Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Solidification in the mold of a continuous billet caster Prasad, Bommaraju V. S. S. Rama 1984

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1984_A7 P73.pdf [ 25.7MB ]
Metadata
JSON: 831-1.0078963.json
JSON-LD: 831-1.0078963-ld.json
RDF/XML (Pretty): 831-1.0078963-rdf.xml
RDF/JSON: 831-1.0078963-rdf.json
Turtle: 831-1.0078963-turtle.txt
N-Triples: 831-1.0078963-rdf-ntriples.txt
Original Record: 831-1.0078963-source.json
Full Text
831-1.0078963-fulltext.txt
Citation
831-1.0078963.ris

Full Text

SOLIDIFICATION IN THE MOLD OF A CONTINUOUS BILLET CASTER B.Tech., Regional Engineering C o l l e g e , Warangal, I n d i a , 1975. M.Tech., Banares Hindu U n i v e r s i t y , V a r a n a s i , I n d i a 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES Department Of M e t a l l u r g i c a l Engineering We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March 1984 © Bommaraju V.S.S.Rama Prasad, 1984 by BOMMARAJU V.S.S.RAMA PRASAD MASTER OF APPLIED SCIENCE i n 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 o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t 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 r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r 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 t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f HoJ^nljCJS- f W y ^ o / v x ^ The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 3 6 | K ^ Q C |<?H DE-6 (3/81) i i A b s t r a c t B i l l e t s a mples o b t a i n e d from a l a r g e number of h e a t s were examined t o s t u d y d i f f e r e n t a s p e c t s o f s o l i d i f i c a t i o n i n t h e mould; c o l u m n a r - e q u i a x e d t r a n s i t i o n , s e g r e g a t i o n bands, o f f -c o r n e r c r a c k s , r h o m b o i d i t y , s u b - s u r f a c e s t r u c t u r e and o s c i l l a t i o n marks. W i t h t h e a i d of two m a t h e m a t i c a l models th e i n f l u e n c e of h e a t e x t r a c t i o n on e a r l y s o l i d i f i c a t i o n and b i l l e t q u a l i t y has been e l u c i d a t e d . In t h e c a r b o n r a n g e 0.13-0.36%, i n c r e a s i n g c a r b o n and p h o s p h o r o u s (0.020-0.060%) were seen t o c a u s e an e a r l y c o l u m n a r -e q u i a x e d t r a n s i t i o n . H i g h e r h e a t t r a n s f e r r a t e s as c a r b o n i s i n c r e a s e d from 0.13 t o 0.36% were seen t o promote t h e g r o w th of e q u i a x e d z o n e . A s t e e p r i s e i n t h e l e n g t h of c o l u m n a r zone i n s t e e l s w i t h c a r b o n c o n t e n t more t h a n 0.38% was o b s e r v e d f o r t h e f i r s t t i m e i n b i l l e t c a s t i n g i n s p i t e of h i g h h e a t t r a n s f e r r a t e s . E x a m i n a t i o n of m a c r o e t c h e s r e v e a l e d t h a t t h e r e e x i s t two d i s t i n c t bands w i t h i n 10-11. mm o f f t h e edge o f t h e b i l l e t s . T h e se were f o u n d t o d e l i n e a t e t h e s h e l l p r o f i l e a t an i n s t a n t of t i m e . W i t h t h e h e l p of t h e o n e - d i m e n s i o n a l model i t was shown t h a t t h e w h i t e band w h i c h i s c l o s e r t o t h e s u r f a c e forms 450-550 mm below t h e m e n i s c u s where t h e s o l i d i f y i n g s h e l l i s s u b j e c t e d t o a c o n s t a n t c o o l i n g r a t e and t h e s o l i d i f i c a t i o n f r o n t p r o c e e d s a t a m i n i m a l s p e e d . The s h e l l t h i c k n e s s c o r r e s p o n d i n g t o t h e s e c o n d band w h i c h was d a r k or d e e p l y e t c h e d was f o u n d t o have d e v e l o p e d a t t h e m i d f a c e v e r y c l o s e t o t h e b o t t o m o f t h e mould. B a s e d on t h e p o o l p r o f i l e e x h i b i t e d by t h e s e bands i n t h e t r a n s v e r s e s e c t i o n s i t has been c o n c l u d e d t h a t w h i l e s o l i d i f i c a t i o n p r o c e e d s s l o w l y a t t h e m i d f a c e a f t e r t h e f o r m a t i o n o f t h e w h i t e band 450-550 mm below t h e m e n i s c u s i t a p p e a r s t o have v i r t u a l l y s t o p p e d a t t h e o f f - c o r n e r / c o r n e r a r e a . T h i n w h i t e bands a t t h e o b t u s e a n g l e c o r n e r s of t h e b i l l e t s w h i c h a r e a r e s u l t of i m p r o p e r h e a t e x t r a c t i o n c o u l d l e a d t o r h o m b o i d b i l l e t s i n t h e sub-mould r e g i o n s b e c a u s e of d i f f e r e n t i a l c o n t r a c t i o n of a d j a c e n t f a c e s . S t u d y of t h e s e bands has shown t h a t i n t h e m a j o r i t y o f t h e b i l l e t s h e a t t r a n s f e r on t h e f o u r f a c e s i s not i d e n t i c a l i n d i c a t i n g t h a t t h e mould b i l l e t gap i s d i f f e r e n t on t h e f o u r s i d e s . When one of t h e mould w a l l s was c o n s t r a i n e d , r h o m o i d i t y was f o u n d t o be minimum showing t h a t w a l l movement g i v e s r i s e t o d i f f e r e n t c o o l i n g c a p a c i t i e s o f c u r v e d and s t r a i g h t w a l l s . O f f - c o r n e r c r a c k i n g was s e e n t o be a g g r a v a t e d by mould w a t e r v e l o c i t i e s of 6.5-7.2 m/s and Mn/S r a t i o s l o w e r t h a n 22-25. Absence of s e c o n d a r y c o o l i n g water was f o u n d t o a c c e n t u a t e c r a c k i n g a t t h e o f f - c o r n e r s . I t was s u g g e s t e d t h a t t h e s e c r a c k s would form when t h e m i d f a c e b u l g e s i n t h e l o w e r p a r t s o f t h e mould o r i m m e d i a t e l y a f t e r t h e b i l l e t ' s e x i t from t h e mould w h e r e i n a c c o r d i n g t o t h e o n e - d i m e n s i o n a l u n s t e a d y s t a t e model t h e s u r f a c e was f o u n d t o u n d e r g o e x t e n s i v e amount of r e h e a t i n g . A t w o - d i m e n s i o n a l u n s t e a d y - s t a t e m a t h e m a t i c a l model was d e v e l o p e d t o s t u d y t h e p o s s i b i l i t y of m e n i s c u s s o l i d i f i c a t i o n o b s e r v e d i n t h e t h e s t r u c t u r e a r o u n d t h e o s c i l l a t i o n marks on t h e b i l l e t s u r f a c e s . I t was f o u n d t h a t t h e h e a t f l u x n e c e s s a r y i v t o a l l o w growth o f s o l i d o v e r t h e m e n i s c u s has t o be much l a r g e r t h a n t h o s e o b t a i n e d f r o m e x p e r i m e n t s c o n d u c t e d i n a s e p a r a t e s t u d y . A new mechanism has been f o r m u l a t e d t o e x p l a i n m e n i s c u s s o l i d i f i c a t i o n and t h e f o r m a t i o n o f o s c i l l a t i o n marks i n b i l l e t c a s t i n g . The n e g a t i v e t a p e r i n b i l l e t moulds c o n s t r a i n e d o n l y a t t h e t o p was t h o u g h t t o be t h e p o t e n t i a l c a u s e o f t h e s e p e r i o d i c d e p r e s s i o n s . V T a b l e o f C o n t e n t s A b s t r a c t i i L i s t of T a b l e s v i i L i s t of F i g u r e s v i i i Acknowledgement x i C h a p t e r I INTRODUCTION 1 C h a p t e r II LITERATURE SURVEY 4 2.1 G e n e r a l D e s c r i p t i o n o f B i l l e t M o u l d s 4 2.2 B i l l e t M o u l d - h e a t T r a n s f e r 8 2.2.1 I n f l u e n c e of S t e e l C o m p o s i t i o n on Heat T r a n s f e r ..9 2.2.2 I n f l u e n c e of O p e r a t i n g V a r i a b l e s on Heat T r a n s f e r 13 2.2.3 I n f l u e n c e of Mou l d D e s i g n on Heat T r a n s f e r 14 2.2.4 I n f l u e n c e Of Mou l d B e h a v i o u r On Heat T r a n s f e r ...14 2.3 M o u l d - r e l a t e d D e f e c t s 17 2.3.1 R h o m b o i d i t y 17 2.3.2 C r a c k F o r m a t i o n 19 2.4 O s c i l l a t i o n Marks 23 2.5 Scope o f t h e P r e s e n t S t u d y 27 C h a p t e r I I I EXPERIMENTAL PROCEDURES 31 3.1 Heat F l u x D a t a G e n e r a t i o n 32 3.2 C o l l e c t i o n of B i l l e t Samples 35 3.3 S u l p h u r P r i n t i n g and C r a c k R a t i n g 35 3.4 M a c r o e t c h i n g 37 3.5 S u r f a c e and S u b - s u r f a c e F e a t u r e s 37 3.6 Measurement o f R h o m b o i d i t y 39 C h a p t e r IV MATHEMATICAL MODELS 40 4.1 O n e - D i m e n s i o n a l Heat T r a n s f e r Model 40 4.1.1 F o r m u l a t i o n 40 4.1.2 D e r i v a t i o n of t h e F i n i t e D i f f e r e n c e E q u a t i o n s ...42 4.1.3 C h a r a c t e r i s a t i o n of I n p u t C o n d i t i o n s 43 4.2 M e n i s c u s S o l i d i f i c a t i o n Model 44 4.2.1 F o r m u l a t i o n 46 4.2.2 C h a r a c t e r i s a t i o n Of I n p u t C o n d i t i o n s 48 C h a p t e r V RESULTS 53 5.1 M e t a l l o g r a p h i c E x a m i n a t i o n 53 5.1.1 Columnar Zone L e n g t h 53 5.1.2 S u b s u r f a c e S t r u c t u r e 59 5.2 Band F o r m a t i o n 71 5.3 C r a c k F o r m a t i o n 82 5.4 R h o m b o i d i t y 94 5.5 O s c i l l a t i o n Marks 94 5.6 Model P r e d i c t i o n s 105 5.6.1 S o l i d i f i c a t i o n o f B i l l e t S e c t i o n 105 5.6.2 M e n i s c u s S o l i d i f i c a t i o n In B i l l e t C a s t i n g 117 C h a p t e r VI DISCUSSION 125 6.1 M a c r o s t r u c t u r e Of B i l l e t S e c t i o n s 125 6.2 Band F o r m a t i o n 128 6.3 C r a c k F o r m a t i o n 130 6.4 R h o m b o i d i t y 133 6.5 O s c i l l a t i o n Marks 135 C h a p t e r V I I SUMMARY AND CONCLUSIONS 142 BIBLIOGRAPHY 151 APPENDIX A - TABLES 155 APPENDIX B - DERIVATION OF FINITE-DIFFERENCE EQUATIONS FOR ONE-DIMENSIONAL UNSTEADY STATE HEAT TRANSFER MODEL FOR BILLETS 178 APPENDIX C - ONE DIMENSIONAL B I L L E T CASTING HEAT TRANSFER MODEL IMPLICIT F I N I T E DIFFERENCE METHOD 182 APPENDIX D - TWO-DIMENSIONAL UN-STEADY STATE MENISCUS SOLIDIFICATION MODEL .' 201 v i i L i s t of T a b l e s I . C h a r a c t e r i s t i c s of t h e mold a t W e s t e r n Canada S t e e l 155 I I . S t e e l C o m p o s i t i o n 156 I I I . Columnar zone l e n g t h and o t h e r d e t a i l s 158 IV. S u b - s t r u c t u r e d e t a i l s 163 V. L o c a t i o n of s o l i d i f i c a t i o n bands 164 V I . L o c a t i o n of o f f - c o r n e r c r a c k s and t h e i r number i n e a c h b i l l e t sample 167 V I I . S e v e r i t y o f o f f - c o r n e r c r a c k s 168 V I I I . R h o m b o i d i t y d a t a 170 IX. V i s u a l e s t i m a t i o n of o s c i l l a t i o n marks 173 X. D e p t h of o s c i l l a t i o n marks m e a s u r e d by p r o f i l o m e t e r 174 X I . E x i t s h e l l t h i c k n e s s f o r h i g h c a r b o n s t e e l 176 X I I . E x i t s h e l l t h i c k n e s s f o r low c a r b o n s t e e l 177 v i i i L i s t o f F i g u r e s 1. Mould r e l a t e d d e f e c t s i n B i l l e t c a s t i n g 3 2. T y p i c a l mould a s s e m b l y f o r a b i l l e t c a s t e r 5 3. T y p i c a l h e a t f l u x p r o f i l e i n b i l l e t moulds 10 4. P l o t o f c o n t i n u o u s c a s t i n g d a t a o f o v e r - a l l h e a t f l u x i n t h e mould a g a i n s t c a r b o n c o n t e n t of t h e s t e e l . A f t e r S i n g h and B a l z e k 1 and G r i l l and B r i m a c o m b e 1 3 11 5. T y p i c a l mould d i s t o r t i o n p r o f i l e of b i l l e t moulds w i t h c o n s t r a i n t s on two s t r a i g h t s i d e s 16 6. Mechanism o f c r a c k f o r m a t i o n a t o f f - c o r n e r s 2 2 21 7. A r r a n g e m e n t of t h e r m o c o u p l e s i n mould w a l l 33 8. T h e r m o c o u p l e d e s i g n 34 9. T r a n s v e r s e and l o n g i t u d i n a l s e c t i o n s of t h e b i l l e t sample 36 10. A r r a n g e m e n t of nodes i n t h e m e n i s c u s s o l i d i f i c a t i o n model 45 11. O b s e r v e d and c a l c u l a t e d m e n i s c u s p r o f i l e a f t e r Tomono e t a l 6 9 49 12. P l o t o f c a r b o n v e r s u s t h e l e n g t h of c o l u m n a r zone (measured from t h e o u t s i d e r a d i u s ) 54 13. P l o t o f p h o s p h o r o u s v e r s u s t h e l e n g t h of c o l u m n a r zone (measured from t h e o u t s i d e r a d i u s ) i n s t e e l s w i t h c a r b o n r a n g i n g f r o m 0.13-0.20 56 14. P l o t o f p h o s p h o r o u s v e r s u s t h e l e n g t h of c o l u m n a r zone (measured from t h e o u t s i d e r a d i u s ) i n s t e e l s w i t h c a r b o n r a n g i n g f r o m 0.28-0.36 57 15. P l o t o f s u p e r h e a t i n t h e t u n d i s h v e r s u s t h e l e n g t h of c o l u m n a r zone (measured from t h e o u t s i d e r a d i u s ) i n s t e e l s w i t h c a r b o n r a n g i n g f r o m 0.28-0.36 58 16. P l o t o f a v e r a g e mould h e a t f l u x v e r s u s l e n g t h of c o l u m n a r zone (measured from t h e o u t s i d e r a d i u s ) 60 17. S t r u c t u r e a r o u n d o s c i l l a t i o n mark of t h e s e c o n d b i l l e t f r o m h e a t 24276. (1 1X) 61 i x 18. S t r u c t u r e a r o u n d o s c i l l a t i o n mark of t h e l a s t b i l l e t from h e a t 24276. ( 1 1X) 62 19. S t r u c t u r e a r o u n d o s c i l l a t i o n mark of t h e f i r s t b i l l e t f rom h e a t 24277. ( 1 1X) ..63 20. S t r u c t u r e a r o u n d o s c i l l a t i o n mark of t h e l a s t b i l l e t f rom h e a t 24277. ( 1 1X) 64 21. S t r u c t u r e a r o u n d o s c i l l a t i o n mark of t h e s e c o n d b i l l e t from h e a t 24272; o f f - c o r n e r r e g i o n on t h e l e f t and m i d f a c e r e g i o n on t h e r i g h t . (11X) 65 22. S t r u c t u r e a r o u n d o s c i l l a t i o n marks i n t h e l o n g i t u d i n a l s e c t i o n of t h e l a s t b i l l e t from h e a t 24269. (2.9X) ...67 23. C l o s e up of t h e s t r u c t u r e a r o u n d o s c i l l a t i o n mark o f t h e l a s t b i l l e t from h e a t 24269. (1 1X) 68 24. S t r u c t u r e a r o u n d o s c i l l a t i o n marks i n t h e l o n g i t u d i n a l s e c t i o n s from t h e h e a t 24276; from l e f t t o r i g h t . F i r s t b i l l e t , s e c o n d b i l l e t and l a s t b i l l e t . (2.9X) 69 25. C l o s e - u p o f t h e s t r u c t u r e a r o u n d o s c i l l a t i o n mark o f t h e s e c o n d b i l l e t from h e a t 24276. ( 17X) 70 26. W h i t e band i n t h e o b t u s e a n g l e c o r n e r of t h e s e c o n d b i l l e t f r o m h e a t 24259. (2.5X) 72 27. White and d a r k bands a t t h e edge o f t h e l a s t b i l l e t from h e a t 24242 73 28. T h i n and wavy w h i t e band a t t h e m i d f a c e of t h e s e c o n d b i l l e t f r o m h e a t 24251 74 29. P r o m i n e n t d a r k band i n t h e s e c o n d b i l l e t f r o m h e a t 24221 76 30. Dark band i n t h e l a s t b i l l e t f r o m h e a t 24490 77 31. Dark band m e r g i n g i n t o w h i t e band a t t h e o f f - c o r n e r r e g i o n ; l a s t b i l l e t f r o m h e a t 24251 78 32. Dark band m e r g i n g i n t o w h i t e band a t t h e o f f - c o r n e r r e g i o n ; s e c o n d b i l l e t from h e a t 23471 79 33. M a c r o - e t c h of t h e t r a n s v e r s e s e c t i o n of t h e s e c o n d b i l l e t f r o m h e a t 24249 81 34. O f f - c o r n e r c r a c k s f r o m t h e h e a t 24107 83 35. L o c a t i o n o f o f f - c o r n e r c r a c k s and w h i t e band a t t h e o b t u s e a n g l e c o r n e r of s e c o n d b i l l e t 24277 84 36. P l o t o f water v e l o c i t y v e r s u s t h e number o f o f f - c o r n e r c r a c k s p e r b i l l e t . C a r bon r a n g e 0.13-0.41 85 37. P l o t of water v e l o c i t y v e r s u s t h e number of o f f - c o r n e r c r a c k s p e r b i l l e t . C a r b o n r a n g e 0.13-0.20 86 38. P l o t o f water v e l o c i t y v e r s u s t h e number of o f f - c o r n e r c r a c k s p e r b i l l e t . C a r b o n r a n g e 0.28-0.41 87 39. P l o t o f Mn/S v e r s u s t h e s e v e r i t y of o f f - c o r n e r c r a c k s water v e l o c i t y r a n g e 6.15-7.2 m/s 88 40. P l o t o f Mn/S v e r s u s t h e s e v e r i t y of o f f - c o r n e r c r a c k s w ater v e l o c i t y <6 m/s 89 41. P l o t of Mn/S v e r s u s t h e s e v e r i t y of o f f - c o r n e r c r a c k s w ater v e l o c i t y >8.65 m/s 90 42. O f f - c o r n e r c r a c k s i n t h e h e a t 24109 92 43. M o l d d i s t o r t i o n a f t e r Campaign 12 93 44. C o n c a v i t y i n t h e s t r a i g h t f a c e i n t h e s e c o n d b i l l e t f r o m 24090 95 45. B i l l e t s u r f a c e from t h e h e a t 24270 (0.13%C, l a s t b i l l e t ) c a s t i n g d i r e c t i o n down 97 46. B i l l e t s u r f a c e from t h e h e a t 24271 (0.20%C, s e c o n d b i l l e t ) c a s t i n g d i r e c t i o n down 98 47. B i l l e t s u r f a c e from t h e h e a t 24276 (0.30%C, f i r s t b i l l e t ) c a s t i n g d i r e c t i o n down 99 48. B i l l e t s u r f a c e from t h e h e a t 24265 (0.35%C, s e c o n d b i l l e t ) c a s t i n g d i r e c t i o n down 100 49. B i l l e t s u r f a c e from t h e h e a t 24273 (0.41%C, t h i r d b i l l e t ) c a s t i n g d i r e c t i o n down 101 50. P l o t of c a r b o n v e r s u s v i s u a l e s t i m a t i o n of t h e d e p t h o f o s c i l l a t i o n marks ...102 51. P l o t o f p h o s p h o r o u s v e r s u s d e p t h o f o s c i l l a t i o n marks, wa t e r v e l o c i t y >8.65 m/s 104 52. Heat f l u x p r o f i l e f o r low c a r b o n s t e e l , water v e l o c i t y 9.7 m/s 1 06 53. Heat f l u x p r o f i l e f o r low c a r b o n s t e e l , w a ter v e l o c i t y 6.85 m/s 107 54. Heat f l u x p r o f i l e f o r low c a r b o n s t e e l , w a t e r v e l o c i t y 5.55 m/s 108 x i 55. Heat f l u x p r o f i l e f o r h i g h c a r b o n s t e e l , water v e l o c i t y 9.7 m/s 1 09 56. Heat f l u x p r o f i l e f o r h i g h c a r b o n s t e e l , water v e l o c i t y 7.2 m/s 110 57. Heat f l u x p r o f i l e f o r h i g h c a r b o n s t e e l , w a ter v e l o c i t y 5. 55 m/s 111 58. L o c a t i o n o f d a r k band and e x i t s h e l l t h i c k n e s s i n h i g h c a r b o n s t e e l h e a t s 113 59. C o o l i n g r a t e s of t h r e e d i f f e r e n t nodes down t h e mould, h i g h c a r b o n s t e e l . Water v e l o c i t y 9.7 m/s 114 60. C o o l i n g r a t e s o f t h r e e d i f f e r e n t nodes down t h e mould, h i g h c a r b o n s t e e l . Water v e l o c i t y 7.2 m/s 115 61. C o o l i n g r a t e s o f t h r e e d i f f e r e n t nodes down t h e mould, h i g h c a r b o n s t e e l . Water v e l o c i t y 5.55 m/s. 116 62. C o o l i n g r a t e s o f t h r e e d i f f e r e n t nodes down t h e mould, low c a r b o n s t e e l . Water v e l o c i t y 9.7 m/s 118 63. C o o l i n g r a t e s of t h r e e d i f f e r e n t nodes down t h e mould, low c a r b o n s t e e l . Water v e l o c i t y 6.85 m/s 119 64. C o o l i n g r a t e s o f t h r e e d i f f e r e n t nodes down t h e mould, low c a r b o n s t e e l . Water v e l o c i t y 5.55 m/s .120 65. M o d e l p r e d i c t e d s o l i d i f i c a t i o n a t t h e m e n i s c u s - l o w c a r b o n s t e e l 122 66. Model p r e d i c t e d s o l i d i f i c a t i o n a t t h e m e n i s c u s - h i g h c a r b o n s t e e l 123 67. S c h e m a t i c d i a g r a m o f t h e mechanism o f f o r m a t i o n o f o s c i l l a t i o n marks i n b i l l e t c a s t i n g 141 x i i Acknowledgement I would l i k e t o thank Dr.J.K.Brimacombe and D r . I . V . S a m a r a s e k e r a f o r t h e i r s u p e r v i s i o n and encouragement t h r o u g h o u t t h e c o u r s e of t h i s p r o j e c t . My t h a n k s a r e due t o D r . F . W i e n b e r g , head of t h e Department 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 , f o r h i s many v a l u a b l e s u g g e s t i o n s and f o r p r o v i d i n g t h e l a b o r a t o r y and workshop f a c i l i t i e s . I am g r a t e f u l t o t h e W e s t e r n Canada S t e e l Company f o r s u p p l y i n g t h e b i l l e t s a m p l e s . I would l i k e t o e x t e n d a p e r s o n a l n o t e of t h a n k s t o Mr.Ian B a k s h i f o r h i s h e l p . The a s s i s t a n c e r e n d e r e d by M r . N e i l W a l k e r i s g r a t e f u l l y a c k n o w l e d g e d . 1 I . INTRODUCTION As an a l t e r n a t i v e t o t h e c o n v e n t i o n a l i n g o t c a s t i n g - s o a k i n g p i t - b l o o m i n g m i l l r o u t e , c o n t i n o u s c a s t i n g has emerged as a f a r more e c o n o m i c a l and e f f i c i e n t p r o c e s s t o p r o d u c e s e m i - f i n i s h e d s h a p e s from l i q u i d s t e e l . I t o f f e r s i m p r oved p r o d u c t q u a l i t y and h o m o g e n e i t y and i s b e i n g e x t e n s i v e l y employed by b o t h t h e m i n i - m i l l s and t h e major i n t e g r a t e d s t e e l works t o c a s t b i l l e t s , blooms and s l a b s . C o n t i n o u s c a s t i n g y i e l d s g r e a t e s t economic b e n e f i t s when s e c t i o n s i z e s c l o s e s t t o t h e shape of f i n i s h e d p r o d u c t a r e c a s t , e l i m i n a t i n g t h e need f o r s e v e r a l p r o c e s s i n g and r e d u c t i o n s t a g e s . F o r r o d and b a r p r o d u c t i o n , b i l l e t c a s t i n g w i t h w h i c h t h i s t h e s i s i s c o n c e r n e d , i s g a i n i n g momentum be y o n d i t s a l r e a d y w i d e s p r e a d use i n m i n i - m i l l s . But l o w e r r e d u c t i o n r a t i o s , h i g h e r c a s t i n g s p e e d s and t h e r e c e n t a p p l i c a t i o n o f i n - l i n e r e d u c t i o n have b r o u g h t a b o u t s t r i n g e n t q u a l i t y r e q u i r e m e n t s i n b i l l e t c a s t i n g . V a r i o u s machine d e s i g n c r i t e r i a and o p e r a t i n g p r a c t i c e s a r e u n d e r s c r u t i n y t o o b t a i n a b e t t e r u n d e r s t a n d i n g of b i l l e t c a s t i n g and t h e p r o b l e m s i n v o l v e d . The h e a r t o f t h e c o n t i n u o u s c a s t i n g i n s t a l l a t i o n i s t h e w a t e r - c o o l e d c o p p e r mould w h i c h g i v e s shape t o t h e l i q u i d s t e e l i s s u i n g f r o m t h e l a d l e t h r o u g h an i n t e r m e d i a t e v e s s e l , t h e t u n d i s h . R a p i d h e a t e x t r a c t i o n i n t h e mould and e a s y w i t h d r a w a l f r om i t , f a c i l i t a t e d by o s c i l l a t i o n o f t h e mould c o u p l e d w i t h l u b r i c a t i o n , a r e f e a t u r e s t h a t make t h e c o n t i n u o u s c a s t i n g p r o c e s s p h y s i c a l l y v i a b l e . A 9-12mm s h e l l d e v e l o p s i n t h e 2 mould; growth o f t h e s h e l l i s d e p e n d e n t on h e a t f l o w a c r o s s t h e gap t o t h e mould w a l l w h i c h i n t u r n i s a f f e c t e d by s t e e l c o m p o s i t i o n , 1 mould d i s t o r t i o n 2 , e t c . C h a r a c t e r i s t i c s of t h e p r i m a r y s h e l l formed have a p r o f o u n d i n f l u e n c e on t h e q u a l i t y o f t h e b i l l e t s c a s t : m o u l d - r e l a t e d d e f e c t s i n c l u d e , r h o m b o i d i t y , l o n g i t u d i n a l c o r n e r c r a c k s , m i d f a c e c r a c k s as shown i n F i g 1. An a t t e m p t i s made i n t h e p r e s e n t s t u d y t o u n d e r s t a n d t h e n a t u r e of some of t h e m o u l d - r e l a t e d p r o b l e m s v i z , o f f - c o r n e r c r a c k s , and r h o m b o i d i t y and t h e mechanisms i n v o l v e d . E m p h a sis i s p l a c e d on t h e r e l a t i o n s h i p between h e a t t r a n s f e r i n t h e mould and t h e q u a l i t y of t h e b i l l e t s . M ould h e a t f l u x d a t a o b t a i n e d from i n - p l a n t mould t e m p e r a t u r e measurements, t o g e t h e r w i t h b i l l e t s a mples from c o r r e s p o n d i n g h e a t s a r e u t i l i s e d t o e l u d i c a t e t h e e f f e c t of c a s t i n g v a r i a b l e s s u c h as s t e e l c o m p o s i t i o n , s u p e r h e a t and mould water v e l o c i t y on t h e i n c i d e n c e and s e v e r i t y o f some of t h e mould r e l a t e d d e f e c t s . The p r e s e n t s t u d y has a l s o been aimed a t e x p l a i n i n g some of t h e b a s i c f e a t u r e s o f t h e b i l l e t s c a s t t h r o u g h o s c i l l a t i n g c o p p e r m o u l d s . Mould r e l a t e d phenomena l i k e o s c i l l a t i o n marks and s o l i d i f i c a t i o n band f o r m a t i o n have been a n a l y s e d . A f r e s h l o o k was t a k e n a t t h e m a c r o s t r u c t u r a l f e a t u r e s of a p o p u l a t i o n of b i l l e t s i n o r d e r t o e s t i m a t e t h e i n f l u e n c e o f h e a t t r a n s f e r as w e l l a s o f s t e e l c o m p o s i t i o n . 3 i I 1 Rhomboidity 2 Off-corner cracks (internal) 3 Mid-face cracks 4 Internal corner cracks 5 Longitudinal corner cracks 6 Surface cracks 7 Transverse corner cracks Figure 1 - Mould r e l a t e d defects i n B i l l e t c a s t i n g 4 I I . LITERATURE SURVEY The i n i t i a l s e c o n d s when l i q u i d s t e e l p a s s e s t h r o u g h t h e b i l l e t mould and a c q u i r e s a t h i n s h e l l a r e of g r e a t i m p o r t a n c e , b e c a u s e i t i s d u r i n g t h i s t i m e t h a t c r a c k s o f t h e o f f - c o r n e r , m i d f a c e , d i a g o n a l and l o n g i t u d i n a l c o r n e r t y p e i n i t i a t e ( F i g 1 ) . A l s o , as t h e mould i s o s c i l l a t e d s i n u s o i d a l l y t o f a c i l i t a t e w i t h d r a w a l of t h e b i l l e t d e p r e s s i o n s on t h e s u r f a c e a r e formed p e r i o d i c a l l y . B e f o r e t h e s t u d y of t h e r e l a t i o n between t h e s e phenomena and h e a t t r a n s f e r i n t h e mould i s p r e s e n t e d , i t i s c o n s i d e r e d a p t t o g i v e a b r i e f d e s c r i p t i o n o f t h e b i l l e t moulds and t h e h e a t t r a n s p o r t p r o c e s s e s i n v o l v e d a l o n g w i t h t h e v a r i a b l e s w h i c h a r e known t o a f f e c t t h e l a t t e r i n a s i g n i f i c a n t way. A r e v i e w of t h e l i t e r a t u r e p u b l i s h e d r e g a r d i n g t h e mould-r e l a t e d d e f e c t s and o s c i l l a t i o n marks i s p r e s e n t e d t h e r e a f t e r . 2.1 G e n e r a l D e s c r i p t i o n o f B i l l e t M o u l d s B i l l e t moulds a r e made up of 9-12mm, t h i c k c o p p e r t u b e , u s u a l l y o f s q u a r e c r o s s - s e c t i o n w i t h round c o r n e r s . F i g 2 shows a t y p i c a l b i l l e t mould a s s e m b l y . The mould t u b e i s h e l d i n a s t e e l l i n e r and i s c o n s t r a i n e d e i t h e r a t b o t h ends o r o n l y a t t h e t o p . C o n s t r a i n t s may be on a l l f o u r s i d e s o r o n l y on two o p p o s i t e f a c e s . C o o l i n g water f l o w s n o r m a l l y i n an upward d i r e c t i o n , t h r o u g h t h e a n n u l u s between t h e t u b e and t h e l i n e r . A back p r e s s u r e of a b o u t 240KPa i s a l s o a p p l i e d i n c o m b i n a t i o n 5 1 Mould 2 Steel jocket 3 Housing 4 Support plote 5 Lubricotor plote 6 Cover plote 7 Water channel Figure 2 - T y p i c a l mould assembly fo r a b i l l e t c a s t e r 6 w i t h t h e upward f l o w t o e n s u r e t h a t t h e a n n u l u s i s f u l l o f w a t e r . An a r r a y of s t e e l s p a c e r s w e l d e d t o t h e mould l i n e r a r e u s e d t o e n s u r e t h a t t h e mould i s mounted c o n c e n t r i c a l l y and t h e wat e r gap i s u n i f o r m a r o u n d t h e p e r i p h e r y of t h e mould. Water v e l o c i t i e s r a n g e from as low as 5m/s t o 14m/s. U s u a l l y 60-80 cm l o n g , b i l l e t moulds a r e t a p e r e d t o compensate f o r s h r i n k a g e o f t h e s o l i d i f y i n g s h e l l . T a p e r s v a r y from a s i n g l e s t r a i g h t t a p e r o f 0.7-0.8%/m t h a t e x t e n d s o v e r t h e e n t i r e mould l e n g t h t o d o u b l e t a p e r s t h a t change down t h e mould. D e o x i d i s e d c o p p e r i s most commonly employed o f t h e mould m a t e r i a l s owing t o i t s h i g h t h e r m a l c o n d u c t i v i t y . In some i n s t a n c e s s i l v e r - b e a r i n g c o p p e r s have been u s e d as w e l l . T h e s e mould c o p p e r s do have a t e n d e n c y t o d i s t o r t and t h i s h as l e d t o t h e d e v e l o p m e n t of a whole new ran g e of h i g h s t r e n g t h low d u c t i l i t y c o p p e r - a l l o y s . The i n s i d e s u r f a c e of t h e moulds i s u s u a l l y chrome p l a t e d t o improve wear r e s i s t a n c e . L u b r i c a t i n g o i l s s u c h as r a p e s e e d o i l and o t h e r v e g e t a b l e o i l s a r e most commonly u s e d i n b i l l e t m o u l d s . 3 O i l i s pumped t o a s p l i t t e r t h a t d i s t r i b u t e s i t t o c h a n n e l s i n an o i l i n g p l a t e f r o m w h i c h i t i s t r a n s m i t t e d u n i f o r m l y a r o u n d t h e mould w a l l . " J 5 The mould i s o s c i l l a t e d t o p r e v e n t s t i c k i n g b e c a u s e e a r l y i n t h e d e v e l o p m e n t of c o n t i n u o u s c a s t i n g t h e r a p i d w i t h d r a w l of t h e i n g o t from t h e f i x e d mould c a u s e d an immediate r u p t u r e o f i n g o t s k i n i n t h e mould l e a d i n g t o a b r e a k o u t . 6 mould o s c i l l a t i o n was i n t r o d u c e d by Junghans 7 ; t h e o s c i l l a t i o n c o n s i s t e d of a d o w n s t r o k e a t a r a t e e q u a l t o t h e s t r a n d w i t h d r a w l s p e e d f o l l o w e d by a h i g h s p e e d upward s t r o k e . But i t 7 was f o u n d t h a t t h e b i l l e t was s t u c k t o t h e mould d u r i n g t h e d o w n s t r o k e and was t o r n on t h e u p s t r o k e . In o r d e r t o e l i m i n a t e t h i s p o t e n t i a l s o u r c e of t r o u b l e BISRA 8 p r o p o s e d a s l i g h t o v e r t a k i n g a c t i o n o f t h e mould r e l a t i v e t o t h e i n g o t s u r f a c e d u r i n g t h e d o w n s t r o k e . T h i s t e c h n i q u e of moving t h e mould downward a t f a s t e r s p e e d s t h a n t h e s t r a n d i s c a l l e d ' N e g a t i v e s t r i p ' . Not o n l y does i t p r e v e n t s t i c k i n g , i n t h e downward s t r o k e , i t i s c l a i m e d t h a t i f s t i c k i n g o c c u r s i t r e l e a s e s t h e a t t a c h e d b i l l e t c o m p r e s s i v e l y . 9 C o m p r e s s i v e f o r c e s e x e r t e d on t h e s o l i d i f y i n g s h e l l seem t o be v e r y b e n e f i c i a l t o t h e smooth o p e r a t i o n of c a s t e r s . T h e s e f o r c e s p r e v e n t f o r m a t i o n of c r a c k s on t h e b i l l e t s u r f a c e i n t h e d o w n s t r o k e . I f any c r a c k s do form by d r a g d u r i n g t h e upward s t r o k e of t h e mould, i t i s c l a i m e d t h e y a r e c l o s e d on t h e d o w n s t r o k e by p r e s s i n g i n t o s t i l l s o f t e r p a r t s of t h e s t r a n d s h e l l . 9 ' 1 0 B a s e d on t h i s p a r t i c u l a r f e a t u r e of n e g a t i v e s t r i p , U . S . S t e e l 1 1 d e v e l o p e d an o s c i l l a t i o n c y c l e t o p r o v i d e a d o w n s t r o k e t i m e t h a t i s g r e a t e r t h a n t h e t i m e r e q u i r e d f o r t h e u p s t r o k e whereby t h e s k i n i s s u b j e c t e d t o t h e d e s i r a b l e c o m p r e s s i v e s t r e s s e s f o r a l o n g e r t i m e t h a n can be o b t a i n e d w i t h r e g u l a r s i n u s o i d a l cams. However, mould o s c i l l a t i o n t h a t f o l l o w s a s i n e c u r v e i s b e i n g e x t e n s i v e l y u s e d i n many of t h e modern i n s t a l l a t i o n s and u s u a l l y t h e m o t i o n i s e l e c t r i c a l l y s y n c h r o n i s e d w i t h t h e c a s t i n g s p e e d . F o r o s c i l l a t i o n u s i n g a s i n e wave p r i n c i p l e , t h e n e g a t i v e s t r i p t i m e , i . e , t h e t i m e f o r w h i c h t h e mould has an o v e r t a k i n g a c t i o n o v e r t h e b i l l e t i n t h e down 8 s t r o k e i s given by the f o l l o w i n g equation. fcn = ^ ^ ^ ( W - ) ] 2.1 where f=freguency of o s c i l l a t i o n s v=casting speed 2a=stroke l e n g t h 2.2 B i l l e t Mould-heat Transfer Although simple i n d e s i g n , the b i l l e t mould i s very complex from the standpoint of heat t r a n s f e r . Heat i s t r a n s f e r r e d from the s t r a n d surface to the mould c o o l i n g water v i a a s e r i e s of thermal r e s i s t a n c e s , v i z ; i . r e s i s t a n c e across the a i r gap between mould and strand where heat flow takes place both by conduction and r a d i a t i o n i i . conductive r e s i s t a n c e through the mould w a l l . i i i . convective r e s i s t a n c e at the mould/cooling water i n t e r f a c e . Using mould w a l l temperature measurements Watanabe 1 2 has found t h a t the a i r gap accounts f o r 84% of the t o t a l r e s i s t a n c e to heat flow. Thus the p a t t e r n of heat removal i n the mould i s l a r g e l y a f u n c t i o n of the dynamics of gap formation. E v o l u t i o n of the gap depends on the shrinkage of the s o l i d i f y i n g s h e l l and 9 i t s a b i l i t y t o w i t h s t a n d f e r r o s t a t i c p r e s s u r e . As t h e b i l l e t makes i t s j o u r n e y t h r o u g h t h e mould, t h e c o r n e r s l o s e c o n t a c t w i t h t h e mould f i r s t owing t o t h e e f f e c t s of t w o - d i m e n s i o n a l h e a t f l o w and t h e h e a t f l u x a t t h e c o r n e r s d r o p s o f f . As t h e s h e l l t h i c k n e s s i n c r e a s e s down t h e mould, t h e i n f l u e n c e of t h e i n c r e a s e d gap i s m a n i f e s t e d as a r e d u c t i o n i n h e a t f l u x . 1 From t h e h e a t f l u x measurements r e p o r t e d i n t h e l i t e r a t u r e , i t i s c l e a r t h a t t h e b u l k of t h e h e a t i s e x t r a c t e d i n t h e upper h a l f of t h e mould where t h e gap i s s m a l l e s t and s t r a n d s u r f a c e i s h o t t e s t . The l o w e r p a r t of t h e mould i s t h e r m a l l y i n e f f i c i e n t s i n c e t h e s h e l l i s s t r o n g enough t o w i t h s t a n d t h e f e r r o s t a t i c p r e s s u r e r e s u l t i n g i n an i n c r e a s e d a i r gap. F i g 3 shows a t y p i c a l a x i a l h e a t f l u x p r o f i l e f r o m b i l l e t m o u l d s . 2.2.1 I n f l u e n c e of S t e e l C o m p o s i t i o n on Heat T r a n s f e r Many of t h e p a r a m e t e r s t h a t i n f l u e n c e h e a t t r a n s f e r i n t h e mould , do so by c h a n g i n g gap f o r m a t i o n on h e a t t r a n s f e r . The w i d t h of t h e gap a t a g i v e n l o c a t i o n i s d e p e n d e n t on t h e s t r e n g t h of t h e s h e l l t o w i t h s t a n d t h e f e r r o s t a t i c p r e s s u r e . The c o m p o s i t i o n and t e m p e r a t u r e o f t h e s t e e l d e t e r m i n e i t s s t r e n g t h and have a s t r o n g i n f l u e n c e on h e a t t r a n s f e r . D a t a from an e x p e r i m e n t a l c o n t i n u o u s c a s t e r by S i n g h and B l a z e k 1 and from i n d u s t r i a l m a c h i n e s by G r i l l and B r i m a c o m b e 1 3 shows a d i s t i n c t minimum i n h e a t f l u x a t 0.1%C, F i g 4. A l s o t h e c o r r e s p o n d i n g s t e e l s h e l l s have r i p p l e d s u r f a c e s compared w i t h t h e smoother s u r f a c e s o b t a i n e d i n h i g h e r c a r b o n s t e e l s . 10 gure 3 - T y p i c a l heat f l u x p r o f i l e i n b i l l e t moulds I I I I I 0 0-3 0-6 0-9 1-2 1-5 Corbon (%) Figure 4 - Plot of continuous casting data of over-all heat flux in the mould against carbon content of the steel. After Singh and Balzek 1 and G r i l l and Brimacombe13 12 The i n c r e a s e d gap w i d t h r e s u l t i n g f r o m s u c h a s u r f a c e seems t o be t h e o b v i o u s c a u s e f o r l o w e r e d h e a t t r a n s f e r r a t e s . As t o t h e mechanism i n v o l v e d i n f o r m a t i o n of s u c h a wavy s h e l l , Wolf and K u r z 1 " a t t r i b u t e i t t o t h e m i c r o s e g r e g a t i o n o f p h o s p h o r o u s ( s u l p h u r i s t a k e n c a r e of by Mn i n s t e e l ) and i t s e f f e c t on t h e m e c h a n i c a l p r o p e r t i e s o f s t e e l . I t was shown t h a t p h o s p h o r o u s s e g r e g a t i o n i s a minimum f o r 0.1% C s t e e l s , and t h i s g i v e s t h e s o l i d i f i e d s h e l l i n t h e c o n t i n u o u s c a s t i n g mould a h i g h e r h o t t e n s i l e s t r e n g t h . T h i s s t r o n g s h e l l , as i t s h r i n k s forms a r o u g h o r r i p p l e d s u r f a c e . The s p a c i o u s F.C.C c r y s t a l l a t t i c e o f a u s t e n i t e i n h i g h e r c a r b o n s t e e l s , on t h e o t h e r hand a l l o w s f o r more s e g r e g a t i o n of P h o s p h o r o u s and t h e r e s u l t i n g weaker s h e l l r e m a i n s i n b e t t e r c o n t a c t w i t h t h e mould w a l l under t h e f e r r o -s t a t i c p r e s s u r e . T h i s l e a d t o u n i f o r m s h e l l g r o w t h . They a l s o have shown t h a t h i g h e r p h o s p h o r o u s l e v e l s i n a 0.12% C s t e e l g i v e r i s e t o i n c r e a s e d h e a t f l u x e s and t h i c k e r s h e l l s . I t s h o u l d , however, be of i n t e r e s t t o see i f s u c h wavy s h e l l s do form i n t h e a b s e n c e of e l e m e n t s l i k e p h o s p h o r o u s w h i c h have a low d i s t r i b u t i o n c o e f f i c i e n t . G r i l l and B r i m a c o m b e 1 3 e x p l a i n e d t h e u n u s u a l s u r f a c e r o u g h n e s s of low c a r b o n s t e e l i n t h e l i g h t o f t h e s h r i n k a g e i n h e r e n t i n t h e 6—>y t r a n s f o r m a t i o n of t h e p e r i t e c t i c r e a c t i o n . These a u t h o r s p o i n t o u t t h a t compared t o h i g h e r c a r b o n s t e e l s , 0.1%C s t e e l u n d e r g o e s g r e a t e r s o l i d s t a t e t r a n s f o r m a t i o n w h i c h i s a c c o m p a n i e d by a c o n t r a c t i o n of 0.38%. Thus 0.1%C s t e e l e x p e r i e n c e s g r e a t e r s h r i n k a g e t h a n h i g h e r c a r b o n s t e e l s and t h e l a r g e a i r gaps t h a t a p p e a r r e p e a t e d l y b e c a u s e o f e n c h a n c e d 13 s h r i n k a g e c a u s e t h e o b s e r v e d low h e a t e x t r a c t i o n r a t e s . R e c e n t i n v e s t i g a t i o n s by H u r t u k and T z a v a r a s 1 5 show t h a t t h e s e wavy s u r f a c e s a p p e a r not o n l y i n Fe-C s y s t e m , but a l s o i n F e - N i a l l o y s where t h e p e r i t e c t i c r e a c t i o n p r e v a i l s . H u r t u k 1 6 has a l s o shown t h a t p l a i n c a r b o n s t e e l s c o n t a i n i n g no p h o s p h o u r o u s showed s i m i l a r wavy s u r f a c e s when t h e c a r b o n was 0.1%. 2.2.2 I n f l u e n c e of O p e r a t i n g V a r i a b l e s on Heat T r a n s f e r I t has been shown by many i n v e s t i g a t o r s t h a t h i g h e r c a s t i n g s p e e d s i n c r e a s e h e a t f l u x i n t h e mould. N e v e r t h e l e s s , t h e s p e c i f i c amount of h e a t e x t r a c t i o n (J/Kg) d e c r e a s e s 1 7 and so does t h e n e t s h e l l t h i c k n e s s 1 8 when t h e b i l l e t e x i t s t h e mould. Though e a r l i e r p u b l i c a t i o n s 1 ' 1 7 ' 1 9 r e p o r t t h a t p o u r i n g t e m p e r a t u r e has n e g l i g i b l e e f f e c t on h e a t t r a n s f e r i n t h e mould, r e c e n t i n v e s t i g a t i o n s 2 0 by S a m a r a s e k e r a and Brimacombe r e v e a l t h a t h i g h e r s u p e r h e a t s g i v e r i s e t o h i g h e r h e a t f l u x e s i n t h e mould. C o o l i n g w a ter v e l o c i t y seems t o a f f e c t t h e h e a t e x t r a c t i o n from t h e t h e b i l l e t mould s u b s t a n t i a l l y 2 0 . Use o f o i l l u b r i c a n t l e a d s t o h i g h e r h e a t f l u x e s n e a r t h e m e n i s c u s compared t o mould powders, w h i l e t h e l a t t e r y i e l d h i g h e r h e a t f l u x e s i n t h e l o w e r r e g i o n s of t h e m o u l d . 2 1 1 4 2.2.3 I n f l u e n c e of Moul d D e s i g n on Heat T r a n s f e r As t h e b u l k o f t h e h e a t i s e x t r a c t e d i n t h e u p p e r p o r t i o n o f t h e mould and t h e low e r p a r t a c t s m a i n l y as a s u p p o r t i n g s y s t e m f o r t h e s t r a n d s , l o n g e r moulds do not seem t o have any a d v a n t a g e . B r i m a c o m b e 2 2 has s u g g e s t e d t h a t mould l e n g t h s h o u l d be i n c r e a s e d p r o p o r t i o n a t e l y t o i n c r e a s e i n t h e c a s t i n g s p e e d t o r e d u c e t h e i n c r e a s e d d a nger of b r e a k o u t s . mould t a p e r seems t o improve h e a t t r a n s f e r p r e s u m a b l y b e c a u s e i t r e d u c e s t h e gap w i d t h o v e r t h e low e r r e g i o n s 1 9 ' 2 3 . E x c e s s i v e mould t a p e r however c a u s e s i n c r e a s e d mould wear and r e s i s t a n c e t o w i t h d r a w a l . 2 3 ' 2 * W i t h d i r e c t i o n a l c o r r u g a t i o n s and t a p e r e d r i b s i n moulds i t i s c l a i m e d t h a t 8-9% i n c r e a s e i n h e a t t r a n s f e r r a t e s c a n be o b t a i n e d , p o s s i b l y b e c a u s e of r e d u c e d a i r g a p s . 2 5 ' 2 6 D e s p i t e t h e f r e q u e n t use o f s u c h d e s i g n s i n t h e S o v i e t U n i o n t h e h i g h e r m a c h i n i n g c o s t s i n v o l v e d make t h e i r a p p l i c a t i o n i n t h e west i m p r a c t i c a l . 2.2.4 I n f l u e n c e Of Moul d B e h a v i o u r On Heat T r a n s f e r When s t e e l i s c a s t t h r o u g h b i l l e t m oulds t h e mou l d w a l l s h e a t up n o n u n i f o r m l y b e c a u s e h e a t f l o w i s m a i n l y gap d e p e n d e n t a s was shown i n F i g 3. The r e s u l t i n g t e m p e r a t u r e d i s t r i b u t i o n g i v e s r i s e t o d i f f e r e n t i a l t h e r m a l e x p a n s i o n as h o t t e r r e g i o n s expand more t h a n c o l d e r r e g i o n s so t h a t t h e mould w a l l c h a n g e s i t s s h a p e . In a r e a s of h i g h t h e r m a l g r a d i e n t s , s u c h as n e a r t h e m e n i s c u s where t h e y i e l d s t r e s s i s l o c a l l y r e d u c e d due t o h i g h e r t e m p e r a t u r e s , p l a s t i c f l o w t a k e s p l a c e t o r e s u l t i n permanent 15 d e f o r m a t i o n . F i g 5 shows a t y p i c a l mould d i s t o r t i o n p r o f i l e f o r b i l l e t m o ulds. Though t h e e x a c t e x t e n t of t h e d i s t o r t i o n v a r i e s from mould t o mould and f o r d i f f e r e n t t y p e o f c o n s t r a i n t s , t h e p r e d o m i n a n t b u l g e a t t h e t o p of t h e mould i s c h a r a c t e r i s t i c o f b i l l e t m o ulds. S a m a r a s e k e r a e t a l 2 7 have c a l c u l a t e d t h e e x t e n t of d i s t o r t i o n from an e l a s t o - p l a s t i c f i n i t e - e l e m e n t a n a l y s i s . W i t h t h e h e l p o f d a t a c o l l e c t e d from t h e N o r t h A m e r i c a n S t e e l I n d u s t r y Brimacombe e t a l 2 8 have shown t h a t t h e t y p e and e x t e n t of d i s t o r t i o n depends on t h e m e n i s c u s l e v e l , d e p o s i t i o n of s o l i d s on t h e c o l d f a c e and p o s s i b l y t h e t y p e o f c o n s t r a i n t . A l s o w a l l t h i c k n e s s c a n have s i g n i f i c a n t e f f e c t on t h e r m a l d i s t o r t i o n . I n c r e a s i n g w a l l t h i c k n e s s r e s u l t s i n an i n c r e a s e i n b o t h peak w a l l t e m p e r a t u r e s and t h e r m a l g r a d i e n t s l e a d i n g t o more permament d e f o r m a t i o n . 2 7 S a m a r a s e k e r a and B r i m a c o m b e 2 9 showed t h a t u n d e r c o n d i t i o n s of i n t e r m i t t e n t b o i l i n g i n t h e mould-water c h a n n e l s , i t i s p o s s i b l e t o have dynamic d i s t o r t i o n . W a l l t h i c k n e s s , w a ter v e l o c i t y , w a ter back p r e s s u r e and s t e e l c a r b o n c o n t e n t a r e some of t h e f a c t o r s t h a t c o u l d t r i g g e r b o i l i n g and l e a d t o mould w a l l movement. The p o s s i b i l i t y of a s y n c h r o n o u s i n t e r m i t t e n t b o i l i n g on d i f f e r e n t f a c e s of t h e mould r e s u l t i n g i n a dynamic d i s t o r t i o n has been c o n s i d e r e d t o be an i m p o r t a n t f a c t o r i n c a u s i n g r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k s . 3 0 16 Distance Between Opposite Faces (in) 5 5 5 0 5 6 5 0 5 5 5 0 5 6 5 0 1 l ~f" 1 Strtii^ ht Wall Curved Wall 0 O A • AO • 0 • A • • Ao • As rAo 100 - 4 QfS A G ATO 2 0 0 - a AO 8 •Is 3 0 0 rito A m 12 O A>D o 4 0 0 AOD 16 a. o 1- Aon E o 5 0 0 •3D A d 2 0 •b A o c o 6 0 0 nV> AO 2 4 Tn Q CAD CA • ? 0 0 • AO cA • 28 • Ao • A) o A • A CD A 8 0 0 32 i i i i i • i i i 140 142 144 142 144 Distance Between Opposite Faces (mm) 3 O CL o E o a> o c p Tn O Figure 5 -moulds T y p i c a l mould d i s t o r t i o n p r o f i l e with c o n s t r a i n t s on two s t r a i g h t of b i l l e t sides 1 7 2.3 M o u l d - r e l a t e d D e f e c t s Q u a l i t y i n b i l l e t c a s t i n g as r e l a t e d t o t h e mould i s d e t e r m i n e d by i n t e r n a l s o u n d n e s s , s u r f a c e t o p o g r a p h y and t h e shape of t h e b i l l e t s . T y p i c a l d e f e c t s have been s c h e m a t i c a l l y shown i n F i g 1. W h i l e r h o m b o i d i t y and c r a c k f o r m a t i o n a r e r e l a t e d t o t h e mould b e h a v i o u r d u r i n g c a s t i n g , s u r f a c e r o u g h n e s s i s g e n e r a t e d by i m p r o p e r s e l e c t i o n of p a r a m e t e r s of t h e mould o s c i l l a t o r . In t h i s s e c t i o n t h e i n f l u e n c e of mould d e s i g n and o p e r a t i n g v a r i a b l e s on r h o m b o i d i t y and c r a c k f o r m a t i o n i s r e v i e w e d . F a c t o r s t h a t c o n t r o l t h e d e p t h , s p a c i n g and s e v e r i t y o f t h e o s c i l l a t i o n marks a r e c o n s i d e r e d f o l l o w e d by an o v e r v i e w of t h e mechanisms p r o p o s e d t o e x p l a i n t h e f o r m a t i o n of o s c i l l a t i o n marks. 2.3.1 R h o m b o i d i t y R h o m b o i d i t y i s o f t e n c h a r a c t e r i s e d by t h e d i f f e r e n c e i n d i a g o n a l s o f t h e c r o s s - s e c t i o n b e i n g c a s t . F r e q u e n t l y r h o m b o i d i t y i s a c c o m p a n i e d by l o n g i t u d i n a l c o r n e r c r a c k s . In e xtreme c a s e s d i a g o n a l c r a c k s may r e s u l t . E x c e s s i v e r h o m b o i d i t y r e n d e r s t h e b i l l e t u n f i t f o r r o l l i n g . R h o m b o i d i t y i s c a u s e d by n o n - s y m m e t r i c a l c o o l i n g of t h e b i l l e t e i t h e r i n t h e mould o r i n t h e s p r a y s . F r e q u e n t l y r h o m b o i d i t y has been o b s e r v e d t o c hange o r i e n t a t i o n d u r i n g t h e c a s t i n g o f a h e a t 2 6 . S t e e l c o m p o s i t i o n a p p e a r s t o i n f l u e n c e r h o m b o i d i t y . S t e e l s c o n t a i n i n g 0.18 t o 0.25% c a r b o n 3 1 ' 3 2 and h i g h e r c a r b o n (>0.4%) g r a d e s 2 9 ' 3 3 ' 3 " g i v e r i s e t o more r h o m b o i d i t y . S m a l l e r c r o s s - s e c t i o n s , 3 1 h i g h e r p o u r i n g 18 t e m p e r a t u r e s and h i g h e r c a s t i n g s p e e d s 1 7 ' 2 3 a g g r a v a t e r h o m b o i d i t y . Poor a l i g n m e n t between t h e mould and submould a s s e m b l y 2 6 ' 3 1 ' 3 3 ' 3 5 i s a s i g n i f i c a n t f a c t o r c o n t r i b u t i n g t o r h o m b o i d i t y . I t a l s o has been p o s t u l a t e d 2 9 ' 3 0 t h a t when i n t e r m i t t e n t b o i l i n g o c c u r s a s y n c h r o n o u s l y on d i f f e r e n t f a c e s o f t h e mould, t h e mould i t s e l f can assume a rhomboid shape and c o u l d g i v e r i s e t o g r e a t e r r h o m b o i d i t y i n t h e s o l i d i f y i n g s h e l l . R h o m b o i d i t y and c r a c k i n g i n c r e a s e w i t h i n c r e a s i n g s e r v i c e l i f e o f t h e mould and i n c r e a s i n g wear a t t h e bottom of t h e m o u l d . 3 1 In p r a c t i c e , d i f f e r e n t c o r r e c t i v e measures have been s o u g h t t o m i n i m i s e r h o m b o i d i t y and t h e a s s o c i a t e d c r a c k s . E m p h asis has been p l a c e d on m o d i f i c a t i o n s t o c o o l i n g i n t h e mould as a means t o r e d u c e t h e s e d e f e c t s . H i g h e r h e a t t r a n s f e r r a t e s o b t a i n e d w i t h c o r r u g a t e d m o u l d s 2 6 seem t o r e d u c e r h o m b o i d i t y . At F u n a b a s h i s t e e l w o r k s 3 6 a " s o f t - c o o l i n g " p r a c t i c e has p r o v e d t o be e f f e c t i v e i n r e d u c i n g r h o m b o i d i t y . T h i s i s a c c o m p l i s h e d by m a c h i n i n g h o r i z o n t a l s e r r a t i o n s on t h e o u t s i d e s u r f a c e of t h e mould i n c o n t a c t w i t h t h e c o o l i n g w a t e r . As S a m a r a s e k e r a and B r i m a c o m b e 2 1 have s u g g e s t e d t h e r e d u c t i o n i n r h o m b o i d i t y by t h i s measure may be due t o s u p p r e s s i o n of i n t e r m i t t e n t b o i l i n g . A t e c h n i q u e c u r r e n t l y u s e d t o c o r r e c t c o r n e r c r a c k i n g i n h i g h -c a r b o n s t e e l s i s t o r e d u c e water f l o w r a t e s r e l a t i v e t o t h o s e employed i n l o w - c a r b o n s t e e l s . 2 9 ' 3 " I t has been s u g g e s t e d 2 9 t h a t l o w e r water f l o w r a t e s c a u s e l e s s i n t e r m i t t e n t and more v i g o r o u s b o i l i n g whereby c o o l i n g o f t h e mould i s more u n i f o r m and rhomboid mould c o n d i t i o n s a r e l e s s l i k e l y . B o i l i n g , however, c a u s e s e x c e s s i v e s c a l e d e p o s i t i o n on t h e mould w a l l s w h i c h t h e n 19 become h o t t e r and d i s t o r t . T hus, w i t h t h i s p r a c t i c e moulds have t o be r e p l a c e d f r e q u e n t l y t o m a i n t a i n a c c e p t a b l e b i l l e t q u a l i t y . S a m a r a s e k e r a and B r i m a c o m b e 2 9 s u g e s t t h a t s u p p r e s s i o n r a t h e r t h a n e n h a n c i n g b o i l i n g s h o u l d i mprove t h e b i l l e t q u a l i t y t o a c o n s i d e r a b l e e x t e n t . T h i s c o u l d be a c c o m p l i s h e d by i n c r e a s i n g water v e l o c i t y , r a i s i n g water p r e s s u r e , e n h a n c i n g s u r f a c e r o u g h n e s s o r i n c r e a s i n g w a l l t h i c k n e s s . I t i s a l s o i m p o r t a n t t o c o n s i d e r sub-mould c o n d i t i o n s t o r e d u c e r h o m b o i d i t y . P r o p e r m a i n t e n a n c e of a l i g n m e n t between mould and submould a s s e m b l y i s e s s e n t i a l t o m i n i m i z e r h o m b o i d i t y . 3 3 Tokuyama and S u z u k i 3 7 show t h a t i f t h e s p r a y a n g l e i n t h e s e c o n d a r y c o o l i n g zone i s d e c r e a s e d so t h a t w ater does n o t r e a c h t h e c o r n e r s , r h o m b o i d i t y may be m i t i g a t e d . C o r n e r r o l l s may be i n s t a l l e d a t t h e t o p end o f t h e r o l l e r a p r o n i n o r d e r t o c o r r e c t r h o m b o i d i t y 3 ' m e c h a n i c a l l y . 2.3.2 C r a c k F o r m a t i o n C r a c k s i n c o n t i n o u s c a s t i n g a r i s e due t o a c o m b i n a t i o n of t h e r m a l s t r e s s e s c a u s e d by a d v e r s e t h e r m a l g r a d i e n t s and m e c h a n i c a l l y i n d u c e d s t r e s s e s a c t i n g on r e g i o n s o f low d u c t i l i t y of s t e e l a t h i g h t e m p e r a t u r e s . T a k e n t o g e t h e r t h e r e a r e t h r e e d i s t i n c t t e m p e r a t u r e r a n g e s v i z ; above 1340°C, 8 0 0 - 1 2 0 0 ° C , 700-900°C i n w h i c h s t e e l has low s t r e n g t h and / o r d u c t i l i t y . mould r e l a t e d c r a c k s a r e a t t r i b u t e d t o t h e low d u c t i l i t y of s t e e l above 1340°C. T h e s e c r a c k s a r e e s s e n t i a l l y i n t e r d e n d r i t i c w i t h smooth s u r f a c e s i n d i c a t i v e o f h o t t e a r i n g n e a r t h e s o l i d i f i c a t i o n f r o n t . 2 6 ' 3 8 " " 0 T h e s e c r a c k s a r e f i l l e d w i t h 20 l i q u i d r i c h i n s o l u t e e l e m e n t s as w e l l a s i n c l u s i o n s . 2 6 ' " 0 The h i g h t e m p e r a t u r e zone o f low d u c t i l i t y i s b r o u g h t a b o u t by t h e p r e s e n c e of l i q u i d f i l m s i n t h e i n t e r d e n d r i t i c r e g i o n s whereby f r e e z i n g i s d e l a y e d u n t i l t e m p e r a t u r e s w e l l below t h e s o l i d u s a r e r e a c h e d . When t h e s t r a i n t o f a i l u r e e x c e e d s , 0.2 t o 0.3 p e r c e n t a c c o r d i n g t o Vom Ende and V o g t " 1 , c r a c k s can a p p e a r . O f f - c o r n e r i n t e r n a l c r a c k s a r e o b s e r v e d i n t r a n s v e r s e s e c t i o n s n o r m a l t o a g i v e n f a c e and w i t h i n 1-2 cm of t h e c o r n e r . They may f o r m a t one or more o f f - c o r n e r l o c a t i o n s i n a g i v e n s e c t i o n . In some i n s t a n c e s , t h e c r a c k s a r e a s s o c i a t e d w i t h l o n g i t u d i n a l s u r f a c e d e p r e s s i o n and i n more s e v e r e c a s e s t h e i n n e r m o s t p a r t of t h e c r a c k may c u r v e i n w a r d t o f o l l o w a d i a g o n a l l i n e ( o r a d i a g o n a l c r a c k ) j o i n i n g o p p o s i t e c o r n e r s of t h e b i l l e t . Brimacombe e t a l 3 8 p r o p o s e d a mechanism t o e x p l a i n t h e f o r m a t i o n of t h e s e c r a c k s b a s e d on t h e t e n s i l e s t r a i n s imposed on t h e s o l i d i f i c a t i o n f r o n t a t t h e o f f - c o r n e r r e g i o n s . These s t r a i n s a r i s e b e c a u s e o f b u l g i n g o f t h e m i d f a c e a g a i n s t t h e c o l d c o r n e r , F i g 6. Randomness i n t h e l o c a t i o n of t h e s e c r a c k s was a c c o u n t e d f o r by t h e i r argument t h a t b u l g i n g c o u l d a l s o be r a n d o m l y o c c u r i n g on d i f f e r e n t s i d e s o f t h e b i l l e t owing t o w o b b l i n g o f t h e mould, p o o r s e t t i n g o f f o o t r o l l s o r m i s a l i g n m e n t o f t h e m a c h i n e . They s u g g e s t e d t h a t h i g h e r w a t e r f l u x e s i n t h e upper s p r a y s c o u l d r e d u c e t h e i n t e n s i t y o f t h e c r a c k i n g p r o b l e m by i m p o s i n g c o m p r e s s i v e s t r a i n s n e a r t h e s o l i d i f i c a t i o n f r o n t . R e c e n t i n v e s t i g a t i o n s by Z e t t u r l a n d and K r i s t i a n s s o n * 2 have 21 Mould wall :ic diagram of a billet bulging in the mould and forming off-corner cracks. Simplified sketch to illustrate calculation of strain in off-corner region caused by bulging. F i g u r e 6 - Mechanism of crack formation at o f f - c o r n e r s 2 2 22 shown t h a t mould wear, e s p i c i a l l y i n t h e lower p a r t t o be an i m p o r t a n t f a c t o r i n c a u s i n g o f f - c o r n e r c r a c k i n g . They have l a i d e m p h a s i s on t h e r e d u c t i o n o f h e a t e x t r a c t i o n i n t h e l o w e r p a r t of t h e mould l e a d i n g t o s u b s t a n t i a l r e h e a t i n g o f t h e s u r f a c e . C o m p r e s s i o n a t t h e s u r f a c e and t e n s i l e s t r e s s e s a t t h e s o l i d i f i c a t i o n f r o n t c o u l d l e a d t o o f f - c o r n e r c r a c k i n g . L o n g i t u d i n a l c o r n e r c r a c k s as seen i n t r a n s v e r s e s e c t i o n a r e g e n e r a l l y 1 t o 2 mm deep, a l t h o u g h under u n f a v o u r a b l e c o n d i t i o n s , c r a c k s w i t h a d e p t h of 6 t o 7 mm have been o b s e r v e d . 4 3 O f t e n c r a c k w i d t h i s seen t o be g r e a t e r i n t h e i n t e r i o r t h a n n e a r th e s u r f a c e . C o r n e r r a d i u s has a s i g n i f i c a n t i n f l u e n c e on t h e l o c a t i o n o f t h e s e c r a c k s . W i t h a l a r g e c o r n e r r a d i u s l o n g i t u d i n a l c r a c k s a p p e a r on t h e c o r n e r w h i l e w i t h s m a l l e r r a d i i , the. c r a c k s f o r m more f r e q u e n t l y o f f t h e c o r n e r 3 0 . C o r n e r c r a c k s m o s t l y o c c u r i n a s s o c i a t i o n w i t h r h o m b o i d i t y , a t t h e o b t u s e c o r n e r s of a r h o m b o i d b i l l e t . E a r l i e r s t u d i e s have r e v e a l e d t h a t when r h o m b o i d i t y i s r e d u c e d t h r o u g h c o r r e c t i v e m e a s ures f o r a d v e r s e c o n d i t i o n s i n t h e mould t h e r e i s a marked d e c r e a s e i n t h e s e v e r i t y of l o n g i t u d i n a l c o r n e r c r a c k s . 2 6 ' 3 1 ' 3 3 ' 3 5 As t o t h e r e l a t i o n between r h o m b o i d i t y and c o r n e r c r a c k i n g , S a m a r a s e k e r a and B r i m a c o m b e 2 9 p o i n t o u t t h a t t e n s i l e s t r a i n s a c t i n g p a r a l l e l t o t h e d i a g o n a l j o i n i n g t h e a c u t e a n g l e c o r n e r s c o u l d c a u s e i n t e r n a l c r a c k i n g a l o n g t h e d i a g o n a l j o i n i n g t h e o b t u s e - a n g l e c o r n e r s . T e n s i l e s t r a i n s a r e g e n e r a t e d a t t h e s o l i d i f i c a t i o n f r o n t due t o s u r f a c e r e h e a t i n g i f t h e o b t u s e - a n g l e c o r n e r o f t h e b i l l e t p u l l s away from th e b i l l e t . D e p e n d i n g on t h e c r a c k d e p t h , t h e e x t e n t o f r e h e a t i n g 23 and t h e m a g n i t u d e of t e n s i l e s t r a i n s g e n e r a t e d by t h e e n s u i n g s h r i n k a g e of t h e s h e l l as i t c o o l s d e e p e r i n t h e mould, t h e c r a c k may p e n e t r a t e t o t h e s u r f a c e and become a v i s i b l e d e f e c t a t t h e c o r n e r . 2 9 D i a g o n a l c r a c k s a r e a l s o a s s o c i a t e d w i t h r h o m b o i d i t y . D i a g o n a l c r a c k s u s u a l l y r u n between o b t u s e c o r n e r s o f t h e rhomboid s e c t i o n . These c r a c k s f o r m 3 9 i n i t i a l l y i n t h e h i g h t e m p e r a t u r e zone of low d u c t i l i t y , b ut may grow o u t w a r d t o w a r d s t h e c o r n e r s d e p e n d i n g on t h e m a g n i t u d e o f t h e s t r a i n . C o n t r o l of r h o m b o i d i t y s h o u l d e l i m i n a t e t h e s e c r a c k s . 2.4 O s c i l l a t i o n Marks S t e e l c a s t c o n t i n o u s l y a t r e l a t i v e l y h i g h s p e e d s i n r e c i p r o c a t i n g moulds i s u s u a l l y c h a r a c t e r i s e d by t h e p r e s e n c e of f i n e l a t e r a l s u r f a c e m a r k i n g s . Though i n d i v i d u a l l y of somewhat i r r e g u l a r shape t h e r e i s a r e g u l a r s p a c i n g betwen t h e s e marks. W h i l e t h e s e marks do not n o r m a l l y g i v e r i s e t o d i f f i c u l t i e s i n r o l l i n g , l o w e r h e a t e x t r a c t i o n r a t e s a t t h e o s c i l l a t i o n marks"" l o c a l l y r e d u c e t h e s h e l l t h i c k n e s s i n c r e a s i n g t h e r i s k o f a b r e a k o u t below t h e mould w h i c h c o u l d p r e v e n t t h e a t t a i n m e n t o f optimum c a s t i n g s p e e d s . T h e s e marks a r e a l s o a s s o c i a t e d w i t h b l e e d e r - t y p e s l i v e r s w h i c h , when s e v e r e , r e s u l t i n u n a c c e p t a b l e s u r f a c e q u a l i t y . In s l a b c a s t e r s l u b r i c a t e d by mould powders t h e s e marks a r e d e e p e r and a r e p o t e n t i a l s i t e s f o r s u r f a c e c r a c k s t o f o r m . W o l f " 5 r e p o r t s t h a t l o c a l c o a r s e n i n g o f t h e s o l i d i f i c a t i o n s t r u c t u r e c a u s e d by r e d u c e d h e a t f l o w n e a r t h e o s c i l l a t i o n mark i s d e t r i m e n t a l t o s t e e l d u c t i l i t y and c r e a t e s 24 t h e h a z a r d of t r a n s v e r s e c r a c k s d u r i n g s t r a n d s t r a i g h t e n i n g , p a r t i c u l a r l y i n t h e c a s e o f s t e e l s w i t h f i n e p r e c i p i t a t e s of n i t r i d e s o f A l , V o r Nb S u b s u r f a c e s t r u c t u r e s a t an o s c i l l a t i o n mark i n s l a b s u s u a l l y e x i b i t a h o o k - l i k e mark c o v e r e d by e x t r a m e t a l which when v i e w e d i n c r o s s s e c t i o n a p p e a r s as a l a p . Emi e t a l " 6 from t h e i r i n v e s t i g a t i o n s on o s c i l l a t i o n marks i n s l a b s r e p o r t t h a t t h e p i t c h o f t h e o s c i l l a t i o n marks i s i d e n t i c a l t o t h e d i s t a n c e between t h e s e hook m a r k i n g s . T h i s hook forms i n t h e v a l l e y of t h e o s c i l l a t i o n mark" 7 and Tanaka e t a l 4 * ' " 8 have shown t h a t p o s i t i v e and n e g a t i v e s e g r e g a t i o n i s a s s o c i a t e d w i t h t h e mark. H i g h e r s e g r e g a t i o n i s o b s e r v e d i n d e e p e r o s c i l l a t i o n marks. Saucedo e t a l 4 9 have shown t h a t d e n d r i t e s n e a r t h e o s c i l l a t i o n mark o r i e n t d i f f e r e n t l y compared t o t h e i r o r i e n t a t i o n p e r p e n d i c u l a r t o t h e mould w a l l e l s e w h e r e . I t emerges c l e a r l y f r o m t h e l i t e r a t u r e t h a t t h e p i t c h of t h e o s c i l l a t i o n marks i s n e a r l y t h e r a t i o of c a s t i n g s p e e d t o t h e o s c i l l a t i o n f r e q u e n c y . 4 4 ' 4 8 ' 4 6 ' 5 0 ~ 5 3 S t r o k e l e n g t h seems t o have a s i g n i f i c a n t i n f l u e n c e on t h e o s c i l l a t i o n marks. Many i n v e s t i g a t o r s 9 ' 4 6 ' 5 4 have shown t h a t s h o r t s t r o k e s d e c r e a s e t h e d e p t h of o s c i l l a t i o n m arks. H i g h e r f r e q u e n c y o s c i l l a t i o n a l s o has a s i m i l a r e f f e c t . 1 0 ' 5 5 S h o r t and f r e q u e n t s t r o k e s c a u s e s h a l l o w but a g r e a t e r number of o s c i l l a t i o n marks; but l o n g e r t h e n e g a t i v e s t r i p l a s t s t h e o s c i l l a t i o n marks t e n d t o be d e e p e r . 1 0 ' 5 6 However, S c h o f f m a n n 1 0 and T a k e u c h i e t a l 4 7 r e p o r t t h a t o s c i l l a t i o n marks a r e f l a t t e r and l e s s n o t i c e a b l e when t h e d i f f e r e n c e i n s p e e d between t h e mould and t h e s t r a n d d u r i n g t h e 25 n e g a t i v e s t r i p p e r i o d i s s m a l l . Kawakami e t a l 5 3 p o i n t out t h a t t h e maximum d e p t h t o w h i c h t h e m e n i s c u s s h e l l i s b e n t a l s o i n c r e a s e s w i t h i n c r e a s i n g n e g a t i v e s t r i p t i m e . J a c o b i e t a l 5 7 s u g g e s t an optimum n e g a t i v e s t r i p t i m e between 0.15 t o 0.25 s e c o n d s whereby s u f f i c i e n t c o m p r e s s i v e s t r e s s e s a r e g e n e r a t e d i n t h e s h e l l t o ' h e a l ' t h e damage done d u r i n g t h e u p s t r o k e ; t h i s n e g a t i v e s t r i p t i m e a l s o p r o d u c e s o s c i l l a t i o n marks t h a t a r e not t o o deep. Komatsu e t a l 5 8 have o b s e r v e d t h a t t h e c o m p r e s s i v e f o r c e a p p l i e d t o t h e s h e l l i n t h e n e g a t i v e s t r i p p i n g p e r i o d d e c r e a s e d c o n s i d e r a b l y w i t h t h e n e g a t i v e s t r i p r a t i o (NSR) w h i c h i s t h e r a t i o of n e g a t i v e s t r i p t i m e t o t h e t o t a l down s t r o k e t i m e e x p r e s s e d as a p e r c e n t a g e . When t h e NSR was d e c r e a s e d from 62 t o 35%, c o m p r e s s i v e f o r c e was n o t a p p l i e d t o t h e s h e l l a t a l l . I t s h o u l d be n o t e d t h a t t h i s c o r r e s p o n d s r o u g h l y t o a n e g a t i v e s t r i p t i m e of 0.14 s e c o n d s where a c c o r d i n g t o J a c o b i e t a l 5 7 o s c i l l a t i o n marks a r e s h a l l o w . Rape s e e d o i l l u b r i c a t i o n g i v e s s h a l l o w e r marks compared t o t h e o s c i l l a t i o n marks c h a r a c t e r i s t i c o f mould powder l u b r i c a t i o n . 5 " J 5 5 B r o w n 5 9 e x p e r i m e n t i n g w i t h amounts of l u b r i c a t i o n r e p o r t s t h a t t h e o s c i l l a t i o n marks showed a s e r i e s o f t r a n s v e r s e c r a c k s when no o i l was u s e d . H i g h e r c a r b o n s t e e l s do not e x h i b i t t h e deep o s c i l l a t i o n marks t h a t low c a r b o n s t e e l s show. W o l f 5 5 s u g g e s t s t h a t most p r o n o u n c e d o s c i l l a t i o n mark f o r m a t i o n i n t h e s t e e l s w i t h c a r b o n a r o u n d 0.1% C c a n be a t t r i b u t e d t o t h e maximum s h e l l s t r e n g t h ( a s s o c i a t e d w i t h a minimum i n m i c r o - s e g r e g a t i o n ) and maximum 26 s h r i n k a g e of t h e m e n i s c u s s h e l l . A l s o a u s t e n i t i c s t a i n l e s s s t e e l w i t h a N i / C r r a t i o of a b o u t 0.55 have deep o s c i l l a t i o n marks. T h e r e a r e many models i n t h e l i t e r a t u r e a t t e m p t i n g t o e x p l a i n t h e f o r m a t i o n of o s c i l l a t i o n marks i n c o n t i n u o u s l y c a s t b i l l e t s and o t h e r s h a p e s . One o f t h e e a r l i e s t and b r o a d l y a c c e p t e d mechanism was p r o p o s e d by S a v a g e 5 0 i n 1961 f o r b i l l e t c a s t i n g and was l a t e r a p p l i e d t o s l a b c a s t i n g by S a t o 5 1 . I t i s assumed t h a t t h e t h i n and f r a g i l e s h e l l t h a t forms d u r i n g t h e d o w n s t r o k e i s b r o k e n and c a r r i e d upward i n t h e u p s t r o k e . In t h e e n s u i n g d o w n s t r o k e c o m p r e s s i v e s t r e s s e s of n e g a t i v e s t r i p w eld t h e p i e c e of s t e e l s h e l l t o t h a t below r e s u l t i n g i n an o s c i l l a t i o n mark. L a p s u s u a l l y a s s o c i a t e d w i t h t h e marks may be s u c h j o i n t s . T h i s model s u f f e r s from many q u e s t i o n a b l e , a s s u m p t i o n s and has n e v e r been e x p e r i m e n t a l l y v e r i f i e d . W i t h t h e h e l p o f a mould s i m u l a t o r Kawakami e t a l 5 3 have examined an o l d t h e o r y s u g g e s t e d by S c h o f f m a n n 1 0 and Emi e t a l " 6 . T h i s t h e o r y was f o r m u l a t e d m a i n l y t o e x p l a i n o s c i l l a t i o n marks i n s l a b s where m o l t e n s l a g l u b r i c a t e s t h e mould. I t i s p o s t u l a t e d t h a t i n t h e n e g a t i v e s t r i p t i m e mould s l a g f l o w s i n t o t h e gap betwen mould and s l a b and p u s h e s t h e i n g o t s u r f a c e away from t h e mould w a l l . At t h e end of t h e n e g a t i v e s t r i p o r t h e b e g i n n i n g o f t h e p o s i t i v e s t r i p p e r i o d , t h e v i s c o u s l a y e r of powder t e n d s t o d e t a c h from t h e t i p of t h e s h e l l and m o l t e n s t e e l b e g i n s t o f l o o d o v e r t h e b e n t t i p so t h a t o s c i l l a t i o n mark f o r m s . I t i s n e e d l e s s t o say t h a t t h e e x i s t e n c e of s l a g g y m a t e r i a l on t h e s t e e l s u r f a c e i s a n e c e s s a r y 27 c o n d i t i o n f o r t h e f o r m a t i o n o f o s c i l l a t i o n marks by t h i s mechanism. F r e e z i n g o f t h e m e n i s c u s and e n s u i n g o v e r f l o w i s a c l a s s i c a l t h e o r y s u g g e s t e d by T h o r t o n 6 0 t o e x p l a i n s u r f a c e r i p p l e s i n s t a t i c i n g o t c a s t i n g . W r a y 6 1 has shown t h a t many t y p e s o f r i p p l e s a p p e a r on s t a t i c i n g o t s but l a p s a r e a t t r i b u t a b l e t o m e n i s c u s f r e e z i n g . Saucedo and B e e c h " 9 have a p p l i e d m e n i s c u s s o l i d i f i c a t i o n a s i n s t a t i c i n g o t c a s t i n g s t o e x p l a i n o s c i l l a t i o n mark f o r m a t i o n i n c o n t i n u o u s c a s t i n g s . W i t h t h e h e l p of a m a t h e m a t i c a l m o d e l 6 2 t h e y have shown t h a t p a r t of th e m e n i s c u s c a n f r e e z e i n a m a t t e r of 0.1 t o 0.2 s e c o n d s . They s u g g e s t 4 9 t h a t s t r o n g r i p p l i n g o c c u r s when t h e v e l o c i t y of t h e mould e q u a l s t h e v e l o c i t y o f t h e b i l l e t . However t h i s i s c o n t r a r y t o t h e o b s e r v a t i o n t h a t o s c i l l a t i o n mark d e p t h s and t h e d e p t h of t h e m e n i s c u s s h e l l s i n c r e a s e w i t h i n c r e a s i n g d i f f e r e n c e between t h e downward mould s p e e d and e a t i n g s p e e d . A l s o t h i s model does n o t a c c o u n t f o r t h e v a r i o u s s h a p e s of t h e m e n i s c u s s h e l l s . 2.5 Scope o f t h e P r e s e n t S t u d y I t s h o u l d be n o t e d t h a t t h e p r i n c i p a l c o n n e c t i o n between the mould and t h e b i l l e t i s t h r o u g h h e a t t r a n s f e r . T h e r e f o r e t h e i n f l u e n c e o f t h e mould on b i l l e t q u a l i t y s h o u l d be r e f l e c t e d i n t h e h e a t e x t r a c t i o n . Though mould g e n e r a t e d r h o m b o i d i t y has been i n v e s t i g a t e d i n terms o f s e v e r a l o p e r a t i n g v a r i a b l e s , s t e e l c o m p o s i t i o n and 28 mould geometry, t h e l o c a t i o n i n t h e mould where b i l l e t s become rhomboid i s unknown. I t i s not c l e a r how r h o m b o i d i t y c o u l d be g e n e r a t e d h i g h up i n t h e mould where t h e f e r r o s t a t i c p r e s s u r e i s q u i t e low. The r e l a t i o n between r h o m b o i d i t y and mould b e h a v i o u r a s r e f l e c t e d i n h e a t t r a n s f e r and mould d i s t o r t i o n needs t o be examined more c a r e f u l l y . O f f - c o r n e r c r a c k i n g has been s u g g e s t e d t o be m a i n l y due t o t h e b u l g i n g o f t h e mid f a c e a g a i n s t t h e c o l d c o r n e r . I t i s not c l e a r where i n t h e mould t h i s o c c u r s and how s u c h b u l g i n g c o u l d t a k e p l a c e and what f a c t o r s i n f l u e n c e i t . C l e a r l y t e m p e r a t u r e and t h e m e c h a n i c a l p r o p e r t i e s of t h e s t e e l s h o u l d p l a y a d e c i s i v e r o l e i n c r a c k f o r m a t i o n . The i n f l u e n c e of f a c t o r s l i k e m o u l d - m e t a l gap and h e a t t r a n s f e r a r e i m p o r t a n t i n l o c a t i n g t h e p o s i t i o n i n t h e mould where o f f - c o r n e r c r a c k i n g c o u l d t a k e p l a c e . T h i s would be q u i t e u s e f u l i n e s t a b l i s h i n g p r o c e d u r e s t o a v e r t c r a c k i n g o r t o i n h i b i t c r a c k g r o w t h . O s c i l l a t i o n marks i n b i l l e t s have n e v e r been s y s t e m a t i c a l l y e xamined i n t h e l i t e r a t u r e . I t i s unknown a s t o how t h e mould i n t e r a c t s w i t h t h e b i l l e t t o form t h e s e u n d u l a t i o n s on t h e s u r f a c e w h i c h when s e v e r e c a u s e t h e s u r f a c e q u a l i t y t o d e t e r i o r a t e . The r e l a t i o n between h e a t f l u x a t t h e m e n i s c u s l e v e l i n b i l l e t moulds on t h e o s c i l l a t i o n mark d e p t h s h o u l d be examined. F o c u s s h o u l d be l a i d on t h e i n f l u e n c e o f o s c i l l a t i o n marks on h e a t t r a n s f e r and s o l i d i f i c a t i o n . The p r i n c i p a l a i m o f t h e p r e s e n t s t u d y i s t o r e l a t e h e a t t r a n s f e r c o n d i t i o n s i n t h e mould t o s u r f a c e , s u b s u r f a c e and i n t e r n a l c h a r a c t e r i s t i c s o f t h e b i l l e t b e i n g c a s t . More 29 s p e c i f i c a l l y , t h e o b j e c t i v e s o f t h e s t u d y a r e as f o l l o w s . i . t o s t u d y t h e i n f l u e n c e o f c a s t i n g v a r i a b l e s on t h e s t r u c t u r e of t h e b i l l e t s i i . t o examine r h o m b o i d i t y from t h e s t a n d p o i n t of c a s t i n g p a r a m e t e r s , s t e e l c o m p o s i t i o n and t h e d i m e n s i o n a l s t a b i l i t y of t h e mould and t o c l a r i f y t h e c o n t r i b u t i o n o f t h e mould t o r h o m b o i d i t y i i i . t o e s t a b l i s h t h e mechanism o f f o r m a t i o n o f o f f -c o r n e r c r a c k s i v . t o examine t h e t o p o g r a p h y and s u b s u r f a c e s t r u c t u r e of t h e b i l l e t s t o s h e d l i g h t on t h e n a t u r e and c a u s e o f o s c i l l a t i o n marks. Heat f l u x d a t a and b i l l e t samples from i n d u s t r i a l e x p e r i m e n t s were o b t a i n e d t o a c h i e v e t h e s e o b j e c t i v e s . B i l l e t s e c t i o n s were c u t and examined f o r t h e c o l u m n a r zone l e n g t h , r h o m b o i d i t y and o f f - c o r n e r c r a c k s . B i l l e t s u r f a c e s and sub-s u r f a c e s t r u c t u r e s were c a r e f u l l y a s s e s s e d and r e l a t e d t o mould o s c i l l a t i o n . The i n f l u e n c e of s u p e r h e a t , c a r b o n c o n t e n t and mould water f l o w r a t e was i n v e s t i g a t e d . S h e l l g r o w t h p r o f i l e s and t h e r m a l f i e l d s i n t h e s h e l l were e s t a b l i s h e d w i t h t h e a i d o f a o n e - d i m e n s i o n a l u n s t e a d y - s t a t e h e a t - t r a n s f e r m odel. The model was employed t o c a l c u l a t e t h e d i s t a n c e below t h e m e n i s c u s a t w h i c h t h e c r a c k s f o r m e d and t h e t h e r m a l c o n d i t i o n s a t s u c h l o c a t i o n s were examined t o d e t e r m i n e a mechanism f o r t h e i r f o r m a t i o n . A t w o - d i m e n s i o n a l h e a t -t r a n s f e r model was d e v e l o p e d t o s t u d y t h e e x t e n t o f m e n i s c u s s o l i d i f i c a t i o n . From t h e model p r e d i c t i o n s and on e x a m i n a t i o n 30 o f t h e s u r f a c e and s u b s u r f a c e f e a t u r e s of t h e b i l l e t s , mechanism has been f o r m u l a t e d t o e x p l a i n t h e f o r m a t i o n o s c i l l a t i o n m arks. 31 I I I . EXPERIMENTAL PROCEDURES The c u r r e n t s t u d y i n v o l v i n g e v a l u a t i o n o f b i l l e t q u a l i t y , a s a f f e c t e d by mould c o n d i t i o n s , i s an e x t e n s i o n o f a major p r o j e c t s u p p o r t e d l a r g e l y by AISI t o s t u d y h e a t t r a n s f e r c o n d i t i o n s and d i s t o r t i o n i n b i l l e t m o u l d s . A l l t h e e x p e r i m e n t s were c o n d u c t e d a t We s t e r n Canada S t e e l l o c a t e d i n V a n c o u v e r , B.C. The b i l l e t c a s t e r a t W e s t e r n Canada S t e e l i s a r e c e n t l y c o m m i s s i o n e d R o k o p - c u r v e d mould machine h a v i n g f o u r s t r a n d s . B i l l e t s e c t i o n s i z e s i n a ran g e o f 140mm (5.5") t o 203mm (8") s q u a r e a r e p r o d u c e d a l t h o u g h 140mm b i l l e t s a r e most t y p i c a l . The c a s t i n g s p e e d i s n o r m a l l y 1.5 t o 2.0 m/min (60 t o 80 i p m ) . The s p e c i f i e d mould c h a r a c t e r i s t i c s a r e g i v e n i n T a b l e I T e m p e r a t u r e measurements were made i n t h e b i l l e t mould f o r s e l e c t e d t i m e i n t e r v a l s d u r i n g a h e a t and t i m e - a v e r a g e d h e a t -f l u x p r o f i l e s were o b t a i n e d f o r t h e s e t i m e i n t e r v a l s u s i n g a m a t h e m a t i c a l model i n a s e p e r a t e s t u d y . B i l l e t s e c t i o n s t h a t f ormed i n t h e mould w h i l e t e m p e r a t u r e d a t a was b e i n g g a t h e r e d were c o l l e c t e d a t t h e gas c u t t i n g u n i t a l l o w a n c e b e i n g made f o r t h e ti m e t a k e n t o t r a v e l between t h e two l o c a t i o n s . To e s t a b l i s h l i n k s between t h e measured h e a t f l u x e s and b i l l e t q u a l i t y t h e s t r u c t u r e , s u r f a c e f e a t u r e s , r h o m b o i d i t y and c r a c k f o r m a t i o n i n t h e b i l l e t s e c t i o n were examined. 32 3.1 Heat F l u x D a t a G e n e r a t i o n The t h e r m a l r e s p o n s e o f t h e mould w a l l was m o n i t e r e d by t h e r m o c o u p l e s l o c a t e d a t t h e m i d f a c e and o f f - c o r n e r p o s i t i o n s of t h e o u t s i d e c u r v e d w a l l and on one of t h e s t r a i g h t w a l l s . The a r r a n g e m e n t o f t h e r m o c o u p l e s i s shown i n F i g 7. The t h e r m o c o u p l e s were of i n t r i n s i c c o n s t a n t a n w i r e t y p e and were embedded i n 3.1mm deep h o l e s i n t h e mould w a l l as shown i n F i g 8. S i g n a l s from t h e t h e r m o c o u p l e s were f e d t o a H e w l e t t -P a c k a r d 3485A s c a n n i n g D i g i t a l V o l t m e t e r w h i c h has t h e c a p a b i l i t y of s c a n n i n g up t o 50 c h a n n e l s a t v a r i o u s p r e s e t s p e e d s . The d a t a was r e c o r d e d on a H e w l e t t - P a c k a r d 5055A D i g i t a l R e c o r d e r t h a t was a t t a c h e d t o t h e V o l t m e t e r . The o u t p u t was r e c o r d e d on b o t h p a p e r t a p e and punch t a p e . The l a t t e r p e r m i t t e d t h e d a t a t o be f e d d i r e c t l y t o t h e UBC main frame computer f o r a n a l y s i s . The t h e r m a l d a t a was c o n v e r t e d t o a x i a l h e a t f l u x p r o f i l e s i n a s e p e r a t e s t u d y 2 0 . A two d i m e n s i o n a l h e a t f l o w model of t h e mould w a l l d e v e l o p e d e a r l i e r by S a m a r a s e k e r a and Brimacombe was u s e d s e p a r a t e l y t o d e t e r m i n e h e a t f l u x p r o f i l e s f o r d i f f e r e n t h e a t s 2 0 . The model was employed t o p r e d i c t t h e t e m p e r a t u r e f i e l d f o r d i f f e r e n t h e a t f l u x p r o f i l e s a t t h e h o t f a c e o f t h e mould. By c o m p a r i n g t h e c a l c u l a t e d a x i a l t e m p e r a t u r e p r o f i l e a t t h e d e p t h of t h e r m o c o u p l e s t o t h e measured t e m p e r a t u r e p r o f i l e , t h e h e a t f l u x w h i c h gave t h e b e s t f i t was o b t a i n e d . Heat f l u x d a t a , t h u s g e n e r a t e d by S a m a r a s e k e r a and Brimacombe 2 0 was u s e d i n t h e p r e s e n t i n v e s t i g a t i o n as an i n p u t t o a o n e - d i m e n s i o n a l h e a t t r a n s f e r model t o c a l c u l a t e t h e s h e l l 33 02 18 20^ HO I5_ I7N 19 21 rr CD g m J_? 5B •f) ro CM CVJ 0> O ro in co ro in CVJ m CVJ Figure 7 Arrangement of thermocouples i n mould w a l l 34 Water Gap Figure 8 - Thermocouple design 35 t h i c k n e s s p r o f i l e s arid t e m p e r a t u r e f i e l d s i n t h e s h e l l . 3.2 C o l l e c t i o n o f B i l l e t Samples As t h e t e m p e r a t u r e measurements a r e made i n t h e mould t h r e e b i l l e t samples were o b t a i n e d from e a c h h e a t one i n t h e b e g i n n i n g , one i n t h e m i d d l e and one t o w a r d t h e end of t h e h e a t . The t e m p e r a t u r e of t h e s t e e l i n t h e t u n d i s h was measured i n e a c h h e a t . In some h e a t s two measurements o f s u p e r h e a t were made one i n t h e b e g i n n i n g and one t o w a r d t h e end of h e a t . E f f o r t s were made t o o b t a i n a ra n g e of water f l o w r a t e s i n t h e mould from 16 t o 34 1/s. B i l l e t s a mples o b t a i n e d were a p p r o x i m a t e l y 20cm i n l e n g t h and were l a t e r t r a n s p o r t e d t o t h e l a b o r a t o r y t o be c u t i n t o t r a n s v e r s e s e c t i o n s o f 10mm t h i c k n e s s a s shown i n F i g 9. A t o t a l of 12 cam p a i g n s c o n s i s t i n g o f a t o t a l o f 70 h e a t s were m o n i t e r e d a t W e s t e r n Canada S t e e l . T a b l e II g i v e s t h e c o m p o s t i o n of t h e s t e e l s i n a l l t h e c a m p a i g n s . I t s h o u l d be n o t e d t h a t d u r i n g Heat 24109 t h e s p r a y c o o l i n g s y s t e m f a i l e d t o be a p p l i e d t o t h e b i l l e t . A b i l l e t sample from t h i s h e a t was o b t a i n e d f o r a n a l y s i s ; c o m p o s i t i o n o f t h e h e a t i s g i v e n i n T a b l e I I . 3.3 S u l p h u r P r i n t i n g and C r a c k R a t i n g S u l p h u r p r i n t s were t a k e n o f t h e t r a n s v e r s e s e c t i o n s o f a b o u t 150 b i l l e t s a f t e r t h e y were s u r f a c e g r o u n d and examined f o r c r a c k s . Among t h e m o u l d - r e l a t e d c r a c k s , o f f - c o r n e r c r a c k s were o b s e r v e d i n many c a s e s . L o c a t i o n of t h e c r a c k i n t h e 36 Figure 9 - Transverse and l o n g i t u d i n a l s e c t i o n s of the b i l l e t sample 37 o f f - c o r n e r r e g i o n ( d i s t a n c e s between t h e c r a c k and t h e two s u r f a c e s of t h e b i l l e t t h a t form t h e c o r n e r ) i s n o t e d from s u l p h u r p r i n t . The t o t a l number o f c r a c k s p r e s e n t i n e a c h s u l p h u r p r i n t was r e c o r d e d . S e v e r i t y o f c r a c k i n g was e s t i m a t e d on an a r b i t r a r y s c a l e of 0 t o 6 b a s e d o n t h e number as w e l l as t h e e x t e n t of c r a c k i n g . 3.4 M a c r o e t c h i n q B i l l e t s e c t i o n s from campaigns 3, 5, 6 and 7 were i n i t i a l l y m a c r o e t c h e d i n a 50% h y d r o c h l o r i c a c i d s o l u t i o n a t 70°C. T h e s e m a c r o s t r u c t u r e s were examined and t h e l o c a t i o n o f w h i t e bands was r e c o r d e d . About 100 b i l l e t s e c t i o n s from r e m a i n i n g c a mpaigns were l a t e r m a c r o e t c h e d t o r e v e a l t h e p r e s e n c e of s o l i d i f i c a t i o n bands and t o e s t i m a t e t h e c o l u m n a r zone l e n g t h as a f u n c t i o n of c o m p o s i t i o n , s u p e r h e a t and mould w a t e r f l o w r a t e . S e l e c t e d m a c r o s t r u c t u r e s were p h o t o g r a p h e d u s i n g a 10 mmX 13 mm n e g a t i v e a f t e r i m mersing t h e b i l l e t s e c t i o n i n w a t e r t o o b t a i n b e t t e r d e t a i l . 3.5 S u r f a c e and S u b - s u r f a c e F e a t u r e s In most of t h e h e a t s t h e r e was e v i d e n c e t h a t t h e s t r a n d was b e i n g e x c e s s i v e l y s q u e e z e d by t h e p i n c h r o l l s b e c a u s e t h e 140 X 140 sq.mm s e c t i o n was d e f o r m e d i n t o a s l i g h t l y r e c t a n g u l a r b i l l e t . F l a t t e r o s c i l l a t i o n marks were o b s e r v e d on t h e o u t e r and i n n e r r a d i u s f a c e s w h i c h were i n c o n t a c t w i t h t h e r o l l s and c r a c k s a p p e a r e d t o w a r d t h e c e n t e r p e r p e n d i c u l e r t o t h e c u r v e d 38 f a c e s of t h e b i l l e t . Thus o n l y s u r f a c e f e a t u r e s on t h e s t r a i g h t f a c e s were a n a l y s e d i n t h e p r e s e n t s t u d y . The s p a c i n g between t h e o s c i l l a t i o n marks was measured a t s e v e r a l l o c a t i o n s i n a s e r i e s of h e a t s . The d e p t h of t h e o s c i l l a t i o n marks was i n i t i a l l y c h a r a c t e r i s e d v i s u a l l y by an i n d e x of 1 t o 5 of i n c r e a s i n g s e v e r i t y i n 10 h e a t s c o v e r i n g a c a r b o n r a n g e of 0.13 t o 0.41 and a s u p e r h e a t r a n g e of 6°C t o 51°C i n t h e t u n d i s h . The d e p t h and e x t e n t of o s c i l l a t i o n marks on h i g h c a r b o n s t e e l b i l l e t s was l a t e r c h a r a c t e r i s e d more p r e c i s e l y u s i n g a p r o f i l o m e t e r . T h i s p r o f i l o m e t e r , a s s e m b l e d i n our l a b o r a t o r y , c o n s i s t e d of a l i n e a r d i s p l a c e m e n t t r a n s d u c e r c l a m p e d t o a t r a v e l l i n g m i c r o s c o p e t o f a c i l i t a t e smooth movement of t h e f o r m e r on t h e s p e c i m e n s u r f a c e . The s i g n a l c o r r e s p o n d i n g t o t h e v e r t i c a l d i s p l a c e m e n t of t h e t r a n s d u c e r stem was p r o c e s s e d by a s i g n a l c o n d i t i o n e r ( d i g i t a l i n d i c a t o r of t h e 3000 i n s t r u m e n t s e r i e s made by t h e D a y t r o n i c C o r p o r a t i o n ) and f e d t o a c h a r t r e c o r d e r . The a c c u r a c y o f t h e measurement was t o w i t h i n 0.02032 mm. The s u r f a c e of a b o u t 30 b i l l e t s (0.20 t o 0.40% c a r b o n ) was measured i n t h i s way r e g a r d i n g t h e d e p t h of t h e marks. P r o f i l e s of t h e two o p p o s i t e s i d e s of t h e b i l l e t on t h e o f f - c o r n e r as w e l l as m i d f a c e r e g i o n s were o b t a i n e d . An a v e r a g e of t h e f i r s t 5 major d e p r e s s i o n s was t a k e n t o r e p r e s e n t t h e d e p t h o f o s c i l l a t i o n mark on e a c h s u r f a c e . L o n g i t u d i n a l s e c t i o n s of 10 mm X 10 mm X 200 mm ( as shown i n F i g 9) were c u t from 12 b i l l e t s e c t i o n s a c r o s s t h e o s c i l l a t i o n marks. T h e s e s e c t i o n s were s u r f a c e g r o u n d and l a t e r p o l i s h e d t o 0.3 m i c r o n l e v e l s . E a c h two of t h e s e s e c t i o n s were 39 b o l t e d t o g e t h e r so as not t o r o u n d o f f t h e edges o f t h e s u r f a c e p e r p e n d i c u l a r t o t h e t h e o s c i l l a t i o n mark. P o l i s h e d s a m p l e s were e t c h e d f o r c a s t s t r u c t u r e m o s t l y u s i n g t h e O b e r h o f f e r ' s e t c h . Hot p i c r i c a c i d was a l s o t r i e d on some s a m p l e s . 3.6 Measurement of R h o m b o i d i t y The r h o m b o i d i t y o f 150 b i l l e t s was c h a r a c t e r i s e d by t h e d i f f e r e n c e between the d i a g o n a l s . The s h apes o f t h e b i l l e t s were drawn on t r a n s p a r e n c i e s and t h e d i s t a n c e between o p p o s i t e f a c e s was m easured. 40 IV. MATHEMATICAL MODELS 4.1 One-Dimensional Heat Transfer Model In order to p r e d i c t the s h e l l t h i c k n e s s p r o f i l e and temperature f i e l d s i n the growing s h e l l , a one-dimensional heat-t r a n s f e r model developed by H i b b i n s 6 3 was employed a f t e r a few minor m o d i f i c a t i o n s . Heat-flux p r o f i l e s developed by Samarasekera and Brimacombe 2 0 were employed as an input to the model a f t e r f i t t i n g the data with polynomials. Since these h e a t - f l u x p r o f i l e s were mainly obtained from the midface area where corner e f f e c t s would be n e g l i g i b l e , i t was considered that t h i s model would s u f f i c e . 4.1.1 Formulation A one-dimensional unsteady s t a t e heat conduction equation given below d e s c r i b e s heat t r a n s f e r i n a cont i n u o u s l y cast b i l l e t where p = den s i t y C = S p e c i f i c heat T = Temperature t = Time k = Thermal C o n d u c t i v i t y If k i s assumed constant over small time i n t e r v a l s t h i s equation 41 becomes m - -si s i — • S o l u t i o n of t h i s second-order p a r t i a l d i f f e r e n t i a l equation r e q u i r e s knowledge of one i n i t i a l c o n d i t i o n and two boundary c o n d i t i o n s . For the case of continuous c a s t i n g , the i n i t i a l c o n d i t i o n i s At t = 0 and 0£x£x m , T = T 4.3 pour where ^ = Mid p o i n t of t r a n s v e r s e s e c t i o n Tpour = P o u r " i n 9 temperature The two boundary c o n d i t i o n s may be determined by c o n s i d e r i n g c o n t i n u i t y of heat t r a n s f e r at the s u r f a c e and centre plane of the b i l l e t . Thus at the surface of the s l a b . At t > 0 and x=x m ( H ) - - - - 4 . 4 aold where Q 1,=Heat f l u x out of mould mold At the c e n t r e plane, i f symmetry of heat t r a n s f e r on each face i s assumed, the net r a t e 6 of heat t r a n s f e r i s 0, so that At x=0, t £ 0 J I - 0 - - - 4 . 5 S o l u t i o n of the d i f f e r e n t i a l equations f o r heat t r a n s f e r has i n 42 t h e p a s t been a c c o m p l i s h e d by a n a l y t i c a l and n u m e r i c a l methods. The n u m e r i c a l methods have p r o v e d t h e m s e l v e s t o be most v e r s a t i l e i n t h e s o l u t i o n o f s o l i d i f i c a t i o n p r o b l e m s b e c a u s e t h e y p e r m i t t h e use o f v a r y i n g b o u n d a r y c o n d i t i o n s , r e l e a s e of l a t e n t h e a t o v e r a b r o a d f r e e z i n g r a n g e and t e m p e r a t u r e d e p e n d e n t t h e r m o - p h y s i c a l p r o p e r t i e s . Of t h e v a r i o u s n u m e r i c a l methods t h e e x p l i c i t and i m p l i c i t f i n i t e - d i f f e r e n c e t e c h n i q u e s have been employed w i d e l y i n c a s t i n g a p p l i c a t i o n s . The i m p l i c i t method has an a d v a n t a g e i n t h a t i t i s f r e e of a s t a b i l i t y c r i t e r i o n t h a t r e s t r i c t s t h e i n d e p e n d e n t c h o i c e of t i m e and d i s t a n c e i n t e r v a l s as r e q u i r e d by t h e e x p l i c i t t e c h n i q u e 6 * . F o r t h e p u r p o s e s o f t h i s model t h i s was t h e j u s t i f i c a t i o n f o r use of t h e i m p l i c i t - f i n i t e d i f f e r e n c e method. 4.1.2 D e r i v a t i o n o f t h e F i n i t e D i f f e r e n c e E q u a t i o n s The most e l e g a n t s o l u t i o n of t h e u n s t e a d y - s t a t e , h e a t -c o n d u c t i o n e q u a t i o n i n v o l v e s t h e a p p r o x i m a t i o n of t h e p a r t i a l d i f f e r e n t i a l s o f E q u a t i o n ( 4 . 2 ) by t h e use of t h e T a y l o r s e r i e s . 6 " Though t h e d e r i v a t i o n of t h e f i n i t e d i f f e r e n c e e q u a t i o n s i n t h i s way i s more r i g o r o u s i n t h a t t h e o r d e r o f t h e e r r o r s i n t h e a p p r o x i m a t i o n i s known, i d e n t i c a l e q u a t i o n s can be d e r i v e d by p e r f o r m i n g a s i m p l e h e a t b a l a n c e f o r e a c h node. D e r i v a t i o n of t h e f i n i t e - d i f f e r e n c e e q u a t i o n s i s p r e s e n t e d i n A p p e n d i x B. 43 4.1.3 C h a r a c t e r i s a t i o n of Input C o n d i t i o n s An important aspect of mathematical model f o r m u l a t i o n i s the manner i n which thermo-physical p r o p e r t i e s and boundary c o n d i t i o n s are handled. In p a r t i c u l a r c h a r a c t e r i s a t i o n of the thermal c o n d u c t i v i t y of the l i q u i d and the r e l e a s e of l a t e n t heat must be considered. Turbulent mixing i n the l i q u i d pool makes i t d i f f i c u l t to a s s i g n proper values to the thermal c o n d u c t i v i t y . Brimacombe 2 2 has shown th a t the exact value of thermal c o n d u c t i v i t y employed i n the l i q u i d pool does not a f f e c t the c a l c u l a t e d temperature f i e l d s i g n i f i c a n t l y . The e f f e c t i v e thermal c o n d u c t i v i t y i n the l i q u i d pool has been estimated from the l i t e r a t u r e to be seven times the thermal c o n d u c t i v i t y of the l i q u i d metal at the p a r t i c u l a r temperature. Latent-heat release i s accomplished by a technique commonly employed by other w o r k e r s 6 5 " 6 7 i . e , a d j u s t i n g the s p e c i f i c heat between the l i q u i d u s and s o l i d u s temperatures as shown below. C = C + L 4.6 m T l " T s where = S p e c i f i c heat i n the mushy zone C = S p e c i f i c heat evaluated at the known temperature of the node T^  = L i q u i d u s temperature Tg = S o l i d u s temperature 44 4.2 Meniscus S o l i d i f i c a t i o n Model Of the proposed mechanisms f o r o s c i l l a t i o n - m a r k formation i n b i l l e t c a s t i n g , s o l i d i f i c a t i o n of the meniscus has been given considerable importance i n recent years. During the mould downstroke, i t has been speculated that the meniscus i s allowed to remain i n contact with the mould w a l l for a f r a c t i o n of a second under c o n d i t i o n s s i m i l a r to those i n s t a t i c c a s t i n g . From mathematical model p r e d i c t i o n s Saucedo et a l " 9 proposed that the strong c h i l l i n g power of the mould cr e a t e s steep temperature qr a d i e n t s a l l o w i n g p a r t i a l s o l i d i f i c a t i o n of the meniscus. They assumed that the p a r t i a l l y s o l i d i f i e d s t e e l behaved as a s o l i d when only 20 percent s o l i d i f i e d , and used t h i s as a c r i t e r i o n for the formation of o s c i l l a t i o n marks. They employed an a r b i t r a r y heat t r a n s f e r c o e f f i c i e n t for c h a r a c t e r i s i n g the heat f l u x from the s t e e l to the mould. Because measured values of meniscus heat f l u x were a v a i l a b l e 2 0 for the present study, i t was considered worthwhile to examine the p o s s i b i l i t y of meniscus s o l i d i f i c a t i o n i n b i l l e t c a s t i n g . A two-dimensional h e a t - t r a n s f e r model was chosen to c a l c u l a t e heat t r a n s f e r from a curved-boundary l o s i n g heat from the top by r a d i a t i o n and convection and from the s t r a i g h t side to the mould. F i g 10 shows a schematic of the arrangement of nodes. Considering no r e l a t i v e v e l o c i t y between the mould and the b i l l e t i n the downstroke, the heat flow due to bulk motion i s neglected. A l s o , l i k e i n the previous model, heat flow toward the corners i s considered n e g l i g i b l e . Expecting r a p i d s o l i d i f i c a t i o n i n a short time i n t e r v a l , the model was 45 Figure 10 - Arrangement of nodes i n the meniscus s o l i d i f i c a t i o n model 46 formulated to simulate heat flow i n unsteady-state. 4.2.1 Formulation The equation governing such unsteady s t a t e heat t r a n s f e r two d i r e c t i o n s can be w r i t t e n as k [ " ( I f ] + h [-(If where A = Area of the node V = Volume of the node To s o l v e t h i s equation one i n i t i a l c o n d i t i o n and four boundary c o n d i t i o n s are r e q u i r e d . As shown before i n Eq.(4.2) the i n i t i a l c o n d i t i o n i s t=0, 0<x<x , 0<z£z , ' max' max ' T = T 4.8 pour The boundary c o n d i t i o n for t>0, x=0 plane i s ~ k(fx") = ^ o l d At z=0 the r a d i a t i v e and convective heat f l u x i s given by Q = h ( T - T . ) + oe ( T 4 - T 4 . ) 4.9 s conv amb amb 47 where h = Convective heat t r a n s f e r c o e f f i c i e n t conv o = Stefan Boltzmann constant e = E m i s s i v i t y T = Ambient Temperature amb At an a r b i t r a r y d i s t a n c e i n the x=0 plane and z=0 plane, two a d i a b a t i c boundaries were assumed to e x i s t . At z=z , 0<x£x max max k[iz] 4.10 At x=x , O^z^z max max 4 3T 8x1 = 0 4 .11 where X J J ^ = Maximum -distance i n X - d i r e c t i o n = Maximum d i s t a n c e i n Z - d i r e c t i o n S o l u t i o n t o the d i f f e r e n t i a l equation was done by the i m p l i c i t technique as i n the case of the previous model. However, to av o i d the problems a s s o c i a t e d with s o l v i n g a la r g e number of simultaneous equations i n the i m p l i c i t method a s p e c i a l procedure, o f t e n employed by other workers, 6 7 c a l l e d the a l t e r n a t i n g - d i r e c t i o n i m p l i c i t method was used. I t has an u n c o n d i t i o n a l s t a b i l i t y and converges w i t h d i s c r e t i z a t i o n e r r o r of the order of ( ( A x ) 2 + ( A t ) 2 ) (where Ax i s the node s i z e i n X-d i r e c t i o n and At i s the time s t e p ) . In t h i s method, each time i n t e r v a l At i s subdivided i n t o At/2 and c a l c u l a t i o n s are performed s u c c e s s i v e l y over the h a l f 48 time st e p s . In the f i r s t s t e p c a l c u l a t i o n s are made i m p l i c i t i n the X - d i r e c t i o n and e x p l i c i t i n the Z - d i r e c t i o n to o b t a i n an determined i n the second h a l f time step by doing a s i m i l a r c a l c u l a t i o n , but, i m p l i c i t i n Z - d i r e c t i o n and e x p l i c i t i n X-d i r e c t i o n u s i n g the intermediate temperatures obtained i n the f i r s t s t e p. Thus, i n each h a l f time a system of l i n e a r d i f f e r e n t i a l equations s i m i l a r t o those i n the pr e v i o u s model are obtained and solved using the Gaussian e l i m i n a t i o n technique. • « 4.2.2 C h a r a c t e r i s a t i o n Of Input C o n d i t i o n s The observed meniscus p r o f i l e s reported by Tomono et a l 6 8 i s shown i n F i g 11.For the purpose of the present model the height Z and with X are taken from the observed meniscus p r o f i l e . Node s i z e i n the X - d i r e c t i o n i . e . , Ax i s kept constant wh i l e i n the Z - d i r e c t i o n i t i . e . , Az i s v a r i e d using a root curve of the form shown below and the meniscus shape i s a r b i t r a r i l y d e f i n e d . i n t e r m e d i a t e temperature T . The f i n a l temperatures are AZ(N) = 2/X . min - { X - [ AX (N) ] } 4.12 min 20 -e—r 30 mm 1 O Observed — Calculated via interfacial tension (y steel/air 516 dyn/cm) Mould wall Figure 11 - Observed and c a l c u l a t e d meniscus p r o f i l e a f t e r Tomono et a l 6 9 50 where X . = Width of the meniscus min min AZ * Depth of the meniscus = Node s i z e i n Z - d i r e c t i o n N * Node number No attempt was made to o b t a i n a f i t w i t h the observed p r o f i l e . While r e c t a n g u l r nodes are s e l e c t e d elsewhere, t r i a n g u l a r nodes are assigned t o the curved boundary as shown i n F i g 10. L i b e r a t i o n of l a t e n t heat i s accomplised i n t h i s model using a d i f f e r e n t technique. Instead of a r t i f i c i a l l y i n c r e a s i n g the s p e c i f i c heat to i n c o r p o r a t e the l a t e n t heat r e l e a s e d , i t i s added as a heat generation term as shown i n the equation. The heat occumalation (Q ) i n any node i n h a l f time step i s given acc by acc = -pVC N * T - T 'pVLAf* ~ I ft)] [ft) J A . 1 3 where T* = L = At = * • Af = Temperature of node at the present time step Temperature of node a f t e r h a l f time step Latent heat of s o l i d i f i c a t i o n Time step F r a c t i o n of s o l i d that w i l l appear i n the h a l f time step under c o n s i d e r a t i o n , 51 Because Af i s unknown, i t has been approximated by Af which i s £ S the f r a c t i o n of s o l i d formed d u r i n g the previous time s t e p t o r e s u l t i n the present temperature. Thus eg.(4.13) becomes Q = -pVC acc p where Af N= f r a c t i o n s o l i d that formed i n the previous s time step t o r e s u l t i n the p r e s e n t l y T . known temperature T N. But such a formulation l e d to an o s c i l l a t i n g s o l u t i o n l e a d i n g to meaningless temperatures. Eg.(4.14) was l a t e r m o dified as TN - T*" pVLAfg 0f) fAtX v 2) A.14 *acc = -pVC T - T • (4) J N N faVLAf "I s r N * - i T - T * JT - T 4.15 Approximating T -T as T -T where T i s the temperature at the previous h a l f time step, eg.(4.15) becomes Q = -pVC acc p frN-T*1 rpvLAfNir s LTp- TNJL I M - T*1 4.16 With t h i s technigue there w i l l be no l a t e n t heat missed even when the nodal temperature jumps the l i g u i d u s or s o l i d u s temperatures. Latent heat was r e l e a s e d both using the Fe-C diagram and c o n s i d e r i n g l i n e a r s o l i d i f i c a t i o n over the mushy zone. A f t e r each time step the f r a c t i o n of s o l i d i n every node i s c a l c u l a t e d using one of the above technigues and no 52 d i f f e r e n c e was observed. One of the problems w i t h the a l t e r n a t i n g - d i r e c t i o n i m p l i c i t f i n i t e d i f f e r e n c e technique i s the treatment of the r a d i a t i o n boundary c o n d i t i o n . I n t r o d u c t i o n of the r a d i a t i o n c o n d i t i o n i n t o the heat balance equations renders the equations non-l i n e a r . In order to over come the problem, the boundary c o n d i t i o n Eq.(4.14) can be l i n e a r i s e d as = H total 4.17 where the over heat t r a n s f e r c o e f f i c i e n t H i s given by total H total = h conv + 4.18 53 V. RESULTS 5.1 M e t a l l o g r a p h i c E x a m i n a t i o n A f t e r m a c r o e t c h i n g of b i l l e t s e c t i o n s , t r a n s v e r s e as w e l l as l o n g i t u d i n a l , v a r i o u s f e a t u r e s - l i k e t h e c o l u m n a r zone l e n g t h , and s o l i d i f i c a t i o n bands were e x a m i n e d and t h e r e s u l t s a r e d e s c r i b e d b elow. 5.1.1 Columnar Zone L e n g t h T y p i c a l o f t h e r a d i a l c a s t i n g machine from w h i c h t h e b i l l e t s were o b t a i n e d , t h e c o l u m n a r d e n d r i t e s g r o w i n g from t h e o u t s i d e r a d i u s f a c e were seen t o be s h o r t e r t h a n t h o s e g r o w i n g from t h e i n n e r r a d i u s f a c e ( f r o m t h i s f a c e t h e d e n d r i t e s f r e q u e n t l y p e n e t r a t e d t o t h e c e n t r e o f t h e b i l l e t ). The c o l u m n a r zone l e n g t h was measured from t h e m i d f a c e on a l l s i d e s e x c e p t f o r t h e i n s i d e r a d i u s f a c e and t h e r e s u l t s a r e t a b u l a t e d i n T a b l e I I I . The i n f l u e n c e of c a r b o n , p h o s p h o r o u s , s u p e r h e a t , a v e r a g e mould h e a t f l u x , m e n i s c u s h e a t f l u x and Mn/S r a t i o on t h e c o l u m n a r zone l e n g t h measured f r o m t h e o u t s i d e r a d i u s were examined c a r e f u l l y . F i g 12 shows a p l o t of c o l u m n a r zone l e n g t h s of a l l t h e b i l l e t s a mples as a f u n c t i o n of t h e c a r b o n i n s t e e l . I t c a n be seen from t h e p l o t t h a t as c a r b o n i s i n c r e a s e d f r o m 0.13 t o 0.38, t h e l e n g t h of t h e c o l u m n a r zone d e c r e a s e s . Between 0.28% and 0.38% c a r b o n , t h e a v e r a g e c o l u m n a r zone l e n g t h ( a v e r a g e o f t h e l e n g t h of c o l u m n a r zone i n t h e t h r e e b i l l e t s o b t a i n e d d u r i n g t h e h e a t ) r e m a i n s unchanged. However, s t e e l b i l l e t s w i t h 0 Inside Radius J * Columnar Zone Length Outside Radius • O B A • ° 4 ° a pa DO o 8 A A Sample During Heat Q/ First • Second A Third O A Ol 6 •Q O • X • D 0 O AA A AO Q 0 ° 0 0 A A n ° A g ° A Aa 0-2 0-3 Carbon (%) ° 8 o§ 8 • A i O O o • A 8 i) 8 o 04 0-5 Figure 12 - P l o t of carbon versus the length of columnar zone (measured from the outside r a d i u s ) 55 c a r b o n c o n t e n t o f 0.38-0.45% e x h i b i t much l o n g e r c o l u m n a r zone l e n g t h s compared t o 0.28-0.36 c a r b o n s t e e l s . T h e r e a p p e a r s t o be a sudden change i n t h e c o l u m n a r zone l e n g t h i n t h e v i c i n i t y o f 0.37%C. As shown i n F i g s 13 and 14 P h o s p h o r o u s seems t o i n f l u e n c e t h e c o l u m n a r zone l e n g t h . In l o w - c a r b o n s t e e l s (0.13-0.20) h i g h e r p h o s p h o r o u s t e n d s t o promote t h e growth o f t h e e q u i a x e d zone ( F i g 1 3 ) . In 0.28-0.36% c a r b o n s t e e l s , F i g 14, P h o s p h o r o u s a p p e a r s t o r e d u c e t h e l e n g t h o f c o l u m n a r zone o n l y b eyond a l e v e l o f 0.025%P. In s t e e l s c o n t a i n i n g g r e a t e r t h a n 0.36% c a r b o n t h e c o l u m n a r zone r e m a i n e d l a r g e i r r e s p e c t i v e of t h e p h o s p h o r o u s c o n t e n t o f s t e e l . C o n t r a r y t o t h e g e n e r a l b e l i e f t h a t h i g h e r s u p e r h e a t s i n f l u e n c e t h e c o l u m n a r zone l e n g t h s s i n g u l a r l y , t h e m a c r o s t r u c t u r e o f t h e b i l l e t s e c t i o n s showed m i n i m a l dependance on s u p e r h e a t . C o n s i d e r i n g t h a t t h e s u p e r h e a t s h o u l d d r o p from th e b e g i n n i n g t o t h e end o f h e a t , t h e c o l u m n a r zone l e n g t h of th e t h i r d b i l l e t s h o u l d be s h o r t e r t h a n t h a t of t h e f i r s t b i l l e t o b t a i n e d i n t h e b e g i n n i n g o f t h e h e a t . I t has been o b s e r v e d t h a t i n a b o u t ' 4 0 % of t h e h e a t s , t h e t h i r d b i l l e t shows a l a r g e r or an e q u a l c o l u m n a r zone l e n g t h compared t o t h e f i r s t b i l l e t . In l o w - c a r b o n s t e e l s , no c o r r e l a t i o n c o u l d be o b t a i n e d between th e s u p e r h e a t and c o l u m n a r zone l e n g t h . In s t e e l s w i t h 0.28-0.36% c a r b o n , as shown i n F i g 15, t h e r e a p p e a r s t o be a t r e n d w h e r e i n h i g h e r s u p e r h e a t s gave r i s e t o s l i g h t l y h i g h e r c o l u m n a r zone l e n g t h s . But p a r t o f t h i s e f f e c t may a g a i n be due t o t h a p r e s e n c e o f p h o s p h o r o u s . Thus w h i l e h i g h e r p h o s p h o r o u s i n 56 60 | 50 c o M o c o O 40 c 30 20 (0-13,22) O, O (015,22) (OI324) • o Low Carbon (%C,AT) 0(014,19) O (013,34) 0(016,28) (015,46) O O(0i6,28) (0 20,46) O (014,19)0 O (014,18) O (01422) (014,25)0 0010 0020 0030 Phosphorus {%) 0040 Figure 13 - P l o t of phosphorous versus the length of columnar zone (measured from the outside radius) i n s t e e l s with carbon ranging from 0.13-0.20 45 40 if) •^35 ro 0 3 0) c 30 o c I § 2 5 §20 0) o o o o o o o Carbon Range 0 2 8 - 0 3 6 % 15 '8 i i i i 010 0020 O030 0 040 0 050 0 060 Phosphorus (%) Figure 14 - Plo t of phosphorous versus the le n g t h of columnar zone (measured from the outside radius) i n s t e e l s with carbon ranging from 0.28-0.36 40 30 I • i i o (0020) -0(00.7) 0(0028) O ( 0 0 , 7 , • (0022)0 O (0026) 0(0 018) O (0-028) 0(0014) * O (0016) 0(0 028) O (0016) O (0022) O (0035) (0023)0 O(00l6) O(O026) -0 (0 035) -- Carbon Range 0 28-036% -(% Phosphorus) -O (0 061) -i i i » 10 20 30 40 50 60 Superheat (°C) Figure 15 - Plo t of superheat i n the tundish_versus_the length of columnar zone (measured from the o u t s i d e r a d i u s ) i n s t e e l s with carbon ranging from 0.28-0.36 59 s t e e l f a v o u r s a l a r g e e q u i a x e d zone, h i g h e r s u p e r h e a t a p p e a r s t o oppose i t . In t h e range o f s u p e r h e a t s under c o n s i d e r a t i o n , p h o s p h o r o u s seem t o have a more dominant i n f l u e n c e i n d e t e r m i n i n g t h e e x t e n t of c o l u m n a r z o n e . Beyond 0.38%C, t h e r e a p p e a r s t o be no i n f l u e n c e o f e i t h e r t h e p h o s p h r o u s c o n t e n t o r t h e s u p e r h e a t s i n c e t h e c o l u m n a r zone l e n g t h i s v e r y l a r g e , e x t e n d i n g a l m o s t t o t h e c e n t r e o f t h e b i l l e t . F i g 16 shows t h e p l o t of c o l u m n a r zone l e n g t h from t h e t h r e e b i l l e t s o f e a c h h e a t p l o t t e d a g a i n s t t h e measured a v e r a g e mould h e a t f l u x . In s p i t e of t h e s c a t t e r t h e r e i s a good c o r r e l a t i o n s h o wing h i g h e r mould h e a t f l u x e s g i v i n g r i s e t o s h o r t e r c o l u m n a r zone l e n g t h s . I t s h o u l d be n o t e d , however, t h a t s t e e l s w i t h c a r b o n g r e a t e r t h a n 0.36% showed h i g h e r h e a t f l u x e s as w e l l as l a r g e c o l u m n a r z o n e s . The m e n i s c u s h e a t f l u x o b t a i n e d f r o m t h e e x p e r i m e n t s d i d n o t c o r r e l a t e w i t h t h e s t r u c t u r e i n t h e i n d i v i d u a l c a r b o n r a n g e s examined. mm. V i z : 0.13-0.20, 0.28-0.36, 0.38-0.41. 5.1.2 S u b s u r f a c e S t r u c t u r e When s u b s u r f a c e e t c h e s were examined, m e n i s c u s - s h a p e d 'hooks' a r e e v i d e n t a t t h e l o c a t i o n o f e a c h o s c i l l a t i o n mark i n t h e c a s e o f h i g h - c a r b o n s t e e l b i l l e t s . F i g s . 17 t o 20 show t y p i c a l ' h o o k s ' p r e s e n t i n s a m p l e s 24276-2, 24276-3, 24277-1 and 24277-3. F i g 21 shows no d i f f e r n c e i n t h e l e n g t h o r t h e shape of 'hooks' p r e s e n t a t t h e m i d f a c e and o f f - c o r n e r r e g i o n s of t h e b i l l e t 24277-2. Of t h e 10 samples examined, o n l y two s a m p l e s 24276-1 and 24266-1 d i d n o t e x h i b i t t h e f o r m a t i o n 60 70, • 6 0 £ £ | 50 65 ro c o NI • O c E O o 40 c OJ 30 O IJ • o o o o • • O O A Water Velocity %C O >865m/s <020 • <8j65 " <020 A >865 " 028-036 O <8£5 " II • 48-S8 " Q38-04I o A A A A 8 % OA A A A O A A O A O A A 20 1300 1500 1700 1900 Aver/age Heat Flux (kW/m ) 2100 gure 16 - P l o t of average mould heat f l u x versus length of columnar zone (measured from the outside radius) 61 F i g u r e 17 - S t r u c t u r e a r o u n d o s c i l l a t i o n mark o f t h e s e c o n d b i l l e t from h e a t 24276. (11X) 62 Figure 18 - Structure around o s c i l l a t i o n mark of the l a s t b i l l e t from heat 24276. (1IX) 63 Figure 19 - Structure around o s c i l l a t i o n mark of the f i r s t b i l l e t from heat 24277. (11X) 64 Figure 20 - S t r u c t u r e around o s c i l l a t i o n mark of the l a s t b i l l e t from heat 24277. (11X) 65 F i g u r e 21 - S t r u c t u r e a r o u n d o s c i l l a t i o n mark o f t h e s e c o n d b i l l e t from h e a t 24272; o f f - c o r n e r r e g i o n on t h e l e f t and m i d f a c e r e g i o n on t h e r i g h t . (11X) 66 of m e n i s c u s shaped 'hooks'. On t h e o t h e r hand l o w - c a r b o n s t e e l b i l l e t s (24261-2, 24270-3, 24269-3) d i d not show t h e p r e s e n c e o f h o o k s . F i g 22 shows the s u b s t r u c t r u e o b t a i n e d f o r 24269-3 and F i g 23 shows a c l o s e - u p o f one of t h e a r e a s c o r r e s p o n d i n g t o an o s c i l l a t i o n mark. T a b l e IV summarises t h e i n f o r m a t i o n o b t a i n e d from t h e s u b s u r f a c e e t c h e s r e g a r d i n g t h e p r e s e n c e of h o o k s . In h i g h - c a r b o n s t e e l s , t h e a p p e a r a n c e of hooks seems t o be r e l a t e d t o s u p e r h e a t , water v e l o c i t y and t h e d e p t h of o s c i l l a t i o n marks. In e a c h of t h e h e a t s i n v e s t i g a t e d , d e e p e r o s c i l l a t i o n marks seem t o be a s s o c i a t e d w i t h 'hooks'. F i g 24 shows t h e s u b s u r f a c e e t c h e s of H e a t s 24276-1, 24276-2, 24276-3. The f i r s t b i l l e t sample has no hooks; but t h e e n s u i n g b i l l e t s do. The l e n g t h of t h e hooks i n t h e t h i r d sample i s a b o u t d o u b l e of t h o s e seen i n t h e s e c o n d sample. T h i s would, a t a f i r s t g l a n c e , s u g g e s t s t h a t as t h e s u p e r h e a t of t h e s t e e l d r o p s from 1st t o 3 r d sample hooks a p p e a r t o grow l o n g e r . Heat number 24266 shows a s i m i l a r t r e n d . However, h e a t s 24277 and 24272 show t h e o p p o s i t e b e h a v i o u r . T h i s s u g g e s t s t h a t t h e r e i s y e t a n o t h e r f a c t o r i n f l u e n c i n g t h e f o r m a t i o n of h o o k s . In h e a t s where mould water v e l o c i t y was 6.7 t o 7.2 m/s h o o k s seem t o a p p e a r i r r e s p e c t i v e o f t h e s u p e r h e a t . S i m i l a r l y a t h i g h e r water v e l o c i t i e s o f 9.7 m/s e i t h e r no hooks o r o n l y m i n o r hooks a p p e a r d e p e n d i n g on t h e s u p e r h e a t . I t s h o u l d a l s o be n o t e d t h a t i n h e a t 24272, w i t h water v e l o c i t y of 9.7 m/s t h e f o r m a t i o n o f 'hooks' was g r e a t e r t h a n when a v e l o c i t y of 3.1 m/s was e m p l o y e d . F i g 25 p r e s e n t s a c l o s e up o f t h e ' meniscus s h a p e d hook' 67 F i g u r e 22 - S t r u c t u r e l o n g i t u d i n a l s e c t i o n o f t h e a r o u n d o s c i l l a t i o n marks i n t h e l a s t b i l l e t f r o m h e a t 24269. (2.9X) 68 F i g u r e 23 - C l o s e up of t h e s t r u c t u r e a r o u n d o s c i l l a t i o n mark o f t h e l a s t b i l l e t from h e a t 24269. (11X) F i g u r e 24 - S t r u c t u r e a r o u n d o s c i l l a t i o n marks i n t h e l o n g i t u d i n a l s e c t i o n s from t h e h e a t 24276; from l e f t t o r i g h t . F i r s t b i l l e t , s e c o n d b i l l e t and l a s t b i l l e t . (2.9X) g u r e 25 - C l o s e - u p of t h e s t r u c t u r e a r o u n d o s c i l l a t i o n mark of t h e s e c o n d b i l l e t from h e a t 24276. (17X) 71 f r o m t h e s e c o n d b i l l e t o f t h e h e a t 24276. The o r i e n t a t i o n of th e d e n d r i t e s a d j a c e n t t o t h e hook, i . e . , p e r p e n d i c u l a r t o t h e c u r v a t u r e , c l e a r l y r e v e a l s t h a t t h e d i r e c t i o n o f s o l i d i f i c a t i o n i s d i f f e r e n t from t h a t e l s e w h e r e ( c l o s e t o t h e edge o f t h e b i l l e t ) . The f i n e ' c h i l l ' t y p e of s t r u c t u r e a r o u n d t h e o s c i l l a t i o n marks i n d i c a t e s t h a t d u r i n g t h e f o r m a t i o n o f t h e s e marks h e a t e x t r a c t i o n r a t e s were v e r y h i g h . T h i s c o u l d c l e a r l y be s e e n by c o m p a r i n g t h i s c h i l l s t r u c t u r e t o t h e p r o n o u n c e d d e n d r i t i c g r o w t h n e a r t h e s u r f a c e i n between t h e o s c i l l a t i o n m arks. U s u a l l y t h e s e hooks a r e a s s o c i a t e d w i t h some amount of o v e r f l o w of m e t a l on t o p . The s t r u c t u r e i n t h i s p o r t i o n o f s t e e l i s f i n e w i t h s m a l l c r y s t a l l i t e s . T h e s e o v e r f l o w s l o o k l i k e ' b l e e d s ' when v i e w e d n o r m a l l y from t h e b i l l e t s u r f a c e ; t h e r e i s no e v i d e n c e of ' t e a r s ' c a u s i n g m e t a l t o b l e e d as s u g g e s t e d o r i g i n a l l y by S a v a g e 5 0 . 5.2 Band F o r m a t i o n When m a c r o e t c h e s of t r a n s v e r s e s e c t i o n s a r e c a r e f u l l y e x amined, a w h i t e band can be seen r u n n i n g a l l a r o u n d e a c h s e c t i o n a b o u t 4-7 mm from t h e edge of t h e b i l l e t . I t i s o f t e n not c l e a r l y v i s i b l e making p h o t o g r a p h y d i f f i c u l t . However, F i g 26 shows t h e w h i t e bands i n a c o r n e r a r e a o f t h e b i l l e t f r o m Heat 24249. F i g 27 shows t h e w h i t e band i n t h e m i d f a c e a r e a o f t h e b i l l e t 24242-3. I t was o b s e r v e d t h a t when t h e c o r r e s p o n d i n g s u r f a c e has a deep o s c i l l a t i o n mark, t h e w h i t e band i s v e r y wavy and i s p o s i t i o n e d c l o s e r t o t h e edge o f t h e b i l l e t . F i g 28 72 F i g u r e 26 - W h i t e band i n t h e o b t u s e a n g l e c o r n e r of t h e s e c o n d b i l l e t from h e a t 24259. (2.5X) 73 Figure 27 - White and dark bands at the edge of the l a s t b i l l e t from heat 24242. 74 Figure 28 - Thin and wavy white band at the midface of the second b i l l e t from heat 24251. 75 d e p i c t s s u c h an o c c u r r a n c e . T h i s c o u l d be more c l e a r l y s e e n i n th e l o n g i t u d i n a l e t c h e s f o r example i n F i g 24. The band draws e s p e c i a l l y c l o s e t o d e e p e r o s c i l l a t i o n marks. C o n v e r s e l y , t h e w h i t e band i s s t r a i g h t i n sample 24276-1 w h i c h had no hooks and v e r y s h a l l o w o s c i l l a t i o n marks. At t h i s w h i t e band, t h e r e seems t o be a f l u c t u a t i o n i n t h e c o m p o s i t i o n of s t e e l c a u s i n g i t t o e t c h d i f f e r e n t l y . A t t e m p t s have been made t o a n a l y s e t h e c o m p o s i t i o n a l v a r i a t i o n a c r o s s t h e band u s i n g t h e m i c r o p r o b e . B e c a u s e t h e d i f f e r e n c e i n c o m p o s i t i o n i s below t h e r e s o l u t i o n of the p r o b e no p o s i t i v e i d e n t i f i c a t i o n o f e i t h e r t h e e x t e n t o r t h e t y p e o f s e g r e g a t i o n c o u l d be made. In most o f t h e t r a n s v e r s e s e c t i o n t h e r e i s y e t a n o t h e r v i s i b l e ' d a r k ' band a b o u t 9-12mm from t h e edge of t h e b i l l e t ( t h i s d i s t a n c e i s a t t h e m i d f a c e ) . Deeper e t c h e s show t h i s band more d i s t i n c t l y . F i g 29 i s a u n i q u e example o f s u c h a d a r k band f o r m a t i o n a l l a r o u n d t h e t r a n s v e r s e s e c t i o n . F i g 30 a l s o shows a d a r k band i n B i l l e t 23490-3. In t h e m a j o r i t y of b i l l e t s e c t i o n s , t h e d a r k band a p p e a r s as seen i n F i g 31 showing t h e t r a n s v e r s e s e c t i o n o f B i l l e t 24251-3. I t has been o b s e r v e d t h a t i n many b i l l e t s e c i o n s t h e ' w h i t e ' and 'dark' bands o v e r l a p n e a r th e c o r n e r / o f f - c o r n e r a r e a . The d a r k band e i t h e r a p p e a r s d i s t i n c t l y c l o s e t o t h e w h i t e band o r d i s a p p e a r s g r a d u a l l y as one p r o c e e d s t o l o o k from m i d f a c e t o t h e c o r n e r . F i g s 27 and 32 a r e t y p i c a l examples o f t h i s o b s e r a v a t i o n . I t s h o u l d be p o i n t e d o u t t h a t t h e maximum d i s t a n c e between the d a r k band and t h e s u r f a c e o f t h e b i l l e t i s n o t a l w a y s a t t h e m i d p o i n t o f t h e f a c e ( s e e F i g 3 1 ) . On d i f f e r e n t f a c e s t h i s 76 Figure 29 - Prominent dark band in the second b i l l e t from heat 24221 . 77 F i g u r e 30 - Dark band i n t h e l a s t b i l l e t f r o m h e a t 24490. 78 F i g u r e 31 - Dark band m e r g i n g i n t o w h i t e band a t t h e o f f -c o r n e r r e g i o n ; l a s t b i l l e t f r o m h e a t 24251. 79 F i g u r e 32 - Dark band m e r g i n g i n t o w h i t e band a t t h e o f f -c o r n e r r e g i o n ; s e c o n d b i l l e t f r o m h e a t 23471. 80 maximum d i s t a n c e i s o f f - c e n t e r by v a r y i n g l e n g t h s . T h i s c l e a r l y p o i n t s o u t t h a t i n any g i v e n c r o s s - s e c t i o n under c o n s i d e r a t i o n h e a t e x t r a c t i o n i s not t h e same on a l l f o u r f a c e s and t h e h i g h e s t h e a t f l u x need n o t n e c e s s a r i l y be a t t h e m i d - p o i n t of t h e f a c e . The d i s t a n c e from t h e edge of t h e b i l l e t where t h e s e bands a p p e a r a t t h e m i d f a c e and o f f - c o r n e r a r e a s of t h e t r a n s v e r s e s e c t i o n s was measured i n a s e r i e s o f b i l l e t s a m p l e s . The r e s u l t s a r e p r e s e n t e d i n T a b l e V. I t has been o b s e r v e d t h a t i n l o w - c a r b o n s t e e l b i l l e t s t h e s e bands a r e not v i s i b l e e x c e p t i n a few i n s t a n c e s ( m a i n l y b e c a u s e l o w - c a r b o n b i l l e t s e t c h p o o r l y w i t h o u t much d e t a i l ) where as i n h i g h - c a r b o n b i l l e t s t h e y a r e r e a d i l y v i s i b l e . However, where t h e s t r u c t u r e i s c o m p l e t e l y e q u i a x e d , band f o r m a t i o n i s n o t a l w a y s c l e a r l y v i s i b l e t o make any measurements; but a g a i n , c o m p l e t e l y e q u i a x e d s t r u c t u r e s show v e r y l i t t l e d e t a i l when m a c r o e t c h e d . One of t h e i n t e r e s t i n g o b s e r v a t i o n s i s t h a t i n rhomboid b i l l e t s e c t i o n s , t h e w h i t e band i s c l o s e r t o t h e edge o f t h e b i l l e t a t t h e o b t u s e - a n g l e c o r n e r s t h a n a t t h e a c u t e - a n g l e c o r n e r s . In o t h e r words, t h e s h o r t d i a g o n o l o f t h e rhomboid b i l l e t i n v a r i a b l y has t h e w h i t e band c l o s e s t t o t h e edge o f a t l e a s t one o f f - c o r n e r a r e a . T h i s c h a r a c t e r i s t i c o f t h e w h i t e band r e v e a l s a t y p i c a l r e / e n t r a n t c o r n e r , eg., F i g 26 shows m a g n i f i e d v i e w o f w h i t e band f o r m a t i o n c l o s e r t o t h e edge of t h e b i l l e t a t t h e o b t u s e a n g l e c o r n e r . F i g 33 shows t h e whole t r a n s v e r s e s e c t i o n of t h e b i l l e t p r e s e n t e d i n F i g 26. 81 Figure 33 - Macro-etch of the transverse s e c t i o n of the second b i l l e t from heat 24249. 82 5.3 C r a c k Format i o n O f f - c o r n e r c r a c k s were examined i n b i l l e t s f r o m H e a t s 24090 t o 24278. T h e s e c r a c k s o f t e n a r e a s s o c i a t e d w i t h a s u r f a c e d e p r e s s i o n ( F i g 34) and a r e p o s i t i o n e d about 4 t o 6 mm from t h e b i l l e t s u r f a c e . They a p p e a r r a n d o m l y on any one, o r many, o f t h e e i g h t o f f - c o r n e r s i t e s . T a b l e VI shows t h e p l a c e m e n t o f n e a r e s t ( t o c o r n e r ) o f f - c o r n e r c r a c k s and t h e t o t a l number of c r a c k s i n e a c h b i l l e t sample. O f f - c o r n e r c r a c k s a p p e a r t o form i n t h e v i c i n i t y o f t h e w h i t e b a n d . F i g 35 shows t h e p r o x i m i t y o f t h e w h i t e band and t h e c r a c k ( s ) a t t h e o f f - c o r n e r r e g i o n . T a b l e V I I p r e s e n t s t h e s e v e r i t y o f c r a c k i n g i n t h e v a r i o u s b i l l e t s e x a mined t o g e t h e r w i t h s e l e c t e d o p e r a t i n g d a t a . When th e number o f c r a c k s p e r sample a r e p l o t t e d a g a i n s t t h e water f l o w r a t e employed, F i g s 36 t o 38, i t c a n be s e e n t h a t t h e s e v e r i t y o f c r a c k i n g i s h i g h e s t a t 26.50 - 27.76 1/s ( e q u i v a l e n t t o a water v e l o c i t y o f 7.25 - 7.65 m/s). A l s o a r o u n d 17.66 -18.93 1/s ( e q u i v a l e n t t o a water v e l o c i t y of 3.35 - 3.85 m/s) o f water f l o w r a t e a l s o t h e r e a p p e a r s t o be e x t e n s i v e c r a c k i n g i n t h e b i l l e t s . T h i s seems t o be t r u e b o t h i n t h e c a s e of low-(<0.20%C) and h i g h - c a r b o n s t e e l s . In t h e r a n g e s t u d i e d n e i t h e r s t e e l c h e m i s t r y nor s u p e r h e a t i n f l u e n c e d t h e c r a c k s e v e r i t y w i t h t h e e x c e p t i o n o f t h e s u l p h u r i n t h e s t e e l . Though t h e amount o f s u l p h u r d i d n o t c o r r e l a t e w e l l w i t h t h e c r a c k s e v e r i t y , Mn/S r a t i o had a d e f i n i t e i n f l u e n c e . F i g 39, 40, 41 show t h e r e l a t i o n 83 Figure 34 - Off-corner cracks from the heat 24276. 84 F i g u r e 35 - L o c a t i o n of o f f - c o r n e r c r a c k s and w h i t e band a t t h e o b t u s e a n g l e c o r n e r o f s e c o n d b i l l e t 24277. 85 T 1 I r 8 I 6 in 0) Q. is 4 O OJ -Q e All Carbons J I I I L T- I r 31 3 6 4 1 4 6 5 2 5 9 64 75 8 2 8 9 Water Velocity (m/s) F i g u r e 36 - P l o t of water v e l o c i t y versus the number of o f f - c o r n e r cracks per b i l l e t . Carbon range 0.13-0.41. Carbon Range 0 1 3 - 0 2 0 % CO 447 57 73 86 Water Velocity (m/s) 98 Figure 37 - P l o t of water v e l o c i t y versus the number of o f f - c o r n e r cracks per b i l l e t . Carbon range 0.13-0.20. 87 8 Carbon Range >0 20 % O) GO 0) a. o o o E 3 32 4 47 5 9 7 3 Water Velocity (m/s) 8 6 9 8 Figure 38 - P l o t of water v e l o c i t y versus the number of o f f - c o r n e r cracks per b i l l e t . Carbon range 0.28-0.41. 88 6 -g 4 ° 3 i O o o o Severity Rating 1 No cracks 2 Very mild 3 Mild 4 Medium 5 Extensive 6 Serious Water Velocity 615-72 m/s 2 - O O 10 20 30 Mn/S Ratio 40 F i g u r e 39 - P l o t of Mn/S versus the s e v e r i t y of o f f - c o r n e r c racks water v e l o c i t y range 6 .15-7.2 m/s. 6 1 ' T— Water Velocity <60m/s O O O O Severity Roting 1 No cracks 2 Very mild 3 Mild 4 Medium 5 Extensive 6 Serious o o co VO OO O OO 10 20 30 Mn/S Ratio 40 50 Figure 40 - P l o t of Mn/S versus the s e v e r i t y of o f f - c o r n e r cracks water v e l o c i t y <6 m/s. Water Velocity >865m/s o o Severity Rating 1 No cracks 2 Very mild 3 Mild 4 Medium 5 Extensive 6 Serious o o o o o o o o o o OA 10 20 30 Mn/S Ratio 40 50 Figure 41 - P l o t of Mn/S versus the s e v e r i t y of o f f - c o r n e r cracks water v e l o c i t y >8.65 m/s. 91 between c r a c k s e v e r i t y and Mn/S r a t i o i n t h r e e r a n g e s of w ater f l o w r a t e s employed. R a t i o s above 25-30 seem t o p r e v e n t c r a c k f o r m a t i o n i r r e s p e c t i v e o f t h e water f l o w r a t e . In t h e water v e l o c i t y r ange 6.7 t o 8 m/s ( water f l o w r a t e o f 25-28 1 / s ) , as t h e Mn/S i n c r e a s e d from 14 t o 36, c r a c k s e v e r i t y showed a s t e a d y d r o p , F i g 39. However, i n o t h e r water v e l o c i t y r a n g e s i n v e s t i g a t e d , i . e , above 9.7 m/s( water f l o w r a t e of 32 1/s ) and below 6.7 m/s ( water f l o w r a t e of 25 1/s ) c r a c k s e v e r i t y seems t o v a r y randomly when t h e Mn/S r a t i o i s l o w e r t h a n 25-30, F i g 40 and 41 r e s p e c t i v e l y . In Heat 24109 w h i c h was c a s t w i t h o u t s e c o n d a r y c o o l i n g water i t was f o u n d t h a t t h e o f f - c o r n e r c r a c k s d e v e l o p e d on a l l f o u r s i d e s of t h e b i l l e t and p e n e t r a t e d q u i t e d e e p l y t o w a r d t h e c e n t r e o f t h e s e c t i o n a l o n g t h e d i a g o n a l ( F i g 4 2 ) . I t can be seen t h a t t h e r e i s b u l g i n g a t t h e m i d f a c e on t h e s t r a i g h t s i d e s and a s u r f a c e d e p r e s s i o n n e a r t h e c r a c k . As e x p l a i n e d e a r l i e r t h e o u t e r and i n n e r r a d i u s f a c e s o f t h e b i l l e t were r o l l e d so t h a t b u l g i n g c o u l d n o t be s e e n . The i n f l u e n c e of mould wear ( t o w a r d s t h e bottom) on c r a c k i n g i s c o n s i d e r e d h e r e . F i g 5 shows t h e mould d i s t o r t i o n o f t h e mould i n w h i c h b i l l e t s f r o m Campaigns 5 and 6 were c a s t . In s p i t e of t h e b u l g e on t h e c u r v e d s i d e s t o w a r d t h e b o t t o m o f t h e mould, b i l l e t s c a s t i n t h i s mould d i d n o t show any o f f -c o r n e r c r a c k i n g . F i g 43 shows t h e d i s t o r t i o n p r o f i l e o f t h e mould i n w h i c h a l l t h e b i l l e t s of Campaign 12 were c a s t . I t i s q u i t e e v i d e n t t h a t d e s p i t e n e g l i g i b l e mould wear a t t h e e x i t , c r a c k i n g was o b s e r v e d i n t h e samples c a s t t h r o u g h t h i s mould. 92 F i g u r e 42 - Off-corner cracks i n the heat 24109. 93 Distance Between 5600 5650 Opposite Faces (in) 5600 5650 • i Straight Wall i i Curved Wall 0 - • A a o - • A • ak 100 - ryffr •A • & A3 i 200 ZO E Ok AD I 300 Oft 3 in O as 400 "o a. Ok £ 500 ah ZC E o i t a> u 600 on zn c o is b 700 •A ZD • A • Qt\ ftm i i i i i i 141 142 143 144 Distance Between 141 Opposite 142 143 144 Faces (mm) F i g u r e 43 - M o l d d i s t o r t i o n a f t e r Campaign 12. 94 T h i s s u g g e s t s t h a t mould wear seems t o be not a p r e - r e q u i s i t e f o r o f f - c o r n e r c r a c k i n g . 5.4 R h o m b o i d i t y R h o m b o i d i t y , as c h a r a c t e r i s e d by t h e d i f f e r e n c e i n d i a g o n a l s , was measured i n a number of b i l l e t s . T a b l e V I I I shows t h e r e s u l t s . V a r i a t i o n i n r h o m b o i d i t y r a n g e s from 0.0 t o as h i g h as 9.5mm i n t h e d i f f e r e n c e i n d i a g o n a l s . C a r b o n , water f l o w r a t e and s u p e r h e a t o f t h e s t e e l showed no d i s c e r n a b l e i n f l u e n c e on r h o m b o i d i t y . Over a Campaign o f 40 h e a t s , r h o m b o i d i t y f l u c t u a t e d r a n d o m l y . In Campaign 8 when t h e mould was c o n s t r a i n e d a t two t h e r m o c o u p l e l o c a t i o n s on t h e s t r a i g h t w a l l i n Heat 24090, r h o m b o i d i t y was n e g l i g i b l e . I t s h o u l d be n o t e d t h a t even a f t e r r e l e a s i n g t h e c o n s t r a i n t s i n t h e h e a t 24100 (Campaign 9 ) , w h i c h r a n o n l y f o r 20 m i n u t e s n e g l i g i b l e r h o m b o i d i t y was o b s e r v e d i n t h e t h r e e b i l l e t s t h a t were c o l l e c t e d . S l i g h t c o n c a v i t y was o b s e r v e d i n t h e b i l l e t f a c e i n c o n t a c t w i t h t h e c o n s t r a i n e d s i d e of t h e mould as shown i n t h e F i g 44. 5.5 O s c i l l a t i o n Marks The c o n t i n u o u s - c a s t i n g mould a t W e s t e r n Canada S t e e l i s o s c i l l a t e d v e r t i c a l l y a t a s p e e d d e p e n d i n g on t h e c a s t i n g v e l o c i t y t o r e s u l t i n a p r e d e t e r m i n e d n e g a t i v e s t r i p t i m e d u r i n g ure 44 - Concavity i n the s t r a i g h t face i n the second b i l l e t from 24090. 96 t h e d o w n s t r o k e . The v a r i o u s c a s t i n g s p e e d s a nd t h e c o r r e s p o n d i n g o s c i l l a t i o n f r e q u e n c i e s e m p l o y e d a t t h e W e s t e r n Canada S t e e l a r e p r e s e n t e d i n T a b l e 3.1 o f C h a p t e r 3. A t an a v e r a g e c a s t i n g s p e e d o f 36mm/sec ( 8 5 i p m ) t h e o s c i l l a t i o n s p e e d e m p l o y e d i s 75 m i n - 1 t o r e s u l t i n a n e g a t i v e s t r i p t i m e o f 0.27 s. T h i s i s c a l c u l a t e d u s i n g t h e E q u a t i o n 2.1 ( i n c h a p e r . 2 ) f o r s i n u s o i d a l o s c i l l a t i o n o f m o u l d . The s t r o k e l e n g t h p r e s e n t l y i 18.44mm. B i l l e t s u r f a c e s a r e c h a r a c t e r i s e d by p e r i o d i c u n d u l a t i o n w i t h a s p a c i n g o f 29-30mm. T h i s c o i n c i d e s w i t h t h e p i t c h o f o s c i l l a t i o n m a r k s c a l c u l a t e d a s t h e r a t i o o f c a s t i n g s p e e d t o o s c i l l a t i o n . I t s h o u l d be e m p h a s i s e d t h a t t h e sha p e o f t h e s e m a r k s i s q u i t e i r r e g u l a r a n d t h e s p a c i n g v a r i e s by ±2mm. B u t c o n s i d e r i n g t h e v a r i a t i o n i n t h e c a s t i n g s p e e d a n d o s c i l l a t i o n f r e q u e n c y s u c h v a r i a t i o n i s r e l a t i v e l y s m a l l . F i g s 45 t o 49 show some o f t h e t y p i c a l s u r f a c e s o f t h e b i l l e t s o f l o w - and h i g h - c a r b o n s t e e l s . B a s e d on v i s u a l o b s e r v a t i o n t h e s e v e r i t y o f o s c i l l a t i o n m arks was i n d e x e d a n d t h e r e s u l t s a r e p r e s e n t e d i n T a b l e I X . From F i g 50 i t i s c l e a r t h a t l o w - c a r b o n s t e e l s show d i s t i n c t l y d e e p e r m a r k s c o m p a r e d t o h i g h e r c a r b o n s t e e l s . P r o f i l o m e t e r m e a s u r e m e n t s o f t h e o s c i l l a t i o n m a r k s i n h i g h c a r b o n s t e e l b i l l e t s a r e p r e s e n t e d i n T a b l e X. T h e s e r e v e a l t h a t i n most o f t h e h e a t s t h e d e p t h o f o s c i l l a t i o n m a r k s i n c r e a s e s f r o m t h e b e g i n n i n g o f c a s t i n g t o t h e end p r o b a b l y a s r e s u l t o f d e c r e a s i n g s u p e r h e a t . H owever, i n h e a t s where w a t e r v e l o c i t y h a s been c h a n g e d t h e i n f l u e n c e o f s u p e r h e a t i s n o t 97 F i g u r e 45 - B i l l e t s u r f a c e from t h e h e a t 24270 l a s t b i l l e t ) c a s t i n g d i r e c t i o n down. (0.13%C, 98 F i g u r e 4 6 - B i l l e t s u r f a c e from t h e h e a t 2 4 2 7 1 ( 0 . 2 0 % C , s e c o n d b i l l e t ) c a s t i n g d i r e c t i o n down. 99 Figure 47 - B i l l e t surface from the heat 24276 (0.30%C, f i r s t b i l l e t ) c a s t i n g d i r e c t i o n down. 100 gure 48 - B i l l e t s u r f a c e from the heat second b i l l e t ) c a s t i n g d i r e c t i o n 24265 down. (0.35%C, F i g u r e 49 - B i l l e t s u r f a c e f r o m t h e h e a t 24273 t h i r d b i l l e t ) c a s t i n g d i r e c t i o n d o w n . (0.41%C, 102 OOO c o u 10 O OOOO Visual Estimation of Depth 1 None 2 Mild 3 Deep 4 Very Deep 5 Severely rDeep o 3 OOO O OOOO o o -> co o. 2 a  v Q o ooo o o -0-1 0-2 03 04 Carbon (% Figure 50 - P l o t of carbon versus v i s u a l e s t i m a t i o n of the depth of o s c i l l a t i o n marks. 103 e v i d e n t . In h e a t s 24266, 24263, 24277 as t h e water v e l o c i t y i s i n c r e a s e d f r o m a low v a l u e o f 4 - 7.5 m/s (water f l o w r a t e o f 20 - 27.5 1/s) t o above 9.2 m/s (water f l o w r a t e of 31.5 1/s, o s c i l l a t i o n marks t e n d t o become s h a l l o w e r i n s p i t e of a p o s s i b l e d r o p i n s u p e r h e a t . However, i n c a s e of t h e Heat 24265, t h e r e a p p p e a r s t o be no change i n t h e d e p t h of o s c i l l a t i o n marks a f t e r i n c r e a s i n g t h e water v e l o c i t y f r o m 7.1 -8.1 m/s ( water f l o w r a t e of 21.45 t o 32.80 1 / s ) . B i l l e t s a mples from Heat 24272 p r e s e n t a d i f f e r e n t s i t u a t i o n t o t h e a b o v e . As t h e water v e l o c i t y i s d e c r e a s e d from 9.7 m/s (water f l o w r a t e of 32.80 1/s) t o 3.1 m/s ( w a t e r f l o w r a t e of 16.72 1/s ), t h e o s c i l l a t i o n marks seem t o become s h a l l o w e r . E x c l u d i n g t h i s sample a t 3.1 m/s (water f l o w r a t e o f 16.72 1 / s ) , one can g e n e r a l i s e , t h a t i r r e s p e c t i v e of s u p e r h e a t , l o w e r water v e l o c i t i e s 4 - 7.2 m/s( water f l o w r a t e s of 20 - 26.5 1/s) g i v e r i s e t o d e e p e r o s c i l l a t i o n marks t h a n t h o s e above 9.15 m/s (water f l o w r a t e s of 31.54 1 / s ) . Above 9.15 m/s (water f l o w r a t e o f 31.54 1 / s ) , p h o s p h o r a u s seems t o have a s t r o n g i n f l u e n c e on t h e d e p t h o f o s c i l l a t i o n marks ( o b t a i n e d as an a v e r a g e of t h e t h r e e s a m p l e s a v a i l a b l e f o r t h e h e a t ) . T h i s can be seen i n F i g 51. Any s u c h d e p e n d e n c y of c o m p o s i t i o n c o u l d not be o b s e r v e d a t water v e l o c i t i e s l o w e r t h a n 9.15 m/s ( w a t e r f l o w r a t e of 31.54 1/s) i n t h e h i g h - c a r b o n s t e e l b i l l e t s . T a b l e X p r e s e n t s t h e d e p t h o f o s c i l l a t i o n marks b o t h a t t h e m i d f a c e and o f f - c o r n e r r e g i o n s from e a c h b i l l e t . I t i s Woter Velocity >ft65*n/s A- Average Superheat During Length of Heat Q43A O 58A O 28 0 43A O 4 0 A 26A O 26 Q <29 0 29 Q48A 8 52 0014 0018 0022 Phosphorus (%) 0-026 0030 Figure 51 - P l o t of phosphorous versus depth of o s c i l l a t i o n marks, water v e l o c i t y >8.65 m/s. 105 i n t e r e s t i n g t o n o t e t h a t t h e o s c i l l a t i o n mark a p p e a r s t o be d e e p e r i n t h e o f f - c o r n e r r e g i o n t h a n i n t h e m i d f a c e . T h i s can be o b s e r v e d i n F i g 45 t o 49. F o r t h e u n a i d e d eye l o w - c a r b o n s t e e l b i l l e t s p r e s e n t deep o f f - c o r n e r v a l l e y s ( F i g 45) 5.6 Model P r e d i c t i o n s C h a r a c t e r i s t i c s of s h e l l f o r m a t i o n b a s e d on h e a t - f l u x d a t a d e r i v e d from t e m p e r a t u r e measurements i n t h e mould a r e p r e s e n t e d h e r e . A f t e r e s t a b l i s h i n g t h i c k n e s s p r o f i l e and t h e r m a l f i e l d s i n t h e s h e l l w i t h t h e h e l p of a o n e - d i m e n s i o n a l h e a t - t r a n s f e r model, t h e p o s s i b i l i t y o f m e n i s c u s s o l i d i f i c a t i o n i s examined u s i n g t h e t w o - d i m e n s i o n a l h e a t - f l o w model d e s c r i b e d e a r l i e r . 5.6.1 S o l i d i f i c a t i o n of B i l l e t S e c t i o n The s h e l l p r o f i l e and s h e l l t h i c k n e s s a t t h e mould e x i t was c a l c u l a t e d f o r H e a t s 24258 t o 24278 u s i n g t h e 1-D u n s t e a d y s t a t e h e a t t r a n s f e r m o d e l . Heat f l u x d a t a e s t a b l i s h e d by S a m a r a s e k e r a and Brimacombe were f i t t e d w i t h p o l y n o m i a l s and u s e d as an i n p u t t h e m o d e l . F i g s 52 t o 54 show s u c h h e a t - f l u x p r o f i l e s f o r h i g h -c a r b o n s t e e l s a t 9.1 m/s, 8.1 m/s and 7.1 m/s r e s p e c t i v e l y . A l s o F i g s 55 t o 57 show h e a t - f l u x p r o f i l e s f o r l o w - c a r b o n s t e e l s a t 9.1, 8.1 and 7.1 m/s. The e x i t s h e l l t h i c k n e s s c a l c u l a t e d f o r v a r i o u s h i g h - c a r b o n h e a t s i s t a b u l a t e d i n T a b l e XI and i s seen t o be r a n g i n g from F i g u r e 52 - Heat f l u x p r o f i l e f o r low carbon s t e e l , water v e l o c i t y 9.7 m/s. 107 Figure 53 Heat f l u x p r o f i l e f o r low carbon s t e e l , water v e l o c i t y 6.85 m/s. 108 Figure 54 - Heat f l u x p r o f i l e f o r low carbon s t e e l , water v e l o c i t y 5.55 m/s. 109 0 IOO -F i g u r e 55 - Heat f l u x p r o f i l e f o r high carbon s t e e l , water v e l o c i t y 9.7 m/s. Figure 56 - Heat f l u x p r o f i l e f o r high carbon s t e e l , water v e l o c i t y 7.2 m/s. 111 Heat f l u x p r o f i l e f o r high carbon s t e e l , water v e l o c i t y 5.55 m/s. 1 1 2 8.31 t o 11.2mm. The l o c a t i o n o f t h e d a r k band i s a l s o p r e s e n t e d i n t h e T a b l e X I . E x i t s h e l l t h i c k n e s s and t h e p o s i t i o n o f d a r k band a r e compared f o r t h e h e a t s under c o n s i d e r a t i o n i n F i g 58. T a b l e X I I p r e s e n t s t h e e x i t s h e l l t h i c k n e s s f o r t h e low-c a r b o n h e a t s . The s h e l l t h i c k n e s s o f t h e b i l l e t s a t t h e mould e x i t t h u s v a r i e s f r o m 7.59-11.2mm. The c h i l l band c o u l d be seen o n l y i n H e a t s 24259 and 24269 and i t s l o c a t i o n i s g i v e n i n T a b l e X I I . From T a b l e X I , T a b l e XII and F i g 54 i t can be s a i d t h a t t h e c h i l l band a p p e a r s i n t h e b i l l e t a f t e r t h e l a t t e r e x i t s the mould. The measured d i s t a n c e from t h e s u r f a c e o f t h e b i l l e t t o t h e w h i t e band i n t h e m i d f a c e a l s o a r e p r e s e n t e d i n T a b l e s 5.9 and 5.10. The d i s t a n c e below t h e m e n i s c u s where t h e s h e l l t h i c k n e s s c o r r e s p o n d i n g t o t h e w h i t e band d e v e l o p s i s a l s o p r e s e n t e d i n t h e s e t a b l e s . Thus on an a v e r a g e i t can be c o n c l u d e d t h a t i n h i g h - c a r b o n s t e e l , t h e w h i t e band f o r m s a r o u n d 450 - 550 mm below t h e m e n i s c u s . In l o w - c a r b o n s t e e l s , t h i s p o s i t i o n s h i f t s upwards t o 300 - 450 mm. W h i l e f o r m u l a t i n g t h e s h e l l t h i c k n e s s p r o f i l e s t h e t e m p e r a t u r e g r i d a f t e r e a c h t i m e s t e p has been employed t o f i n d t h e c o o l i n g r a t e s o f i n d i v i d u a l l o c a t i o n s . F i g s 55 t o 59 show some t y p i c a l c o o l i n g - r a t e p l o t s f o r h i g h - c a r b o n s t e e l s . C o o l i n g r a t e s o f t h e s u f a c e , t h e l o c a t i o n c o r r e s p o n d i n g a p p r o x i m a t e l y t o th e w h i t e band and t h e l o c a t i o n a h e a d o f t h e w h i t e band a r e p l o t t e d a g a i n s t t h e d i s t a n c e and t i m e below t h e m e n i s c u s . I t ca n be seen t h a t a t a b o u t 400 mm below t h e -• o I I o o o I I CD O I » o o 9 o i o I O Model Predicted Exist Shell Thickness Meosured Dark Bond Location J I I I I 24277 24276 24266 0-29 030 031 I I 24272 033 24278 24263 Heat Number 033 034 Corbon (%) I I f I I I I I I I I I L 24265 24264 24274 035 036 0-38 24273 041 Figure 58 - Location of dark band and e x i t s h e l l t hickness in high carbon s t e e l heats. 114 Oj—i : — i 1 1 1 1 r 100 -E Cooling Rate (°C/s) Figure 59 - Cooling rates of three d i f f e r e n t nodes down the mould, high carbon s t e e l . Water v e l o c i t y 9.7 m/s. 115 Figure 60 - Cooling r a t e s of three d i f f e r e n t nodes down the mould, high carbon s t e e l . Water v e l o c i t y 7.2 m/s. 116 20 40 Cooling Rate Figure 61 - Cooli n g r a t e s of three d i f f e r e n t nodes down the mould, high carbon s t e e l . Water v e l o c i t y 5.55 m/s. 1 17 m e n i s c u s , t h e s u r f a c e node g e t s r e h e a t e d . F u r t h e r m o r e a t a b o u t 480 - 640 mm from t h e m e n i s c u s , w h i l e t h e s u r f a c e node r e h e a t s , t h e p o s i t i o n c o r r e s p o n d i n g t o t h e s h e l l t h i c k n e s s of 7mm a c o n s t a n t c o o l i n g r a t e i n t h e p r e s e n c e of a d e c r e a s i n g h e a t e x t r a c t i o n r a t e . S i m i l a r phenomena were o b s e r v e d i n t h e c a s e o f l o w - c a r b o n s t e e l s but lo w e r i n t h e mould ( 560 -720 mm ) as i l l u s t r a t e d i n F i g s 62 and 64. I t s h o u l d a l s o be n o t e d t h a t below 640 mm from t h e m e n i s c u s , t h e s u r f a c e u n d e r g o e s e x t e n s i v e r e h e a t i n g some t i m e s a t a r a t e of -25 °C/s 5.6.2 M e n i s c u s S o l i d i f i c a t i o n In B i l l e t C a s t i n g S o l i d i f i c a t i o n a t t h e m e n i s c u s has been examined w i t h t h e h e l p o f t h e s e 2-D u n s t e a d y s t a t e m e n i s c u s m o d e l . C o n s i d e r i n g t h e r e l a t i o n between t h e o s c i l l a t i o n marks and n e g a t i v e s t r i p t i m e , i t was t h o u g h t an a v e r a g e e s t i m a t e o f 0.3seconds of n e g a t i v e s t r i p t i m e c o u l d s e r v e v e r y w e l l a s t h e maximum time a v a i l a b l e f o r s o l i d i f i c a t i o n . I t i s i m p l i e d t h a t m e t a l a t t h e m e n i s c u s does n o t move down t h e mould d u r i n g t h i s t i m e o r moves w i t h no m e t a l b e i n g added a t t h e m e n i s c u s and t h e h e a t f l u x i s c o n s t a n t o v e r t h e i n t e r v a l of c a l c u l a t i o n . F o r a l l p r a c t i c a l p u r p o s e s , s o l i d i f i c a t i o n of a s t a t i c m e t a l m e n i s c u s has been u n d e r t a k e n t o s i m u l a t e c a s t i n g c o n d i t i o n s i n t h e n e g a t i v e s t r i p t i m e o f a c o n t i n u o u s c a s t i n g mould. In c o n s i d e r i n g s u c h s o l i d i f i c a t i o n , i t i s n e c e s s a r y t o have a c r i t e r i o n a s t o what f r a c t i o n of s o l i d i n t h e s t e e l would make i t behave a s a r i g i d body. Saucedo and B e e c h 6 9 have c o n s i d e r e d 118 Fi g u r e 62 - Cooling r a t e s of three d i f f e r e n t nodes down the mould, low carbon s t e e l . Water v e l o c i t y 9.7 m/s. Figure 63 - Cool i n g rates of three d i f f e r e n t nodes down the mould, low carbon s t e e l . Water v e l o c i t y 6.85 m/s. 120 Figure 64 - Cooling r a t e s of three d i f f e r e n t nodes down the mould, low carbon s t e e l . Water v e l o c i t y 5.55 m/s. 121 t h a t a f r a c t i o n of s o l i d g r e a t e r t h a n 0.2 i s s u f f i c i e n t t o p r o d u c e a d e n d r i t i c a r r a n g e m e n t t h a t w i l l behave r i g i d l y . C o n s i d e r i n g t h e u n c e r t a i n i t y i n v o l v e d i n d e c i d i n g t h e e x t e n t o f mushy zone b e c a u s e of t h e l a c k of i n f o r m a t i o n a b o u t t h e l i q u i d t h e r m a l c o n d u c t i v i t y a s a f f e c t e d by f l u i d m o t i o n e t c . , i t was t h o u g h t b e s t t o employ 100% s o l i d i n s t e e l a s t h e r i g i d i t y c r i t e r i o n . The model was run e s s t e n t i a l l y f o r 0.1%C s t e e l and 0.3%C s t e e l e m p l o y i n g v a r i o u s h e a t f l u x e s and a t d i f f e r e n t s u p e r h e a t . T h e r e i s no d a t a a v a i l a b l e i n t h e l i t e r a t u r e f o r h e a t -e x t r a c t i o n r a t e s i n t h e n e g a t i v e s t r i p t i m e . M e n i s c u s h e a t f l u x e s r e p o r t e d by S a m a r a s e k e r a and Brimacombe a r e o n l y a v e r a g e v a l u e s o v e r t h e e n t i r e o s c i l l a t i o n c y c l e . T h e s e range from 3200-5000 KW/M2 f o r h i g h - c a r b o n s t e e l s and 3200-4000 KW/M2 f o r l o w - c a r b o n s t e e l s f o r s u p e r h e a t s w e l l above 1°C. A t t h e s u p e r h e a t s e n c o u n t e r e d i n t h e e x p e r i m e n t s a t W e s t e r n Canada S t e e l and u s i n g m e n i s c u s h e a t f l u x e s e x t a b l i s h e d by S a m a r a s e k e r a ( w h i c h a r e p r e s e n t e d i n T a b l e I I I ) s o l i d i f i c a t i o n was not f o u n d t o o c c u r a t t h e m e n i s c u s even a f t e r 0.3 s e c o n d s of s i m u l a t i o n . I t was c o n s i d e r e d w o r t h w h i l e t o examine a s t o what v a l u e of peak h e a t f l u x would be needed f o r s o l i d i f i c a t i o n a t t h e m e n i s c u s . W i t h 1°C s u p e r h e a t i n t h e m e l t , a minimum of 4186 KW/M2 (100 C a l / c m 2 s e c ) was f o u n d t o be n e c e s s a r y t o f r e e z e t h e p o r t i o n o f m e n i s c u s c l o s e r t o t h e mould f o r t h e 0.1% c a r b o n s t e e l whereas 5233 KW/M2 (125 C a l / c m 2 s e c ) i s t h e minimum h e a t f l u x f o r 0.3%C s t e e l . F i g s 65 and 66 show t h e g r o w t h o f s o l i d i n t h e 0.1%C and 0.3%C s t e e l r e s p e c t i v e l y a t t h e s e minimum I I • I I » I I I » ' » ' I U J—I 1 1 1 1 0 2: 4 0 2 4: Distance From Mould Wall (mm) Figure 65 - Model p r e d i c t e d s o l i d i f i c a t i o n at the meniscus-low carbon s t e e l . £ E IS) 3 O </> C 0) 0 10 • 15 o m 20 4) U C o ."5 25 Q 03s Carbon 0 3 % Superheat l°C Simulation Time 0-30s Heat Flux 5232 kW/m 30 •01s 0 2s -0-3s Carbon 0-3% Superheat l°C Simulation Time 030s Heat Flux 8372 kW/m -L 4 Distance 0 From Mould 2 Wall (mm) to CO Figure 66 - Model p r e d i c t e d s o l i d i f i c a t i o n at the menisc high carbon s t e e l . us-1 24 h e a t f l u x v a l u e s . W h i l e w i t h 4186 KW/M2 s o l i d i f i c a t i o n o f t h e m e n i s c u s j u s t b e g i n s , F i g 65 shows e x t e n s i v e s o l i d g r o w t h a t 6279 KW/M2 f o r t h e l o w - c a r b o n s t e e l . F i g 66 shows s i m i l a r c a l c u l a t i o n f o r t h e 0.3%C s t e e l a t 1325 KW/M2 (175 C a l / c m 2 s e c ) and 8372 KW/M2 (200 C a l / c m 2 s e c ) . Not s u r p r i s i n g l y a s t h e h e a t f l u x i s i n c r e a s e d t h e e x t e n t t o whi c h t h e m e n i s c u s s o l i d i f i e s i n c r e a s e s i n a d d i t i o n t o t h e s h e l l becoming t h i c k e r a d j a c e n t t o th e mould. H i g h e r s u p e r h e a t s d e f i n i t e l y need h i g h e r h e a t f l u x e s f o r t h e same e x t e n t of f r e e z i n g . I t was f o u n d t h a t w h i l e 4186 KW/M2 (100 C a l / c m 2 s e c ) a t 1°C o f s u p e r h e a t g i v e s r i s e t o s o l i d i f i c a t i o n a t t h e m e n i s c u s , a t 20°C no s o l i d was p r e s e n t . A l s o , a s t h e t i m e f o r s o l i d i f i c a t i o n i s d e c r e a s e d , h i g h e r h e a t f l u x e s would be needed, f o r t h e same e x t e n t o f s o l i d i f i c a t i o n . However, p r o v i d e d e i t h e r w i t h h i g h e r h e a t f l u x e s i n t h e n e g a t i v e s t r i p t i m e o r c o l d e r s t e e l a d j a c e n t t o t h e mould w a l l s , i t i s p h y s i c a l l y p o s s i b l e t o have p a r t o f t h e m e n i s c u s f r o z e n . 1 25 V I . DISCUSSION 6.1 M a c r o s t r u c t u r e Of B i l l e t S e c t i o n s As was shown i n s e c t i o n 5.1.1 o f t h e l a s t c h a p t e r , f a c t o r s w h i c h i n f l u e n c e t h e e x t e n t of t h e c o l u m n a r zone a r e p r i m a r i l y t h e c a r b o n and p h o s p h o r o u s c o n t e n t s . As c a r b o n i s i n c r e a s e d from 0.13 t o 0.36%, c o l u m n a r zone l e n g t h d e c r e a s e s f r o m a h i g h v a l u e of 65mm t o below 44mm a t 0.29%C and r e m a i n s a l m o s t c o n s t a n t . However, a t 0.38%C and beyond t h e r e a p p e a r s t o be a s t e e p r i s e i n t h e c o l u m n a r zone l e n g t h . S i m i l a r o b s e r v a t i o n s have been made i n bloom c a s t i n g by M i y a h a r a e t a l 6 9 . However, t h e r i s e i n c o l u m n a r zone l e n g t h was a f t e r 0.42%C. T i w a r i and B e e c h 7 0 o b s e r v e d t h a t a t h i g h s u p e r h e a t s , t h e e q u i a x e d a r e a i n c r e a s e s t o a h i g h v a l u e a t 0.35%C and s t e a d i l y d e c r e a s e s t h e r e o n as t h e c a r b o n c o n t e n t i n c r e a s e s . The i n f l u e n c e o f i n c r e a s i n g c a r b o n c o n t e n t on t h e e a l i e r c o l u m n a r - e q u i a x e d t r a n s i t i o n may be due, a t l e a s t i n p a r t t o i n c r e a s e d c o n s t i t u t i o n a l s u p e r c o o l i n g . T h e r e may be a s i m i l a r e x p l a n a t i o n f o r t h e i n f l u e n c e of p h o s p h o r o u s on t h e c o l u m n a r zone l e n g t h . I t s h o u l d a l s o be p o i n t e d o ut t h a t d e n d r i t e arm r e m e l t i n g becomes e a s i e r a s t h e a l l o y c o n t e n t of t h e s t e e l i n c r e a s e s and t h e r e s u l t i n g n u c l e i i f r o m t h e e a r l y s t a g e s o f s o l i d i f i c a t i o n d r i f t away and t h e n d e s c e n d t h r o u g h t h e b u l k l i q u i d t o l a t e r c o n t r i b u t e t o t h e f o r m a t i o n of t h e e q u i a x e d 1 26 z o n e . The s e t t l i n g of d e n d r i t e d e b r i s i n t o t h e l o w e r p a r t o f t h e l i q u i d p o o l where t h e y i n t e r f e r e w i t h c o l u m n a r g r o w t h c e r t a i n l y i s r e s p o n s i b l e f o r t h e n o n - s y m m e t r i c a l s t r u c t u r e o b s e r v e d i n t h e b i l l e t s e c t i o n s , i . e s h o r t e r c o l u m n a r zone a d j a c e n t t o t h e o u t s i d e r a d i u s f a c e . Flow of c o l d e r s t e e l t o t h e lower p a r t o f t h e l i q u i d c r a t e r owing t o d e n s i t y d i f f e r e n c e s 7 1 would enhance s u c h a n o n - s y m m e t r i c i t y o f t h e s t r u c t u r e . A n o t h e r i m p o r t a n t e f f e c t of c a r b o n i s t h e s h r i n k a g e a s s o c i a t e d w i t h t h e 5—>y t r a n s f o r m a t i o n w h i c h c o n t r i b u t e s t o t h e w a v i n e s s o f t h e b i l l e t s u r f a c e . T h e w a v i n e s s would r e d u c e w i t h t h e amount of p r i m a r y - 5 beyond c a r b o n l e v e l s 0.16%. I n c r e a s e d smoothness of t h e b i l l e t s u r f a c e a t h i g h e r c a r b o n i n s t e e l i n c r e a s e s t h e h e a t t r a n s f e r r a t e s b e c a u s e t h e a i r gap i s r e d u c e d . T h i s s h o u l d l e a d t o more e f f e c t i v e r e m o v a l o f s u p e r h e a t i n t h e mould and t h e r e b y c o n t r i b u t e t o t h e growth of t h e n u c l e i i n c o m p e t i t i o n w i t h t h e c o l u m n a r d e n d r i t e s by r e d u c i n g t h e g r a d i e n t s a t t h e c o l u m n a r d e n d r i t e t i p s . The s u r v i v a l of t h e n u c l e i i i s as w e l l e n h a n c e d by r e m o v a l o f more s u p e r h e a t . A i d e d by i n c r e a s e d c o n s t i t u t i o n a l s u p e r c o o l i n g , c o l u m n a r zo n e s a r e s h o r t e r a s c a r b o n o r p h o s p h o r o u s a r e i n c r e a s e d . The sudden i n c r e a s e i n t h e c o l u m n a r zone l e n g t h a r o u n d 0.38%C i n s t e e l s i s a new f i n d i n g i n b i l l e t c a s t i n g . C o n s i d e r i n g t h e Fe-C e q u i l i b r i u m p h a s e d i a g r a m , s u c h a t r a n s i t i o n a p p e r s t o be more l i k e l y a t a c a r b o n l e v e l o f 0.51%. Upto 0.51% C i n s t e e l , s o l i d i f i c a t i o n p r o c e e d s t h r o u g h t h e p e r i t e c t i c r e a c t i o n i n v o l v i n g t h e 6—phase w i t h a s o l u b i l i t y o f 1 27 0.1% f o r c a r b o n . Beyond 0.51% C t h e l i q u i d f r e e z e s d i r e c t l y t o 7-Fe w h i c h c a n d i s s o l v e u p t o 2.0wt% of c a r b o n . Thus, t h e e x t e n t of c o n s t i t u t i o n a l s u p e r c o o l i n g would be l i m i t e d i n t h e c a s e o f s t e e l s w i t h g r e a t e r t h a n 0 . 5 l w t % C . In t h e c a s e o f t h e p e r i t e c t i c r e a c t i o n , t h e r e would be more c o n t r i b u t i o n t o t h e e q u i a x e d zone from d e n d r i t e arm r e m e l t i n g b e c a u s e t h e s e c r y s t a l l i t e s w i l l be i n t h e form 8-Fe and would t h e r e f o r e p r o v i d e g r e a t e r r e s i s t a n c e t o r e - s o l u t i o n (due t o t h e i r h i g h e r m e l t i n g p o i n t s ) compared t o 7-Fe c r y s t a l l i t e s i n s t e e l s i n t h e 0.51%C. T i w a r i and B e e c h 7 0 s u g g e s t t h a t t h e r e i s y e t a n o t h e r way t h e p e r i t e c t i c r e a c t i o n p r o m o t e s e q u i a x e d z o n e s i . e . , by i n c r e a s i n g t h e s u r v i v a l r a t e of t h e ' f a l l i n g c r y s t a l l i t e s ' . T h e s e c r y s t a l l i t e s would t r a n s f o r m f r o m 6 t o 7 on t h e mould w a l l b e f o r e moving i n t o t h e l i q u i d . On r e m e l t i n g of s u c h n u c l e i , t h e change from 7 t o 6 must o c c u r and t h e r e f o r e an e n v e l o p e o f 5 w i l l form i n i t i a l l y a r o u n d t h e 7-phase. T h i s e n v e l o p e may p r o v i d e a g r e a t e r b a r r i e r t o r e - s o l u t i o n t h a n i f t h e 7 and t h e l i q u i d were i n d i r e c t c o n t a c t . The d e l a y so c a u s e d i n r e m e l t i n g would r e s u l t i n an i n c r e a s e i n t h e number of s u r v i v i n g c r y s t a l l i t e s and t h e r e f o r e a l a r g e r e q u i a x e d z o n e . I t i s n o t c l e a r , however, why t h i s t r a n s i t i o n t o p r o n o u n c e d c o l u m n a r d e n d r i t e g r o w t h t a k e s p l a c e a t a c a r b o n l e v e l of 0.38% r a t h e r t h a n 0.51%. Under t h e i n f l u e n c e o f t h e a l l o y i n g e l e m e n t s p r e s e n t i n t h e s t e e l ( v i z . , Mn, S i , C r , A l , P, S) i t i s n o t known as t o how t h e p e r i t e c t i c r e a c t i o n would t a k e p l a c e . N e e d l e s s t o say t h e s o l i d i f i c a t i o n c o n d i t i o n s a r e f a r from e q u i l i b r i u m . 128 I t i s i n t e r e s t i n g t o n o t e t h a t a l t h o u g h t h e c o l u m n a r zone l e n g t h goes up a f t e r c a r b o n l e v e l s o f 0.38% t h e a v e r a g e h e a t e x t r a c t i o n r a t e s r e m a i n h i g h and c o m p a r a b l e t o t h e h e a t s w i t h c a r b o n s l e v e l s of 0.29-0.36. T h i s shows t h a t t h e i n f l u e n c e of h e a t t r a n s f e r i s o n l y s e c o n d a r y , t h e p r i m a r y c o n t r i b u t i o n a r i s i n g from c o m p o s i t i o n a l f a c t o r s . S i m i l a r l y t h e s u p e r h e a t s under c o n s i d e r a t i o n seem t o be g e n e r a l l y q u i t e h i g h b ut a g a i n t h e c o m p o s i t i o n of t h e s t e e l seem t o d i c t a t e t h e c o l u m n a r -e q u i a x e d t r a n s i t i o n . 6.2 Band F o r m a t i o n D e t a i l s a b o u t t h e w h i t e and d a r k bands i n t r a n s v e r s e s e c t i o n s a r e p r e s e n t e d i n S e c t i o n 5.1.3 of t h e l a s t c h a p t e r . C o n s i d e r i n g t h e c o n t i n u i t y of t h e s e bands as shown i n F i g s 26 t o 33, i t i s r e a s o n a b l e t o c o n s i d e r f o r m a t i o n of e a c h band as a o n e - t i m e e v e n t i . e . , h a p p e n i n g s i m u l t a n e o u s l y a c r o s s th e c r o s s s e c t i o n a t t h e same d i s t a n c e below t h e m e n i s c u s . I t was shown i n s e c t i o n 5.5 t h a t t h e d a r k band a p p e a r s t o o c c u r a p p r o x i m a t e l y soon a f t e r t h e b i l l e t e x i t s t h e mould and t h e w h i t e band a p p e a r s a t a b o u t 450-550 mm below t h e m e n i s c u s i n t h e c a s e of h i g h - c a r b o n s t e e l s and 300-450 mm below t h e m e n i s c u s i n the c a s e o f l o w - c a r b o n s t e e l s . I t was a l s o shown t h a t a r o u n d t h e s e a r e a s o f t h e mould t h e b i l l e t seems t o e x h i b i t p e c u l i a r t h e r m a l b e h a v i o u r . W e i n b e r g 7 1 s u g g e s t s t h a t t h e p r e s e n c e o f s u c h w h i t e ( o r 1 29 l i g h t l y e t c h e d ) bands i n i n g o t s o l i d i f i c a t i o n i s due t o t h e w a s h i n g a c t i o n o f l i q u i d on t h e s o l i d i f i c a t i o n f r o n t . In t h e p r e s e n t s t u d y of c o n t i n u o u s c a s t i n g i t i s w e l l known t h a t b e c a u s e o f t h e i n p u t s t r e a m t h e r e i s s i g n i f i c a n t c i r c u l a t i o n of l i q u i d s t e e l i n t h e mould a r e a . W h i l e e x p e r i m e n t i n g w i t h a d d i t i o n o f r a d i o a c t i v e i s o t o p e s t o d e l i n e a t e t h e s h e l l p r o f i l e , s e v e r a l i n v e s t i g a t o r s i n c l u d i n g M o r t o n and W e i n b e r g 7 2 and L a i t e t a l 7 3 have shown t h a t t h e f l u i d f l o w i s s u f f i c i e n t l y t u r b u l e n t t o mix t h e i s o t o p e f a i r l y q u i c k l y i n t o t h e l i q u i d p o o l a s i t i s b e i n g a d d e d . I t has been shown t h a t a r o u n d 450-550 mm below t h e m e n i s c u s , t h e s h e l l c o r r e s p o n d i n g a p p r o x i m a t e l y t o t h e l o c a t i o n of t h e w h i t e band u n d e r g o e s a c o n s t a n t c o o l i n g r a t e ( F i g s 59 t o 6 4 ) . T h i s c o u l d mean t h a t t h e s p e e d o f t h e s o l i d i f i c a t i o n f r o n t i s r e t a r d e d . C o n s i d e r i n g t h e f l u i d m o t i o n i n t h e l i q u i d c r a t e r of t h e c o n t i n o u s l y c a s t s t r a n d , t h e f l u i d s t r e a m t h e n has more t i m e t o wash t h e i n t e r d e n d r i t i c - r i c h l i q u i d f r o m t h e s o l i d i f i c a t i o n f r o n t . C o n s i d e r i n g t h e ' d e e p l y e t c h e d ' or d a r k n a t u r e of t h e o t h e r band i t c a n be s p e c u l a t e d t h a t t h e d a r k band r e p r e s e n t s an a r e a of e n h a n c e d s o l i d i f i c a t i o n . The d a r k band seems t o c o r r e s p o n d t o a sub-mould s h e l l t h i c k n e s s . l t has been shown ( i n F i g s 59 t o 64) t h a t t h e b i l l e t s h e l l u n d e r g o e s s u b s t a n t i a l r e h e a t i n g a s t h e b i l l e t i s e x i t i n g t h e m o u l d . T h i s r e h e a t i n g c o u l d be a t t r i b u t e d t o i n c r e a s e d gaps r e s u l t i n g from t h e s t r o n g s h e l l and t h e c o r r e s p o n d i n g l y l o w e r e d h e a t f l u x e s t o w a r d t h e e x i t o f t h e mould. I t i s q u i t e p o s s i b l e t h a t t h e impingement o f th e f i r s t sub m o u l d - s p r a y s would a c c e l e r a t e t h e s o l i d i f i c a t i o n 1 30 f r o n t w h i c h was g r o w i n g s l o w l y up t o t h i s p o i n t . I t i s known t h a t l o c a t i o n of t h e f i r s t s p r a y n o z z l e below t h e mould i s w i t h i n 1.25 cms below t h e m ould. T h i s would l e a d t o e n h a n c e d f r e e z i n g of t h e i n t e r d e n d r i t i c l i q u i d r i c h i n s o l u t e e l e m e n t s w i t h o u t g i v i n g t i m e f o r t h e c i r c u l a t i n g f l u i d s t r e a m t o wash i t away. The i m p o r t a n c e o f t h e s t u d y of t h e s e bands l i e s i n t h e f a c t t h a t t h e w h i t e and dark bands w h i c h a r e wide a p a r t a t t h e m i d f a c e seem t o o v e r l a p a t t h e o f f - c o r n e r . In t h e a b s e n c e of h e a t f l u x d a t a a t t h e c o r n e r / o f f c o r n e r a r e a , t h i s w ould s e r v e as a g u i d e - l i n e t o e s t i m a t e t h e s h e l l t h i c k n e s s a t t h e o f f -c o r n e r a r e a . More i m p o r t a n t l y t h e above o b s e r v a t i o n s u g g e s t s t h a t a f t e r t h e w h i t e band has formed, s o l i d i f i c a t i o n i n t h e o f f -c o r n e r / c o r n e r a r e a s v i r t u a l l y s t o p s even t h o u g h i t c o n t i n u e s i n t h e m i d f a c e r e g i o n . As w i l l be shown l a t e r t h i s a p p e a r s t o be an i m p o r t a n t c l u e t o e x p l a i n one of t h e r e a s o n s f o r mould-g e n e r a t e d r h o m b o i d i t y . 6. 3 C r a c k F o r m a t i o n The p r o x i m i t y of t h e o f f c o r n e r c r a c k s t o t h e w h i t e band s u g g e s t s t h a t t h e s e c r a c k s c o u l d have formed o n l y a f t e r t h e b i l l e t has t r a v e l l e d a t l e a s t 450-550 mm below t h e m e n i s c u s . However, as has been s u g g e s t e d e a r l i e r , t h e r e a p p e a r s t o be p r a c t i c a l l y v e r y l i t t l e s h e l l q r o w t h a f t e r t h i s s t a g e a t t h e o f f - c o r n e r a r e a s . A l s o t h e r e i s r e h e a t i n g a s s o c i a t e d w i t h t h e 131 b i l l e t s h e l l a f t e r i t s e x i t from t h e mould. Because o f l a c k of s u p p o r t i n g r o l l s f o r t h e e x i t i n g b i l l e t , i t i s p o s s i b l e f o r t h e m i d f a c e of t h e b i l l e t t o b u l g e a g a i n s t t h e c o l d c o r n e r . As was m e n t i o n e d e a r l i e r when r e h e a t i n g was e n h a n c e d i n t h e a b s e n c e o f s e c o n d a r y c o o l i n g w a t e r , o f f - c o r n e r c r a c k s r e s u l t e d a t a l l f o u r c o r n e r s . T h i s f a c t f u r t h e r c o n f i r m s t h e i n f l u e n c e of r e h e a t i n g on t h e o c c u r r e n c e o f t h e s e c r a c k s . The l o c a t i o n of t h e o f f -c o r n e r c r a c k and t h e b u l g e i n t h e a d j a c e n t f a c e shown i n F i g 42 c l e a r l y p o i n t s o u t t h a t t h e s e c r a c k s would o r i g i n a t e b e c a u s e o f , as s u g g e s t e d by Brimacombe e t a l 2 2 , t h e t e n s i l e s t r a i n s g e n e r a t e d a t t h e s o l i d i f i c a t i o n f r o n t due t o t h e m i d f a c e moving out a g a i n s t a c o l d c o r n e r . The p r e s e n t s t u d y i n d i c a t e s t h a t t h e s e c r a c k s f o r m o u t s i d e t h e mould or i n t h e bottom-most p a r t of t h e mould where t h e m i d f a c e c o u l d f r e e l y b u l g e . . However, i t i s not c l e a r as t o how t h e mould c o o l i n g w ater a f f e c t s t h e o f f - c o r n e r c r a c k i n g . S a m a r a s e k e r a and B r i m a c o m b e 2 0 s u g g e s t t h a t i n t e r m i t t e n t b o i l i n g t a k e s p l a c e n e a r t h e m e n i s c u s a r e a of t h e mould and c o u l d l e a d t o dynamic d i s t o r t i o n of t h e m ould. As t o how d i s t o r t i o n c a u s e d h i g h e r up i n t h e mould c o u l d enhance o f f - c o r n e r c r a c k i n g i s unknown ; but i t may l e a d t o an u n u s u a l l y t h i n s h e l l t h i c k n e s s i n t h e c o r n e r r e g i o n w h i c h may e x a c e r b a t e b u l g i n g below t h e mould. In t h e l i g h t of t h e above o b s e r v a t i o n t h e d a t a was a n a l y s e d s i m p l y i n t e r m s o f t h e Mn/S r a t i o . T h i s r a t i o d e t e r m i n e s t h e a v a i l a b l e s u l p h u r t o form low m e l t i n g FeS w h i c h a l l o w s d e n d r i t e s t o s e p e r a t e under t e n s i l e s t r a i n t o f o r m t h e o f f - c o r n e r c r a c k s . I t was o b s e r v e d t h a t when t h i s r a t i o i s l o w e r t h a n 25-30, c r a c k 132 s e v e r i t y was h i g h . T h i s a p p e a r s t o be much more c l e a r when water v e l o c i t i e s were lo w e r t h a n 9.7 m/s (water f l o w r a t e s of 32 1 / s ) . I t i s i n t e r e s t i n g t o n o t e t h a t t h e b i l l e t f r o m t h e Heat 24109 ( w h i c h had no sub-mould w a t e r ) has a Mn/S r a t i o of o n l y 21 and t h u s i t was p r o n e t o c r a c k i n g . The a b s e n c e of t h e e f f e c t s of mould wear ( a t t h e bottom ) on t h e o f f - c o r n e r c r a c k i n g p o i n t s out t h a t i t i s n o t a n e c e s s a r y c o n d i t i o n . T h e r e i s s u b s t a n t i a l r e h e a t i n g as t h e b i l l e t e x i t s t h e mould and mould wear c o u l d o n l y make i t worse. C o m p o s i t i o n seems t o a major f a c t o r i n d e c i d i n g whether c r a c k i n g would o c c u r . However, s e v e r e r e h e a t i n g c o u l d promote t h e c r a c k i n g t e n d e n c y . A l s o , from t h e s e r e s u l t s i t would a p p e a r b e s t t o a v o i d water v e l o c i t i e s l o w e r t h a n 9.7 m/s (water f l o w r a t e of 32 1 / s ) . As t h e above a n a l y s i s would s u g g e s t , o f f - c o r n e r c r a c k s c o u l d be p r e v e n t e d by a c o m b i n a t i o n of t h e f o l l o w i n g m e a s u r e s : • e m p l o y i n g Mn/S r a t i o s h i g h e r t h a n 24 • e n s u r i n g s p r a y c o o l i n g on t h e b i l l e t as i t e x i t s t h e b i l l e t . T h i s would g i v e r i s e t o c o m p r e s s i v e s t r e s s e s a t t h e s o l i d i f i c a t i o n f r o n t whereby c r a c k i n g would be p r e v e n t e d • p l a c i n g f o o t r o l l s t o g u i d e and s u p p o r t t h e b i l l e t so t h a t m i d f a c e b u l g i n g would n o t o c c u r . • e m p l o y i n g w a t e r v e l o c i t i e s above 9.7 m/s. • i n c r e a s i n g mould t a p e r t o r e d u c e r e h e a t i n g . 1 33 6.4 R h o m b o i d i t y C o n s i d e r i n g t h e f a c t t h a t t h e c u r v e d f a c e s o f most o f t h e b i l l e t s were r o l l e d s l i g h t l y as t h e y were e x t r a c t e d from t h e c a s t e r by t h e p i n c h r o l l a s s e m b l y , t h e measured r h o m b o i d i t y i s n o t a t r u e r e p r e s e n t a t i o n of t h e o f f - s q u a r e n e s s of t h e b i l l e t . T h u s t h e i n f l u e n c e o f c a r b o n , s u p e r h e a t , l i f e o f t h e mould and water v e l o c i t y on r h o m b o i d i t y , i f any, c o u l d not be s e e n . In t h e f o l l o w i n g d i s c u s s i o n on r h o m b o i d i t y an extreme c a s e has been s e l e c t e d where r h o m b o i d i t y was minimum and a p p r o x i m a t e l y r e m a i n e d so d u r i n g t h e c o u r s e of t h e c a s t i n g . The c o n s t r a i n e d mould e x p e r i m e n t d e s c r i b e d i n t h e s e c t i o n 5.4 o f l a s t C h a p t e r c l e a r l y p o i n t s out t h e i m p o r t a n c e of mould w a l l movement and i t s i n f l u e n c e on r h o m b o i d i t y . I t i s known t h a t t h e p r e s e n t mould a t W e s t e r n Canada S t e e l i s h e l d a t t h e t o p on o n l y two s t r a i g h t f a c e s . M o u l d d i s t o r t i o n owing t o d i f f e r e n t i a l t h e r m a l e x p a n s i o n i n t h e t r a n s v e r s e d i r e c t i o n w ould be s i g n i f i c a n t l y l e s s on t h e c u r v e d s i d e s of t h e mould as compared t o t h e s t r a i g h t f a c e s , owing t o t h e a b s e n c e of c o n s t r a i n t s . T h i s c o u l d l e a d t o n o n - s y m m e t r i c a l c o o l i n g o f t h e b i l l e t c a u s i n g n o n - u n i f o r m s h r i n k a g e w h i c h c o u l d g e n e r a t e r h o m b o i d i t y . As c a n be s e e n , i n t h e c a s e of Heat 24090, where t h e s i d e w a l l was c o n s t r a i n e d from moving away from t h e b i l l e t , r h o m b o i d i t y was r e d u c e d t o a minimum. Permanant d e f o r m a t i o n o f mould w a l l was minimum d u r i n g t h e Heat 24100 r e s u l t i n g i n m i n i m a l r h o m b o i d i t y . The p r e s e n c e of r h o m b o i d i t y i n a l l o t h e r h e a t s c a s t i n t h e 1 34 u n - c o n s t r a i n e d mould i r r e s p e c t i v e of c a r b o n , s u p e r h e a t , w ater f l o w r a t e o f mould p o i n t s o ut t h e o v e r r i d i n g i n f l u e n c e of d i f f e r e n t c o o l i n g c a p a c i t i e s o f t h e s t r a i g h t and c u r v e d w a l l s . The o b s e r v a t i o n t h a t a t o b t u s e a n g l e s of t h e b i l l e t t h e w h i t e band g e t s i s q u i t e t h i n ( F i g s 26 and 33) i n d i c a t e s t h a t a s t o how r h o m b o i d i t y c o u l d r e s u l t f r o m mould - b i l l e t i n t e r a c t i o n . In a s q u a r e s e c t i o n i f one of t h e f o u r c o r n e r s i s t h i n n e r t h a n o t h e r s , t h e n d u r i n g s p r a y c o o l i n g t h e d i f f e r e n c e i n t h e c o n t r a c t i o n of a d j a c e n t f a c e s would make t h e b i l l e t r h o m boid. T h i s c l e a r l y p o i n t s o ut t h a t even i f t h e measured r h o m b o i d i t y i s not t o t a l l y due t o mould e v e n t s , t h e g e n e s i s of i t i s c e r t a i n l y i n t h e mould. I t i s not c l e a r , however, as t o why s u c h t h i n n e r c o r n e r s form i n s o l i d i f y i n g s e c t i o n . As p o i n t e d out e a r l i e r , t h e f a c t t h a t t h e m i d - p o i n t of a f a c e i s not where t h e maximum s h e l l t h i c k n e s s f orms ( b a s e d on o b s e r v a t i o n s of t h e d a r k band) i n many b i l l e t s e c t i o n s ) i n d i c a t e t h a t t h e h e a t t r a n s f e r i s not u n i f o r m on a d j a c e n t f a c e s . I t i s p o s s i b l e t h a t s u c h v a r i a t i o n s p r e c e d e a change i n t h e gap, c o n t r o l l e d t o a g r e a t e x t e n t by t h e mould d i s t o r t i o n f o r a g i v e n s e t of c o n d i t i o n s ( e g . , w a t e r f l o w r a t e , c a r b o n i n s t e e l e t c . ) . I t i s known t h a t t h e d i s t o r t i o n i s d i f f e r e n t on t h e s t r a i g h t and c u r v e d s i d e s . C o n s i d e r i n g t h a t t h e d i s t o r t i o n i n t h e mould i s non-symmetric i n a t r a n s v e r s e -s e c t i o n i t i s p o s s i b l e t h a t one o f t h e f o u r c o r n e r s l o s e s t h e r m a l c o n t a c t e a r l i e r i n t h e b i l l e t ' s j o u r n e y t h r o u g h t h e mould. T h i s would p l a c e t h e w h i t e band c l o s e r t o t h e s u r f a c e a t t h a t c o r n e r . 1 3 5 6.5 O s c i l l a t i o n Marks I t emerges q u i t e c l e a r l y from t h e l i t e r a t u r e ( p r e s e n t e d i n C h a p t e r I I ) t h a t o s c i l l a t i o n marks form d u r i n g t h e down s t r o k e o f t h e o s c i l l a t i o n c y c l e . Dependence of t h e i r s e v e r i t y on t h e n e g a t i v e s t r i p t i m e i n d i c a t e s t h a t any mechanism w h i c h a t t e m p t s t o e x p l a i n t h e f o r m a t i o n of t h e s e marks s h o u l d m a n i f e s t i t s e l f i n t h i s f r a c t i o n o f a s e c o n d d u r i n g w h i c h t h e mould o v e r t a k e s t h e b i l l e t . C o m p r e s s i v e f o r c e s a r e s u g g e s t e d " 5 ' " 6 ' 5 6 ' " 7 t o a c t on t h e s o l i d i f y i n g t h i n s h e l l d u r i n g t h e n e g a t i v e s t r i p p e r i o d . F o r s u c h f o r c e , w h i c h p r e s u m a b l y i s due t o m o u l d - b i l l e t f r i c t i o n , t o a c t on a s h e l l , t h e n o r m a l f o r c e s h o u l d a r i s e f r o m a f e r r o s t a t i c head p r e s e n t near t h e m e n i s c u s ; but t h i s i s u n l i k e l y b e c a u s e t h e head a t t h i s p o i n t i s e x c e e d i n g l y s m a l l " 2 . Many of t h e t h e o r i e s r e v i e w e d i n C h a p t e r I I a p p l y t o s l a b c a s t i n g where i n t h e l i q u i d s l a g i n t h e M o u l d - m e t a l gap c o u l d a c t as a pump p u s h i n g a g a i n s t t h e s o l i d i f y i n g t h i n s k i n i n e a c h n e g a t i v e s t r i p t i m e . K a w a k a m i ' s 5 1 argument t h a t same a p p l i e s t o b i l l e t c a s t i n g w h e r e i n scum r e s u l t i n g from o x i d e s and o t h e r i n c l u s i o n s t h a t f l o a t up seems r a t h e r f a r f e t c h e d . An even s c u m - l a y e r o v e r t h e e n t i r e s u r f a c e o f t h e m e t a l p o o l i s non-e x i s t a n t . I t i s not known whether t h e y have t e s t e d t h e e f f e c t s o f s u p e r h e a t s and c a r b o n l e v e l s a t a l l . One o f t h e t h e o r i e s t h a t i s more r e l e v a n t t o b i l l e t c a s t i n g , s u g g e s t e d by S a v a g e " 8 p r o p o s e s t h a t t h e s h e l l i s b r o k e n i n e a c h n e g a t i v e s t r i p p e r i o d and i s c a r r i e d above t h e m e t a l 1 36 l e v e l o f t h e mould and i s s u b s e q u e n t l y l a p p e d on t o s u r f a c e of t h e b i l l e t . T h i s c o u l d happen o c c a s s i o n a l l y a t some l o c a t i o n s a c r o s s t h e s e c t i o n . The p r o b a b i l i t y f o r i t t o o c c u r a l w a y s and a l l a r o u n d th e s u r f a c e of t h e b i l l e t a t a g i v e n i n s t a n t i s v e r y s m a l l . I t i s n e e d l e s s t o say t h a t a c c o r d i n g t o t h i s t h e o r y upcoming p i e c e s of s t e e l s h o u l d be v i s i b l e above t h e m e t a l l e v e l d u r i n g e a c h o s c i l l a t i o n s t r o k e . However t h e r e a r e no s u c h r e p o r t s i n t h e l i t e r a t u r e . I t i s known t h a t t h e b i l l e t mould w i t h c o n s t r a i n t s a t t h e t o p d i s t o r t s l e a d i n g t o a 2% n e g a t i v e t a p e r i n t h e f i r s t few i n c h e s o f t h e mould. F i g 44 shows t h e mould d i s t o r t i o n o b s e r v e d i n t h e mould employed i n campaign 12 of t h e p r e s e n t e x p e r i m e n t s . T a p e r i s more on t h e s t r a i g h t w a l l w h i c h i s c o n s t r a i n e d t h a n on t h e c u r v e d w a l l . I t i s a l s o d u r i n g ' n e g a t i v e s t r i p ' t h e mould moves f a s t e r s p e e d t h a n t h e b i l l e t . As t h e mould o v e r t a k e s t h e b i l l e t i n i t s downward t r a v e l , i t i s l i k e l y t h a t owing t o t h e n e g a t i v e t a p e r t h e m o u l d - m e t a l gap c l o s e s down l e a d i n g t o h i g h e r h e a t t r a n s f e r r a t e s . At t h e m e t a l m e n i s c u s (where t h e gap a r e n e g l i g i b l e t o s t a r t w i t h ) , t h e h e a t e x t r a c t i o n w o u l d be much more t h a n i n t h e p o s i t i v e s t r i p p e r i o d of t h e c y c l e . D u r i n g t h e e n t i r e n e g a t i v e s t r i p p e r i o d , as t h e m e t a l a t t h e m e n i s c u s s o l i d i f i e s and t r i e s t o s h r i n k and c r e a t e a gap, t h e mould c o n t i n u a l l y c l o s e s t h e gap. In o t h e r words a v e r y h i g h h e a t f l u x of t h e o r d e r o f 7000- 8000 KW/M2 c o u l d e a s i l y be r e a l i s e d i n t h i s s h o r t d u r a t i o n of t i m e w h i c h a s p e r t h e m e n i s c u s -s o l i d i f i c a t i o n model would l e a d t o s o l i d i f i c a t i o n as shown i n F i g 65 t o 66. F i g 67 shows s c h e m a t i c a l l y t h e f o r m a t i o n o f t h e 1 37 o s c i l l a t i o n marks (hooks, d e p r e s s i o n and o v e r f l o w ) . W h i l e t h e m e n i s c u s i s b e i n g f r o z e n , i t i s p o s s i b l e f o r t h e mould t o e x e r t a s t r o n g n ormal f o r c e and c a u s e t h e s h e l l t o b u c k l e a t i t s weakest p o i n t . The l o w - c a r b o n s t e e l b i l l e t s due t o t h e i r wavy-s u r f a c e s ( F i g s 45 and 46) o f f e r c o n v e n i e n t ' h i n g e s ' r e s u l t i n g i n g r e a t e r d e p r e s s i o n s w h i l e i n h i g h - c a r b o n s t e e l s , t h e d e p r e s s i o n i s s m a l l and i s l o c a t e d i n t h e t h i n n e s t a r e a c l o s e t o t h e m e n i s c u s . The a b s e n c e of m e n i s c u s - s h a p e d hooks i n l o w - c a r b o n s t e e l s ( F i g 23) c o u l d be e x p l a i n e d by c o n s i d e r i n g t h e p e r i t e c t i c r e a c t i o n and e x t e n s i v e 6—>y s o l i d s t a t e t r a n s f o r m a t i o n . T h i s t r a n s f o r m a t i o n , a s has been m e n t i o n e d e a r l i e r , i s a s s o c i a t e d w i t h a b o u t 0.38% s h r i n k a g e . I t i s a l s o known t h a t a t t h e s e v e r e c o o l i n g r a t e s u nder c o n s i d e r a t i o n , t h e s o l i d s t a t e t r a n s f o r m a t i o n t a k e s p l a c e q u i t e r e a d i l y . No s o o n e r a t h i n f i l m of l o w - c a r b o n s t e e l s o l i d i f i e s i t p u l l s o f f from t h e mould due t o t h e s h r i n k a g e . T h i s would r e s u l t i n t h e o b s e r v e d l o w e r m e n i s c u s h e a t f l u x e s r e p o r t e d i n T a b l e I I I and m e n i s c u s s o l i d i f i c a t i o n w ould n o t t a k e p l a c e . T h u s , t h o u g h model p r e d i c t i o n s d e s c r i b e d i n s e c t i o n 5.6 e m p h a s i s e t h a t i t i s e a s i e r t o s o l i d i f y low C s t e e l m e n i s c i i , t h i s d o e s n o t seem t o be a p h y s i c a l r e a l i t y . Towards t h e end o f t h e n e g a t i v e s t r i p p e r i o d , as t h e mould s l o w s down and r e l e a s e s t h e b i l l e t , new m e t a l o v e r f l o w s f r o m t h e t o p of t h e s o l i d m e n i s c u s . In t h e e n s u i n g p o s i t i v e s t r i p p e r i o d , t h e mould f l u x d r o p s o f f b e c a u s e o f t h e i n c r e a s e d gap r e s u l t i n g f r o m t h e mould moving away from t h e b i l l e t . T i l l t h e 1 38 b e g i n n i n g of t h e n e x t n e g a t i v e s t r i p p e r i o d , t h e b i l l e t would have t r a v e l l e d a d i s t a n c e e q u a l t o t h e r a t i o of c a s t i n g -speed t o o s c i l l a t i o n s p e e d and would r e c e i v e a n o t h e r o s c i l l a t i o n mark. I t s h o u l d be e m p h a s i s e d t h a t i t i s not n e c e a s s a r y f o r t h e mould t o g r a b and h o l d t h e m e n i s c u s f o r a l l t h e a v a i l a b l e t i m e of n e g a t i v e s t r i p . The t i m e of c o n t a c t would be d e p e n d a n t on how soon t h e mould would c l o s e t h e gap c r e a t e d by s h r i n k i n g s h e l l a t t h e m e n i s c u s . However, t h e c h i l l e d s t r u c t u r e o f t h e o v e r f l o w n m e t a l s u g g e s t s t h a t t h e o v e r f l o w must have t a k e n p l a c e d u r i n g t h e n e g a t i v e s t r i p i . e . , d u r i n g a t i m e of e x t e n s i v e h e a t t r a n s f e r by t h e mould. The e x t e n t of s o l i d i f i c a t i o n f o r a g i v e n n e g a t i v e s t r i p p e r i o d depends on t h e h e a t f l u x r e a l i s e d and t h e s u p e r h e a t . C a l c u l a t i o n s f r o m t h e 2-D h e a t t r a n s f e r model s u p p o r t t h i s v i e w . I t was shown t h a t w h i l e 4186 KW/M2 a t 1°C of s u p e r h e a t g i v e s r i s e t o s o l i d i f i c a t i o n a t t h e m e n i s c u s a t 20°C no s o l i d i f i c a t i o n was o b s e r v e d . Thus f o r a g i v e n n e g a t i v e s t r i p and f o r a g i v e n t i m e of e x t e n s i v e h e a t e x t r a c t i o n f r o m t h e m e n i s c u s , d e p e n d i n g on t h e h e a t f l u x m e n i s c u s s o l i d i f i c a t i o n c o u l d o c c u r o n l y below a c e r t a i n s u p e r h e a t . At p r e s e n t t h e m a g n i t u d e and t h e r e q u i r e d d w e l l t i m e o f s u c h a h e a t f l u x i s not known. C o n s i d e r i n g t h e s h o r t d u r a t i o n of s u c h h i g h h e a t f l u x e s a t t h e m e n i s c u s , i t i s no s u r p r i s e t h a t t h e a v e r a g e h e a t f l u x o b t a i n e d f r o m t h e s e t r a i l s was much l o w e r . The t e c h n i q u e employed c o u l d not p i c k up t h e i n s t a n t a n e o u s s u r g e s o f h e a t on t h e h o t f a c e of t h e mould one has t o r e s o r t t o much more s o p h i s t i c a t e d m e a s u r i n g s y s t e m w h i c h would m o n i t o r t h e 1 39 t e m p e r a t u r e o f mould c o n t i n u o u s l y a t t h e 'moving m e n i s c u s ' . A l s o c a r e has t o be t a k e n a b o u t a c o n s t a n t m e t a l l e v e l as i t would have an i n f l u e n c e on t h e t y p e and e x t e n t of t a p e r a t t h e m e n i s c u s a r e a . I t i s not c l e a r , however, a s t o why t h e o f f - c o r n e r r e g i o n s had d e e p e r marks compared t o t h e m i d f a c e . As F i g 21 i n d i c a t e s , t h e r e i s no d i f f e r e n c e i n t h e e x t e n t of m e n i s c u s s o l i d i f i c a t i o n between t h e m i d f a c e and o f f - c o r n e r . The d i f f e r e n c e i s o n l y i n t h e d e p r e s s i o n . One p o s s i b i l i t y c o u l d be t h a t t h e c o r n e r of t h e b i l l e t i s m e c h a n i c a l l y more r i g i d t h a n any o t h e r a r e a o f t h e s e c t i o n and a s t h e mould t r i e s t o s q u e e z e on t h e b i l l e t ' s t h i n s h e l l t h e a r e a a d j a c e n t t o t h e r i g i d mould c o r n e r would undergo more d e f o r m a t i o n . I t was shown t h a t p h o s p h o r o u s has a s t r o n g i n f l u e n c e on t h e d e p t h o f o s c i l l a t i o n marks when t h e w ater f l o w r a t e s a r e h i g h e r t h a n 31 1/s. T h i s c o u l d be e x p l a i n e d by c o n s i d e r i n g t h e e f f e c t of p h o s p h o r o u s on m e c h a n i c a l p r o p e r t i e s o f s t e e l . P h o s p h o r o u s has been shown t o t h e d e c r e a s e t h e h i g h t e m p e r a t u r e f r a c t u r e s t r e n g t h of s t e e l s 7 " . When t h e mould e x e r t s a n o r m a l f o r c e i n t h e n e g a t i v e s t r i p t i m e a weaker s h e l l w o u l d u n d e r g o more d e f o r m a t i o n o r b u c k l i n g . Water v e l o c i t i e s lower t h a n 9.7 m/s were shown t o c a u s e d e e p e r o s c i l l a t i o n marks. S a m a r a s e k e r a and B r i m a c o m b e 2 0 p o i n t out a t t h e s e l o w e r f l o w r a t e s i n t e r m i t t e n t o r c o m p l e t e b o i l i n g t a k e s p l a c e l e a d i n g t o more dynamic d i s t o r t i o n . Thus l a r g e r t a p e r s a t t h e mould t o p would l e a d t o s e v e r e d e f o r m a t i o n o f t h e b i l l e t s u r f a c e i n q u e s t i o n . Howvever, c o n s i d e r i n g e x c e p t i o n s 1 40 l i k e sample 24272 ( a t 3.1 m/s) and 24278 ( a t 7.2 m/s) more samp l e s a r e needed t o be examined t o o b t a i n any d e f i n i t e c o r r e l a t i o n . T h us, t h i s mechanism e x p l a i n s most of t h e o b s e r v a t i o n s made d u r i n g t h i s s t u d y . However, i t r e m a i n s t o be s e e n t h a t i n c a s e of c o n s t r a i n e d mould where s u c h mould d i s t o r t i o n i s l i m i t e d , w hether c h a r a c t e r i s t i c s of o s c i l l a t i o n marks w o u l d r emain t h e same. I t i s e x p e c t e d t h a t i n s u c h a mould p r e s e n c e of hooks would be a minimum and so would be t h e d e p t h o f o s c i l l a t i o n marks. I t s h o u l d be c l e a r l y p o i n t e d o u t t h a t t h e p r o p o s e d mechanism does n o t e x p l a i n t h e c h a r a c t e r i s t i c s o f o s c i l l a t i o n marks i n a l l c a s t e r s . In s l a b c a s t i n g , use of powder f l u x and a b s e n c e of t h e n e g a t i v e t a p e r i n t h e p l a t e moulds t h a t a r e u s e d t o c a s t s l a b s p r e s e n t s an e n t i r e l y d i f f e r e n t s i t u a t i o n and d i f f e r e n t mechanisms s h o u l d be l o o k e d i n t o . 1 42 V I I . SUMMARY AND CONCLUSIONS B i l l e t samples o b t a i n e d from a l a r g e number of h e a t s were examined t o s t u d y d i f f e r e n t a s p e c t s of s o l i d i f i c a t i o n ; c o l u m n a r -e q u i a x e d t r a n s i t i o n , s e g r e g a t i o n bands, o f f - c o r n e r c r a c k s , r h o m b o i d i t y , s u b - s u r f a c e s t r u c t u r e and o s c i l l a t i o n marks. C o r r e s p o n d i n g h e a t f l u x e s o b t a i n e d from measured t e m p e r a t u r e s were employed i n a i d o f two m a t h e m a t i c a l models t o c a l c u l a t e t h e s h e l l t h i c k n e s s i n t h e mould a t t h e m e n i s c u s as w e l l as down t h e mould l e n g t h . The f i n d i n g s of t h e s t u d y a r e as f o l l o w s . 1. C o l u m n a r - e q u i a x e d t r a n s i t i o n : C o m p o s i t i o n a l f a c t o r s were f o u n d t o p r o f o u n d l y i n f l u e n c e t h e e x t e n t o f c o l u m n a r zone i n t h e b i l l e t s e x a mined. i . E f f e c t o f c a r b o n : In t h e c a r b o n r a n g e 0.13-0.36%, i n c r e a s i n g c a r b o n was seen t o c a u s e an e a r l y c o l u m n a r -e q u i a x e d t r a n s i t i o n p a r t l y by e n h a n c i n g c o n s t i t u t i o n a l s u p e r c o o l i n g and f a c i l i t a t i n g e a s i e r m e l t i n g o f s e c o n d a r y d e n d r i t e arms. A sudden r i s e i n t h e l e n g t h o f c o l u m n a r zone i n s t e e l s w i t h c a r b o n c o n t e n t more t h a n 0.38% was o b s e r v e d f o r t h e f i r s t t i m e i n b i l l e t c a s t i n g . I t was t h o u g h t t h a t t h e s o l i d i f i c a t i o n p r o c e e d s i n t h e s e s t e e l s by d i r e c t l y f o r m i n g 7 f r o m t h e l i q u i d r a t h e r t h a n t h r o u g h t h e p e r i t e c t i c 1 43 / / r e a c t i o n w h i c h e n a b l e s e n h a n c e d s u r v i v a l of n u c l e i g e n e r a t e d a t t h e mould w a l l and c o n s t i t u t i o n a l s u p e r c o o l i n g . However, as p e r t h e Fe-C e q u i l i b r i u m d i a g r a m s u c h a s h i f t i n t h e s o l i d i f i c a t i o n p a t t e r n s h o u l d o n l y t a k e p l a c e beyond 0.52% c a r b o n i n s t e e l . The r e a s o n f o r t h e t r a n s i t i o n a t c a r b o n l e v e l s o f 0.38% and beyond i s unknown. E f f e c t of p h o s p h o r o u s : P h o s p h o r o u s was f o u n d t o have s t r o n g i n f l u e n c e on t h e c o l u m n a r - e q u i a x e d t r a n s i t i o n . In low-c a r b o n s t e e l s i n c r e a s i n g p h o s p h o r o u s l e v e l s f r o m 0.012 t o 0.036 was seen t o d e c r e a s e t h e c o l u m n a r zone l e n g t h . In h i g h - c a r b o n s t e e l s (0.28-0.36% C) p h o s p h o r o u s l e v e l s of g r e a t e r t h a n 0.025 were f o u n d t o g i v e r i s e t o p r o n o u n c e d g r o w t h o f t h e e q u i a x e d z o n e . T h i s e f f e c t o f p h o s p h o r o u s has been a t t r i b u t e d t o t h e v e r y low d i s t r i b u t i o n c o e f f i c e n t of p h o s p h o r o u s . W h i l e m e l t i n g o f t h e s e c o n d a r y d e n d r i t e arms i s e n h a n c e d , i n c r e a s e d c o n s t i t u t i o n a l s u p e r c o o l i n g would f a c i l i t a t e s u r v i v a l and g r o w t h o f t h e n u c l e i i p r e s e n t i n t h e m e l t . However i n s t e e l s w i t h more t h a n 0.38% c a r b o n no i n f l u e n c e o f p h o s p h o r o u s was o b s e r v e d . The i n f l u e n c e o f h e a t e x t r a c t i o n : 1 44 Heat t r a n s f e r e f f e c t s were f o u n d t o be of s e c o n d a r y i m p o r t a n c e i n c o l u m n a r - e q u i a x e d t r a n s t i o n . H i g h e r h e a t t r a n s f e r r a t e s , as c a r b o n i s i n c r e a s e d from 0.13 t o 0.36%, were shown t o be a s s o c i a t e d w i t h a l a r g e r e q u i a x e d zone. I n c r e a s e d smoothness of b i l l e t s u r f a c e a s s o c i a t e d w i t h h i g h e r c a r b o n i n s t e e l i s due t o a d e c r e a s e i n t h e amount o f p r i m a r y 6-phase, t h e e x t e n t o f 5 t o 7 s o l i d s t a t e t r a n s f o r m a t i o n and t h e a s s o c i a t e d s h r i n k a g e . T h i s l e a d s t o s m a l l e r m o u l d - b i l l e t gap and t h e r e s u l t i n g h i g h e r h e a t f l u x e x t r a c t s more s u p e r h e a t i n t h e mould f a c i l i t a t i n g s u r v i v a l and growth of n u c l e i w h i c h g i v e s r i s e t o l a r g e e q u i a x e d zone. The f a c t t h a t t h e l a r g e c o l u m n a r z o n e s e x i s t i n s p i t e of h i g h h e a t t r a n s f e r r a t e s i n s t e e l s w i t h more t h a n 0.38% C i n d i c a t e s t h a t t h e i n f l u e n c e o f h e a t t r a n s f e r i s o n l y o f s e c o n d a r y i m p o r t a n c e and i t i s t h e c o m p o s i t i o n of t h e s t e e l t h a t p l a y s a s i g n i f i c a n t r o l e i n p r o m o t i n g t h e g r o w t h o f e q u i a x e d z o n e . E f f e c t o f s u p e r h e a t : I t i s r a t h e r s u r p r i s i n g t o f i n d t h a t t h e s u p e r h e a t i n t h e s t e e l showed n e g l i g i b l e i n f l u e n c e . C o n s i d e r i n g t h a t t h e s u p e r h e a t s employed a r e g e n a r a l l y h i g h , i t a p p e a r s l o g i c a l t h a t o t h e r f a c t o r s d o m i n a t e d i n c o n t r o l l i n g t h e 1 45 e x t e n t of c o l u m n a r z o n e . A d d i t i o n a l work has t o be done t o u n d e r s t a n d t h e n a t u r e o f t h e c o l u m n a r - e q u i a x e d t r a n s i t i o n and t h e f a c t o r s t h a t c o n t r o l i t . From t h e p r e s e n t s t u d y i t c a n , however, be s u g g e s t e d t h a t b e s i d e s use of v e r y low s u p e r h e a t s and e m p l o y i n g e x p e n s i v e e l e c t r o -m a g n e t i c s t i r r e r s , s t r u c t u r e c o n t r o l c o u l d be a c c o m p l i s h e d by c o n t r o l o f c o m p o s i t i o n , i . e a d d i t i o n of a l l o y i n g e l e m e n t s w i t h low d i s t r i b u t i o n c o e f f i c e n t . R e - p h o s p h o r i s i n g and a d d i t i o n of b o r o n not o n l y would g i v e r i s e t o b e t t e r s t r u c t u r e s but a l s o s t r e n g t h e n l o w - c a r b o n s t e e l s . However, c a r e has t o be e x e r c i s e d i n s e l e c t i n g t h e e l e m e n t a s some of t h e s e , e s p i c i a l l y p h o s p h o r o u s i s known t o c a u s e temper e m b r i t t l e m e n t . B e t t e r molds s h o u l d be d e s i g n e d t o e x t r a c t more h e a t from t h e b i l l e t and improve t h e s t r u c t u r e . W h ite and d a r k s o l i d i f i c a t i o n b a n d s: E x a m i n a t i o n of m a c r o e t c h e s r e v e a l e d t h a t t h e r e e x i s t two d i s t i n c t bands w i t h i n 10-11 mm of t h e edge of t h e b i l l e t s . T h e s e were f o u n d t o d e l i n e a t e t h e s h e l l p r o f i l e a t an i n s t a n t o f t i m e , i . W h i t e band: W i t h t h e h e l p o f t h e o n e - d i m e n s i o n a l model i t was shown t h a t t h e w h i t e band w h i c h i s c l o s e r t o t h e s u r f a c e forms 450-550 mm below t h e m e n i s c u s where t h e s o l i d i f y i n g s h e l l u n d e r g o e s 146 a c o n s t a n t c o o l i n g r a t e and t h e s o l i d i f i c a t i o n f r o n t p r o c e e d s a t a m i n i m a l s p e e d . Dark band: The s h e l l t h i c k n e s s c o r r e s p o n d i n g t o t h e s e c o n d band w h i c h was d a r k o r d e e p l y e t c h e d was f o u n d t o have d e v e l o p e d a t t h e m i d f a c e v e r y c l o s e t o t h e b o t t o m o f t h e mould. Mechanism of f o r m a t i o n of t h e b a n d s : The w h i t e band has been a t t r i b u t e d t o t h e d e p l e t i o n of i n t e r d e n d r i t i c s o l u t e owing t o t h e w a s h i n g a c t i o n of f l u i d f l o w i n t h e l i q u i d p o o l a g a i n s t t h e s l o w l y moving s o l i d i f i c a t i o n f r o n t . However, t h e d a r k band by a s i m i l a r a n a l o g y was s u g g e s t e d as r e s u l t i n g from t h e enhancement of s o l i d i f i c a t i o n , p r o b a b l y , when t h e f i r s t s p r a y s h i t ' t h e b i l l e t and t h e i n a b i l i t y of t h e t u r b u l e n t l i q u i d p o o l t o wash t h e q u i c k l y g r o w i n g s o l i d . B a s e d on t h e p o o l p r o f i l e e x i b i t e d by t h e s e bands i n t h e t r a n s v e r s e s e c t i o n s i t can be c o n c l u d e d t h a t w h i l e s o l i d i f i c a t i o n p r o c e e d s s l o w l y a t t h e m i d f a c e a f t e r t h e f o r m a t i o n o f t h e w h i t e band 450-550 mm below t h e m e n i s c u s i t a p p e a r s t o have v i r t u a l l y s t o p p e d a t t h e o f f -c o r n e r / c o r n e r a r e a . S t u d y of t h e s e bands has shown t h a t i n t h e m a j o r i t y o f t h e b i l l e t s h e a t t r a n s f e r on t h e 1 47 f o u r f a c e s i s f a r from b e i n g i d e n t i c a l i n d i c a t i n g t h a t t h e mould b i l l e t gap i s d i f f e r e n t on t h e f o u r s i d e s . v i . T h i n w h i t e bands a t t h e o b t u s e a n g l e c o r n e r s o f t h e b i l l e t s , w h i c h i s a r e s u l t o f i m p r o p e r h e a t e x t r a c t i o n i n t h e mould, c o u l d e x p l a i n t h e o r i g i n o f r h o m b o i d a l b i l l e t s , an e v e n t o c c u r i n g i n t h e s p r a y s b e c a u s e o f d i f f e r e n t i a l c o n t r a c t i o n of a d j a c e n t f a c e s . • From t h e above o b s e r v a t i o n s i t c a n be s u g g e s t e d t h a t i t w ould be a d v a n t a g e o u s t o use s h o r t e r moulds i n c o n j u n c t i o n w i t h p l a c e m e n t o f s u p p o r t i n g r o l l s and s p r a y - c o o l i n g i n t h e sub-mould r e g i o n . 3. R h o m b o i d i t y : i . R h o m b o i d i t y , as c h a r a c t e r i s e d by t h e d i f f e r e n c e i n d i a g o n a l s was f o u n d t o v a r y between 0 t o 9 mm. i i . No e f f e c t o f c a r b o n , l i f e o f t h e mould, or t h e water f l o w r a t e was f o u n d t o i n f l u e n c e r h o m b o i d i t y , p o s s i b l y b e c a u s e t h e b i l l e t s a r e s l i g h t l y r o l l e d as t h e y were w i t h d r a w n by t h e p i n c h r o l l s . i i i . M i n i m a l r h o m b o i d i t y was f o u n d when one o f t h e s t r a i g h t w a l l s of t h e mould was c o n s t r a i n e d i n t h e m e n i s c u s a r e a . I t was c o n c l u d e d t h a t d i f f e r e n c e i n t h e d i s t o r t i o n o f t h e c u r v e d and 148 s t r a i g h t w a l l s i s an i m p o r t a n t f a c t o r i n c a u s i n g r h o m b o i d i t y . • I t i s s u g g e s t e d t h a t new mould d e s i g n s s h o u l d be s o u g h t w h e r e i n t h e mould w a l l s a r e c o n s t r a i n e d f r o m d i s p l a c i n g f r o m t h e b i l l e t s u r f a c e . 4. O f f - c o r n e r c r a c k i n g : i . O f f - c o r n e r c r a c k i n g was o b s e r v e d i n a number of b i l l e t s a mples i n one o r many o f t h e e i g h t o f f -c o r n e r l o c a t i o n s . The s u r f a c e s p e r p e n d i c u l a r t o t h e c r a c k a r e o f t e n a s s o c i a t e d w i t h a d e p r e s s i o n a t t h e o f f - c o r n e r and a b u l g e a t t h e m i d f a c e . i i . The p r e s e n t s t u d y i s i n agreement w i t h t h e l i t e r a t u r e i n t h a t t h e c r a c k s have fo r m e d b e c a u s e of t e n s i l e s t r a i n s g e n e r a t e d a t t h e s o l i d i f i c a t i o n f r o n t when t h e m i d f a c e b u l g e s . R i g i d i t y of t h e c o l d c o r n e r would l o c a t e t h e s t r a i n i n t h e o f f - c o r n e r r e g i o n t o r e s u l t i n a c r a c k . i i i . T h e se c r a c k s were f o u n d i n c l o s e p r o x i m i t y t o t h e w h i t e band i n d i c a t i n g t h a t t h e y c o u l d have formed t o w a r d s t h e m o u l d - e x i t . i v . R e h e a t i n g of t h e b i l l e t s u r f a c e was f o u n d t o be a key f a c t o r i n c a u s i n g t h e s e c r a c k s . I t was f o u n d from t h e o n e - d i m e n s i o n a l h e a t t r a n s f e r 1 49 model t h a t as t h e b i l l e t e x i t s t h e mould w h e r e i n o f f - c o r n e r c r a c k i n g o c c u r s , t h e b i l l e t s u r f a c e u n d e r g o e s e x t e n s i v e r e h e a t i n g . A l s o , a b s e n c e of s e c o n d a r y c o o l i n g w a t e r was f o u n d t o a c c e n u a t e c r a c k i n g a t t h e o f f - c o r n e r s . v. O f f - c o r n e r c r a c k i n g was a g g r a v a t e d by mould water v e l o c i t i e s o f 6.5-7.2 m/s. Such low water v e l o c i t i e s a r e known t o c a u s e i n t e r m i t t e n t b o i l i n g i n t h e u p p e r r e g i o n s o f t h e mould. I t i s unknown as t o how e v e n t s a t t h e m e n i c u s c o u l d a f f e c t c r a c k i n g w h i c h o c c u r s i n t h e l o w e r p a r t s of t h e mould. v i . Mn/S r a t i o s l o w e r t h a n 22-25 were f o u n d t o i n c r e a s e t h e s e v e r i t y o f c r a c k i n g , p o s s i b l y by i n c r e a s i n g t h e a v a i l a b l e s u l p h u r t o form t h e l o w - m e l t i n g F e s . R e m e d i a l m e a s u r e s must i n c l u d e , e m p l o y i n g Mn/S r a t i o s h i g h e r t h a n 24, e n s u r i n g s p r a y c o o l i n g on t h e b i l l e t as i t e x i t s t h e b i l l e t , p l a c i n g f o o t r o l l s t o g u i d e and s u p p o r t t h e b i l l e t so t h a t m i d f a c e b u l g i n g would n o t o c c u r , e m p l o y i n g water v e l o c i t i e s above 9.7 m/s and i n c r e a s i n g mould t a p e r t o r e d u c e r e h e a t i n g . O s c i l l a t i o n marks: i . C a r b o n c o n t e n t o f s t e e l was f o u n d t o a f f e c t t h e d e p t h o f and s t r u c t u r e below t h e o s c i l l a t i o n marks. S t e e l s w i t h c a r b o n l o w e r t h a n 0.20% 1 50 were f o u n d t o have much d e e p e r o s c i l l a t i o n marks t h a n t h e h i g h e r c a r b o n s t e e l s . i i . M e n i s c u s - s h a p e d hooks were f o u n d t o l i e below t h e o s c i l l a t i o n marks i n h i g h - c a r b o n s t e e l s . Low-carbon s t e e l s d i d not e x h i b i t any hook f o r m a t i o n . i i i . Hooks were f o u n d t o be l o n g e r a t lo w e r s u p e r h e a t s . i v . A t w o - d i m e n s i o n a l u n s t e a d y - s t a t e m a t h e m a t i c a l model was d e v e l o p e d t o s t u d y t h e p o s s i b i l i t y of m e n i s c u s s o l i d i f i c a t i o n . I t was f o u n d t h e n e c e s s a r y h e a t f l u x n e c e s s a r y t o a l l o w g r o w th of s o l i d o v e r t h e m e n i s c u s has t o be much l a r g e r t h a n t h o s e o b t a i n e d f r o m e x p e r i m e n t s c o n d u c t e d i n a s e p e r a t e s t u d y . v . A new mechanism h a s been f o r m u l a t e d t o e x p l a i n m e n i s c u s s o l i d i f i c a t i o n , t h e f o r m a t i o n of o s c i l l a t i o n marks i n b i l l e t c a s t i n g . The n e g a t i v e t a p e r i n b i l l e t m oulds c o n s t r a i n e d o n l y a t t h e t o p was t h o u g h t t o be t h e p o t e n t i a l c a u s e of t h e s e p e r i o d i c d e p r e s s i o n s . S e v e r i t y of o s c i l l a t i o n marks c o u l d be r e d u c e d by c o n s t r a i n i n g t h e mould from b u l g i n g below t h e c o n s t r a i n t and e m p l o y i n g low n e g a t i v e s t r i p t i m e . 151 BIBLIOGRAPHY 1. S i n g h S.N and B l a z e k K.E; J l o f M e t a l s , 1974, ( 1 0 ) , P.17. 2. S a m a r a s e k e r a I.V and Brimacombe J.K; I r o n m a k i n g S t e e l m a k i n g , 1982" , ( 1 ) , P.1. 3. Nemoto H; T r a n s . I S I J , 1976, j_6, ( 1 ) , P.51. 4. M o r t o n J.S and P r i t c h a r d W.H; ' C o n t i n u o u s C a s t i n g of S t e e l ' 1965, London, The I r o n and S t e e l I n s t i t u t e , P.145. 5. J o h n s o n R, M i d d l e t o n J.W and F o r d D; i b i d P.34. 6. Savage J and P r i t c h a r d W.H; J I S I 1954, ( 1 1 ) , P.178. 7. M e s s i n g w e r k e Schwazwald B r i t i s h p a t e n t No:437,064. 8. BISRA P a t e n t No:806,803. 9. H o l l i d a y I.M.D; 'The main i s s u e s of C o n t i n u o u s c a s t i n g . ' T M S - S p e c i a l R e p o r t 89, P.6. 10. S c h o f f m a n n R; I r o n and S t e e l Y e a r book, 1972, P.369. 11. Kashay A.M; I r o n and S t e e l Y e a r book, 1973, P.180. 12. Watanabe S,Harada K, F a j i t a N, Tamura Y, and Naro K; ' T e s t u - t o - H a g n e ' , J l . I r o n S t e e l I n s t . J p n , 1972, 5_8, ( 2 ) , P.107. 13. G r i l l A, Brimacombe J.K; I r o n m a k i n g S t e e l m a k i n g 1976, 3 ( 2 ) , P.107 14. Wolf M, K u r z W; M e t . T r a n s . , 1981, J_2B, P.85 15. H u r t u k D.J and T z a v a r a s A; J l . o f M e t a l s , 1982, ( 2 ) , 40 16. H u r t u k D.J; Ph.D T h e s i s , C a s e W e s t e r n R e s e r v e U n i v e r s i t y , J a n u a r y , 1981 17. U s h i j i m a K; ' T e s t u - t o - H a g n e ' , 1961, 47, ( 3 ) , P.390 18. I b i d ; ' C o n t i n u o u s C a s t i n g o f S t e e l ' , 1964, London, The I r o n and S t e e l i n s t i t u t e , P.59 19. E v t e e v D.P; S t a l i n E n g l i s h , 1969, ( 8 ) , P.708 20. S a m a r a s e k e r a I.V and Brimacombe J.K; U n p u b l i s h e d r e s e a r c h 21. i b i d . ; I n t e r n a t i o n a l M e t a l s Review 1978 ( 6 ) , P.286. 22. Brimacombe J.K.,Hawbolt E.B and W e i n b e r g F; Can. Met. 1 52 Q u a r t . , 1976, J_5, ( 2 ) , P.163. 23. A k e t a Y and U s h i j u m a K; ' T e s t u - t o - H a g n e ' , 1962, 2, ( 4 ) , p.334. 24. G r i l l A, S o r i m a c h i K and Brimacombe J.K; Met. T r a n s . , 1976, 7B ( 2 ) , P.177. 25. D y u d k i n D.A; e t a l ; S t e e l USSR, 1972, 2, ( 2 ) , P.122. 26. P e r m i n o v V.P. e t a l ; S t a l i n E n g l i s h , 1968, ( 7 ) , P.560. 27. S a m a r a s e k e r a I.V, A n d e r s o n D.L and Brimacombe J.K; Met. T r a n s . , 1982, 13B, P.91. 28. Brimacombe J.K e t a l ; ISS T r a n s a c t i o n ; 1982,J_, p.29. 29. S a m a r a s e k e r a I.V and Brimacombe J.K; I r o n m a k i n g S t e e l m a k i n g , 1982, ( 1 ) , P.1 30. i b i d . ; M e t . T r a n s . , 1982, J J B , P.105. 31. M o r i H; ' T e s t u - t o - H a g n e ' , 1972, 58, P.1511 32. F u j i n a m i K; ' R a p i d M e l t i n g i n UHP E l e t r i c A r c f u r n a c e t o p e r m i t l o n g s e q u e n c e c a s t i n g ' . P aper p r e s e n t e d a t C o n f . of M e t a l l u r g i s t , C a n a d i a n I n s t . o f M i n i n g and M e t a l l u r g i s t , 1971. 33. Young W.P and W h i t f i e l d W.T; Open H e a r t h P r o c , AIME, 1 968, 51_, P. 1 27. 34. M c C o n n e l l J . E ; Open h e a r t h p r o c , AIME, 1972, 55, P.56 35. Matsunaga K; Open H e a r t h P r o c , AIME, 1976, 59, P.228. 36. F u j i n a n i K, F u n a b a s h i S t e e l works L t d , F u m a b a s h i , J a p a n -u n p u b l i s h e d r e s e a r c h . 37. Tokuyama, S u z u k i ; The 2 4 t h J o i n t Symposium by t h e Kyushu C h a p t e r s of I S I J and NKG, J u l y 1970. 38. Brimacombe J.K.,Hawbolt E.B and W e i n b e r g F; Can. Met. Q u a r t . , 1980, J_9, P.215. 39. Brimacombe J.K, S o r i m a c h i ; Met. T r a n s . , 1977, 8B, P.489. 40. Baumann H.G and Lopmann W.J; Wire W o r l d I n t , 1974, 16, P.1733. 41. Vom Ende H and Vo g t G; J I S I , 1972, 2J_0, P . 8 8 9 . 42. Z e t t e r l u n d E.H and K r i s t i a n s s o n J.O; S c a n d i n a v i a n J l . of M e t a l l u r g y , 1983, J_2, P.211. 1 53 43. L a i t J . E , Brimacombe J.K, W e i n b e r g F and M u l t i t t F.C; Open H e a r t h P r o c e e d i n g s , AIME, 1973, 56, P.269. 44. Tanaka S e t a l ; L e c t u r e a t 101st I S I J m e e t i n g , A p r i l , 1981, L e c t u r e S172. 45. Wolf M; M e t a l l u r g i c a l P l a n t and T e c h n o l o g y , 1983, 2, P.46. 46. Emi T e t a l ; S t e e l M aking P r o c . , 6_[, 1978, P.350. 47. T a k e u c h i S e t a l ; L e c t u r e p r e s e n t e d a t 102nd I S I J m e e t i n g , Nov., 1981, L e c t u r e S906. 48. Tanaka S e t a l ; ' T e s t u - t o - H a g n e ' , 1981, 67, s852. 49. Saucedo I , Beech J and D a v i e s G.J; Conf P r o c e e . of 6 t h I n t e r n a t i o n a l Vacuum M e t a l l u r g y ' S p e c i a l M e l t i n g and M e t a l l u r g i c a l C o a t i n g s ' San D i e g o . C a l i f . Apr 1979, A m e r i c a n Vacuum S o c i e t y . P.885 50. Savage J ; I r o n and C o a l T r a d e s Review, London 1961, 182, ( 4 ) , P.787 51. S a t o R; 62nd N a t i o n a l OPH and BOS C o n f - P r o c e e d i n g s , 1979, 62, P.48. 52. R i b o u d P.V and L a r r e c q M; S t e e l M a k i n g P r o c , 1979, 62, P.78. 53. Kawakamii K e t a l ; N i p p o n Kokan T e c h n i c a l R e p o r t ( O v e r s e a s ) No:36, 1982, P.1. 54. Boemer W.C.K and Raper A.G; J I S I , 1970, P.18. 55. Wolf M; P r o c e e 4 0 t h E l e c t r i c C o n f . P r o c e e d i n g s , Kansas C i t y , P.41 56. Wolf M; 102nd I S I J m e e t i n g Nov 1981, L e c t u r e S-904. 57. J a c o b i H e t a l ; M e t a l l u r g i c a l P l a n t and T e c h n o l o g y , 1982 , ( 4 ) , P.43. 58. Komatsu M E t a l ; 104th I S I J m e e t i n g Sep 1982, l e c t u r e S-927. 59. Brown D.I; J l . of M e t a l s , 1965, ( 4 ) , P.426. 60. T h o r t o n D.R; J I S I , 1956, ( 7 ) , P.300. 61. Wray P . J ; Met. T r a n s . , 1 9 8 1 , J[2B, ( 3 ) , P.167 62. Saucedo I , Beech J and D a v i e s G.J; M e t a l s T e c h n o l o g y , 1982, 9, ( 7 ) , P.282. 154 63. H i b b i n s s; M.A.Sc T h e s i s , U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , B.C., Canada, 1982 64. Carnahan B, L u t h e r H.A and W i l k e s J ; ' A p p l i e d N u m e r i c a l Methods' John W i l e y and Sons I n c . 1969. 65. P e e l D.A, P e n g e l l y A.E; M a t h e m a t i c a l models i n M e t a l l u r g i c a l P r o c e s s D e v e l o p m e n t , 1970, London, I S I , P. 186. 66. B a l l a n t y n e A.S; Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , B.C., Canada, 1978. 67. V e n k a t e s w a r a n ; Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , B.C., Canada, 1978. • 68. Tomono H, K u r z W and Heinemann W; Met. T r a n s . , 1981, 13B, ( 6 ) , P.409. 69. M i y a h a r a S . , T s u c h i t a H . , S u z u k i M.,Kuwano.S and S h i r o y a m a A; T e t s u - t o - h a g n e , 1979, S273 70. T i w a r i S.N and Beech J ; T r a n s . I S I J , 1981, 2J_, P.564 71. W e i n b e r g F; P r o f e s s o r , D e p t . o f M e t a l l u r g y , U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , B.C., Canada, P r i v a t e Commun i c a t i o n . 72. Morton S.k and W e i n b e r g F; J I S I , 1973, ( 1 ) , P.13. 73. L a i t J . E . , Brimacombe J.K. and W e i n b e r g F.W; I r o n m a k i n g S t e e l m a k i n g , 1974, ( 1 ) , P.35. 74. Sopher R.P; W e l d i n g J o u r n a l , 1958, 44, P.481s. 1 5 5 APPENDIX - I TABLES Table - I CHARACTERISTICS OF THE MOLD AT WESTERN CANADA STEEL Curved Mold Copper Grade: DHP 122 taper: 0.1064% Length: 812.8 mm Stroke: 18.44 mm Water Gap: 4.76 mm Negative S t r i p : 0.254 s Wall Thickness: 7.9375 mm Insid e Corner Radius: 6.35 mm Chrome P l a t e Thickness: 0.127 mm Insid e Bottom Dimensions: 143.129 mm2 Location of the f i r s t spray nozzle below the mold:1.27cm Co n s t r a i n t Type 2: (Co n s t r a i n t on the two s t r a i g h t sides near the t o p ) . OSCILLATION PARAMETERS Casting speed (meters per minute) O s c i l l a t i o n speed ( o s c i l l a t i o n per minute) 1 .397 49 1 .702 58 1.956 68 2.159 74 2.540 85 1 56 Table II - S t e e l Composition C.NO Heat No. Composition in Wt% C Mn P S Cu Cr Si 1 22697 0.28 0.65 0.022 0.029 0.38 0.14 0.18 3 22847 0.25 1 .22 0.011 0.029 0.31 0.09 0.18 22848 0.25 0.21 0.013 0.029 0.26 0.10 0.21 22853 0.43 1 .38 0.015 0.049 0.30 0.23 0.29 22854 0.39 1 .05 0.035 0.047 0.32 0.23 0.20 22855 0.39 0.96 0.013 0.032 0.29 0.13 0.12 4 23369 0.40 1.13 0.013 0.021 0.35 0.13 0.22 23370 0.36 1.10 0.01 1 0.023 0.40 0.10 0.14 23372 0.39 1 .05 0.015 0.030 0.47 0.17 0.14 23373 0.15 0.61 0.013 0.035 0.41 0.13 0.12 23374 0.33 0.97 0.029 0.041 0.49 0.15 0.23 5 23470 0.24 1.15 0.012 0.017 0.33 0.15 0.10 23471 0.50 0.75 0.013 0.030 0.43 0.13 0.18 23472 0.48 0.55 0.012 0.024 0.41 0.12 0.13 23474 0.50 0.68 0.016 0.020 0.34 0.18 0.07 6 23490 0.43 0.96 0.014 0.049 0.44 0.18 0.21 23491 0.23 1.11 0.030 0.038 0.39 0.20 0.15 23492 0.33 1 .20 0.020 0.039 0.45 0.24 0.17 7 23528 0.41 1 .04 0.010 0.047 0.52 0.17 0.11 23529 0.34 1 .12 0.015 0.037 0.51 0.22 0.18 23530 0.16 0.63 0.012 0.025 0.40 0.15 0.13 8 24090 0.13 0.65 0.010 0.029 0.27 0.26 0.11 9 24100 0.45 0.86 0.025 0.026 0.36 0.16 0.15 ** 24109 0.21 0.58 0.025 0.028 0.34 0.16 0.14 10 241 11 0.33 1 .25 0.025 0.047 0.40 0.22 0.26 241 12 0.26 1 .46 0.029 0.050 0.46 0.24 0.27 241 13 0.17 0.38 0.012 0.042 0.39 0.12 0.04 241 14 0.27 1.31 0.011 0.047 0.34 0.11 0.27 241 15 0.34 1.25 0.028 0.048 0.40 0.15 0.22 241 16 0.33 1 .20 0.014 0.035 0.42 0.16 0.22 11 24217 0.20 0.71 0.020 0.045 0.42 0.19 0.18 24218 0.18 0.64 0.034 0.044 0.41 0.17 0.10 24219 0.20 0.75 0.025 0.032 0.26 0.15 0.17 24220 0.21 0.74 0.021 0.043 0.37 0.15 0.20 24221 0.38 0.91 0.031 0.071 0.44 0.20 0.38 **-This heat was cast without secondery cooling water. 1 5 7 Table II - continued C N o Heat Composition i n Wt% No. c Mn P S Cu Cr S i 12 24242 0.29 1 .18 0.017 0.039 0.35 0.20 0.25 24243 0.32 1.12 0.014 0.034 0.28 0.14 0.20 24244 0.26 1 .20 0.017 0.039 0.42 0.15 0.25 24245 0.30 1 .24 0.020 0.040 0.43 0.16 0.26 24246 0.34 1 .09 0.016 0.051 0.43 0.14 0.18 24247 0.38 1 .07 0.012 0.033 0.35 0.14 0.15 24248 0.28 1 .00 0.022 0.036 0.36 0.16 0.18 24249 0.28 0.86 0.018 0.039 0.34 0.14 0.15 24250 0.36 0.86 0.035 0.034 0.27 0.23 0.13 24251 0.29 1 .07 0.028 0.046 0.34 0.15 0.18 24252 0.28 1 .08 0.035 0.046 0.40 0.20 0.26 24253 0. 14 0.64 0.022 0.042 0.36 0. 18 0.21 24254 0.36 0.84 0.026 0.028 0.35 0.17 0.12 24255 0.14 0.58 0.036 0.036 0.45 0.12 0.15 24256 0.36 0.73 0.061 0.038 0.45 0.13 0.12 24257 0.16 0.51 0.020 0.033 0.45 0.12 0.15 24258 0.14 0.63 0.028 0.035 0.31 0.15 0.15 24259 0.16 0.51 0.021 0.033 0.32 0.16 0.12 24260 0.15 0.62 0.01 1 0.033 0.29 0.12 0.16 24261 0.13 0.46 0.009 0.031 0.37 0.12 0.09 24262 0.13 0.55 0.008 0.031 0.34 0.10 0.03 24263 0.34 1 .06 0.016 0.035 0.33 0.18 0.19 24264 0.36 1 .30 0.017 0.028 0.35 0.22 0.19 24265 0.35 1.11 0.022 0.036 0.32 0.21 0.20 24266 0.31 0.88 0.016 0.037 0.37 0.39 0.15 24268 0.14 0.55 0.020 0.039 0.35 0.29 0. 14 24269 0.15 0.63 0.017 0.031 0.34 0.20 0.03 24270 0.13 0.55 0.012 0.027 0.36 0.20 0.10 24271 0.20 0.71 0.016 0.030 0.30 0.16 0.17 24272 0.33 0.92 0.028 0.033 0.30 0.41 0.17 24273 0.41 1 .04 0.017 0.029 0.34 0.26 0.18 24274 0.38 1 .39 0.015 0.027 0.37 0.11 0.45 24275 0.14 0.69 0.011 0.024 0.30 0.18 0.15 24276 0.30 0.97 0.023 0.044 0.35 0.35 0.17 24277 0.29 1 .02 0.023 0.047 0.32 0.20 0.18 24278 0.33 1 .26 0.026 0.043 0.41 0.21 0.36 TABLE -III Columnar zone l e n g t h and o t h e r d e t a i l s C . N o . Heat B.No. Compos 111 on W.F.R W. V S.H Av .H . F M.H. F Co lumnar Zone Leng th (mm) No (1/s ) m/s C c ) (KW/M*) c% P% S. S1 S. S2 O u t s i d e Rad1 us A c t u a l Av. L o c a l Av . 1 32 .80 9 . 70 70 70 65 XII 24261 2 0 .13 0 .009 32 .80 9 .70 22 1566 3350 63 56 54 61 60 3 21 .76 4 .95 1342 3300 60 60 65 1 21 . 76 4 .95 65 67 ( 52 XII 24262 2 0 .13 0 .008 21 .76 4 . 95 24 1507 330O 62 64 55 56 59 3 32 .80 9 .70 1704 3200 62 60 50 1 31 .54 9. 15 60 - 50 XII 24270 2 0 .13 0.012 31 . 54 9. 15 34 1509 3500 65 64 54 53 52 3 25 .23 6 . 70 1402 3400 61 61 56 1 32 .80 9 .70 3000 60 60 45 XII 24253 2 0 . 14 0 .022 32 .80 9 .70 18 1487 57 56 42 41 3 32 .80 9. 70 37 34 36 1 32 .80 9 .70 25 1790 4000 54 55 43 XII 24255 2 0. 14 0 .036 32 .80 9 .70 56 48 36 40 3 32 .80 9 .70 55 42 40 1 22 . 7 1 5.55 22 56 60 44 XII 24258 2 0 .14 0 .028 32 .49 9 .55 50 58 52 43 3 32 .49 9 .55 47 47 34 1 32 .80 9 .70 19 1749 3300 52 55 40 XII 24268 2 0 .14 0.02 27 . 13 7 .45 1654 66 50 44 42 43 3 27 . 13 7.45 3300 53 58 41 1 25 .23 6 . 70 19 1575 3300 60 60 47 XII 24275 2 0 .14 0.011 25.54 6 .85 62 65 57 53 52 3 31 .54 9.15 1728 3200 63 62 54 1 21 .76 4 .95 22 1520 3300 65 67 59 XII 24260 2 0 .15 O.01 1 21 .76 4 .95 64 64 60 61 59 3 32 .80 9 .70 1682 3200 65 65 64 1 18.93 3.85 NA 1561 4300 60 62 42 XII 24269 2 0 .15 0 .017 18 .93 3.85 55 65 44 45 43 3 31 . 23 9 .05 1756 3500 58 64 50 TABLE - I I I (Cont) C.No. Heat B.No. Com p o s i t i o n W.F.R W. V S.H Av.H.F M.H.F Columnar Zone Length (mm) No (1/S) m/s C c ) (KW/M!) C% P% S.51 S.S2 Outs 1de Radius A c t u a l Av. L o c a l Av. 1 32 . 80 9 . 70 48 50 40 XII 24257 2 0. 16 0.02 32.80 9 . 70 50 50 46 45 3 32.80 9 . 70 54 49 48 1 22.71 5.55 28 1658 3500 50 52 48 XII 24259 2 0. 16 0.021 22.71 5 . 55 49 52 41 46 45 3 31 .54 9.15 1997 3500 58 62 49 1 20.82 4 .60 64 59 59 48 XII 24271 2 0.2 0.016 32.80 9 . 70 1996 3500 51 51 39 44 3 17.98 3.45 1739 4100 60 59 46 1 32. 17 9.40 63 4600 XII 24244 2 0. 26 0.017 32 . 17 9.40 1971 3 32 . 17 1 31 .85 9 . 30 27 4000 46 43 35 XII 24248 2 0. 28 0.022 31 .85 9 . 30 1896 32 3 31 .85 9 . 30 31 34 29 1 31 .54 9.15 37 4200 40 40 35 XII 24249 2 0. 28 0.018 31.54 . 9. 15 1913 38 43 38 36 3 31 .54 9.15 40 36 34 1 32.49 9 . 55 36 4000 42 45 33 XII 24252 2 0. 28 0.035 32.49 9.55 1776 42 42 40 24 3 32.49 9.55 -- -- --1 32.80 9.70 54 3850 46 45 40 XII 24242 2 0.29 0.017 32.80 9.70 1815 45 48 40 40 3 32.80 9.70 45 45 40 1 31 .85 9.30 43 3600 46 44 33 XII 24251 2 0.29 0.028 31 .85 9.30 1844 36 38 36 36 3 31 .85 9 . 30 40 40 39 1 26.50 7.20 23 2012 3800 42 43 32 XII 24277 2 0.29 0.023 26.50 7.20 35 35 29 31 3 32 .80 9.70 6 1854 3800 36 37 32 (_ri TABLE - III (Cont) C . No. Heat B .No. Compos i t i on W.F.R W. V S.H Av.H.F M.H.F Columnar Zone Length (mm) No (1/s) m/s C O (KW/M') c°/= P% S.S1 S.S2 Outside Radius Actual Av. Local Av . 1 32 .80 9 . 70 48 4000 45 46 44 XII 24245 2 0.3 0.02 32.80 9 . 70 1926 44 3 32 .80 9 . 70 55 40 45 1 32.80 9.70 52 2056 4400 43 43 37 XII 24276 2 0.3 0.023 25.86 6 .95 1925 35 37 31 34 3 25.86 6.95 4200 35 33 35 1 32.80 9 . 70 26 1973 4400 35 28 33 XII 24266 2 0.31 0.016 20. 19 4 . 30 37 37 38 34 3 20. 19 4 . 30 32 33 30 1 32.49 9.55 55 48 52 40 XII 24243 2 0.32 0.014 32.49 9.55 1976 50 48 34 36 3 32.49 9.55 44 40 35 1 32.80 9.70 40 2085 3700 45 50 35 XII 24272 2 0. 33 0.028 32.80 9.70 45 4 1 39 39 3 16 . 72 3 . 10 26 1520 3900 40 47 44 1 34 .07 10. 3 43 4300 38 41 33 XII 24278 2 0. 33 0.026 34 .07 10. 3 38 39 31 31 3 26.50 7.20 9.8 1875 4200 36 36 30 1 32.49 9.55 58 4000 40 39 31 XII 24246 2 0. 34 0.016 32.49 9.55 1923 34 3 32.49 9.55 38 40 37 1 22.71 5.55 29 1956 4700 41 40 33 XII 24263 2 0. 34 0.016 22 . 71 5 . 55 43 44 31 30 32 3 31 .54 9. 15 2163 4200 33 30 25 1 32.80 9.70 27.8 2086 3700 45 48 39 XII 24265 2 0.35 0.022 21 .76 4 .95 2221 5000 47 47 39 38 3 32.80 9.70 27.8 47 49 37 1 31 .54 9. 15 41 4000 35 35 27 XII 24250 2 0. 36 0.035 31 .54 9. 15 1963 38 38 36 32 3 31 .54 9. 15 33 36 33 TABLE - III (Cont ) C .No . Heat No B.No. Compos i t i on W.F.R (1/S) W.V m/s S.H C C ) Av .H . F M.H.F (KW/M !) Co lumnar Zone Leng th (mm) C% P% S.S1 S . S2 Ou t s I de R a d i u s A c t u a l Av. Loca l Av. XII 24254 1 2 3 0 . 36 0 .026 32 .80 32 .80 32 .80 9 .70 9 .70 9 .70 32 2047 4600 52 59 12 62 57 12 43 55 12 37 XII 24256 1 2 3 0 .36 0.061 33 . 12 33 . 12 33. 12 9.75 9.75 9.75 28 34 34 27 14 XII 24264 1 2 3 0 .36 0 .017 31 .54 31 . 54 22.71 9. 15 9. 15 5.55 29 1972 1974 4700 4800 50 50 38 62 54 35 43 46 30 39 44 XII 24247 1 2 3 0 .38 0.012 32 .80 32 .80 32 .80 9 .70 9 .70 9 .70 26 1923 4000 59 54 56 55 50 50 50 XII 24274 1 2 3 0 .38 0 .015 30.28 30. 28 30. 28 8 .65 8 .65 8.65 45 2129 3200 57 52 57 56 53 55 54 49 46 50 XI 24221 1 2 3 0 .38 0.031 31 .54 31 .54 31 . 54 9. 15 9. 15 39 58 61 59 59 II I 22854 1 2 3 0 . 39 0 .035 31 .54 31 .54 31 .54 9. 15 9. 15 9. 15 ? 62 55 42 66 53 40 59 54 32 48 III 22855 1 2 3 0 . 39 0 .013 31 .54 31 .54 31 .54 9.15 9. 15 9.15 ? 60 61 60 63 65 64 56 54 55 55 IV 23372 1 2 3 0 .39 0 .015 31 .54 31 .54 31 . 54 9.15 9. 15 9. 15 ? 62 62 58 62 57 56 54 58 42 51 XII 24273 1 2 3 0.41 0 .017 23.97 23.97 23 .97 6. 15 6. 15 6.15 26 1585 3900 53 53 58 56 53 58 50 51 51 51 TABLE - III (Cont) C.No. Heat No B.No. Composition W.F.R (1/s) W. V m/s S.H C c ) Av.H.F M.H.F (KW/M') Columnar Zone Length (mm) c% P% S.S1 S.S2 Outside Radius Actual Av. Local Av. VII 23528 1 2 3 0.41 0.01 31 . 54 31 . 54 31 .54 9. 15 9. 15 9.15 52* 60 55 55 59 53 60 55 55 48 53 VI 23490 1 2 3 0.43 0.014 31 .54 31 .54 31 . 54 9. 15 9.15 9. 15 57* 60 58 59 64 62 61 48 55 52 52 IX 24100 1 2 3 0.45 0.025 31 . 54 31 . 54 31 .54 9. 15 9. 15 9. 15 28 58 60 60 61 58 50 46 45 58 50 V 23472 1 2 3 0.48 0.012 31 . 54 31 .54 31 . 54 9. 15 9. 15 9. 15 19 + 48 55 47 59 41 48 45 V 23471 1 2 • 3 0.5 0.013 31 .54 31 .54 31 .54 9.15 9. 15 9. 15 59* 61 58 59 61 61 57 58 50 49 52 V 23474 1 2 3 0.5 0.016 31 . 54 31 . 54 31 .54 9. 15 9. 15 9. 15 61 + 60 60 62 60 56 55 56 Note: C.No.- Campaign Number, B.No.- B i l l e t Number, W.F.R.- Water Flow rate, S.H.- Super Heat Av.H.F.- Average Heat Flux, M.H.F.- Meniscus Heat Flux S.S1.- Straight Side -1 S.S2 . - Straight Side -2 Av.- Average Table - IV Sub-structure D e t a i l s Heat No. B.No. Carbon wt% W.F.R (1/s) W.V m/s S.H C O Whether Hooks are present Columnar zone length (mm) Depth of o s d 1 l a t l o n marks (mm) S.S1 S.S2 r a d i i Outside JS 24261 2 0.13 32.80 9.70 22 No 63 56 54 -24270 3 0. 13 24.92 6.55 «34 No 61 61 56 -24266 3 0. 15 31 .23 9.05 «46 No 58 64 50 -24277 24277 1 3 0.29 0.29 26.50 32.80 7.20 9.70 23 7 Yes Minor Hooks 42 36 43 37 32 32 0.206 0. 176 24276 24276 24276 • 1 2 3 0.30 0.30 0.30 32.80 25.86 25.86 9.70 6.95 6.95 51 7-51 7 No Yes Long hooks 43 35 35 43 37 33 37 31 35 0. 174 0. 194 0.210 24266 24266 1 3 0.31 0.31 32 .80 20. 19 9.70 4.30 26 «26 No Minor hooks 35 32 28 33 33 30 0. 176 0. 192 24272 24272 2 3 0.33 0.33 32.80 16.72 9.70 3. 10 <40 26 Yes | 45 Minor hooks. 40 41 47 39 44 0.219 0. 178 Table - V Location of S o l i d i f i c a t i o n bands Heat B.No cy. Dark band at mid face White band at mid White band at of f-corner Remarks No. face O.C S.S1 I . c S.S2 O.C S.S1 I .C S.S2 A-1 A-4 B-1 B-2 C-2 C-3 D-3 D-4 1 _ _ - - 3-4* - - - 3 3 - - - - - - •wavy 2426i 2 0. 13 - - - - - - - - - - - - - - - -3 - - - - - - - - - - - - - - 3-4 3-4 1 _ _ _ _ _ 2 _ 2 3 _ - - -24270 2 0. 13 - - - - 2* - - - 2 - 2 3 - - - - • w h i t e . t h i n and wavy 3 - - - - - - - - 4 4 2 4 - - - -1 _ _ _ _ 5-6 4 _ - _ - - -24253 2 0. 14 - • - - - + - - - - - - - - - - - •white band Is t w i s t e d 3 10 10 10 10 1 8 9 8 10 l o t of 24255 2 0. 14 7 8 8 7 - - - - - - 4 4 - - - - o f f corner 3 7 7 - 7 - - - - 4 4 - - - — - - c r a c k i n g 1 - - - - 5-6* 6 - 7 5.5 6 3.5 3.5 - - - - •wavy,corner crack obtuse angli 24258 2 0. 14 - - - - 3-5* - - - 4 4 - - - - - - •wavy 3 - - - 9 3-4 - 7 6 4 3 3 4 - - - - no thinness l e s s rhombldlty 1 6 3-4 5 5 6 5 24269 2 0. 15 - - - - 3-4 4-5 - - - - - - - - - -3 - - - - - - - - - - 4 3* - - - - • t h i n corner 1 6-7 6-7 6-7 6-7 4! _ 5 3! 4-5 3 7 _ _ 24257 2 0. 16 8 8 10 3! - 5 5 - - 5 6 - - - -3 10 10 10 10 6-7 - - - - - 5 5 - • - - -1 7 7-7.5 - _ _ 4-5 _ 5 _ _ _ _ _ _ _ 24259 2 0. 16 8 9 8 8 - 5 - 5 - - - _ - _ _ 3 7 7 - 7-8 5-6 5-6 - 6-7 - - - - • - - - -1 3 3 24271 2 0.20 - - - - - - - - - - - 4 _ - _ _ 3 - - - - 6 5 5 6 4 4 6 5 - - - -1 1 1 11 11 11 7 5-6 5-6 5-6 _ _ _ _ _ _ _ _ dark etc h 24244 2 0.26 - - - - - - - - - - - - - • - - - dark etc h 3 — — - - - - - - - - - •- - - dark etch 1 11 10 12 _ 6 _ _ _ _ • 24248 2 0. 28 10 10 10 - 6-7 8 5 5-6 5-6 5 5-6 4-5 5 4 3-4 _ 3 10 1 1 12 10 - - - - - - - - - - -Table - V (cont) Heat B.No C% Dark band at mid face White band at mid White band at of f - c o r n e r Remarks No. face O.C S.S1 I .C S.S2 0 .C S.S1 I .C S.S2 A-1 A-4 B-1 B-2 C-2 G-3 D-3 D-4 1 11 12 _ _ _ _ _ 3-4 3-4 - - - - - -24249 2 0. 28 11 9 8-9 - 8 7 - - - - 4* 4* - - 4* 4** re-entrant type 3 9 11 12 - - - - - - - 3-4 3-4 - - - -1 13 13 13 13 9 7 6-7 6 5 5 5-6 5-6 5-6 3* - - •Re-entrant type 24252 2 0 28 - - - - - 6 - - - - - - - - - -3 - - - - - - - - - - - - - - - - equtaxed 1 _ _ 6 -7 7 7-8 5-6 4 4-5 4-5 5-6 7 3 5-6 5-6 24242 2 0 29 10 11 10.5 10.5 8 8 - - 6 6 5 5 - - 4 4 corner crack at obtuse angle 3 11 10-11 - - 7 5 6-7 - 5-6 3-4 5-6 - - - - - corner crack at obtuse angle 1 10 10 9 10 6 8 8 6 5 3 6 7 5 4 4 5 24251 2 8 9 8 9 1 * 5-6 3-4 6-7 6-7 6-7 4 2 4 6 3 3 *very short white band 3 10 11 9 1 1 - - - - 3* 3* 7 7 4* 4* 5 5 *< obtuse angle 1 - | 1 _ 11 7 3-4 7 7 6 5 5-6 6-7 _ _ 7 8 24277 2 0 29 11 10 9 11 7 6-7 7 7 4-5* 4-5* 8* 8-9* 4 3 - -3 10 1 1 - 10 6 -7 - 8 6 7-8 7 - - - - 6 5-6 1 11 11 11 1 1 6 7 7 6 3-4 5-7 5-6 7 5-6 7 4 4 24245 2 0 30 11 10 10 12 5 -6 5-6 - - 3.5* 3.5* 4-5 - 4* 3-4* 5 - *obtuse angle 3 8 9-10 10-11 10- 1 1 6 4 5 5 3* 4* 4 5-6 5-6 - - - *obtuse angle 1 9 9 - - 5 -6 6 _ 6 4 6 4 _ _ 4 4 2427G 2 0 30 - - 1 1 1 1 7 - - - 7* 8* 5 - - - 5 4 •t h i c k 3 10.5 1 1 11.5 11 7 7 - 8 - 3* 5-6 5 4-5 5 4 3-4* *off-corner cracks 1 - - - 11 7 4 7 7-8 _ _ _ _ 4.5 5 5-6 5-6 *on 8C-wh1te band i s uniform 24266 2 0 31 - 9-10 - - 8 7 8 - 5 5 6-7 7-5 4 4 - - from corner to mid face on CD 3 - 9 - 10 7+ 5.5+ 7 - 3.5 3.5 3.5 5 - - 5 5 midface 1s t h i c k and drops o f f 1 1 1 11 1 1 11 _ _ _ _ _ 24243 2 0 32 8 9 1 1 11 - - - - - - - - - - - -3 11 10 10.5 11 7 6 8 - 3-4* 4* 5 6 2* 7* - - •short diagonal 1 10 - 8.5 - _ - _ _ 6 8 _ 2-4 5 5-6 _ _ 24272 2 0 33 - 8 8 7 - - - 7 - - 4 3 - _ -3 - - 9 9 - - 9 9 6 6 3-4 4 - ' - -1 - - - - 9 5-6 10 7 5-6 6 8 6 _ _ 3 4 24278 2 0 33 11.5 1 1 11 9.5 7 -8 - - 2-3* - - - - - - 3 7 *very t w i s t e d 3 11 10 — 8 7-8 3-5* - - - - - 3 - •very t w i s t e d CJ1 Table - V (Cont) Heat B.No C% Dark band at mid face White band at mid White band at of f-corner Remarks No. face O.C S.S1 I .C S. S2 O.C 5.S1 I. C 5.S2 A-1 A-4 B-1 B-2 C-2 C-3 D-3 D-4 1 12 11 12 - - - 5 5 2 2 - - 2 2 2 2 24246 2 0.34 - - - - - - - - - - - - - - - -3 10 10 11 11 - - - - 3-4 2 5-6 6-7 5-6 6-7 5 5-6 1 14 11 12.5 6-7 9 10 8-9 5 5-6 7 6 3-4 5 3 6 24263 2 0.34 1 1 - • 12 13.5 - - 9 - - 4-5 - - 5 5 3 3 3 6-8 + 7 8 8 - - 9 - 4 5 7 8 - - - - •wavy 1 11 10 8 11 5 5 - 6 3 3 4 3 4 3 3 4 24265 2 0.35 11 12 1 1 10 7 7 7 6 6 6 4-5 5 5 5-6 4 9 3 11 12 12 13 8 7 8 7 6 5-6 4-5 4-5 8 8-9 5 5 1 11 11 11 12 5-6 - - 8 5-6 - 4 4-5 - - - -24250 2 0.36 10 10 10 10 6-7 - - - - - - - - - - -3 9 10 10 11 6 6-7 - - - - - - - - 3 3 1 10 11 11 12 6-7 7-8 7-8 9 - - • 4* 4' 6-7 6-7 5* 5 • •short diagonal 24254 2 0.36 10 10 10 10 6-7 6-7 6-7 7 - - - - - - - -3 10-11 10-11 10-11 10-11 - - - - - - 5* 5* - - 5* 5» • t h i n c h i l l e d s t r u c t u r e 1 - - - - - - - - 3-4 3-4 - - - - - -24256 2 0.36 - - - - - - - - - - - - - - - -3 - - - - - - - - - - - - - - - equiaxed 1 12 10 _ 14 6 7 8 8 5 7 5.5 _ _ 6 5-6 24264 2 0.36 11 12.5 11 1 1 6-7 - - - - - - - 3 3-4 - -3 11 11 11 10 5 - - 8 6 6 6 6 • - - - -1 12 13 10 - 6-7 6-7 6-7 6-7 4-5 4-5 2-3 2-3 6-7 6-7 4-3 4-3 24247 2 3 0.38 10 11 13 10 7-8 7-8 7-8 7-8 "- : : - — : -1 10 11 11 10 8 8 8 7-8 - _ _ _ _ 3 3-4 24274 2 0.38 10 10 11 10 7 6 8 8 - - - - - 5 4 3 10 10 10 11 1 10 13 57 11 6 6-7 6 _ 7 5 5 6 _ _ 24273 2 0.41 10 12 11 11 - 6-7 - - - - 4 4 5 4 6 4 3 10 10 11 10 - - - 7 5 4 6-7 7 6 5 4 5 Table - VI Location of off-corner cracks and t h e i r number i n each b i l l e t sample Heat No. S.H Cc) C% (Wt%) B 1 1 l e t n 1 B i l l e t ¥ 2 811 l e t # 3 | W.F.R (1/s) 01St. from corner Total no. of cracks w.F.R (1/s) D i s t . from corner Total no. of cracks W.F.R (1/s) D i s t . from corner Total no. of cracks 24258 22 0. 14 22.71 5.0 6 32.49 8.0 1 32.49 6.0 2 24259 28 0. 16 22.71 4.0 3 22.71 5.0 4 • 31 .54 3.0 8 24260 22 0. 15 21 .45 - 0 21 .45 - 0 32.80 - 0 24261 25 0. 13 32.80 - 0 32.80 - 0 350 - 0 24262 24 0. 13 21 .76 - 0 21 .76 • - 0 32.80 - 0 24263 29 0.34 22.71 - 0 22.71 4.0 1 31 .54 - 0 24264 29 0.36 31 .85 - 0 21 .76 4.0 2 21 .76 4.0 1 24265 28 0.35 32.80 - 0 21 .45 - 0 32 .80 - 0 24266 26 0.31 32.80 4.0 3 20. 19 6.0 3 32.80 4.0 6 24268 19 0. 14 32.80 6.0 6 27. 13 4.0 7 27. 13 4.0 12 24269 46 0. 15 18.93 3.0 6 18.93 4.0 5 31 .54 4.0 1 24270 34 . 0. 13 31 .54 - 0 31 .54 - 0 24.92 3.0 5 24271 63 0.20 20.82 - 0 32.80 4.0 3 17.98 4.0 8 24272 39 0.33 32 .80 - 0 32 .80 - 0 16.72 4.0 1 24273 26 0.41 23.97 - 0 23.97 - 0 23.97 - 0 24274 45 0.38 30. 28 - 0 30.28 - 0 30.28 - 0 24275 19 0. 14 25.54 5.0 1 25.54 6.0 1 31 .54 5.0 1 24276 51 0.30 32.80 4.0 5 25.86 5.0 13 25 .86 5.0 8 24277 23 0.29 26.50 4.0 6 26.50 5.0 10 32.80 5.0 2 24278 43 0.33 34.07 5.0 3 34.07 4.0 2 32.80 4.0 5 Table - VII Sev e r i t y of Off-corner cracks B i l l e t Composition (wt%) W.F . R W. V Total No. Of Se v e r i t y No. C P S Mn Mn/S (1/S) o f f - c o r n e r of cracks c r a c k i n g -1 22. 71 5 55 6 4 24258-2 0. 14 0.028 0.035 0.63 18 32. 49 9 55 1 2 -3 32. 49 9 55 2 2 -1 22. 71 5 55 3 4 24259-2 0. 16 0.021 0.033 0.54 16 22. 71 5 55 4 4 -3 31 . 54 9 15 8 5 -1 21 45 4 80 0 1 24260-2 0. 15 0.01 1 0.033 0.44 19 21 45 4 80 0 1 -3 32 80 9 70 0 1 -1 32 80 9 70 0 1 24261-2 0. 13 0.009 0.031 0.40 15 32 80 9 70 0 1 -3 22 08 5 10 0 1 -1 21 76 4 95 0 1 24262-2 0. 13 0.008 0.031 0.39 18 21 76 4 95 -3 32 80 9 70 --1 22 71 5 55 0 1 24263-2 0. 34 0.016 0.035 0.51 30 22 71 5 55 1 2 -3 31 54 9 15 1 2 -1 31 85 9 30 1 2 24264-2 0. 36 0.017 0.028 0.45 46 21 76 4 95 2 2 -3 21 76 4 95 1 2 -1 32 80 9 70 0 1 24265-2 0. 35 0.022 0.036 0.58 31 21 45 4 80 0 1 -3 32 80 9 70 0 . 1 -1 32 80 9 70 3 2 24266-2 0. 31 0.016 0.037 0.53 24 20 19 4 30 3 2 -3 32 80 9 70 6 3 -1 32 80 9 70 6 3 24268-2 0. 14 0.020 0.039 0.59 14 27 13 7 45 7 5 -3 27 13 7 45 12 6 Table VII (Cont) B11let Composition (wt%) W.F.R W.V Total No. Of S e v e r i t y No. C P S Mn Mn/S (1/s) o f f - c o r n e r of cracks c r a c k i n g -1 18.93 3.85 6 3 24269-2 0. 15 0.017 0.031 0.48 20 18.93 3.85 5 4 -3 31 .54 9. 15 1 2 -1 31 .54 9. 15 0 1 24270-2 0. 13 0.012 0.027 0.39 20 31 .54 9. 15 .0 1 -3 25.20 6.70 5 5 -1 20.82 4.60 0 1 24271-2 0. 20 0.016 0.030 0.46 24 32.80 9.70 3 2 -3 17.98 3.45 8 5 -1 32.80 9.70 0 1 24272-2 0. 33 0.028 0.033 0.61 28 32.80 9.70 0 1 -3 16.72 3. 10 1 2 -1 23.97 6. 15 0 1 24273-2 0. 41 0.017 0.029 0.46 36 23.97 6. 15 0 1 -3 23.97 6. 15 0 1 -1 30.28 8.65 0 1 24274-2 0. 38 0.015 0.027 0.42 51 30.28 8.65 0 1 -3 30.28 8.65 0 1 -1 25.54 6.85 1 2 24275-2 0. 14 0.011 0.024 0.35 29 25.54 6.85 1 2 -3 31 .54 9. 15 1 2 -1 32.80 9.70 5 3 24276-2 0. 30 0.023 0.044 0.67 22 25.86 6.95 13 4 -3 25.86 6.95 8 5 -1 26.50 7.20 6 5 24277-2 0. 29 0.023 0.047 0.70 22 26.50 7.20 10 5 -3 32.80 9.70 2 2 -1 34.07 10.3 3 2 24278-2 0. 33 0.026 0.043 0.69 29 34.07 10.3 2 2 -3 26.50 7.20 1 • 2 170 Table VII I - Rhomboidity data Heat No. B .No. C% W.F.R (1/s ) W. V m/s D i agona1 D1 (cm) D(agona 1 D2 (cm) Rhomb 1di ty D=D1-D2 (mm) 24090 1 2 3 0 .13 32 .49 9.55 19.0 18.95 18.9 19. 15 19 18.95 1 . 5 -0 .5 -0 .5 24 100 1 2 3 0 .45 3 2 . 8 0 9 . 70 19 19 18.9 19 19 19 . 0 . 0 0 . 0 - 1 . 0 241 1 1 1 2 3 0. 33 31 .54 9. 15 18.7 19.2 19.1 19.2 18.65 18.85 - 5 . 0 5.5 4 .5 241 12 1 2 3 0 .26 31 .54 9.15 18.7 19.15 19 . 1 19.15 18.8 18.9 -4 . 5 3.5 2 .0 241 13 1 2 3 0. 17 32 .80 9 .70 19.35 19.2 19 . 25 18.4 18.6 18 . 5 7 . 5 S O 7 . 5 241 14 1 2 3 0.21 32 . 17 9. 40 18 .95 18.95 19 19 .05 19 19 - 1 . 0 -0.5 0 . 0 241 15 1 0. 34 32 .80 9 .70 19. 1 18.8 3 .0 24 1 16 1 2 3 O. 33 32 .49 9 . 55 18.55 18.6 19.25 19 . 35 19 . 3 18 .65 - 8 . 0 - 7 . 0 6 . 0 242 17 1 2 0 .2 34 .07 10.3 18.9 18.75 19 .00 19.3 - 1 . 0 -5 . 5 24218 1 2 O. 18 34 .07 10.3 19.2 19.5 18.85 18.6 3 . 5 9 .0 24219 1 2 0 . 20 34 .07 10. 3 19.2 18.9 18 . 8 19 4 . 0 - 1 . 0 24220 1 2 0.21 33 . 12 9.75 19 . 1 19. 15 18.9 18.85 2 .0 3 .0 24221 1 2 0. 38 3 2 . 8 0 9 .70 19.2 19 . 1 18.95 19 2 . 5 1 .0 24242 1 2 3 0. 29 32 .80 9 .70 18.85 19. 1 19.1 19 18 . 85 18 . 85 -1 .5 2.5 2 . 5 24243 1 2 3 0. 32 32 .49 9 . 55 19 19 . 25 19 . 25 18.9 18.7 18 .65 1 5 . 5 6 . 0 24244 1 3 0 . 26 32 . 17 9 .40 18.4 18.7 19.4 19.2 - 1 0 . 0 - 5 . 0 24245 1 2 3 0. 30 32 . 80 9 .70 18.95 19.2 18.8 18.9 1 . 5 3 .0 24246 1 2 3 0. 34 32 .49 9 .55 19.0 18.8 18 . 85 19.05 1 .5 -2 .5 24247 1 2 3 0 . 38 32 .80 9 .70 19 . 3 19.3 18.6 18.2 7.0 11.0 17 I i Heat B .No. W . F . R W V D iagona1 D i agona1 Rhomb i d i ty No. ( 1 ' S ) m/ D 1 cm) D2 (cm) D = D1 -D2 (mm) 1 19 . 2 18 . 75 5 5 24248 2 0. 28 3 1 . 85 a 30 18 . 75 19 . 25 -5 .0 3 18 . 75 19 . 25 -5 .0 1 18 . 35 19 . 55 - 12 0 24249 2 0. 28 31 . 54 9 15 19 .5 18 .55 9 5 3 19 35 18 . 5 7 5 1 19 . 2 18 .75 4 .5 24250 2 0. 36 31 . 54 9 15 19 .3 18 .85 4 5 3 19 . 25 18 . 2 10 5 1 19 25 18 . 7 5 5 24251 2 0. 29 31 .85 9 30 19 . 15 18 .7 3 5 3 18 . 6 19 . 3 -6 0 1 18 6 19 . 25 -6 5 24252 2 0. 28 32 .49 9 55 18 .5 19 . 3 -8 0 3 19 1 18 .9 2 0 1 19 .0 19 .0 0 0 24253 2 0 .14 32 .80 9 70 19 3 18 .55 7 5 3 19 35 18 .65 7 0 1 19 56 18 .65 8 0 24254 2 0 . 36 32 . 80 9 . 70 19 56 18 . 55 9 0 3 18 65 19 35 -7 0 1 19 2 18 8 4 0 24255 2 0. 14 32 .80 9 70 19 35 18 65 7 0 3 18 55 19 4 -8 5 1 18 .65 19 . 25 -5 5 24256 2 0. 36 33 . 12 9 75 - - -3 19 3 18 65 6 5 1 18 7 19 35 -6 5 24257 2 0 .16 32 .80 9. 70 18 9 19 05 - 1 5 3 18 85 •19 1 -2 5 1 22 . 7 1 5 55 19 3 18 8 5 0 24258 2 0.14 32 . 49 9 55 18 .7 19 3 6 0 3 32 . 49 9. 55 19 1 18 9 2 0 1 22 . 7 1 5 55 19 1 19 1 0 24259 2 0 . 1 6 22 . 7 1 5 55 18 .8 19 2 -4 0 3 3 1 . 54 g 15 19 4 18 7 7 0 1 2 1 . 4 5 80 18 7 19 25 -6 0 24260 2 0 . 1 5 2 1 . 4 5 4 . 80 19 18 8 2 0 3 32 . 80 9 70 19 18 95 -0 05 1 32 . 80 9 . 70 19 3 18 6 -7 0 24261 2 0.13 32 .80 9 70 19 25 18 7 5 5 3 22 .08 5 10 18 7 19 25 -5 5 1 21 . 76 4 95 19 25 18 7 5 5 24262 2 0.13 21 . 76 4 95 18 9 19 -1 0 3 32 .80 9 . 70 19 1 18 9 2 0 1 22 .71 5 55 19 1 18 95 1 5 24263 2 0.34 22 . 7 1 5 55 18 95 19 20 -2 5 3 3 1 . 54 9 15 19 05 19 95 1 0 1 31 . 85 9 . 30 19 0 19 0 0 24264 2 0. 36 21 . 76 4 . 95 19 1 19 1 0 3 21 . 76 4 95 19 1 18 9 2 0 1 32 .80 9 70 19 1 18 8 3 0 24265 2 0. 35 21 .45 4 80 18 9 19 1 -2 0 3 32 .80 9 70 18 9 19 05 - 1 5 1 72 Heat B . No . C% W. F . R W V D i agona1 D i agona1 Rhomb d i ty No. (1/s ) m/s D1 (cm) 02 cm) D = D 1 -D2 (mm ) 1 32 80 9 70 19 2 18 8 4 0 24266 2 0 31 20 19 4 30 18 8 19 0 -4 0 3 32 . 80 9 70 18 95 19 0 -0 5 1 32 80 9 70 18 75 19 2 -5 5 24268 2 0 14 27 13 7 45 19 35 18 7 6 5 3 27 13 7 45 19 05 18 8 2 5 1 18 93 3 85 18 9 19 0 - 1 0 24269 2 0 15 18 93 3 85 19 0 19 1 - 1 0 3 31 54 9 15 19 3 18 7 6 0 1 31 54 9 15 18 75 19 . 2 -4 5 24270 2 0 13 31 54 9 15 18 7 19 25 -5 5 3 24 92 6 55 18 8 19 2 -4 0 1 20 82 4 60 18 6 19 3 -7 0 24271 2 0 2 32 80 9 70 18 9 19 . 1 -2 0 3 17 98 3 45 18 8 19 1 -3 0 1 32 80 9 70 19 15 19 1 5 24272 2 0 33 32 80 9 70 19 .2 18 .9 3 0 3 16 72 3 10 19 .0 19 .0 0 0 1 23 97 6 15 19 1 18 .9 2 0 24273 2 0 4 1 23 97 6 15 18 8 19 . 3 -5 0 3 23 97 6 15 18 .9 19 . 15 -2 5 1 30 28 8 65 18 .8 19 . 15 -3 5 24274 2 0 38 30 28 8 65 19 2 18 .81 4 0 3 30 28 8 65 19 0 19 0 0 0 1 25 54 6 85 19 . 1 18 .9 2 0 24275 2 0 14 25 54 6 85 19 . 3 18 . 7 6 0 3 31 54 9 15 18 7 19 . 2 -5 0 1 32 80 9 70 18 .9 19 . 1 -2 0 24276 2 o 30 25 86 6 95 18 .9 19 .2 -3 0 3 25 86 6 95 18 .8 19 . 1 -3 0 1 26 50 7 20 19 25 18 . 7 5 5 24277 2 0 29 26 50 7 20 19 1 18 .9 2 0 3 32 80 9 70 18 .8 19 . 2 -4 0 1 34 07 10. 3 18 .9 19 .0 - 1 0 24278 2 0 33 34 07 10. 3 18 7 19 2 -5 0 3 32 80 9 70 18 7 19 25 -5 5 173 Table - IX VISUAL ESTIMATION OF OSCILLATION MARKS 1-No marks 2-Very M i l d 3-Mild 4-Deep 5-Very Deep Heat Carbon B i l l e t B i l l e t B i l l e t No. Wt% No. 1 No. 2 No. 3 24261 0.13 4 4 4 24270 0.13 4 3 5 24255 0.14 3 4 4 24275 0.14 5 5 5 24260 0.15 5 4 4 24269 0.15 5 4 5 24257 0.16 4 4 4 24259 0.16 4 4 4 241 13 0.17 4 5 3 24218 0.18 4 3 -24271 0.20 - 3 3 24244 0.26 2 2 2 24248 0.28 3 3 3 24242 0.29 2 2 2 24277 0.29 3 3 3 24276 0.30 2 3 3 24266 0.31 2 3 3 24246 0.34 2 3 2 24265 0.35 2 3 3 24274 0.38 2 3 3 24247 0.38 3 3 3 Table - X Depth of O s c i l l a t i o n Marks Measured By Prof 11ometer Heat No. B1 n e t No. Compos 11ion Water f 1 ow rate (1/s) W.V m/s Super Heat CC) Menlscus Heat Flux KW/M« Average Heat Flux KW/M* Depth of O s c i l l a t i o n Marks (mm) C% P% Mid-face Off Corner Average Depth Local Average 24273 1 2 3 0.41 0.017 23.97 6. 15 26 3900 1585 0.181 0.203 0.277 0. 165 0.309 0.288 0.220 X X X 24274 1 2 3 0.38 0.015 30.28 8.65 26 3200 2129 0. 161 0. 194 0. 149 0. 190 0. 178 24264 1 2 3 0.36 0.017 31 .54 22.71 22.71 9. 15 5.55 5.55 29 4700 4800 1972 1974 0. 171 0.201 0.226 0. 150 0.251 0.213 0. 199 0. 171 0.214 24265 1 2 3 0.35 0.022 32.80 21 .45 32.80 9.70 4.80 9.70 28 28 3700 5000 2085 2220 0.206 0.226 0.226 0.222 0. 183 0.235 0.219 24263 1 2 3 0.34 0.016 22 . 71 22.71 31 .54 5. 55 5.55 9. 15 29 4700B 4200 1956 2163 0.23-8 0.226 0. 17.4 0.277 0.235 0. 188 0.213 0.232 0. 174 24246 1 2 3 0.34 0.016 32.49 9.55 58 4000 1923 0. 185 0.25 0.226 0. 178 0.218 0. 185 0.25 24272 1 2 3 0.33 0.028 32.80 32.80 16.72 9.70 9.70 3. 10 40 26 3700 3900 2085 1520 0.245 0.219 0. 178 0. 165 0.210 0. 169 0.214 0.232 0. 178 Table - X (Cont) Heat No. Bl1 l e t No. Compos 11ion Water flow rate (1/s) W.V m/s S.H CO Meniscus Heat Flux KW/M' Average Heat Flux KW/M* Depth of O s c i l l a t i o n Marks (mm) C% py. Mid-face Off Corner Average Depth Local Average 24278 1 2 3 0.33 0.026 34 .07 34 .07 26.50 10.3 10.3 7.20 43 10 4300 4200 1875 0. 199 0.233 0.265 0.230 0.295 0.318 0.232 0.216 0.265 242G6 1 2 3 0.31 0.016 32.80 20. 19 32.80 9.70 4.30 9.70 26 4400 1973 0. 176 0.229 0. 192 0. 187 0.327 0.233 0.20O 24245 1 2 3 0.3 0.02 32.80 9.70 48 4000 1926 0.24! 0. 149 0. 156 i o.: 0. 19 0. 165 06 0.185 24276 1 2 3 0.30 0.023 32.80 25.86 25.86 9.70 6.95 6.95 52 7 4400 4200 2056 1925 O. 174 0. 194 0.210 0.274 0.297 0.306 0.193 0. 174 0.202 24251 1 2 3 0.29 0.028 31 .85 9.30 43 3600 1844 0.23: 0.242 0.252 I o.: 0.224 0.277 182 0.242 24277 1 2 3 0.29 0.023 26.50 26.50 32.80 7.20 7.20 9.70 23 6 3800 3800 2012 1854 0.206 0.220 0. 176 0.233 0.268 0.213 24271 1 2 3 0.2 0.016 20.82 32.80 17.98 4; 60 9.70 3.45 64 30 4100 3500 1996 1739 0.594 0.343 0.224 0.503 0.453 0.261 0.387 I Table - X1 E x i t s h e l l thickness f o r High Carbon Steel Heat Carbon W.F . R W V E x i t shel 1 Dark band at White band at midface midface No. Wt% (1/s) m/s thickness Range Average Range Average Model (mn (mm) (mm) (mm) (mm) P r e d i c t i o n • (cm) 24277 0.29 26. 50 7 20 9 76 9-11 10.4 3-7 6.5 48 32 . 80 9 70 9 03 10-11 10. 3 6.5-8 6.8 55 2427G 0.30 32. 80 9 70 9 76 9 9 7 7 52 25. 86 6 95 9 76 10.5-11 5 11 7-8 7.3 55 24266 0.31 32. 80 9 70 9 76 11 11 4-7.5 6.4 45 20. 19 4 30 9-10 9.5 5.5-7 6.5 52 24272 0.33 32 80 9 70 9 76 7-10 8.3 7 7 53 16 72 3 10 8 31 9 9 9 9 >80 24278 0.33 34 07 10.3 9 76 9.5-11 5 10.8 2.5r7.5 5 41 26 50 7 20 9 03 10-11 10.5 3.5-7.5 6.3 51 24263 0.34 22 71 5 55 10 48 11-14 12.3 6.5-10 8.6 61 31 54 9 15 11 20 7-8 7.5 9 9 58 24265 0.35 32 80 9 70 1 1 20 8-11 10 5-6 5.3 38.5 21 45 4 80 11 20 10-12 11 6-7 6.8 42 24264 0.36 31 54 9 15 9 76 10-14 11.6 6-8 6.9 . 52 22 71 5 55 9 76 10-11 10.8 5-8 6.5, 47 24274 0.38 30 28 8 65 11 20 10-11 10.3 6-8 6.8 41 24273 0.41 23 97 6 15 8 31 5-13 10.3 6-7 6.4 60 ON Table - X11 E x i t s h e l l thickness f o r Low Carbon s t e e l s Heat No. Carbon Wt% W.F.R (1/s) W.V m/s E x i t s h e l l thickness (mm) Dark band at midface White band at midface Range (mm) Average (mm) Range • (mm) Average (mm) Model P r e d i c t ion (cm) 242G1 0.13 32.80 21 .45 9.70 4.80 9.03 7.59 - -3.5 3.5 34 24262 24270 0. 13 0.13 21 .76 32.80 31 .54 24 .92 4 .95 9 .70 9. 15 6.55 8.31 9.76 7.59 7 .59 - -2 2 31 24268 0. 14 32.80 27. 13 9.70 7.45 9.76 9.03 -- - - -24275 0. 14 25.23 31 .54 6.70 9. 15 9.03 9.76 - - ' • - - -24260 0. 15 21 .45 32.80 4.80 9.70 8.31 9.76 -- - - -24269 0.15 18.93 31 .23 3.85 9.05 8.31 9.03 6 6 3.5-5 4.5 41 24259 0. 16 22.71 31 .54 5.55 9. 15 9.76 11 .20 7-9 7-7.5 7.9 7.2 4.5-5 5.5-6.5 4.9 5.8 43 42 24271 0.20 32.80 17.98 9.70 3.45 10.48 9.03 - - 5.5 42 - • * I / O APPENDIX B - DERIVATION OF FINITE-DIFFERENCE EQUATIONS FOR ONE- DIMENSIONAL UNSTEADY STATE HEAT TRANSFER MODEL FOR BILLETS If a one dimensional section through the b i l l e t thickness i s divided into elements or nodes containing a nodal point at the centre of each heat balances on the interior nodes may be derived. Similarly heat balances for half nodes at the surface and midplane as shown. In the absence of heat generation or consumption, the lav of conservation states: Rate of heat input - Rate of heat output •= Rate of heat accumalation q i n " q o u t At 1 For the general case of an interior node, i , the f i n i t e difference approximation incorporating Fourier's Law of heat conduction leads to where T' e Unknown nodal temperatures at the end of a particular time step At T^  «= Known temperature of node i at the beginning of each time step. The thermal conductivities have been calculated as the average values between adjascent nodes, where 179 Equation 2 can be rearranged to give Heat balance f o r surface h a l f node i s T , —T B - l S Ax Void T -T s s At Rearranging, "WES1-1 +tpcpE ]^is = T-"Py In t h i s d e r i v a t i o n , the surface heat f l u x i s approximated by i t s value at the beginning of the time step. For the case of the mid plane h a l f , node the heat balance i s -k2A Rearranging = T Because each of the equations which have been d e r i v e d c o n t a i n at l e a s t two unknown temperature, none can be solved independently. However, when a l l the equations i n 'n' unknowns r e s u l t s . 1 8 0 I f we d e f i n e r lr 1 At A P J ( A x ) 2 LPCpJ (Ax) M 2 10 C « 1+A+B — — H D <= 1+2*X : 12 E «= 1+2*B 1 3 the f o l l o w i n g t r i d i a g o n a l system of equations i s o b t a i n e d . ET'-2BT' =F o 1 o -AT'+CT'-BT' =F o 1 2 • . 1 •AT' +CT' -BT i - 2 i - 1 -AT' +CT i - 1 -AT =F i - 1 -BT' =F i+1 i +CT' -BT' =F i+1 i+2 i+1 -AT' +CT' -BT' =F s - 2 s - 1 s s - 1 -2AT' +DT' =F s - 1 s s where f =T -2A _Ax s s "^1 f = T , i = 0 s-1 i i This system of equations i s e a s i l y s o l v e d by Gaussian e l i m i n a t i o n . Comparision of model-predicted temperatures w i t h an a n a l y t i c a l s o l u t i o n to Eq 4.1 showed very good agreement f o r the case of a s l a b of i n i t i a l uniform temperature w i t h 181 constant s u r f a c e boundary c o n d i t i o n and thermophysical p r o p e r t i e s . 182 APPENDIX C ~ ONE DIMENSIONAL BILLET CASTING HEAT TRANSFER MODEL IMPLICIT F I N I T E DIFFERENCE METHOD C- ONE DIMENSIONAL B I L L E T CASTING HEAT TRANSFER MODEL c _ C- IMPLICIT F I N I T E DIFFERENCE METHOD c-C C- THIS MODEL FI T S 6 POLYNOMIALS FOR THE HEATFLUX (KW/SQ.M) C- HEAT NUMBER SHOULD BE SPECIFIED. C C-HEAT FLUX , TEMPERATURES OF NODES , SHELL THICKNESS PROFILE AND COOLING RATES OF SPECIFIED NODES ARE PLOTTED BY THE MODEL c c-************************************ * INPUT DATA F I L E =5 GENERAL OUTPUT =6 * c- * OUTPUT TEMPERATURES =7 * c- * FLUX /DISTANCE OUTPUT =8 * c- * GENERAL PLOT OUTPUT =9 * c-c ************************************************** c c c-c KEY TO SYMBOLS:-L, c- CP SP E C I F I C HEAT CAPACITY (J/KG/DEG C) c- DT TIME STEP (SEC) c- DX NODE SIZE (CM) c- END SIMULATION LENGTH (M) c- K THERMAL CONDUCTIVITY (W/M/DEG C) c- MOLD - MOLD LENGTH (M) c- QDOTX - SURFACE HEAT FLUX (W/M**2) c- NNODE - NO. OF NODES c- ROW STEEL DENSITY(KG/M**3) c- SPEED - CASTING SPEED (M/MIN) c- STEEL - STEEL GRADE c- THICK - B I L L E T SIZE(M) c- TLIQ - LIQUIDUS TEMP (DEG C) c- TPOR - POURING TEMP (DEG C) c- TSOL - SOLIDUS TEMP (DEG C) c-c T ( I ) -NODE TEMP (DEG C) REAL K,MOLD,K1 ,K 2,K 3 REAL MOLY,MANGY REAL ZL( 9 0 ) REAL LIQUID INTEGER STEEL INTEGER OR1,OR2,OR3,OR4,OR5,OR6 LOGICAL LK C COMMON /CO/ TW(200),NZON,T(100) COMMON / C 1 / TNEW(100),AC(100),BC(100),CC(100),DC(100) COMMON /C2/ STEEL,NNODE,TLIQ,TSOL 183 COMMON /C3/ C ( 5 0 ) , D ( 5 0 ) , E ( 5 0 ) , F ( 5 0 ) COMMON /C4/ S(1 5 0 ) f S I G M A ( 1 5 0 ) , A A ( 1 5 0 ) , B B ( 1 5 0 ) COMMON /C5/ YF(200),YD(200),WT(200) COMMON /C6/ P 1 ( 5 0 ) , P 2 ( 5 0 ) , P 3 ( 5 0 ) , P 4 ( 5 0 ) , P 5 ( 5 0 ) , P 6 ( 5 0 ) COMMON /C7/ KU1,KU2,KU3,KU4,KU5,KU6 COMMON /C8/ G 1 ( 5 0 ) , G 2 ( 5 0 ) , G 3 ( 5 0 ) , G 4 ( 5 0 ) COMMON /CC8/ H 1 ( 5 0 ) , H 2 ( 5 0 ) , H 3 ( 5 0 ) , H 4 ( 5 0 ) COMMON /C9/ Z A ( 5 0 ) , Z B ( 5 0 ) , Z C ( 5 0 ) , Z D ( 5 0 ) , Z E ( 5 0 ) , Z F ( 5 0 ) COMMON /C10/ Z A E ( 5 0 ) , Z B E ( 5 0 ) , Z C E ( 5 0 ) , Z D E ( 5 0 ) , Z E E ( 5 0 ) , Z F E ( 5 0 COMMON /C11/ T1,T2,T3,T4,T5 COMMON /C12/ TAL L ( 3 0 0 , 1 0 0 ) , O U T ( 3 0 0 ) , S L O P E ( 3 0 0 ) 1,TPOR,DT,MOLD,END,SLEVEL,ENDCM1,ENDCM2,ENDCM3,DSTEP,TlSTEP COMMON /CI 3/ TNNODE(300),SSOLID(300),SLIQID(300),QSCALE(300 COMMON /CI 4/ DISREV(300),DISSCA(300),NSTEPS COMMON / C 1 5 / A B C ( 5 ) , B C D ( 5 ) , C D E ( 5 ) , D E F ( 5 ) , E F G ( 1 1 ) C CALL P L C T R L ( ' M E T R I C , 1 ) C C- DATA IS INPUT C RE A D ( 5 , 1 ) ( A B C ( I ) , 1 = 1 , 5 ) READ(5,1)(BCD(I),1=1,5) R E A D ( 5 , 1 ) ( C D E ( I ) , 1 = 1 , 5 ) R E A D ( 5 , 1 ) ( D E F ( I ) , 1 = 1 , 5 ) R E A D ( 5 , 1 ) ( E F G ( I ) , 1 = 1 , 5 ) C READ(5,2)STEEL,THICK,SPEED,DT,SLEVEL,NNODE READ(5,3)ROW READ(5,4)MOLD READ(5,5)CARBON,SULFUR,PHOS,GMN,SI READ(5,6)CROME,YMOLY,COPPER,SICKLE READ(5,7)SUPER READ(5,9)N1,N2,N3,N4,N5,N6 R E A D ( 5 , 1 0 ) ( C ( I ) , D ( I ) , I = 1 , N 1 ) R E A D ( 5 , 1 0 ) ( E ( I ) , F ( I ) , I = 1 , N 2 ) R E A D ( 5 , 1 0 ) ( G 1 ( I ) , H 1 ( I ) , I = 1 , N 3 ) R E A D ( 5 , 1 0 ) ( G 2 ( I ) , H 2 ( I ) , 1 = 1 , N 4 ) READ(5,10)(G 3 ( I ) , H 3 ( I ) , 1 = 1,N5) R E A D ( 5 , 1 0 ) ( G 4 ( I ) , H 4 ( I ) , 1 = 1 , N 6 ) READ(5,11)T1,T2,T3,T4,T5 C 1 FORMAT(5A4) 2 F0RMAT(I5,2X,F5.4,2X,F5.3,2X,F3.1,2X,F6.5,2X,I 3) 3 FORMAT(F6.0) 4 FORMAT(F6.4) 5 FORMAT(F6.3,1X,F6.3,1X,F6.3,1X,F6.3,1X,F6.3) 6 FORMAT(F6.3,1X,F6.3,1X,F6.3,1X,F6.3) 7 FORMAT(F4.1) 9 FORMAT(I2,1X,I2,1X,I2,1X,I2,1X,I2,1X,I2) 10 FORMAT(F5.2,F6.1) 11 FORMAT(F5.2,F5.2,F5.2,F5.2,F5.2) C C-SIMULATION DISTANCE IS'CALCULATED-C 184 TMOLD=0.8000 DIFF=MOLD-TMOLD DLEVEL=SLEVEL-DIFF C MOLD=TMOLD SLEVEL=DLEVEL END=MOLD-SLEVEL C DSTEP=(MOLD*10 0)/2 0.0 ENDCM1=22.0-((SLEVEL*100)/DSTEP) ENDCM2=22.0 ENDCM3=ENDCM2+0.5 C C- LIQUIDUS/SOLIDUS/POURING TEMPERATURES ARE CALCULATED C TLIQ=1540.0-59.2*CARBON-22.5*(CARBON**0.5) 1- 31.2*(SULFUR+PHOS)-11.5*SI-3.6*GMN 2- 3.8*SICKLE-1.8*CROME-2.3*YMOLY-4.3*COPPER C C C C TSOL=TFEC(CARBON)-(20.5*SI+6.0*GMN+500*PHOS+7 00*SULFUR) TPOR=TLIQ+SUPER C C- PERTINENT INFORMATION IS PRINTED C WRITE(6,80)STEEL 80 FORMAT( 11',//,5X,'HEAT FLOW MODEL TEMPERATURE PREDICTIONS', -///,4X,'CASTING CONDITIONS : STEEL GRADE : ',15) W R I T E ( 6 , 8 1 ) ( E F G ( I ) , I = 1 , 5 ) 81 FORMAT(2OX,'MOLD WATER ',5A4) WRITE(6,82)SPEED 82 FORMAT(25X,'CASTING SPEED = WRITE(6,83)SUPER 83 FORMAT(25X,'SUPERHEAT WRITE(6,84)TPOR 84 FORMAT(25X,'POURING TEMP. = WRITE(6,85)THICK 85 FORMAT(25X,'BILLET SIZE WRITE(6,86)MOLD 86 FORMAT(25X,'MOLD LENGTH WRITE(6,87)SLEVEL 87 FORMAT(25X,'MENISCUS LEVEL = WRITE(6,88)END 88 FORMAT(23X,'SIMULATION LENGTH= ',F6.4,3X,'M.') TFF=TFEC(CARBON) WRITE(6,8 9)CARBON,SULFUR,PHOS,GMN,SI 1,SICKLE,CROME,YMOLY,COPPER,TFF 89 FORMAT(//,2X,'COMPOSITION OF STEEL : CARBON',9X,'=',F6.3, 1/25X,'SULFUR',9X,'=',F6.3,/25X,'PHOSPHOROUS',4X,'=',F6.3, 1/25X,'MANGANESE',6X,'=',F6.3,/25X,'SILICON',8X,'=',F6.3,/25 1'NICKLE',9X,'=',F6.3,/25X,'CHROMIUM',7X,'=',F6.3,/25X, 1'MOLYBDYNUM',5X,'=',F6.3,/25X,'COPPER',9X,'=',F6.3,//1X, ,F5.2,4X,'M./MIN.') ,F4.1,3X,'DEG C ) ,F5.0,4X,'DEG. C.') ,F5.4,4X,'M.') ,F6.4,3X,'M.') ,F6.5,3X,'M.') 185 T S O L I D U S TEMPERATURE FOR PURE FE-C ALLOY=',F6.1) WRITE(6,9 0)TLIQ,TSOL 90 FORMAT(//,5XMODEL PARAMETERS : LIQUIDUS TEMP. = ',F5.0,4 -'DEG. C.',/25X,'SOLIDUS TEMP. = ',F5.0,4X,'DEG. C ) WRITE(6,91)NNODE 91 FORMAT(25X,'NO.OF NODES = ',13) C C- NODE SIZE IS CALCULATED -C DX=THICK/(FLOAT(NNODE-1)*2.) SIZE=100.*DX C WRITE(6,92)SIZE,DT 92 FORMAT(25X,'NODE S I Z E = ' ,F7.5,2X,'CM.' ,/25X, 'TIME STEP = ',F5.2,4X,'SEC.',//) C C- INPUT HEAT FLUX DATA IS FITTED WITH POLYNOMIALS C WRITE(6,100) 100 FORMAT(//,'TIME',3X,'HEATFLUX',3X,'COMPUTED', 13X,'%ERROR',3X,'COEFFICENTS OF',7X,'OR OF') WRITE(6,101) 101 FORMAT( 1'(SEC)',2X,'(W/M*M) ',3X,'HEATFLUX',12X, 1'POLYNOMIAL',11X,'THE POLYNOMIAL' 1,/,' ',3X,' ',3X,' ',3X, 1 ' ' ,3X, ' ' ,8X, '--' ,/) C CALL CURFIT(N1,C,D,P1,KU1,ZA,ZAE,OR1) CALL CURFIT(N2,E,F,P2,KU2,ZB,ZBE,OR2) CALL CURFIT(N3,G1,H1,P3,KU3,ZC,ZCE,OR3) CALL CURFIT(N4,G2,H2,P4,KU4,ZD,ZDE,OR4) CALL CURFIT(N5,G3,H3,P5,KU5,ZE,ZEE,OR5) CALL CURFIT(N6,G4,H4,P6,KU6,ZF,ZFE,OR6) C WRITE(6,222)T1,T2,T3,T4,T5 222 FORMAT(//,'TIME BELOW MENISCUS WHERE POLYNOMIALS ARE DIFFER 1',/,'T1=',F4.1,3X,'T2=',F4.1,3X,'T3=',F4.1 1 ,3X,'T4=',F4.1,3X,'T5=',F4.1,//) C WRITE(6,233) 233 FORMATCTIME BELOW' , 5X, ' POSITION OF ' , 3X, ' SHELL (CM) ' , 2X, 1'DISTANCE(CM)',4X,'SURFACE',6X,'HEATFLUX',/, 1'MENISCUS(SEC)',2X,'LIQUIDUS(CM)',2X,'THICKNESS',2X, 1'BELOW MENISCUS',2X,'TEMP(DEG C)',2X,'(KW/M*M)',/, 1' ',3X,'---',2X,'',2X, 1 ' ' ,2X,' — ' ,2X,' ' ,/) C C-HEAT FLUX POINTS ARE REPRESENTED ON THS PLOT C CALL AXCTRLCXORIGIN',0.0) CALL AXCTRL('YORIGIN',ENDCM2) C WWWW=((l00.0*SPEED)/60.0)/4.0 186 DO 300 1=1,N1 CWWW=ENDCM1-C(I)*WWWW DWWW=(D(I)-500.0)/250.0 CALL SYMBOL(DWWW,CWWW,0.28,11,0.0,-1) 300 CONTINUE C DO 301 1=2,N2 EWWW=ENDCM1-E(I)*WWWW FWWW=(F(I)-500.0)/250.0 CALL SYMBOL(FWWW,EWWW,0.28,11,0.0,-1) 301 CONTINUE C DO 302 1=2,N3 G1WWW=ENDCM1-G1(I)*WWWW IF(G1WWW.LE.2.0)GO TO 306 H1WWW=(H1(I)-500.0)/250.0 CALL SYMBOL(H1WWW,G1WWW,0.28,11,0.0,-1) 302 CONTINUE C DO 303 1=2,N4 G2WWW=ENDCM1-G2(I)*WWWW IF(G2WWW.LE.2.0)GO TO 306 H2WWW=(H2(I)-500.0)/250.0 CALL SYMBOL(H2WWW,G2WWW,0.28,11,0.0,-1) 303 CONTINUE C DO 304 1=2,N5 G3WWW=ENDCM1~G3(I)*WWWW IF(G3WWW.LE.2.0)GO TO 306 H3WWW=(H3(I)-500.0)/250.0 CALL SYMBOL(H3WWW,G3WWW,0.28,11,0.0,-1) 304 CONTINUE C DO 305 1=2,N6 G4WWW=ENDCM1"G4(I)*WWWW IF(G4WWW.LE.2.0)GO TO 306 H4WWW=(H4(I)-500.0)/250.0 CALL SYMBOL(H4WWW,G4WWW,0.28,11,0.0,-1) 305 CONTINUE C 306 CONTINUE C C- DISTANCE, TIME AND NODE TEMPERATURES.ARE I N I T I A L I Z E D C NSTEP=1 QTOTAL=0. TOTRAD=0. DIST=0. TIME=0. DTZ=DT IF = 0 C DO 110 1=1,NNODE T(I)=TPOR 187 1 1 0 CONTINUE C WRITE(7,1103)DIST,TIME WRITE(7,1102)(I,T(I),1=1,NNODE) C C C C@ START OF THE LOOP @(l@@(a(a@@@(a(a C @ @ @ @ @ @ @ § @ @ @ @ @ C 5000 XDIST=DIST XTIME=TIME CALL POS(TIME,IF,DTZ,DT) DIST=TIME*SPEED/60. XLAM=DTZ/(DX*DX) C IF(DIST.GT.END)GO TO 9999 c c C- THE TRIDIAGONAL MATRIX IS CONSTRUCTED C DO 230 1=1,NNODE D C ( I ) = T ( I ) I F ( I . E Q . 1 ) G O TO 180 IF(I.EQ.NNODE)GO TO 190 C C INTERIOR NODE C K1=K(I) K2=K(I-1 ) K3=K(I + 1 ) I F ( T ( I - 1 ) . G E . T L I Q ) K 1 = K 2 I F ( T ( I ) . G E . T L I Q ) K 3 = K 1 A=(K1+K2)*XLAM/(2.*ROW*CP(I)) B=(K1+K3)*XLAM/(2.*R0W*CP(I)) AC(I)=-A BC(I)=1.+A+B C C ( I ) = - B GO TO 230 C C MIDPLANE HALF NODE C 180 K1=K(1) K2=K(2) IF ( T ( 1 ) . G E . T L I Q ) K 2 = K 1 B=(K1+K2)*XLAM/(2.*ROW*CP(1)) A C ( I ) = 0 . BC(I)=1.+2.*B C C ( I ) = - 2 . * B GO TO 230 C C SURFACE HALF NODE 188 C 190 K1=K(NNODE) K2=K(NNODE) A=(K1+K2)*XLAM/(2.*ROW*CP(NNODE)) AC(I)=-2.*A BC(I)=1.+2.*A C C ( I ) = 0 . QNOWW=QDOTX(IZ,NNODE,TIME) QTOTAL=QTOTAL+QNOWW*DTZ DC(I)=DC(I)-4.*A*(DX/(K1+K2))*QNOWW C 2 30 CONTINUE C C- THE TRIDIAGONAL MATRIX IS SOLVED FOR THE NEW C- TEMPERATURE DISTRIBUTION. C- THE NEW TEMPERATURES ARE ADJUSTED TO COMPENSATE C- FOR LATENT HEAT RELEASE MISSED DUE TO THE C- DISCRETE TIME STEP. C CALL TRIDAG(1,NNODE) DO 1200 J=1,NNODE CALL CHECK(J,TNEW(J)) T(J)=TNEW(J) 1200 CONTINUE C C-DISTANCE AND FLUX UNITS ARE CHANGED TO CM AND KILO WATTS C DISTCM=DIST*100. QN0WKW=QN0WW/1000.0 C C- TO CALCULATE SHELL THICKNESS AT EACH TIME STEP C NNNODE=NNODE-1 IF(T(NNODE).LT.TSOL) GO TO 999 SHELL=0 GO TO 997 999 SHELL=SIZE/2.0 NNNODE=NNODE-1 DO 998 J=1,NNNODE IF(T(NNODE-J) .GT.TSODGO TO 997 SHELL=SHELL+SIZE 998 CONTINUE C C- TO CALCULATE POSITON OF LIQUIDUS ~ C 997 IF(T(NNODE).LT.TLIQ) GO TO 996 LIQUID=0 GO TO 994 996 LIQUID=SIZE/2.0 DO 995 J=1,NNNODE IF(T(NNODE-J).GT.TLIQ)GO TO 994 LIQUID=LIQUID+SI ZE 995 CONTINUE C 189 C-RELEVENT OUTPUT IS COLLECTED IN F I L E 6 C 994 WRITE(6,991)TIME,LIQUID,SHELL,DISTCM,T(NNODE),QNOWKW 991 FORMAT(F5.2,1 OX,F6.3,8X,F6.3,5X,F6.3,11X,F6.1,7X,F6.1) C C-THE ALL TEMPERATURE MATRIX IS GENERATED--C-ALL OTHER PLOTTABLE PARAMETERS ARE SCALED AND STORED C DO 1101 1=1,NNODE T A L L ( N S T E P , I ) = T ( I ) 1101 CONTINUE C DISSCA(NSTEP)=DISTCM/4.0 QSCALE(NSTEP)=(QNOWKW-500.0)/250.0 TNNODE(NSTEP)=(T(NNODE)-l000.0)/50.0 SSOLID(NSTEP)=25.0+SHELL*10.0 SLIQID(NSTEP)=25.0+LIQUID*10.0 TNNODE(NSTEP)=44.0+((T(NNODE)-1000.0)/50.0) C C-NEW C-TEMPERATURE DISTRIBUTION IS PRINTED IN F I L E 7 C WRITE(7,1103)DISTCM,TIME 1103 FORMAT(///,5X,'TEMPERATURE DISTRIBUTION AT',F8.2,' CM OR ', -F8.2,' SECONDS BELOW MENISCUS') C WRITE(7,1102)(I,T(I),1=1,NNODE) 1102 FORMAT(10(' T ( ' ,I 3,' ) = ' ,F5.0)) C C-HEAT FLUX /TIME IS PRINTED IN F I L E 8 C WRITE(8,1001)QNOWW,TIME C1001 FORMAT(F12.2,',',F8.2) C C HEAT FLUX AND TOTAL HEAT REMOVED ARE PRINTED AFTER C- PRINTING THE TEMPERARURE PROFILE IN F I L E 7 C 321 WRITE(7,323)QNOWW,QTOTAL 323 FORMAT(5X,'HEAT FLUX = ',F11.2,' W/M*M',/, 5X,'TOTAL HEAT REMOVED IN ZONE = ',F12.2,' J/M*M') C NSTEP=NSTEP+1 GO TO 5000 C C@@@(a@@@@@@@@@ C @ @ § @ @ @ END OF LOOP @@@@@@ C c 9999 NSTEPS=NSTEP-1 WRITE(6,8888)NSTEPS 8888 FORMAT('TOTAL NUMBER OF TIMES THE MODEL CALCULATED THE HEAT 1,' FLUX PROFILE=',14) C CALL CREATE(TIME) 1 90 STOP END C SUBROUTINE TRIDAG(IF,L) c-C C- SUBROUTINE TO SOLVE TRIDIAGONAL MATRIX FOR C- THE NEW TEMPERATURE DISTRIBUTION C COMMON /C1/ V ( 1 0 0 ) , A C ( 1 0 0 ) , B C ( 1 0 0 ) , C C ( 1 0 0 ) , D C ( 1 0 0 ) DIMENSION BETA(101),GAMMA(101) B E T A ( I F ) = B C ( I F ) GAMMA(IF)=DC(IF)/BETA(IF) IFP1=IF+1 DO 10 I=IFP1,L B E T A ( I ) = B C ( I ) - A C ( I ) * C C ( I - 1 ) / B E T A ( I - 1) GAMMA(I)=(DC(I)~AC(I)*GAMMA(l-1))/BETA(I) 10 CONTINUE V(L)=GAMMA(L) LAST=L-IF DO 20 KT=1,LAST I=L-KT V ( I ) = G A M M A ( I ) - C C ( I ) * V ( I + 1 ) / B E T A ( I ) 20 CONTINUE RETURN END C SUBROUTINE CHECK(J,TP) c-C C- SUBROUTINE TO COMPENSATE FOR LATENT HEAT RELEASE C- MISSED DUE TO DISCRETE TIME STEP C COMMON /CO/ TW(200),NZON,T(100) COMMON /C2/ STEEL,NNODE,TL,TS I F ( T P . G T . T ( J ) ) G O TO 50 IF(T P . G T . T L ) G O TO 90 I F ( T P . L E . T S ) G O TO 10 IF (T ( J ) . LE . TL) GO TO 90 CPL=CP(J) T( J)=TP CPM=CP(J) TP=TL-CPL/CPM*(TL-TP) GO TO 90 10 I F ( T ( J ) . L E . T S ) G O TO 90 I F ( T ( J ) . G E . T L ) G O TO 20 CPM=CP(J) T ( J ) = T P CPS=CP(J) TP=TS-CPM/CPS*(TS-TP) RETURN 20 CPL=CP(J) TOLD=T(J) T ( j ) = ( T L + T S ) / 2 . 191 CPM=CP(J) T ( J ) = T P CPS=CP(J) A=CPL* (TOLD-TD+CPM* (TL-TS) B=CPS*(TS-TP) I F ( A . L T . B ) G O TO 30 TP=TL-CPL/CPM*(TL-TP) RETURN 30 TP=TS+CPM/CPS*(TL-TS)-CPL/CPS*(TL-TP) GO TO 90 50 I F ( T P . L E . T S ) G O TO 90 IF(TP.GT.TL)GO TO 60 I F ( T ( J ) . G E . T S ) G O TO 90 TP=TL-CPL/CPM*(TL-TP) RETURN 60 I F ( T ( J ) . G E . T L ) G O TO 90 I F ( T ( J ) . L E . T S ) G O TO 70 CPM=CP(J) T ( J ) = T P CPL=CP(J) TP=TL+CPM/CPL*(TP-TL) GO TO 90 70 T0LD=T(J) CPS=CP(J) T ( J ) = ( T L + T S ) / 2 . CPM=CP(J) T ( J ) = T P CPL=CP(J) A=CPS*(TOLD-TS)+CPM*(TS-TL) B=CPL*(TL-TP) IF ( A . L E . B ) G O TO 80 TP=TL+CPS/CPL*(TP-TS)+CPM/CPL*(TS-TL) GO TO 90 80 TP=TS+CPS/CPM*(TP-TS) 90 RETURN END C FUNCTION QDOTX(IZ,I,TIME) c-C C- SUBPROGRAM TO CALCULATE SURFACE HEAT FLUX C COMMON /CO/ TW(200),NZON,T(100) COMMON /C3/ C ( 5 0 ) , D ( 5 0 ) , E ( 5 0 ) , F ( 5 0 ) COMMON /C4/ S(1 5 0 ) , S I G M A ( 1 5 0 ) , A A ( 1 5 0 ) , B B ( 1 5 0 ) COMMON /C5/ YF(200),YD(200),WT(200) COMMON /C6/ P 1 ( 5 0 ) , P 2 ( 5 0 ) , P 3 ( 5 0 ) , P 4 ( 5 0 ) , P 5 ( 5 0 ) , P 6 ( 5 0 ) COMMON /C7/ KU1,KU2,KU3,KU4,KU5,KU6 COMMON /C8/ G 1 ( 5 0 ) , G 2 ( 5 0 ) f G 3 ( 5 0 ) , G 4 ( 5 0 ) COMMON /C C 8 / H 1 ( 5 0 ) , H 2 ( 5 0 ) , H 3 ( 5 0 ) , H 4 ( 5 0 ) COMMON /C11/T1,T2,T3,T4,T5 C C- MOLD HEAT FLUX IS CALCULATED HERE C 1 92 10 IF(TIME.GT.T1)GO TO 1 QDOTX=0 DO 11 J=1,KU1 Q D O T X X = 1 0 0 0 * ( P 1 ( J ) * ( T I M E * * ( J - 1 ) ) ) 11 QDOTX=QDOTX+QDOTXX RETURN C 1 IF(TIME.GT.T2)GO TO 2 QDOTX=0 DO 12 J=1,KU2 Q D O T X X = 1 0 0 0 * ( P 2 ( J ) * ( T I M E * * ( J - 1 ) ) ) 12 QDOTX=QDOTX+QDOTXX RETURN C 2 IF(TIME.GT.T3)GO TO 3 QDOTX=0 DO 13 J=1,KU3 Q D O T X X = 1 0 0 0 * ( P 3 ( J ) * ( T I M E * * ( J - 1 ) ) ) 13 QDOTX=QDOTX+QDOTXX RETURN C 3 IF(TIME.GT.T4)GO TO 4 QDOTX=0 DO 14 J=1,KU4 Q D O T X X = 1 0 0 0 * ( P 4 ( J ) * ( T I M E * * ( J - 1 ) ) ) 14 QDOTX=QDOTX+QDOTXX RETURN C 4 IF(TIME.GT.T5)GO TO 5 QDOTX=0 DO 15 J=1,KU5 Q D O T X X = 1 0 0 0 * ( P 5 ( J ) * ( T I M E * * ( J - 1 ) ) ) 15 QDOTX=QDOTX+QDOTXX RETURN C 5 QDOTX=0 DO 16 J=1,KU6 Q D O T X X = 1 0 0 0 * ( P 6 ( J ) * ( T I M E * * ( J - 1 ) ) ) 16 QDOTX=QDOTX+QDOTXX C RETURN END C FUNCTION C P ( I ) c-c C- SUBPROGRAM TO CALCULATE S P E C I F I C HEAT CAPACITY C COMMON /CO/ TW(200),NZON,T(100) COMMON /C2/ STEEL,NNODE,TLIQ,TSOL C CP = 1 0 0 0 * ( 0 . 5 4 0 + ( 0 . 0 0 0 0 9 4 1 * T ( I ) ) ) C I F ( T ( I ) . L T . T L I Q ) G O TO 1 193 GO TO 1 1 C 1 I F ( T ( I ) . L T . T S O L ) GO TO 11 CPP=CP CP=CPP+((1000*272.09)/(TLIQ-TSOL)) C C 11 W R I T E ( 1 5 , 1 0 ) T ( I ) , C P C 10 FORMAT('TEMPERATURES,F6.1,2X,F12.2) C 11 RETURN END C REAL FUNCTION K ( I ) C C C SUBPROGRAM TO CALCULATE THERMAL CONDUCTIVITY C COMMON /CO/ TW(200),NZON,T(100) COMMON /C2/ STEEL,NNODE,TLIQ,TSOL C K= 1 0 0 0 * ( 0 . 0 1 7 + ( 0 . 0 0 0 0 1 2 * T ( I ) ) ) c I F ( T ( I ) . L T . T L I Q ) G O TO 12 WK=K K=7.0*WK C 12 RETURN END C SUBROUTINE POS(A,I,B,C) C C C SUBROUTINE TO DETERMINE NEW TIME C A=A+B 1 = 1 + 1 C RETURN END C SUBROUTINE CURFIT(N,X,Y,P,KU,Z,Z1,NUMBER) c-c C THIS SUBROUTINE F I T S A POLYNOMIAL TO THE HEAT FLUX / TIME C-BELOW THE MENISCUS DATA -THIS WILL BE USED IN MOLD C- HEAT FLUX ESTIMATIONS AT EACH TIME STEP C DIMENSION X ( 1 0 0 ) , Y ( 1 0 0 ) , P ( 5 0 ) DIMENSION Z ( 5 0 ) , Z 1 ( 5 0 ) , Z Z ( 5 0 ) COMMON /C4/ S(1 5 0 ) , S I G M A ( 1 5 0 ) , A A ( 1 5 0 ) , B B ( 1 5 0 ) COMMON /C5/ YF(200),YD(200),WT(200) C LOGICAL LK LK=.FALSE. 1 94 NWT=0 C NUMBER=N-2 IF(NUMBER.GE.4 0)NUMBER=4 0 C CALL OLSF (NUMBER,N,X,Y,YF,YD,WT,NWT,S 1,SIGMA,AA,BB,SS,LK,P) C KU=NUMBER+1 C C DO 8 I=1,N Z ( I ) = P ( 1 ) DO 6 J=2,KU Z Z ( I ) = P ( J ) * ( X ( I ) * * ( J - 1 ) ) Z ( I ) = Z Z ( I ) + Z ( l ) 6 CONTINUE Z 1 ( I ) = ( Z ( I ) - Y ( l ) ) / Y ( l ) * 1 0 0 . 0 8 CONTINUE C WRITE(6,10)NUMBER 10 FORMAT(60X,' ',1X,I2,1X,' ',/) W R I T E ( 6 , 7 ) ( X ( I ) , Y ( I ) , Z ( I ) , Z 1 ( I ) , 1 , P ( l ) , 1 = 1 , K U ) KU=KU+1 W R I T E ( 6 , 9 ) ( X ( I ) , Y ( I ) , Z ( I ) , Z 1 ( I ) , I = K U , N ) C 7 FORMAT(F5.1,4X,F6.1,4X,F6.1,4X,F6.1,4X,'P(',12,')=',F12.2) 9 FORMAT(F5.1,4X,F6.1,4X,F6.1,4X,F6.1) C RETURN END C FUNCTION T F E C ( C ) C IF(C.GE.0.1)GO TO 100 TFEC=-(C*410.0)+1534.0 RETURN 100 IF(C.GE.O.16)G0 TO 200 TFEC=-( 0 . 3 3 3 3 * 0 + 1493.3333 RETURN 200 TFEC=-(182.10526*C)+1522.1368 RETURN END C C SUBROUTINE CREATE(TOTIME) C COMMON /C2/ STEEL,NNODE,TLIQ,TSOL COMMON /C12/ T A L L ( 3 0 0 , 1 0 0 ) , O U T ( 3 0 0 ) , S L O P E ( 3 0 0 ) 1,TPOR,DT,MOLD,END,SLEVEL,ENDCM1,ENDCM2,ENDCM3,DSTEP,TISTEP COMMON /C13/ TNNODE(300),SSOLID(300),SLIQID(300),QSCALE(300 COMMON /C14/ DISREV(300),DISSCA(300),NSTEPS COMMON / C 1 5 / A B C ( 5 ) , B C D ( 5 ) , C D E ( 5 ) , D E F ( 5 ) , E F G ( 1 1 ) C 195 CALL DASHLN(0.2,0.2,0.2,0.2) C C DISTNCE BELOW THE MENISCUS IS SCALED C DO 100 1=1,NSTEPS DISREV(I)=ENDCM1-DISSCA(I) 100 CONTINUE C ENDCM=(END*100)/DSTEP TlSTEP=TOTIME/ENDCM ENDCMM=ENDCM- 1 .0 ENDCM=ENDCMM EMENSC=ENDCM1+0.25 N1STEP=NSTEPS-1 C C C HEAT FLUX PROFILE IS PLOTTED * C c CALL AXCTRL('DIGITS' , 1 ) CALL AXCTRL('XORIGIN',0.0) CALL AXCTRL('YORIGIN',ENDCM2) CALL AXCTRL('SIDE',-1) C A L L A X P L O T C ;',270.0,20.0,0.0,DSTEP) CALL SYMBOL(-1.0,7.0,0.42,'DISTANCE DOWN THE MOLD (CM) Q,90.0,31) C CALL AXCTRL('SIDE' , + 1 ) CALL AXCTRL('XORIGIN',21.0) CALL AXCTRL('YORIGIN',ENDCM1) CALL A X P L O T C ;',270.0,ENDCM,0.0,TISTEP) CALL WHERE(XX,YY) CALL PLOT(XX,YY,3) CALL PLOT(XX,2.0,2) CALL PLOT(0.0,0.0,3) CALL SYMBOL(22.35,7.0,0.42,'TIME BELOW THE MENISCUS(SEC)' Q,90.0,28) C CALL AXCTRL('XORIGIN',0.0) CALL AXCTRL('YORIGIN',2.0) CALL AXCTRL('SIDE',-1) CALL A X P L O T C ; ' , 0 . 0 , 20 . 0 , 50 . 0 , 25. 0 ) CALL SYMBOL(5.5,0.65,0.42,'MOLD HEAT FLUX(KW/SQ.M)(X10)' Q,0.0,28) C CALL PLOT(0.0,ENDCM2,3) CALL PL0T(21.0,ENDCM2,2) CALL PL0T(21.0,2.0,3) CALL PLOT(20.0,2.0,2) CALL PLOT(0.0,ENDCM1,3) CALL PLOT(21.0,ENDCM1,4) CALL PLOT(21.0,ENDCM1,3) CALL PLOT(21.0,ENDCM2,2) CALL PLOT(0.0,2.0,3) 1 96 C C CALL SYMBOL( 1 7 . 0 ,EMENSC,0.42,'MENISCUS',0.0,8) CALL P L O T ( Q S C A L E ( 1 ) , D I S R E V ( 1 ) , 3 ) DO 200 1=1,N1STEP CALL P L O T ( Q S C A L E ( I ) , D I S R E V ( I ) , 2 ) 200 CONTINUE C CALL PSYM(12.0,10.0,0.42,ABC,0.0,20,22) CALL PSYM(12.0,9.0,0.42,BCD,0.0,20,22) CALL PSYM(12.0,8.0,0.42,CDE,0.0,20,22) CALL PSYM(12.0,7.0,0.42,DEF,0.0,20,22) CALL PSYM(12.0,6.0,0.42,EFG,0.0,20,22) C CALL PSYM(6.5,ENDCM3,0.56,'HEAT FLUX PROFILE',0.0,17,22) C C C SHELL THICKNESS PROFILE IS PLOTTED** C C CALL AXCTRL('DIGITS',1) CALL AXCTRL('XORIGIN',25.0) CALL AXCTRL('YORIGIN',ENDCM2) CALL AXCTRL('SIDE',-1) CALL AXPLOTC ;',270.0,20.0,0.0,DSTEP) CALL SYMBOL(24.0,7.0,0.42,'DISTANCE DOWN THE MOLD (CM) Q,90.0,31) C CALL AXCTRL('SIDE',+1j CALL AXCTRL('XORIGIN',40.0) CALL AXCTRL('YORIGIN',ENDCM1) CALL AXPLOTC ;',270.0,ENDCM,0.0,TISTEP) CALL WHERE(XX,YY) CALL PLOT(XX,YY,3) CALL PLOT(XX,2.0,2) CALL PLOT(0.0,0.0,3) CALL SYMBOL(41.35,7.0,0.42,'TIME BELOW THE MENISCUS(SEC)' Q,90.0,28) C CALL AXCTRL('XORIGIN',25.0) CALL AXCTRL('SIDE',-1) CALL AXCTRL('YORIGIN',2.0) CALL AXPLOT ( ' ,-',0.0,14.0,0.0,1.0) CALL SYMBOL(29.0,0.65,0.42,'SHELL THICKNESS(MM)' Q,0.0,19) CALL P L O T ( S S O L I D ( 1 ) , D I S R E V ( 1 ) , 3 ) C DO 300 1=1,N1STEP IPLUS=I+1 IF(I.EQ.N1STEP) GO TO 301 I F ( S S 0 L I D ( I P L U S ) . G T . S S O L I D d ) ) G O TO 299 GO TO 300 299 CALL PLOT(SSOLID(lPLUS),DISREV(I PLUS),2) 300 CONTINUE 197 C C 301 CALL PLOT(SSOLID(N1STEP),2.0,2) CALL PLOT(0.0,0.0,3) CALL PSYM(31.0,20.0,0.42,ABC,0.0,20,22) CALL PSYM(31.0,19.0,0.42,BCD,0.0,20,22) CALL PSYM(31.0,18.0,0.42,CDE,0.0,20,22) CALL PSYM(31.0,17.0,0.42,DEF,0.0,20,22) CALL PSYM(31.0,16.0,0.42,EFG,0.0,20,22) CALL PSYM(27.0,ENDCM3,0.56,'SHELL THICKNESS PROFILE',0.0,23 CALL CALL CALL CALL CALL CALL CALL CALL CALL PLOT(25. PLOT(40. PLOT(40. PLOT(39. PLOT(25. PLOT(30. PLOT(40. PLOT(40. PLOT(0.0 0,ENDCM2,3) 0,ENDCM2,2) 0,2.0,3) 0,2.0,2) 0,ENDCM1 0,ENDCM1 0,ENDCM1 0,ENDCM2 ,0.0,3) 3) 4) 3) 2) CALL SYMBOL(26.0,EMENSC,0.42,'MENISCUS',0.0,8) C C C TEMPERATURES OF ALL NODES ARE PLOTTED C C CALL AXCTRL('DIGITS',1) CALL AXCTRL('XORIGIN',44.0) CALL AXCTRL('YORIGIN',ENDCM2) CALL AXCTRL('SIDE',-1) CALL A XPLOTC ;',270.0,20.0,0.0,DSTEP) CALL SYMBOL(43.0,7.0,0.42,'DISTANCE DOWN THE MOLD (CM) Q,90.0,31) C CALL AXCTRL('SIDE',+1) CALL AXCTRL('XORIGIN',61.0) CALL AXCTRL('YORIGIN',ENDCM1) CALL A XPLOTC ;',270.0,ENDCM,0.0,TlSTEP) CALL WHERE(XX,YY) CALL PLOT(XX,YY,3) CALL PLOT(XX,2.0,2) CALL PLOT(0.0,0.0,3) CALL SYMBOL(62.35,7.0,0.42,'TIME BELOW THE MENISCUS(SEC)' Q,90.0,28) C CALL AXCTRL('XORIGIN',44.0) CALL AXCTRL('SIDE',-1) CALL AXCTRLCYORIGIN',2.0) CALL A X P L O T C ; ' , 0 . 0 , 1 6 . 0 , 800 . 0 , 50 . 0 ) CALL SYMBOL(48.0,0.65,0.42,'TEMPERATURE OF NODES(DEG.C)' Q,0.0,27) 198 C C-LIQIDUS AND SOLIDUS LINES ARE DRAWN * C TLIQX=44.0+((TLIQ-800)/50.0) TSOLX=44.0+((TSOL-800)/50.0) CALL PLOT(TLIQX,2.0,3) CALL PLOT(TLIQX,ENDCM1,2) CALL PLOT(TSOLX,ENDCM1,2) CALL PLOT(TSOLX,2.0,2) -TEMPERATURES OF INDIVIDUAL NODES ARE PICKED UP AND PLOTTED ** DO 500 INODE=85,NNODE O U T ( l ) = 4 4 . 0 + ( ( ( T A L L ( 1 , I N O D E ) _ 8 0 0 . 0 ) / 5 0 . 0 ) ) CALL P L O T ( O U T ( 1 ) , D I S R E V ( 1 ) , 3 ) 00 00 c--DO 400 JTIMES=1,N1STEP OUT(JTIMES)=44.0+(((TALL(JTIMES,INODE)-800.0)/50, CALL P L O T ( OUT( J TIMES),DISREV(JTIMES),2) CONTINUE CONTINUE CALL PSYM(45.0,20.0,0.42,ABC,0.0,20,22) CALL PSYM(45.0,19.0,0.42,BCD,0.0,20,22) CALL PSYM(45.0,18.0,0.42,CDE,0.0,20,22) CALL PSYM(45.0,17.0,0.42,DEF,0.0,20,22) CALL PSYMU5.0, 1 6 . 0 , 0 . 42 , EFG, 0 . 0 , 20 , 22 ) CALL PSYM(46.5,ENDCM3,0.56,'TEMPERATURES OF ALL NODES',0.0, CALL PLOT(44.0,ENDCM2,3) CALL PLOT(61.0,ENDCM2, 2) CALL PLOT(61.0,2.0,3) CALL PLOT(60.0,2.0,2) TSSOLX=TSOLX-2.5 CALL PLOT(TSSOLX,ENDCM1,3) CALL PLOT(61.0,ENDCM1,4) CALL PLOT(61.0,ENDCM1,3) CALL PLOT(61.0,ENDCM2,2) CALL PLOT(0.0,0.0,3) C C c c c CALL SYMBOL(57.0,EMENSC,0.42,'MENISCUS',0.0,8) COOLING RATES OF ALL NODES ARE PLOTTED ** CALL CALL CALL CALL CALL Q,90.0,31 ) AXCTRL('XORIGIN',65.0) AXCTRL('YORIGIN',ENDCM2) AXCTRL('SIDE',-1) AXPLOT(' ;',270.0,20.0,0.0,DSTEP) SYMBOL(64.0,7.0,0.42,'DISTANCE DOWN THE MOLD (CM) 199 C CALL AXCTRL('SIDE',+1) CALL AXCTRL('XORIGIN',91.0) CALL AXCTRL('YORIGIN',ENDCM1) CALL AXPLOTC ;*,270.0,ENDCM,0.0,TISTEP) CALL WHERE(XX,YY) CALL PLOT(XX,YY,3) CALL PLOT(XX,2.0,2) CALL PLOT(0.0,0.0,3) CALL SYMBOL(92.35,7.0,0.42,'TIME BELOW THE MENISCUS(SEC)' Q,90.0,28) C CALL AXCTRL('SIDE',-1) CALL AXCTRL('XORIGIN',70.0) CALL AXCTRL('YORIGIN',2.0) CALL AXPLOTC ;',0.0,20.0,0.0,5.0) CALL SYMBOL(73.0,0.65,0.42,'COOLING RATES (DEG.PER.SEC)' Q,0.0,27) C CALL AXCTRL('XORIGIN',65.0) CALL AXPLOTC ;',0.0,5.0,-25.0,5.0) C C-SLOPES (COOLING RATES) ARE CALCULATED * C DO 1900 INODE=80,NNODE C IF(INODE.EQ.80)GO TO 1899 IF(INODE.EQ.90)GO TO 1899 IF(INODE.EQ.NNODE)GOTO 1899 C GO TO 1900 C 1899 WRITE(14,127)INODE 127 FORMAT(1 OX,' ',14,' ') c-DO 1800 JTIME=1,NSTEPS C IF(JTIME.GT.1)GO TO 120 S L O P E ( 1 ) = ( ( ( T P O R - T A L L ( 1 , I N O D E ) ) / D T ) / 5 . 0 ) + 7 0 . 0 GO TO 121 120 JMINUS=JTIME-1 " S L O P E ( J T I M E ) = ( ( ( T A L L ( J M I N U S , I N O D E ) - T A L L ( J T I M E , I N O D E ) ) / D T ) ,/5.0)+70.0 121 1705 1 706 I F ( S L O P E ( J T I M E ) . G T . 9 0 . 0 ) G O TO 1705 I F ( S L O P E ( J T I M E ) . L T . 6 5 . 0 ) G O TO 17 06 GO TO 1707 SLOPE(JTIME)=90.0 GO TO 1707 SLOPE(JTIME)=65.0 1 707 1 29 WRITE0 4, 1 29)SLOPE(JTIME) , JTIME,IN FORMAT(F 5.1,2X,I3,2X,'INODE=' ,13) 1800 CONTINUE 200 C C CALL P L O T ( S L O P E ( 1 ) , D I S R E V ( 1 ) , 3 ) DO 1850 JTIME=1,N1STEP JNEXT=JTIME+1 IF(JTIME.EQ.N1STEP) GO T01851 I F ( ( S L O P E ( J N E X T ) - S L O P E ( J T I M E ) ) . G T . 3 . 0 ) G O TO 1850 I F ( ( S L O P E ( J N E X T ) - S L O P E ( J T I M E ) ) . L T . - 3 . 0 ) G O TO 1850 IF(INODE.EQ.90)GO TO 1849 CALL PLOT(SLOPE(JNEXT),DISREV(JNEXT) , 2) GO TO 1850 1849 CALL PLOT(SLOPE(JNEXT),DISREV(JNEXT),4) 1850 CONTINUE C 1851 CALL PLOT(0.0,0.0,3) C 1900 CONTINUE c-CALL PSYM(72.0,ENDCM3,0.56,'COOLING RATES OF NODES',0.0,22, CALL PSYM(82.0, 1 0 . 0 , 0 . 42', ABC, 0 . 0 , 20 , 22 ) CALL PSYM(82.0,9.0,0.42,BCD,0.0,20,22) CALL PSYM(82.0,8.0,0.42,CDE,0.0,20,22) CALL PSYM(82.0,7.0,0.42,DEF,0.0,20,22) CALL PSYM(82.0,6.0,0.42,EFG,0.0,20,22) C CALL PLOT(65.0,ENDCM2,3) CALL PLOT(91.0,ENDCM2,2) CALL PL0T(91.0,2.0,3) CALL PLOT(90.0,2.0,2) C CALL PLOT(70.0,ENDCM1,3) CALL PLOT(70.0,2.0,2) CALL PLOT(0.0,0.0,3) C CALL PLOT(65.0,ENDCM1,3) CALL PLOT(91.0,ENDCM1,4) CALL PLOT(91.0,ENDCM1,3) CALL PLOT(91.0,ENDCM2,2) CALL PLOT(0.0,0.0,3) CALL SYMBOL(66.0,EMENSC,0.42,'MENISCUS ',0.0,8) 22 CALL PLOTND RETURN END C 201 APPENDIX D - TWO-DIMENSIONAL UN-STEADY STATE MENISCUS SOLIDIFICATION MODEL C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c * * C * 6=-6 (FINAL OUTPUT) * C * 7=-7 (TO CHECK HOW LATENT HEAT IS LIBERATED TN S/R ESTMAT)* C * 9=-9 (GIVES THE PLOTTER OUTPUT) C * 10=-10 (TEMPERATURES IN THE HALF TIME STEPS ONLY) * C * * C * 1=-1 2=-2 3=-3 AND 11=-1 12=-2 13=-3 ARE ONLY * C * TO GIVE PLOTTER REQUIRED INFORMATION-NEED NEVER BE PRINTED * C * * C * * C * ALL UNITS ARE C.G.S * C * * C * * c * * C * THIS PROGRAM USES HEAT FLUX VALUES CAL/(SQ.CM)(SEC) * C * * C * * c * **** DOUBLE PRECITION * C * * c * **** F E _ C DIAGRAM IS INCORPORATED * C * * c * **** THEMAL CONDUCTIVITY AND S P E C I F I C HEAT ARE VARIED * C * WITH TEMPERATURE * C * * C * * c * **** A L L NODES ARE CONSIDERED AT ANY TIME IN TRIDAG **** * C * * C * * C * THIS IS A TWO-DIMENSIONAL UNSTEADY STATE HEAT TRANSFER MODEL* C * FOR SOLIDIFICATION AT THE MENISCUS OF THE LIQUID STEEL POOL * C * IN A CONTINUOUS CASTING MOLD.IT IS ASSUMED THAT THE MENISCUS* C * IS STAGNANT FOR A VERY SHORT TIME.THE SOLUTION OF THE SIMUL-* C * TANEOUS EQUATIONS DEVELOPED FROM HEAT BALANCES IS DONE BY * C * FlNETE-DIFFERENCE APPROXIMATIONS AND EMPLOYING THE ALTERNAT-* C * ING DIRECTION METHOD OVER SUCCESSIVE TIME STEPS. * C * * C * AT TIME=0.0 ALL THE NODES ARE SET EQUAL TO POURING TEMPERAT-* C * URES.THE FOLLOWING BOUNDARY CONDITIONS ARE APPLIED * C * * C * * C * ( 1 ) X=0 Z=0 TO Z2...HEAT IN=-(QMOLD*AREA OF NODE) * C * ( 2 ) X=X2 Z=0 TO Z2...HEAT IN=0 * C * ( 3 ) Z=0 X=0 TO X2...HEAT IN=RADIATION+CONVECTION LOSS * C * ( 4 ) Z=Z2 X=0 TO X2...HEAT IN=0 * C * * C * NOMENCLATURE * c * * C * AK =THERMAL CONDUCTIVITY OF STEEL (CAL/CM*SEC*DEG O * C * AKL =THERMAL CONDUCTIVITY OF LIQUID STEEL * C * AKS =THERMAL CONDUCTIVITY OF SOLID STEEL * 202 C * AKMU =THERMAL CONDUCTIVITY OF STEEL IN THE MUSHY ZONE * C * * C * BETA INTERMEDIATE SOLUTION VECTOR USED IN S/R TRIDAG * C * * C * CP =SPECFIC HEAT (CAL/GM*DEG C) * C * CPLIQ =SPECIFC HEAT OF LIQUID * C * CPSOL =SPECIFC HEAT OF SOLID STEEL * C * CPMU =SPECIFC HEAT IN THE MUSHY ZONE * C * * C * DBOT =SUB DIAGNOL COEFFICENT VECTOR USED IN S/R TRIDAG * C * D =DIAGNOL COEFFICENT VECTOR USED IN S/R TRIDAG * C * DTOP =SUPER DIAGNOL COEFFICENT VECTOR USED IN S/R TRIDAG * C * * C * DT =TIME STEP (SEC) * C * DT=TIMESTEP (SEC) * C * DX(CM)=WIDTH OF NODE IN X-D/N =X1/M1 (CONSTANT) AND X2=DX*M2* C * DZ(CM)=WIDTH OF NODE IN Z-D/N =Z1/N1 (VARIES) * C * * C * END =SIMULATION TIME (SEC) * C * * C * FSLANT=VIEW FACTOR FOR THE TRIANGULER NODES * C * FHORIZ=VIEW FACTOR FOR FLAT BOUNDRU NODES ON THE MENISCUS * C * * C * GAMMA =INTERMEDIATE SOLUTION VECTOR USED IN S/R TRIDAG * C * * C * HEAT =LATENT HEAT RELEASE (CAL/GM) * C * * C * M1 =NO. OF NODES IN X-D/N IN MENISCUS * C * M2 =NO. OF NODES IN X-D/N AFTER MENISCUS * C * NI =NO. OF NODES IN Z-D/N IN MENISCUS * C * N2 =NO. OF NODES IN Z-D/N AFTER MENISCUS * C * * C * QMOLD =MOLD HEAT FLUX CAL/(SQ. CM) *SEC * C * * C * R =RIGHT HAND SIDE OF THE LINEAR SIMULTANEOUS EQUATIONS * C * ROW =DENSITY OF STEEL (GM/CUBIC CM) * C * ROWL =DENSITY OF LIQUID STEEL * C * ROWS =DENSITY OF SOLID STEEL * C * ROWMU =DENSITY OF STEEL IN THE MUSHY ZONE * C * * C * TT =TEMPERATURE OF STEEL (DEG C) * C * TLIQID=LIQUIDUS TEMPERATURE * C * TPRIME=SOLUTION VECTOR EMPLOYED IN S/R'S 'TRIDAG','SOLVE' * C * TOLD =ALL TEMPERATURES FROM PREVIOUS TIME STEP ARE STORED * C * TPOUR =POURING TEMPERATURE * C * TS =TEMPERATURE OF STEEL AFTER ONE HAFF TIME STEP * C * TSOLID=SOLIDUS TEMPERATURE * C * * C * SUHEAT=SUPER HEAT IN THE STEEL * C * * Q ********************************* C IMPLICIT REAL*8 (A-H,0-Z) LOGICAL LK 203 COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON 1KK,KCP COMMON COMMON COMMON /CI /C2 /C3 /C4 /C5 /C6 /C7 /C8 /C9 /CIO/ /CI 3/ /CI 4/ /C15/ /C16/ /CI 7/ /C18/ / / / / / / / / / T T ( 7 5 , 1 0 0 ) , T S ( 7 5 , 1 DZ(75) AK(75,100),CP(75,1 SOLID(75,100) DMID(7600),DTOP(76 BETA(7600),GAMMA(7 N1,N2,M1,M2,Z1,Z2, ROWS,ROWL MTIMES,NTIMES,END, CARBON,TLIQID,TSOL FSLANT,FHORIZ,TAMB NTOTAL,MTOTAL,NX TFORK(15),TFORCP(1 AKD1(15),AKD2(15), C P D 1 ( 1 5 ) , C P D 2 ( 1 5 ) , Y K ( 1 5 ) , Y C P ( 1 5 ) , P K ( 00) 00),ROW(75,100) 00),DBOT(7600),R(7600),DX,L 600),TPRIME(7600) XI,X2,N10,N12,M10,M12 THEEND,TIME,DT ID,SUHEAT,TPOUR,HEAT,QMOLD 5),NFORK,NFORCP AKD3(15),AKD4(15),AKD5(15) CPD3(15),CPD4(15),CPD5(15) 75),PCP(15),ORDERK,ORDERC, /C19/ C1,C2,C3,C4,C5 /C20/ S H E L L F ( 7 5 ) , X M I N S C ( 7 5 ) , D I S T ( 7 5 ) /C21/ FS(75,100),FRACTN,NREPET Q * * * c CALL PLCTRL('METR',1) CALL READ CALL XANDY CALL SHAPE(M1,N1,89,61) 1 CALL START * * * * * * * * * * * * * * * * * * * * * * * 2 DO 8 NTIME=1,NREPET IF(MTIMES.EQ.0)GO TO 4 CONTINUE TIME=TIME+DT CALL OUTPUT(6) C C 4 5 C Q-k * * 10 CALL SOLVE(53,55) TIME=NTIME*END CALL OUTPUT(6) IF(NTIME.GT.1)GO TO 7 CALL MSHAPE(N1) CALL MENISC CALL PROFLE(TIME) CALL S H E L L ( N l ) 8 CONTINUE * * * * * * * * * * * * * * * * * * * * * * * * WRITE(6,100) 0 FORMAT(////////////,'THE FINAL RESULT',/, f / / ) CALL OUTPUT(6) CALL PLOTND 9 STOP END 204 C SUBROUTINE READ INPUTS.THE DATA FROM F I L E 5 SUBROUTINE READ C IMPLICIT REAL*8 (A-H,0~Z) DIMENSION A B C ( 6 ) , B C D ( 6 ) , C D E ( 6 ) , D E F ( 6 ) , E F G ( 6 ) , F G H ( 6 ) COMMON /C7 / N1,N2,M1,M2,Z1,Z2,X1,X2,N10,N12,M10,M12 COMMON /C8 / ROWS,ROWL COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /C13/ FSLANT,FHORIZ,TAMB COMMON /C15/ TFORK(15),TFORCP(15),NFORK,NFORCP COMMON /C16/ AKD1(15),AKD2(15),AKD3(15),AKD4(15),AKD5(15) COMMON /C17/ CPD1(15),CPD2(15),CPD3(15),CPD4(15),CPD5(15) COMMON /C19/ C1,C2,C3,C4,C5 COMMON /C21/ FS(75,100),FRACTN,NREPET C C RE A D ( 5 , 1 ) ( A B C ( I ) , 1 = 1 , 6 ) READ(5,1)(BCD(I),1=1,6) R E A D ( 5 , 1 ) ( C D E ( I ) , 1 = 1 , 6 ) R E A D ( 5 , 1 ) ( D E F ( I ) , 1 = 1 , 6 ) READ ( 5 , 1 ) (EFG (I )', I = 1 , 6 ) RE A D ( 5 , 1 ) ( F G H ( I ) , I = 1 , 6 ) C READ(5,500) DT,NTIMES,NREPET READ(5,501) CARBON,QMOLD,SUHEAT,FRACTN READ(5,502) HEAT,ROWL,ROWS,TAMB,FSLANT,FHORIZ READ(5,503) N1,N2,M1,M2 READ(5,504) Z1,Z2,X1 C READ(5,505)NFORK,NFORCP READ(5,506)C1,C2,C3,C4,C5 DO 100 1=1,NFORK 100 R E A D ( 5 , 5 0 7 ) T F O R K ( I ) , A K D 1 ( I ) , A K D 2 ( I ) , A K D 3 ( I ) , A K D 4 ( I ) , A K D 5 ( I ) DO 200 1=1,NFORCP 200 R E A D ( 5 , 5 0 7 ) T F O R C P ( I ) , C P D 1 ( I ) , C P D 2 ( I ) , C P D 3 ( I ) , C P D 4 ( I ) 1,CPD5(I) C 1 FORMAT(6A4) C 500 FORMAT(F10.5,I4,6X,I4) 501 FORMAT(4F10.5) 502 FORMAT(6F10.6) 503 F0RMAT(I4,6X,I4,6X,I4,6X,I4) 504 FORMAT(3F10.5) C 505 FORMAT(12,1X,I2) 506 FORMAT(7X,F4.2,2X,F4.2,2X,F4.2,2X,F4.2,2X,F4.2) 507 FORMAT(F6.1,1X,F5.3,1X,F5.3,1X,F5.3,1X,F5.3,1X,F5.3) C WRITE(11,1)(ABC(I),1=1,6) WRITE(11,1)(BCD(I),1=1,6) 205 C c c-WRITE(11,1)(CDE(I ),1 = 1 ,6) WRIT E ( 1 1 , 1 ) ( D E F ( I ),1 = 1 ,6) W R I T E ( 1 1 , 1 ) ( E F G ( I ) , I = 1 , 6 ) WRITE(11,1)(FGH(I),1=1,6) RETURN END SUBROUTINE SHAPE MAKES SURE THAT M1=N1 SUBROUTINE SHAPE(Ml,N1,*,*) IMPLICIT REAL*8 (A-H,0~Z) IF(M1-N1) 800,802,800 800 WRITE(6,801) 801 FORMAT(1 HO,2OH M1=N1 NOT EQUAL ) RETURN 1 802 RETURN 2 END SUBROUTINE START I N I T I A T E S VARIOUS PARAMETERS C C c- SUBROUTINE START IMPLICIT REAL*8 (A-H,0-Z) COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON 1,KK,KCP COMMON /C19/ COMMON /C20/ COMMON /C21/ /CI /C2 /C3 /C4 /C5 /C7 /C8 /C9 /C10/ /CI 4/ /CI 5/ /CI 6/ /C17/ /C18/ / / / / / / / / T T ( 7 5 , 1 0 0 ) , T S ( 7 5 , 1 DZ(75) AK(75,100),CP(75,1 SOLID(75,100) DMID(7600),DTOP(76 N1,N2,M1,M2,Z1,Z2, ROWS,ROWL MTIMES,NTIMES,END, CARBON,TLIQID,TSOL NTOTAL,MTOTAL,NX TFORK(15),TFORCP(1 AKD1(15),AKD2(15), C P D 1 ( 1 5 ) , C P D 2 ( 1 5 ) , Y K ( 1 5 ) , Y C P ( 1 5 ) , P K ( CI,C2,C3,C4,C5 SHELLF(75),XMINSC( F S ( 7 5,100),FRACTN, 00) 00),ROW(75,100) 00),DBOT(7600),R(7600),DX,L X1,X2,N10,N12,M10,M12 THEEND,TlME,DT ID,SUHEAT,TPOUR,HEAT,QMOLD 5),NFORK,NFORCP AKD3(15),AKD4(15),AKD5(15) CPD3(15),CPD4(15),CPD5(15) 75),PCP(15),ORDERK,ORDERC 7 5 ) , D I S T ( 7 5 ) NREPET C C c c DIMENSION DOOO(75) ********TIME IS I N I T I A L I S E D TIME=0.0D0 END=NTIMES*DT THEEND=NREPET*END MTIMES=0 C C*********DEFINITION OF THE GRID C 206 NTOTAL=N1+N2 MTOTAL=M1+M2 N10=N1- 1 N12=N1+1 M10=M1-1 M12=M1+1 DX=X1/FLOAT(M1) X2=DX*FLOAT(M2) IF(MTOTAL.GE.23) GO TO 100 NX=MTOTAL GO TO 200 100 NX=23 C********DEFINITION OF THE MENISCUS C 200 DO 300 N=1,N1 300 DZ(N)=DSQRT(Z1**2/X1*(X1-DX*FLOAT(N-1))) A-DSQRT(Z1**2/X1*(X1-DX*FLOAT(N))) C DO 400 N=N1,NTOTAL 400 DZ(N)=Z2/FLOAT(N2) D I S T ( 1 ) = D Z ( 1 ) DO 500 N=2,NTOTAL 500 DIST(N)=DIST(N-1)+DZ(N) C DO 600 N=1,NTOTAL DOOO=20.0D0-(5.0D0*DIST(N)) DIST(N)=DOOO 600 CONTINUE C C ***** * * * LIQUIDUS,SOLIDUS,POURING TEMPERATURES ARE CALCULATED C TLIQID=TLIQ(CARBON) C TSOLID=TSOL(CARBON) C TPOUR=TLIQID+SUHEAT C CALL INPUT C C C THERMAL CONDUCTIVITY 8 S P E C I F I C HEAT ARE VARIED WITH TEMP. C CC1=DABS(CARBON-C1) CC2=DABS(CARBON-C2) CC3=DABS(CARBON-C3) CC4=DABS(CARBON-C4) CC5=DABS(CARBON-C5) C IF(CC1.GT.CC2)GO TO 111 DO 1 1=1,NFORK 1 Y K ( I ) = A K D 1 ( I ) 207 DO 2 1=1,NFORCP 2 Y C P ( I ) = C P D 1 ( I ) GO TO 555 C 111 IF(CC2.GT.CC3)GO TO 222 DO 3 1=1,NFORK 3 Y K ( I ) = A K D 2 ( I ) DO 4 1=1,NFORCP 4 Y C P ( I ) = C P D 2 ( I ) GO TO 555 C 222 IF(CC3.GT.CC4)GO TO 333 DO 5 1=1,NFORK 5 Y K ( I ) = A K D 3 ( I ) DO 6 1=1,NFORCP 6 Y C P ( I ) = C P D 3 ( I ) GO TO 555 C 333 IF(CC4.GT.CC5)GO TO 444 DO 7 1=1,NFORK 7 Y K ( I ) = A K D 4 ( I ) DO 8 1=1,NFORCP 8 Y C P ( I ) = C P D 4 ( I ) GO TO 555 C 444 DO 9 1=1,NFORK 9 YK(I)= A K D 5 ( I ) DO 10 1=1,NFORCP 10 Y C P ( I ) = C P D 5 ( I ) C 555 WRITE(6,11) 11 FORMAT(//, 'TEMP',3X,'TH.CONDC',3X,'COMPUTED', 13X,'%ERROR',3X,'COEFFICENTS OF',7X,'ORDER OF') WRITE(6,12) 12 FORMAT( 1'( C )',2X,' ',3X,'TH.COND',12X, 1'POLYNOMIAL',11X,'THE POLYNOMIAL' 1 , /, ' •- ' , 3X,' ' , 3X,' ' , 3X, 1 ' ' ,3X, ' ' ,8X, ' ' ,/) C CALL CURFIT(NFORK,TFORK,YK,PK,KK,ORDERK) C WRITE(6,13) 13 FORMAT(//,'TEMP',3X,'SP.HEAT ',3X,'COMPUTED', 13X,'%ERROR',3X,'COEFFICENTS OF',7X,'ORDER OF') WRITE(6,14) 14 FORMAT( 1'( C )',2X,' ',3X,'SP.HEAT',12X, 1'POLYNOMIAL',11X,'THE POLYNOMIAL' 1,/,' ',3X,' ',3X,' ',3X, 1 ' ' ,3X, ' ' ,8X, ' ' ,/) C CALL CURFIT(NFORCP,TFORCP,YCP,PCP,KCP,OEDERC) 208 C C TEMPERATURES6THERMO-PHYSICAL PROPERTIES ARE INITIATED C DO 800 N=1,NTOTAL DO 700 M=1,MTOTAL L=0 TT(N,M)=TPOUR TS(N,M)=TPOUR SOLID(N,M)=0.0D0 FS(N,M)=0.0D0 AK(N,M)=7*(AKSOL(TPOUR)) CP(N,M)=CPSOL(TPOUR) ROW(N,M)=ROWL 700 CONTINUE 800 CONTINUE C RETURN END C C SUBROUTINE WRITE PRINTS PERTINENT INFORMATION IN F I L E 6 c  SUBROUTINE INPUT C IMPLICIT REAL*8 (A-H,0-Z) COMMON /C2 / DZ(75) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C7 / Nl,N2,M1,M2,Z1,Z2,X1,X2,N10,N12,M10,M12 COMMON /C8 / ROWS,ROWL COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /C13/ FSLANT,FHORIZ,TAMB COMMON /C14/ NTOTAL,MTOTAL,NX COMMON /C21/ FS(75,100),FRACTN,NREPET C WRITE(6,5 0 0)CARBON 500 FORMAT(5X,'HEAT FLOW MODEL TEMPERATURE PREDICTIONS' -,/,5X,'FOR SOLIDIFICATION OF A PURE IRON-C ALLOY' -,/,5X, ' ' -,/,5X, ' ' -,///,4X,'WEIGHT % OF CARBON IN THE STEEL : ',F6.3 -,/,4X, ' ' ) C WRITE(6,502)TLIQID,TSOLID 502 FORMAT(/,5X,'MODEL PARAMETERS : LIQUIDUS TEMP. : ' -,F5.0,4X -,'DEG C ,/,5X, ' SOLIDUS TEMP : ' -,F5.0,4X,'DEG C ) WRITE(6,50 3)SUHEAT 503 FORMAT(25X,'SUPERHEAT : ' ,F5.0,4X,'DEG C' ) WRITE(6,504)TPOUR 504 FORMATC25X,'POURING TEMP. : ',F5.0,4X,'DEG C ) WRITE(6,505)TAMB 505 FORMAT(25X,'AMBIENT TEMP : ',F5.0,4X,'DEG C ) C 209 WRITE(6,506)QMOLD 506 FORMAT(23X,'MOLD HEAT FLUX :',F8.2,1X 1,'CAL/SEC*SQ.CM') WRITE(6,508)FRACTN 508 FORMAT(18X,'RIGIDITY CRITERION IS >',F3.1,1X 1,'FRACTION OF SOLID') WRITE(6,511)HEAT 511 FORMAT(1 OX,'LATENT HEAT OF SOLIDIFICATION :',F10.5,4X 1,'CAL/GM',/) WRITE(6,512)N1,N2 512 FORMAT(/,5X,'DEFINITION OF THE CONTINUAM : N1: ',14,/ -,5X, ' : N2: ' ,14) WRITE(6,513)M1,M2,Z1,Z2,X1,X2 513 FORMAT(38X,'M1: ',I4,/,38X 1,'M2: ',14,/ 1,38X,'Z1: ',F6.4,/,38X,'Z2: ',F6.4,/,38X,'X1: ',F6.4,/,38X 1,'X2: ',F6.4,/) WRITE(6,514)DX 514 FORMAT(3 8X,'DX:' ,F10.7) DO 516 N=1,N1 WRITE(6,515)N,DZ(N) 515 FORMAT(34X,'DZ(' ,I 4 , ' ) : ' ,F10.7) 516 CONTINUE C ' WRITE(6,517) 517 FORMAT(/,5X,'TIME STEPS EMPLOYED IN THE PROGRAM :',/ -,5X,' ' ) WRITE(6,518)DT 518 F0RMAT(31X,'TIME STEP:',F10.8) WRITE(6,519)END 519 FORMAT(19X,'TOTAL SIMULATION TIME:',F10.8,/) C WRITE(6,520)FSLANT,FHORIZ 520 FORMAT(9X,'VIEW FACTOR FOR INCLINED NODES :',F5.3,/ 1,7X,'VIEW FACTOR FOR HORIZONTAL NODES :' , F 5 . 3 , / / / / 1,'DETALS OF THE POLYNOMIALS FITTED TO THE THERMAL 1 CONDUCTIVITY' 1,/,8X,'AND S P E C I F I C HEAT DATA OBTAINED FROM LITERATURE',/ 1 i C C RETURN END C C SUBROUTINE SOLVE ACTUALLY DEVELOPS COEFFICENT ARRAYS FOR THE C THE GRID AND CALLS TRIDAG TO SOLVE THEM c . C SUBROUTINE SOLVE(*,*) C IMPLICIT REAL*8 (A-H,0~Z) COMMON /C1 / TT(75,100),TS(75,100) 210 COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON COMMON C £*** C C 30 20 /C2 /C3 /C5 /C6 /C7 /C8 /C9 /C10/ /CI 3/ /CI 4/ /C21/ / / / / / / / DZ(7.5) AK(75,100),CP(75,100),ROW(75,100) DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L BETA(7 600),GAMMA(7600),TPRIME(7600) N1,N2,M1,M2,Z1,Z2,X1,X2,N10,N12,M10,M12 ROWS,ROWL MTIME S,NTIME S,END,THEEND,TIME,DT CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD FSLANT,FHORIZ,TAMB NTOTAL,MTOTAL,NX FS(75,100),FRACTN,NREPET 40 40 40 8 10 1 0 10 10 10 **** COMPUTE TEMPERATURES OVER THE FIRST HALF TIME STEP **** IMPLICIT IN THE Z-DIRECTION DO 200 N=1,NTOTAL DO 300 M=1,MTOTAL CALL FIND(N , M,1) 0 CONTINUE 0 CONTINUE CALL TRIDAG(1,L) L0=1 DO 400 N=1,NTOTAL DO 402 M=1,MTOTAL IF(TS(N,M).EQ.0.OD0) GO TO 401 TT(N,M)=TPRIME(L0) L0=L0+1 GO TO 402 1 TT(N,M)=0.0D0 2 CONTINUE 0 CONTINUE WRITE(10,89) 9 FORMAT(//,'AFTER THE FIRST HALF TIME STEP') WRITE(10,101)TIME 1 FORMAT(1 HO,1 OH TIME =,F10.7) WRITE(10,102) 2 FORMAT(1 HO,5H T-1 ,5H T-2 ,5H T-3 ,5H T-4 ,5H T-5 A,5H T-6,5H T-7 A,5H T-8 ,5H T-9 ,5H T-10,5H T-11,5H T-12,5H T-13 A,5H T-14,5H T-15 AH T-16,5H T-17,5H T-18,5H T-19,5H T-20,5H T-21 A,5H T-22,5H T-23) DO 104 N=1,NTOTAL WRITE(10,103) (TT(N,J),J=1,NX) 3 FORMAT(23F5.0) 4 CONTINUE WRITE(10,105) 5 FORMAT(1 HO,5H F-1 ,5H F-2 ,5H F-3 ,5H F-4 ,5H F-5 A,5H F-6 ,5H F-7 21 1 A,5H F-8 , 5H F-9 ,5H F-10,5H F-11,5H F-12,5H F-13 A,5H F-14,5H F-15,5 AH F-16,5H F-17,5H F-18/5H F~19,5H F-20,5H F-21 A,5H F-22,5H F-23) C DO 107 N=1,NTOTAL WRITE(10,106) (FS(N,J),J=1,NX) 106 FORMAT(23F5.3) 107 CONTINUE C L=0 C c C****** COMPUTE TEMPERATURES AT THE END OF A WHOLE TIME STEP c * * * * * * IMPLICIT IN X-DIRECTION C DO 700 M=1,MTOTAL DO 500 N=1,NTOTAL CALL FIND(N,M,0) 500 CONTINUE 700 CONTINUE CALL TRIDAG(1,L) L1 = 1 DO 600 M=1,MTOTAL DO 602 N=1,NTOTAL IF(TT(N,M).EQ.0.0D0)GO TO 601 TS(N,M)=TPRIME(L1) L1=L1+1 GO TO 602 601 TS(N,M)=0.0D0 602 CONTINUE 600 CONTINUE L=0 WRITE(10,79) 79 FORMAT(/,'AFTER THE WHOLE TIME STEP') WRITE(10,101)TIME WRITE(10,102) C DO 201 N=1,NTOTAL WRITE(10,103) (TS(N,J),J=1,NX) 201 CONTINUE C WRITE(10,105) C DO 202 N=1,NTOTAL WRITE(10,106) (FS(N,J),J=1,NX) 202 CONTINUE C C******* RESET ALL THERMO PHYSICAL PROPERTIES FOR EACH NODE C DO 222 M=1,MTOTAL DO 222 N=1,NTOTAL CALL CHECK(N,M,TS(N,M)) 212 222 CONTINUE Q************** * ********* ********* * c MTIMES=MTIMES+1 IF(MTIMES.EQ.NTIMES) GOTO 3000 C RETURN 1 C 3000 MTIMES=0 WRITE(6,802) 802 FORMAT(///,1 OX,'THE END',//) RETURN 2 END C C SUBROUTINE CHECK GIVES THE RIGHT THERMO-PHYSICAL PROPERTIES C AS PER THEIR TEMPERATURES c  SUBROUTINE CHECK(N,M,TNODE) C IMPLICIT REAL*8 (A-H fO-Z) COMMON /C3 / AK(75,100),CP(75,100),ROW(75,100) COMMON /C8 / ROWS,ROWL COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /C21/ FS(75,100),FRACTN,NREPET C IF(TNODE.GT.0.0D0) GO TO 10 RETURN 10 CONTINUE C IF(TNODE.LT.TLIQID)GO TO 1 CP(N,M)=CPSOL(TNODE) ROW(N,M)=ROWL AK(N,M)=7*(AKSOL(TNODE)) GO TO 3 C 1 IF(TNODE.LT.TSOLID)GO TO 2 CP(N,M)=CPSOL(TNODE) ROW(N,M)=(ROWS+ROWL)/2.0D0 AK(N,M)=(1+(6*((1-FS(N,M))**2)))*AKSOL(TNODE) GO TO 3 C 2 CP(N,M)=CPSOL(TNODE) ROW(N,M)=ROWS AK(N,M)=AKSOL(TNODE) C 3 RETURN END C C SUBROUTINE OUTPUT PRINTS OUT TIME STEMPERATURES IN F I L E 6 OR 7 C AS DESIRED c  SUBROUTINE OUTPUT(KFILE) C 213 IMPLICIT REAL*8 (A-H,0-Z) COMMON /CI / T T ( 7 5 , 1 0 0 ) , T S ( 7 5 , 1 0 0 ) COMMON /C2 / DZ(75) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON / C U / NTOTAL,MTOTAL,NX COMMON /C20/ SHE L L F ( 7 5 ) , X M I N S C ( 7 5 ) , D I S T ( 7 5 ) COMMON /C21/ FS(75,100),FRACTN,NREPET WRITE(KFILE,100)TIME 100 FORMAT(1 HO,1 OH TIME =,F10.7) WRITE(KFILE,1000) 1000 FORMAT(1 HO,5H T-1 ,5H T-2 ,5H T-3 ,5H T-4 A,5H T-5 ,5H T-6 ,5H T-7 A,5H T-8 ,5H T-9 ,5H T-10,5H T-11,5H T-12 A,5H T-13,5H T-14,5H T-15,5 AH T-16,5H T-17,5H T-18,5H T-19,5H T-20 A,5H T-21,5H T-22,5H T-23) C DO 200 N=1,NTOTAL WRITE(KFILE,2000) (TS(N,J),J=1,NX) 2000 FORMAT(23F5.0) 200 CONTINUE C WRITE(KFILE,3000) 3000 FORMAT(1 HO,5H F-1 ,5H F-2 ,5H F-3 ,5H F-4 A,5H F-5 ,5H F-6 ,5H F-7 A,5H F-8 ,5H F-9 ,5H F-10,5H F-11,5H F-12 A,5H F-13,5H F-14,5H F-15,5 AH F-16,5H F-17,5H F-18,5H F-19,5H F-20 A,5H F-21,5H F-22,5H F-23) C DO 300 N=1,NTOTAL WRITE(KFILE,4000) (FS(N,J),J=1,NX) 4000 FORMAT(23F5.3) 300 CONTINUE C RETURN END C C SUBROTINE FIND DESIGNATES THE TYPE OF NODE AND FORMULATES THE C COEFFICENT ARRAYS NEEDED FOR TRIDAG MATRIX c _ C SUBROUTINE FlND(N,M,KSTEP) C IMPLICIT REAL*8 (A-H,0-Z) COMMON /CI / T T ( 7 5 , 1 0 0 ) , T S ( 7 5 , 1 0 0 ) COMMON /C2 / DZ(75) • COMMON /C3 / AK(75,100),CP(75,100),ROW(75,100) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C7 / N1,N2,M1,M2,Z1,Z2,X1,X2,N10,N12,M10,M12 COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /C13/ FSLANT,FHORIZ,TAMB 2 1 4 COMMON / C 1 4 / N T O T A L , M T O T A L , N X C C * * * * * * T O F I N D T H E T Y P E O F N O D E C I F ( N - 1 ) 1 , 1 , 1 0 0 C 1 I F ( M - M 1 0 ) 1 0 0 0 , 1 0 0 0 , 2 2 I F ( M - M 1 2 ) 1 1 0 0 , 1 2 0 0 , 3 3 I F ( M - M T O T A L ) 1 3 0 0 , 1 4 0 0 , 1 4 0 0 C 100 I F ( N - N I O ) 1 0 , 2 0 0 , 3 0 0 C 10 M C 1 = M 1 - N M C 2 = M C 1 + 2 I F ( M - 1 ) 1 0 0 0 , 1 0 0 0 , 1 1 11 I F ( M - M C 1 ) 1 0 0 0 , 1 0 0 0 , 1 2 12 I F ( N . G T . 2 ) G O T O 15 I F ( M - M C 2 ) 1 5 0 0 , 1 6 0 0 , 1 3 13 I F ( M - M T O T A L ) 1 7 0 0 , 1 8 0 0 , 1 8 0 0 15 C O N T I N U E I F ( M - M C 2 ) 1 5 0 0 , 1 9 0 0 , 1 6 16 I F ( M - M T O T A L ) 2 0 0 0 , 2 1 0 0 , 2 1 0 0 C C C C C 2 0 0 C O N T I N U E C I F ( M - 1 ) 1 0 0 0 , 1 0 0 0 , 2 1 21 I F ( M - 3 ) 1 5 0 0 , 1 9 0 0 , 2 2 22 I F ( M - M T O T A L ) 2 0 0 0 , 2 1 0 0 , 2 1 0 0 C 3 0 0 I F ( N - N 1 2 ) 3 1 , 4 0 0 , 5 0 0 C 31 I F ( M - 2 ) 2 2 0 0 , 2 3 0 0 , 3 2 32 I F ( M - M T O T A L ) 2 0 0 0 , 2 1 0 0 , 2 1 0 0 C 4 0 0 C O N T I N U E C I F ( M - 2 ) 2 4 0 0 , 2 5 0 0 , 4 1 41 I F ( M - M T O T A L ) 2 0 0 0 , 2 1 0 0 , 2 1 0 0 C 5 0 0 I F ( N - N T O T A L ) 5 1 , 6 0 0 , 6 0 0 C 51 I F ( M - 2 ) 2 6 0 0 , 2 5 0 0 , 5 2 5 2 I F ( M - M T O T A L ) 2 0 0 0 , 2 1 0 0 , 2 1 0 0 C 6 0 0 C O N T I N U E C I F ( M-2) 2700,2800,61 61 I F ( M - M T O T A L ) 2900,3000,3000 C * * * * * * * T 0 C A L C U L A T E T H E M A T R I X D I A G N O L S F O R T H E N O D E 215 1000 IF(KSTEP.EQ.1) GO TO 1001 CALL DUMMY(TT(N-,M) ,0, 699) 1001 CALL DUMMY(TS(N,M),1,599) 1100 IF(KSTEP.EQ.1) GO TO 1101 CALL TOPTRI(TT(N,M),TT(N,M+1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N+1,M),AK(N,M+1),DZ(N),0,699,N,M) 1101 CALL TOPTRI(TS(N,M),TS(N+1,M),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N+1,M),AK(N,M+1),DZ(N),1,699,N,M) 1200 IF(KSTEP.EQ.1) GO TO 1201 CALL TBONE(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M) 1,CP(N,M) 1,AK(N,M),AK(N,M+1),AK(N,M-1),AK(N+1,M),DZ(N),0,699,N,M) 1201 CALL TBONE(TS(N,M),TS(N+1,M),TS(N+1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N,M+1),AK(N,M-1),AK(N+1,M),DZ(N),1,699,N,M) 1300 IF(KSTEP.EQ.1) GO TO 1301 CALL TB(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N,M+1),AK(N,M-1),AK(N+1,M),DZ(N),0,599,N,M) 1301 CALL TB(TS(N,M),TS(N+1,M),TS(N+1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N,M+1),AK(N,M-1),AK(N+1,M),DZ(N),1,699,N,M) 1400 IF(KSTEP.EQ.1) GO TO 1401 CALL TLCORN(TT(N,M),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N+1,M),AK(N,M-1),DZ(N),0,599,N,M) 1401 CALL TLCORN(TS(N , M),TS(N+1,M),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N+1,M),AK(N,M-1),DZ(N),1,699,N,M) 1500 IF(KSTEP.EQ.1) GO TO 1501 CALL TRIANG(TT(N,M),TT(N,M+1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N,M+1),AK(N+1,M),DZ(N-1),DZ(N),0,699,N,M) 1501 CALL TRIANG(TS(N,M),TS(N+1,M),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N,M+1),AK(N+1,M),DZ(N-1),DZ(N),1,599,N,M) 1600 IF(KSTEP.EQ.1) GO TO 1601 CALL INONE(TT(N,M),TT(N,M+1),TT(N,M~1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M~1) 1,DZ(N-1),DZ(N),0,699,N,M) 1601 CALL INONE(TS(N,M),TS(N+1,M),TS(N-1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),1,699,N,M) 1700 IF(KSTEP.EQ.1) GO TO 1701 CALL INTOP(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),0,699,N,M) 1701 CALL INTOP(TS(N,M),TS(N+1,M),TS(N-1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M~1) 216 1,DZ(N-1),DZ(N),1,699,N,M) 1800 IF(KSTEP.EQ.1) GO TO 1801 CALL RBONE(TT(N,M),TT(N,M-1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M-1),DZ(N-1),DZ(N),0,599, 1801 CALL RBONE(TS(N,M),TS(N-1,M),TS(N+1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M-1),DZ(N-1),DZ(N),1,699, 1900 IF(KSTEP.EQ.1) GO TO 1901 CALL INNEXT(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),0,599,DZ(N-2),N,M) 1901 CALL INNEXT(TS(N,M),TS(N+1,M),TS(N~ 1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),1,599,DZ(N-2),N,M) 2000 IF(KSTEP.EQ.1) GO TO 2001 CALL IN(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M~1) 1,DZ(N-1),DZ(N),0,599,N,M) 2001 CALL IN(TS(N,M),TS(N+1,M),TS(N-1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),1,699,N,M) 2100 IF(KSTEP.EQ.1) GO TO 2101 CALL RB(TT(N,M),TT(N,M-1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M-1),DZ(N-1),DZ(N),0,599, 2101 CALL RB(TS(N,M),TS(N-1,M),TS(N+1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M-1),DZ(N-1),DZ(N),1,599, 2200 IF(KSTEP.EQ.1) GO TO 2201 CALL BOTTRI(TT(N,M),TT(N,M+1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N+1,M),AK(N,M+1),DZ(N-1),DZ(N),0,599,N,M) 2201 CALL BOTTRI(TS(N,M),TS(N+1,M),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N+1,M),AK(N,M+1),DZ(N-1),DZ(N),1,599,N,M) 2300 IF(KSTEP.EQ.1) GO TO 2301 CALL INBOT(TT(N,M).,TT(N,M+1 ) ,TT(N,M-1 ) ,ROW(N,M) ,CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),0,599,DZ(N-2),N,M) 2301 CALL INBOT(TS(N,M),TS(N+1,M),TS(N-1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),1,599,DZ(N-2),N,M) 2400 IF(KSTEP.EQ.1) GO TO 2401 CALL LBONE(TT(N,M),TT(N,M+1),TT(N,M+1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),DZ(N-1),DZ(N),0 1,699,DZ(N-2),N,M) 2401 CALL LBONE(TS(N,M),TS(N+1,M),TS(N-1,M),ROW(N,M),CP(N,M) 217 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),DZ(N-1),DZ(N),1 1,699,DZ(N-2),N,M) C 2500 IF(KSTEP.EQ.1) GO TO 2501 CALL INLEF(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M) ,AK(N-1 ,M) ,AK(N+1 ,M) ,AK(N,M+1),AK(N,M~1) 1,DZ(N-1),DZ(N),0,599,N,M) 2501 CALL INLEF(TS(N,M),TS(N+1,M),TS(N-1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),AK(N,M-1) 1,DZ(N-1),DZ(N),1,599,N,M) C 2600 IF(KSTEP.EQ.1) GO TO 2601 CALL LB(TT(N,M),TT(N,M+1),TT(N,M+1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),DZ(N-1),DZ(N) 1,0,699,N,M) 2601 CALL LB(TS(N,M),TS(N+1,M),TS(N-1,M),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N+1,M),AK(N,M+1),DZ(N-1),DZ(N) 1,1,699,N,M) C 2700 IF(KSTEP.EQ.1) GO TO 2701 CALL BLCORN(TT(N,M),TT(N,M+1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N,M+1),DZ(N-1),0,699,N,M) 2701 CALL BLCORN(TS(N,M),TS(N-1,M),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N,M+1),DZ(N-1),1,599,N,M) C 2800 IF(KSTEP.EQ.1) GO TO 2801 CALL BBONE(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N,M+1),AK(N,M-1),DZ(N-1),0,699,N,M) 2801 CALL BBONE(TS(N,M),TS(N-1,M),TS(N-1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N,M+1),AK(N,M-1),DZ(N-1),1,599,N,M) C 2900 IF(KSTEP.EQ.1) GO TO 2901 CALL BB(TT(N,M),TT(N,M+1),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N,M+1),AK(N,M-1),DZ(N-1),0,699,N,M) 2901 CALL BB(TS(N,M),TS(N-1,M),TS(N-1,M) 1,ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N,M+1),AK(N,M-1),DZ(N-1),1,599,N,M) C 3000 IF(KSTEP.EQ.1) GO TO 3001 CALL BRCORN(TT(N,M),TT(N,M-1),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N-1,M),AK(N,M-1),DZ(N-1),0,599,N,M) 3001 CALL BRCORN(TS(N,M),TS(N-1,M),ROW(N,M),CP(N,M) 1,AK(N,M),AK(N,M-1),AK(N-1,M),DZ(N-1),1,599,N,M) C 99 RETURN END C C SUBROUTINE TRIDAG SOLVES THE SYSTEM OF LINEAR SIMULTANEOUSONS C EQUATIONSHAVING A TRIDIAGONAL COEFFICENT MATRIX.THE EQUATIONS C ARE NUMBERED FROM 'NFIRST' THRUOGH 'NLAST' AND THEIR SUBDIAGNOL C DIAGNOL AND SUPERDIAGNOL COEFFICENTS ARE STORED IN ARRAYS C 'DBOT','D'AND 'DTOP'.THE COMPUTED SOLUTION VECTOR 218 C 'TPRIME(NFIRST)' 'TPRIME(NLAST)' IS STORED IN THE ARRAY 'TP c  C SUBROUTINE TRIDAG(NFIRST,NLAST) C IMPLICIT REAL*8 (A-H,0-Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C6 / BETA(7600),GAMMA(7600),TPRIME(7600) C C******** COMPUTE INTERMEDIATE ARRAYS BETA AND GAMMA C BETA(NFIRST)=DMID(NFIRST) GAMMA(NFIRST)=R(NFIRST)/BETA(NFIRST) NEXT=NFIRST+1 DO 100 I=NEXT,NLAST B E T A ( I ) = D M I D ( I ) - D B O T ( I ) * D T O P ( I - 1 ) / B E T A ( l ~ 1 ) GAMMA(I)=(R(I)-DBOT(I)*GAMMA(I-1))/BETA(I) 100 CONTINUE C C******** COMPUTE FINAL SOLUTION VECTOR C TPRIME(NLAST)=GAMMA(NLAST) LAST=NLAST-NFIRST DO 200 K=1,LAST J=NLAST-K TPRIME(J)=GAMMA(J)-DTOP(J)*TPRIME(J+1)/BETA(J) 200 CONTINUE C RETURN END C SUBROUTINE DUMMY(T,KSTEP,*) c _ IMPLICIT REAL*8 (A-H,0"Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD C IF(KSTEP.EQ.1)GO TO 100 C******************* FRIRST HALF TIME STEP T=0.0D0 GO TO 200 C******************* NEXT HALF TIME STEP 100 T=0.0D0 C 200 RETURN 1 C END C SUBROUTINE IN(T,TLATER,TBEFOR,ROW,CP 1,AK,AKTOP,AKBOT,AKRIG,AKLEF,DZ0,DZ1,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A-H,0~Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(760Q),DX,L COMMON / C 9 / MTIMES,NTIMES,END,THEEND,TIME,DT 219 COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD C L=L+1 VOL=DX*((DZ0+DZ1)/2) A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) B=DX*(AK+AKTOP)/(2*DZ0) C=DX*(AK+AKBOT)/(2*DZ1) D=(DZ0+DZ1)*(AK+AKRIG)/(4*DX) E=(DZ0+DZ1)*(AK+AKLEF)/(4*DX) C IF(KSTEP.EQ.1)GO TO 100 ( 2 * * * * * * * * * * * * * * * * * * * FIRST HALF TIME STEP DMID(L)=A+B+C DTOP(L)=-C DBOT(L)=-B R(L)=(T*(A-D-E))+(D*TLATER)+(E*TBEFOR) C GO TO 200 ( 2 * * * * * * * * * * * * * * * * * * * NEXT HALF TIME STEP 100 DMID(L)=A+D+E DTOP(L)=-D DBOT(L)=-E R( L ) = ( T * ( A - B - C ) ) + ( C * T L A T E R ) + ( B * T B E F O R ) C 2 00 RETURN 1 END C SUBROUTINE I NONE(T,TLATER,TBEFOR,ROW,CP 1,AK,AKTOP,AKBOT,AKRIG,AKLEF,DZ0,DZ1,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A-H,0-Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD C L=L+1 VOL=DX*((DZ0+DZ1)/2) A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) B=(3*DX*(AK+AKTOP))/(5*DZ0) C=DX*(AK+AKBOT)/(2*DZ1) . D=(DZ0+DZ1)*(AK+AKRIG)/(4*DX) E=(3*(DZ0+DZ1)*(AK+AKLEF) ) / ( 10*DX) C IF(KSTEP.EQ.1)GO TO 100 C******************* FIRST HALF TIME STEP DMID(L)=A+B+C DTOP(L)=-C DBOT(L)=-B R(L)=(T*(A-D-E))+(D*TLATER)+(E*TBEFOR) C GO TO 200 C******************* NEXT HALF TIME STEP 2 2 0 100 D M I D ( L ) = A + D + E D T O P ( L ) = - D D B O T ( L ) = - E R ( L ) = ( T * ( A - B - C ) ) + ( C * T L A T E R ) + ( B * T B E F O R ) C 2 0 0 R E T U R N 1 E N D C C S U B R O U T I N E I N N E X T ( T , T L A T E R , T B E F O R , R O W , C P 1 , A K , A K T O P , A K B O T , A K R I G , A K L E F , D Z 0 , D Z 1 , K S T E P , * , D Z T , N , M ) C  I M P L I C I T R E A L * 8 ( A - H , 0 ~ Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D C L = L + 1 V O L = D X * ( ( D Z 0 + D Z 1 ) / 2 ) A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = ( 3 * D X * ( A K + A K T O P ) ) / ( D Z T + ( 4 * D Z 0 ) ) C = D X * ( A K + A K B O T ) / ( 2 * D Z 1 ) D = ( D Z 0 + D Z 1 ) * ( A K + A K R I G ) / ( 4 * D X ) E = ( 3 * ( D Z 0 + D Z 1 ) * ( A K + A K L E F ) ) / ( 1 0 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 Q******************* P I R S T H A L F T I M E S T E P D M I D ( L ) = A + B + C D T O P ( L ) = - C D B O T ( L ) = - B R ( L ) = ( T * ( A - D - E ) ) + ( D * T L A T E R ) + ( E * T B E F O R ) C G O T O 2 0 0 Q******************* N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A + D + E D T O P ( L ) = - D D B O T ( L ) = - E R ( L ) = ( T * ( A - B - C ) ) + ( C * T L A T E R ) + ( B * T B E F O R ) 2 0 0 R E T U R N 1 E N D C S U B R O U T I N E I N T O P ( T , T L A T E R , T B E F O R , R O W , C P 1 , A K , A K T O P , A K B O T , A K R I G , A K L E F , D Z 0 , D Z 1 , K S T E P , * , N , M ) C  I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D C L = L + 1 V O L = D X * ( ( D Z 0 + D Z 1 ) / 2 ) A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) 221 C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = ( 2 * D X * ( A K + A K T O P ) ) / ( 3 * D Z 0 ) C = D X * ( A K + A K B O T ) / ( 2 * D Z 1 ) D = ( D Z 0 + D Z 1 ) * ( A K + A K R I G ) / ( 4 * D X ) E = ( D Z 0 + D Z 1 ) * ( A K + A K L E F ) / ( 4 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 Q******************* F I R S T H A L F T I M E S T E P D M I D ( L ) = A + B + C D T O P ( L ) = - C D B O T ( L ) = - B R ( L ) = ( T * ( A - D - E ) ) + ( D * T L A T E R ) + ( E * T B E F O R ) C G O T O 2 0 0 Q******************* N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A + D + E D T O P ( L ) = - D D B O T ( L ) = - E R ( L ) = ( T * ( A - B - C ) ) + ( C * T L A T E R ) + ( B * T B E F O R ) 2 0 0 R E T U R N 1 E N D C S U B R O U T I N E I N B O T ( T , T L A T E R , T B E F O R , R O W , C P 1 , A K , A K T O P , A K B O T , A K R I G , A K L E F , D Z 0 , D Z 1 , K S T E P , * , D Z T , N , M ) C  I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D C L = L + 1 V O L = D X * ( ( D Z 0 + D Z 1 ) / 2 ) A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = ( 3 * D X * ( A K + A K T O P ) ) / ( 4 * ( D Z 0 + D Z T ) ) C = D X * ( A K + A K B O T ) / ( 2 * D Z 1 ) D = ( D Z 0 + D Z 1 ) * ( A K + A K R I G ) / ( 4 * D X ) E = ( 6 * ( D Z 0 + D Z 1 ) * ( A K + A K L E F ) ) / ( 1 7 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 Q******************* F I R S T H A L F T I M E S T E P D M I D ( L ) = A + B + C D T O P ( L ) = - C D B O T ( L ) = - B R ( L ) = ( T * ( A - D - E ) ) + ( D * T L A T E R ) + ( E * T B E F O R ) C G O T O 2 0 0 Q******************* N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A + D + E D T O P ( L ) = - D D B O T ( L ) = - E R ( L ) = ( T * ( A - B - C ) ) + ( C * T L A T E R ) + ( B * T B E F O R ) 2 0 0 R E T U R N 1 E N D 2 2 2 C . S U B R O U T I N E I N L E F ( T , T L A T E R , T B E F O R , R O W , C P 1 , A K , A K T O P , A K B O T , A K R I G , A K L E F , D Z O , D Z 1 , K S T E P , * , N , M ) C  I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D C L = L + 1 V O L = D X * ( ( D Z 0 + D Z 1 ) / 2 ) A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = D X * ( A K + A K T O P ) / ( 2 * D Z 0 ) C = ( D X * ( A K + A K B O T ) ) / . ( 2 * D Z 0 ) D = ( D Z 0 + D Z 1 ) * ( A K + A K R I G ) / ( 4 * D X ) E = ( D Z 0 + D Z 1 ) * ( A K + A K L E F ) / ( 3 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 Q******************* F I R S T H A L F T I M E S T E P D M I D ( L ) = A + B + C D T O P ( L ) = - C D B O T ( L ) = - B R ( L ) = ( T * ( A - D - E ) ) + ( D * T L A T E R ) + ( E * T B E F O R ) C G O T O 2 0 0 Q******************* N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A + D + E D T O P ( L ) = - D D B O T ( L ) = - E R ( L ) = ( T * ( A - B - C ) ) + ( C * T L A T E R ) + ( B * T B E F O R ) 2 0 0 R E T U R N 1 E N D C S U B R O U T I N E T R I A N G ( T , T L A T E R , R O W , C P 1 , A K , A K R I G , A K B O T , D Z O , D Z 1 , K S T E P , * , N , M ) C  I M P L I C I T R E A L * 8 ( A - H F O - Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D COMMON / C I 3 / F S L A N T , F H O R I Z , T A M B C L = L + 1 V O L = D X * ( D Z 0 + D Z 1 ) / 4 . 0 D 0 A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) A R E A = D S Q R T ( ( ( ( D Z 0 + D Z 1 ) / 2 ) * * 2 ) + D X * * 2 ) H C = ( 2 / ( 1 0 . 0 D 0 * * 4 ) ) * ( ( T - T A M B ) * * ( 1 / 4 ) ) H R = ( 1 . 3 3 5 D 0 / ( 1 0 . 0 D 0 * * 1 2 ) ) * F S L A N T * ( ( ( T + 2 7 3 ) * * 4 ) - ( ( T A M B + 2 7 3 ) * 1 / ( T - T A M B ) H = H C + H R R A T I O = ( ( D Z 0 + D Z 1 ) / ( 2 * D X ) ) * 0 . 0 1 7 4 5 3 D 0 T H E T A = D A T A N ( R A T I O ) 223 B=-H*AREA*(DCOS(THETA)) C=(3*DX*(AK+AKBOT))/(DZ0+(4*DZ1)) D=(5*(DZ0+DZ1)*(AK+AKRIG))/(24*DX) E=-H*AREA*(DSIN(THETA)) C IF(KSTEP.EQ.1)GO TO 100 Q******************* FIRST HALF TIME STEP DMID(L)=A-B+C DTOP(L)=-C DBOT(L)=0.0D0 R(L)=(T*(A+E-D))+(D*TLATER)-(TAMB*(B+E)) C GO TO 200 Q******************* NEXT HALF TIME STEP 100 DMID(L)=A-E+D DTOP(L)=-D DBOT(L)=0.0D0 R(L)=(T*(A-C+B))+(C*TLATER)-(TAMB*(B+E)) 200 RETURN 1 END C SUBROUTINE TOPTRI(T,TOTHER,ROW,CP 1,AK,AKBOT,AKRIG,DZ1,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A-H,0~Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /CI 3/ FSLANT,FHORIZ,TAMB C VOL=DX*DZ1*3/8 A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) AREA=DSQRT(((DX/2)**2)+((DZ1/2)**2)) H C = ( 2 / ( 1 0 . 0 D 0 * * 4 ) ) * ( ( T - T A M B ) * * ( 1 / 4 ) ) HR=(1.335D0/(10.0D0**12))*FSLANT*(((T+273)**4)-((TAMB+273)* 1/(T-TAMB) H=HR+HC RATIO=(DZ1/DX)*0.017453DO THETA=DATAN(RATIO) B=-H*((AREA*(DCOS(THETA)))+(DX/2)) C=-H*AREA*(DSIN(THETA)) D=(6*DZ1*(AK+AKRIG))/(23*DX) E=(12*DX*(AK+AKBOT))/(17*DZ1) IF(KSTEP.EQ.1)GO TO 100 Q******************* FIRST HALF TIME STEP L=L+1 DMID(L)=A-B+E DTOP(L)=-E DBOT(L)=0.0D0 R(L)=(T*(A+C-D))+(D*TOTHER)-(TAMB*(B+C)) 224 C GO TO 200 C********* * * * * * * * * * * NEXT HALF TIME STEP 100 L=L+1 DMID(L)=A-C+D DTOP(L)=-D DBOT(L)=0.0D0 R(L)=(T*(A+B-E))+(E*TOTHER)-(TAMB*(B+C)) 200 RETURN 1 END C SUBROUTINE BOTTRI(T,TOTHER,ROW,CP 1,AK,AKBOT,AKRIG,DZO,DZ1,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A-H,0~Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /C13/ FSLANT,FHORIZ,TAMB C VOL=DX*(DZ0+(2*DZ1))/8.0D0 A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) AREA=DSQRT(((DX/2)**2)+((DZ0/2)**2)) H C = ( 2 / ( 1 0 . 0 D 0 * * 4 ) ) * ( ( T - T A M B ) * * ( 1 / 4 ) ) HR=(1.335D0/(10.0D0**12))*FSLANT*(((T+273)**4)-((TAMB+273)* 1/(T-TAMB) H=HR+HC RATIO=(DZ0/DX)*0.017453D0 THETA=DATAN(RATIO) B=-H*AREA*(DCOS(THETA)) C=-H*AREA*(DSIN(THETA)) D=-(QMOLD*DZ1)/2 E=(6*DX*(AK+AKBOT))/((21*DZ1)+(2*DZ0)) F=(6*(DZ0+DZ1)*(AK+AKRIG))/(17*DX) C IF(KSTEP.EQ.1)GO TO 100 C******************* FIRST HALF TIME STEP L=L+1 DMID(L)=A-B+E DTOP(L)=-E DBOT(L)=0.0D0 R(L)=(T*(A+C-F))+(F*TOTHER)-(TAMB*(B+C))+D C GO TO 200 C******************* NEXT HALF TIME STEP 100 L=L+1 DMID(L)=A-C+F DTOP(L)=-F DBOT(L)=0.0D0 R(L)=(T*(A+B-E))+(E*TOTHER)-(TAMB*(B+C))+D 200 RETURN 1 C END 225 C SUBROUTINE TB(T,TLATER,TBEFOR,ROW,CP 1,AK,AKRIG,AKLEF,AKBOT,DZ1,KSTEP,*,N,M) C IMPLICIT REAL*8 (A-H,0-Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /CI 3/ FSLANT,FHORIZ,TAMB L=L+1 C VOL=DX*DZ1/2.0DO A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) AREA=DX HC=( 2 / ( 1 0 . 0 D 0 * * 4 ) ) * ( ( T - T A M B ) * * ( 0 . 2 5 D 0 ) ) HR=(1.335D0/(10.0D0**12))*FHORIZ*(((T+273)**4)-((TAMB+273)* 1/(T-TAMB) H=HC+HR B=-H*AREA C=(2*DX*(AK+AKBOT))/(3*DZ1) D=DZ1*(AK+AKRIG)/(4*DX) E=DZ1*(AK+AKLEF)/(4*DX) C IF(KSTEP.EQ.1)GO TO 100 Q******************* FIRST HALF TIME STEP DMID(L)=A-B+C DTOP(L)=-C DBOT(L)=0.0D0 R(L)=(T*(A-D-E))+(D*TLATER)+(E*TBEFOR)-(B*TAMB) C GO TO 200 Q******************* NEXT HALF TIME STEP 100 DMID(L)=A-B+D+E DTOP(L)=-D DBOT(L)=-E R ( L ) = ( T * ( A - C ) ) + ( C * T L A T E R ) - ( B * T A M B ) 200 RETURN 1 END C SUBROUTINE TBONE(T,TLATER,TBEFOR,ROW,CP 1,AK,AKRIG,AKLEF,AKBOT,DZ1,KSTEP,*,N,M) C L ^ IMPLICIT REAL*8 (A-H,0-Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TlME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /C13/ FSLANT,FHORIZ,TAMB L=L+1 C VOL=DX*DZ1/2.0DO A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) AREA=DX f 2 2 6 H C = ( 2 / ( 1 0 . 0 D 0 * * 4 ) ) * ( ( T - T A M B ) * * ( 0 . 2 5 D 0 ) ) H R = ( 1 . 3 3 5 D 0 / ( 1 0 . 0 D 0 * * 1 2 ) ) * F H O R I Z * ( ( ( T + 2 7 3 ) * * 4 ) - ( ( T A M B + 2 7 3 ) * 1 / ( T - T A M B ) H=HC+HR B = - H * A R E A C = ( 2 * D X * ( A K + A K B 0 T ) ) / ( 3 * D Z 1 ) D = D Z 1 * ( A K + A K R I G ) / ( 4 * D X ) E = ( 6 * D Z 1 * ( A K + A K L E F ) ) / ( 2 3 * D X ) C I F ( K S T E P . E Q . 1 ) G 0 T O 100 C * * * * * * * * * * * * * * * * * * * F I R S T H A L F T I M E S T E P D M I D ( L ) = A - B + C D T O P ( L ) = - C D B O T ( L ) = 0 . 0 D 0 R ( L ) = ( T * ( A - D - E ) ) + ( D * T L A T E R ) + ( E * T B E F O R ) - ( B * T A M B ) C G O T O 2 0 0 C * * * * * * * * * * * * * * * * * * * N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A ~ B + D + E D T O P ( L ) = - D D B O T ( L ) = - E R ( L ) = ( T * ( A - C ) ) + ( C * T L A T E R ) - ( B * T A M B ) 2 0 0 R E T U R N 1 E N D C S U B R O U T I N E R B ( T , T B E F O R , T L A T E R , R O W , C P 1 , A K , A K T O P , A K B O T , A K L E F , D Z O , D Z 1 , K S T E P , * , N , M ) C  I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D C L = L + 1 V O L = D X * ( D Z 0 + D Z 1 ) / 4 . 0 D 0 A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = D X * ( A K + A K T O P ) / ( 4 * D Z 0 ) C = D X * ( A K + A K B O T ) / ( 4 * D Z 1 ) D = ( D Z 0 + D Z 1 ) * ( A K + A K L E F ) / ( 4 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 O * * * * * * * * * * * * * * * * * * * F I R S T H A L F T I M E S T E P D M I D ( L ) = A + B + C D T O P ( L ) = - C D B O T ( L ) = - B R ( L ) = ( T * ( A - D ) ) + ( D * T B E F O R ) C G O T O 2 0 0 C * * * * * * * * * * * * * * * * * * * N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A + D D T O P ( L ) = 0 . 0 D 0 D B O T ( L ) = - D R ( L ) = ( T * ( A - B - C ) ) + ( B * T B E F O R ) + ( C * T L A T E R ) 227 200 RETURN 1 END C SUBROUTINE RBONE(T,TBEFOR,TLATER,ROW,CP 1,AK,AKTOP,AKBOT,AKLEF,DZ0,DZ1,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A~H,0~Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TlME,DT COMMON /CIO/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD C L=L+1 VOL=DX*(DZ0+DZ1)/4.0D0 A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) B=DX*(AK+AKTOP)/(3*DZ 0) C=DX*(AK+AKBOT)/(4*DZ1) D=(DZ0+DZ1)*(AK+AKLEF)/(4*DX) C IF(KSTEP.EQ.1)GO TO 100 Q******************* FIRST HALF TIME STEP DMID(L)=A+B+C DTOP(L)=-C DBOT(L)=-B R(L)=(T*(A-D))+(D*TBEFOR) C GO TO 200 Q******************* NEXT HALF TIME STEP 100 DMID(L)=A+D DTOP(L)=0.0D0 DBOT(L)=-D R( L ) = ( T * ( A - B - C ) ) + ( B * T B E F O R ) + ( C * T L A T E R ) 200 RETURN 1 END C SUBROUTINE LB(T,TLATER,TBEFOR,ROW,CP 1,AK,AKTOP,AKBOT,AKRIG,DZ0,DZ1,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A-H,0~Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD L=L+1 C VOL=DX*(DZ0+DZ1)/4.0D0 A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) B=DX*(AK+AKTOP)/(4*DZ 0) C=DX*(AK+AKBOT)/(4*DZ1) D=(DZ0+DZ1)*(AK+AKRIG)/(3*DX) E=-(QMOLD*(DZ0+DZ1))/2 C IF(KSTEP.EQ.1)GO TO 100 C******************* FIRST HALF TIME STEP 228 DMID(L )=A+B+C DTOP(L)=-C DBOT(L)=-B R ( L ) = ( T * ( A - D ) ) + ( D * T L A T E R ) + E C GO TO 200 Q******************* NEXT HALF TIME STEP 100 D M I D ( L ) =A + D DTOP(L ) = - D DBOT(L)=0.0D0 R(L)=(T*(A-B-C))+(B*TBEFOR)+(C*TLATER)+E 2 00 RETURN 1 END C SUBROUTINE LBONE(T,TLATER,TBEFOR,ROW,CP 1,AK,AKTOP,AKBOT,AKRIG,DZO,DZ1,KSTEP,*,DZT,N,M) c _ IMPLICIT REAL*8 (A-H,0"Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD L=L+1 C VOL=DX*(DZ0+DZ1)/4.0D0 A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) B=(6*DX*(AK+AKTOP))/((21*DZ0)+(2*DZT)) C=DX*'(AK+AKBOT)/(4*DZ1 ) D=(DZ0+DZ1)*(AK+AKRIG)/(3*DX) E=-(QMOLD*(DZ0+DZ1))/2 C IF(KSTEP.EQ.1)GO TO 100 Q******************* p i R S T HALF TIME STEP DMID(L)=A+B+C DTOP(L)=-C DBOT(L)=-B R ( L ) = ( T * ( A - D ) ) + ( D * T L A T E R ) + E C GO TO 200 Q******************* NEXT HALF TIME STEP 100 DMID(L)=A+D DTOP(L)=-D DBOT(L)=0.0D0 R(L)=(T*(A-B-C))+(B*TBEFOR)+(C*TLATER)+E 200 RETURN 1 END C SUBROUTINE BB(T,TLATER,TBEFOR,ROW,CP 1 ,AK,AKTOP,AKRIG,AKLEF,DZ 0,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A-H,0"Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TIME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD 2 2 9 C L = L + 1 V O L = D X * D Z 0 / 2 . 0 D 0 A = R O W * C P * V O L / ( D T / 2 . O D O ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = D X * ( A K + A K T O P ) / ( 2 * D Z 0 ) C = D Z 0 * ( A K + A K R I G ) / ( 4 * D X ) D = D Z 0 * ( A K + A K L E F ) / ( 4 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 O * * * * * * * * * * * * * * * * * * * F I R S T H A L F T I M E S T E P D M I D ( L ) = A + B D T O P ( L ) = 0 . 0 D 0 D B O T ( L ) = - B R ( L ) = ( T * ( A - C - D ) ) + ( D * T B E F O R ) + ( C * T L A T E R ) C G O T O 2 0 0 O * * * * * * * * * * * * * * * * * * * N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A + C + D D T O P ( L ) = - C D B O T ( L ) = - D R ( L ) = ( T * ( A - B ) ) + ( B * T B E F O R ) 2 0 0 R E T U R N 1 E N D C S U B R O U T I N E B B O N E ( T , T L A T E R , T B E F O R , R O W , C P 1 , A K , A K T O P , A K R I G , A K L E F , D Z 0 , K S T E P , * , N , M ) C  I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D C L = L + 1 V O L = D X * D Z 0 / 2 . 0 D 0 A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = D X * ( A K + A K T O P ) / ( 2 * D Z 0 ) C = D Z 0 * ( A K + A K R I G ) / ( 4 * D X ) D = D Z 0 * ( A K + A K L E F ) / ( 3 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 O * * * * * * * * * * * * * * * * * * * F I R S T H A L F T I M E S T E P D M I D ( L ) = A + B D T O P ( L ) = 0 . 0 D 0 D B O T ( L ) = - B R ( L ) = ( T * ( A - C - D ) ) + ( D * T B E F O R ) + ( C * T L A T E R ) C G O T O 2 0 0 O * * * * * * * * * * * * * * * * * * * N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A + C + D D T O P ( L ) = - C D B O T ( L ) = - D R ( L ) = ( T * ( A - B ) ) + ( B * T B E F O R ) 2 3 0 2 0 0 R E T U R N 1 E N D C S U B R O U T I N E T L C O R N ( T , T O T H E R , R O W , C P , A K , A K B O T , A K L E F , D Z 1 , K S T E P , C  I M P L I C I T R E A L * 8 (A-H , 0-Z) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D COMMON / C 1 3 / F S L A N T , F H O R I Z , T A M B L = L + 1 C V O L = D X * D Z 1 / 4 . 0 D O A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) A R E A = D X / 2 H C = ( 2 / ( 1 0 . 0 D 0 * * 4 ) ) * ( ( T - T A M B ) * * ( 0 . 2 5 D 0 ) ) H R = ( 1 . 3 3 5 D 0 / ( 1 0 . 0 D 0 * * 1 2 ) ) * F H O R I Z * ( ( ( T + 2 7 3 ) * * 4 ) - ( ( T A M B + 2 7 3 ) * 1 / ( T - T A M B ) H=HC+HR B = - H * A R E A C = D X * ( A K + A K B O T ) / ( 3 * D Z 1 ) D = D Z 1 * ( A K + A K L E F ) / ( 4 * D X ) C I F ( K S T E P . E Q . 1 ) G O T O 100 Q******************* F I R S T H A L F T I M E S T E P D M I D ( L ) = A - B + C D T O P ( L ) = - C D B O T ( L ) = 0 . 0 D 0 R ( L ) = ( T * ( A - D ) ) + ( D * T O T H E R ) - ( B * T A M B ) C G O T O 2 0 0 Q******************* N E X T H A L F T I M E S T E P 100 D M I D ( L ) = A - B + D D T O P ( L ) = 0 . 0 D 0 D B O T ( L ) = - D R ( L ) = ( T * ( A - C ) ) + ( C * T O T H E R ) - ( B * T A M B ) 2 0 0 R E T U R N 1 E N D C S U B R O U T I N E B R C O R N ( T , T B E F O R , R O W , C P , A K , A K T O P , A K L E F , D Z 0 , K S T E P , C _ I M P L I C I T R E A L * 8 ( A - H , 0 " Z ) COMMON / C 5 / D M I D ( 7 6 0 0 ) , D T O P ( 7 6 0 0 ) , D B O T ( 7 6 0 0 ) , R ( 7 6 0 0 ) , D X , L COMMON / C 9 / M T I M E S , N T I M E S , E N D , T H E E N D , T I M E , D T COMMON / C 1 0 / C A R B O N , T L I Q I D , T S O L I D , S U H E A T , T P O U R , H E A T , Q M O L D C L = L + 1 V O L = D X * D Z 0 / 4 . 0 D 0 A = R O W * C P * V O L / ( D T / 2 . 0 D 0 ) C A L L E S T M A T ( T , K S T E P , V O L , A , N , M ) B = D X * ( A K + A K T O P ) / ( 4 * D Z 0 ) . C = D Z 0 * ( A K + A K L E F ) / ( 4 * D X ) C 231 IF(KSTEP.EQ.1)G0 TO 100 Q******************* FIRST HALF TIME STEP DMID(L)=A+B DTOP(L)=0.0D0 DBOT(L)=-B R ( L ) = ( T * ( A - C ) ) + ( C * T B E F O R ) C GO TO 200 Q******************* NEXT HALF TIME STEP 100 DMID(L)=A+C DTOP(L)=0.0D0 DBOT(L)=-C R ( L ) = ( T * ( A ~ B ) ) + ( B * T B E F O R ) 200 RETURN 1 END C SUBROUTINE BLCORN(T,TOTHER,ROW,CP,AK,AKTOP,AKRIG 1,DZ0,KSTEP,*,N,M) c  IMPLICIT REAL*8 (A-H,0-Z) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C9 / MTIMES,NTIMES,END,THEEND,TlME,DT COMMON /C10/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD L=L+1 C VOL=DX*DZ0/4.0D0 A=ROW*CP*VOL/(DT/2.0D0) CALL ESTMAT(T,KSTEP,VOL,A,N,M) B=DX*(AK+AKTOP)/(4*DZ0) C=DZ0*(AK+AKRIG)/(3*DX) D=-(QMOLD*DZ0)/2 C IF(KSTEP.EQ.1)GO TO 100 Q******************* FIRST HALF TIME STEP DMID(L)=A+B DTOP(L)=0.0D0 DBOT(L)=-B R(L)=(T*(A-C))+(C*TOTHER)+D GO TO 200 Q******************* NEXT HALF TIME STEP 100 DMID(L)=A+C DTOP(L)=-C DBOT(L)=0.0D0 R(L)=(T*(A-B))+(B*TOTHER)+D 200 RETURN 1 END C REAL FUNCTION C L I Q * 8 ( T ) c  IMPLICIT REAL*8 (A-H,0-Z) C IF(T. L E . 1 4 9 3 . 0 D 0 ) GO TO 100 CLIQ=(-0.012439D0*T)+19.081463D0 RETURN 232 100 CLIQ=167.7523373565D0-T*0.36470799D0 A+(T**2.0D0)*0.00027556876D0-(T**3.0D0) A*0.00000007120875D0 RETURN END C REAL FUNCTION CSOL*8(T) c  IMPLICIT REAL*8 (A-H,0-Z) C IF(T.LE.1493.0D0) GO TO 100 CSOL=(-0.002439D0*T)+3.7414634D0 RETURN 100 IF(T.LE.1492.98D0)GO TO 200 CSOL=(-3.0D0*T)+447 9.1D0 RETURN 200 CSOL=(-0.0054913D0*T)+8.3 585549D0 RETURN END C C REAL FUNCTION T L I Q * 8 ( C ) c  IMPLICIT REAL*8 (A-H rO-Z) C IF(C.GT.0.51D0) GO TO 100 TLIQ=(-80.392157D0*C)+1534.0D0 RETURN 100 TLIQ=1506.36955637144D0-17.4 073 3 404564D0*C A-27.01800384717D0*(C**2.0D0)+(C**3.0D0) A*2.83618411819D0 RETURN END C REAL FUNCTION TSOL*8(C) c  IMPLICIT REAL*8 (A~H,0-Z) C IF(C.GE.0.10D0)GO TO 100 TSOL=-(C*410.0D0)+1534.0D0 RETURN 100 IF(C.GE.0.16D0)GO TO 200 TSOL=-(0.3333D0*C)+1493.3333D0 RETURN 200 TSOL=-(182.10526D0*C)+1522.1368D0 RETURN END C SUBROUTINE ESTMAT(T,KSTEP,VOL,A,N,M) c  IMPLICIT REAL*8(A-H,0-Z) COMMON /C3 / AK(75,100),CP(75,100),ROW(75,100) COMMON /C1 / T T ( 7 5 , 1 0 0 ) , T S ( 7 5 , 1 0 0 ) COMMON /C4 / SOLID(75,100) 233 COMMON /CIO/ CARBON,TLIQID,TSOLID,SUHEAT,TPOUR,HEAT,QMOLD COMMON /C20/ SHE L L F ( 7 5 ) , X M I N S C ( 7 5 ) , D I S T ( 7 5 ) COMMON /C21/ FS(75,100),FRACTN,NREPET COMMON /C9 / MTIMES,NTIMES,END,THEEND,TlME,DT C WEIGHT=VOL*ROW(N,M) TOHEAT=WEIGHT*HEAT FSOLID=0.0D0 CARLQ=0.0D0 CARSL=0.0D0 ELHEAT=0.0D0 ELRATE=0.0D0 C IF(KSTEP.EQ.0)GO TO 1000 C IF(T.GE.TLIQID)GO TO 800 C IF(T.LT.TSOLID)GO TO 500 C c _ _ _ -F S ( N , M ) = ( T L I Q I D - T ) / ( T L I Q I D - T S O L I D ) FSOLID=FS(N,M)*WEIGHT ELHEAT=DABS((FSOLID-SOLID(N,M)))*HEAT ELRATE=ELHEAT/(DT*DABS(TS(N,M)-TT(N,M))) SOLID(N,M)=FSOLID GO TO 900 c _ 500 IF(FS(N,M).GE.1.0D0)GO TO 1000 FS(N,M)=1.0D0 FSOLID=WEIGHT ELHEAT=DABS((FSOLID-SOLID(N,M)))*HEAT ELRATE=ELHEAT/(DT*DABS(TS(N,M)-TT(N,M))) SOLID(N,M)=FSOLID GO TO 900 c  800 IF(FS(N,M).GT.0.0D0) GO TO 801 GO TO 1000 801 ELHEAT=HEAT*WEIGHT*FS(N,M) ELRATE=ELHEAT/(DT*DABS(TS(N,M)-TT(N,M))) A=A-ELRATE FS(N,M)=0.0D0 SOLID(N,M)=0.0D0 GO TO 1000 c  c  900 A=A+ELRATE c  1000 IF(N.EQ.25)GO TO 1001 GO TO 1100 1001 IF(M.EQ.1)GO TO 1002 GO TO 1100 1002 I F ( T l M E . L T . D T ) G O TO 1003 GO TO 1005 1003 IF(KSTEP.EQ.O) GO TO 1005 234 WRITE(7,10 0 4)N,VOL,HEAT,M,ROW(N , M),TLIQID,TOHEAT,WEIGHT,TSO 1004 FORMAT(9X,'N=',12,6X,'VOL ..=',F 6.5,2X,'HEAT..=',F4.1,/, 19X,'M=',I2,6X,'ROW ..=',F6.2,2X,'TLIQID=',F6. 1 , / , 1'TOTAL HEAT=',F6.4,2X,'WEIGHT^',F6.5,2X,'TSOLID= ',F6.1,/, 1 ' ' ,/) WRITE(7,1007) 1007 FORMAT('K',2X,'TIME',2X, 1'C-LIQ',2X,'C-SOL',2X,' FS ',2X,'SOLID',2X, 1'LATENT HT',2X,' TEMP ',2X,'R',7X,'PERCEN',2X,'RATE',/) C 1005 PERCEN=(ELHEAT/TOHEAT)* 100.0D0 IF(KSTEP.EQ.0)GO TO 1009 WRITE(7,1006)KSTEP,TIME,CARLQ,CARSL,FS(N,M), 1SOLID(N,M),ELHEAT,T,A,PERCEN,ELRATE GO TO 1100 1009 WRITE(7,1006)KSTEP,TIME,CARLQ,CARSL,FS(N,M), 1SOLID(N,M),ELHEAT,T,A,PERCEN,ELRATE 1006 FORMAT(I 1 ,2X,F4.2,2X,F5.4,2X,F5.4,2X,F5.3,2X,F5.4,2X,F8.6, 12X,F6.1,2X,F6.1,2X,F5.1,2X,F10.4) C 1100 RETURN END C SUBROUTINE CURFIT(N,X,Y,P,KURVE,ORDER) c  IMPLICIT REAL*8 (A-H,0-Z) DIMENSION X ( 1 5 ) , Y ( 1 5 ) , Z ( 1 5 ) , Z Z ( 1 5 ) , P ( 1 5 ) , Z E ( 1 5 ) DIMENSION S ( 7 5 ) , S I G M A ( 7 5 ) , S S ( 7 5 ) , A A ( 7 5 ) , B B ( 7 5 ) , Y F ( 2 0 0 ) 1,YD(200),WT(200) LOGICAL LK LK=.FALSE. NWT=0 C NUMBER=N-2 IF(NUMBER.GE.4 0)NUMBER=4 0 C CALL DOLSF (NUMBER,N,X,Y,YF,YD,WT,NWT,S 1,SIGMA,AA,BB,SS,LK,P) C KURVE=NUMBER+1 C DO 2 I = 1 , N Z ( I ) = P ( 1 ) DO 1 J=2,KURVE Z Z ( I ) = P ( J ) * ( X ( I ) * * ( J - 1 ) ) Z ( I ) = Z Z ( I ) + Z ( l ) 1 CONTINUE Z E ( I ) = ( Z ( I ) - Y ( l ) ) / Y ( l ) * 1 0 0 . 0 D 0 2 CONTINUE C WRITE(6,3)NUMBER 3 FORMAT(60X,' ',1X,I2,1X,' ',/) W R I T E ( 6 , 4 ) ( X ( I ) , Y ( I ) , Z ( l ) , Z E ( l ) , 1 , P ( I ) , 1 = 1 , K U R V E ) KU=KURVE+1 235 W R I T E ( 6 , 5 ) ( X ( I ) , Y ( I ) , Z ( I ) , Z E ( I ) , I = K U , N ) C 4 FORMAT(F5.0,3X,F7.5,3X,F7.5,4X,F5.2,4X,'P(',I2, ,)=',F20.15) 5 FORMAT(F5.0,3X,F7.5,3X,F7.5,4X,F5.2) C RETURN END C REAL FUNCTION AKSOL*8(T) c  IMPLICIT REAL*8 (A-H,0"Z) C COMMON /C18/ YK(15),YCP(15),PK(75),PCP(15),ORDERK,ORDERC,KK AKSOL=0.0D0 DO 100 1=1,KK A K S S S = P K ( I ) * ( T * * ( I - 1 ) ) AKSOL=AKSOL+AKSSS 100 CONTINUE C RETURN END C REAL FUNCTION CPSOL*8(T) c  IMPLICIT REAL*8 (A~H,0-Z) C COMMON /C18/ Y K ( 1 5 ) , Y C P ( 1 5 ) , P K ( 7 5),PCP(15),ORDERK,ORDERC,KK CPSOL=0.0D0 DO 100 1=1,KCP C P S S S = P C P ( I ) * ( T * * ( I - 1 ) ) CPSOL=CPSOL+CPSSS 100 CONTINUE C RETURN END C C SUBROUTINE PROFILE PRINTS THE MINISCUS PROFILE IN F I L E - 8 8 C AND THE SHELL PROFILE IN THE F I L E - 9 C SUBROUTINE PROFLE(TIME) c  IMPLICIT REAL*8(A-H,0-Z) COMMON /C1 / T T ( 7 5 , 1 0 0 ) , T S ( 7 5 , 1 0 0 ) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C14/ NTOTAL,MTOTAL,NX COMMON /C20/ SH E L L F ( 7 5 ) , X M I N S C ( 7 5 ) , D I S T ( 7 5 ) COMMON /C21/ FS(75,100),FRACTN,NREPET C c  DO 1000 N=1,NTOTAL £************************ ******** DO 400 M=1,MTOTAL IF(TT(N,M).GT.0.0D0) GO TO 100 GO TO 300 236 100 IF(FS(N,M).GT.0.0D0) GO TO 200 GO TO 300 Q ***************** 200 MNOW=M SHELLF(N)=0.0D0 GO TO 500 Q ***************** 300 SHELLF(N)=0.0D0 400 CONTINUE c  GO TO 1000 c  500 DO 900 M=MNOW,MTOTAL IF(FS(N,M).GT.FRACTN)GO TO 800 GO TO 1000 800 SHELLF(N)=(20*DX*M)-(20*DX*0.5) 900 CONTINUE O******************************** 1000 CONTINUE c  C WRITE(13,1)NTOTAL 1 FORMAT(14) WRITE(6,2)TIME 2 FORMAT(//,10X,'TIME=',1X,F7.4,1X,'SEC,/ 1,5X,'SHELL THICKNESS(CM*10)',2X,'DISTANCE BELOW MENISCUS') DO 5 1=1,NTOTAL W R I T E ( 6 , 3 ) S H E L L F ( I ) , D I S T ( I ) WRITE(13,4) S H E L L F ( I ) , D I S T ( I ) , X M I N S C ( I ) 3 FORMAT(11X,F10.5,5X,F10.5) 4 FORMAT(3F5.2) 5 CONTINUE C RETURN END SUBROUTINE MSHAPE(N1) IMPLICIT REAL*8(A-H,0-Z) COMMON /C1 / T T ( 7 5 , 1 0 0 ) , T S ( 7 5 , 1 0 0 ) COMMON /C5 / DMID(7600),DTOP(7600),DBOT(7600),R(7600),DX,L COMMON /C14/ NTOTAL,MTOTAL,NX COMMON /C20/ SHE L L F ( 7 5 ) , X M I N S C ( 7 5 ) , D I S T ( 7 5 ) COMMON /C21/ FS(75,100),FRACTN,NREPET DO 200 N=1,NTOTAL DO 100 M=1,MTOTAL IF(TT(N,M).GT.0.0D0)GO TO 300 GO TO 100 300 XMINSC(N)=(DX*M*20.0D0)-(DX*0.5*20.0D0) IF(N.GT.NI) XMINSC(N)=0.0D0 GO TO 200 100 CONTINUE 200 CONTINUE 237 C WRITE(12,500)NTOTAL 500 FORMAT(14) WRITE(6,600) 600 FORMAT(///,7X,'MENISCUS SHAPE DESCRIPTION',/,7X 1 f . , f / / ) DO 900 1=1,NTOTAL WRI T E ( 6 , 7 0 0 ) X M I N S C ( I ) , D I S T ( I ) W R I T E ( 1 2 , 8 0 0 ) X M I N S C ( I ) , D I S T ( I ) 700 FORMAT('X-COORDINATE=',1X,F10.5,'Y-COORDINATE=',1X,F10.5) 800 FORMAT(2F5.2) 900 CONTINUE C RETURN END C SUBROUTINE XANDY c :  C DIMENSION A B C ( 6 ) , B C D ( 6 ) , C D E ( 6 ) , D E F ( 6 ) , E F G ( 6 ) , F G H ( 6 ) C READ(1, 1 ) ( A B C ( I ) ,1=1,6) READ(1, 1 ) ( B C D ( I j ,1=1,6) READ(1, 1 ) ( C D E ( I ) ,1=1,6) READ(1, 1 ) ( D E F ( I ) ,1=1,6) READ(1, 1 ) ( E F G ( I ) ,1=1,6) READ(1, 1 ) ( F G H ( I ) ,1=1,6) FORMAT < 6A4) c CALL AXCTRL('XORIGIN',0.0) CALL AXCTRL('YORIGIN',20.0) CALL AXCTRL('SIDE',-1) CALL AXPLOT('DISTANCE BELOW THE MENISCUS(CM);' 1 ,270.0,15.0,0.0,0.2) CALL AXCTRL('YORIGIN',5.0) CALL AXPLOT('DISTANCE FROM THE MOLD WALL(CM);' 1,0.0,20.0,0.0,0.05) C CALL PLOT(0.0,20.0,3) CALL PLOT(0.0,22.0,2) CALL PLOT(20.0,22.0,2) CALL PLOT(20.0,5.0,2) C CALL PSYM(2.0,23.0,0.6,'SOLIDIFICATION AT THE MENISCUS',0.0 1,100) CALL PSYM(11.0,11.0,0.4,ABC,0.0,24,100) CALL PSYM(11.0,10.0,0.4,BCD,0.0,24,100) CALL PSYM(11.0,9.0,0.4,CDE,0.0,24,100) CALL PSYM(11.0,8.0,0.4,DEF,0.0,24,100) CALL PSYM(11.0,7.0,0.4,EFG,0.0,24,100) CALL PSYM(11.0,6.0,0.4,FGH,0.0,24,100) CALL PSYM(4.0,22.5,0.3, 1'LATENT HEAT IS RELEASED AS PER FE-C DIAGRAM',0.0,45,100) CALL P L O T O 0 . 5 , 12.0,3) 238 GALL PLOT(10.5,5.5,2) CALL PLOT(19.5,5.5,2) CALL PLOT(19.5,12.0,2) CALL PLOT(10.5,12.0,2) CALL PLOT(0.0,0.0,3) C 100 RETURN END C SUBROUTINE MENISC c  DIMENSION S C U S ( 7 5 ) , D I S T ( 7 5 ) , S H E L ( 7 5 ) C Q * * * * * * * * * * * THE MENISCUS PROFILE IS PLOTTED * * * * * * * * * * * * * * C READ(2,100)NTOTAL 100 FORMAT(14) C DO 300 N=1,NTOTAL READ(2,200)SCUS(N),DIST(N) 200 FORMAT(2F5.2) 300 CONTINUE C CALL LINE(SCUS,DIST,NTOTAL,1) CALL P L O T ( S C U S ( 1 ) , D I S T ( 1 ) , 3 ) CALL P L O T ( 2 0 . 0 , D I S T ( 1 ) , 2 ) CALL PLOT(0.0,0.0,3) C RETURN END C SUBROUTINE SHELL(N1) c  DIMENSION S C U S ( 7 5 ) , D I S T ( 7 5 ) , S H E L ( 7 5 ) C C * * * * * * * * * * * THE SHELL PROFILE IS PLOTTED * * * * * * * * * * * * * * * * * C READ(3,400)NTOTAL 400 FORMAT(14) C DO 600 N=1,NTOTAL READ(3,500)SHEL(N),DIST(N),SCUS(N) 500 FORMAT(3F5.2) 600 CONTINUE C CALL PLOT(SHEL(NTOTAL),DIST(NTOTAL),3) C DO 800 N=1,NTOTAL NKPD=NTOTAL-N IF(NKPD.EQ.0)GO TO 1000 IF(SHEL(NKPD).GT.0.0)GO TO 699 IF(SCUS(NKPD).GT.0.0)GO TO 699 CALL PLOT(0.0,DIST(NKPD),2) GO TO 1000 699 IF(SHEL(NKPD).GT.SCUS(NKPD))GO TO 700 239 GO TO 900 700 CALL PLOT(SHEL(NKPD),DIST(NKPD),2) 800 CONTINUE C 900 NKPD1=NKPD-1 CALL PLOT(SCUS(NKPD1),DIST(NKPD1),2) 1000 CALL PLOT(0.0,0.0,3) CALL PLOT(SHEL(NTOTAL),DIST(NTOTAL),3) CALL PLOT(SHEL(NTOTAL),5.0,2) CALL PLOT(0.0,0.0,3) C 1100 RETURN END 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0078963/manifest

Comment

Related Items