n i t i a l stages of s o l i d i f i c a t i o n , the s o l i d / l i q u i d region can be represented s c h e m a t i c a l l y by s o l i d pyramids p r o j e c t i n g i n t o the l i q u i d as shown i n F i g 2a. F o l l o w i n g the above d i s c u s s i o n , the microsegregation i s governed by equation (A-27) which give the s o l i d and l i q u i d composition p r o f i l e s along the c h i l l face as shown i n F i g 2b. When the d e n d r i t e t i p reaches L, the f r a c t i o n s o l i d i n the volume element away from the c h i l l face ( f ) can be r e l a t e d to that i n the volume element s adjacent to the c h i l l face (f ) by, s •P » — l * ~ x c fS " ~ L ~ fS CA-28) and the composition p r o f i l e i s given by the S c h e i l equation, T k -1 C q = k C ( i - i r * f ) ° S o o \ L SJ (A-29a) and T k - 1 L-x L F i g 2c shows the d i s t r i b u t i o n of i n t e r d e n d r i t i c l i q u i d composition i n the growing d i r e c t i o n c a l c u l a t e d from equation (A-29b). I t can be seen that the flow of s o l u t e enriched l i q u i d to feed shrinkage r e s u l t s i n inverse segregation. The next step i s to e s t a b l i s h the amount of f l u i d flow, which i s r e l a t e d to the amount of volume c o n t r a c t i o n . As described i n the previous p a p e r s 5 ' 9 , the volume change occuring 1 6 i n the process of s o l i d i f i c a t i o n i s made up of three mechanisms. 1) The change due to the l i q u i d / s o l i d phase change. 2) The change due to the d e n s i t y change i n the r e s i d u a l l i q u i d a s s o c i a t e d with composition changes. 3) Thermal c o n t r a c t i o n a s s o c i a t e d with temperature changes. The f i r s t two volume changes can be c a l c u l a t e d from the equations for microsegregation, equations (A-27) and (A-29). The thermal c o n t r a c t i o n i s determined by the temperature drop. 2.1.1 A l l o y D e n s i t i e s The c a l c u l a t i o n of the t o t a l amount of c o n t r a c t i o n due to s o l i d i f i c a t i o n and temperature changes r e q u i r e s accurate information of the d e n s i t i e s of the l i q u i d and s o l i d a l l o y s as a f u n c t i o n of both composition and temperature. The data reported by Sauerwald 3 (see F i g 1) and P r e z e l et a l 6 are not a p p l i c a b l e i n the present a n a l y s i s s i n c e the d e n s i t i e s they gave assume that the a l l o y composition at a given temperature are d e f i n e d by the e q u i l i b r i u m phase diagram, which i s not the case. This i s the reason that the previo'us m o d e l s 1 ' 2 ' 8 were forced t o neglect thermal c o n t r a c t i o n during s o l i d i f i c a t i o n . The only r e l i a b l e data which can be used i n the present case are the published values f o r the temperature dependence of the d ensity for pure metals. F i g 3 shows the temperature dependence of the d e n s i t i e s of copper and aluminium compiled by 1 7 S m i t h e l l s 1 6 and E l l i o t et a l 1 7 . The corresponding d e n s i t i e s f o r z i n c , antimony and bismuth are shown i n F i g 4. Provided that a bin a r y e x i s t s as a simple mixture of each component, the d e n s i t y of the a l l o y can be determined from the percentage of the components i n the a l l o y and the components d e n s i t i e s at a given temperature. C a l c u l a t e d values of a l l o y d e n s i t i e s f o r the Al-Cu system using the component d e n s i t e s are compared to those measured d i r e c t l y or compiled by Bornemann et a l 1 8 and S m i t h e l s 1 6 i n Table I . For both s o l i d s and l i q u i d s e x c e l l e n t agreement i s obtained between the c a l c u l a t e d d e n s i t i e s determined from the c o n s t i t u e n t s and the d i r e c t measurements w i t h d i f f e r e n c e s of l e s s than 5% at most. S i m i l a r comparisons were made f o r the Al-Zn and Pb-Sn systems using the data of Bornemann et a l 1 9 and Thresh et a l 2 0 (Table I ) . Very good agreement i s again evident w i t h e r r o r s , l e s s than 1.6% and 1.2% i n the Al-Zn and the Pb-Sn systems, r e s p e c t i v e l y . These e r r o r s are n e g l i g i b l e i n the present case. 2.1.2 Shrinkage Along The C h i l l Face Consider the re p r e s e n t a t i v e volume adjacent to the c h i l l face ( F i g 2a). The s o l i d / l i q u i d i n t e r f a c e i s plane and the s o l i d i f i c a t i o n advances u n i d i r e c t i o n a l l y because of the microsegregation discussed p r e v i o u s l y . In the numerical c a l c u l a t i o n , the volume was subdivided i n t o n elements. When the i - t h element i s s o l i d i f i e d , composition of the remaining 18 l i q u i d i s : i k o - 1 CL± = C o ( 1 ~ n } ° (A-30a) and the average s o l i d composition of the i - t h element i s : 5 S i - J L l V * • -=.{(1- ^ ) k ° - (1- i)k°j <A-30b> u n t i l C L reaches the e u t e c t i c composition. The corresponding i temperature of the r e p r e s e n t a t i v e volume (T^ ) can be obtained -" i from the phase diagram as the l i q u i d u s temperature for the composition CT . Once the e u t e c t i c composition i s reached, no s o l u t e r e j e c t i o n occurs at the s o l i d / l i q u i d i n t e r f a c e . From the d i s c u s s i o n i n the previous s e c t i o n , the d e n s i t y of each element i s c a l c u l a t e d f or the average s o l i d composition (C ) or l i q u i d i composition (C ) at temperature T . To c a l c u l a t e the volume T . L. change i n each element, a constant mass w i t h i n the element i s assumed. Then the shrinkage r a t i o i n each element i s : P l ~ p o SH. = — - - ( A _ 3 1 ) and t h e . t o t a l shrinkage of the whole volume i s : n e i = Z S H i (A-32) 1=1 The shrinkage formed during the s o l i d i f i c a t i o n of the i - t h element i s : A e ± = e. - e i _ 1 (A-33) 19 2.1.3 Segregation i n the F i r s t Column Adjacent To The C h i l l Face Consider the system shown i n F i g 5. I f the number of s u b d i v i s i o n i n the x and y d i r e c t i o n s are equal then the assumption (3) d i c t a t e s that the s o l i d i f i c a t i o n proceeds one element by one element i n both d i r e c t i o n s . The growth of the dendrite i n the x d i r e c t i o n r e s u l t s from the u n i d i r e c t i o n a l heat t r a n s f e r and the growth i n y d i r e c t i o n from the microsegregation expressed by equation (A-27). The f i r s t column i n F i g 5 coresponds to the r e p r e s e n t a t i v e volume discussed i n the previous s e c t i o n . The shrinkage formed i n the f i r s t column i s completely fed by the l i q u i d i n the second column. During the s o l i d i f i c a t i o n of the i - t h element i n column 1, the l i q u i d composition i n column 2 increases from C to C . Taking the average f o r the l i q u i d composition i n column 2, the segregation caused by the feeding flow during t h i s p e r i o d i s : and the f i n a l segregation a f t e r complete s o l i d i f i c a t i o n of the f i r s t column i s : (A-34) n AC 1 " 2 A C L i , l (A-35) i=l 20 2.1.4 P o s i t i o n a l Segregation Based on the assumption that shrinkage due to s o l i d i f i c a t i o n and thermal shrinkage i s completely fed w i t h r e s i d u a l l i q u i d , i n t e r d e n d r i t i c f l u i d flow should occur c o n t i n u o u s l y during s o l i d i f i c a t i o n . The maximum segregation occurs at the c h i l l face since the l i q u i d r e q u i r e d to feed shrinkage i s concentrated i n s o l u t e and there i s no l i q u i d . flow across the c h i l l face (note that we neglect surface exudations). At an i n t e r i o r p o s i t i o n of the i n g o t , the composition of l i q u i d f l o w i n g out of the elemental volume i s higher than that of l i q u i d flowing i n . A c c o r d i n g l y , the f l u i d flow across the volume element away from the c h i l l face d i l u t e s the l i q u i d c o n centration i n the volume, decreasing the amount of segregation. F i g 6a shows the feeding sequence s c h e m a t i c a l l y . The system i s subdivided i n t o t h i n columns i n which the i n t e r d e n d r i t i c l i q u i d i s assumed to be completely homogeneous. The dotted l i n e i n F i g 6a denotes the a c t u a l l o c a t i o n of s o l i d / l i q u i d i n t e r f a c e . Note that the r a t i o of length to width f o r the dendrite remains unchanged on the b a s i s of the assumption that dendrite shape does not change during s o l i d i f i c a t i o n . The corresponding p r o f i l e of l i q u i d composition i n the s o l i d i f y i n g d i r e c t i o n i s shown i n F i g 6b. F i g 7 summarizes the occurance of a l l the i n t e r d e n d r i t i c f l u i d flow i n the model with 10X10 s u b d i v i s i o n s by the time the p o s i t i o n L reaches e u t e c t i c temperature, that i s when the e n t i r e model volume has s o l i d i f i e d . Each dotted l i n e denotes the a c t u a l 21 l o c a t i o n of s o l i d / l i q u i d i n t e r f a c e , and the d i g i t i n each element the number of columns whose shrinkages cause f l u i d flow across a p a r t i c u l a r column at the corresponding l o c a t i o n of the i n t e r f a c e . The number 5 appearing at (b,6), f o r example, means that the shrinkages i n f i v e columns, 1-5, induce the f l u i d flow across column 6 because of the f l u i d flow to feed shrinkages i n elements (9,1),(8,2),(7,3),(6,4) and (5,5). Thus the f l u i d i n - ~ flow to feed i t s own shrinkage i s not included i n t h i s f i g u r e . When the element ( i , j ) s o l i d i f i e s , ( j-1) f l u i d flows w i l l occur across the j - t h column, which are induced by the flows to feed shrinkages i n the column ( j - 1 ) , ( j - 2 ) , ( j - 3 ) , - - - - , which i s d e p i c t e d i n F i g 8 (Note t h i s d e s c r i p t i o n a p p l i e s for i+j ^ 11). Then, the l i q u i d compositions increase from C to C L i - 1 L i i n the j - t h column and from C to C i n the ( j + l ) - t h L i - 2 L ± _ ! column. Taking the averages f o r the compositions, the l i q u i d to flow out of the column j i s (C +C )/2 and the l i q u i d to flow L i - 1 L i i n t o i s (C +C )/2 so that the d i f f e r e n c e between them i s L i - 2 L i - 1 given by (C -C )/2. Since the microsegregation i s not L i L i - 3 a f f e c t e d by f l u i d flow, the s o l i d i f i c a t i o n shrinkages formed i n inner columns are the same as that i n the f i r s t column c a l c u l a t e d by equation (A-33). The d i l u t i o n e f f e c t of the above flow induced by the shrinkage i n the ( j - 1 ) - t h column i n which ( i + l ) - t h element s o l i d i f i e s i s : c — c - (-V** - c > « 22 The sum of d i l u t i o n e f f e c t s f o r (j - 1 ) flow occurances i s : AC i+j . V C L j L . .. ( Z **±)( \ ^ -Co) CA-37) i-i+1 ' 2 When the element (5,4) s o l i d i f i e s i n F i g 7 f o r example, the shrinkages i n the f i r s t , second and t h i r d columns - w i l l cause f l u i d flow across the column 4. The composition d i f f e r e n c e of the l i q u i d f l owing i n t o and out of the column 4 i s (C T -C, )/2 and the sum of d i l u t i o n e f f e c t s i s : CL ~ CL A C L 5 > 4 = <AB6 + AB7 + A P 8 ) ( - i — - c J This kind of d i l u t i o n flow continues u n t i l the whole column s o l i d i f i e s completely, t h e r e f o r e , the t o t a l d i l u t i o n e f f e c t f o r the j - t h column i^s given by: A C L X a c L ( A _ 3 8 ) i i=i Since the feeding flow i n t o a column to compensate i t s own shrinkage increase the composition of the column by the an amount equal to the increase i n the f i r s t column, the r e s u l t i n g segregation i n the j - t h column i s : AC. - A C l - AC L_ ( A _ 3 9 ) 23 2.1.5 The Length of Inverse S e g r e t a t i o n Zone On the assumption that the l o c a l e q u i l i b r i u m c o n d i t i o n s e x i s t at the l i q u i d / s o l i d i n t e r f a c e , the temperature at a dend r i t e t i p corresponds to the l i q u i d u s temperature of the a l l o y . Hence using the d e n d r i t e t i p temperature, the steady s t a t e length of s o l i d / l i q u i d zone, L , can be def i n e d as the s d i s t a n c e between l i q u i d u s isotherm and the c h i l l face when the metal at the c h i l l face reaches the e u t e c t i c temperature. The value of L„ can be c a l c u l a t e d n u m e r i c a l l y . The heat t r a n s f e r S equation i s given by: 3 T = k 3 2 T (A-40) 9 t p C . 2 p 3 x for a s e m i - i n f i n i t e mold and ingot. During s o l i d i f i c a t i o n the metal s h r i n k s from the copper c h i l l producing an a i r gap which lowers the heat t r a n s f e r between the metal and mould. The heat t r a n s f e r c o e f f i c i e n t through the a i r gap can be determined from the f o l l o w i n g e xpression: k (TT+ 273) 4 - (T + 273) 4 h = -f- + OF T - T ( A" 4^) I I I The l a t e n t heat of s o l i d i f i c a t i o n i s taken to be released uniformly between the l i q u i d u s and s o l i d u s temperatures. Values for the thermal p r o p e r t i e s used i n the numerical c a l c u l a t i o n are l i s t e d i n Table I I . 24 2.2 Computer Programming Based on the mathematical formul a t i o n and information d i s c u s s e d i n the previous chapter, c a l c u l a t i o n s of segregation were performed numerically with a computer. F i g 9 shows the flow chart f o r the c a l c u l a t i o n of inverse segregation at and adjacent to the c h i l l face. To c a l c u l a t e the l e n g t h of the inverse segregation zone, the p a r t i a l d i f f e r e n t i a l equation f o r heat t r a n s f e r , equation (A-40), was solved i m p l i c i t l y with the f i n i t e d i f f e r e n c e method. The nodal equations for i n t e r n a l nodes are: * * * * T. - T. . T. N - 2T. + T. .a At p p (Ax) 2 and f o r surface h a l f node: At each time step, the Gauss-Jordan e l i m i n a t i o n method was a p p l i e d to solve the t r i d i a g o n a l matrix. The a c t u a l c a l c u l a t i o n sequence i s shown i n F i g 10. 25 A - I I I . RESULTS AND DISCUSSION The mathematical model e s t a b l i s h e d i n the previous chapter was a p p l i e d to the c a l c u l a t i o n of inverse segregation i n Al-Cu, Al-Zn and Sb-Bi systems. 3.1 The Aluminium Copper System The Al-Cu system i s a t y p i c a l e u t e c t i c a l l o y with nearly l i n e a r l i q u i d u s and s o l i d u s l i n e s i n the phase diagram ( F i g 1). The l i n e s are assumed to be l i n e a r i n the c a l c u l a t i o n g i v i n g a constant p a r t i t i o n r a t i o during s o l i d i f i c a t i o n . 3.1.1 Volume Change during S o l i d i f i c a t i o n I t i s seen i n F i g 1 that expansion instead of c o n t r a c t i o n at l i q u i d / s o l i d phase change appears over most stages of s o l i d i f i c a t i o n of the Al-Cu a l l o y . The c a l c u l a t e d r e s u l t s of volume change fo r A1-10%C U, F i g 11, a l s o r e v e a l t h i s phenomena c l e a r l y , where the s o l i d l i n e denotes the sequence of volume change i n each s o l i d i f i e d element and the dotted l i n e i n the r e s i d u a l l i q u i d . Here the e q u i l i b r i u m p a r t i t i o n r a t i o was taken to have a constant value, k =0.171. Each element with l e s s than e u t e c t i c composition shows expansion during s o l i d i f i c a t i o n . The 26 volume expansion r a t i o increases with an increase of c o n c e n t r a t i o n because the volume change i s h i g h l y dependent on the d i f f e r e n c e i n copper conce n t r a t i o n i n the l i q u i d and s o l i d . On the other hand, we can observe volume c o n t r a c t i o n d u r i n g e u t e c t i c s o l i d i f i c a t i o n because no composition change takes place during s o l i d i f i c a t i o n a f t e r the e u t e c t i c composition i s reached. The t o t a l shrinkage of the whole r e p r e s e n t a t i v e volume" i s a l s o p l o t t e d as a dot dashed l i n e i n F i g 11. Of p a r t i c u l a r importance here i s t h a t , even i n the s o l i d i f i c a t i o n process of the region with composition l e s s than the e u t e c t i c , where s o l i d i f i c a t i o n expansion occurs, the t o t a l shrinkage r a t i o f o r the whole system continues to increase almost s t e a d i l y . This phenomena can be a t t r i b u t e d to the increase i n l i q u i d d e n s i t y as a r e s u l t of solute enrichment i n the r e s i d u a l l i q u i d , and to thermal c o n t r a c t i o n which i s observed i n F i g 11 as the s l i g h t increase of the shrinkage i n each element a f t e r i t s s o l i d i f i c a t i o n . Hence we can conclude t h a t , even though the volume expansion occurs during the l i q u i d / s o l i d phase t r a n s f o r m a t i o n , the sum of the c o n t r a c t i o n due to the s o l u t e c o n c e n t r a t i o n in the l i q u i d and the temperature drop i s always greater than the expansion. F i g 12 shows the c a l c u l a t e d r e s u l t s of volume change of each element i n Al-5%Cu a l l o y together with the composition p r o f i l e of remaining l i q u i d . Note that the volume c o n t r a c t i o n f o l l o w s the l i q u i d / s o l i d phase transformation i n the beginning of s o l i d i f i c a t i o n , decreasing with increase of a l l o y composition. The volume change e v e n t u a l l y reverses from 27 c o n t r a c t i o n to expansion between elements 4 and 5. The corresponding composition of the r e v e r s a l p o i n t i s found t o be approximately 7%Cu which i s l e s s than the c r i t i c a l value of 9%Cu, obtained by Sauerwald 3. The c a l c u l a t e d t o t a l shrinkage r a t i o s are p l o t t e d a g a i n s t the a l l o y composition i n F i g 13. These can be compared with the volume change r a t i o s at the l i q u i d / s o l i d phase change obtained from the s p e c i f i c volume data of F i g 1, a l s o shown i n Fig-13. The t o t a l shrinkage r a t i o , d i f f e r i n g from the l o c a l volume change, i s always p o s i t i v e with values ranging between 5.2% f o r Al-25%Cu to 7.2% for nearly pure A l . The abrupt increase i n the t o t a l shrinkage r a t i o near pure A l r e s u l t s from the thermal c o n t r a c t i o n . N e g l e c t i n g t h i s volume c o n t r a c t i o n as was done i n the previous i n v e s t i g a t i o n 1 ' 2 ' 8 would c l e a r l y r e s u l t i n a s i g n i f i c a n t e r r o r i n the c a l c u l a t e d values of shrinkage and segregation. 3.1.2 Segregation at the C h i l l Face Segregation p r o f i l e s of the a l l o y s can now be c a l c u l a t e d using the complete volume changes discussed above. P o s i t i o n a l inverse segregation p r o f i l e s r e s u l t i n g from these c a l c u l a t i o n s are given i n F i g 14 f o r Al-Cu a l l o y s as f u n c t i o n s of f r a c t i o n a l d i s t a n c e from the c h i l l face. The i n i t i a l slopes of the curves decrease with i n c r e a s i n g a l l o y composition. The maximum segregation i s obtained at the c h i l l face. The c a l c u l a t e d 28 inverse segregation at the c h i l l face of the Al-Cu a l l o y s as a f u n c t i o n of composition i s shown by the s o l i d l i n e i n F i g 15. As the co n c e n t r a t i o n of copper i n the a l l o y i n c r e a s e s , the c h i l l face inverse segregation increases r a p i d l y to a peak value at about A l - l 2 % C u , then decreases to zero at the e u t e c t i c composition. Also p l o t t e d i n F i g 15 are the experimental measurements by S c h e i l 1 and K i r k a l d y et a l 2 and t h e o r e t i c a l -p r e d i c t i o n s by S c h e i l 1 and Flemings et a l 8 . The agreement of the model p r e d i c t i o n s with the experimental values i s e x c e l l e n t f o r copper concentrations l e s s than 15%. At concentrations above 15% the c a l c u l a t e d r e s u l t s are greater than the experimental values. 3.1.3 P o s i t i o n a l Segregation The length of the s o l i d / l i q u i d zone adjacent to the c h i l l face i s dependent on the rate of heat e x t r a c t i o n from the metal at the c h i l l face. This i n turn can be markedly i n f l u e n c e d by the presence of an a i r gap between the metal and the c h i l l surfaces due to the c o n t r a c t i o n of the metal. The length of the s o l i d / l i q u i d region i s defined as the d i s t a n c e between the l i q u i d u s and the c h i l l at the moment when the metal at the c h i l l face reaches the e u t e c t i c temperature. The c a l c u l a t e d zone leng t h i s p l o t t e d in F i g 16 as a f u n c t i o n of the width of a i r gap for three melt superheats, as i n d i c a t e d . The length of the s o l i d / l i q u i d zone i s observed to increase r a p i d l y with small 29 increases i n the a i r gap. The length goes from n e a r l y zero to 15mm with the formation of a small gap of lO^m wi t h a superheat of 4 0 ° C . The rate of increase i s highest with the lowest melt superheat. There are r e l a t i v e l y few r e l i a b l e measurements of the leng t h of the s o l i d / l i q u i d region, p a r t i c u l a r l y with-the length being very s e n s i t i v e to the s o l i d i f i c a t i o n c o n d i t i o n s . R e l i a b l e measurements have been reported by Prabhakar and Weinberg 7 using r a d i o a c t i v e t r a c e r techniques to measure the inverse segregation. Their r e s u l t s show that the length of s o l i d / l i q u i d region with 4 1 ° C superheat i s approximately 5mm f o r A 1 - 1 0 % C U . I t i s i n d i c a t e d from F i g 16 that there was an a i r gap of 3MITI between the a l l o y and the c h i l l in t h e i r measurements. Inverse segregation adjacent to the c h i l l face ( p o s i t i o n a l segregation) c a l c u l a t e d from the present theory f o r A 1 - 1 0 %C U i s shown i n F i g 17. For comparison the r e s u l t s reported by Prabhakar et a l 7 and the t h e o r e t i c a l p r e d i c t i o n of K i r k a l d y et a l 2 are a l s o shown. The inverse segregation c a l c u l a t e d from the present theory i s a l i t t l e lower than the experimental r e s u l t s of Prabhakar et a l . The r e s u l t s by K i r k a l d y et a l deviate f u r t h e r from the experimental r e s u l t s near the c h i l l and maintain a p o s i t i v e value a f t e r 5mm when the measured value i s zero. Inverse segregation i s reported by K i r k a l d y et a l w e l l away from the c h i l l face, d i f f e r i n g from the r e s u l t s of the present model and the experimental r e s u l t s of Prabhakar et a l . Prabhakar and Weinberg 7 have a l s o found that with melt 30 superheats above 40°C the c o n c e n t r a t i o n at the c h i l l face drops r a p i d l y with i n c r e a s i n g superheat against a copper c h i l l . T h i s occurance can be accounted f o r from the r e s u l t s given i n F i g 16. The superheat reduces the t h i c k n e s s of a i r gap at mold/metal i n t e r f a c e which lowers the l e n g t h of the s o l i d / l i q u i d r e g i o n . I t i s q u i t e d i f f i c u l t to measure the c h i l l face segregation p r o p e r l y w i t h i n a very narrow inverse segregation zone. 3.2 The Aluminium Zinc System The phase diagram of the Al-Zn a l l o y system i s shown i n F i g 18. In t h i s case the s o l i d u s and l i q u i d u s l i n e s cannot be assumed to be l i n e a r . To apply the segregation model to t h i s system, the phase diagram was d i v i d e d i n t o nineteen p o r t i o n s , every 5% along l i q u i d u s . The s o l i d u s and l i q u i d u s l i n e s were assumed to be l i n e a r i n each p o r t i o n g i v i n g a constant p a r t i t i o n r a t i o . The c a l c u l a t e d t o t a l shrinkage r a t i o i s p l o t t e d as a f u n c t i o n of composition i n F i g 19. The shrinkage r a t i o decreases with increase of composition i n the low a l l o y r e g i o n , then increases s l i g h t l y as i t approaches the e u t e c t i c composition. The increase of t o t a l shrinkage near pure A l , which i s much more prominent than that i n Al-Cu ( F i g 12), i s a l s o caused by the thermal shrinkage. In the region between 10% to 40%Zn, the d e c l i n e of shrinkage i s markedly a c c e l e r a t e d . 31 F i g 20 shows the c a l c u l a t e d r e s u l t s of volume change f o r Al-lO%Zn. S i m i l a r to the Al-Cu a l l o y , the r e v e r s a l from shrinkage to expansion i n volume change during l i q u i d / s o l i d t ransformation i s observed i n the low composition r e g i o n . This c r i t c i a l p oint l i e s between the elements 7 and 8. However, independent of shrinkage or expansion during the s o l i d i f i c a t i o n of each element, the t o t a l shrinkage keeps i n c r e a s i n g s t e a d i l y -u n t i l the s o l i d / l i q u i d reaches the f i n a l part of s o l i d i f i c a t i o n . During the s o l i d i f i c a t i o n of the f i n a l element, the shrinkage for both, each element and t h e i r t o t a l , are s i g n i f i c a n t l y r a i s e d . This occurrence can be accounted f o r by the l a r g e thermal shrinkage r e s u l t i n g from the long f r e e z i n g range of the element 10. The l i q u i d composition i n t h i s element increases from 31% to 95%Zn because of the s o l u t e r e j e c t i o n away from the s o l i d / l i q u i d i n t e r f a c e during i t s s o l i d i f i c a t i o n and so the temperature of the system, as seen i n the same f i g u r e , i s required to drop from 600°C to 382°C. Considering that the i n t e r d e n d r i t i c f l u i d to feed shrinkage i s s o l u t e enriched during s o l i d i f i c a t i o n and has the highest composition ( e u t e c t i c ) at the end of s o l i d i f i c a t i o n , the shrinkage formed at the l a t e stage of s o l i d i f i c a t i o n makes f a r more c o n t r i b u t i o n to macrosegregation than that at the i n i t i a l stage. F i g 21 summarizes the temperature p r o f i l e s i n v a r i o u s compositions of Al-Zn. I t can be c l e a r l y seen that the thermal c o n t r a c t i o n at the end of s o l i d i f i c a t i o n r a p i d l y decreases with increase of composition i n the low a l l o y region. I f we neglect t h i s thermal c o n t r a c t i o n as Youdelis et a l " d i d , the r e s u l t i n g 32 segregation f o r low a l l o y Al-Zn must be f a r l e s s than that c a l c u l a t e d i n the present work. The r a p i d decrease i n ~ the - t o t a l shrinkage observed i n F i g 19 can be a t t r i b u t e d to the decrease i n thermal c o n t r a c t i o n , proving that the thermal c o n t r a c t i o n plays an important r o l e i n shrinkage and macrosegregation i n Al-Zn a l l o y . -The inverse segregation at the c h i l l face, i s p l o t t e d against composition i n F i g 22. U n l i k e the smooth curve f o r Al-Cu a l l o y , the curve f o r Al-Zn c o n s i s t s of a r i s e and f a l l i n composition up to 40%Zn, followed by a slow progressive decrease i n composition to the e u t e c t i c . The d e c l i n e of the inverse segregation a f t e r the peak at 20%Zn i s due to the a c c e l e r a t e d decrease i n t o t a l shrinkage. The prog r e s s i v e slower decrease i n segregation a f t e r 40%Zn i s due to a nearly constant and very low shrinkage r a t i o ( F i g 19). Also p l o t t e d i n F i g 22 are experimental and t h e o r e t i c a l r e s u l t s by Youdelis and C o l t o n * . The present model p r e d i c t i o n agrees with the reported r e s u l t s i n the region up to 23%Zn even though thermal c o n t r a c t i o n was neglected i n the previous i n v e s t i g a t i o n . Beyond 23%Zn there i s a strong discrepancy between the present r e s u l t s and the r e s u l t s o f ' Y o u d e l i s and Colton". I t should be noted that there i s considerable s c a t t e r i n the reported data which makes the comparison w i t h the measured r e s u l t s and the models only approximate. As has been demonstrated e a r l i e r , the inverse segregation i s very s e n s i t i v e to the c a s t i n g c o n d i t i o n s . 33 3.3 The Antimony Bismuth System This a l l o y i s - unique - i n - that both metals expand during s o l i d i f i c a t i o n (see F i g 4). The phase diagram f o r the Sb-Bi system i s shown i n F i g 23. There i s complete s o l i d s o l u b i l i t y of B i i n Sb over the e n i t r e composition range, d i f f e r i n g from the other two systems i n v e s t i g a t e d . F o l l o w i n g the same procedure used i n the Al-Zn system w i t h non l i n e a r s o l i d u s : and l i q u i d u s l i n e s , the phase diagram was d i v i d e d i n t o twenty s e c t i o n s , each of 5%Bi, and the s o l i d u s and l i q u i d u s l i n e s assumed l i n e a r i n each s e c t i o n with a constant p a r t i t i o n r a t i o . The t o t a l shrinkage r a t i o f o r Sb-Bi, as a f u n c t i o n of a l l o y composition i s given i n F i g 24. Two d i f f e r e n t shrinkage sequences are c l e a r l y observed, one region between zero and 30%Bi where the shrinkage r a t i o decreased r a p i d l y , and above 30%Bi where the shrinkage decreases s l o w l y . The abrupt increase i n the shrinkage near pure Sb i s a t t r i b u t e d to the thermal c o n t r a c t i o n during s o l i d i f i c a t i o n , which s h i f t s the negative value of shrinkage f o r the pure Sb to the p o s i t i v e s i d e . The d e t a i l e d volume change and temperature p r o f i l e f o r Sb-lO%Bi i s shown i n F i g 25. The s o l i d l i n e denotes the volume change i n each s o l i d i f i e d element and the dotted l i n e the volume change i n r e s i d u a l l i q u i d . Every element shows extensive volume expansion on s o l i d i f i c a t i o n except f o r the l a s t element whose volume expansion i s compensated by the thermal c o n t r a c t i o n . Although the remaining l i q u i d continues to shrink during 34 s o l i d i f i c a t i o n because of the den s i t y change due to s o l u t e c o n c e n t r a t i o n and a s l i g h t temperature drop, the whole volume expands s t e a d i l y u n t i l the s o l i d / l i q u i d i n t e r f a c e reaches the f i n a l element to s o l i d i f y . The volume expansion as a t o t a l system must cause the i n t e r d e n d r i t i c flow of s o l u t e enriched f l u i d to move away from the volume elememt r e s u l t i n g i n normal seg r e g a t i o n . On the other hand, the c o n t r a c t i o n occurs d u r i n g -the s o l i d i f i c a t i o n of the l a s t element r e v e r s i n g the flow. The temperature p r o f i l e shown i n F i g 25 i n d i c a t e s that the temperature drops from 530°C to 271°C during s o l i d i f i c a t i o n , which r e s u l t s i n a marked c o n t r a c t i o n , as c l e a r l y observed i n the volume change. This thermal c o n t r a c t i o n at the l a t e stage of s o l i d i f i c a t i o n reverses the t o t a l shrinkage from expansion to c o n t r a c t i o n , which r e s u l t s i n extensive macrosegregation since the i n t e r d e n d r i t i c f l u i d which feeds the shrinkage has the maximum composition, lOO%Bi, at the end of s o l i d i f i c a t i o n . The c a l c u l a t e d inverse segregation p r o f i l e at the c h i l l face f o r the Sb-Bi a l l o y i s shown i n F i g 26, as w e l l as the experimental and t h e o r e t i c a l r e s u l t s reported by Y o u d e l i s 5 . Maximum segregation at the c h i l l face was observed at 13%Bi, and minimum segregation at 70%Bi. The r e v e r s a l from inverse segregation at the c h i l l face to normal segregation occurs at approximately 30%Bi. The present r e s u l t s agree reasonably- w e l l with the experimental r e s u l t s , c o n s i d e r i n g the u n c e r t a i n i t y of the measured data. Comparing the segregation p r o f i l e with the t o t a l shrinkage 35 r a t i o ( F i g 24) shows the composition i n the segregation p r o f i l e at which the segregation i s reversed (30%Bi) does not c o i n c i d e w i t h the point at which the shrinkage r a t i o i s zero ( 2 0 % B i ) . Between 20% and 30%Bi, there i s expansion a s s o c i a t e d with s o l i d i f i c a t i o n and yet inverse segregation i s observed which i s anomalous. This can be accounted f o r by the volume change and temperature sequence "for Sb-20%Bi which i s found to have arf almost zero s o l i d i f i c a t i o n shrinkage r a t i o . In F i g 27, s i m i l a r d i s t r i b u t i o n of volume change to that of Sb-!0%Bi ( F i g 25) can be observed with large expansion at f r e e z i n g of each element and s l i g h t decrease of t o t a l shrinkage during s o l i d i f i c a t i o n . The i n t e g r a t e d negative shrinkage i s made up by the thermal c o n t r a c t i o n due to the temperature drop at the l a t e stage of s o l i d i f i c a t i o n . The same t h i n g must occur during the s o l i d i f i c a t i o n of Sb-Bi a l l o y up to 30%Bi. I t can be now confirmed t h a t , even though the t o t a l shrinkage r a t i o i s zero or negative, the p o s i t i v e segregation r e s u l t i n g from the thermal c o n t r a c t i o n at the l a t e stage of s o l i d i f i c a t i o n always overcomes the negative segregation formed at the primary stage i n the Sb-B i a l l o y with 20% to 30%Bi. Again i t i s reminded that the r e s i d u a l l i q u i d i s enriched i n s o l u t e by the advance of the s o l i d / l i q u i d i n t e r f a c e . Of p a r t i c u l a r importance i s t h a t , i n order to c a l c u l a t e the macrosegregation i n the a l l o y a c c u r a t e l y , i t i s e s s e n t i a l to obtain the information of volume change over the e n t i r e s o l i d i f i c a t i o n process as w e l l as the t o t a l volume change. The behaviour of i n t e r d e n d r i t i c f l u i d flow at the l a t e stage of s o l i d i f i c a t i o n c o n t r o l s the o v e r a l l segregation. 36 3.4 Examination Of Previous Model P r e d i c t i o n s In the d i s c u s s i o n of- the : present computer c a l c u l a t e d r e s u l t s i t has been pointed out that the thermal c o n t r a c t i o n p l a y s a much more s i g n i f i c a n t r o l e i n the formation of inverse segregation than i s g e n e r a l l y accepted. This i s p a r t i c u l a r l y the case f o r the Al-Zn and Sb-Bi a l l o y s at low compositions. Previous c a l c u l a t i o n s d i d not include thermal c o n t r a c t i o n i n the m o d e l s 1 ' 2 ' 8 " 1 1 . Without thermal c o n t r a c t i o n the c a l c u l a t i o n s w i l l give low values for the inverse segregation. To c l a r i f y the c o n t r a d i c t i o n s found i n the comparison of present work with the previous c a l c u l a t i o n s , c a l c u l a t i o n s of c h i l l face inverse segregation values were repeated using the models and assumptions used p r e v i o u s l y 1 ' 2 ' " ' 5 . The repeated c a l c u l a t i o n s f o r the three a l l o y systems d i d not c o i n c i d e with the published curves, as shown i n Appendix A. I t i s not. c l e a r what i s the cause of the discrepancy. 37 A-IV. CONCLUSIONS The present r e s u l t s show inverse segregation observed near the c h i l l face i s governed by the back flow of s o l u t e enriched l i q u i d through d e n d r i t e channels r e s u l t i n g from volume c o n t r a c t i o n during s o l i d i f i c a t i o n . A mathematical model i s developed which q u a n t i t a t i v e l y determines the inv e r s e segregation at the c h i l l face, and adjacent to the c h i l l f ace. The model includes the e f f e c t of thermal c o n t r a c t i o n due to the temperature changes as w e l l as c o n t r a c t i o n due to composition changes. Comparing the values of segregation determined from the model f o r the Al-Cu, Al-Zn and Sb-Bi a l l o y s w i t h experimental values shows reasonable agreement between the pr e d i c t e d and observed values. The f o l l o w i n g conclusions have been reached. 1) A binary a l l o y may be considered as a simple mixture of the two componets i n order to determine the d e n s i t y of both s o l i d and l i q u i d . 2 ) The volume change during s o l i d i f i c a t i o n c o n s i t s of three p a r t s . a) . The volume change due to the l i q u i d / s o l i d phase t r a n s f o r m a t i o n . b) . C o n t r a c t i o n i n the r e s i d u a l l i q u i d due to solute c o n c e n t r a t i o n . c) . Thermal c o n t r a c t i o n due to temperature changes. 38 3) Volume expansion occurs over most stages of s o l i d i f i c a t i o n i n both the Al-Cu and Al-Zn systems due to l i q u i d / s o l i d change but t h i s i s compensated f o r by the other two e f f e c t s producing a net c o n t r a c t i o n . 4) Thermal c o n t r a c t i o n p l a y s a s i g n i f i c a n t r o l e i n the inverse segregation of Al-Zn and Sb-Bi i n low a l l o y composition regions s i n c e la r g e temperature d r o p s -occur i n the l a s t stage of s o l i d i f i c a t i o n . 5) To c a l c u l a t e macrosegregation i t i s e s s e n t i a l to know the volume change over the e n t i r e s o l i d i f i c a t i o n process since the behaviour of i n t e r d e n d r i t i c f l u i d flow at the l a t e stage of s o l i d i f i c a t i o n c o n t r o l s the o v e r a l l segregation. 6) The s o l i d i f i c a t i o n shrinkage r a t i o of an a l l o y i s a f u n c t i o n of the volume change on s o l i d i f i c a t i o n , compositional changes i n the l i q u i d and temperature changes, a l l of which must be considered i n any s i g n f i c a n t model f o r segregation. 7) The length of s o l i d / l i q u i d region adjacent to the c h i l l face i s h i g h l y s e n s i t i v e to the contact thermal r e s i s t a n c e at mold/metal i n t e r f a c e a s s o c i a t e d with the formation of an a i r gap at the c h i l l face. 39 Table I - The comparison of a l l o y d e n s i t i e s with those c a l c u l a t e d from data f o r pure metals A l l o y Temp(°C) p-s o l v e n t 1 p - s o l u t e 1 p-mean pmeasured E r r o r ( % ) Al-4.5%Cu 20 2.70 8.93 2.82 2.75 2 2.5 Al-4.5%Cu 800 2.34 8.15 2.45 2.45 3 0 Al-4.5%Cu 1000 2.29 7.99 2.40 2.40 3 0 Al-8%Cu 20 2.70 8.93 2.92 2.83 2 3.2 Al-10%Cu 800 2.34 8.15 2.60 2.55 3 2.0 A1-10%C U 1000 2.29 7.99 2.53 2.50 3 1 .2 Al-12%Cu 20 2.70 8.93 3.04 2.93 2 3.8 Al-20%Cu 1000 2.29 7.99 2.84 2.75 3 3.3 Al-40%Cu 800 2.34 8.15 3.62 3.45 3 4.9 Al-80%Cu 1200 2.25 7.84 5.76 5.60 3 2.9 Al-5%Zn 700 2.36 6.45 2.45 2.45" 0 Al-5%Zn 900 2.31 6.35 2.39 2.40* 0.4 Al-10%Zn 700 2.36 6.45 2.54 2.55" 0.4 Al-10%Zn 900 2.31 6.35 2.49 2.45* 1.6 Al-20%Zn 700 2.36 6.45 2.74 2.70* 1 .5 Al-40%Zn 700 2.36 6.45 3.24 3.20* 1 .5 Pb-5%Sn 315 10.68 6.91 10.36 10. 40 5 0.4 Pb-10%Sn 300 10.70 6.92 10.09 10. 15 5 0.6 Pb-20%Sn 275 10.73 6.94 9.58 9.70 5 1 .2 Pb-40%Sn 235 10.78 6.97 8.73 8.80 5 0.8 Note: 1. S m i t h e l l s 1 6 and E l l i o t et a l 1 7 2. S m i t h e l l s 1 6 3. Bornemann et a l 1 8 4. Bornemann et a l 1 9 5. Thresh et a l 2 0 40 Table II - The thermal p r o p e r t i e s employed f o r the c a l c u l a t i o n of temperature d i s t r i b u t i o n A 1 - 1 0 %C U Cu mold thermal c o n d u c t i v i t y , k(cal/cm.s.°C) 0.43 0 .94 s p e c i f i c heat, Cp (cal/g.°C) 0.259 0 .093 d e n s i t y , p (g/cm 3) v a r i a b l e 8 .89 l a t e n t heat of s o l i d i f i c a t i o n , Hs (cal/g) 93.0 -e m i s s i v i t y , e 0.1.8 0 .18 thermal c o n d u c t i v i t y of a i r gap, ka shape f a c t o r through a i r gap 1.0X10-* cal/cm 0.10 .s.°C 41 5401 SPECIFIC VOLUME, cmVqm 0.42 0.38 0.54 0.30 0 5 10 15 20 25 30 33 CONCENTRATION, wt-<& Cu Figure 1 - E q u i l i b r i u m phase diagram and s p e c i f i c volume f o r the Al-Cu a l l o y s (a) koC< LI W-4 * " " (b) (c) CL B Figure 2 - Schematic d e s c r i p t i o n of s o l i d i f y i n g zone? (a) dendrite morphology, (b) s o l i d and l i q u i d compositions along c h i l l face, and (c) composition of i n t e r d e n d r i t i c l i q u i d X 4 3 T E M P E R A T U R E CC) Figure 3 - D e n s i t i e s of copper and aluminium vs temperature 44 200 400 600 800 T E M P E R A T U R E C O F i g u r e 4 - D e n s i t i e s of Zinc, Aluminium and Bismuth vs temperature 45 y Figure 5 - Schematic c o n f i g u r a t i o n of the model i n v e s t i g a t e d and the s u b d i v i s i o n f o r the numerical s i m u l a t i o n 46 Figure 6 - Schematic c o n f i g u r a t i o n of the feedinq sequence; (a) F l u i d flow, and (b) p r o f i l e of the l i q u i d composition 1 2 3 i i i 1 J " ^ 3 " " - . 5 " v - ^ 6 " -^9 " ^ 10. N ^ y y - 2 , y y ' y y y y ."1 y -"9 y y y y y y y y y y y ' y' y -"1 . y y 2 - 3 . y . y y — . " 7 • * y ^ y y y'\ y y y y y y -is . y y y y ''\ y y ^ 2 „ y y y - " 5 y y'7 . y " 7 y ^ 1 y y LU 6 y y >' 1 y y y y -®. y " 6 . " 6 - 6 y y y y y y y y y y y <- 5 y - 5 . y y ^ 1 . y " 2 y y y " V „ y ' A . y - "4 y • 8 ° 9 10 : y y\ . y ^ 2 , y " 3 -y y — 3 . y ' 3 . — 3 . ^ \ y y y y " 2 y <'2 „ y ''1 „ y - '2 y < 2 y y y ''1 y -"1 y y y y 1 - " 1 . y y i y ^ ^ ^ ^ ^ ^ -^ ^ ^ ..f^ ^ f ACTUAL SOLIDIFICATION FRONT Figure 7 - Summary of the f l u i d flow induced by the shrinkage during the s o l i d i f i c a t i o n of the model with 1 0 X 1 0 s u b d i v i sions Figure 8 - Schematic d e s c r i p t i o n of the d i l u t i o n e f f e c t of the i n t e r d e n d r i t i c f l u i d flow during s o l i d i f i c a t i o n 49 (START) n r (READ INPUT DATA 1CALCULATE MICROSEGREGATION] CALCULATE MEAN SOLID COMPOSITION AND LIQUIDUS TEMPERATURE FOR EACH ELEMENT IN IT IAL IZE VARIABLES LET SOLIDIFY ONE ELEMENT ] CALCULATE SHRINKAGE | I CALCULATE SEGREGATION IN THE F IRST COLUMN CALCULATE DILUTION EFFECT DUE TO FLUID FLOW CALCULATE POSITIONAL SEGREGATION (STOP) F i g u r e 9 - Flow chart for the c a l c u l a t i o n of inverse segregation 50 (START) READ INPUT DATA INITIALIZE VARIABLES TIME-TIME+At CALCULATE TEMPERATURE FIELD USING IMPLICIT FTfl F i g u r e 10 - Flow chart f o r the c a l c u l a t i o n of the le n g t h of s o l i d / l i q u i d zone adjacent to the c h i l l face 51 Figure 11 - C a l c u l a t e d volume change during s o l i d i f i c a t i o n of Al-10%Cu. The number on the curve denotes each element s o l i d i f i e d . 52 Figure 12 - C a l c u l a t e d volume change and l i q u i d composition in the r e p r e s e n t a t i v e volume for Al-5%Cu. The number on the curve denotes each element s o l i d f i e d . 53 0 5 10 15 20 25 30 C O N C E N T R A T I O N (wrtCu) Figure 13 - S o l i d i f i c a t i o n shrinkage r a t i o vs composition of the Al-Cu a l l o y s . The dotted l i n e was derived from F i g 1. 54 FRACTIONAL DISTANCE FROM THE CHILL FACE, x/L Figure 14 - P o s i t i o n a l segregation p r o f i l e s f o r v a r i o u s compositions of the Al-Cu a l l o y s Figure 15 - Comparison of the inverse segregation at the c h i l l face for the Al-Cu a l l o y s 56 0 1 1 ' 1 1 r 2 4 6 8 10 A I R G A P (/M) Figure 16 - E f f e c t of a i r gap at the metal/mold i n t e r f a c e on the length of s o l i d / l i q u i d zone adjacent to the c h i l l face 57 Figure 17 - Comparison of p o s i t i o n a l segregation f o r A l -10%C U 58 0 10 20, 30 40 50 60 70 80 90 100 C o n c e n t r a t i o n CLtCs (%Zn) Figure 18 - E q u i l i b r i u m phase diagram f o r Al-Zn a l l o y s 59 10 1 H I 1 1 1 1 1 0 20 HO 60 80 100 C O N C E N T R A T I O N (wtfZn) Figure 19 - T o t a l shrinkage r a t i o vs composition f o r the Al-Zn a l l o y s 60 POSITION OF SOLID/LIQUID INTERFACE ( ELEMENT NO.) Figure 20 - C a l c u l a t e d volume change and temperature i n the r e p r e s e n t a t i v e volume f o r Al-lO%Zn. The number on the curve denotes each element s o l i f i e d . 61 650 • 600 f 550 = 3 500 450 -400 -POSITION OF SOLID/LIQUID INTERFACE (ELEfOfT NO.) Figure 21 - C a l c u l a t e d p r o f i l e s of temperature i n the r e p r e s e n t a t i v e volume f o r va r i o u s compositions of the Al-Zn a l l o y s i n d i c a t e d . Figure 22 - Comparison of the inverse segregation at the c h i l l face for the Al-Zn a l l o y s 63 0 20 4 0 60 8 0 100 C o n c e n t r a t i o n (wt % Bi ) Figure 23 - E q u i l i b r i u m phase diagram for Sb-Bi a l l o y s 64 C O N C E N T R A T I O N ( wrt Bi ) Figure 24 - T o t a l shrinkage r a t i o vs composition f o r the Sb-Bi a l l o y s 1 I 2 I 3 I » I 5 I « I 7 I 8 I « I 10 POSITION OF SOLID/LIQUID INTERFACE ( ELEMENT NO.) Figure 25 - C a l c u l a t e d volume change and temperature i n the re p r e s e n t a t i v e volume for Sb-lO%Bi. The number on the curve denotes each element s o l i d i f i e d . 66 Figure 26 - Comparison of the inverse segregation at the c h i l l face for the Sb-Bi a l l o y s Figure 27 - C a l c u l a t e d volume change and temperature i n the rep r e s e n t a t i v e volume f o r Sb-20%Bi. The number on the curve denotes each element s o l i d i f i e d . 68 PART - B CENTERLINE SHRINKAGE IN STEEL PLATE CASTINGS 69 B-I. INTRODUCTION A unique advantage of metal c a s t i n g s i s that h i g h l y complex shapes can r e a d i l y be f a b r i c a t e d . This inherent advantage of c a s t i n g s , however, i s oft e n l o s t because of the d i f f i c u l t y of c o n t r o l l i n g c a s t i n g d e f e c t s , among which shrinkage p o r o s i t y i s one of the major d e f e c t s . Shrinkage p o r o s i t y i s c o n t r o l l e d by proper design of the c a s t i n g , using s u i t a b l e g a t i n g and r i s e r i n g techniques and moulding m a t e r i a l s . Since the c a s t i n g process i s complex, the design of these systems i s only approximate and e m p i r i c a l so that p o r o s i t y does occur i n c a s t i n g s . 1 .1 Gross Shrinkage Recent development of the numerical s i m u l a t i o n of the c a s t i n g p r o c e s s 2 1 " 2 5 has enabled the s o l i d i f i c a t i o n sequence of a simple c a s t i n g to be c a l c u l a t e d . Jeyarajan and P e h l k e 2 6 a p p l i e d a mathematical model to sand mold c a s t i n g f o r the p r e d i c t i o n of gross shrinkage i n a r a i l wheel c a s t i n g due to i n s u f f i c i e n t mass feeding. Comparing t h e i r numerical c a l c u l a t i o n s w i t h experimental observations i t was found that the observed gross shrinkage c a v i t y agreed very w e l l with the c a l c u l a t i o n s which p r e d i c t e d the shape and p o s i t i o n of the l a s t l i q u i d to s o l i d i f y . The remaining problem i n p r e d i c t i n g gross shrinkage i n c a s t i n g s i s to extend the Pehlke a n a l y s i s to the 70 h i g h l y complex shapes used i n normal foundry p r a c t i c e . 1.2 C e n t e r l i n e Shrinkage Towards the end of s o l i d i f i c a t i o n , the s o l i d i f i c a t i o n shrinkage must be fed by l i q u i d flowing down h i g h l y i r r e g u l a r narrow i n t e r d e n d r i t i c channels. I f complete back flow does not occur, small s c a t t e r e d i n t e r d e n d r i t i c pores forms, c a l l e d c e n t e r l i n e p o r o s i t y or c e n t e r l i n e shrinkage. A heat t r a n s f e r model gives i s o c h r o n a l s o l i d i f i c a t i o n l i n e s assuming simple heat flow p a t t e r n s . In the l a t e r part of s o l i d i f i c a t i o n the heat flow l o c a l l y i s complex and ther e f o r e uncertain when a p p l i e d to the c e n t e r l i n e p o r o s i t y problem. In a d d i t i o n f l u i d flow of the i n t e r d e n d r i t i c l i q u i d must a l s o be considered. P e l l i n i et a l 2 7 " 3 0 c a r r i e d out an extensive s e r i e s of measurements i n which the feeding distance of s t e e l c a s t i n g s i n green-sand moulds was determined. I t was found f o r a p l a t e c a s t i n g , 2.5 to 5.0cm t h i c k ( F i g 28a), t h a t : 1) the maximum p l a t e length cast completely sound by one r i s e r i s 4.5s (s=plate t h i c k n e s s ) , 2) the length D of the end of the p l a t e f r e e of p o r o s i t y i s always 2.5s f o r the p l a t e c a s t i n g more than 4.5s long, and 3) the sound area due to r i s e r e f f e c t i s 2s. S i m i l a r l y the maximum leng t h cast completely sound i s 9.56 s and the sound region due t o end e f f e c t s i s 1.5s-2s i n square bar 71 c a s t i n g s w i t h 5 to 20cm s e c t i o n s i z e ( F i g 28b). They a l s o examined the e f f e c t of end c h i l l on the feeding d i s t a n c e and found that c h i l l s c o n t r i b u t e to i n c r e a s i n g the sound region by 5cm and 1s i n p l a t e and square bar c a s t i n g s r e s p e c t i v e l y . Johnson and L o p e r 3 1 examined the c e n t e r l i n e p o r o s i t y of t h i n s t e e l s e c t i o n s cast i n green-sand mould. Sections s t u d i e d i n c l u d e d f l a t p l a t e s and square bars of t h i c k n e s s e s l e s s than 2.5cm which i s smaller than the range considered by Bishop et a l 2 7 ' 2 8 . They found that the maximum length which c o u l d be cast without p o r o s i t y was given by (16.1/s-14)cm fo r p l a t e c a s t i n g s and (12.7/s-8.4)cm fo r square bar c a s t i n g s . Niyama et a l 3 2 p r e d i c t e d c e n t e r l i n e shrinkage on the bases of a numerical s i m u l a t i o n of the process g i v i n g the i s o c h r o n a l contours during s o l i d i f i c a t i o n . They showed that the temperature gradient at the end of s o l i d i f i c a t i o n i s a simple and e f f e c t i v e parameter to p r e d i c t p o r o s i t y . However the c r i t i c a l value of the gradient i n the p r e d i c t i o n can only be determined e m p i r i c a l l y f o r each c a s t i n g as shown by P e l l i n i 3 0 . A f u r t h e r step was taken by Niyama et a l 3 3 i n proposing a new parameter G//R to p r e d i c t p o r o s i t y with the c r i t i c a l value of 1.0 /min.°C/cm for t h i s parameter, a p p l i c a b l e to both gross and c e n t e r l i n e p o r o s i t i e s . This would be. independent of the a l l o y composition and the s i z e and shape of the s t e e l c a s t i n g considered. This has been challenged by C h i j i i w a and Imafuku 3* who questioned the general a p p l i c a b i l i t y of t h i s parameter. As an a l t e r n a t i v e they proposed that the gradient of f r a c t i o n s o l i d 72 at the end of s o l i d i f i c a t i o n has the c r i t i c a l value of 0.25 per cm for c e n t e r l i n e shrinkage. The three parameters proposed to p r e d i c t p o r o s i t y , G, G//R and the gradient of f r a c t i o n s o l i d , might be e f f e c t i v e i n p r e d i c t i n g the formation of c e n t e r l i n e shrinkage i n d u s t r i a l l y . However, they are a l l e f f e c t i v e l y e m p i r i c a l parameters l a c k i n g a c l e a r p h y s i c a l b a s i s i n which to e s t a b l i s h a q u a n t i t a t i v e measure of the p o r o s i t y i n a c a s t i n g . 1.3 Previous Models P o r o s i t y i s a s s o c i a t e d with feeding d i s t a n c e i n the c a s t i n g as w e l l as the s o l i d i f i c a t i o n s t r u c t u r e . Feeding of l i q u i d through i n t e r d e n d r i t i c channels i s considered by D a v i e s 3 5 using numerical s i m u l a t i o n . The c a p i l l a r y flow through a channel i s given by: dv = nr 4 P dt 8y 1 (B-1) The amount of l i q u i d which must flow through the c a p i l l a r y to compensate f o r s o l i d i f i c a t i o n c o n t r a c t i o n i s ; " v r 2* (B"2) Combining equations (B-1) and (B-2), gives the c a p i l l a r y feeding d i s t a n c e , l f , < P r 2 s F r i c t i o n reduces the e f f e c t i v e pressure, P=P Q-AP, where, for 73 laminar flow (r<<l): A P 2 (B-4) The flow v e l o c i t y i s given by, v = Bv s Combining equations (B-3),(B-4) and (B-5) gives: (B-5) (P -AP)r 2 P r 2 h = = ~ 1 (B_6) s s When shrinkage appears, l = l f and: P r 2 The c a p i l l a r y feeding d i s t a n c e was c a l c u l a t e d using equation (B-7), i n c o r p o r a t i n g an e m p i r i c a l c o r r e c t i o n f a c t o r B which includes the f a c t o r s that are d i f f i c u l t to consider: BYP r 2 1 = - — i _ (B-8) f yBvs The c o r r e c t i o n f a c t o r , B, was chosen to be 0.22 i n order that the c a l c u l a t e d feeding d i s t a n c e for a s t e e l p l a t e cast i n green sand mould has a feeding d i s t a n c e equal to that e m p i r i c a l l y determined value of 4.5s 3 0. Based on the type of a l l o y , assumptions were made for the f r a c t i o n s o l i d at which p o r o s i t y due to incomplete feeding would occur. For example, D a v i e s 3 5 estimated that incomplete feeding could occur above s o l i d f r a c t i o n s of 95%, for 0.6%C s t e e l s and 99.8%Al; above s o l i d f r a c t i o n of 90% for nodular cast i r o n and 99% f o r pure A l . The above model (Davies') was a p p l i e d to a wide range of m a t e r i a l s and good agreement was obtained between 74 the c a l c u l a t e d r e s u l t s and the experimentally determined v a l u e s . Shortcomings i n the Davies' model are that an e m p i r i c a l c o r r e c t i o n f a c t o r B i s included i n the model, and the assumption of when incomplete feeding s t a r t s which s t r o n g l y i n f l u e n c e s the presence of c e n t e r l i n e shrinkage. The value for B used i n f i t t i n g the model to experimental r e s u l t s was d e r i v e d from the experiments on cast s t e e l . One would expect d i f f e r e n t values for B would be required f o r other m a t e r i a l s s i n c e B i n equation (B-8) i n v o l v e s a l o t of f a c t o r s not involved i n h i s model. Another t h e o r e t i c a l model f o r p o r o s i t y was proposed by F l e m i n g s 3 6 i n c o r p o r a t i n g Darcy's Law f o r i n t e r d e n d r i t i c f l u i d flow. Assuming that the mold/metal i n t e r f a c e r e s i s t a n c e i s of o v e r r i d i n g importance and that the mold, being i n f i n i t e i n extent, remains at i t s i n i t i a l temperature ( T Q ) , the rat e of heat flow across t h i s i n t e r f a c e i s : q = -h(T-TQ) (B-9) The heat e n t e r i n g the mold comes nearly e n t i r e l y from the heat of f u s i o n of the s o l i d i f y i n g metal. Thus: q = ( p H ) - ± V 3 G L s V A 8t (B-10) c Combining equation (B-9) with (B-10) g i v e s : 3 t " ~ C (B-11) where A.T=T-TQ and c i s a constant. Since the heat flow i s c o n t r o l l e d by the heat t r a n s f e r c o e f f i c i e n t h, there i s no s i g n i f i c a n t thermal gradient i n the s o l i d i f y i n g metal and a^/St i s independent of p o s i t i o n i n the c a s t i n g . Darcy's law i s given 75 by: v = - (VP + ( B _ 1 2 ) For one dimensional f l u i d flow, equation (B-12) reduces t o : „ _ K dP . . v x ~ " ( dx" + p L g r } (B-13) S i m i l a r l y , the mass balance equation i n the volume element, equation (A-13), reduces t o : 3 t o~k~ ( p L g L V x ^ (B-14) where P = P s g s + p L g L (B-15) Equation (B-15) i s rearranged for constant d e n s i t i e s i n l i q u i d and s o l i d as: _ _ _ - _ ( B_ 1 6 ) Equations (B-11), (B-14) and (B-16) y i e l d , 3 x ( 8 L V ' " 1=3 C (B-17) Since g i s independent of p o s i t i o n , i n t e g r a t i n g equation (B-17) gives the i n t e r d e n d r i t i c flow v e l o c i t y as a fu n c t i o n of p o s i t i o n ; P cx " 1-3 g T (B-18) v X S u b s t i t u t i n g equation (B-18) i n t o equation (B-13) 76 and i n t e g r a t i n g y i e l d s the pressure at x; ? = P a + A S ( D 2 - x 2 ) + p L 8 r Y (B-19) This expression i s t h e o r e t i c a l l y enough to de s c r i b e the formation of c e n t e r l i n e shrinkage, but i s v a l i d only f o r the system w i t h homogeneous temperature a l l over the c a s t i n g and without superheat, which i s not s a t i s f i e d under normal c a s t i n g c o n d i t i o n s . 1.4 P e r m e a b i l i t y Towards the end of s o l i d i f i c a t i o n when the f r a c t i o n s o l i d i n the a l l o y i s high, f l u i d flow to compensate for volume shrinkage occurs through narrow tortuous i n t e r d e n d r i t i c channels. This flow i s analogous to flow through packed beds and, i f the flow occurs under low pressure c o n d i t i o n s , the flow rate w i l l obey the "Darcy's Law" expressed i n equation (B-12). For one dimensional f l u i d flow without a m e t a l l o s t a t i c head, equation (B-12) reduces t o : v- = - ^ ( B - 2 0 ) y 1 where v' i s s u p e r f i c i a l flow v e l o c i t y and v* = g Tv (B-21) L x Flow v e l o c i t i e s through a s t r a i g h t c a p i l l a r y tube are given by 77 Hargen-Poisulle' s equation: = _ ILL (B-22) dt 8y 1 Introducing a " t o r t u o s i t y f a c t o r , T" i n the case where the c a p i l l a r y i s not s t r a i g h t and for n c a p i l l a r i e s per u n i t area: v' = - (B-23) 8y T l Equating equations (B-20) and (B-23): 4 K = n 7 r r 8T (B-24) and from geometry: 2 (B-25) g L = mrr x rearranging g i v e s : 2 r = 2 2 2 (B-26) n IT x S u b s t i t u t i n g equation (B-26) i n t o (B-24) y i e l d s : 2 K = — Q 3 (B-27) This shows that f o r a given d e n d r i t e morphology, the p e r m e a b i l i t y i s d i r e c t l y p r o p o r t i o n a l to the square of the volume f r a c t i o n l i q u i d 3 7 " 3 9 . The v e r i f i c a t i o n of equation (B-27) was f i r s t attempted by Piwonka and F l e m i n g s 3 7 using Al-4.5%Cu porous media and molten lead. The theory p r e d i c t s ln(K) i s p r o p o r t i o n a l to l n ( g ) f o r constant T with a slope of 2. This was found to be the case experimentally when the f r a c t i o n l i q u i d was l e s s than 0.3. Ape l i a n , Flemings and Mehrabian" 0 noted that the p e r m e a b i l i t y i n equation (B-27) i s i n v e r s e l y p r o p o r t i o n a l to the number of flow 78 channels per u n i t area. They compared K values of A l - 4 % S i w i t h those of A l - 4 % S i - 0 . 2 5 % T i , and found that the p e r m e a b i l i t y i s c o n s i s t e n t l y lower for the g r a i n r e f i n e d a l l o y , the l a t t e r , which has more c a p i l l a r i e s i n a u n i t area. I n t e r d e n d r i t i c f l u i d flow was c a r e f u l l y measured by S t r e a t and Weinberg 3 9 i n a p a r t i a l l y s o l i d f i e d columnar d e n d r i t i c c a s t i n g . Assuming the number of flow channels i n the c a s t i n g i s equal t o the number of channels between the columnar primary d e n d r i t e branches and the spacing between these channels equals the primary dendrite arm spacing, equation (B-27) i s reduced t o : g 2 ( D A S ) 2 K 3 - (B-28) 8ITT Measurements of p e r m e a b i l i t y f o r a wide range of dendrite arm spacing and a constant f r a c t i o n l i q u i d , 0.19, showed the p e r m e a b i l i t y K obeys equation (B-28) during the e a r l y part of i n t e r d e n d r i t i c flow. A f t e r flow has occured the flow rate changes due to the i n t e r a c t i o n of the flowing l i q u i d with the dend r i t e which changes the s i z e and c o n f i g u r a t i o n of the channels. However d e v i a t i o n from equation (B-28) i n the system examined occured a f t e r a r e l a t i v e l y long time p e r i o d , i n comparison with the s o l i d i f i c a t i o n times a s s o c i a t e d with a normal c a s t i n g . Values of p e r m e a b i l i t y as a f u n c t i o n of f r a c t i o n l i q u i d reported i n the l i t e r a t u r e are summarized i n F i g 29. The r e s u l t s show that equation (B-27) i s v a l i d for i n t e r d e n d r i t i c f l u i d flow under steady s t a t e c o n d i t i o n s for volume l i q u i d 79 f r a c t i o n s l e s s than about 0.3. 1.5 Present O b j e c t i v e s Even though shrinkage p o r o s i t y i s a major f a c t o r i n the q u a l i t y of a c a s t i n g , no a n a l y t i c a l method has been developed which c l e a r l y p r e d i c t s whether c e n t e r l i n e shrinkage w i l l occur i n a c a s t i n g . This i s p r i m a r i l y because the mechanism fo r c e n t e r l i n e p o r o s i t y i s complex, governed by i n t e r d e n d r i t i c f l u i d flow. To p r e d i c t the formation of c e n t e r l i n e shrinkage, a proper mathematical s i m u l a t i o n i s r e q u i r e d combining a heat t r a n s f e r model with an i n t e r d e n d r i t i c f l u i d flow model. 80 B - I I . MODELLING PROCEDURE The formation of c e n t e r l i n e shrinkage i s a s s o c i a t e d with i n s u f f i c i e n t flow of i n t e r d e n d r i t i c l i q u i d to compensate f o r s o l i d i f i c a t i o n shrinkage. The mathematical model, t h e r e f o r e , has to involve both heat t r a n s f e r and f l u i d flow. 2.1 Mathematical Formulation 2.1.1 Temperature C a l c u l a t i o n s The c a s t i n g and sand mold geometry i n v e s t i g a t e d i s shown i n F i g 30. The r i s e r was chosen to be the same s i z e as the one used i n the experiments reported by P e l l i n i et a l 2 7 ' 3 0 , that i s 3s wide and 5s high. This r i s e r s i z e i s large enough to avoid gross shrinkages i n the p l a t e c a s t i n g . The basic mathematical r e l a t i o n used to c a l c u l a t e the temperature d i s t r i b u t i o n i n the system was the general equation that governs heat conduction i n a s o l i d ; fCpf^) = V(kAT) (B-29) For constant heat c o n d u c t i v i t y and two dimensional heat flow, equation (B-29)reduces t o ; |T ( + l i l y (B-30) 3 t 3 x 3 y Several key assumptions concerning the c a s t i n g process are 81 re q u i r e d to adjust equation (B-30) to a form s u i t a b l e f o r numerical s o l u t i o n ; 1) The mold i s ins t a n t a n e o u s l y f i l l e d with l i q u i d metal at the pouring temperature. 2) Once the mold i s f i l l e d , the l i q u i d metal i s stagnant. 3) The thermal contact r e s i s t a n c e at the sand/metal i n t e r f a c e i s n e g l i g i b l e . 4) Segregation i n the s o l i d and so l u t e enrichment i n the l i q u i d are n e g l i g i b l e . 5) The l a t e n t heat of s o l i d i f i c a t i o n i s re l e a s e d uniformly between the l i q u i d u s and s o l i d u s temperatures. 6) L i q u i d u s and s o l i d u s are l i n e a r f u n c t i o n s of temperature so that the weight f r a c t i o n s o l i d i f i e d i s given by, T L " T f s = £-Tir (B-31) L s The i n i t i a l c o n d i t i o n s at t=0 are, T M = T (B-32) M P T = T (B-33) s o The boundary c o n d i t i o n at the v e r t i c a l sand mold surface i s : - ks ( r r > = V T s - V (B-34) and at the h o r i z o n t a l mold surface i s : " k s ( ^ ) = h s ( T s - T a > < B" 3 5> and at metal surface i s : 3 T _ kM ' (3"T ) = ( ^ " T a > (B-36) Convective heat t r a n s f e r from the metal to ambient a i r i s 82 neglected giving (T + 273)A - (T + 273)4 \ = ° ^ ^ - T a 8 ( B " 3 7 ) There i s continuity of heat flux across the sand/metal interface, which i s expressed at the v e r t i c a l interface by 3 T 3 T M k ( — - ) = k ( — ) (B-38) s l 3 x ' *M V 3 x ; and at the horizontal interface by; 9 T s 9 TM (B-39) 2.1.2 Pressure Required To Feed Shrinkage It has been shown in the l i t e r a t u r e 3 8 ' * 0 ' ' 1 that i n t e r d e n d r i t i c f l u i d flow obeys Darcy's Law at steady state when the volume fr a c t i o n l i q u i d i s less than 0.3. To calculate the pressure drop due to i n t e r d e n d r i t i c f l u i d flow in s o l i d i f y i n g casting, Darcy's Law was applied along the centerline of the plate on the assumptions that the Law i s also v a l i d for the flow through the media with higher l i q u i d f r a c t i o n than 0.3 and for a temperature gradient present in the a l l o y . The governing equation without a metallostatic head i s , K dP v x = ~ ^ 7 dx" (B-40) 83 rearranging g i v e s , yg Ldx dP = - K x v x (B-41) where u i s constant, K i s given i n F i g 29, and g L can be c a l c u l a t e d from, ( 1 - f s ) / p L * L - f 8 / p . - + ( 1 - f . ) / ^ <-"> Equation (B-41) expresses a pressure drop w i t h i n an i n f i n i t e s i m a l element as a f u n c t i o n of f l u i d v e l o c i t y . The boundary c o n d i t i o n for equation (B-41) appears at s o l i d / l i q u i d i n t e r f a c e where, i f the shrinkage i s fed by the l i q u i d , the r a t e of pore formation i s equal t o the rate of f l u i d flow. This g i v e s : ( v s A g L ) 6 = - v A g L , ( B _ 4 3 ) y i e l d i n g , V x - " V (B-44) and the c o n t i n u i t y of f l u i d flow g i v e s , V L = 3 c o n ^ a n t ( B _ 4 5 ) The t o t a l pressure drop, c a l c u l a t e d by, AP = Z AP (B-46) j 3 denotes the pressure r e q u i r e d to feed the s o l i d i f i c a t i o n shrinkage. Shrinkage p o r o s i t y w i l l form when the t o t a l pressure r e q u i r e d to feed the shrinkage exceeds the pressure a c t i n g on the system, that i s the atmospheric pressure plus the m e t a l l o s t a t i c head pressure. 84 2.2 Computer Programming The e x p l i c i t f i n i t e d i f f e r e n c e method i s used for the s o l u t i o n of heat t r a n s f e r equations since a small time step i s required to c a l c u l a t e small changes during s o l i d i f i c a t i o n . The d e r i v a t i o n of the nodal equation i s given i n Appendix B. The c a l c u l a t i o n was f i r s t performed to determine the temperature d i s t r i b u t i o n and then to determine the pressure drop. Values of p e r m e a b i l i t y were taken from the Piwonka and F l e m i n g s 3 7 data i n F i g 29 since they cover a wide range of l i q u i d f r a c t i o n , which i s r e q u i r e d i n the present case. The a c t u a l c a l c u l a t i o n sequence used i s shown i n the flow c h a r t , F i g 31, and a sample of the Fortran program i s given i n Appendix C. The computer program a l s o involves the c a l c u l a t i o n of temperature gradient and the s o l i d i f i c a t i o n parameter, G/^R, at the end of s o l i d i f i c a t i o n . 2.3 V a l i d a t i o n Of The Heat Transfer Model To v a l i d a t e the present model for c a l c u l a t i n g the temperature d i s t r i b u t i o n i n the c a s t i n g , some c a l c u l a t i o n s were performed and the r e s u l t s were compared to the experimental measurements of Bishop and P e l l i n i 2 7 . F i g 32 shows t y p i c a l c a l c u l a t e d values f o r the temperature d i s t r i b u t i o n s along the c e n t e r l i n e at d i f f e r e n t time i n t e r v a l s a f t e r pouring, for the 85 p l a t e c a s t i n g with 5cm t h i c k and 33cm long. A lso p l o t t e d i n the f i g u r e s are the temperature p r o f i l e s determined e x p e r i m e n t a l l y . The measured l i q u i d u s and s o l i d u s temperatures f o r the low carbon s t e e l , having the f o l l o w i n g composition Fe-0.34%C-0.68%Si-0.89%Mn were 1507°C and 1463°C r e s p e c t i v e l y . Reasonable agreement can be observed between the model p r e d i c t i o n s and the experimental r e s u l t s over the e n t i r e length of the p l a t e , except at p o s i t i o n s c l o s e to the end surface and to the r i s e r . The p r e d i c t e d temperatures are q u i t e accurate e s p e c i a l l y i n the region of the l i q u i d / s o l i d c o - e x i s t i n g zone which i s most re l e v a n t to the formation of c e n t e r l i n e shrinkage. I t w i l l be noted from the curves i n F i g 32 t h a t the temperature g r a d i e n t s , e s t a b l i s h e d soon a f t e r s o l i d i f i c a t i o n s t a r t s by heat flow i n t o the p l a t e from the r i s e r and heat l o s s e s at the end of the p l a t e , tend to progress toward each other during s o l i d i f i c a t i o n . The movement of the s o l i d u s temperature along the c e n t e r l i n e i s shown i n F i g 33. The c a l c u l a t e d values are i n good agreement with the measured temperature p r o f i l e s except f o r a small d e v i a t i o n . The f r e e z i n g r a t e , given by the slope of the curve i n F i g 33 i s observed to be very r a p i d i n the c e n t r a l region of the c a s t i n g and r e l a t i v e l y slow at e i t h e r end. Values of the p h y s i c a l p r o p e r t i e s used i n the above c a l c u l a t i o n s are l i s t e d i n Table I I I together with other p h y s i c a l data f o r the c a l c u l a t i o n of pressure drop due to i n t e r d e n d r i t i c f l u i d flow. 86 B - I I I . RESULTS AND DISCUSSION The mathematical model described i n the previous s e c t i o n was a p p l i e d to a range of s t e e l p l a t e s of d i f f e r e n t s i z e , to p r e d i c t the formation of c e n t e r l i n e shrinkage. C a l c u a t i o n s were a l s o c a r r i e d out to evaluate the e f f e c t of the end c h i l l on the c a s t i n g soundness. Table IV l i s t s the p l a t e s s i z e s considered. 3.1 S o l i d i f i c a t i o n Sequence The temperature d i s t r i b u t i o n along the c e n t e r l i n e , F i g 32, can be d i v i d e d i n t o three p a r t s . These are the r i s e r end with a r e l a t i v e l y steep temperature g r a d i e n t , the centre with e f f e c t i v e l y zero temperature gradient and the p l a t e end where the temperature gradient i s again steep. The c e n t e r l i n e shrinkage must form i n the c e n t r a l region where 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 does not occur. T y p i c a l contour l i n e s during s o l i d i f i c a t i o n determined from the model are shown i n F i g 34. The v e r t i c a l dotted l i n e s i n d i c a t e the c r i t i c a l p o i n t s f o r the formation of c e n t e r l i n e shrinkage determined experimentally by Bishop and P e l l i n i 2 8 . At the i n t i a l stage of s o l i d i f i c a t i o n ( F i g 34a) contour l i n e corresponding to 70% s o l i d p r o j e c t s out along the c e n t e r l i n e of the p l a t e toward the s o l i d u s . The remaining l i q u i d w i t h i n the 70% s o l i d region w i l l c o n t a in i n t e r d e n d r i t i c channels broad 87 enough to feed the s o l i d i f i c a t i o n c o n t r a c t i o n at the bottom of s o l i d / l i q u i d r e g i o n . At the middle stage of s o l i d i f i c a t i o n ( F i g 34b), the s o l i d u s s t i l l p r o j e c t s out along the c e n t e r l i n e whereas the 70% and 90% contour l i n e s move back toward the r i s e r end of the c a s t i n g . The region between 90% s o l i d and the s o l i d u s c o n t a i n s l i t t l e l i q u i d making i t very d i f f i c u l t f o r l i q u i d to flow through the e x i s t i n g channel to feed volume shrinkage at the s o l i d u s along the c e n t e r l i n e . The bottom of the s o l i d / l i q u i d region i s l o c a t e d i n the area where c e n t e r l i n e shrinkage was found e x p e r i m e n t a l l y 2 7 . Near the end of s o l i d i f i c a t i o n the s o l i d u s and 70% s o l i d contour l i n e approach each other ( F i g 34c). Thus, the shape and p o s i t i o n of the s o l i d i f i c a t i o n contour l i n e s i n d i c a t e whether c e n t e r l i n e p o r o s i t y i s l i k e l y to occur. However the contour l i n e c o n f i g u r a t i o n cannot q u a n t i t a t i v e l y p r e d i c t the region along the c e n t e r l i n e where shrinkage p o r o s i t y w i l l occur. 3.2 P r e d i c t i o n Of C e n t e r l i n e Shrinkage The c a l c u l a t e d pressure required to feed the shrinkage along the c e n t e r l i n e f o r a 5cm t h i c k p l a t e c a s t i n g i s shown i n F i g 35. The pressure a c t i n g on the system i s d e f i n e d as the sum of atmospheric pressure and the m e t a l l o s t a t i c head pressure. When the pressure r e q u i r e d to feed the shrinkage formed at the end of s o l i d i f i c a t i o n i s greater than the pressure a c t i n g on the system, the shrinkage w i l l not be fed completely. 88 The pressure required to feed a 4.6s(23cm) long c a s t i n g i s always l e s s than that a c t i n g on the system. The pressure r e q u i r e d to feed long c a s t i n g s , greater than 4.8s(24cm) i s greater than the a c t i n g p r e s s u r e . The maximum length which can be cast completely sound using one r i s e r , determined from the model, i s 4.7s for 5cm t h i c k p l a t e . This value agrees w e l l with the experimentally determined value of 4.5s 3 0. In F i g 35 the sound region determined e x p e r i m e n t a l l y adjacent t o the free end of the p l a t e i s shown for c a s t i n g s greater than 4.8s i n length. The length f r e e of p o r o s i t y shown i n the f i g u r e ranges from 2.35s to 2.7s, which agrees w e l l with the experimental value of 2.5s 3 0. Of p a r t i c u l a r i n t e r e s t i s that the maximum required pressure increases e x p o n e n t i a l l y with c a s t i n g l e n g t h . I f the c a s t i n g was subjected to 2 atmosphere pressure during s o l i d i f i c a t i o n , f or example, the e f f e c t would be s m a l l , i n c r e a s i n g the region f r e e of p o r o s i t y by 0.2s(1cm). C a l c u l a t e d p r o f i l e s of the pressure required to feed shrinkage along the c e n t e r l i n e of 2.5 and 0.5cm t h i c k p l a t e c a s t i n g s are shown i n F i g s 36 and 37, r e s p e c t i v e l y . The pressure p r o f i l e s for both p l a t e thickness are s i m i l a r to those i n F i g 35. I t i s found f o r 2.5cm t h i c k p l a t e that the maximum c a s t i n g l e n g t h which i s f r e e of p o r o s i t y i s 4.6s and that the sound region at the free end of the p l a t e i s 2.5s to 2.7s. These values agree with experimental data, 4.5s and 2.5s r e s p e c t i v e l y . For 0.5cm t h i c k c a s t i n g (Fig 37), the maximum sound len g t h i s a l s o 4.6s. There i s no i n d i c a t i o n that the maximum feeding distance to t h i c k n e s s r a t i o decreases with a 89 decrease i n p l a t e t h i c k n e s s i n the region l e s s than 2.5cm, as Johnson and Loper r e p o r t e d 3 1 . P r e d i c t e d maximum feeding d i s t a n c e s are p l o t t e d against p l a t e t h i c k n e s s i n F i g 38. Experimental r e s u l t s of the maximum feeding d i s t a n c e s are a l s o p l o t t e d i n the f i g u r e 2 7 ' 3 1 . For p l a t e s which are t h i c k e r than 2cm the agreement between the c a l c u l a t e d and measured values i s e x c e l l e n t . For the p l a t e s thinner than 2cm the experimental values of the feeding d i s t a n c e s are l e s s than those p r e d i c t e d . This can be accounted for by the larg e drop i n temperature of the l i q u i d metal at the free end of the p l a t e as the l i q u i d metal i s f i l l i n g the mold during pouring. This temperature drop i s not considered i n the present model. The r a p i d c o o l i n g at the end lowers the e f f e c t i v e superheat of the melt and e f f e c t i v e l y reduces the feeding d i s t a n c e . 9.3 The E f f e c t of an End C h i l l on the Length of the P o r o s i t y  Free Region The c o n t r i b u t i o n of an end c h i l l to i n c r e a s i n g the sound region i n a p l a t e c a s t i n g was a l s o examined by ap p l y i n g a s t e e l c h i l l at the c a s t i n g end of a 5cm t h i c k p l a t e . The c h i l l s i z e , 5cm t h i c k , was chosen since i t i s reported by Myscowski et a l 2 9 that Is c h i l l s are of s u f f i c i e n t t h i c k n e s s f o r p l a t e c a s t i n g s when i t s cross s e c t i o n i s the same as that of the ca s t i n g s ( F i g 30). 90 The c a l c u l a t e d pressures f o r the c a s t i n g w i t h an end c h i l l as a f u n c t i o n of dis t a n c e from the edge along the c e n t e r l i n e i s shown i n F i g 39. Comparing t h i s f i g u r e to F i g 35, the maximum pressure i n t h i s case i s s h i f t e d towards the r i s e r because of the c h i l l . The maximum length of the p l a t e c a s t sound i s 5.7s and the sound region due to end and c h i l l e f f e c t i s approximately 3.5s. This shows that the end c h i l l increases the -sound region by 1s (5cm) which c o i n c i d e s w i t h the value, 5cm determined e x p e r i m e n t a l l y . This distance i s r e l a t i v e l y s m a l l , i n d i c a t i n g that end c h i l l s w i l l not s i g n f i c a n t l y improve the soundness of p l a t e s cast i n sand molds. 3.4 Proposed S o l i d i f i c a t i o n Parameters D e f i n i n g the  T r a n s i t i o n from Porous to Nonporous Castings As described i n the previous s e c t i o n three s o l i d i f i c a t i o n parameters have been proposed to def i n e when c e n t e r l i n e p o r o s i t y w i l l occur i n a c a s t i n g . These are the temperature gradient ( G ) 3 0 , the temperature gradient d i v i d e d by the square root of c o o l i n g r a t e (G/vfc) 3 3 and the gradient of f r a c t i o n s o l i d ( P ^ ^ ) 3 5 . These parameters were c a l c u l a t e d i n the present computer program ( F i g 31) and the c r i t i c a l values f o r c e n t e r l i n e shrinkage were obtained as the values at the p o s i t i o n corresponding to the boundary f o r c e n t e r l i n e shrinkage p r e d i c t e d by the present model. The c r i t i c a l values f o r each parameter are l i s t e d i n Table V f o r each p l a t e t h i c k n e s s . The c r i t i c a l 91 values of both G and P are observed to change with p l a t e t h i c k n e s s . The proposed values are not a p p l i c a b l e g e n e r a l l y . On the other hand, the c r i t i c a l values f o r (G/^R) are very c l o s e to the proposed value, u n i t y , independent of c a s t i n g s i z e i n the range examined. A c c o r d i n g l y , the parameter G/ y/R i s a simple and e f f e c t i v e parameter to p r e d i c t roughly the l o c a t i o n of c e n t r e l i n e shrinkage i n a p l a t e c a s t i n g . 92 B-IV. CONCLUSIONS The l o c a t i o n of c e n t r e l i n e p o r o s i t y i n the p l a t e s cast i n sand molds has been examined using c a l c u l a t e d values of the thermal f i e l d during s o l i d i f i c a t i o n and Darcy's Law to de f i n e the i n t e r d e n d r i t i c f l u i d flow. The c a l c u l a t i o n s were compared with experimental r e s u l t s reported i n the l i t e r a t u r e from which the f o l l o w i n g conclusions may be made. 1) There i s good agreement between the c a l c u l a t e d and observed values of c e n t r e l i n e p o r o s i t y . This c l e a r l y i n d i c a t e s the c e n t r e l i n e shrinkage can be a t t r i b u t e d to i n s u f f i c i e n t i n t e r d e n d r i t i c f l u i d flow to feed the volume c o n t r a c t i o n during s o l i d i f i c a t i o n . 2) Darcy's Law can be used to c a l c u l a t e the i n t e r d e n d r i t i c f l u i d flow i n a c a s t i n g even though the system i s not at steady s t a t e during flow. 3) The pressure r e q u i r e d to produce a sound p l a t e c a s t i n g increases e x p o n e n t i a l l y as the c a s t i n g length i s increased. As a r e s u l t c a s t i n g under pressure w i l l not a p p r e c i a b l y reduce c e n t r e l i n e p o r o s i t y . 93 Table I I I - P h y s i c a l data employed i n c a l c u l a t i o n s S t e e l Mold s p e c i f i c heat, Cp (cal/g°C) 0.20 0.25 dens i t y of l i q u i d , p i (g/cm 3) 7.10 density of s o l i d , ps (g/cm 3) 7.50 1 .65 thermal c o n d u c t i v i t y , k (cal/cm.s. °C) 0.074 0.0037 heat t r a n s f e r c o e f f i c i e n t , hs (cal/cm2.s.°C) 5.0X10-* l a t e n t heat of s o l i d i f i c a t i o n , Hs (cal/g) 65.0 s o l i d i f i c a t i o n shrinkage r a t i o , 0 0.03 l i q u i d u s temperature, T l (°C) 1507 s o l i d u s temperature, Ts (°C) 1463 pouring temperature, Tp _(°C) 1595 i n i t i a l temperature, To (°C) 20 e m i s s i v i t y , e 0.45 v i s c o s i t y of l i q u i d , M (poise) 0.05 ambient temperature, Ta (°C) 20 94 Table IV- Dimenstion of the st e e l plate casting examined thickness(cm) length (cm) s*5.0 4. 6s, 4 .8s, 5.0s, 6 6s, 9 0s, 12.0s 2.5 4. 4S, 4 .6s, 4.8s, 5 0s, 6 6s 1.25 4. 4s, 4 .6s, 5.0s, 6 6s, 0.5 4. 0s, 4 .4s, 4.6s, 4 8s, 5 8s 5.01 5. 6s, 5 .8s, 6.6s, 9 ,0s Mote: 1. end c h i l l e d by 5cm thi c k s t e e l Table v ~ Comparison of the c r i t i c a l values of s o l i d f i c a t i o n parameters for cen t e r l i n e shrinkage parameters proposed value obtained values for each p l a t e thickness 5cm 2.5cm 1.25cm 0.5cm G(°C/cm) Pj(/min°C/cm) PJJ(1/cm) 0.22-0.44" 1 . 0 " 0.25 1 5 1.8-2.2 0.92-1.10 0.037-0.042 3.6-4.4 0.93-1.08 0.072-0.082 6.6-8.0 0.83-0.98 0.12-0.14 14.6-19.7 0.94-1.07 0.37-0.41 Kote: P . - G/fR 95 r v i MAXIMUM LENGTH CAST SOUND V 4.5s s r \ / i LENGTH GREATER THAN MAXIMUM V 2s 2.5s * al u. -J s VARIABLE rvi MAXIMUM LENGTH CAST SOUND Y •> rvi LENGTH GREATER THAN MAXIMUM Y 2s 2s j I , -'1 1' c 1 S VARIABLE . . . VARIABLE F i g u r e i n the 28 - Feeding r e l a t i o n s h i p determined e x p e r i m e n t a l l y s t e e l c a s t i n g s ; (a) P l a t e s 3 0 , and (b) Square b a r s 3 1 96 Figure 29 - Comparison of measured p e r m e a b i l i t i e s vs volume f r a c t i o n l i q u i d * • • « * • . 1. . RISER • - • • » . • • » 3s * * .GREEN SAND MOLD * • • • • * # . * SOUND \ • DUE TO RISER EFFECT! *'v CENTERLINE. * " -SHRINKAGE.'. . :• SOUND * ; •••*•• DUE TO 1 *. END EFFECT • . ' • p. • n i 4 s - 1 2 s . a ' 1 CHILL f • l a — I * - — • •• • • i S = PLATE THICKNESS t> X CO to CO vo Figure 30 - Co n f i g u r a t i o n of the system i n v e s t i g a t e d 98 I START) READ INPUT DATA INITIALIZE VARIABLES ^TIMF=TlME+*t|* CALCULATE TEMPERATURE FIELD USING EXPLICIT FDM CALCULATA 9u*& AND Vx I CALCULATE THE REQUIRED PRESSURE TO FEED SHRINKAGE YES CALCULATE -SOLIDIF ICATION PARAMETERS (STOP) Figure 31 - Flow chart of the computer program f o r the p r e d i c t i o n of c e n t e r l i n e shrinkage 99 LIQUIDUS 0 5 10 15 20 25 30 DISTANCE FROM RISER ( CM ) Figure 32 - Temperature d i s t r i b u t i o n along the c e n t e r l i n e of the p l a t e c a s t i n g RISER " 30 20 -c MN SI .34 .89 .68 CASTING SIZE : 2" x 13" o o 10 • MPASII&FT) nv BISHOP AND PELLINI 5 7 • CALCULATED RESULTS 0 -r-10 TIME ( MIN ) 15 2u" Figure 33 Soli d u s movement along the c e n t e r l i n e of p l a t e c a s t i n g 101 CASTING SIZE ! 5 X 33 CM SOLIDUS SOUND DUE TO END EFFECT' 7055 SOLID Z7 SOUND DUE TO RISER EFFECT (b) (c) Figure 34 - The d i s t r i b u t i o n of s o l i d i f i c a t i o n contour l i n e s at the ( a ) i n i t i a l , (b) middle and (c) l a s t stages of s o l i d i f i c a t i o n 102 Figure 35 - D i s t r i b u t i o n of the pressure required to feed shrinkage at the end of s o l i d i f i c a t i o n (s=5cm) Figure 36 - D i s t r i b u t i o n of the pressure r e q u i r e d to feed shrinkage at the end of s o l i d i f i c a t i o n (s=2.5cm) 104 Figure 37 - D i s t r i b u t i o n of the pressure r e q u i r e d to feed shrinkage at the end of s o l i d i f i c a t i o n (s=0.5cm) 1 0 5 40-35 -30 -- 25 • 20 -15 -10 -5 -° • EXPERIMENT BY BISHOP AND PELLINl' 7 ° • EXPERIMENT BY JOHNSON AND LOPER31 EXPERIMENTAL BOUNDARY PRESENT WORK I 1 1 r 2 3 4 5 PLATE THICKNESS, s ( CM ) T -6 Figure 38 - Soundness of s t e e l p l a t e c a s t i n g s 106 Figure 39 - D i s t r i b u t i o n of the pressure r e q u i r e d to feed shrinkage at the end of s o l i d i f i c a t i o n -end c h i l l e d c a s t i n g 1 0 7 BIBLIOGRAPHY 1 . E . S c h e i l ; M e t a l l f o r s c h u n g , 2 ( 1 9 4 7 ) , p 6 9 . 2 . J . K i r k a l d y and W.V.Youdelis; Trans. AIME, 2 1 2 ( 1 9 5 8 ) , p 8 3 3 . -3 . F.Sauerwald; M e t a l l w i r s c h a f t , 2 2 ( 1 9 4 3 ) , p 5 4 3 . 4 . W.V.Youdelis and D.R.Colton; Trans. AIME, 2 1 8 ( 1 9 6 0 ) , p 8 0 9 . 5 . W.V.Youdelis; "The S o l i d i f i c a t i o n of Metals" (Brighton Conference), The Iron and S t e e l I n s t . London ( 1 9 6 7 ) , p i 1 2 . 6 . E . P l e z e l and Z.Schneider; Z. Metallkunde, 3 5 ( 1 9 4 3 ) , p i 2 1 . 7 . B.Prabhakar and F.Weinberg; Met. Trans. ASM, 9 B ( 1 9 7 8 ) , P 1 5 0 . 8 . M.C.Flemings and G.E.Nereo; Trans. AIME, 2 3 9 ( 1 9 6 7 ) , p 1 4 4 9 . 9 . M.C.Flemings, R.Mehrabian and G.E.Nereo; i b i d , 2 4 2 ( 1 9 6 8 ) , p 4 1 . 1 0 . M.C.Flemings and G.E.Nereo; i b i d , 2 4 2 ( 1 9 6 8 ) , p 5 0 . 1 1 . R.Mehrabian, M.A.Keane and M.C.Flemings; Met Trans. ASM, 1 ( 1 9 7 0 ) , p 3 2 3 8 1 2 . E . S c h e i l ; Z.Metallkunde, 3 4 ( 1 9 4 2 ) , p 7 0 . 1 3 . W.G.Pfann; Trans. AIME, 1 9 4 ( 1 9 5 2 ) , p 7 4 7 . 1 4 . H.D.Brody and M.C.Flemings; i b i d , 2 3 6 ( 1 9 6 6 ) , p 6 l 5 . 1 5 . T.F.Bower, H.D.Brody and M.C.Flemings; i b i d , 2 3 6 ( 1 9 6 6 ) , p 6 2 4 . 1 6 . C . J . S m i t h e l l s ; "Metals Reference Book" 4 t h Ed., Butterworths, London ( 1 9 6 7 ) , p 6 8 5 . . 1 7 . J . F . E l l i o t and M.Gleiser; "Thermochemistry of Steelmaking", Addison-Wesley Pub. Co. ( i 9 6 0 ) . 1 8 . K.Bornemann and F.Sauerwald; Z.Metallkunde, 1 4 ( 1 9 2 2 ) , p l O . 1 9 . K.Bornemann and F.Sauerwald; i b i d , 1 4 ( 1 9 2 2 ) , p ! 5 4 . 108 20. H.R.Thresh, A.F.Crawley and D.W.G.White; Trans. AIME, 242 (1968), p8l9. 21. R.E.Morrone, J.O.Wilkes and R.D.Pehlke; AFS Cast Metals Research J o u r n a l , Dec (1970), pi84. 22. R.E.Morrone, J.O.Wilkes and R.D.Pehlke; i b i d , Dec (1970), pl88. 23. G.Sciama; i b i d , Mar (1972), p20. 24. G.Sciama; i b i d , Dec (1972), p145 25. I.Ohnaka, Y.Yashima and T.Fukusako; Imono, 52 (1980), No 1, plO. 26. A.Jeyarajan and R.D.Pehlke; AFS Trans." 86 ( 1 978), p457. 27. H.F.Bishop and W . S . P e l l i n i ; AFS Trans. 58 (1950), p l 8 5 . 28. H.F.Bishop, E.T.Myskowski and W . S . P e l l i n i ; i b i d , 59 (1951) , p171. 29. E.T.Myskowski, H.F.Bishop and W . S . P e l l i n i ; i b i d , 60 (1952) , p389. 30. W . S . P e l l i n i ; i b i d , 61 (1953), p 6 l . 31. S.B.Johnson and C.R.Loper,Jr; i b i d , 77 (1969), p360. 32. E.Niyama, T.Uchida, M.Morioka and S.Saito; AFS I n t e r n a t i o n a l Cast Metals J o u r n a l ; 6 (1981), No.2, p16. 33. E.Niyama, T.Uchida, M.Morioka and S.Saito; i b i d , 7 (1982), No.3, p52. 34. K . C h i j i i w a and I.Imafuku; Imono, 55 (1983), No.5, p 2 7 l . 35. V.L.Davies; AFS Cast Metals Research J o u r n a l , Jun (1975), p33. 36. M.C.Flemings; " S o l i d i f i c a t i o n P r o c e s s i n g " , McGraw-Hill Inc. (1974), p234. 37. T.S.Piwonka and M.C.Flemings; Trans. AIME, 236 (1966), p1157. 38. R.Mehrabian, M.Kreane and M.C.Flemings; Met. Trans. ASM, 1 (1970), p1209. 39. N.Streat and F.Weinberg; Met. Trans. ASM, 7B (1976), P417. 40. D.Apelian, M.C.Flemings and R.Mehrabian; Met. Trans. 109 ASM, 5 (1974), p2533. 110 APPENDIX A ~ RECALCULATION OF THE SCHEIL-YOUDELIS MODEL PREDICTIONS The S c h e i l - Y o u d e l i s ' model p r e d i c t i o n s have been r e c a l c u l a t e d f o r the Al-Cu, Al-Zn and Sb-Bi systems and are shown i n F i g s 40, 41 and 42, r e s p e c t i v e l y . C a l c u l a t i o n s from t h i s i n v e s t i g a t i o n plus c a l c u l a t i o n s published and the experimental r e s u l t s reported are inc l u d e d . For the Al-Cu a l l o y s F i g 40 the r e c a l c u l a t e d values are s i g n i f i c a n t l y lower than the curve published by S c h e i l and K i r k a l d y et a l 2 , although the general shape of the curve i s the same. I t i s not c l e a r why the r e c a l c u l a t e d curve d i f f e r s from the published one. In the Al-Zn and Sb-Bi a l l o y s there are very large d i f f e r e n c e s between the r e c a l c u l a t e d curves and those reported i n the l i t e r a t u r e , i n p a r t i c u l a r at the low a l l o y compositions. 111 APPENDIX B ~ DERIVATION OF NODAL EQUATIONS The p a r t i a l d i f f e r e n t i a l equation f o r two dimensional t r a n s i e n t problem (equation (B-30)) and boundary equations (equations B-34 to B-36) were solved e x p l i c i t l y using f i n i t e d i f f e r e n c e method. F i g 43 shows types of nodes appearing i n the model examined. The nodal d i f f e r e n c e equations were d e r i v e d by apply i n g heat balance for each element as f o l l o w i n g s ; I n t e r i o r nodes (node type 10) ' T . . , - T . . T . T . . T . . . - T . . i k k L A y + k -JbJ±i i i i A y + k ^ l l t l i L L ^ Ax J Ax J Ay * T . . - T . . T . - T . . + k 1 + 1 >j Ax = AxAypC ^ (1) Ay J *p At T* . = Fo (T. . ,+ T . . . . ) + Fo (T. . .+ T . . . .) i , J x i , j - l i .J+1 y i - l , J i + l , J + (1 - 2Fo - 2Fo ) T . . (1) ' x y ' i , j Corner node i n the mold (node type 1) T* = 2Fo T . . . . + 2Fo T. . ._ .+ 2(Fo B i + Fo B i )T i , j x i , j + l y i + l , j x x y y ' a + (1 - 2F6 - 2Fo - 2Fo B i - 2Fo B i ) T . . (2) x y x x y y ' i , j Surface nodes i n the mold (node type 2) + (1 - 2Fo - 2Fo - 2Fo B i )T x y y y I , J (3) 1 12 Boundary node i n the mold (node type 4) 1 T - T . T (T, T, OAy + k c ^ Ay + k c ^Ax ±AX ^ ''I-1 ^ " S A X ' S * T. - T. T . - T. + k c 1 + 1 >l id. Ax = AxAyp C ± > J S Ay J p At * 2 1, . = T - ~ F6 T. . - + Fo T. ... + Fo (T, . .+ T, ,) i , j k s + kM x x y i + 1 » J 2 kM + ( 1 ~ TTTT F o ~ Fo - 2Fo ) T. kS + kM X X y Surface boundary nodes i n the metal (node type 6) - (T _ T \*L+ v T i > j + l " T l , j Ay_ ( _ 1. 1. U i , j - 1 ±,j ; 2 + *M Ax 2 " U a 1 yAx yAx ' J •=r— + S T - T T - T ^d id- & = C .^1 + *M "Ay " *< * ^ -p At * 2 k 9 T. . = T—-P- Fo T. A , + Fo T. . + 2Fo T._,_. . + 2Fo Bi T i , j kg+kj^ x i , j - l x i,j + l y l+l,j y y a 2 k s + ( 1 - V — r f Fo - Fo - 2Fo - 2Fo Bi ) T. . v k s + k M x x y y y X » J Corner nodes i n the metal (node type 9) 2 k s T = (Fo T ,+ Fo T J M .) .+ Fo T 3 .,,+ Fo T. , . A i , j k c + k M « i . j - i y i + i . j x y 2k i 1 - < I + V 4 ' < f V F V 1 T i . a 113 APPENDIX C ~ FORTRAN PROGRAM FOR THE PREDICTION OF CENTRELINE SHRINKAGE C C r * * * * * * * * * * * * * * * * * * * * * * * * * * > * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C FILE NAME : CENTO91 CALC. OF SOLIDUS PRESSURE AT GL= C C FEEDING DISTANCE DETERMINATION OF HORIZONTAL PLATE CA C MOLD BY NUMERICAL SIMULATION.(APPLICATION OF 1 ~D DARC C 2-D UNSTEADY-STATE PROBLEM. C HANDLING OF LATENT HEAT = EQUIVALLENT SPECIFIC HEAT M C C ASSUMPTIONS; 1)CONST.SPECIFIC HEAT,CONDUCTIVITY A DEN C 2)LINEAR SOLIDUS A LIQUIDUS C 3)NO H.T.RESISTANCE"AT S/M INTERFACE C 4)THE MOLD IS INSTANTANEOUSLY FILLED WIT C METAL WHICH IS STAGNANT C SYMBOLS; DX =GRID DISTANCE DY =GRID DISTANCE C DT =TIME STEP H =HEAT TRANSF.COE C TA =AMBIENT TEMP. TO =INITIAL TEMP.OF C TP =POURING TEMP. N =NO.OF NODES(Y) C M =NO.OF NODES(X) OTI=OUTPUT TIME INT C OT =OUTPUT TIME CPM=SPECIFIC HEAT 0 C CPS=SPECIFIC HEAT OF SAN DM =DENSITY OF META C DS =DENSITY OF SAND KM =CONDUCTIVITY OF C KS =CONDUCTIVITY OF SAND TL =LIQUIDUS TEMP. C TS =SOLIDUS TEMP. HS =LATENT HEAT OF C TM =TIME TMS=SOLIDN.START T l C TM7=70% SOLIDN.TIME TM9=90% SOLIDN. TIM C TMF=SOLIDN.COMPLETE TIME TMLS=LOCAL SOLIDN.T C CR =MEAN COOLING RATE GR =TEMP.GRADIENT C P =SOLIDN.PARAMETER C AM,AS =THERMAL DIFFUSIVITY C FMX,FMY=FOURIER NO.OF METAL C FSX,FSY=FOURIER NO.OF SAND C BSX,BSY=BIOT NO IN SAND C DDX,DDY=1/DX**2,1/DY**2 C A1,A2 =2*(KS,KM)/(KS+KM) C CPME =EQUIVALLENT SPECIFIC HEAT OF METAL C T,TB =NEW A OLD TEMP. C VSX=SOLIDUS VELOCITY VX =FLUID VELOCITY C VXI,VXO=IN- A OUT-FLOW VELOCITY C GL,GLB =NEW A OLD VOLUME FRACTION LIQUID C TA =ATOM. PRESSURE AK = PERMEABILITY C DP =PRESSURE DROP PR =PRESSURE AT SOL C B =CONTRACTION RATIO C Q ********************************************************** c c REAL KM,KS DIMENSION T(100,100),TB(100,100),AM(100,100),FMX(100,100), 1 14 1 CP( 100, 100),TMS( 100, 100),PRR( 100, 100), 2TM98(100,100), GR(100,100),P(100,100), 3FMY(100,100),TMF(100,100),VSX(100,100),VX(100,100), 4GL(100,100),GLB(100,100),DP(100,100),PR(100,100) C C ' -C ***** DATA INPUT READ(5,1000) N,LC,DX,DY,DT,OTI 1000 FORMAT(//,2I12,4F12.5) READ(5,1020) CPM,DM,KM,HS,B READ(5,1020) TL,TS,TP,TA,TO 1020 FORMAT(/,5F12.5) READ(5,1030) CPS,DS,KS,H1,E,VIS 1030 FORMAT(/,6F12.5) C M=25+LC MM5=M-5 MM6=M-6 C c ***** DETERMINATION OF INITIAL VALUES DO 20 J=1,M DO 10 1=1,N 10 TB(I,J)=T0 20 CONTINUE C DO 40 J=6,20 DO 30 1=1,25 GLB(I,J)=1.0 30 TB(I,J)=TP 40 CONTINUE C DO 60 J=21,MM5 DO 50 1=21,25 GLB(I,J)=1.0 50 TB(I,J)=TP 60 CONTINUE C TM=0.0 OT=OTI JF=MM6 C c ***** ITERATION START 70 TM=TM+DT/60.0 C C ***** CHECK OF SPECIFIC HEAT OF METAL CALL CPCHK(MM5,TB,TL,TS,CP,CPM,CPME,HS) C C ***** CALCULATION OF CONSTANTS DDX=1/(DX**2) DDY=1/(DY**2) A1=2.0*KM/(KS+KM) A2=2.0*KS/(KS+KM) C DO 130 J=6,20 115 DO '120 1 = 1 ,25 AM(I,J)=KM/(DM*CP(I,J)) FMX(I,J)=AM(I,J)*DT*DDX 120 FMY (I , J) =AM(I , J) *DT.*DDY 130 CONTINUE C - . DO 150 J=21,MM5 DO 140 1=21,25 AM(I,J)=KM/(DM*CP(I,J)) FMX(I,J)=AM(I,J)*DT*DDX 140 FMY(I,J)=AM(I,J)*DT*DDY 150 CONTINUE C AS=KS/(DS*CPS) FSX=AS*DT*DDX FSY=AS*DT*DDY BSX=H1*DX/KS BSY=H1*DY/KS PA=1.0l3E+06 G=981.0 C c C ***** CALCULATION OF T'S AT TIME=TM CALL TCALC(N,M,T,TB,TA,FSX,FSY,FMX,FMY,BSX,BSY,E,DY,KM,A1,A C c ***** CALCULATION OF SOLIDIFICATION TIMES CALL TMSOL(TL,TS,MM5,T,TB,TM,TMS, TM98,TMF) C C ***** CALC. OF FLUID VELOCITY C IF(JF.LT.21) GO TO 235 1=23 DO 165 J=21,JF IF(T(I,J).GT.TL) GO TO 155 IF(T(I,J).LT.TS) GO TO 160 FS=(TL-T(I,J))/(TL-TS) GL(I,J)=((1.0-FS)/7.1)/(FS/7.5+(1.0-FS)/7.1) GO TO 165 155 GL(I,J)=1.0 GO TO 165 160 GL(I,J)=0.0 165 CONTINUE C C 185 IF(GL(23,JF).GT.0.02) GO TO 228 DO 190 L=1,30 IF((TM98(23,JF)-TM98(23,JF+L)).LE.0.0) GOTO 190 VSX(23,JF)=DX*L/ ((TM98(23,JF)-TM98(23,JF+L))*60.0) GO TO 195 190 CONTINUE C 195 VX(23,JF)=B*VSX(23,JF) IF(JF.LE.21) GO TO 220 DO 200 L=1,50 VX(23,JF-L)=VX(23,JF)*GLB(23,JF)/GLB(23,JF-L) 116 IF((JF-L).EQ.21) GO TO 220 200 CONTINUE C c ***** APPLICATION OF DARCY'S LAW 220 DPTX=0.0 1=23 - . IF(GL(I,21).LE.0.02) GO TO 235 DO 225 J=21, JF IF(GLB(I,J).GT.0.3) GO TO 222 AK=6.89E-09*GLB(I,J)**2 GO TO 223 222 AK=1.0E-05*GLB(I,J)**8 223 DP(I,J)=VIS*GLB(I,J) *VX(I,J)*DX/AK DPTX=DPTX+DP(I , J) 225 CONTINUE C PR(23,JF)=DPTX PRR(23,JF)=PR(23,JF)/PA WRITE(6,1032) JF,TM 1 032 FORMAT(//,' (23, 1 ,12,') IS SOLIDIFIED AT',F10.5,' (MIN)',/) WRITE(6,1033) (T(23,J),J=21,MM5) 1033 FORMAT(20F6.0) WRITE(6,1034) 1034 FORMAT(/,'NO',5X,'T',7X,'GL',9X,'Vf',7X,'P.DROP') WRITE(6,1035) 1035 FORMAT(7X,'C,16X,'cm/sec',5X,'dyn/cm2',/) WRITE(6,1036) (J,T(23,J),GLB(23,J),VX(23,J),DP(23,J),J=21,J 1036 FORMAT(12,F2.1,F3.4,F11.5,E12.4) WRITE(6,1040) PRR(23,JF) 1040 FORMAT*/,5X,'PRESSURE REQUIRED =',F11.4,' atm',//) C WRITE(6,1032) JF,TM C1032 FORMAT(//,' (23, ' ,12,') IS SOLIDIFIED AT',F10.5,' (MIN)',/) C WRITE(6,1034) (VX(23,J),J=21,JF) C1034 FORMATO0E10.3,' cm/sec') C WRITE(6,1036) (DP(23,J),J=21,JF) C1036 FORMAT(10E10.3,' dyn/cm2') C WRITE(6, 1038) DPTX,PR(2 3,JF) C1038 FORMAT(/,5X, E12.4,' PRESSURE =',E12.4,' (dyn/cm2)') C JF=JF-1 IF(JF.LT.21) GO TO 235 C IF(GL(23,JF).LE.0.02) GO TO 185 C 228 1=23 DO 229 J=21,JF 229 GLB(I,J)=GL(I,J) C C ***** CALCULATION OF TEMP.GRADIENTS 235 CALL TGR(MM5,T,TB,TS,DX,DY,GR) C c ***** CHECK OF COMPLETE SOLIDIFICATION IF(T(23,21).GT.TS) GO TO 239 1 17 C c ***** SOLIDN. PARAMETERS 1=23 DO 237 J=21,MM6 TMLS =TMF(I , J)-TMS(I , J) CR =(TL-TS)/(TMLS *"-60..0) 237 P(I,J)=GR(I,J)/SQRT(CR) C c ***** OUTPUT OF T'S AT COMPLETE SOLIDIFICATION WRITE(6,1050) TM 1050 FORMATC////,'TEMP DISTRIBUTION AT',F10.4,' MIN ',//) WRITE(6,1080) ( ( T ( I , J),I=16,N), J= 1,M) 1080 FORMAT(15F5.0) C WRITE(6,1170) 1170 FORMAT(////,'COMPLETE SOLIDN. TIME : TMF (MIN)',//) WRITE(6,1180) ((TMF(I,J),1=21,25),J=21,MM5) 1180 FORMAT(5F10.3) C WRITE(6,1190) 1190 FORMAT(////,'NO',6X,'T.GRAD',8X,'P',7X,'V(SOLID.M)',2X, 1'PRESSURE',7X,'PR/PA', 2/,9X,'C/cm',19X,'cm/sec',5X,'dyn/cm',9X,'atm',/) 1=23 WRITE(6,1200) ( J , G R ( I , J ) , P ( I , J ) , V S X ( I , J ) , P R ( I , J ) , P R R ( I , J ) , 1J=21,MM6) 1200 F0RMAT(I2,3F12.4,E12.3,F12.4) C C STOP C C c ***** OUTPUT TIME CHECK DURING ITERATION 239 IF(TM.LT.OT) GO TO 240 C c ***** OUTPUT OF T'S WRITE(6,1245) TM 1245 FORMAT(////,'TEMP. DISTRIBUTION AT',F10.4,'MIN ',//) WRITE(6,1246) ((T(I,J),I=16,N),J=1,M) 1246 FORMAT(15F5.0) C OT=OT+OTI C C C ***** SUBSTITUTION OF TEMP. A ITERATION 240 DO 260 J=1,M DO 250 1=1,N 250 TB(I,J)=T(I,J) 260 CONTINUE C GO TO 70 C END C 118 C C c c Q ************************************ C : . SUBROUTINE CPCHK(MM5,TB,TL,TS,CP,CPM,CPME,HS) C C ASSIGN THE EQUIVALLENT SPECIFIC HEAT TO METAL IN THE RAN C Q ********************************************************* c c DIMENSION TB(100,100),CP(100,100) C C TLS=(TL+TS)/2.0 DO 30 J=6,20 DO 20 1=1,25 IF (TB (I , J).GT.TL.OR.TB(I,J).LT.TS) GO TO 10 C IF(TB(I,J).GT.TLS) GO TO 5 C CP(I,J)=CPM+(TB(I,J)-TS)*(4.0*HS/(TL-TS)**2) C GO TO 20 C 5 CP(I,J)=CPM+(TL-TB(I,J))*(4.0*HS/(TL-TS)**2) 5 CP(I,J)=CPM+HS/(TL-TS) GO TO 20 10 CP(I,J)=CPM 20 CONTINUE 30 CONTINUE C DO 60 J=21,MM5 DO 50 1=21,25 IF(TB(I,J).GT.TL.OR.TB(I,J).LT.TS) GO TO 40 C IF(TB(I,J).GT.TLS) GO TO 35 C CP(I,J)=CPM+(TB(I,J)-TS)*(4.0*HS/(TL-TS)**2) C GO TO 50 C 35 CP(I,J)=CPM+(TL-TB(I,J))*(4.0*HS/(TL-TS)**2) 35 CP(I,J)=CPM+HS/(TL-TS) GO TO 50 40 CP(I,J)=CPM 50 CONTINUE 60 CONTINUE C RETURN END C C C C C Q *********************************************************** c SUBROUTINE TCALC(N,M,T,TB,TA,FSX,FSY,FMX,FMY,BSX,BSY,E,DY,K 1A1,A2) C 119 C CALCULATION NODES' TEMPS. BY EXPLICIT FINITE DIFFERENCE C Q ************************************* C c REAL KM -DIMENSION T(100,100),TB(100,100),FMX(100,100),FMY(100,100) MM1=M-1 MM2=M-2 MM3=M-3 MM4=M-4 MM5=M-5 MM6=M-6 C C ***** TEMPS. IN SAND C C NODE 1 DO 15 J=2,4 DO 10 1=2,26 10 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*(TB(I- 1,J)+TB(I + 1,J)) 1 +(1.0-2.0*FSX-2.0*FSY)*TB(l,J) 15 CONTINUE C 1 = 26 J=5 20 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*(TB(I- 1,J)+TB(I + 1 ,J)) 1 +(1.0~2.0*FSX-2.0*FSY)*TB(I,J) C DO 40 1=27,29 DO 30 J=2,MM1 30 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*(TB(I- 1,J)+TB(I+1 ,J) ) 1 +(1.0-2.0*FSX~2.0*FSY)*TB(I,J) 40 CONTINUE C DO 45 1=2,19 DO 42 J=22,MM1 42 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*(TB(I-1,J)+TB(I+1,J)) 1 +(1.0-2.0*FSX~2.0*FSY)*TB(I,J) 45 CONTINUE C 1 = 20 J=MM4 50 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*(TB(I-1,J)+TB(I+1,J)) 1 + (1.0-2.0*FSX-2.0*FSY)*TB(I,J) C 1 = 26 J=MM4 55 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*(TB(I-1,J)+TB(I+1,J)) 1 +(1.0-2.0*FSX-2.0*FSY)*TB(I,J) C DO 70 1=20,26 DO 60 J=MM3,MM1 60 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*(TB(I-1,J)+TB(I+1,J)) 1 +(1.0-2.0*FSX-2.0*FSY)*TB(I,J) 120 70 CONTINUE C C NODE 2 1 = 1 J=1 1 00 T(l,J)=FSX*TB(I , J+1 )+FSY*TB-(l+J , J) + (FSX*BSX+FSY*BSY) *TA 1 +(1 .0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(l ,.J) C C NODE 3 1 = 1 DO 110 J = 2,4 110 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB(1+1,J)+FSY*BSY*TA 1 +(1.0-2.0*FSX-FSY~FSY*BSY)*TB(I,J) C 1 = 1 DO 115 J=22,MM1 115 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB (1+1,J)+FSY*BSY*TA 1 +(1.0-2.0*FSX-FSY-FSY*BSY)*TB(I,J) C C NODE 4 1 = 1 J=5 120 T(I,J)=FSX*TB(I,J-1)+FSY*TB(l+1,J)+A1*FSX*TB(I,J+1)+FSY*BSY 1 +(1.0-FSX-FSY-A1*FSX-FSY*BSY)*TB(I,J) C C NODE 5 1 = 1 J=21 125 T(I,J)=FSX*TB(I ,J+1 )+FSY*TB(I + 1 , J)+A1 *FSX*TB('l ,J-1 )+FSY*BSY 1 +(1.0-FSX-FSY-A1*FSX-FSY*BSY)*TB(l,J) C C NODE 6 1 = 1 J=M 128 T(I,J)=FSX*TB(I,J-1)+FSY*TB(l+1,J)+(FSX*BSX+FSY*BSY)*TA 1 +(1.0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(I,J) C C NODE 7 J=1 DO 130 1=2,29 1 30 T(I,J)=FSY*(TB(I-1,J)+TB(I + 1 ,J))+FSX*TB(I,J+1 )+FSX*BSX*TA 1 +(1.O-FSX-2.0*FSY~FSX*BSX)*TB(I,J) C C NODE 8 J=5 DO 180 1=2,25 180 T(I,J)=FSY*(TB(I-1,J)+TB(l+1,J))+FSX*TB(I,J-1)+A1*FSX*TB(I, 1 +(1.0-2.0*FSY-FSX-A1*FSX)*TB(l,J) C C NODE 9 J=21 DO 185 1=2,19 185 T(I,J)=FSY*(TB(I-1,J)+TB(I+1,J))+FSX*TB(I,J+1)+A1*FSX*TB(I, 1 +(1.0-2.0*FSY-FSX-A1*FSX)*TB(l,J) 121 C J=MM4 DO 187 1=21,25 187 T(I,J)=FSY*(TB(I-1,J)+TB(I+1,J))+FSX*TB(I,J+1)+A1*FSX*TB(I, 1 +(1.0-2.0*FSY-FSX-A1*FSX)*TB(l,J) C -C NODE 10 -J=M DO 190 1=2,29 190 T(I,J)=FSY*(TB(I-1,J)+TB(I+1,J))+FSX*TB(I,J-1)+FSX*BSX*TA 1 +(1.0-FSX-2.0*FSY-FSX*BSX)*TB(l,J) C C NODE 11 1 = 20 J=21 220 T(I,J)=FSX*TB(I,J+1)+FSY*TB(I-1,J)+A1*FSX*TB(I,J-1) 1 +A1*FSY*TB(I + 1 ,J) + ( 1 .0-FSX-FSY-A1*"(FSX+FSY) )*TB(I ,J) C C NODE 12 1=20 DO 225 J=22,MM5 225 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB(l-1,J)+A1*FSY*TB(I+ 1 +(1.0-2.0*FSX-FSY-A1*FSY)*TB(l,J) C C NODE 13 1 = 26 DO 230 J=6,MM5 230 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB(l+1,J)+A1*FSY*TB(I-1 +(1.0-2.0*FSX-FSY-A1*FSY)*TB(I,J) C C NODE 14 1 = 30 J=1 240 T(I,J)=FSX*TB(I,J+1)+FSY*TB(I-1,J)+(FSX*BSX+FSY*BSY)*TA 1 +(1,0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(l,J) C C NODE 15 1 = 30 DO 250 J=2,MM1 250 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB (1-1,J)+FSY*BSY*TA 1 +(1.0-2.0*FSX-FSY-FSY*BSY)*TB(I,J) C C NODE 16 1 = 30 J=M 260 T(I,J)=FSX*TB(I,J-1)+FSY*TB(I-1,J)+(FSX*BSX+FSY*BSY)*TA 1 +(1.0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(I,J) C C C ***** TEMPS. IN METAL C C NODE 17 DO 280 1=2,24 DO 270 J=7,19 122 270 T(I,J)=FMX(I,J)*(TB(I,J-1)+TB(l,J+1))+FMY(I,J)*(TB(I-1,J) 1 +TB(I+1,J))+(1.0-2.0*FMX(l,J)-2.0*FMY(I,J))*TB(I,J) 280 CONTINUE C 1=21 J=20 - . 290 T(I,J)=FMX(I,J)*(TB(I,J-1)+TB(l,J+1))+FMY(I,J)*(TB(I-1,J) 1 +TB(I+1,J))+(1.0-2.0*FMX(l,J)-2.0*FMY(I,J))*TB(I,J) C DO 310 1=22,24 DO 300 J=20,MM5 300 T(I,J)=FMX(I,J)*(TB(I,J-1)+TB(l,J+1))+FMY(l,J)*(TB(I-1,J)+T 1 (I+1,J))+(1.0-2.0*FMX(I,J)-2.0*FMY(I,J))*TB(I,J) 310 CONTINUE C C NODE 18 1 = 1 J = 6 H=1.37E-12*E*((TB(I,J)+273.0)**4-(TA+273.0)**4)/(TB(I,J)-TA BMY=H*DY/KM 320 T(I,J)=FMX(I,J)*TB(I,J+1)+FMY(I,J)*TB(I+1,J)+A2*FMX(I,J)*TB 1 (I,J-1)+FMY(l,J)*BMY*TA+(1.0-FMX(I,J)-FMY(I,J)-A2*FMX(I 2 -BMY*FMY(I,J))*TB(I,J) C C NODE 19 1 = 1 DO 330 J=7,19 H=1.37E-12*E*((TB(I,J)+273.0)**4-(TA+273.0)**4)/(TB(I,J)-TA BMY=H*DY/KM 330 T(I,J)=FMX(I,J)*(TB(I,J-1)+TB(l,J+1))+FMY(l,J)*TB(I+1,J) 1 +BMY*FMY(I,J)*TA 2 +(1.0-2.0*FMX(I,J)-(1.0+BMY)*FMY(I,J))*TB(I,J) C C NODE 20 1 = 1 J=20 H=1.37E-12*E*((TB(I,J)+273.0)**4-(TA+273.0)**4)/(TB(I,J)-TA BMY=H*DY/KM 340 T(I,J)=FMX(I,J)*TB(I,J-1)+FMY(I,J)*TB (1+1,J)+A2*FMX(I,J)*TB 1 (I,J+1)+FMY(l,J)*BMY*TA+(1.0-FMX(I,J)-FMY(I,J)-A2*FMX(I 2 -BMY*FMY(I,J))*TB(I,J) C C NODE 21 J=6 DO 350 1=2,24 350 T(I,J)=FMX(I,J)*TB(I,J+1)+FMY(I,J)*(TB(I-1,J)+TB(I+1,J)) 1 +A2*FMX(I,J)*TB(I,J-1) 2 +(1.0-2.0*FMY(I,J)-(1.0+A2)*FMX(I,J))*TB(I,J) C C NODE 22 J=20 DO 355 1=2,20 355 T(I,J)=FMX(I,J)*TB(I,J-1)+FMY(I,J)*(TB(I-1,J)+TB(I+1,J)) 1 +A2*FMX(I,J)*TB(I,J+1) 123 2 +(1.0-2.0*FMY(l,J)-(1.0+A2)*FMX(l,J))*TB(I,J) C J=MM5 DO 360 1=22,24 360 T(I , J)=FMX(I ,J)*TB(I , J-1 )+FMY(l , J)*(TB(I-1 , J)+TB(I + 1 , J.) ) 1 +A2*FMX(I,J)*TB(I,J+1)-2 +(1.0-2.0*FMY(l,J)-(1.0+A2)*FMX(I,J))*TB(I,J) C C NODE 23 1=21 DO 370 J=21,MM6 370 T(I,J)=FMY(I,J)*TB(I+1,J)+FMX(I,J)*(TB(I,J-1)+TB(l,J+1)) 1 +A2*FMY(I,J)*TB(1-1,J) 2 +(1,0-2.0*FMX(l,J)-(1.0+A2)*FMY(I,J))*TB(I,J) C C NODE 24 1=21 J=MM5 380 T(I,J)=FMX(I,J)*TB(I,J -1)+FMY(I,J)*TB ( 1+1,J) 1 +A2*FMX(I,J)*TB(I,J+1)+A2*FMY(l,J)*TB(I-1,J) 2 +(1.0-(1.0+A2)*(FMX(l,J)+FMY(I,J)))*TB(I,J) C C NODE 25 1=25 J = 6 390 T(I,j)=FMX(l,J)*TB(I,J+1)+FMY(l,J)*TB(I-1,J) 1 +A2*FMX(I,J)*TB(I,J-1)+A2*FMY(l,J)*TB(I+1,J) 2 +(1.0-(1.0+A2)*(FMX(I,J)+FMY(I,J)))*TB(I,J) C C NODE 26 1 = 25 DO 400 J=7,MM6 400 T(I,J)=FMX(I,J)*(TB(I,J-1)+TB(l,J+1))+FMY(I,J)*TB(I- 1,J) 1 +A2*FMY(I,J)*TB(I+1,J) 2 +(1.0-2.0*FMX(l,J)-(1.0+A2)*FMY(l,J))*TB(I,J) C C NODE 27 1=25 J=MM5 410 T(I,J)=FMX(I,J)*TB(I,J-1)+FMY(I,J)*TB(1-1,J) 1 +A2*FMX(I,J)*TB(I,J+1)+A2*FMY(l,J)*TB(I+1,J) 2 +(1.0-(1.0+A2)*(FMX(I,J)+FMY(I,J)))*TB(I,J) C RETURN END C C c c Q **************************************** c SUBROUTINE TMSOL(TL,TS,MM5,T,TB,TM,TMS, TM98,TMF) C C CALCULATION OF SOLIDN. TIMES 124 C SYMBOLS : TMS=START AT TL TM7=70% AT T7 C TM9=90% AT T9 TMF=COMPLETE AT TS C TM95=95% AT T95 TM98=98% AT T98 C TM99=99% AT T99 C Q ************* * *********** *'jk ******************************* c c DIMENSION T(100.100),TB(100,100),TMS(100,100), 1 TMF(100,100),TM98(100,100) C T98=TL-0.98*(TL-TS) C DO 60 J=21,MM5 DO 50 1=21,25 C IF(T(I,J).GT.TL.OR.TB(I,J).LE.TL) GO TO 34 TMS(I,J)=TM 34 IF(T(I,J).GT.T98.OR.TB(I,J).LE.T98) GO TO 40 TM98(I,J)=TM 40 IF(T(I,J).GT.TS.OR.TB(I,J).LE.TS) GO TO 50 TMF(I,J)=TM 50 CONTINUE 60 CONTINUE C RETURN END C C c c Q *********************************************************** c SUBROUTINE TGR(MM5,T,TB,TS,DX,DY,GR) C C CALC. OF TEMP. GRADIENTS(GR) AT THE END OF SOLIDN. C GR IS DEFINED AS THE MAXIMUM POSITIVE VALUE AMONG THE GR C FROM THE CENTER NODE TO THE 8 SURROUNDING NODES. C Q *********************************************************** C c DIMENSION T(100,100),TB(100,100),GR(100,100) C DO 20 J=21,MM5 DO 10 1=22,24 C IF(T(I,J).GT.TS.OR.TB(I,J).LE.TS) GO TO 10 G1 =(T(I-1,J-1)-T(l,J))/SQRT(DX**2+DY**2) G2 =(T(I,J-1)-T(l,J))/DX G3 =(T(I+1,J-1)-T(l,J))/SQRT(DX**2+DY**2) G4 =(T(I-1,J)-T(I,J))/DY G5 =(T(I+1,J)-T(I,J))/DY G6 =(T(I-1,J+1)-T(l,J))/SQRT(DX**2+DY**2) 125 G7 =(T(I,J+1)-T(l,J))/DX G8 =(T(I+1,J+1)-T(l,J))/SQRT(DX**2+DY**2) 5 GR(I,J)=AMAX1(G1,G2,G3,G4,G5,G6,G7,G8) 10 CONTINUE 20 CONTINUE RETURN " END C O N C E N T R A T I O N • (.WTXCU) Figure 40 - Comparison of the inverse segregation at the c h i l l face for the Al-Cu a l l o y s Figure 41 128 Figure 42 - Comparison of the inverse segregation at the c h i l l face f o r the Sb-Bi a l l o y s © © © © RISER © ® © © ® GREEN SAND MOLD ® CASTING Figure 43 

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