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UBC Theses and Dissertations

Initial solidification phenomena in the continuous casting slab mould Takeuchi, Eiichi 1984

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I N I T I A L S OLIDIFICATION PHENOMENA IN THE CONTINUOUS CASTING SLAB MOULD by E I I C H I TAKEUCHI M.E. KYUSHU UNIVERSITY, 1977 THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES METALLURGICAL ENGINEERING We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1984 © E i i c h i T a k e u c h i , 1984 I n 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 t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e 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 , I a g r e e t h a t t h e 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 a n d s t u d y . I f u r t h e r a g r e e 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g 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 n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . E i i c h i TAKEUCHI D e p a r t m e n t Of M e t a l l u r g i c a l Engineering The 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 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 D a t e Oct. 12, 1984 E-6 (3/81) • 4 11 ABSTRACT In a s t u d y of i n i t i a l s o l i d i f i c a t i o n d u r i n g t h e c o n t i n u o u s c a s t i n g o f s t e e l s l a b s , 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 and t h e i r e f f e c t on t h e s u r f a c e q u a l i t y o f t h e s l a b s have been e x a m i n e d by m e t a l l o g r a p h i c a l i n v e s t i g a t i o n o f s l a b s a m p l e s and by p e r f o r m i n g a s e t of m a t h e m a t i c a l a n a l y s e s . The m e t a l l o g r a p h i c s t u d y o f t h e o s c i l l a t i o n marks has r e v e a l e d t h a t t h e a d j a c e n t s u b s u r f a c e s t r u c t u r e may e x h i b i t " h o o k s " . The d e p t h of o s c i l l a t i o n marks e x h i b i t i n g s u b s u r f a c e hooks i s a f f e c t e d by t h e c a r b o n c o n t e n t of t h e s t e e l , w h i l e o s c i l l a t i o n marks w i t h o u t a d j a c e n t hooks do not show t h e c a r b o n d e p e n d e n c e . A n o t h e r i m p o r t a n t f a c t o r w h i c h a f f e c t s t h e d e p t h o f o s c i l l a t i o n marks i s v a r i a t i o n o f t h e m e n i s c u s l e v e l . Q u i c k upward movement of t h e m e n i s c u s l e v e l i n c r e a s e s t h e d e p t h of o s c i l l a t i o n m arks. The t h e o r e t i c a l a n a l y s i s of h e a t f l o w a t t h e m e n i s c u s i n d i c a t e s t h a t t h e m e n i s c u s may p a r t i a l l y f r e e z e w i t h i n t h e p e r i o d o f a t y p i c a l mould o s c i l l a t i o n c y c l e . L u b r i c a t i o n t h e o r y has shown t h a t a s i g n i f i c a n t p r e s s u r e c a n be g e n e r a t e d i n t h e f l u x c h a n n e l by t h e r e c i p r o c a t i n g m o t i o n o f t h e mould r e l a t i v e t o t h e s h e l l . The shape of t h e m e n i s c u s has been computed as a f u n c t i o n o f t h e p r e s s u r e d e v e l o p e d i n t h e mould f l u x . T h i s has d e m o n s t r a t e d t h a t t h e " c o n t a c t " p o i n t o f t h e m e n i s c u s w i t h t h e mould w a l l moves o u t of phase w i t h t h e mould d i s p l a c e m e n t by 7r/2, and has a g r e a t e r a m p l i t u d e t h a n t h e s t r o k e o f mould i i i o s c i l l a t i o n . T h us n e a 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 m o l t e n s t e e l c a n o v e r f l o w a t t h e m e n i s c u s when a r i g i d h o o k - l i k e s h e l l e x i s t s , w h i l s t t h e m e n i s c u s i n t h e a b s e n c e of a r i g i d s h e l l , c a u s e d by h i g h s u p e r h e a t a n d / o r s t e e l c o n v e c t i o n a t t h e s o l i d i f i c a t i o n f r o n t , i s drawn t o w a r d t h e mould w a l l t o f o r m t h e o s c i l l a t i o n marks w i t h o u t a s u b s u r f a c e hook. C o n s e q u e n t l y t h e e f f e c t o f v a r i o u s c a s t i n g v a r i a b l e s on t h e d e p t h o f o s c i l l a t i o n marks can be e x p l a i n e d on t h e o r e t i c a l g r o u n d s . P o s i t i v e s e g r e g a t i o n o f p h o s p h o r u s has been o b s e r v e d a t t h e b o t t o m o f t h e o s c i l l a t i o n marks and has been c l a s s i f i e d m a i n l y i n t o two t y p e s . One t y p e i s o b s e r v e d a t t h e end of t h e o v e r f l o w r e g i o n on t h e s u b s u r f a c e hook. A h e a t - f l o w model w h i c h t a k e s i n t o a c c o u n t t h e shape o f t h e o s c i l l a t i o n marks has r e v e a l e d t h a t t h i s t y p e o f p o s i t i v e s e g r e g a t i o n i s c a u s e d by l o c a l d e l a y of s o l i d i f i c a t i o n a t t h e b o t t o m o f t h e o s c i l l a t i o n m arks. A n o t h e r t y p e of p o s i t i v e s e g r e g a t i o n has been f o u n d i n a l a y e r on t h e b o t t o m o f o s c i l l a t i o n marks w i t h o u t s u b s u r f a c e h o o k s . T h i s f o r m of s e g r e g a t i o n c a n n o t be e x p l a i n e d by t h e h e a t ^ f l o w m o d e l , b u t i s l i k e l y due t o a p e n e t r a t i o n mechanism i n w h i c h t h e n e g a t i v e p r e s s u r e i n t h e f l u x c h a n n e l g e n e r a t e d d u r i n g t h e upward m o t i o n o f t h e mould draws out i n t e r d e n d r i t i c l i q u i d f r o m t h e s e m i - s o l i d i f i e d s h e l l . T r a n s v e r s e c r a c k s a r e f o u n d a l o n g t h e b o t t o m o f o s c i l l a t i o n m a r k s . The s u r f a c e o f t h e t r a n s v e r s e c r a c k s e x h i b i t s an i n t e r d e n d r e t i c a p p e a r a n c e i n t h e v i c i n i t y of t h e s l a b s u r f a c e , w h i c h i m p l i e s t h a t t h e c r a c k s a r e h o t t e a r s i n i t i a t e d i n t h e i i i i mou ld r e g i o n . A h e a t - f l o w a n a l y s i s p r e d i c t s t h a t deep o s c i l l a t i o n marks cause n o n u n i f o r m i t y of t h e s h e l l i n t h e m o u l d , w h i c h was a l s o o b s e r v e d i n t h e m e t a l l o g r a p h i c i n v e s t i g a t i o n . A c c o r d i n g t o t h e h e a t - f l o w a n a l y s i s no t o n l y t h e d e p t h but a l s o t h e p i t c h of o s c i l l a t i o n marks a f f e c t s t h e s h e l l p r o f i l e . T h e r e f o r e i n c r e a s i n g t h e f r e q u e n c y of mould o s c i l l a t i o n e f f e c t i v e l y r e d u c e s t r a n s v e r s e c r a c k s , by d e c r e a s i n g b o t h t h e d e p t h and t h e p i t c h of o s c i l l a t i o n m a r k s . V T a b l e of C o n t e n t s T a b l e o f C o n t e n t s v L i s t o f T a b l e s v i i i L i s t o f F i g u r e s i X L i s t o f Symbols x v i i i A cknowledgement x x l 1 . INTRODUCTION 1 2. PREVIOUS WORK 6 2.1 Moul d O s c i l l a t i o n And M o u l d F l u x 6 2.2 O s c i l l a t i o n Mark F o r m a t i o n 8 2.2.1 The Shape Of O s c i l l a t i o n M a rks 8 2.2.2 S u b s u r f a c e S t r u c t u r e Of O s c i l l a t i o n M a r k s 9 2.2.3 Mechanism Of O s c i l l a t i o n Mark F o r m a t i o n 10 2.3 Heat T r a n s f e r In The M o u l d Near The M e n s i s c u s ...13 2.3.1 F a c t o r s A f f e c t i n g M o u l d H e a t F l u x 13 2.3.2 M e n i s c u s S o l i d i f i c a t i o n 14 2.4 L u b r i c a t i o n In The Mo u l d 16 2.4.1 C o n s u m p t i o n Of Mo u l d F l u x 16 2.4.2 Moul d F r i c t i o n 18 2.5 S u r f a c e Q u a l i t y Of S l a b s R e l a t e d To O s c i l l a t i o n Marks 1 9 2.5.1 T r a n s v e r s e C r a c k s 19 2.5.2 P o s i t i v e S e g r e g a t i o n 22 2.6 Summary - I n d u s t r i a l N e c e s s i t y F o r The P r e s e n t Work 23 3. SCOPE AND OBJECTIVES OF THE PRESENT WORK 51 3.1 Scope Of The P r e s e n t Work 51 3.2 The O b j e c t i v e s Of The P r e s e n t Work 53 4. FORMATION OF OSCILLATION MARKS 55 4.1 M e t a l l u r g i c a l I n v e s t i g a t i o n 55 4.1.1 C a s t i n g C o n d i t i o n s Of S l a b Samples 55 4.1.2 I n v e s t i g a t i o n P r o c e d u r e 56 4.1.3 A p p e a r a n c e And P i t c h Of O s c i l l a t i o n M a r k s 56 4.1.4 S u b s u r f a c e S t r u c t u r e Of O s c i l l a t i o n M a r k s 58 4.1.5 D e p t h Of O s c i l l a t i o n Marks 61 4.2 Heat Flow A n a l y s i s Of The M e n i s c u s R e g i o n 64 4.2.1 A x i a l P r o f i l e Of Mould Heat F l u x At The M e n i s c u s 64 4.2.2 T e m p e r a t u r e D i s t r i b u t i o n In The Mould F l u x And S t e e l 67 4.3 F l u i d P r e s s u r e In The M o u l d F l u x A t The M e n i s c u s 73 4.4 M e n i s c u s Shape ..78 4.4.1 S t a t i c Shape Of M e n i s c u s ..78 4.4.2 Change Of The M e n i s c u s Shape By Mould O s c i l l a t i o n 79 4.5 Mechanism Of O s c i l l a t i o n - M a r k F o r m a t i o n 82 4.6 M e n i s c u s M o d e l P r e d i c t i o n s 86 v i 5 . THE EFFECT OF OSCILLATION MARKS ON THE SURFACE QUALITY OF SLABS 149 5.1 I n t r o d u c t i o n 149 5 .2 M e t a l l u r g i c a l S tudy Of T r a n s v e r s e C r a c k s . . . . . . . 1 5 1 5 . 2 . 1 C a s t i n g C o n d i t i o n s 151 5 . 2 . 2 A p p e a r a n c e Of T r a n s v e r s e C r a c k s 151 5 . 2 . 3 S u b s u r f a c e S t r u c t u r e I n The V i c i n i t y Of T r a n s v e r s e C r a c k s 152 5 . 2 . 4 S u r f a c e Of T r a n s v e r s e C r a c k s 155 5 . 2 . 5 The E f f e c t Of O s c i l l a t i o n - m a r k s Shape On L o c a l S h e l l T h i c k n e s s . 157 5 . 2 . 6 Summary Of M e t a l l u r g i c a l I n v e s t r i g a t i o n Of T r a n s v e r s e C r a c k s 158 5 . 3 M e t a l l u r g i c a l S t u d i e s Of P o s i t i v e S e g r e g a t i o n Near O s c i l l a t i o n Marks 159 5 . 3 . 1 C a s t i n g C o n d i t i o n s 159 5 . 3 . 2 M e t a l l o g r a p h i c C l a s s i f i c a t i o n Of P o s i t i v e S e g r e g a t i o n 160 5 . 3 . 3 M i c r o a n a l y s i s Of P o s i t i v e S e g r e g a t i o n By CMA . . . 1 6 4 5 . 3 . 4 P o s i t i v e S e g r e g a t i o n Caused By O v e r f l o w At The S l a b C o r n e r 165 5 . 3 . 5 Summary Of M e t a l l u r g i c a l I n v e s t i g a t i o n s Of P o s i t i v e S e g r e g a t i o n 167 5 .4 Heat T r a n s f e r A n a l y s i s Of S o l i d i f i c a t i o n In The M o u l d I n The V i c i n i t y Of O s c i l l a t i o n M a r k s 168 5 . 4 . 1 O b j e c t i v e s And D e s c r i p t i o n Of The P h y s i c a l Sys tem 1 68 5 . 4 . 2 M a t h e m a t i c a l M o d e l i n g 169 5 . 4 . 3 C a l c u l a t e d R e s u l t s 172 5 . 4 . 3 . 1 T e m p e r a t u r e D i s t r i b u t i o n I n M o u l d F l u x And S t e e l 1 72 5 . 4 . 3 . 2 The E f f e c t Of O s c i l l a t i o n - M a r k Shape On The N o n u n i f o r m i t y Of S h e l l T h i c k n e s s 174 5 . 4 . 3 . 3 C o o l i n g Rate D i s t r i b u t i o n Near The O s c i l l a t i o n M a r k s 176 5 .5 D i s c u s s i o n On The F o r m a t i o n Of P o s i t i v e S e g r e g a t i o n 1 78 6 . CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 234 6.1 C o n c l u s i o n s 234 6 .2 S u g g e s t i o n s For F u t u r e Work 238 BIBLIOGRAPHY 240 APPENDIX I : NODAL EQUATIONS FOR THE HEAT FLOW CALCULATION IN THE MOULD 248 APPENDIX I I : NODAL EQUATIONS FOR THE TEMPERATURE DISTRIBUTION IN THE MENISCUS REGION 250 APPENDIX I I I : SHAPE OF MENISCUS 253 APPENDIX I V : DYNAMIC PRESSURE IN THE MOULD FLUX CHANNEL 257 APPENDIX V : COMPUTER PROGRAM FOR THE TEMPERATURE PREDICTION IN THE MENISCUS REGION 260 APPENDIX V I : COMPUTER PROGRAM FOR THE CALCULATION OF FLUID PRESSURE IN THE FLUX CHANNEL 278 APPENDIX V I I : COMPUTER PROGRAM FOR THE CALCULATION OF THE CHANGE OF MENISCUS SHAPE 280 v i i APPENDIX V I I I : COMPUTER PROGRAM FOR THE CALCULATION OF NONUNIFORM!TY OF SHELL PROFILE IN THE MOULD .285 v i i i L i s t of T a b l e s I . P r o p e r t i e s of M o u l d F l u x 25 I I . C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n o f t h e S l a b Samples from Company A 91 I I I . C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n of t h e S l a b Samples f rom Company B 92 I V . C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n of t h e S l a b Samples from Company C 93 V . C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n o f t h e S l a b Samples from Company D 94 V I . C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n of t h e S l a b Samples from Company E 95 V I I . T h e r m a l P r o p e r t i e s and C o n d i t i o n s f o r C a l c u l a t i o n of M o u l d Heat F l u x 96 V I I I . Assumed S t e e l C o m p o s i t i o n and T h e r m o p h y s i c a l P r o p e r t i e s f o r C a l c u l a t i o n s of T e m p e r a t u r e D i s t r i b u t i o n a t t h e M e n i s c u s 97 I X . C a s t i n g C o n d i t i o n s of Samples f o r M e t a l l u r g i c a l S t u d y of T r a n s v e r s e C r a c k s 182 X . C a s t i n g C o n d i t i o n s of Samples f o r M e t a l l u r g i c a l S t u d y of P o s i t i v e S e g r e g a t i o n 183 L i s t o f F i g u r e s 1-1 V a r i a t i o n o f w o r l d c r u d e - s t e e l p r o d u c t i o n and p e r c e n t a g e o f t h e s t e e l c o n t i n u o u s l y c a s t (1965-81) f r o m M c P h e r s o n e t a l . 1 4 1- 2 C a u s e s and e f f e c t s of t h e i n i t i a l s o l i d i f i c a t i o n i n t h e s l a b mould 5 2- 1 M o u l d d i s p l a c e m e n t due t o o s c i l l a t i o n 26 2-2 M a c r o s c o p i c s t r u c t u r e and t e m p e r a t u r e d i s t r i b u t i o n i n t h e m e l t i n g mould f l u x l a y e r on t h e m e n i s c u s . 1 2 ....27 2-3 E f f e c t of o s c i l l a t i o n s t r o k e on 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 . 9 28 2-4 E f f e c t o f o s c i l l a t i o n f r e q u e n c y on 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 . 1 6 29 2-5 R e l a t i o n s h i p between t h e d e p t h of o s c i l l a t i o n marks and t h e n e g a t i v e s t r i p t i m e . 2 3 " 2 5 30 2-6 D e n d r i t e s t r u c t u r e i n t h e v i c i n i t y o f o s c i l l a t i o n m a r k s . 9 (1) H o o k - l i k e s t r u c t u r e , (2) N o n - h o o k - l i k e s t r u c t u r e , (3) D i r e c t i o n o f g r o w t h o f p r i m a r y d e n d r i t e , (4) C a s t i n g d i r e c t i o n . 31 2-7 F o r m a t i o n mechanism o f o s c i l l a t i o n marks by S a t o . 3 0 32 2-8 F o r m a t i o n mechansism o f o s c i l l a t i o n marks by Emi e t a l . 9 33 2-9 S c h e m a t i c d i a g r a m of "mould s i m u l a t o r " (a) and f o r m a t i o n mechanism o f o s c i l l a t i o n marks (b) by Kawakami e t a l . 3 6 34 2-10 Change o f d i s p l a c e m e n t o f t h e mould and t h e s h e l l w i t h t ime 35 2-11 C a l c u l a t e d e f f e c t of o s c i l l a t i o n f r e q u e n c y and s t r o k e on t h e d e p t h o f o s c i l l a t i o n marks by N a k a t o ' s m o d e l . 3 9 36 2-12 E f f e c t o f t h e c a s t i n g s p e e d on t h e a x i a l h e a t f l u x p r o f i l e i n t h e mould w a l l . " 0 37 2-13 C a l c u l a t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould w a l l and i n t h e s t e e l . 3 2 38 X 2-14 C a l c u l a t e d s h e l l p r o f i l e w i t h d i f f e r e n t c o n v e c t i v e heat t r a n s f e r c o e f f i c i e n t s . 3 8 39 2-15 E f f e c t of c a s t i n g speed on mould f l u x c o n s u m p t i o n . 5 5 40 2-16 E f f e c t of o s c i l l a t i o n f r e q u e n c y on mould f l u x c o n s u m p t i o n . 8 41 2-17 E f f e c t of v i s c o s i t y o f mould f l u x on t h e f l u x c o n s u m p t i o n . 5 2 42 2-18 E f f e c t of s o f t e n i n g t e m p e r a t u r e of mould f l u x on t h e f l u x c o n s u m p t i o n . 5 2 ' 43 2-19 T r a n s v e r s e c r a c k s o b s e r v e d on s l a b s u r f a c e . 2 9 44 2-20 E f f e c t of n i t r o g e n on t h e hot d u c t i l i t y i n low c a r b o n s t e e l 8 0 ( a ) , and on t h e f r e q u e n c y of t r a n s v e r s e c r a c k f o r m a t i o n i n Nb-V s t e e l 8 2 ( b ) 45 2-21 T e m p e r a t u r e p r o f i l e on t h e s l a b s u r f a c e t o a v o i d t h e b r i t t l e t e m p e r a t u r e range a t t h e u n b e n d i n g p o i n t . " .46 2-22 R e l a t i o n s h i p between t h e maximum c o n t e n t o f P and t h e d e p t h of o s c i l l a t i o n m a r k s . 2 9 47 2-23 Mechanism of p o s i t i v e s e g r e g a t i o n by Tanaka e t a l . 2 9 48 2-24 Types of p o s i t i v e s e g r e g a t i o n of s t a i n l e s s s t e e l s l a b . 8 6 49 2-25 Mechanism of p o s i t i v e s e g r e g a t i o n by H . T a k e u c h i e t a l . 2 7 50 2-26 R e l a t i o n between f r e q u e n c y of p o s i t i v e s e g r e g a t i o n i n s t a i n l e s s s t e e l s l a b s and n e g a t i v e - s t r i p t i m e . 8 6 (Ne : N e g a t i v e - s t r i p r a t i o ) 50 4-1 Immers ion n o z z l e f o r t h e m u l t i - p o r t p r a c t i c e (Company D) 98 4-2 S c h e m a t i c v i e w of t h e e l e c t r o m a g n e t i c s t i r r e r i n t h e m o u l d 8 8 (Company E) 99 4-3 Appearance of o s c i l l a t i o n marks on the n a r r o w f a c e of a s l a b (Company A) 100 4-4 E f f e c t of c a s t i n g speed on t h e p i t c h of o s c i l l a t i o n marks (Company A ) . ' 101 4-5 A p p e a r a n c e of o s c i l l a t i o n marks on t h e n a r r o w f a c e of a s l a b (Company C) 102 XI 4-6 F r e q u e n c y of mould o s c i l l a t i o n i n e a c h c a s t i n g o p e r a t i o n 103 4-7 A p p e a r a n c e o f o s c i l l a t i o n marks on t h e n a r r o w f a c e o f s l a b c a s t w i t h (a) B i f u r c a t e d n o z z l e and (b) M u l t i -p o r t p r a c t i c e (Company D) 104 4-8 A p p e a r a n c e of o s c i l l a t i o n marks on t h e n a r r o w f a c e o f s l a b c a s t (a) w i t h o u t mould EMS and (b) w i t h m o u l d EMS (Company E) 105 4-9 S u b s u r f a c e s t r u c t u r e n e a r o s c i l l a t i o n marks e x h i b i t i n g " h o o k s " i n s t e e l s l a b s c o n t a i n i n g (a) 0.09%C a n d (b) 0.26%C (Company A ) . (x6) 106 4-10 S u b s u r f a c e s t r u c t u r e n e a r o s c i l l a t i o n marks w i t h o u t " h o o k s " i n s t e e l s l a b s c o n t a i n i n g (a) 0.08% C and (b) 0.26% C (Company A ) . (x6) 107 4-11 Change o f s e c o n d a r y d e n d r i t e arm s p a c i n g i n t h e c a s t i n g d i r e c t i o n C=0.09% (Company A) 108 4-12 S u b s t a n c e a t t h e end o f a s u b s u r f a c e hook (a) x 94 (b) x 2000 (Company B) 1 09 4-13 X - r a y s p e c t r o g r a p h of t h e s u b s t a n c e n e a r t h e hook (Company B) 110 4-14 S u b s u r f a c e s t r u c t u r e n e a r o s c i l l a t i o n marks i n s t e e l s l a b s f r o m Company C. (x6) 111 4-15 S u b s u r f a c e s t r u c t u r e n e a r o s c i l l a t i o n marks i n s t e e l s l a b s c a s t by (a) b i f u r i c a t e d n o z z l e and (b) m u l t i -p o r t p r a c t i c e (Company D ) . (x6.5) 112 4-16 S u b s u r f a c e s t r u c t u r e n e a r o s c i l l a t i o n marks i n s t e e l s l a b s c a s t (a) w i t h o u t EMS-M and (b) w i t h EMS-M (Company E ) . (x6) 113 4-17 R e l a t i o n s h i p between d e p t h and p i t c h o f o s c i l l a t i o n marks i n 0.08%-Carbon s l a b s (Company A) 114 4-18 R e l a t i o n s h i p between d e p t h and p i t c h of o s c i l l a t i o n marks on 0.26%-Carbon s l a b s (Company A) 115 4-19 I n f l u e n c e of c a r b o n c o n t e n t o f t h e s l a b s on t h e d e p t h o f o s c i l l a t i o n marks (Company A) 116 4-20 E f f e c t o f A l on t h e shape of o s c i l l a t i o n marks (Company A ) . N o t e : a l l marks a r e h o o k - l i k e s t r u c t u r e s 117 4-21 E f f e c t o f o s c i l l a t i o n s t r o k e on t h e d e p t h o f X l l o s c i l l a t i o n M a r k s (Company C) 118 4-22 E f f e c t o f mould EMS on the d e p t h of o s c i l l a t i o n marks (Company E) 119 4-23 P h y s i c a l s y s t e m f o r t h e c o m p u t a t i o n of h e a t t r a n s f e r i n t h e mould 120 4-24 T e m p e r a t u r e d i s t r i b u t i o n i n t h e mould w a l l and t h e h e a t f l u x d i s t r i b u t i o n near the m e n i s c u s ( C l o s e d c i r c l e s a r e v a l u e s of mould t e m p e r a t u r e r e p o r t e d by N a k a t o e t a l 1 8 ) 121 4-25 Geometry and n o d a l sy s tem employed t o p r e d i c t t e m p e r a t u r e d i s t r i b u t i o n s i n the mould f l u x and s t e e l a t t h e m e n i s c u s 122 4-26 F l o w c h a r t f o r t h e c a l c u l a t i o n of t h e t e m p e r a t u r e d i s t r i b u t i o n i n s t e e l and mould f l u x ( i ) 123 4-26 F l o w c h a r t f o r t h e c a l c u l a t i o n of t h e t e m p e r a t u r e d i s t r i b u t i o n i n s t e e l and mould f l u x ( i i ) 124 4-27 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e o f 0 . 3 s . S u p e r h e a t of s t e e l i s 5 ° C . 125 4-28 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e of 0 . 6 s . S u p e r h e a t of s t e e l i s 5 ° C . 126 4-29 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e of 0 . 3 s . S u p e r h e a t of s t e e l i s 0 ° C . 127 4-30 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e of 0 . 6 s . S u p e r h e a t of s t e e l i s 0 ° C . 128 4-31 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e of 0 . 3 s . S u p e r h e a t of s t e e l i s 2 0 ° C . 1 29 4-32 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e o f 0 . 6 s . S u p e r h e a t of s t e e l i s 2 0 ° C . 1 30 4-35 Geometry of mould f l u x c h a n n e l a t t h e m e n i s c u s . . . 1 3 3 4-36 P r e d i c t e d a x i a l p r e s s u r e p r o f i l e s i n the f l u x c h a n n e l near t h e m e n i s c u s . F l u x c h a n n e l d i m e n s i o n s assumed t o b e : hi=0.35mm, hf=0.05mm, l f=l0mm 134 x i i i 4-37 P r e d i c t e d v e l o c i t y d i s t r i b u t i o n s i n t h e f l u x c h a n n e l n e a r t h e m e n i s c u s a t t i m e o f maximum downward v e l o c i t y o f t h e m o u l d . C o n d i t i o n s a s f o r F i g . 4-36 and assumed f l u x v i s c o s i t y i s 5P ; 135 4-38 S h e a r s t r e s s d i s t r i b u t i o n a l o n g t h e s h e l l 136 4-39 Geometry of t w o - d i m e n s i o n a l m e n i s c u s s y s t e m 137 4-40 P r e d i c t e d c hange of m e n i s c u s shape w i t h t i m e r e s u l t i n g f r o m s i n u s o i d a l mould o s c i l l a t i o n (s=8mm, f=l00cpm, a=l200 dyne/cm, p =7.2 g/cm 3, p.=2.8 g/cm 3) f 5 . 138 4-41 Movement o f " c o n t a c t p o i n t " o f m e n i s c u s w i t h m o u l d w a l l d u r i n g mould o s c i l l a t i o n . See c a p t i o n t o F i g . 4-40 f o r o s c i l l a t i o n c o n d i t i o n s 139 4-42 S c h e m a t i c r e p r e s e n t a t i o n o f t h e f o r m a t i o n o f an o s c i l l a t i o n mark w i t h s u b s u r f a c e hook 140 4-43 S c h e m a t i c r e p r e s e n t a t i o n o f t h e f o r m a t i o n o f an o s c i l l a t i o n mark w i t h o u t s u b s u r f a c e hook 141 • 4-44 I n f l u e n c e o f m e n i s c u s l e v e l v a r i a t i o n (shown a s c h a n g i n g p i t c h o f o s c i l l a t i o n marks) on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . ...142 4-45 I n f l u e n c e of o s c i l l a t i o n s t r o k e on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v s = l m / m i n . 143 4-46 I n f l u e n c e of o s c i l l a t i o n , f r e q u e n c y on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v s=1m/min. 1 44 4-47 I n f l u e n c e o f o s c i l l a t i o n f r e q u e n c y on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v s = l . 2 , 1 . 5 m/min 145 4-48 I n f l u e n c e o f n e g a t i v e - s t r i p t i m e on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l (numbers i n p a r e n t h e s e s i n d i c a t e o s c i l l a t i o n f r e q u e n c i e s ) . v = 1 m/m i n 146 s 4-49 I n f l u e n c e of n e g a t i v e - s t r i p t i m e on b e n d i n g moment due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l s . v s = 1 m/min 147 4-50 I n f l u e n c e o f n e g a t i v e - s t r i p t i m e on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v s=0.8, 1.0, 1.2, 1.5 m/min 148 x i v 5-1 E f f e c t o f N c o n t e n t on t h e f o r m a t i o n of t r a n s v e r s e c r a c k s . N o t e : e q u i l i b r i u m c o n s t a n t K = [ % A l ] r [%N]r by L . S . D a r k e n e t a l . 1 0 8 184 A p p e a r a n c e of t r a n s v e r s e c r a c k s ( i ) (Company B ) . .185 A p p e a r a n c e of t r a n s v e r s e c r a c k s ( i i ) (Company B ) . 186 T r a n s v e r s e c r a c k s formed a t t h e bo t tom of o s c i l l a t i o n m a r k s ; p i c r a l e t c h i n g (Company B ) . ( x 3 . 6 ) 187 T r a n s v e r s e c r a c k s formed a t t h e b o t t o m of o s c i l l a t i o n m a r k s ; n i t a l e t c h i n g (Company B ) . ( x 3 . 6 ) 188 F e r r i t e - p e a r l i t e s t r u c t u r e (a) a t t h e t o p and (b) a t t h e b o t t o m of an o s c i l l a t i o n m a r k ; n i t a l e t c h i n g (Company B ) . (x146) 189 I n t e r d e n d r i t i c c r a c k s a t t h e b o t t o m of o s c i l l a t i o n m a r k s ; p i c r a l e t c h i n g (Company B ) . ( a ) : x 6 . 5 , ( b ) : x 3 2 . 190 5-8 S m a l l c r a c k o b s e r v e d a l o n g and near t h e s u b s u r f a c e h o o k ; p i c r a l e t c h i n g (Company B ) . ( a ) : x 6 . 5 , ( b ) : x 4 3 . 191 5-9 T r a n s v e r s e c r a c k s on t h e s u r f a c e of a s l a b sample from Company B . ( x 1 . 1 8 ) 192 '5-10 S u r f a c e of t r a n s v e r s e c r a c k s [1] from Company B . (x20) 193 5-11 S u r f a c e s of t r a n s v e r s e c r a c k [2] from Company B . (x20) 194 5-12 S u r f a c e of t r a n s v e r s e c r a c k [3] f rom Company B . (x40) 195 5-13 S u r f a c e of t r a n s v e r s e c r a c k f rom Company C . ( a ) : x 4 0 , (b) : x200 196 5-14 X - r a y s p e c t r o g r a p h of a d h e r e n t m a t e r i a l on t h e s u r f a c e o f t r a n s v e r s e c r a c k , Company C 197 5-15 R e l a t i o n s h i p between the d e p t h of o s c i l l a t i o n marks and t h e n o n u n i f o r m i t y o f t h e s h e l l t h i c k n e s s (Company B) 198 5-16 S u b s u r f a c e s t r u c t u r e near p o s i t i v e s e g r e g a t i o n (Sample B 1 ) . ( x 3 8 . 7 ) 199 5-17 S u b s u r f a c e s t r u c t u r e near p o s i t i v e s e g r e g a t i o n (Sample B 2 ) . ( x 3 8 . 7 ) 200 5-2 5-3 5-4 5-5 5-6 5-7 X V 5-18 S u b s u r f a c e s t r u c t u r e near a h o o k ; 0 0 . 1 0 % (Sample B 3 ) . ( x 3 8 . 7 ) 201 5-19 S u b s u r f a c e s t r u c t u r e near a h o o k ; C=0.26% (Sample B 4 ) . ( x 3 8 . 7 ) 202 5-20 S u b s u r f a c e s t r u c t u r e i n t h e v i c i n i t y of p o s i t i v e s e g r e g a t i o n l a y e r w i t h hooks a b s e n t (Sample B 5 ) . ( x 3 8 . 7 ) 203 5-21 R e l a t i o n s h i p between t h e d e p t h of o s c i l l a t i o n marks and t h e t h i c k n e s s of s e g r e g a t i o n l a y e r (Company A ) . 204 5-22 P o s i t i v e s e g r e g a t i o n a t t h e b o t t o m of o s c i l l a t i o n mark i n s t a i n l e s s s t e e l s l a b (Sample B 6 ) . (x6) 205 5-23 S e g r e g a t i o n of Mn and P i n t h e s u b s u r f a c e a r e a o f the o c i l l a t i o n mark d e t e r m i n e d by CMA (Sample B 2 ) . . . . 2 0 6 5-24 S e g r e g a t i o n of Mn and P i n t h e s u b s u r f a c e a r e a of the o s c i l l a t i o n mark d e t e r m i n e d by CMA (Sample B 3 ) . . . 2 0 7 5-25 S e g r e g a t i o n of Mn and P i n t h e s u b s u r f a c e a r e a o f the o s c i l l a t i o n mark d e t e r m i n e d by CMA (Sample B 5 ) . . . 2 0 8 5-26 S e g r e g a t i o n of N i and P i n t h e s u b s u r f a c e a r e a of the o s c i l l a t i o n mark d e t e r m i n e d by CMA (Sample B 6 ) . . . 2 0 9 5-28 A p p e a r a n c e of o v e r f l o w a t t h e s l a b c o r n e r and t h e l o c a t i o n o f c r o s s s e c t i o n f o r m e t a l l o g r a p h i c i n s p e c t i o n , Company A 211 5-29 S u b s u r f a c e s t r u c t u r e i n each l o n g i t u d i n a l c r o s s s e c t i o n shown i n F i g . 5 - 2 8 ; p i c r a l e t c h i n g (Company A ) . ( x 5 . 4 ) 212 5-30 S l a b c o r n e r sample f o r t h e i n v e s t i g a t i o n i n t o t h e s t r u c t u r e i n the h o r i z o n t a l c r o s s s e c t i o n (Company A ) . (x2) 213 5-31 S u b s u r f a c e s t r u c t u r e (a) i n l o n g i t u d i n a l c r o s s s e c t i o n , and ( b ) i n h o r i z o n t a l c r o s s s e c t i o n of t h e sample shown i n F i g . 5 - 3 0 ; O b e r h o f f e r e t c h (Company A ) • (3 . ) » x 6 « 5 f ( ID ) • x 6 • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2 1 4 5-32 C l a s s i f i c a t i o n of p o s i t i v e s e g r e g a t i o n 215 5-33 P h y s i c a l s y s t e m f o r m a t h e m a t i c a l model of h e a t f l o w i n t h e v i c i n i t y of t h e o s c i l l a t i o n mark 216 5-34 C o o r d i n a t e o f the c o n t r o l l e d sy s tem ( s y m m e t r i c sys tem) 217 5-35 T e m p e r a t u r e d i s t r i b u t i o n i n mould f l u x and s t e e l a f t e r 10s . 218 5-36 T e m p e r a t u r e d i s t r i b u t i o n i n mould f l u x and s t e e l a f t e r 48s 219 5-37 Change of s h e l l p r o f i l e w i t h t i m e . . (1=1.0cm, d = 0 . l 0 c m , x? /x * 2 = 0 . 5 ) 220 5-38 Change of s h e l l p r o f i l e w i t h t i m e . (1=1.0cm, d=0.15cm, x f / x * 2 = 0 . 5 ) 220 5-39 Change of s h e l l p r o f i l e w i t h t i m e . (1 = 2 . 0 c m , d = 0 . l 0 c m , x f / x * = 0 . 5 ) 221 5-40 Change of s h e l l p r o f i l e w i t h t i m e . (1 = 2 . 0 c m , d=0.15cm, x f / x * = 0 . 5 ) 222 5-41 Change of s h e l l p r o f i l e w i t h t i m e . (1 = 2 . 0 c m , d = 0 . l 0 c m , x f / x * = 0 . 3 ) 223 5-42 Change of s h e l l p r o f i l e w i t h t i m e . (1 = 2 . 0 c m , d=0 . lOcm, x? /x * 2 = 0 . 7 ) 224 5-43 E f f e c t of t h e shape of o s c i l l a t i n mark on t h e n o n u n i f o r m i t y of s h e l l t h i c k n e s s a f t e r 10s . ( x V ^ 2 = 0 . 5 ) 225 5-44 Change of t e m p e r a t u r e w i t h t i m e a t t h e b o t t o m and a t t h e ' t o p of o s c i l l a t i o n marks 226 5-45 Change of c o o l i n g r a t e w i t h t i m e a t t h e t o p and a t the b o t t o m of o s c i l l a t i o n m a r k . (1=2.0cm, d=0 . l 0cm) . . 2 2 7 5-46 R e l a t i o n s h i p between c o o l i n g r a t e and t h e s e c o n d a r y d e n d r i t e arm s p a c i n g by A . S u z u k i e t a l . 1 0 " N o t e : M e a s u r e d average v a l u e s of s e c o n d a r y arm s p a c i n g a t t h e t o p and a t t h e b o t t o m of o s c i l l a t i o n mark a r e i n d i c a t e d 228 5-47 R e l a t i o n s h i p between s h e l l t h i c k n e s s and t i m e . (1=1.0cm, d=0 . l0cm) 229 5-48 R e l a t i o n s h i p between s h e l l t h i c k n e s s and t i m e . (1 = 2 . 0 c m , d=0 . l0cm) 230 5-49 R e l a t i o n s h i p between s h e l l t h i c k n e s s and t i m e . (1 = 2 . 0 c m , d=0.15cm) 231 5-50 R e l a t i o n s h i p between Pa and n e g a t i v e - s t r i p t i m e . .232 5-51 R e l a t i o n s h i p between p e r m e a b i l i t y and f r a c t i o n l i q u i d f rom P i w o n k a e t a l . 1 0 7 233 x v i i A1 T y p i c a l nodes and t h e i r number o f s u b r o u t i n e s i n computer program f o r t h e c a l c u l a t i o n of t e m p e r a t u r e d i s t r i b u t i o n a t t h e m e n i s c u s 300 A2 T y p i c a l nodes and t h e i r number o f s u b r o u t i n e s i n computer program f o r t h e c a l c u l a t i o n of n o n u n i f o r m i t y of s h e l l p r o f i l e i n t h e mould 301 x v i i i L i s t o f Symbols a 2 C a p i l l a r y c o n s t a n t (-) C £ , C i c S p e c i f i c h e a t of mould f l u x , s t e e l and w a t e r , r e s p e c t i v e l y (J/g°C) C p E S p e c i f i c h e a t between l i q u i d u s and s o l i d u s ( J / g °C) d D e p t h of o s c i l l a t i o n marks (cm) df T h i c k n e s s o f mould f l u x l a y e r between mould and s h e l l (cm) d w T h i c k n e s s o f water c h a n n e l i n t h e mould (cm) Djj H y d r a u l i c d i a m e t e r (cm) f F r e q u e n c y o f mould o s c i l l a t i o n ( c y c l e / s ) f s F r a c t i o n s o l i d (-) f ( x ) Shape o f o s c i l l a t i o n mark g L F r a c t i o n l i q u i d (-) h ( x ) W i d t h of f l u x c h a n n e l (cm) h ,h W i d t h of f l u x c h a n n e l a t i n l e t and o u t l e t , r e s p e c t i v e l y 1 f (cm) h Heat t r a n s f e r c o e f f i c e n t between water and m o u l d w ( W / c m 2 ° C ) h Heat t r a n s f e r c o e f f i c e n t between s t e e l and m o u l d f l u x s " f ( W / c m 2 ° C ) k P e r m e a b i l i t y (cm 2) 1 P i t c h of o s c i l l a t i o n marks (cm) l f L e n g t h o f f l u x c h a n n e l (cm) ]^ , 1 D i s p l a c e m e n t of mould and s h e l l , r e s p e c t i v e l y (cm) A l V a r i a t i o n o f m e n i s c u s l e v e l (cm) m L L a t e n t h e a t of s t e e l d u r i n g s o l i d i f i c a t i o n ( J / g ) XiX L c L e n g t h o f i n t e r d e n d r i t i c c h a n n e l (cm) M b B e n d i n g moment (dyne cm) P A v e r a g e v a l u e of n e g a t i v e p r e s s u r e i n mould f l u x Si ~ ' (dyne/cm 2) / P.,P f P r e s s u r e a t i n l e t and o u t l e t o f f l u x c h a n n e l , r e s p e c t i v e l y ( d y n e / c m 2 ) P ( x ) A x i a l d i s t r i b u t i o n o f p r e s s u r e i n m o u l d - f l u x (dyne/cm 2) AP P r e s s u r e d i f f e r e n c e ( d y n e / c m 2 ) AP S t a t i c p r e s s u r e d i f f e r e n c e ( dyne/cm 2) s q 0 ( x ) H e a t - f l u x d i s t r i b u t i o n a l o n g mould w a l l (W/cm 2) q f ( x ) , q s ( x ) H e a t - f l u x d i s t r i b u t i o n a l o n g mould w a l l i n t h e r e g i o n of d e p r e s s e d p a r t and f l a t p a r t of o s c i l l a t i o n marks, r e s p e c t i v e l y (W/cm 2) Q_ R e l a t i v e c o n s u m p t i o n o f mould f l u x ( c m 3 / s ) R( x ) I n t e g r a t i o n of p r e s s u r e i n f l u x c h a n n e l f r o m x=0 t o x=x (dyne/cm) r D i s t a n c e normal t o m e n i s c u s (cm) S S t r o k e o f mould o s c i l l a t i o n (cm) s D i s t a n c e a l o n g m e n i s c u s (cm) T f , T T e m p e r a t u r e of f l u x and s t e e l , r e s p e c t i v e l y (°C) T f i ' T s i I n i t i a l t e m p e r a t u r e o f f l u x and s t e e l , r e s p e c t i v e l y (°C) T l i q ' T s o l L i q u i d u s and s o l i d u s t e m p e r a t u r e , r e s p e c t i v e l y ( C) T M T e m p e r a t u r e of mould w a l l (°C) T w T e m p e r a t u r e of wa t e r (°C) "*wi , Twf_ T e m p e r a t u r e of water a t i n l e t and o u t l e t o f water c h a n n e l , r e s p e c t i v e l y (°C) t N , t p N e g a t i v e and p o s i t i v e s t r i p t i m e , r e s p e c t i v e l y ( s ) X X u X R e l a t i v e v e l o c i t y o f mould f l u x (cm/s) v V e l o c i t y o f mould f l u x (cm/s) v V e l o c i t y o f i n t e r d e n d r i t i c f l o w (cm/s) v V e l o c i t y o f mould (cm/s) M v V e l o c i t y o f s l a b (cm/s) s v V e l o c i t y of water i n w a t e r c h a n n e l (cm/s) w Av V e l o c i t y of m e n i s c u s l e v e l v a r i a t i o n (cm/s) m X C o n t a c t p o i n t o f m e n i s c u s w i t h mould w a l l (cm) c X^ L e n g t h o f mould (cm) Y H T h i c k n e s s of mould f l u x r e d u c i n g h e a t - f l u x by h a l f (cm) Y.. T h i c k n e s s o f mould w a l l (cm) M X f , X T h e r m a l c o n d u c t i v i t y o f mould f l u x and s t e e l , s r e s p e c t i v e l y (W/cm°C) \^ T h e r m a l c o n d u c t i v i t y o f mould w a l l ( W / c m ° C ) X T h e r m a l c o n d u c t i v i t y of w a t e r (W/cm°C) w •* M f , u V i s c o s i t y of mould f l u x and s t e e l , r e s p e c t i v e l y ( p o i s e ) v V i s c o s i t y of water ( p o i s e ) w p f , p D e n s i t y o f mould f l u x and s t e e l , r e s p e c t i v e l y (g/cm 3) p D e n s i t y o f water (g/cm 3) w a I n t e r f a c i a l t e n s i o n between mould f l u x and s t e e l (dyne/cm) T S h e a r s t r e s s (dyne/cm 2) <p C o n t a c t a n g l e ( r a d i a n ) xx i A c knowledgement I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e t o P r o f e s s o r J.K. Brimacombe f o r h i s i n v a l u a b l e g u i d a n c e , a s s i s t a n c e and e n c o u r a g e m e n t t h a t he p r o v i d e d t h r o u g h o u t t h e c o u r s e o f t h i s s t u d y . I am a l s o g r a t e f u l t o D r . Y. N i s h i d a o f t h e Governement I n d u s t r i a l R e s e a r c h I n s t i t u t e (Nagoya, J a p a n ) and D r . I.V. S a m a r a s e k e r a o f D e p a r t m e n t o f M e t a l l u r g i c a l E n g i n e e r i n g f o r t h e i r a s s i s t a n c e and t o P r o f e s s o r F. W e i n b e r g and P r o f e s s o r E.B. H a w b o l t f o r t h e i r h e l p f u l d i s c u s s i o n s . T h a n k s a r e a l s o e x t e n d e d t o my f e l l o w g r a d u a t e s t u d e n t s and f a c u l t y members o f t h i s D e p a r t m e n t . C o n s i d e r a b l e t h a n k s s h o u l d be e x p r e s s e d t o t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada and N i p p o n S t e e l C o r p o r a t i o n f o r p r o v i d i n g f i n a n c i a l s u p p o r t . S p e c i a l t h a n k s must be e x p r e s s e d t o D r . T. O h a s h i , D r . T. M a t s u m i y a , and Mr. W. Yamada o f N i p p o n S t e e l C o r p o r a t i o n f o r t h e i r CMA measurement. A l a r g e number o f s t e e l c o m p a n i e s a r e g r a t e f u l l y a c k n o w l e d g e d f o r p r o v i d i n g s l a b s a m p l e s and i n f o r m a t i o n . F i n a l l y I w o u l d l i k e t o e x p r e s s a word o f a p p r e c i a t i o n t o my w i f e Yumiko f o r h e r h e a r t y s u p p o r t . 1 1. INTRODUCTION Owing t o a d v a n t a g e s o f h i g h y i e l d , u n i f o r m q u a l i t y , and h i g h p r o d u c t i v i t y , 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 h a s been w i d e l y a c c e p t e d by numerous s t e e l m a k e r s i n t h e w o r l d . The f r a c t i o n of s t e e l t h a t i s c o n t i n u o u s l y c a s t has been s t e a d i l y i n c r e a s i n g , w h i l e t h e g r o w t h o f s t e e l p r o d u c t i o n has c e a s e d a f t e r t h e o i l c r i s i s , s ee F i g . 1-1. 1 P a r t i c u l a r l y i n t h e p a s t d e c a d e , 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 has u n d e r g o n e s i g n i f i c a n t d e v e l o p m e n t s i n b o t h e q uipment and o p e r a t i o n , 2 3 u n t i l now i t h a s become t h e dominant c a s t i n g p r o c e s s i n s t e e l m a k i n g . T oday c o n t i n u o u s c a s t i n g i s e n t e r i n g a new e r a of d e v e l o p m e n t i n w h i c h i t i s l i n k e d d i r e c t l y w i t h t h e h o t r o l l i n g p r o c e s s . I n i t i a l l y t h e h o t - c h a r g i n g p r o c e s s " has been d e v e l o p e d p r i o r t o d i r e c t r o l l i n g , where t h e h i g h t e m p e r a t u r e , a s - c a s t s l a b s a r e t r a n s p o r t e d t o t h e r e h e a t i n g f u r n a c e w i t h o u t s u r f a c e , c o n d i t i o n i n g . T h i s d i r e c t c h a r g i n g of t h e c o n t i n u o u s l y c a s t s l a b s t o t h e h o t r o l l i n g m i l l n o t o n l y m i n i m i z e s t h e e n e r g y l o s s b u t a l s o s i m p l i f i e s t h e i n t e r m e d i a t e p r o c e s s between s t e e l m a k i n g and h o t r o l l i n g , v i z c o o l i n g , i n s p e c t i o n , and hand s c a r f i n g . However h o t c h a r g i n g can o n l y be s u c c e s s f u l l y i m p l e m e n t e d i f t h e c a s t s l a b s a r e f r e e f r o m s u r f a c e d e f e c t s . Thus c o n s i d e r a b l e e f f o r t has been expended t o improve t h e s u r f a c e q u a l i t y o f s l a b s . In t h i s r e g a r d i n i t i a l s o l i d i f i c a t i o n i n t h e c o n t i n u o u s -c a s t i n g mould has been r e c o g n i z e d t o be a s i g n i f i c a n t l y i m p o r t a n t phenomenon t o be u n d e r s t o o d and i f p o s s i b l e c o n t r o l l e d . 2 The f a c t o r s r e l a t e d t o i n i t i a l s o l i d i f i c a t i o n i n t h e mould a r e e x p r e s s e d i n F i g . 1-2. I n i t i a l s o l i d i f i c a t i o n phenomena i n c o n t i n u o u s c a s t i n g a r e c h a r a c t e r i z e d n o t o n l y by r a p i d h e a t e x t r a c t i o n b u t a l s o by i n t e r a c t i o n between t h e i n i t i a l l y s o l i d i f i e d s h e l l and t h e i n f l o w i n g mould f l u x , g i v i n g r i s e t o 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 m a r k s . C o n s u m p t i o n of mould f l u x w h i c h r e s u l t s f r o m t h i s i n t e r a c t i o n i s e s s e n t i a l t o l u b r i c a t i o n and h e a t t r a n s f e r i n t h e mould. F o r example, n o n u n i f o r m i n f l o w of mould f l u x i s t h o u g h t t o be one of t h e m a j o r c a u s e s of l o n g i t u d i n a l s u r f a c e c r a c k s ; 5 d e e p o s c i l l a t i o n marks r e s u l t i n g f r o m i n a p p r o p r i a t e mould o p e r a t i o n o f t e n l e a d t o t h e f o r m a t i o n of t r a n s v e r s e s u r f a c e c r a c k s ; 6 a l s o d e c r e a s i n g t h e c o n s u m p t i o n r a t e o f m o u l d f l u x w i t h i n c r e a s i n g c a s t i n g s p e e d has been one of t h e m a j o r p r o b l e m s w i t h h i g h s p e e d c a s t i n g f r o m t h e s t a n d p o i n t o f l u b r i c a t i o n . 7 R e c e n t l y many i n d u s t r i a l e f f o r t s , f o r example h i g h f r e q u e n c y o s c i l l a t i o n c a s t i n g , 8 2 2 have been i n i t i a t e d t o p r o d u c e s l a b s f r e e from s u r f a c e c o n d i t i o n i n g . However t h e r e s u l t s e x p e c t e d have not been a c h i e v e d y e t . T h i s i s b e c a u s e t h e i n i t i a l 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 n o t been w e l l u n d e r s t o o d . T hus t h e main p u r p o s e of t h e p r e s e n t work i s t o shed l i g h t on t h e i n i t i a l s o l i d i f i c a t i o n i n t h e c o n t i n u o u s - c a s t i n g s l a b mould, e s p e c i a l l y from t h e p r o c e s s s t a n d p o i n t , i n o r d e r t o s u g g e s t t h e optimum mould o p e r a t i o n f o r t h e h i g h s u r f a c e q u a l i t y of s l a b s . The s u b j e c t s t r e a t e d a r e f i r s t l y t h e f o r m a t i o n mechanism o f o s c i l l a t i o n marks and s e c o n d l y t h e e f f e c t of o s c i l l a t i o n marks on t h e s u r f a c e q u a l i t y o f s l a b s . The a p p r o a c h 3 t a k e n has i n v o l v e d m e t a l l u r g i c a l i n v e s t i g a t i o n s o f many s l a b s a m p l e s c o n t i n u o u s l y c a s t u nder d i f f e r e n t c o n d i t i o n s , and t h e f o r m u l a t i o n o f s e v e r a l m a t h e m a t i c a l m o d e l s . 4 o •o O CD CP tn 1000 8 0 0 6 0 0 4 0 0 2 0 0 — i 1 1 r World crude steel production % of world production by continuous casting 5 0 4 0 t£ _ _ «/> 3 0 o CA 3 O 2 0 I c o O 10 I 9 6 0 1970 1980 Year F i g . 1-1 V a r i a t i o n of w o r l d c r u d e - s t e e l p r o d u c t i o n and p e r c e n t a g e o f t h e s t e e l c o n t i n u o u s l y c a s t (1965-81) from M c P h e r s o n e t a l . 1 5 f~ — T e r r i t o r y of the present work ^ ' mould motion superheat mould cooling mould f l u x type of s t e e l i n t e r a c t i o n between the i n i t i a l l y s o l i d i f i e d s h e l l and the flow of mould f l u x at the meniscus I o s c i l l a t i o n marks nonuniform temperature d i s t r i b u t i o n & s h e l l growth I , & s h e l l growth J '- T--T--^JT mould taper l u b r i c a t i o n p o s i t i v e segregation 3 3 = surface i n c l u s i o n s transverse cracks - - • ^ - k ' J s t e e l flow i - nonuniform s h e l l growth I $-? transformation l o n g i t u d i n a l cracks breakout propagation of crack by spray cooling and/or unbending slab free from the surface conditioning high speed casting saving energy * hot-charging * d i r e c t r o l l i n g i n creasing p r o d u c t i v i t y F i g . 1 - 2 Causes and e f f e c t s of t h e i n i t i a l s o l i d i f i c a t i o n i n t h e s l a b m o u l d . 6 2. PREVIOUS WORK 2.1 M o u l d O s c i l l a t i o n And M o u l d F l u x M o u l d o s c i l l a t i o n i n t h e p r e s e n c e o f an a d e q u a t e l u b r i c a n t p l a y s one o f t h e most e s s e n t i a l p a r t s i n t h e c o n t i n u o u s c a s t i n g of s t e e l . I t p r e v e n t s t h e s t i c k i n g o f s t e e l on t h e mould w a l l and makes c o n t i n u o u s c a s t i n g p o s s i b l e . F i g . 2-1 shows a t y p i c a l mould movement as a f u n c t i o n of t i m e . 9 In g e n e r a l t h e mo u l d o s c i l l a t i o n i s s i n u s o i d a l and, hence t h e d i s p l a c e m e n t and t h e s p e e d of t h e mould a r e e x p r e s s e d by E q s . (2-1) and (2-2) r e s p e c t i v e l y . x = | sin(2Trft) (2-1) v = Trsf cos (2nf t) (2-2) M T h e r e i s a c e r t a i n p e r i o d i n e a c h c y c l e o f t h e mould o s c i l l a t i o n i n w h i c h t h e downward v e l o c i t y of t h e mould e x c e e d s t h e w i t h d r a w a l r a t e of t h e s t r a n d . T h i s i s c a l l e d t h e n e g a t i v e -s t r i p t i m e , w h i l e t h e r e m a i n i n g p e r i o d i n e a c h c y c l e i s r e f e r r e d t o a s p o s i t i v e - s t r i p t i m e , e x p r e s s e d by E q s . (2-3) and (2-4) r e s p e c t i v e l y . 60 ^ t = — arc cos(—j) (2-3) N trf Trsf v t = — z - arc cos ( - —j) (2-4) p trf TT sr 7 R a p e s e e d o i l was f i r s t u s e d a s mould l u b r i c a n t i n c o n t i n u o u s s l a b c a s t i n g , b u t l a t e r m ould f l u x was a p p l i e d f o r t h e improvement o f s l a b q u a l i t y . The mould f l u x i n c o n t i n u o u s c a s t i n g p e r f o r m s t h e f o l l o w i n g f u n c t i o n s ; 1 0 1 1 i ) l u b r i c a t i o n ( b e t t e r t h a n r a p e s e e d o i l ) i i ) h e a t t r a n s f e r between mould w a l l and s t e e l i i i ) d i s s o l u t i o n of i n c l u s i o n s f l o a t i n g f r o m t h e m o l t e n s t e e l p o o l i v ) t h e r m a l i n s u l a t i o n a t t h e m e n i s c u s v) p r e v e n t i o n of r e o x i d a t i o n o f s t e e l a t t h e m e n i s c u s M o u l d f l u x i s s y n t h e s i z e d m a i n l y f r o m 1) S i 0 2 - C a O - A l 2 0 3 s y s t e m a s t h e base m a t e r i a l , 2 ) f G a F 2 , N a 2 0 , K 2 0 , e t c . t o c o n t r o l t h e f l u i d i t y , and 3) c a r b o n a n d / o r g r a p h i t e p a r t i c l e s as s k e l e t o n m a t e r i a l . Today mould f l u x i s u s e d m o s t l y i n a p r e f u s e d g r a n u l a r f o r m , w h i c h e n s u r e s t h e most u n i f o r m m e l t i n g b e h a v i o r a t t h e m e n i s c u s , 1 2 see F i g . 2-2. An a d e q u a t e t h i c k n e s s o f m o l t e n f l u x ( 6 - l 5 m m ) 1 2 1 3 i s n e c e s s a r y f o r t h e u n i f o r m i n f l o w o f t h e l u b r i c a n t i n t o t h e "gap" between t h e m o u l d w a l l and newly f o r m i n g s t e e l s h e l l . F o r t h e p u r p o s e of t h e o r e t i c a l a n a l y s i s o f t h e m e l t i n g b e h a v i o r o f mould f l u x on t h e m e n i s c u s and t h e i n f l o w o f mould f l u x i n t o t h e gap between t h e mould w a l l and t h e s h e l l , s e v e r a l t h e r m o p h y s i c a l p r o p e r t i e s a r e r e q u i r e d . However few s t u d i e s have been r e p o r t e d on t h e measurement o f t h e p r o p e r t i e s o f mould f l u x , e x c e p t f o r t h e v i s c o s i t y . 1 0 1 * As f o r t h e o t h e r p r o p e r t i e s , e . g . d e n s i t y , i n t e r f a c i a l t e n s i o n , s p e c i f i c h e a t , and t h e r m a l c o n d u c t i v i t y , t h e p r o p e r t i e s o f b l a s t 8 f u r n a c e s l a g w h i c h a r e b e t t e r c h a r a c t e r i z e d , a r e g e n e r a l l y e m p l o y e d i n p l a c e o f t h e m i s s i n g d a t a f o r mould f l u x . The p r o p e r t i e s of b l a s t f u r n a c e s l a g a r e summarized i n T a b l e 2-1. 2.2 O s c i l l a t i o n Mark F o r m a t i o n 2.2.1 The Shape Of O s c i l l a t i o n M a r k s The use o f mould f l u x has b r o u g h t a b o u t numerous a d v a n t a g e s i n 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 ; however a t t h e same t i m e , o s c i l l a t i o n marks t h e m s e l v e s h a ve become a more s e r i o u s p r o b l e m . 2 0 O s c i l l a t i o n marks a r e f o r m e d p e r i o d i c a l l y and u n i f o r m l y n o r m a l t o t h e c a s t i n g d i r e c t i o n a r o u n d t h e s l a b s u r f a c e . The p i t c h of o s c i l l a t i o n marks c a n be e x p r e s s e d s i m p l y by E q . ( 2 - 5 ) , 9 w h i c h i m p l i e s t h a t t h e marks a r e f o r m e d i n e a c h c y c l e o f t h e mould o s c i l l a t i o n . 1 = 1 * (2-5) f As one of t h e i n d i c e s of s u r f a c e q u a l i t y , t h e d e p t h o f o s c i l l a t i o n marks has been d i s c u s s e d o f t e n i n r e l a t i o n t o t h e m ould o s c i l l a t i o n . The d e p t h o f o s c i l l a t i o n marks i n c r e a s e s w i t h i n c r e a s i n g o s c i l l a t i o n s t r o k e l e n g t h , 9 F i g . 2-3, and d e c r e a s e s w i t h i n c r e a s i n g o s c i l l a t i o n f r e q u e n c y , 8 2 1 2 2 F i g . 2-4. The l o n g e r t h e o s c i l l a t i o n s t r o k e , t h e l a r g e r i s t h e e f f e c t of o s c i l l a t i o n f r e q u e n c y on 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 . 2 1 In more g e n e r a l t e r m s t h e mark d e p t h has been r e l a t e d t o t h e n e g a t i v e - s t r i p t i m e a s e x p r e s s e d by E q . ( 2 - 3 ) . The 9 r e l a t i o n s h i p between t h e d e p t h o f o s c i l l a t i o n marks and t h e n e g a t i v e - s t r i p t i m e i n d i f f e r e n t p l a n t s i s summarized i n F i g . 2 - 5 . 2 3 - 2 5 I t i s seen t h a t t h e d e p t h of o s c i l l a t i o n marks 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 . The e f f e c t o f m o u l d f l u x p r o p e r t i e s and c a s t i n g s p e e d on t h e d e p t h o f o s c i l l a t i o n marks has n o t been d e t e r m i n e d c l e a r l y . 2.2.2 ' S u b s u r f a c e S t r u c t u r e Of O s c i l l a t i o n M a r k s O s c i l l a t i o n marks c a n be c l a s s i f i e d a c c o r d i n g t o t h e a d j a c e n t s o l i d i f i e d s t r u c t u r e . Emi e t a l . 9 have d i s c e r n e d two t y p e s o f o s c i l l a t i o n marks b a s e d on t h e p r e s e n c e o r a b s e n c e o f a s m a l l "hook" i n t h e s u b s u r f a c e s t r u c t u r e a d j a c e n t t o e a c h o s c i l l a t i o n mark, F i g . 2-6. The hooks a r e b e l i e v e d t o be a m a n i f e s t a t i o n o f l i q u i d m e t a l o v e r f l o w i n g t h e t o p o f t h e s h e l l w h i c h i s i n c l i n e d t o w a r d t h e m o l t e n p o o l . The a b s e n c e o f h o o k s , i t h as been s u g g e s t e d , i s due t o t h e t o p of t h e i n c l i n e d s h e l l b e i n g pushed back t o w a r d t h e mould w a l l by t h e f e r r o s t a t i c p r e s s u r e . D e t a i l s of t h e s u b s u r f a c e s t r u c t u r e a d j a c e n t t o o s c i l l a t i o n mark were examined by Saucedo e t a l . 2 6 The s e c o n d a r y d e n d r i t e -arm s p a c i n g was m easured and was f o u n d t o be s m a l l e r a t t h e t o p t h a n a t t h e b o t t o m o f t h e o s c i l l a t i o n marks; t h i s f i n d i n g was t a k e n as e v i d e n c e t h a t t h e a r e a a t t h e b o t t o m of o s c i l l a t i o n marks was s o l i d i f i e d a t a r e d u c e d r a t e away f r o m t h e mould w a l l . Thus t h e y s u g g e s t e d 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 . 10 R e c e n t l y p o s i t i v e s e g r e g a t i o n o f s o l u t e e l e m e n t s i n t h e v i c i n i t y of t h e bo t tom of o s c i l l a t i o n marks has been r e p o r t e d , 2 7  2 9 and t h i s w i l l be d e s c r i b e d l a t e r . 2 . 2 . 3 Mechan i sm Of O s c i l l a t i o n Mark F o r m a t i o n S e v e r a l mechanisms have been 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 o f o s c i l l a t i o m m a r k s . S a t o 3 0 has s u g g e s t e d , as d i d Savage and P r i t c h a r d i n 1 9 5 4 3 1 f o r t h e f o r m a t i o n of r e c i p r o c a t i o n marks i n s t e e l b i l l e t s , t h a t t h e s h e l l " s t i c k s " t o t h e mould w a l l such t h a t on t h e u p s t r o k e of t h e m o u l d , t h e s h e l l r u p t u r e s a l l o w i n g l i q u i d ' s t e e l t o p a r t i a l l y f i l l t h e gap c r e a t e d . T h i s e v e n t i s f o l l o w e d by a " h e a l i n g " p e r i o d w h i l e t h e mould moves downward, see F i g . 2 - 7 . T h i s mechanism i s d i f f i c u l t t o j u s t i f y , h o w e v e r , because t h e o s c i l l a t i o n marks on a s l a b s u r f a c e a r e r e l a t i v e l y s t r a i g h t and h o r i z o n t a l w h i c h w o u l d h a r d l y be p o s s i b l e i f the r u p t u r e mechanism p r e d o m i n a t e d . Emi e t a l 9 have p r o p o s e d a mechan i sm, i n w h i c h t h e t o p edge o f t h e s h e l l i s pushed t o w a r d the m o l t e n s t e e l by l i q u i d f l u x 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 of t h e mould o s c i l l a t i o n , see F i g . 2-8 . The mould f l u x i s "pumped" i n t o t h e c h a n n e l between t h e s t e e l and t h e mould by a f r o z e n s l a g r i m a t t a c h e d t o t h e mould w a l l . A t t h e end of the n e g a t i v e - s t r i p p e r i o d , when t h e mould and s t r a n d a r e moving downward w i t h t h e same v e l o c i t y , t h e f l u x p r e s s u r e i s r e l e a s e d and f e r r o s t a t i c p r e s s u r e e i t h e r c a u s e s m o l t e n s t e e l t o o v e r f l o w t h e p a r t i a l l y s o l i d i f i e d m e n i s c u s t o form a hook or t h e m e n i s c u s i s pushed back t o w a r d t h e mould w a l l such t h a t a hook i s not c r e a t e d . Thus the hook a d j a c e n t t o 11 o s c i l l a t i o n marks i s a m a n i f e s t a t i o n o f m o l t e n s t e e l o v e r f l o w i n g a p a r t i a l l y s o l i d i f i e d m e n i s c u s a s s u g g e s t e d e a r l i e r . S i m i l a r c o n c e p t s have been i n v o k e d by o t h e r s . 2 6 3 2 ~ 3 5 Based on e x p e r m e n t a l r e s u l t s o b t a i n e d w i t h a "mould s i m u l a t o r " , Kawakami e t a l . 3 6 have s u g g e s t e d t h a t o s c i l l a t i o n marks a r i s e from t h e i n t e r a c t i o n o f a v i s c o u s l a y e r o f mould f l u x w h i c h moves w i t h t h e mould w a l l , and d u r i n g t h e n e g a t i v e - s t r i p t i m e , p h y s i c a l l y bends t h e t o p o f t h e s o l i d i f y i n g s h e l l . M o l t e n s t e e l t h e n o v e r f l o w s t h e b e n t s h e l l . The s e q u e n c e o f e v e n t s l e a d i n g t o 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 a c c o r d i n g t o t h i s mechanism i s shown s c h e m a t i c a l l y i n F i g . 2-9. U s i n g t h e same t y p e o f "mould s i m u l a t o r " , H. T a k e u c h i e t a l . 3 7 have p r o p o s e d a s l i g h t l y d i f f e r e n t mechanism. They t h e o r i z e d t h a t a s o l i d i f i e d s l a g r i m a t t h e m e n i s c u s , w h i c h moves w i t h t h e mould w a l l , p u s h e s t h e t o p of t h e s h e l l d i r e c t l y t o w a r d t h e m e n i s c u s . E v e n t h o u g h t h e s o l i d s l a g r i m a n d / o r v i s c o u s f l u x l a y e r a r e a f f e c t e d by h e a t f l o w a t t h e m e n i s c u s , no t h e r m a l a n a l y s i s was done i n t h e s e s t u d i e s . T h e s e r e p o r t e d mechanisms a r e e s s e n t i a l l y c o n c e p t u a l i n n a t u r e and have been a p p l i e d o n l y i n a q u a l i t a t i v e manner t o r a t i o n a l i z e t h e i n f l u e n c e of o s c i l l a t i o n c h a r a c t e r i s t i c s on t h e p i t c h and t h e d e p t h o f o s c i l l a t i o n m arks. Owing t o t h e c o m p l e x i t y o f t h e m e n i s c u s phenomena, o n l y a few t h e o r e t i c a l a p p r o a c h e s have been made on t h e mechanism o f o s c i l l a t i o n - m a r k f o r m a t i o n . B a s e d on r e s u l t s o f an i n v e s t i g a t i o n o f t h e s u b s u r f a c e s t r u c t u r e a d j a c e n t t o o s c i l l a t i o n m a r k s , as m e n t i o n e d b e f o r e , S a u c e d o e t a l . 3 8 f o r m u l a t e d a t w o - d i m e n s i o n a l , h e a t -1 2 t r a n s f e r model of s o l i d i f i c a t i o n i n t h e m e n i s c u s r e g i o n . T h i s h e a t f l o w a n a l y s i s i s t o be d i s c u s s e d i n a l a t e r s e c t i o n . However t h e mould o s c i l l a t i o n e f f e c t was n o t t a k e n i n t o a c c o u n t i n t h e i r m o d e l . Based on a m o d i f i c a t i o n o f t h e i r c o n c e p t u a l m o d e l , F i g . 2-8, N a k a t o e t a l . 3 9 p r o p o s e d a s i m p l e m a t h e m a t i c a l model t o e x p l a i n t h e e f f e c t o f mou l d o s c i l l a t i o n c o n d i t i o n s on t h e d e p t h o f o s c i l l a t i o n m arks. T h e i r a s s u m p t i o n s were a s f o l l o w s ; 1) t h e t o p o f t h e s h e l l i s d e f o r m e d d i r e c t l y by a s o l i d s l a g r i m w h i c h moves w i t h t h e o s c i l l a t i n g mould ( s i m i l a r t o T a k e u c h i ' s m o d e l 3 7 ) , 2) t h e e x t e n t o f d e f o r m a t i o n o f t h e s h e l l , v i z t h e d e p t h of o s c i l l a t i o n mark, i s p r o p o r t i o n a l t o t h e maximum d i s p l a c e m e n t of t h e mould r e l a t i v e t o t h e s h e l l , s ee F i g . 2 - 1 0 . The maximum d i s p l a c e m e n t i s g i v e n by E q . ( 2 - 6 ) . d °= A l = 4 s in(2Trft ) - v t (.2-6) max 2 max s max 1 V s arc cos(——) (2-7) max 2-rrf T r f s A c c o r d i n g t o t h e r e s u l t s p r e d i c t e d w i t h t h i s model shown i n F i g . 2-11, t h e d e p t h o f o s c i l l a t i o n marks i n c r e a s e w i t h i n c r e a s i n g o s c i l l a t i o n s t r o k e . T h i s t e n d e n c y i s c o n s i s t e n t w i t h t h e p l a n t d a t a i n F i g . 2-3. However t h e c a l c u l a t e d e f f e c t o f o s c i l l a t i o n f r e q u e n c y on t h e d e p t h of o s c i l l a t i o n mark, v i z t h e mark d e p t h i n c r e a s e s w i t h i n c r e a s i n g o s c i l l a t i o n f r e q u e n c y , i s o p p o s i t e t o o b s e r v a t i o n s made i n p l a n t , F i g . 2-4. 1 3 2.3 Heat T r a n s f e r In The M o u l d Near The M e n s i s c u s 2.3.1 F a c t o r s A f f e c t i n g M o u l d Heat F l u x M o u l d h e a t t r a n s f e r has been d e t e r m i n e d f r o m t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould m e a s u r e d w i t h a l a r g e number of t h e r m o c o u p l e s embedded i n t h e c o p p e r p l a t e s o f t h e m o u l d w a l l . 1 0 4 ° - a 3 i n a d d i t i o n mould h e a t t r a n s f e r has been c h a r a c t e r i z e d by t h e t e m p e r a t u r e change o f mould c o o l i n g w a ter a n d / o r by t h e c a l c u l a t e d t h e r m a l r e s i s t a n c e between s t e e l s h e l l a n d m o u l d w a l l . 1 8 " 5 From t h e s e t h e r m a l i n v e s t i g a t i o n s , t h e f o l l o w i n g s u b j e c t s have been c l a r i f i e d . The most i m p o r t a n t f a c t o r i n f l u e n c i n g t h e h e a t f l u x e x t r a c t e d f r o m t h e mould w a l l i s t h e c a r b o n c o n t e n t of t h e s t e e l , " 0 "" w h i c h has been d i s c u s s e d f r e q u e n t l y i n r e l a t i o n t o t h e f o r m a t i o n o f s u r f a c e c r a c k s . Near 0.10-0.15%C t h e s o l i d i f i e d s h e l l e x p e r i e n c e s volume s h r i n k a g e due t o t h e 6 - 7 t r a n s f o r m a t i o n , w h i c h g i v e s r i s e t o a i r gaps between t h e mould and s h e l l . F u r t h e r m o r e t h e f e r r i t i c s o l i d i f i c a t i o n s t r u c t u r e , w i t h much l e s s m i c r o s e g r e g a t i o n , has a h i g h s t r e n g t h w h i c h may c o n t r i b u t e t o p a r t i a l c o n t a c t w i t h t h e mould w a l l . " 1 T h e r e f o r e a i r g a p s f o r m e d between t h e s h e l l and t h e mould w a l l r e d u c e l o c a l h e a t f l u x , w h i c h may p r e v e n t l o c a l s h e l l g r o w t h and l e a d t o t h e i n i t i a t i o n o f s u r f a c e c r a c k s . S e l e c t i o n of a s u i t a b l e mould f l u x p a r t i a l l y r e l i e v e s t h e n o n u n i f o r m i t y o f h e a t e x t r a c t i o n . " 7 C a s t i n g s p e e d a l s o has a l a r g e i n f l u e n c e on t h e h e a t f l u x , "° b e c a u s e i t c h a n g e s not o n l y t h e s h e l l t h i c k n e s s , v i z a i r gap f o r m a t i o n , b u t a l s o t h e f l o w 1 4 r a t e o f m o l t e n s t e e l a t t h e s o l i d i f i c a t i o n f r o n t " 8 a n d t h e i n f l o w of t h e mould f l u x i n t o t h e "gap" between m o u l d w a l l and t h e s h e l l . 1 8 F i g . 2-12 shows t h e e f f e c t o f c a s t i n g s p e e d on t h e h e a t f l u x p r o f i l e i n t h e m o u l d . " 0 U n f o r t u n a t e l y , i n o b t a i n i n g t h e s e a x i a l d i s t r i b u t i o n s of h e a t f l u x , f r o m t h e r m o c o u p l e r e a d i n g s , o n e - d i m e n s i o n a l h e a t c o n d u c t i o n t h r o u g h t h e t h i c k n e s s of t h e mould w a l l has been assumed w h e r e a s , as S a m a r a s e k e r a and B r i m a c o m b e " 6 have shown, t h e a x i a l component o f h e a t c o n d u c t i o n i n t h e m e n i s c u s r e g i o n i s l a r g e . I m p l i c i t n e g l e c t o f a x i a l h e a t c o n d u c t i o n means t h a t r e p o r t e d d a t a f o r h e a t f l u x a t t h e m e n i s c u s a r e l o w e r t h a n t h e a c t u a l v a l u e s . A n o t h e r d e f i c e n c y , f r o m t h e s t a n d p o i n t of c h a r a c t e r i z i n g h e a t f l u x i n t h e m e n i s c u s r e g i o n , i s t h e a b s e n c e o f h e a t f l u x d a t a o v e r t h e l e n g t h o f t h e mould w a l l above and a d j a c e n t t o t h e m e n i s c u s . Thus t h e h e a t f l u x p r o f i l e s r e p o r t e d i n t h e l i t e r a t u r e a r e i n a d e q u a t e f o r t h e p r e d i c t i o n of t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and t h e s t e e l a t t h e m e n i s c u s . Knowledge o f mould f l u x t e m p e r a t u r e a t t h e m e n i s c u s i s v e r y i m p o r t a n t b e c a u s e i t a f f e c t s t h e v i s c o s i t y , c o n s e q u e n t l y t h e f l o w o f mould f l u x i n t o t h e gap between t h e mould w a l l and t h e s h e l l . 2.3.2 M e n i s c u s S o l i d i f i c a t i o n S o l i d i f i c a t i o n i n t h e m e n i s c u s r e g i o n has been i n v e s t i g a t e d t h e o r e t i c a l l y by Tomono e t a l . 3 2 and S a u c e d o e t a l . 3 8 The main a s s u m p t i o n s i n t h e f i r s t a u t h o r s ' c o m p u t a t i o n a r e a s f o l l o w s : i ) no mould f l u x and mould o s c i l l a t i o n i i ) 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 f l o w i n t h e s t e e l 15 i i i ) t w o - d i m e n s i o n a l s t e a d y - s t a t e h e a t f l o w i n t h e mould w a l l i v ) t e m p e r a t u r e of mould w a l l - w a t e r i n t e r f a c e a t 100°C v) s p e c i f i e d t h i c k n e s s of a i r gap between m o u l d and s h e l l One of t h e i r c a l c u l a t e d r e s u l t s i s shown i n F i g . 2-13. U n f o r t u n a t e l y , t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould w a l l i s c o m p l e t e l y d i f f e r e n t from a c t u a l measured v a l u e s . " 2 Saucedo e t a l . 3 8 p r e s e n t e d 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 m o d e l , w h i c h i s more a c c e p t a b l e t h a n t h e f o r m e r c a s e . T h e i r c o m p u t a t i o n i s b a s e d m a i n l y on t h e f o l l o w i n g a s s u m p t i o n s ; i ) no mould f l u x and mould o s c i l l a t i o n i i ) 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 h e a t f l o w i n t h e s t e e l i i i ) h e a t t r a n s f e r c o e f f i c i e n t assumed f o r t h e m o u l d / s t e e l b o u n d a r y c o n d i t i o n i v ) s o l i d i f i e d s h e l l i s r i g i d i f f r a c t i o n s o l i d e x c e e d s 0.2. ' F i g . 2-14 shows one of t h e i r c a l c u l a t e d r e s u l t s , t h e e f f e c t of c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t on t h e s h e l l p r o f i l e . A c c o r d i n g t o t h e i r r e s u l t s , p a r t i a l s o l i d i f i c a t i o n of t h e m e n i s c u s may be p o s s i b l e . M o r e o v e r the amount o f h e a t e x t r a c t e d from t h e mould w a l l , c a r b o n c o n t e n t , and s u p e r h e a t o f t h e m o l t e n s t e e l have a s t r o n g e f f e c t on t h i s phenomenon. A l t h o u g h t h e y s u g g e s t e d t h a t 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 i s t h e m a j o r c a u s e of o s c i l l a t i o n mark f o r m a t i o n , t h e i r a s s u m p t i o n s e s p e c i a l l y on t h e r i g i d i t y of t h e s o l i d i f i e d s h e l l l e a v e room f o r argument. 16 A n o t h e r p o i n t t o be n o t e d i s t h a t t h e y o v e r e s t i m a t e d t h e h e a t f l u x f r o m t h e m e n i s c u s by u s i n g s q u a r e nodes i n t h e i r f i n i t e -d i f f e r e n c e a n a l y s i s t o a p p r o x i m a t e t h e c u r v a t u r e o f t h e m e n i s c u s . A more p r e c i s e model s h o u l d be d e v e l o p e d t o e s t i m a t e t h e t e m p e r a t u r e d i s t r i b u t i o n n o t o n l y i n t h e s t e e l b u t a l s o i n t h e m o u l d f l u x . 2.4 L u b r i c a t i o n In The M o u l d 2.4.1 C o n s u m p t i o n Of M o u l d F l u x Numerous i n d u s t r i a l t r i a l s have been c o n d u c t e d t o d e t e r m i n e t h e most s u i t a b l e l u b r i c a t i o n c o n d i t i o n s f o r t h e p r o d u c t i o n o f s l a b s w i t h h i g h s u r f a c e q u a l i t y 9 1 0 * 7 « 9 - 6 5 e s p e c i a l l y i n t h e c a s e o f h i g h s p e e d c a s t i n g . 1 2 5 5 5 6 In g e n e r a l a l a r g e c o n s u m p t i o n o f mould f l u x i s e x p e c t e d t o p r o v i d e good l u b r i c a t i o n i n t h e mould. However i t has been f o u n d t h a t i n c r e a s i n g t h e c a s t i n g s p e e d d e c r e a s e s t h e mould f l u x c o n s u m p t i o n , 5 0 5 5 as shown i n F i g . 2-15. F u r t h e r m o r e i n c r e a s i n g t h e o s c i l l a t i o n f r e q u e n c y , even t h o u g h i t r e d u c e s t h e d e p t h of o s c i l l a t i o n m a r k s , 2 1 a l s o d e c r e a s e s t h e f l u x c o n s u m p t i o n , 8 9 F i g . 2-16. Low f l u x c o n s u m p t i o n c a u s e s a h i g h f r i c t i o n f o r c e 6 6 between t h e s t e e l s h e l l and mould w a l l , w h i c h g i v e s r i s e t o s u r f a c e c r a c k s , 5 1 and i n t h e w o r s t c a s e t h e s t i c k i n g o f t h e s h e l l t o t h e mould w a l l . 6 7 In o r d e r t o c o m p e n s a t e f o r t h e r e d u c e d f l u x c o n s u m p t i o n , i m p r o v e m e n t s t o t h e f l u x p r o p e r t i e s , s u c h as v i s c o s i t y and s o l i d i f y i n g t e m p e r a t u r e , 17 have been made r e c e n t l y i n many c a s t i n g p l a n t s . * 7 5 2 5 5 6 0 6 1 6 8 The e f f e c t of t h e v i s c o s i t y and s o f t e n i n g t e m p e r a t u r e o f m o u l d f l u x on t h e f l u x c o n s u m p t i o n a r e shown i n F i g s . 2-17 and 2-1 8 , 5 2 r e s p e c t i v e l y . Q u a n t i t i e s of L i 2 0 , 5 5 BaO a n d / o r B 2 0 3 5 2 6 1 a r e a d d e d t o c o n v e n t i o n a l mould f l u x e s t o d e c r e a s e t h e i r v i s c o s i t y a nd s o f t e n i n g t e m p e r a t u r e w i t h o u t a t t h e same t i m e i n c r e a s i n g e r o s i o n of i m m e r s i o n n o z z l e s . S e v e r a l t h e o r e t i c a l a p p r o a c h e s have been a t t e m p t e d t o u n d e r s t a n d t h e p r o c e s s o f f l u x c o n s u m p t i o n , 1 6 6 9 - 7 1 a n c ; l u b r i c a t i o n i n t h e mould. K o r , 6 9 f o r example, c a l c u l a t e d t h e v e l o c i t y d i s t r i b u t i o n i n t h e f l u x f i l m between t h e mould w a l l and t h e s h e l l t o e s t i m a t e t h e s h e a r s t r e s s a c t i n g on t h e s h e l l s u r f a c e , v i z f r i c t i o n f o r c e , u s i n g t h e f o l l o w i n g e q u a t i o n . 2 P f ^ - - ^ + ^ 2 ^ f ( 2 " 8 ) The m a j o r a s s u m p t i o n on w h i c h h i s model i s b a s e d i s t h a t t h e mould w a l l and t h e s o l i d i f i e d s h e l l a r e p a r a l l e l , and t h e d i s t a n c e between them, v i z t h e t h i c k n e s s o f mould f l u x l a y e r , i s c o n s t a n t . As a r e s u l t , f o r example, t h e t i m e i n d e p e n d e n t t e r m s of t h e a v e r a g e v e l o c i t y of mould f l u x were e x p r e s s e d by E q . ( 2 -9 ) . v c (p -p,)gd - T- + i 2 * f f <2-9> A c c o r d i n g t o E q . ( 2 - 9 ) , f l u x c o n s u m p t i o n i n c r e a s e s w i t h i n c r e a s i n g c a s t i n g s p e e d whereas t h e o p p o s i t e e f f e c t has been o b s e r v e d i n p l a n t , F i g . 2-15. The p r o b l e m c o u l d be due t o t h e s i m p l e a s s u m p t i o n s a s d e s c r i b e d a b o v e ; a l s o t h e e f f e c t o f 18 o s c i l l a t i o n c o n d i t i o n s and c a s t i n g s p e e d on t h e i n f l o w o f mould f l u x i n t o t h e gap between mould w a l l and s h e l l a t t h e m e n i s c u s , w h i c h w i l l d e t e r m i n e t h e t h i c k n e s s o f f l u x l a y e r , i s n o t t a k e n i n t o a c c o u n t . 2.4.2 M o u l d F r i c t i o n In t h e u p p e r p a r t o f t h e mould, t h e f r i c t i o n f o r c e a c t i n g on t h e s t e e l s h e l l d e p e n d s on t h e v i s c o s i t y o f t h e mould f l u x and v e l o c i t y d i s t r i b u t i o n i n t h e l i q u i d f l u x f i l m . 1 " 6 9 8 u x = - P f - ^ T (2-10) However o v e r most o f t h e mould l e n g t h , s o l i d - s o l i d f r i c t i o n " 8 a d d i t i o n a l l y c o m p l i c a t e s t h e s h e l l b e h a v i o r and makes t h e t h e o r e t i c a l a n a l y s i s o f mould f r i c t i o n much more d i f f i c u l t . R e c e n t l y measurements of t h e mould f r i c t i o n f o r c e u s i n g s t r a i n g a u g e s , e t c . have been c a r r i e d o u t i n many p l a n t s . 5 1 5 " 6 0 6 7 7 2 - 7 « A l t h o u g h t h e a n a l y s i s o f t h i s s o r t of measurement has n o t been w e l l e s t a b l i s h e d , 7 3 " 7 5 i t o f f e r s a method o f d e t e r m i n i n g optimum c o n d i t i o n s of mo u l d o p e r a t i o n . Many e x p e r i m e n t a l r e s u l t s r e v e a l t h a t a . l a r g e f r i c t i o n f o r c e may c a u s e b r e a k o u t s a n d / o r s u r f a c e c r a c k s , s u c h a s l o n g i t u d i n a l c r a c k s and t r a n s v e r s e c r a c k s . The mould f r i c t i o n d e c r e a s e s w i t h d e c r e a s i n g c a s t i n g s p e e d , o s c i l l a t i o n f r e q u e n c y and t h e s o f t e n i n g t e m p e r a t u r e o f mould f l u x , w h i c h i s c o n s i s t e n t w i t h t h e c o n s u m p t i o n o f mould f l u x . 19 2 . 5 S u r f a c e Q u a l i t y Of S l a b s R e l a t e d To O s c i l l a t i o n M a r k s . 2 . 5 . 1 T r a n s v e r s e C r a c k s T r a n s v e r s e c r a c k s a r e one of t h e most common s u r f a c e d e f e c t s on c o n t i n u o u s l y c a s t s l a b s , and t h e r e f o r e have a t t r a c t e d numerous s t u d i e s . T r a n s v e r s e c r a c k s a r e o b s e r v e d b o t h on t h e wide and t h e nar row s i d e a n d / o r even a t t h e c o r n e r of s l a b s , 2 9 see F i g . 2 - 1 9 . They a r e p e r p e n d i c u l a r t o t h e c a s t i n g d i r e c t i o n , and appear u s u a l l y a l o n g the b o t t o m of o s c i l l a t i o n m a r k s . F i n e t r a n s v e r s e c r a c k s a l o n g o s c i l l a t i o n marks a r e d i f f i c u l t t o d e t e c t on t h e a s - c a s t s l a b s u r f a c e . Hence i n t h e ca se of s t e e l s s e n s i s t i v e t o t r a n s v e r s e c r a c k s such as m i c r o - a l l o y g r a d e s , 7 6 " 7 8 a c a r e f u l v i s u a l i n s p e c t i o n i s n e c e s s a r y t o f i n d t h e s e f i n e c r a c k s , and t h i s s e r i o u s l y r e d u c e s t h e p r o d u c t i v i t y of c o n t i n u o u s c a s t i n g . In g e n e r a l t r a n s v e r s e c r a c k s form on t h e i n s i d e r a d i u s ( l o o s e ) s i d e of the s t r a n d i n a c u r v e d c o n t i n u o u s c a s t i n g machine w h i c h has a s i n g l e b e n d i n g p o i n t between t h e c u r v e d r e g i o n and s t r a i g h t e n i n g r o l l s . I n a v e r t i c a l b e n d i n g c o n t i n u o u s c a s t i n g machine h a v i n g an a d d i t i o n a l upper b e n d i n g p o i n t , t h e c r a c k s may form on e i t h e r t h e l o o s e o r t h e f i x e d s i d e of t h e s t r a n d . 7 9 These r e s u l t s sugge s t t h a t t r a n s v e r s e c r a c k s a r e formed by t e n s i l e s t r a i n a p p l i e d on t h e s t r a n d a t t h e s t r a i g h t e n i n g p o i n t . 7 6 7 9 D u c t i l i t y measurements have been p e r f o r m e d i n t h e t e m p e r a t u r e range from 1 0 0 0 ° C t o 6 0 0 ° C and low s t r a i n r a t e e = 5 x 1 0 " 3 S " 1 w h i c h i s c l o s e t o 10~*S" 1 a t t h e 20 b e n d i n g p o i n t , 8 0 and i t has been d e t e r m i n e d t h a t t r a n s v e r s e c r a c k s a r e c l o s e l y r e l a t e d t o i n t e r g r a n u l a r e m b r i t t l e m e n t a s s o c i a t e d w i t h t h e A r 3 t r a n s f o r m a t i o n i n t h e t e m p e r a t u r e range f rom 7 0 0 ° C t o 9 0 0 ° C . 8 ° In t h e ca se of p l a i n c a r b o n s t e e l , t h e e m b r i t t l e m e n t i s c aused by f i l m - l i k e p r o e u t e c t o i d f e r r i t e p r e c i p i t a t i n g a l o n g t h e a u s t e n i t e g r a i n b o u n d a r i e s . Imposed s t r a i n t h e n c o n c e n t r a t e s i n t h e f e r r i t e l a y e r t o cause i n t e r g r a n u l a r c r a c k s . In a d d i t i o n , p r e c i p i t a t i o n of A l N 8 1 and Nb , V c a r b o n i t r i d e 8 2 8 3 f u r t h e r d e c r e a s e s t h e d u c t i l i t y of s t e e l i n t h e same t e m p e r a t u r e r a n g e . N i t r o g e n r e d u c e s t h e d u c t i l i t y of s t e e l and i n c r e a s e s t h e f r e q u e n c y of t r a n s v e r s e c r a c k s as shown i n F i g 2-20 (a) and(b) r e s p e c t i v e l y . I n a c t u a l c a s t i n g o p e r a t i o n s , d e c r e a s i n g t h e s e h a r m f u l a l l o y i n g e l e m e n t s ( A l , N , N b , V ) and c o n t r o l l i n g t h e s l a b s u r f a c e t e m p e r a t u r e above 9 0 0 ° C t o a v o i d t h e b r i t t l e t e m p e r a t u r e range a t t h e s t r a i g h t e n i n g p o i n t have been r e p o r t e d t o be e f f e c t i v e i n t h e p r e v e n t i o n o f t r a n s v e r s e c r a c k f o r m a t i o n , F i g . 2 - 2 1 . " P r i o r t o t h e deve lopment of t h e " h o t c h a r g i n g " p r a c t i c e , t h e s l a b s u r f a c e was c o o l e d down be low t h e b r i t t l e t e m p e r a t u r e r a n g e , v i z about 6 0 0 ° C a t t h e u n b e n d i n g p o i n t . 7 6 S u g i t a n i 8 * has p r o p o s e d t h e f o l l o w i n g mechanism f o r t r a n s v e r s e c r a c k f o r m a t i o n based on t h e i r e x p e r i m e n t a l r e s u l t s . The s u r f a c e r e g i o n of s l a b i s s u b j e c t e d t o a r e p e a t e d r i s e and f a l l o f t e m p e r a t u r e i n t h e s p r a y c o o l i n g zone o w i n g t o c h a n g i n g heat e x t r a c t i o n r a t e s a s s o c i a t i e d w i t h r o l l s and s p r a y s . I f t h i s t e m p e r a t u r e f l u c t u a t i o n i s v e r y l a r g e , i t c a u s e s many f i n e 21 t r a n s v e r s e c r a c k s t o f o r m on t h e s u r f a c e o f s l a b s c o n t a i n i n g h i g h A l , Nb, a n d / o r V. T h e s e f i n e c r a c k s p r o p a g a t e t o g e n e r a t e l a r g e t r a n s v e r s e c r a c k s d u r i n g u n b e n d i n g . T h i s mechanism i s s u p p o r t e d by t h e i n v e s t i g a t i o n r e s u l t s of Yamaki e t a l . 8 5 They have e x a m i n e d s l a b s o b t a i n e d f r o m an i n t e r r u p t e d c a s t and f o u n d t h a t f i n e c r a c k s a p p e a r e d o v e r t h e whole s u r f a c e o f t h e s l a b s b e f o r e t h e u n b e n d i n g p o i n t . W i t h r e s p e c t t o t h e e f f e c t of o s c i l l a t i o n marks on t h e f o r m a t i o n of t r a n s v e r s e c r a c k s , i t has been s u g g e s t e d t h a t t h e b o t t o m o f o s c i l l a t i o n marks a c t s a s a n o t c h t o e n h a n c e c r a c k i n g a t t h e u n b e n d i n g p o i n t , b e c a u s e r e d u c i n g t h e d e p t h o f o s c i l l a t i o n marks d e c r e a s e s t h e f o r m a t i o n of t r a n s v e r s e c r a c k s , and t h e c r a c k s a r e u s u a l l y f o u n d on t h e l o o s e s i d e o f t h e s l a b where t e n s i l e s t r a i n i s g e n e r a t e d a t t h e b e n d i n g p o i n t . 7 9 However i t seems u n r e a s o n a b l e t h a t t h e o s c i l l a t i o n marks s i m p l y c r e a t e a n o t c h e f f e c t , b e c a u s e u s u a l l y t h e b o t t o m o f o s c i l l a t i o n marks i s r o u n d e d and f u r t h e r m o r e t h e d e p t h o f marks i s r e d u c e d by s u p p o r t r o l l s . On t h e o t h e r hand T a n a k a e t a l 2 9 m e t a l l o g r a p h i c a l l y i n v e s t i g a t e d t h e b o t t o m of o s c i l l a t i o n marks, and f o u n d f i n e c r a c k s i n i t i a t e d i n r e g i o n s o f p o s i t i v e s e g r e g a t i o n a l o n g t h e s u b s u r f a c e h o o k s ; t h e s e may o f f e r s i t e s f o r t r a n s v e r s e c r a c k s . 22 2 . 5 . 2 P o s i t i v e S e g r e g a t i o n R e c e n t l y p o s i t i v e s e g r e g a t i o n of s o l u t e e l e m e n t s a t t h e b o t t o m of o s c i l l a t i o n marks has been examined i n r e l a t i o n t o s u r f a c e d e f e c t s on s l a b s . In t h e ca se of a p l a i n - c a r b o n s t e e l , p o s i t i v e s e g r e g a t i o n of P and Mn has been d e t e c t e d a l o n g t h e hook i n t h e s u b s u r f a c e s t r u c t u r e of o s c i l l a t i o n m a r k s . 2 8 2 9 T h i s t y p e of s e g r e g a t i o n i s more s e v e r e a t t h e c o r n e r t h a n a t t h e c e n t e r o f s l a b s . The maximum c o n t e n t of p h o s p h o r u s d e t e r m i n e d by EPMA i n t h e s e g r e g a t i o n r e g i o n i n c r e a s e s w i t h i n c r e a s i n g d e p t h of o s c i l l a t i o n m a r k s , as shown i n F i g . 2 - 2 2 . Below t h e p o s i t i v e s e g r e g a t i o n a l o n g t h e h o o k , n e g a t i v e s e g r e g a t i o n was f o u n d . From t h e s e r e s u l t s t h e mechanism f o r p o s i t i v e s e g r e g a t i o n has been p r o p o s e d as f o l l o w s . I n t e r d e n d r i t i c l i q u i d h a v i n g a h i g h s o l u t e c o n t e n t i n t h e t o p of the s h e l l i s squeezed out as t h e s h e l l i s bent d u r i n g t h e downward m o t i o n of t h e m o u l d , see F i g . 2 - 2 3 . F i n e c r a c k s have been o b s e r v e d i n the zone of p o s i t i v e s e g r e g a t i o n of P , and t h e r e f o r e t h i s r e g i o n was c o n s i d e r e d t o o f f e r a good s i t e f o r t r a n s v e r s e c r a c k f o r m a t i o n . I n a d d i t i o n , t h e same t y p e of p o s i t i v e s e g r e g a t i o n has been f o u n d a t t h e b o t t o m of o s c i l l a t i o n marks i n a u s t e n i t i c s t a i n l e s s - s t e e l s l a b s . 2 7 I n t e n s i v e s e g r e g a t i o n o f t e n g i v e s r i s e t o s u r f a c e d e f e c t s on p r o d u c t s r o l l e d f rom t h e s l a b w i t h o u t s u r f a c e c o n d i t i o n i n g , 8 6 hence i t has become one of the t r o u b l e s o m e d e f e c t s a f f e c t i n g t h e " h o t c h a r g i n g " of s t a i n l e s s -s t e e l s l a b s . S u r f a c e s e g r e g a t i o n on s t a i n l e s s s t e e l s l a b s i s 23 c l a s s i f i e d i n t o t h e f o l l o w i n g two t y p e s : A) s e g r e g a t i o n a t t h e o v e r f l o w p a r t on t h e hook, B) s e g r e g a t i o n l a y e r a t t h e s u r f a c e of o s c i l l a t i o n marks w i t h o u t h o o k s , s e e F i g . 2-24. The s e g r e g a t i o n of N i and S i was d e t e r m i n e d i n b o t h t y p e s of s e g r e g a t i o n z o n e , and t h e s e g r e g a t i o n r a t i o o f t h e s e e l e m e n t s was f o u n d t o be c l o s e t o t h e i n v e r s e of t h e p a r t i t i o n c o e f f i c i e n t o f e a c h . Thus i t has been p r o p o s e d t h a t t h e c o n c e n t r a t e d l i q u i d a t t h e s o l i d i f i c a i t o n f r o n t i s t r a n s p o r t e d t o t h e s u r f a c e of t h e s l a b by o v e r f l o w ( t y p e A) o r by b l e e d i n g ( t y p e B ) , see F i g . 2 - 2 5 . The f r e q u e n c y of t h e p o s i t i v e s e g r e g a t i o n 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 , a s shown F i g . 2-26. 2.6 Summary - I n d u s t r i a l N e c e s s i t y F o r The P r e s e n t Work The o s c i l l a t i o n marks f o r m e d on t h e s u r f a c e o f s l a b s a r e n o t o n l y t h e s i t e o f t r a n s v e r s e c r a c k s 6 b u t t h e m s e l v e s a r e u n f a v o r a b l e f o r t h e e s t a b l i s h m e n t o f h o t - c h a r g i n g . 8 6 H i g h -f r e q u e n c y and s h o r t - s t r o k e mould o s c i l l a t i o n , g i v i n g s h o r t n e g a t i v e - s t r i p t i m e , has been f o u n d t o r e d u c e 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 . 8 2 2 However mould o s c i l l a t i o n o f s h o r t n e g a t i v e - s t r i p d e c r e a s e s t h e c o n s u m p t i o n of mould f l u x , w h i c h i n c r e a s e s t h e mould f r i c t i o n . 5 5 H i g h f r i c t i o n f o r c e g e n e r a t e s t e n s i l e s t r a i n i n t h e s h e l l s o l i d i f y i n g i n t h e mould, w h i c h l e a d s t o l o n g i t u d i n a l a n d / o r t r a n s v e r s e c r a c k s 5 1 and, i n t h e w o r s t c a s e , b r e a k o u t s . 6 7 As t h e c o u n t e r m e a s u r e t o t h i s p r o b l e m , u l t r a - l o w v i s c o s i t y m ould f l u x has been d e v e l o p e d 24 r e c e n t l y . 5 2 5 5 6 1 F u r t h e r i n d u s t r i a l d e v e l o p m e n t c a n be a c h i e v e d more e f f i c i e n t l y i f t h e i n i t i a l - s o l i d i f i c a t i o n phenomena a t t h e m e n s i s c u s a r e b e t t e r u n d e r s t o o d . S i n c e t h e m e n i s c u s phenomena a r e c h a r a c t e r i z e d by 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 m a rks, a s t u d y o f t h e o s c i l l a t i o n - m a r k f o r m a t i o n o u g h t t o p r o v i d e t h e n e c e s s a r y i n f o r m a t i o n . However a l l m o d e l s o f o s c i l l a t i o n - m a r k f o r m a t i o n t h a t have been r e p o r t e d up t o now a r e t o o c o n c e p t u a l t o c o n t r i b u t e t o t h e improvement of mould o p e r a t i o n . 9 3 0 3 6 T h e r e f o r e t h e p r e s e n t s t u d y has been u n d e r t a k e n a s t h e f i r s t t h e o r e t i c a l s t u d y of t h e m e n i s c u s phenomena by m e t a l l o g r a p h i c i n v e s t i g a t i o n o f s u b s u r f a c e s t r u c t u r e o f s l a b s f o l l o w e d by s e v e r a l m a t h e m a t i c a l a n a l y s e s . T a b l e I - P r o p e r t i e s of M o l t e n O x i d e M i x t u r e P r o p e r t y M e a s u r e d V a l u e Ref D e n s i t y 2.8 g/cm 3 C a O - S i 0 2 - A l 2 0 3 , C a 0 / S i 0 2 = 1 a t 1500°C 15 I n t e r -f a c i a l t e n s i o n 1 1 20 a ) , 1250 b) dyn/cm a) CaO=47%, S i 0 2 = 3 7 % , A 1 2 0 3 = 1 6 % b) CaO=45%, S i 0 2 = 3 4 % , A l 2 0 3 = 2 1 % C=0.04%, Mn=0.04%, S i = 0 . 0 3 % , P=0.01%, S=0.02% a t 1500-1580°C 1 5 v i s c o s i t y l o g Mf=0.578xl0"/T-2.979 Mf: v i s c o s i t y of mould f l u x (P) T: t e m p e r a t u r e o f mould f l u x (K) 1 6 Spec i f i c H eat 1 . 1 3 a ) , 1.30 b) J/g°C Ca0=52%, S i 0 2 = 4 3 % , A l 2 0 3 = 5 % a) a t 1000°C, b) a t 1500°C 17 T h e r m a l C o n d u c t i v i t y 1.26 W/mK CaO=40%, S i O 2 = 4 0 % , A l 2 O 3 = 2 0 % a t 1360°C 18 2.34 W/mK mould f l u x ( Kawasaki S t e e l ) a t 1000 - 1500°C 19 27 F i g . 2-2 M a c r o s c o p i c s t r u c t u r e and i n t h e m e l t i n g mould f l u x t e m p e r a t u r e d i s t r i b u t i o n l a y e r on t h e m e n i s c u s . 1 2 E f f e c t o f o s c i l l a t i o n s t r o k e on t h e d e p t h of o s c i l l a t i o n m a r k s . 9 29 Stroke t 1.4 Large • o c Middle O b Small c o ~ 1.2 u o OL a> •o 1.0 0.8 1 60 80 100 120 Frequency (cycle/min) F i g . 2-4 E f f e c t of o s c i l l a t i o n f r e q u e n c y on 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 . 1 6 30 1200 F i g . 2-5 R e l a t i o n s h i p between t h e d e p t h o f o s c i l l a t i o n marks and t h e n e g a t i v e s t r i p t i m e . 2 3 - 2 5 31 F i g . 2-6 D e n d r i t e s t r u c t u r e i n the v i c i n i t y of o s c i l l a t i o n m a r k s . 9 (1) H o o k - l i k e s t r u c t u r e , (2) N o n - h o o k - l i k e s t r u c t u r e , (3) D i r e c t i o n of growth of p r imary d e n d r i t e , (4) C a s t i n g d i r e c t i o n . F i g . 2-7 F o r m a t i o n mechanism of o s c i l l a t i o n marks by S a t o . 3 0 33 34 F i g . 2-9 S c h e m a t i c d i a g r a m o f "mould s i m u l a t o r " (a) and f o r m a t i o n mechanism o f o s c i l l a t i o n marks (b) by Kawakami e t a l . 3 6 35 F i g . 2-10 Change of d i s p l a c e m e n t o f t h e mould and t h e s h e l l w i t h t i m e . 36 F i g . 2-11 C a l c u l a t e d e f f e c t o f o s c i l l a t i o n f r e q u e n c y and s t r o k e on t h e d e p t h of o s c i l l a t i o n marks by N a k a t o ' s m o d e l . 3 9 37 F i g . 2-12 E f f e c t of t h e c a s t i n g s p e e d on t h e a x i a l h e a t f l u x p r o f i l e i n t h e mould w a l l . 0 0 38 1125 1325 1*25 1525 F i g . 2-13 C a l c u l a t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould w a l l and i n t h e s t e e l . 3 2 39 0024 .hcon>kWm~2K1 3 <0O08f UJ O z < u>0-004f Q f s > ° 2 -2-1. h 1 85kWm K carbon content' AT 15 K t,s - - 0-3 0-6 _L 0 002 0 004 0 006 0008 DISTANCE FROM MOULD WALL(x),m F i g . 2-14 C a l c u l a t e d s h e l l p r o f i l e w i t h d i f f e r e n t c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t s ( h ) . 3 8 con 40 OJ CP c o CL E 3 CO c o o X 081 ^ 0.6 0.4 0.2 Fiuxr type B O. O. 1 1 1 S = 6mm f = 60~l20cpm .Q^FIUX ,type E - A Flux , type A O • A - — 0.6 1.0 1.4 1.8 2.2 Casting speed (m/min ) F i g . 2-15 E f f e c t of c a s t i n g s p e e d on mould f l u x c o n s u m p t i o n . 5 5 41 _ 0.7 0) a> Ui c o cn c o Q. E to c o o 0.6 \ 0.5 0.4 60 1 1 • s = 7.8 mm o s = 4.2 mm • s = 3.5 m m V s= 1.0 m/ min JJ f = 2.7P (at I300°C) 1 100 140 180 220 Frequency of mould oscillation (cpm) F i g . 2-16 E f f e c t o f o s c i l l a t i o n f r e q u e n c y on mould f l u x consumpt i o n . 8 42 F i g . 2-17 E f f e c t o f v i s c o s i t y of mould f l u x on t h e f l u x consumpt i o n . 5 2 43 F i g . 2-18 E f f e c t o f s o f t e n i n g t e m p e r a t u r e of m o u l d f l u x on t h e f l u x c o n s u m p t i o n . 5 2 44 F i g . 2-19 T r a n s v e r s e c r a c k s o b s e r v e d on s l a b s u r f a c e . 2 3 45 1 0 0 - 6 0 o < 4 0 " DC 2 0 • 7 0 0 8 0 0 9 0 0 1000 1100 D e f o r m a t i o n t e m p e r a t u r e ( ° C ) c S i M n P S A l N o 0.13 0.07 0.35 0.002 0.003 0.004 0.0015 A ») 1) 0.0033 • „ j; ») 0.003 0.004 — 0.0052 o »5 5J — 0.0079 0 10 20 30 40 50 60 70 80 Nitrogen content in steel, ppm Ca) (b) F i g . 2-20 E f f e c t of n i t r o g e n on t h e hot d u c t i l i t y i n low c a r b o n s t e e l 8 0 ( a ) , and on t h e f r e q u e n c y of t r a n s v e r s e c r a c k f o r m a t i o n i n Nb -V s t e e l B 2 ( b ) . 46 F i g . 2-21 T e m p e r a t u r e p r o f i l e on t h e s l a b s u r f a c e t o a v o i d t h e b r i t t l e t e m p e r a t u r e r a n g e a t t h e u n b e n d i n g p o i n t . " 47 F i g . 2-22 R e l a t i o n s h i p between t h e maximum c o n t e n t of P and 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 . 2 9 48 Mould liquid flux in-flow Bending Dendrite Concentrated liquid liquid flux out-flow — Overflow of liquid steel Positive segregation Negative segregation (a) During negative strip period (b) During positive strip period F i g . 2-23 Mechanism of p o s i t i v e s e g r e g a t i o n by Tanaka e t a l . 2 9 F i g . 2-24 Types of p o s i t i v e s e g r e g a t i o n of s t a i n l e s s s t e e l s l a b . 8 6 50 F i g . 2-25 Mechanism of p o s i t i v e s e g r e g a t i o n by H. T a k e u c h i e t a l . 2 7 O.IO 020 0.30 IN (sec) F i g . 2-26 R e l a t i o n between f r e q u e n c y of p o s i t i v e s e g r e g a t i o n i n s t a i n l e s s s t e e l s l a b s and n e g a t i v e - s t r i p t i m e . 8 (Ne: N e g a t i v e - s t r i p r a t i o ) 51 3. SCOPE AND OBJECTIVES OF THE PRESENT WORK 3.1 Scope Of The P r e s e n t Work The p r e s e n t work has been u n d e r t a k e n t o s h e d l i g h t on t h e i n i t i a l s o l i d i f i c a t i o n phenomena i n t h e c o n t i n u o u s c a s t i n g m ould, t h r o u g h t h e s t u d y o f t h e f o r m a t i o n of o s c i l l a t i o n m arks. An a t t e m p t i s made t o overcome some of t h e d e f i c e n c i e s of e a r l i e r i n v e s t i g a t i o n s and t o e s t a b l i s h a s t r o n g e r t h e o r e t i c a l f o u n d a 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 t h a n h i t h e r t o has been r e p o r t e d . The a p p r o a c h t a k e n has been t o examine o s c i l l a t i o n marks m e t a l l o g r a p h i c a l l y from a l a r g e number of s l a b s a m p l e s and t h e r e b y t o o b s e r v e , a t f i r s t hand, t h e g e o m e t r y of t h e marks, t h e a d j a c e n t s u r f a c e s t r u c t u r e and t h e i n f l u e n c e of c a s t i n g v a r i a b l e s on t h e m e t a l l o g r a p h i c a s p e c t s o f o s c i l l a t i o n m a r k s . Next a s e r i e s o f t h e o r e t i c a l a n a l y s e s have been u n d e r t a k e n t o examine t h r e e of t h e phenomena t h a t a r e i m p o r t a n t i n t h e m e n i s c u s r e g i o n . i ) h e a t f l o w i n v o l v i n g s t e e l , mould f l u x , and t h e mould w a l l i i ) f l o w of m o l t e n mould f l u x i n t o t h e gap between t h e s t e e l and t h e mould w a l l , and t h e g e n e r a t i o n o f p r e s s u r e g r a d i e n t s i n t h e f l u x due t o m o u l d o s c i l l a t i o n i i i ) d e f o r m a t i o n o f t h e m e n i s c u s as a r e s u l t of t h e p r e s s u r e g r a d i e n t i n t h e a d j a c e n t mould f l u x . 52 T h i s a p p r o a c h r e v e a l s t h e i m p o r t a n c e of p r e v i o u s l y u n r e p o r t e d f a c t o r s l i k e t h e g e o m e t r y o f t h e f l u x c h a n n e l n e a r t h e m e n i s c u s , and t h e d e f o r m a t i o n o f t h e m e n i s c u s s h a p e by t h e o s c i l l a t i o n - g e n e r a t e d f l u x p r e s s u r e . F i n a l l y t h e a n a l y s i s o f h e a t t r a n s f e r and f l u i d f l o w a r e c o m b i n e d i n an o v e r a l l model o f t h e m e n i s c u s t h a t p r e d i c t s t r e n d s t h a t a r e i n a g r e e m e n t w i t h i n d u s t r i a l f i n d i n g s . The r e s u l t s o f e a c h s u b - s t u d y a r e f i r s t p r e s e n t e d f o l l o w e d by a d i s c u s s i o n o f mechanism o f o s c i l l a t i o n mark f o r m a t i o n and a p r e s e n t a t i o n of t h e o v e r a l l model p r e d i c t i o n s . I n t h e n e x t s t a g e of t h e work, a d d i t i o n a l s t u d i e s have been made on t h e mechanism of t r a n s v e r s e - c r a c k f o r m a t i o n and p o s i t i v e s e g r e g a t i o n , e s p e c i a l l y i n r e l a t i o n t o o s c i l l a t i o n m arks. F i r s t l y b o t h d e f e c t s have been s t u d i e d m e t a l l o g r a p h i c a l l y w h i c h i n c l u d e d an i n v e s t i g a t i o n of t h e e t c h e d s u b s u r f a c e s t r u c t u r e and o f t h e s u r f a c e o f t h e t r a n s v e r s e c r a c k s by SEM. A l s o t h e t y p e s of s e g r e g a t i o n have been c l a s s i f i e d w i t h t h e a i d o f s e v e r a l e t c h i n g methods; and t h e e x t e n t of p o s i t i v e s e g r e g a t i o n has been d e t e r m i n e d by CMA (Computer a i d e d M i c r o A n a l y z e r ) . S e c o n d l y 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 w h i c h a c c o u n t s f o r t h e shape of t h e o s c i l l a t i o n marks has been d e v e l o p e d t o e x p l a i n t h e m e t a l l u r g i c a l r e l a t i o n s h i p s o f b o t h d e f e c t s t o t h e o s c i l l a t o n m arks. P a r t i c u l a r l y r e g a r d i n g p o s i t i v e s e g r e g a t i o n , a mechanism has been p r o p o s e d b a s e d on t h e mechanism o f 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 i s s t u d y . F i n a l l y a t h e o r e t i c a l d i s c u s s i o n i s p r e s e n t e d w h i c h e x p l a i n s how d e c r e a s i n g t h e d e p t h o f o s c i l l a t i o n marks c o n t r i b u t e s t o r e d u c i n g t h e s u r f a c e d e f e c t s . 53 3 . 2 The O b j e c t i v e s Of The P r e s e n t Work The o b j e c t i v e s of t h e p r e s e n t work can be summar ized as f o l l o w s : I . To d e t e r m i n e t h e i m p o r t a n t f a c t o r s w h i c h a f f e c t t h e shape of o s c i l l a t i o n marks by m e t a l l o g r a p h i c i n v e s t i g a t i o n of s l a b s a m p l e s . I I . To d e v e l o p a t h e o r e t i c a l model of t h e o s c i l l a t i o n mark f o r m a t i o n , w h i c h p r e d i c t s t h e shape of t h e mark as a f u n c t i o n o f p r o c e s s v a r i a b l e s . T h i s i n v o l v e s , (1) C a l c u l a t i o n of t h e h e a t - f l u x p r o f i l e u s i n g 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 and i n p l a n t measurement of mould w a l l t e m p e r a t u r e . (2) Deve lopment of 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 h e a t - t r a n s f e r model t o e s t i m a t e t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e s t e e l and i n t h e mould f l u x c l o s e t o t h e m e n i s c u s . (3) Deve lopment of a f l u i d - f l o w model t o c a l c u l a t e t h e f l u i d p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l a t t h e m e n i s c u s . (4) C a l c u l a t i o n of the change o f t h e m e n i s c u s shape w i t h mould o s c i l l a t i o n t o o b t a i n i n s i g h t i n t o s t e e l o v e r f l o w a t t h e m e n i s c u s and t h e f o r m a t i o n of s u b s u r f a c e hooks a d j a c e n t t o o s c i l l a t i o n m a r k s . (5) T h e o r e t i c a l e x p l a n a t i o n of t h e e f f e c t of p r o c e s s f a c t o r s w h i c h have been a l r e a d y r e p o r t e d and a l s o 54 f o u n d i n m e t a l l o g r a p h i c i n v e s t i g a t i o n i n t h e p r e s e n t work. I I I . To d e t e r m i n e t h e e f f e c t of o s c i l l a t i o n marks on t h e m o r p h o l o g y o f t r a n s v e r s e c r a c k s and p o s i t i v e s e g r e g a t i o n , by m e t a l l o g r a p h i c i n v e s t i g a t i o n s of s l a b s a m p l e s . IV. To e x p l a i n t h e o r e t i c a l l y t h e f o r m a t i o n o f p o s i t i v e s e g r e g a t i o n b a s e d on t h e mechanism o f o s c i l l a t i o n mark f o r m a t i o n p r o p o s e d i n t h i s i n v e s t i g a t i o n . V. To d e v e l o p 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 m o d e l , w h i c h i n c o r p o r a t e s t h e shape o f o s c i l l a t i o n marks, t o e x p l a i n t h e t e m p e r a t u r e h i s t o r y i n e a c h p a r t of o s c i l l a t i o n m arks, and c o n s e q u e n t l y n o n u n i f o r m i t y o f s h e l l g r o w t h i n t h e mould. 55 4. FORMATION OF OSCILLATION MARKS 4 .1 M e t a l l u r g i c a l I n v e s t i g a t i o n 4.1.1 C a s t i n g C o n d i t i o n s Of S l a b Samples The m e t a l l o g r a p h i c s t u d y o f o s c i l l a t i o n marks h a s been made m a i n l y on t h e s l a b s a m p l e s f r o m Company A. The c a s t i n g c o n d i t i o n s and c o m p o s i t i o n o f s l a b s a m p l e s o f Company A a r e p r e s e n t e d i n T a b l e I I . Note t h a t t h e c a r b o n c o n t e n t o f t h e s t e e l r a n g e s from 0.08 t o 0.26% and t h a t t h e s t e e l h as e i t h e r been S i - k i l l e d o r A l - k i l l e d . The same t y p e o f mo u l d f l u x was u s e d f o r a l l t h e h e a t s , and m e t a l - l e v e l c o n t r o l was i m p l e m e n t e d . In a d d i t i o n s l a b s a m p l e s from Company B t o E were e x a m i n e d t o d e t e r m i n e t h e e f f e c t of d i f f e r e n t c a s t i n g c o n d i t i o n s on t h e s u b s u r f a c e s t r u c t u r e o f o s c i l l a t i o n m arks. The c a s t i n g c o n d i t i o n s of s l a b s a m p l e s from t h e s e c o m p a n i e s a r e p r e s e n t e d i n T a b l e I I I t o V I . The c h a r a c t e r i s t i c s o f t h e c a s t i n g c o n d i t i o n s of t h e s e c o m p a n i e s a r e as f o l l o w s : Company B: h i g h n i t r o g e n c o n t e n t i n t h e s t e e l , Company C: l a r g e e x t e n t o f m e n i s c u s l e v e l v a r i a t i o n , Company D: m u l t i - p o r t p r a c t i c e and o r d i n a r y b i f u r c a t e d i m m e r s i o n n o z z l e , s ee F i g . 4-1, Company E : e l e c t r o m a g n e t i c s t i r r i n g i n t h e mould, see F i g . 4-2. 56 4.1.2 I n v e s t i g a t i o n P r o c e d u r e Samples f o r t h e p r e s e n t s t u d y were c u t o u t f r o m t h e n a r r o w f a c e of t h e s l a b s b e c a u s e , u n l i k e t h e b r o a d f a c e , i t i s n o t d e f o r m e d by t h e s u p p o r t r o l l s i n t h e c a s t i n g m a c hine., The s u r f a c e of t h e s l a b s a m p l e s was s a n d - b l a s t e d t o remove t h e o x i d e f i l m b e f o r e e x a m i n a t i o n . F i r s t l y 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 was m e a s u r e d a t t h e c e n t e r and t h e c o r n e r o f t h e a s - c u t s a m p l e s . Then t h e s t e e l s a m p l e s were c u t l o n g i t u d i n a l l y p e r p e n d i c u l a r t o t h e o s c i l l a t i o n marks t o r e v e a l t h e p r o f i l e o f t h e s l a b s u r f a c e a n d t h e s u b s u r f a c e s t r u c t u r e . The l o n g i t u d i n a l s e c t i o n s so o b t a i n e d were m a c h i n e d f l a t , p o l i s h e d and e t c h e d w i t h e i t h e r p i c r i c a c i d o r n i t a l . The s u b s u r f a c e s t r u c t u r e was e x a m i n e d and p h o t o g r a p h e d u s i n g a s t a n d a r d o p t i c a l m i c r o s c o p e . The d e p t h o f o s c i l l a t i o n marks was m e a s u r e d f r o m t h e base o f t h e mark t o t h e l e v e l o f a r u l e r e s t i n g on t h e s l a b s u r f a c e . 4.1.3 A p p e a r a n c e And P i t c h Of O s c i l l a t i o n M a rks The narrow f a c e of a t y p i c a l s l a b , e x h i b i t i n g o s c i l l a t i o n m arks, i s shown i n F i g . 4-3 (Company A ) . The marks a r e s e e n t o be e v e n l y s p a c e d an d c l o s e l y p a r a l l e l a c r o s s t h e f a c e . F i g . 4-4 shows t h e p i t c h of o s c i l l a t i o n marks m e a s u r e d o v e r 20 cm i n t h e c a s t i n g d i r e c t i o n , p l o t t e d a g a i n s t c a s t i n g s p e e d . T h e r e i s n o t much d i f f e r e n c e i n t h e p i t c h between c e n t e r and c o r n e r . The a v e r a g e v a l u e o f 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 a b o u t 16mm w h i c h c o r r e s p o n d s c l o s e l y t o t h a t c a l c u l a t e d f r o m E q . ( 2 - 5 ) , a s e x p e c t e d . The v a l u e was n o t o b s e r v e d t o i n c r e a s e 57 w i t h t h e c a s t i n g speed because mould f r e q u e n c y was l i n k e d t o t h e c a s t i n g s p e e d . However , t h e v a r i a t i o n i n t h e measured p i t c h , e x p r e s s e d as a s t a n d a r d d e v i a t i o n , was found t o i n c r e a s e f rom 0 .7 t o 5 .2 mm w i t h i n c r e a s i n g c a s t i n g speed from 0 . 8 9 t o 1.25 m / m i n . The r e s u l t s were t h e same f o r t h e c e n t e r and c o r n e r of t h e s l a b f a c e . P r e s u m a b l y t h e i n c r e a s e d v a r i a b l i t y of p i t c h i s c a u s e d by sudden s m a l l changes i n m e n i s c u s l e v e l due t o f l o w v a r i a t i o n and t u r b u l e n c e , t h a t a r e more d i f f i c u l t t o c o n t r o l a t t h e h i g h e r c a s t i n g s p e e d s , even though t h e m e n i s c u s l e v e l c o n t r o l l e r was i n u s e . On t h e s u r f a c e of s l a b s ample s f rom Company B , numerous s e v e r e t r a n s v e r s e c r a c k s were o b s e r v e d a l o n g t h e o s c i l l a t i o n m a r k s . The c r a c k s a r e r e l a t e d t o t h e h i g h c o n t e n t of n i t r o g e n i n t h e s t e e l s l a b s . The r e s u l t s of t h e i n v e s t i g a t i o n w i l l be p r e s e n t e d i n t h e f o l l o w i n g c h a p t e r . S l a b s from Company C e x h i b i t i r r e g u l a r o s c i l l a t i o n marks as shown i n F i g . 4 - 5 . T h i s i s c a u s e d by l a r g e v a r i a t i o n i n m e n i s c u s l e v e l due t o s t e e l f l o w i n g t u r b u l e n t l y f rom t h e upward i n c l i n e d p o r t s o f t h e i m m e r s i o n n o z z l e . The measured p i t c h o f t h e o s c i l l a t i o n marks i s s m a l l e r t h a n t h a t c a l c u l a t e d from E q . (2-5) p o s s i b l y because s u r f a c e waves g e n e r a t e d by t h e m e t a l s t r e a m i m p i n g i n g c l o s e t o t h e m e n i s c u s , d i s r u p t e d t h e phenomena r e s u l t i n g f rom mould o s c i l l a t i o n , e g . o v e r f l o w a t t h e m e n i s c u s . The c a s t i n g c o n d i t i o n s of Company D a r e c h a r a c t e r i z e d by a l o w - f r e q u e n c y m c u l d o s c i l l a t i o n , see F i g . 4 - 6 , w h i c h g i v e s r i s e t o o s c i l l a t i o n marks h a v i n g a l o n g p i t c h . I n t h i s c a se 58 i n d i s t i n c t marks were found between t h e r e g u l a r o s c i l l a t i o n marks as shown i n F i g . 4 - 7 . The d e p t h of t h e s e marks i s much l e s s t h a n t h a t of the r e g u l a r o s c i l l a t i o n m a r k s . When a b i f u r c a t e d n o z z l e i s employed t h e o s c i l l a t i o n marks a r e - s t r a i g h t and d i s t i n c t i v e , F i g . 4 - 7 ( a ) . On t h e o t h e r h a n d , t h e y a r e r e l a t i v e l y d i s o r d e r e d , and do not appear as s t r a i g h t l i n e s on t h e s l a b s u r f a c e when a m u l t i - p o r t n o z z l e i s u s e d , F i g . 4 - 7 ( b ) . The i n d i s t i n c t marks between r e g u l a r o s c i l l a t i o n marks show t h e same t e n d e n c y . I n t h e ca se of t h e m u l t i - p o r t p r a c t i c e t h e s e d i s o r d e r e d marks a r e caused by t h e upward d i r e c t e d s t r e a m o f s t e e l w h i c h d i s t u r b s the m e n i s c u s . F i g . 4 - 8 ( a ) and (b) show t h e s u r f a c e s o f s l a b s w h i c h have been p r o d u c e d a t Company E w i t h and w i t h o u t e l e c t r o m a g a n e t i c s t i r r i n g i n t h e mould r e s p e c t i v e l y . I t i s c l e a r l y o b s e r v e d t h a t t h e d e p t h of t h e o s c i l l a t i o n marks was r e d u c e d by t h e e l e c t o m a g n e t i c s t i r r i n g w h i c h p r o d u c e d a l i q u i d f l o w v e l o c i t y of about 0 . 8 m / s . 4 . 1 . 4 S u b s u r f a c e S t r u c t u r e Of O s c i l l a t i o n M a r k s O s c i l l a t i o n marks were found w i t h and w i t h o u t hooks i n t h e a d j a c e n t s u b s u r f a c e s t r u c t u r e , as r e p o r t e d by Emi e t a l . 9 a l t h o u g h t h e m a j o r i t y e x h i b i t h o o k s . A s t u d y o f s l a b samples from Company A r e v e a l e d an e f f e c t of the c a r b o n c o n t e n t of t h e s t e e l on t h e s u b s u r f a c e s t r u c t u r e . T y p i c a l s u b s u r f a c e s t r u c t u r e s a s s o c i a t e d w i t h t h e two t y p e s of o s c i l l a t i o n marks i n l o w - c a r b o n (0 .08-0 .09%) and medium-carbon (0.26%) s l a b s a r e 59 shown i n F i g s . 4-9 and 10. I t i s seen t h a t o s c i l l a t i o n marks a c c o m p a n i e d by hooks i n 0.09%C s l a b s , F i g . 4 ~ 9 ( a ) , a r e deeper and t h e t r o u g h a s s o c i a t e d w i t h each mark i s l o n g e r t h a n t h o s e i n t h e 0.26%C s l a b s , F i g . 4 - 9 ( b ) . T h i s d i f f e r e n c e i s not found i n t h e c a se of o s c i l l a t i o n marks w i t h o u t h o o k s , F i g . 4 - 1 0 . M o r e o v e r , hooks i n t h e s u b s u r f a c e of 0.09%C s l a b s form a s m a l l e r a n g l e w i t h t h e s u r f a c e t h a n i n the 0.26%C s l a b s . The d e n d r i t e o r i e n t a t i o n a d j a c e n t t o t h e o s c i l l a t i o n marks i s s i m i l a r t o t h a t d e s c r i b e d by Emi e t a l . 9 , v i z . t h a t d e n d r i t e s i n i t i a l l y grow n o r m a l e i t h e r t o t h e hooks when they a r e p r e s e n t o r t o t h e c u r v e d s u r f a c e of t h e o s c i l l a t i o n mark when t h e hooks a r e a b s e n t . I n e i t h e r c a s e , t h e d e n d r i t e s t h e n change o r i e n t a t i o n w i t h i n a s h o r t d i s t a n c e t o become r o u g h l y p e r p e n d i u l a r t o t h e mould w a l l . F i g . 4-11 shows t h e change of t h e d e n d r i t e arm s p a c i n g about 1mm from t h e s l a b s u r f a c e a l o n g t h e c a s t i n g d i r e c t i o n i n a 0 . 09%- car bon s l a b . A l t h o u g h t h e r e i s s c a t t e r i n t h e measured v a l u e s , t h e s e c o n d a r y d e n d r i t e arm s p a c i n g s a r e s m a l l e r near t h e t o p of t h e mark t h a n near t h e b o t t o m . T h i s i n d i c a t e s t h a t t h e s o l i d i f i c a t i o n r a t e i s g r e a t e r a t t h e t o p t h a n a t t h e b o t t o m of t h e o s c i l l a t i o n m a r k s . S l a b samples f r e e of s u b s u r f a c e hooks c h a r a c t e r i s t i c a l l y a l s o c o n t a i n e d s m a l l s p h e r i c a l b l o w h o l e s , t h a t a r e l i k e l y t r a p p e d a r g o n gas t h a t was i n j e c t e d t h r o u g h t h e submerged n o z z l e . F i g . 4 - l 2 ( a ) shows t h e s u b s u r f a c e s t r u c t u r e of a s l a b sample f rom Company B . A n o n - m e t a l l i c s u b s t a n c e was found a t t h e end o f t h e s u b s u r f a c e as i n d i c a t e d by t h e a r r o w . Enhanced m a g n i f i c a t i o n of t h e s u b s t a n c e shown i n F i g . 4 - 1 2 ( b ) , s u g g e s t s 60 t h a t i t i s an o x i d e and x - r a y a n a l y s i s u s i n g t h e SEM c l e a r l y i n d i c a t e s i t i s e n t r a p p e d mould f l u x b e c a u s e Na and K p r e s e n t i n t h e mould powder were d e t e c t e d , see F i g . 4-13. T h e s e r e s u l t s s u g g e s t t h a t t h e o u t s i d e of t h e hook once had been c o v e r e d by m o u l d f l u x , a p a r t o f w h i c h was t r a p p e d by o v e r f l o w i n g s t e e l . S l a b s a m p l e s from Company C a r e c h a r a c t e r i z e d by i r r e g u l a r o s c i l l a t i o n marks on t h e s l a b s u r f a c e . The s u b s u r f a c e s t r u c t u r e a d j a c e n t t o t h e s e marks d o e s n o t e x h i b i t t h e p r e s e n c e o f h o o k s , see F i g . 4-14. F i g u r e s 4-15(a) and (b) show t h e s u b s u r f a c e s t r u c t u r e i n s l a b s a m p l e s from Company D, p r o d u c e d w i t h t h e c o n v e n t i o n a l b i f u r c a t e d i m m e r s i o n n o z z l e and w i t h t h e m u l t i - p o r t p r a c t i c e r e s p e c t i v e l y . Long s u b s u r f a c e h o o k s can be s e e n i n t h e s l a b sample c a s t w i t h t h e b i f u r c a t e d i m m e r s i o n n o z z l e (25 d e g r e e downward), F i g . 4 - 1 5 ( a ) , a s compared w i t h t h e sample f r o m t h e s l a b p r o d u c e d u s i n g t h e m u l t i p o r t n o z z l e w i t h t h e p o r t s a n g l e d 15° upward, F i g . 4 - l 5 ( b ) . S uch l o n g hooks a r e u n d e s i r a b l e b e c a u s e t h e y may e n t r a p i n c l u s i o n s r i s i n g i n t h e mould p o o l a n d t h e r e b y r e d u c e t h e s u r f a c e q u a l i t y o f t h e s l a b s . The d i f f e r e n c e i n t h e n a t u r e o f t h e hooks w i t h t h e two t y p e s o f n o z z l e s i s most l i k e l y c a u s e d by d i s s i m i l a r f l o w p a t t e r n s and d i f f e r e n c e s i n t h e c o n v e c t i o n o f s u p e r h e a t t o t h e n ewly s o l i d i f y i n g s h e l l a t t h e m e n i s c u s . In t h e c a s e of t h e b i f u r c a t e d n o z z l e w i t h downward a n g l e d p o r t s , c o n v e c t i o n a t t h e m e n i s c u s s h o u l d be r e l a t i v e l y m i l d so t h a t 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 i s f a v o u r e d . On t h e o t h e r hand, t h e m u l t i p o r t n o z z l e w i t h i t s s i x upward f l o w i n g 61 m e t a l s t r e a m s c r e a t e s g r e a t e r t u r b u l e n t c o n v e c t i o n a t t h e m e n i s c u s and t h e e x t e n t of i n i t i a l s o l i d i f i c a t i o n i s r e d u c e d . As m e n t i o n e d i n t h e p r e v i o u s s e c t i o n , o v e r f l o w marks a r e o b s e r v e d between s u c c e s s i v e o s c i l l a t i o n m arks. The s u b s u r f a c e s t r u c t u r e o f t h e s e marks does n o t e x h i b i t any h o o k s w h i c h s u g g e s t s t h a t t h e s e marks a r e d i f f e r e n t f r o m t h e o s c i l l a t i o n ma r k s . S l a b s a m p l e s f r o m Company E were p r o d u c e d w i t h an i n - m o u l d e l e c t r o m a g n e t i c s t i r r e r (EMS). The s u b s u r f a c e s t r u c t u r e o f s a m p l e s w i t h o u t and w i t h mould EMS a r e p r e s e n t e d i n F i g s . 4 -16(a) and (b) r e s p e c t i v e l y . The t y p e o f s t e e l c a s t i s a p s e u d o -rimmed s t e e l w h i c h c o n t a i n s o n l y a t r a c e o f A l . B e c a u s e o f t h e h i g h oxygen c o n t e n t o f t h e s t e e l , CO b l o w h o l e s , t h a t a r e c l e a r l y d i s t i n g u i s h e d f r o m t h e Ar b l o w h o l e s , were fo r m e d commencing j u s t b e n e a t h t h e s u r f a c e o f s l a b s c a s t w i t h o u t mould EMS, F i g . 4 -1 6 ( a ) . However CO b l o w h o l e s a r e s u p p r e s s e d i n t h e s u b s u r f a c e l a y e r when mould EMS i s employed, F i g . 4 - 1 6 ( b ) . I t i s a l s o o b s e r v e d t h a t t h e hooks become s h o r t e r , and c o n s e q u e n t l y t h e d e p t h of o s c i l l a t i o n marks i s r e d u c e d w i t h mould EMS. 4.1.5 D e p t h Of O s c i l l a t i o n Marks S l a b s a m p l e s from Company A f i r s t were e x a m i n e d f o r t h e shape of o s c i l l a t i o n marks, w h i c h had been p r o d u c e d u n d e r n e a r l y i d e n t i c a l c a s t i n g c o n d i t i o n s w i t h r e s p e c t t o t h e c a s t i n g s p e e d , mould o s c i l l a t i o n , and mould f l u x . F i g s . 4-17 a n d 4 - 1 8 show t h e d e p t h p l o t t e d a g a i n s 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 62 measured i n t h e 0 .08 and 0 . 2 6 % - c a r b o n s l a b s r e s p e c t i v e l y . The d e p t h of o s c i l l a t i o n marks and i t s dependence on t h e p i t c h i s t h e same f o r marks w i t h and w i t h o u t hooks i n t h e l o w - c a r b o n s l a b s , F i g . 4 - 1 7 , and f o r marks w i t h o u t hooks i n t h e h i g h e r c a r b o n s l a b , F i g . 4 - 1 8 . The d e p t h of o s c i l l a t i o n marks i n c r e a s e s w i t h i n c r e a s i n g p i t c h and t h e d e v i a t i o n of t h e p i t c h when s u b s u r f a c e hooks a r e a b s e n t i s l a r g e r t h a n t h a t when hooks a r e p r e s e n t . As m e n t i o n e d b e f o r e , such v a r i a t i o n s of p i t c h i n d i c a t e r a p i d changes i n t h e m e n i s c u s l e v e l . T h e r e f o r e i t can be deduced t h a t the s u b s u r f a c e s t r u c t u r e i s more l i k e l y t o e x h i b i t hooks when the m e n i s c u s l e v e l i s s t a b l e t h a n when i t changes r a p i d l y . The d e p t h of o s c i l l a t i o n marks w i t h and w i t h o u t hooks i s s i m i l a r i n t h e l o w e r c a r b o n s t e e l s l a b (C=0.08%), w h i l e i n t h e h i g h e r c a r b o n s t e e l s l a b (C=0.26%), t h e d e p t h i s g r e a t e r when hooks a r e a b s e n t . The e f f e c t of c a r b o n c o n t e n t of t h e s l a b s on t h e d e p t h of o s c i l l a t i o n marks i s shown i n F i g . 4 - 1 9 . Some s c a t t e r i n t h e d a t a , w h i c h c h i e f l y depends on t h e m e n i s c u s l e v e l v a r i a t i o n , i s e v i d e n t but n o n e t h e l e s s t h e d e p t h of t h e o s c i l l a t i o n marks e x h i b i t i n g s u b s u r f a c e hooks i s seen t o be dependent on c a r b o n c o n t e n t . Of t h e f i v e s t e e l g r a d e s s t u d i e d , t h e d e e p e s t o s c i l l a t i o n marks a r e found i n 0.09% c a r b o n s l a b s and t h e s e a l s o e x h i b i t t h e g r e a t e s t s c a t t e r . I n c o n t r a s t , t h e d e p t h of o s c i l l a t i o n marks w i t h o u t s u b s u r f a c e hooks has no a p p a r e n t dependence on c a r b o n c o n t e n t o v e r t h e range s t u d i e d . The r e a s o n f o r t h i s b e h a v i o u r i s d i s c u s s e d l a t e r . 63 F u r t h e r i n v e s t i g a t i o n was made i n t o t h e shape of o s c i l l a t i o n marks on s l a b s a m p l e s , w h i c h have t h e same c a r b o n c o n t e n t but a r e d e o x i d i z e d d i f f e r e n t l y v i z . A l - o r S i - k i l l e d . The r e s u l t s a r e shown i n F i g . 4-20. Thus t h e same t e n d e n c y i s o b s e r v e d between t h e d e p t h and p i t c h of o s c i l l a t i o n marks f o r t h e A l - k i l l e d and S i - k i l l e d s t e e l s . However t h e a c t u a l d e p t h and t h e d e p e n d e n c e of d e p t h on p i t c h f o r t h e A l - k i l l e d s t e e l i s g r e a t e r t h a n t h a t f o r t h e S i - k i l l e d s t e e l . T h i s f i n d i n g may be due t o t h e i n c r e a s e d v i s c o s i t y of mould f l u x d u r i n g t h e c a s t i n g of A l - k i l l e d s t e e l c a u s e d by a b s o r p t i o n o f a l u m i n a i n c l u s i o n s . 1 4 6 2 8 9 The more v i s c o u s f l u x g i v e s r i s e t o d e e p e r o s c i l l a t i o n m arks. Next s l a b s a m p l e s f r o m Company C were e x a m i n e d t o f i n d t h e r e l a t i o n between t h e d e p t h o f t h e o s c i l l a t i o n marks and mould o s c i l l a t i o n parameters'. The e f f e c t o f o s c i l l a t i o n s t r o k e on t h e mark d e p t h i s shown i n F i g . 4-21. A l t h o u g h m e n i s c u s l e v e l v a r i a t i o n l e a d s t o b r o a d s c a t t e r i n t h e d e p t h o f t h e o s c i l l a t i o n m arks, as e x p e c t e d , a g r e a t e r o s c i l l a t i o n s t r o k e , v i z . l o n g e r n e g a t i v e s t r i p t i m e , g i v e s r i s e t o d e e p e r o s c i l l a t i o n m arks. F i n a l l y t h e e f f e c t o f e l e c t r o m a g n e t i c a l l y d r i v e n f l o w a t t h e m e n i s c u s on t h e d e p t h o f o s c i l l a t i o n marks was d e t e r m i n e d from t h e s l a b s a m p l e s c a s t a t Company E. The mark d e p t h d e c r e a s e s w i t h i n c r e a s i n g f l o w - v e l o c i t y o f s t e e l , a s shown i n F i g . 4-22. A d e c r e a s e i n t h e v a r i a t i o n of t h e mark d e p t h i s a l s o a c h i e v e d a p p a r e n t l y by a p p l y i n g mould EMS. 64 4.2 Heat Flow A n a l y s i s Of The M e n i s c u s R e g i o n The t e m p e r a t u r e d i s t r i b u t i o n i n t h e t h r e e p h a s e s a t t h e m e n i s c u s - s t e e l , mould f l u x , and mould w a l l - has been s t u d i e d m a t h e m a t i c a l l y t o shed l i g h t on v i s c o s i t y v a r i a t i o n i n t h e m o u l d f l u x c l o s e t o t h e m e n i s c u s due t o h e a t f l o w , and t h e e x t e n t t o w h i c h t h e m e n i s c u s may s o l i d i f y . T h e s e phenomena f i g u r e i m p o r t a n t l y i n t h e f o r m a t i o n o f t h e o s c i l l a t i o n marks, a s p r e v i o u s i n v e s t i g a t i o n s 9 2 6 3 2 ~ 3 * 3 6 have p o i n t e d o u t . Some o f t h e s h o r t c o m i n g s of e a r l i e r p r e d i c t i o n s of h e a t f l o w 3 2 3 8 have been overcome by c h a r a c t e r i z i n g t h e a x i a l m ould h e a t - e x t r a c t i o n p r o f i l e i n t h e m e n i s c u s r e g i o n f r o m m e a s u r e d m o u l d t e m p e r a t u r e s , u s i n g a t w o - d i m e n s i o n a l model of t h e mould w a l l w h i c h a c c o u n t s f o r v e r t i c a l a s w e l l as t h r o u g h - t h i c k n e s s h e a t c o n d u c t i o n . M o r e o v e r , t h e m a t h e m a t i c a l model o f h e a t c o n d u c t i o n i n t h e mould f l u x and s t e e l i s a l s o f u l l y two d i m e n s i o n a l . 4.2.1 A x i a l P r o f i l e Of M o u l d Heat F l u x At The M e n i s c u s The mould h e a t - f l u x p r o f i l e h as been c a l c u l a t e d f r o m t h e mould t e m p e r a t u r e s measured by N a k a t o e t a l 1 8 d u r i n g t h e c a s t i n g of s l a b s f o r h e a v y p l a t e . T h e i r v a l u e s o f mould t e m p e r a t u r e c l o s e t o t h e m e n i s c u s a r e shown as c l o s e d c i r c l e s on t h e l e f t hand s i d e o f F i g . 4-24. The mould h e a t - f l u x p r o f i l e was d e t e r m i n e d u s i n g t h e m a t h e m a t i c a l model o f t h e mould w a l l r e p o r t e d by S a m a r a s e k e r a and B rimacombe.' 6 The t w o - d i m e n s i o n a l model i s b a s e d on a s s u m p t i o n s o f 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 and n e g l i g i b l e o s c i l l a t i o n e f f e c t s . F i g . 4-23 shows t h e l o n g i t u d i n a l m i d - p l a n e o f t h e s l a b mould c o n s i d e r e d i n t h e 65 c o m p u t a t i o n . The g o v e r n i n g e q u a t i o n f o r h e a t c o n d u c t i o n i n t h e mou l d w a l l c an be d e s c r i b e d as f o l l o w s : 2 2 3 T„ 3 T — + — = o (*-D 2 2 3x 3y The mould w a l l i s s u f f i c i e n t l y t h i c k , so t h a t n o n u n i f o r m i t y o f t e m p e r a t u r e d i s t r i b u t i o n i n t h e t r a n s v e r s e ( z ) d i r e c t i o n c a n be assumed n e g l i g i b l e . A s s u m i n g t h a t t h e c o o l i n g w a t e r i s i n p l u g f l o w , h e a t t r a n s f e r i n t h e w a t e r c h a n n e l c an be e x p r e s s e d by t h e f o l l o w i n g : 3T p v d C - h (x){T(x,0)-T (x)} = 0 (4-2) w w w pw tfx w M w Eqs (4-1) and (4-2) c a n be s o l v e d n u m e r i c a l l y t o d e t e r m i n e b o t h t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould w a l l and t h e h e a t f l u x a c r o s s t h e mould w a l l . The b o u n d a r y c o n d i t i o n s of t h e s y s t e m a r e a s f o l l o w s : ( i ) x=0, 0 < y < y M ; - * M ~jr = 0 (4-3) ( i i ) x=x M, 0 < y < y M ; - X M — = 0 (4-4) ( i i i ) y=0, 0 < x < x M ; - X M - r - = h (T -T ) (4-5) ' — — M M 3 y w w o 3 TM (iv) y=y M, 0 < x < x M ; - X M — = q Q(x) ( 4 _ 6 ) ( v ) X = X M 5 Tw = T w i ( 4 - 7 ) (vi) x=0 ; T w = T w f ( 4 _ 8 ) 66 A n o t h e r a s s u m p t i o n , on w h i c h t h e model i s b a s e d , i s t h a t r a d i a t i o n i s n e g l e c t e d i n t h e mould f l u x . The h e a t - t r a n s f e r c o e f f i c i e n t between water and mould w a l l i s e x p r e s s e d by t h e f o l l o w i n g e m p i r i c a l e q u a t i o n ; 9 A / p v D \ 0 ' 8 / C y h w = 0.23 ^ w w H p W w ( 4 _ 9 ) H \ / \ w ' An assumed heat f l u x p r o f i l e , w h i c h d e f i n e s t h e boundary c o n d i t i o n s a t t h e hot f a c e of t h e m o u l d , v i z q 0 ( x ) , was i n p u t t o t h e model and t h e t w o - d i m e n s i o n a l t e m p e r a t u r e d i s t r i b u t i o n was c a l c u l a t e d , t h e n compared t o t h e l o c a l v a l u e s measured by N a k a t o e t a l . 1 8 The h e a t - f l u x p r o f i l e n e x t was a d j u s t e d t o o b t a i n a b e t t e r f i t and t h e p r o c e s s was r e p e a t e d u n t i l p r e d i c t e d and measured t e m p e r a t u r e were i n c l o s e a g r e e m e n t . D a t a u sed i n t h i s c a l c u l a t i o n a r e p r e s e n t e d i n T a b l e V I I . The r e s u l t s of t h e t e m p e r a t u r e f i t t i n g and t h e heat f l u x p r o f i l e d e t e r m i n e d a r e shown i n F i g . 4 - 2 4 . The hea t f l u x i s seen t o have a maximum v a l u e of 2500kW/m 2 a t 5mm be low t h e m e n i s c u s , w h i l e i n t h e mould f l u x r e g i o n , i t i s 700 k W / m 2 . I t i s a l s o i n t e r e s t i n g t h a t t h e maximum mould t e m p e r a t u r e i s l o c a t e d a t a b o u t 35mm below t h e m e n i s c u s w h i c h i s 30mm below t h e l e v e l of peak h e a t f l u x . T h i s d i f f e r e n c e r e s u l t s from t h e s i g n i f i c a n t v e r t i c a l h e a t c o n d u c t i o n i n t h e m e n i s c u s r e g i o n . These d a t a a r e used f o r t h e boundary c o n d i t i o n s i n t h e f o l l o w i n g c o m p u t a t i o n . 67 4 . 2 . 2 T e m p e r a t u r e D i s t r i b u t i o n In The M o u l d F l u x And S t e e l To p r e d i c t t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e m e n i s c u s a r e a , an a p p r o a c h s i m i l a r t o t h a t o f Tomono et a l . 3 2 and Saucedo e t a l . 3 8 has been a d o p t e d i n w h i c h , due t o t h e i n f l u e n c e of mould o s c i l l a t i o n on l o c a l f l u i d f l o w , t e m p e r a t u r e g r a d i e n t s i n t h e mould f l u x and s t e e l a r e assumed t o be d e s t r o y e d p e r i o d i c a l l y . T h i s e v e n t i s f o l l o w e d by an i n t e r v a l i n w h i c h h e a t i s c o n d u c t e d i n u n s t e a d y s t a t e t h r o u g h t h e p h a s e s . The c o n c e p t i s a n a l o g o u s t o t h e " s u r f a c e r e n e w a l " t h e o r y p r o p o s e d f o r m a s s - t r a n s f e r sys tems by H i g b i e 9 1 9 2 many y e a r s a g o . D u r i n g t h e u n s t e a d y - s t a t e p e r i o d , the phase s a r e assumed t o be s t a g n a n t w h i c h i n c l u d e s n e g l e c t i n g the downward m o t i o n of s t e e l . In t h i s c a l c u l a t i o n t h i s p e r i o d , i s assumed t o commence a t t h e end of the n e g a t i v e - s t r i p t i m e i n t e r v a l of t h e mould o s c i l l a t i o n c y c l e because " o v e r f l o w " a t t h e m e n i s c u s , w h i c h a c t s t o d e s t r o y l o c a l t e m p e r a t u r e g r a d i e n t s , o c c u r s a t t h i s t i m e as d e s c r i b e d p r e v i o u s l y . 9 3 6 Thus t h e t ime f o r 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 i s t a k e n t o be t h e p o s i t i v e - s t r i p p e r i o d o f t h e mould o s c i l l a t i o n c y c l e g i v e n by E q . ( 2 - 4 ) . T y p i c a l p o s i t i v e - s t r i p t i m e s a r e i n t h e range of 0 .2 t o 0 . 7 s . F i g . 4-25 shows t h e geometry of t h e sy s tem c o n s i d e r e d i n t h e c a l c u l a t i o n s . The shape of t h e m e n i s c u s has been c a l c u l a t e d f rom e q u i l i b r i u m c o n s i d e r a t i o n s ; m e n i s c u s shape w i l l be d i s c u s s e d l a t e r . A s i m p l i f i e d e q u a t i o n has been a d o p t e d f o r the h e a t f l o w c a l c u l a t i o n . x - x = (x - xx) A - Y 1 (4-10) 68 The e q u a t i o n s g o v e r n i n g h e a t c o n d u c t i o n i n t h e s t e e l and mould f l u x a r e r e s p e c t i v e l y , 9T A s _ s , dt ~ p C { 9T A " ( 9 2T 2 8 T 2 + S ) 2 (4-11) 9x 3y 2 8 2T f (4-12) 9x^ 9y 9t p,C f f The i n i t i a l c o n d i t i o n s a r e ; (i ) t=0, x >_ x l t >_ x 2, y >_ 0 ; T s=T s i (4-13) ( i i ) t=0, 0 £ x £ X ; L , x 2, y >_ 0 ; T = T _ (4-14) and t h e b o u n d a r y c o n d i t i o n s a r e ; 9T ( i i i ) t > 0, x 0 < x < x_, y = 0 ; - A = q (x) (4-15) — z — j s dy o (iv) t >_ 0, (x 2,0)+ ( x ^ y j ; 9T 9T - A —5- = - X. = h (T - T J (4-16) s 3r f 8r s-f s f y x < y £ y 2» x = x i J 3T " 3T _ X = _ x _ £ = h . (T - T ) (4-17) s 9x f 9x s-f s f 9T (v) t > 0 , x = x , , 0 < y < y „ ; - X - r f = 0 (4-18) L3' - ' - "2 ' s 9x ^ < x < x 3, y = y 2 ; T s = T s. (vi) t > 0, x _ „  .. ( 4~ 1 9> 9T ( v i i ) t > 0 , 0 < x < x 2 , y = 0 ; ~ ^ f — = q Q(x) ( 4" 2°) q 0 ( x ) c h a n g e s f r o m q 0 ( x , ) t o q 0 ( x 2 ) i n i n v e r s e p r o p o r t i o n t o t h e 69 d i s t a n c e between mould w a l l and m e n i s c u s ; ( v i i i ) t >_ 0, x <_ x _< x 2' y = 0 - X (4-21) f {x /q (x ) + x /q (x ) -2 2 1 o 1 x ( / q ( x j - /q ( x ) ) } 2 o 2 o 1 = 0 (4-22) (x) t _> 0, 0 <_ x <_ x l S y = y± ; f i (4-23) E q u a t i o n s (4-11) and (4-12) were s o l v e d , s u b j e c t t o t h e i n i t i a l and b o u n d a r y c o n d i t i o n s , u s i n g t h e e x p l i c i t f i n i t e - d i f f e r e n c e m e t h o d . 9 3 T r i a n g u l a r volume e l e m e n t s have been e m p l o y e d t o s i m u l a t e t h e c u r v a t u r e of t h e mould f l u x / s t e e l m e n i s c u s , e s p e c i a l l y c l o s e t o t h e mould w a l l ; t h e t y p e o f nodes r e q u i r e d f o r t h e c a l c u l a t i o n a r e shown i n F i g . 4-25. N o d a l e q u a t i o n s f o r e a c h t y p e of node a r e g i v e n i n A p p e n d i x I I . The r e l e a s e of t h e l a t e n t h e a t of s o l i d i f i c a t i o n has been i n c o r p o r a t e d by a d j u s t i n g t h e s p e c i f i c h e a t between t h e s o l i d u s a nd t h e l i q u i d u s t e m p e r a t u r e . C o m p u t a t i o n s have been p e r f o r m e d f o r a p l a t e - g r a d e s t e e l u s i n g t h e h e a t - f l u x p r o f i l e p r e s e n t e d i n F i g . 4-22, w h i c h was c a s t i n g o f t h e same t y p e o f s t e e l . F i g . 4-26 shows t h e s i m p l i f i e d f l o w d i a g r a m o f t h e computer p r o g r a m t h a t i s p r e s e n t e d i n A p p e n d i x V. The c o m p o s i t i o n o f t h e s t e e l and d a t a s o l (4-24) c a l c u l a t e d f r o m t h e mould w a l l t e m p e r a t u r e m e a s u r e d d u r i n g t h e 70 employed i n t h e c o m p u t a t i o n a re g i v e n i n T a b l e V I I I . Note t h a t t h e i n i t i a l t e m p e r a t u r e of the mould f l u x was assumed t o be 1 5 0 0 ° C from t h e r e p o r t e d d a t a w h i c h a r e shown i n F i g . 2 - 4 . Computed r e s u l t s a r e shown i n F i g s . 4-27 - 4 - 3 4 , e x p r e s s e d as i s o t h e r m s i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d , f , i n s t h e s t e e l . The f r a c t i o n s o l i d i s c a l c u l a t e d f rom t h e l e v e r r u l e , a s suming t h a t t h e s o l i d u s and l i q u i d u s l i n e s a r e s t r a i g h t o v e r t h e t e m p e r a t u r e range of i n t e r e s t . T . - T. . f = _ i±3 hil (4-25) l i q s o l F i g u r e s 4-27 and 4-28 show t h e p r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n when the s u p e r h e a t of s t e e l i s t a k e n t o be 5 ° C i n t h e v i c i n i t y of t h e m e n i s c u s , a f t e r a p e r i o d of 0 . 3 s and 0 . 6 s r e s p e c t i v e l y . Thus t h e s o l i d i f y i n g s h e l l i s p r e d i c t e d t o be o n l y p a r t i a l l y s o l i d w i t h f g = 0 . 6 l o c a t e d a t about 0.1mm from t h e mould w a l l and f g = 0 . 2 s i t u a t e d a t about 0 .6 -0 .7mm. These p o s i t i o n s a r e s i m i l a r t o t h o s e r e p o r t e d by Saucedo e t a l . 3 8 a l t h o u g h t h e y a l s o p r e d i c t e d s o l i d i f i c a t i o n f a r t h e r a l o n g the m e n i s c u s . Tha t t h e r e g i o n of f =1 i s n e g l i g i b l y s m a l l does not mean t h a t t h e s e m i -s o l i d s h e l l has no r i g i d i t y . Saucedo e t a l . 3 8 have s u g g e s t e d t h a t t h e s h e l l may e x h i b i t r i g i d i t y i f f i s as low as 0 .2 whereas M a t s u m i y a e t a l . 9 " have used a v a l u e of 0 . 8 5 f o r a r i g i d i t y c r i t e r i o n . The c r i t i c a l f r ema ins unknown but i t i s p o s s i b l e t h a t t h e s h e l l formed i n 0 .3 or 0 .4 s c o u l d a c t as a s o l i d . T h i s s u g g e s t s t h a t the o v e r f l o w mechanism t h a t g i v e s r i s e t o s u b s u r f a c e hooks i s p l a u s i b l e ; but because t h e e x t e n t of t h e p r e d i c t e d m e n i s c u s s o l i d i f i c a t i o n i s c o n s i d e r a b l y l e s s t h a n 71 t h e d e p t h o f t h e o b s e r v e d h o o k s , 1.5 t o 2 .0 mm i n F i g . 4 - 9 , i t does n o t , by i t s e l f , a d e q u a t e l y a c c o u n t 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 m a r k s . The e f f e c t of the s u p e r h e a t of t h e s t e e l on 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 a l s o has been e x a m i n e d . S t e e l n e c e s s a r i l y has a c e r t a i n d e g r e e of s u p e r h e a t i n t h e t u n d i s h , w h i c h i s g e n e r a l l y c o n t r o l l e d t o w i t h i n 2 0 - 3 0 ° C , d e p e n d i n g on t h e c a s t i n g c o n d i t i o n s . H i g h e r s u p e r h e a t s a r e a v o i d e d because t h e i n t e r n a l q u a l i t y of t h e s l a b i s a f f e c t e d d e l e t e r i o u s l y , v i z r e d u c t i o n of e q u i a x e d z o n e . T h i s s u p e r h e a t i s r e d u c e d d u r i n g pas sage of t h e s t e e l t h r o u g h t h e i m m e r s i o n n o z z l e ; and a l s o i n t h e m e n i s c u s a r e a f u r t h e r r e d u c t i o n of s t e e l t e m p e r a t u r e can be e x p e c t e d . I f t h e s t e e l t e m p e r a t u r e d e c r e a s e s be low the l i q u i d u s a t t h e m e n i s c u s , t h e s u r f a c e of the mould p o o l near t h e mould w a l l b e g i n s t o s o l i d i f y . From t h e s e c o n s i d e r a t i o n s t h e s u p e r h e a t a t t h e m e n i s c u s has been e s t i m a t e d t o be 5 ° C f o r n o r m a l c a s t i n g c o n d i t i o n s . F i g s . 4-29 - 4-32 show t h e t e m p e r a t u r e d i s t r i b u t i o n a t t h e m e n i s c u s when t h e s u p e r h e a t i s 0 ° C and 2 0 ° C , a f t e r a p e r i o d o f 0 . 3 s and 0 . 6 s . I n t h e case o f no s u p e r h e a t , F i g s . 4-29 and 4-30 , t h e hook may grow r e l a t i v e l y l o n g e r . I n t h e s u b s u r f a c e s t r u c t u r e o f Company D, l o n g hooks were o b s e r v e d , see F i g . 4-1 5 ( a ) . T a k i n g i n t o a c c o u n t t h a t t h e measured s t e e l t e m p e r a t u r e a t t h e m e n i s c u s i n t h e c a s t i n g o f t h i s sample i s c l o s e t o t h e l i q u i d u s t e m p e r a t u r e of s t e e l , see T a b l e I V , t h e l o n g hooks a r e u n d o u b t e d l y due t o m e n i s c u s s o l i d i f i c a t i o n . As shown i n F i g s . 72 4-31 a n d 4-32 h i g h s u p e r h e a t r e d u c e s t h e l e n g t h o f t h e hook. T h e s e c a l c u l a t i o n s have been p e r f o r m e d a s s u m i n g t h a t t h e m o l t e n s t e e l i s s t a g n a n t w h i c h e f f e c t i v e l y m i n i m i z e s t h e e x t r a c t i o n of s u p e r h e a t and m a x i m i z e s t h e g r o w t h o f t h e s o l i d s h e l l . However c o n v e c t i o n i n t h e m o l t e n s t e e l i s g e n e r a t e d a t t h e m e n i s c u s by e l e c t r o m a g n e t r e a l l y s t i r r i n g , by t h e i n p u t s t r e a m s d i s c h a r g i n g from t h e i m m e r s i o n n o z z l e a n d / o r by i n e r t gas t h a t i s i n j e c t e d i n t o t h e i m m e r s i o n n o z z l e and e n t r a i n e d by t h e f l o w i n g s t e e l . T h e s e e f f e c t s have been m o d e l l e d c r u d e l y by i n c r e a s i n g t h e t h e r m a l c o n d u c t i v i t y o f t h e s t a g n a n t s t e e l by a f a c t o r o f f o u r . The c a l c u l a t i o n o f t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e m e n i s c u s r e g i o n has been r e p e a t e d and t h e r e s u l t s a r e shown i n F i g s . 4-33 and 4-34, when t h e s u p e r h e a t o f s t e e l i s 5 ° C . The f r a c t i o n s o l i d p r e d i c t e d i s seen t o be d i m i n i s h e d s i g n i f i c a n t l y , more t h a n t h a t i n t h e c a s e of h i g h s u p e r h e a t a s shown i n F i g s . 4-31 and 4-32. Thus s t r o n g c o n v e c t i o n w i l l c a u s e t h e s h e l l t o behave more l i k e a l i q u i d t h a n a s o l i d . U nder t h e s e c o n d i t i o n s f l o w o f m o l t e n s t e e l o v e r a p a r t i a l l y s o l i d i f i e d r i g i d m e n i s c u s w i l l n o t t a k e p l a c e , and s u b s u r f a c e h o o k s w i l l n o t f o r m . As m e n t i o n e d p r e v i o u s l y , t h e l e n g t h o f h o o k s i s s i g n i f i c a n t l y d e c r e a s e d by mould EMS (Company E) or by m u l t i - p o r t p r a c t i c e (Company D ) ; r a p i d v a r i a t i o n o f t h e m e n i s c u s l e v e l , w h i c h c a u s e s s t e e l f l o w i n t h e m e n i s c u s r e g i o n , b r i n g s a b o u t o s c i l l a t i o n marks w i t h o u t hooks (Company A ) . 73 Thus f r o m t h e s e c a l c u l a t i o n s , i t can be a r g u e d t h a t t h e p r e s e n c e o r a b s e n c e of s u b s u r f a c e hook d e p e n d s on t h e l o c a l e x t r a c t i o n o f s u p e r h e a t a t t h e m e n i s c u s w h i c h i s g o v e r n e d by t h e m a g n i t u d e o f s u p e r h e a t and t h e c o n v e c t i o n i n t h e m o l t e n s t e e l . T h i s argument w i l l be t a k e n up l a t e r . A n o t h e r i m p o r t a n t a s p e c t of t h e p r e d i c t i o n s shown i n F i g s . 4-27 - 4-34 i s t h e low t e m p e r a t u r e zone of t h e mould f l u x a d j a c e n t t o t h e mould w a l l . S i n c e t h e v i s c o s i t y of t y p i c a l m o uld f l u x e s i n c r e a s e s s h a r p l y w i t h t e m p e r a t u r e below 1000-1 2 0 0 ° C , 1 * a v e r y t h i n zone of h i g h v i s c o s i t y f l u x must e x i s t a g a i n s t t h e mould w a l l . As s h a l l be s e e n , t h i s h i g h v i s c o s i t y f l u x p l a y s an i m p o r t a n t r o l e i n 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 m a rks. 4.3 F l u i d P r e s s u r e In The M o u l d F l u x At The M e n i s c u s I t has a l r e a d y been seen t h a t a s i m p l e o v e r f l o w mechanism c a n n o t e x p l a i n t h e d e p t h of t h e s u b s u r f a c e h o o k s ; and t h e r e f o r e o t h e r f a c t o r s must a l s o p l a y a r o l e i n 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 m a r k s . The most o b v i o u s i s t h e mould f l u x and, i n p a r t i c u l a r , i t s b e h a v i o u r as i t f l o w s , under t h e i n f l u e n c e o f mould o s c i l l a t i o n and s t r a n d w i t h d r a w a l , i n t o t h e gap between t h e s o l i d i f i e d s h e l l a t t h e m e n i s c u s and t h e mould w a l l . P r e v i o u s l y t h e a c t i o n of t h e mould f l u x , s p e c i f i c a l l y i t s l u b r i c i t y , has been c h a r a c t e r i z e d i n t e r m s o f t h e s h e a r s t r e s s 1 " 6 9 t h a t a c t s on t h e s t r a n d . W h i l e t h i s may be i m p o r t a n t f o r o v e r a l l mould l u b r i c a t i o n , i t w i l l be shown h e r e t h a t t h e s h e a r 74 s t r e s s i s n o t v e r y s i g n i f i c a n t a t t h e m e n i s c u s as compared t o t h e p r e s s u r e g e n e r a t e d i n t h e mould f l u x by mould o s c i l l a t i o n . T h i s p r e s s u r e , w h i c h has not been c o n s i d e r e d p r e v i o u s l y , a r i s e s b e c a u s e t h e f l u x c h a n n e l n a r r o w s i n w i d t h as t h e m e n i s c u s c u r v e s t o w a r d t h e mould w a l l , as shown s c h e m a t i c a l l y i n F i g . 4-35. In t h i s p a r t o f t h e s t u d y , t h e f l u x p r e s s u r e has been e s t i m a t e d r o u g h l y f r o m f l u i d f l o w p r i n c i p l e s o r i g i n a l l y a p p l i e d t o l u b r i c a t i o n p r o b l e m s by R e y n o l d s and o t h e r s . 9 6 The f o l l o w i n g a s s u m p t i o n s have been made t o p r e d i c t t h e p r e s s u r e and v e l o c i t y d i s t r i b u t i o n i n t h e f l u x c h a n n e l : [ i ] S t e a d y s t a t e i s assumed w h i c h i s e q u i v a l e n t t o s a y i n g t h a t t h e p r e s s u r e and v e l o c i t y g r a d i e n t a r e i n s t a n t a n e o u s l y e s t a b l i s h e d ; [ i i ] I n e r t i a l f o r c e s a r e n e g l e c t e d b e c a u s e f l u i d v e l o c i t i e s a r e low and t h e f l u x v e l o c i t y i s h i g h ; [ i i i ] The f l u x b e h a v e s as a N e w t o n i a n f l u i d ; [ i v ] The m e n i s c u s i n t h e r e g i o n o f i n t e r e s t i s c o v e r e d w i t h a r i g i d " s o l i d " s k i n ; and t h e shape o f t h e f l u x c h a n n e l does n o t change s i g n i f i c a n t l y ; [ v ] F l u x v e l o c i t i e s i n t h e t r a n s v e r s e d i r e c t i o n (u ) and z n o r m a l t o t h e mould w a l l ( u ) a r e n e g l i g i b l e . u i s y y n e g l i g i b l e b e c a u s e t h e a n g l e f o r m e d by t h e m e n i s c u s w i t h t h e v e r t i c a l i s s m a l l ; [ v i ] The d e n s i t y and v i s c o s i t y o f t h e mould f l u x a r e c o n s t a n t . 75 The l a t t e r a s s u m p t i o n i s , a d m i t t e d l y , v e r y r e s t r i c t i v e b e c a u s e t h e l a r g e t e m p e r a t u r e g r a d i e n t s a c r o s s t h e f l u x c h a n n e l w i l l r e s u l t i n a c o r r e s p o n d i n g l y s t e e p v i s c o s i t y g r a d i e n t . I t wou l d be p r e f e r a b l e t o c o u p l e t h e h e a t and f l u i d f l o w s and s o l v e s i m u l t a n e o u s l y f o r t h e t e m p e r a t u r e , v e l o c i t y and p r e s s u r e d i s t r i b u t i o n s , b u t a t t h i s e a r l y s t a g e i n t h e d e v e l o p m e n t of t h e m o d e l , t h i s a d d e d c o m p l e x i t y i s u n w a r r a n t e d . I t d o e s mean, however, t h a t t h e f l u x p r e s s u r e s p r e d i c t e d a r e o n l y v e r y a p p r o x i m a t e r e l a t i v e t o r e a l v a l u e s . Under t h e s e assumed c o n d i t i o n s , f l u i d f l o w i n t h e f l u x c h a n n e l i s g o v e r n e d by t h e f o l l o w i n g e q u a t i o n o f m o t i o n , „ 2 dx" ~ y f T T + p f 8 ( 4 - 2 6 ) 3y and e q u a t i o n o f c o n t i n u i t y . I' 4 , < t < V « - 0 (A-27) 9x dx ' o x where i s a r e l a t i v e c o n s u m p t i o n r a t e of mould f l u x . I t s h o u l d be n o t e d t h a t , b e c a u s e t h e r i g i d m e n i s c u s s k i n i s moving downward w i t h t h e s t r a n d , u i s a r e l a t i v e v e l o c i t y e x p r e s s e d as ux = v f " v s (4-28) Where v and v a r e f l u x and s t r a n d v e l o c i t y r e s p e c t i v e l y , b o t h f s o f w h i c h a r e d e f i n e d r e l a t i v e t o a f i x e d r e f e r e n c e f r a m e . The b o u n d a r y c o n d i t i o n s , s u b j e c t t o w h i c h E q s . (4-26) and (4-27) a r e s o l v e d , a r e 76 ( i ) 0 < x < 1 , y = 0, u = v - v (4-29) — — f x m s ( i i ) 0 <_ x <_ 1 , y = h(x) , u x = 0 (4-30) ( i i i ) X = 0, 0 < y £ h P = P (4-31) ( iv) X = 1 , 0 < y < h f , P = P f (4-32) Note t h a t B . C . [ i ] and [ i i ] a r e a s t a t e m e n t o f t h e "no s l i p " c o n d i t i o n w h i c h i s c o r r e c t p r o v i d e d t h a t t h e r i g i d s k i n assumed t o c o v e r the m e n i s c u s i n t h e f l u x c h a n n e l , behaves as a s o l i d . The s o l u t i o n t o E q s . (4-26) and (4-27) i s d e s c r i b e d b r i e f l y i n t h e A p p e n d i x I V . For t h e c a s e where t h e p a r t of t h e m e n i s c u s under c o n s i d e r a t i o n i s t a k e n t o be l i n e a r , and bounded by t h e c o o r d i n a t e s ( 0 , h i ) and ( l f , h f ) , see F i g . 4 - 3 5 , t h e p r e s s u r e g r a d i e n t i n t h e mould f l u x i s g i v e n b y : P(x) P f 8 l f + 6y i -l^(v - v ) -w h . - h r f m s 1 hh, I h , - h, h . - h hT - h . I h 6y 1 h ) ( P f g l f + P, - V + (—S^Xr^Cv r f l f , 2 h . m h I (4-33) and t h e v e l o c i t y d i s t r i b u t i o n i s u = ( v - v )(1 - £ ) - J 3 ( v - v ) x m . s h [ m s ( p . g l - + P. - P f ) h 2 h 2 + 6u 1 (v - v ) h . h , 2 f . f i f l f f f m s l _ f | , y y \ ( 4 - 3 4 ) y f l £ h ( h . + h £) , .. h . - te - ^  F i g . 4-36 shows t h e p r e s s u r e d i s t r i b u t i o n c a l c u l a t e d from Eq , 77 (4-33) f o r d i f f e r e n t f l u x v i s c o s i t i e s and f o r b o t h upward and downward mould v e l o c i t i e s . A mould o s c i l l a t i o n f r e q u e n c y o f 100 cpm, a s t r o k e l e n g t h of 8mm and a c a s t i n g s p e e d of 1.0 m/min have been assumed f o r t h e c a l c u l a t i o n . The l e n g t h o f t h e f l u x c h a n n e l has been d e t e r m i n e d f r o m t h e p i t c h of t h e o s c i l l a t i o n marks w h i l e t h e v a l u e of t h e l o w e r c h a n n e l w i d t h , h f , has been e s t i m a t e d from c a l c u l a t e d v a l u e s of t h e minimum t h i c k n e s s o f m ould f l u x b a s e d on measured f l u x c o n s u m p t i o n r a t e s . 9 In F i g . 4-36 t h e mould f l u x i s seen t o d e v e l o p a p o s i t i v e p r e s s u r e on t h e down s t r o k e , and a n e g a t i v e p r e s s u r e on t h e up s t r o k e o f t h e m o u l d . A maximum i n t h e f l u x p r e s s u r e i s p r e d i c t e d t o w a r d t h e b o t t o m o f t h e f l u x c h a n n e l f o r b o t h upward an d downward mould v e l o c i t i e s . As e x p e c t e d t h e m a g n i t u d e of t h e peak p r e s s u r e i n c r e a s e s w i t h i n c r e a s i n g f l u x v i s c o s i t y . I n t e r e s t i n g l y t h e maximum v a l u e o f n e g a t i v e p r e s s u r e on t h e u p s t r o k e i s g r e a t e r t h a n t h a t of t h e p o s i t v e p r e s s u r e on t h e d o w n s t r o k e , b e c a u s e on t h e u p s t r o k e t h e m o t i o n o f t h e mould r e l a t i v e t o t h e s t r a n d i s g r e a t e r . T h e s e p o s i t i v e and n e g a t i v e p r e s s u r e s a r e much l a r g e r t h a n t h e l o c a l f e r r o s t a t i c p r e s s u r e . F i g . 4-37 shows t h e - v e l o c i t y d i s t r i b u t i o n i n t h e f l u x c h a n n e l , a s c a l c u l a t e d from E q . ( 4 - 3 4 ) , a t t h e maximum downward v e l o c i t y o f t h e mould. The c o n d i t i o n s assumed a r e t h e same a s a p p l i e d i n t h e p r e s s u r e c a l c u l a t i o n s ; and t h e f l u x v i s c o s i t y i s 5P. As e x p e c t e d , a downward v e l o c i t y i s p r e d i c t e d n e a r t h e mould w a l l , and a f l o w r e v e r s a l i s s e e n n e a r t h e s t e e l i n t h e u p p e r r e g i o n o f t h e f l u x c h a n n e l . 78 The s h e a r s t r e s s a c t i n g on t h e m e n i s c u s (assumed t o have a r i g i d s o l i d i f i e d s k i n ) has been c a l c u l a t e d f r o m t h e v e l o c i t y d i s t r i b u t i o n i n F i g . 4-37, as f o l l o w s 3u T = - . (4-35) f 9y y=h The c a l c u l a t e d s h e a r s t r e s s e s a r e shown i n F i g . 4-38. T h i s s h e a r s t r e s s i n t h e f l u x l a y e r , between mould w a l l a n d t h e s h e l l , has been d i s c u s s e d as an i n d e x f o r t h e s u r f a c e q u a l i t y o f s l a b s . 6 9 However t h e v a l u e s o b t a i n e d a r e an o r d e r of m a g n i t u d e l e s s t h a n t h e p r e s s u r e shown i n F i g . 4-36; and t h e r e f o r e t h e s h e a r s t r e s s w i l l have o n l y a m i n o r e f f e c t on m e n i s c u s b e h a v i o u r . 4.4 M e n i s c u s Shape 4.4.1 S t a t i c Shape Of M e n i s c u s The shape o f m e n i s c u s , e s p e c i a l l y i n t h e v i c i n i t y o f t h e mould w a l l , i s i m p o r t a n t t o u n d e r s t a n d t h e i n i t i a l s o l i d i f i c a t i o n phenomena i n t h e c o n t i n u o u s - c a s t i n g m ould. Thus a t w o - d i m e n s i o n a l m e n i s c u s , as shown i n F i g . 4-39, has been c o n s i d e r e d . The m e n i s c u s shape c a n be d e t e r m i n e d by a f o r c e b a l a n c e a t t h e i n t e r f a c e between l i q u i d s t e e l and l i q u i d m ould f l u x . E s p e c i a l l y i n t h e c a s e of a s t a t i c m e n i s c u s , t h e m e n i s c u s shape depends on b o t h t h e d e n s i t y o f mould f l u x and m o l t e n s t e e l , a s w e l l as t h e i n t e r f a c i a l t e n s i o n . An a n a l y t i c a l 79 s o l u t i o n f o r t h i s c a s e was g i v e n by M a t i j e v i c 9 7 a n d B i k e r m a n , 9 8 a s f o l l o w s ; ( s e e A p p e n d i x I I I f o r t h e d e r i v a t i o n ) . y = - / 2 a 2 - x 2 + ^ f- m( ^ S) +0.3768a (4-36) where a 2 , t h e c a p i l l a r y c o n s t a n t , i s d e f i n e d a s 2 2o 3 = (P s " P f)g (^-37) " a " g i v e s t h e v e r t i c a l d i s t a n c e between t h e h o r i z o n t a l m e n i s c u s l e v e l and t h e " c o n t a c t " p o i n t o f th e m e n i s c u s w i t h t h e mould w a l l . As m e n t i o n e d b e f o r e , l i t t l e work has been done on t h e p r o p e r t i e s o f mould f l u x . T h e r e f o r e t h e p r o p e r t i e s o f b l a s t f u r n a c e s l a g , w h i c h has s i m i l a r c h e m i c a l components t o t h a t o f mould f l u x , was s u b s t i t u t e d f o r t h e c a l c u l a t i o n o f t h e m e n i s c u s s h a p e . 4.4.2 Change Of The M e n i s c u s Shape By Mo u l d O s c i l l a t i o n The shape of t h e m e n i s c u s depends on t h e p r e s s u r e a c t i n g on th e s o l i d i f i e d s k i n and, more c o n v e n t i o n a l l y , on t h e b a l a n c e o f f o r c e s a c t i n g a t t h e i n t e r f a c e between t h e mould f l u x and m o l t e n s t e e l . I t i s c l e a r f r o m t h e p r e v i o u s s e c t i o n t h a t an i m p o r t a n t f o r c e t h a t c a n n o t be i g n o r e d i s t h e f l u i d p r e s s u r e g e n e r a t e d i n t h e mould f l u x c h a n n e l . T h u s , i n t h i s p a r t o f t h e s t u d y , a m a t h e m a t i c a l r e l a t i o n s h i p has been d e v e l o p e d t o p r e d i c t t h e m e n i s c u s shape r e s u l t i n g f r o m t h e f l u x p r e s s u r e d e v e l o p e d a t d i f f e r e n t s t a g e s of t h e mould o s c i l l a t i o n c y c l e . 80 To s i m p l i f y t h e c a l c u l a t i o n s , t h e m e n i s c u s i s assumed t o be f r e e o f a r i g i d s k i n and t o a t t a i n i t s e q u i l i b r i u m shape i n s t a n t a n e o u s l y . The e q u a t i o n g o v e r n i n g t h e shape of t h e m e n i s c u s under t h e i n f l u e n c e o f f l u i d p r e s s u r e , i s a l s o d e r i v e d i n t h e A p p e n d i x I I I , and i s as f o l l o w s : 2o(p s - p f ) g x 2 - 4 0{R(x) + a) lPB ~ P f ) 2 8 2 x 4 - A(p s - p f){R(x) + o}gx 2 + 4R(x){R(x) + 2 a } ] 1 / 2 ( 4 - 3 8 ) R ( x ) = J X ( P ( x ) - p g x l d x ( 4 - 3 9 ) P ( x ) i s t h e a x i a l p r e s s u r e d i s t r i b u t i o n i n t h e mould f l u x c a l c u l a t e d f r o m E q s . (4-26) and ( 4 - 2 7 ) , a s s u m i n g c o m p l e t e s l i p a t t h e m e n i s c u s , i . e . a t y = h ( x ) , v f = 0 and y=0, v f =vM . i t may be n o t e d t h a t i f R(x)=0, i . e . no dynamic p r e s s u r e , E q . (4-38) r e d u c e s t o t h e a n a l y t i c a l s o l u t i o n f o r a s t a t i c m e n i s c u s , E q . (4-36) . The " c o n t a c t " p o i n t of t h e m e n i s c u s w i t h t h e mould w a l l , X c , a s d e f i n e d by c4=7r/2, was c a l c u l a t e d f r o m E q . ( A - 3 - 1 1 ) . N o t e t h a t x c c o r r e s p o n d s t o " a " when t h e f l u i d p r e s s u r e o f t h e moul d f l u x i s e q u a l t o z e r o . 2 o R ( V ° > 5 ( 4 - 4 0 ) X = ,  1 , 1 + — — c ( p g - P f ) g o The m e n i s c u s shape was c a l c u l a t e d a t s e l e c t e d p o i n t s i n t h e mould o s c i l l a t i o n c y c l e by t h e f o l l o w i n g i t e r a t i v e p r o c e d u r e : [ i ] P ( x ) and R(x) were computed i n i t i a l l y a s s u m i n g a q u a d r a t i c r e l a t i o n s h i p f o r t h e m e n i s c u s , E q . (4-39) 81 a p p r o x i m a t e d from E q . ( 4 - 3 6 ) . [ i i ] . To s i m p l i f y t h e c a l c u l a t i o n , R ( x ) was f i t t e d t o a q u a d r a t i c e q u a t i o n , and t h e m e n i s c u s p r o f i l e and c o n t a c t p o i n t were computed from E q s . (4-38) and (4-40) r e s p e c t i v e l y . [ i i i ] The n e w l y 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 was a p p r o x i m a t e d by a new q u a d r a t i c e q u a t i o n : P ( x ) and R ( x ) were r e c a l c u l a t e d ; then t h e m e n i s c u s p r o f i l e and c o n t a c t p o i n t were r e c o m p u t e d . [ i v ] The p r o c e s s was r e p e a t e d u n t i l s u c c e s s i v e c a l c u l a t i o n s of t h e m e n i s c u s shape d i f f e r e d n e g l i g i b l y . I n t h i s way t h e g e n e r a t i o n of f l u x p r e s s u r e , due t o mould o s c i l l a t i o n , and t h e men i scus shape have been c o u p l e d . F i g u r e 4-40 shows the m e n i s c u s p r o f i l e s p r e d i c t e d a t d i f f e r e n t t i m e s i n a s i n u s o i d a l mould o s c i l l a t i o n c y c l e h a v i n g a f r e q u e n c y of 100 cpm and a s t r o k e l e n g t h of 8mm. The i n i t i a l t i m e c o r r e s p o n d s t o the mould b e i n g a t t h e t o p of i t s s t r o k e when v =0. A t t h i s t i m e , t h e r e i s no o s c i l l a t i o n - g e n e r a t e d p r e s s u r e i n t h e f l u x i f t h e m e n i s c u s i s c o m p l e t e l y l i q u i d because v i s e f f e c t i v e l y z e r o . Thus a t t=0 , t h e m e n i s c u s s p r o f i l e shown i s t h a t p r e d i c t e d f rom E q . ( 4 - 3 6 ) . A t 0 . 1 5 s t h e mould has r e a c h e d i t s maximum downward v e l o c i t y and t h e m e n i s c u s p r o f i l e and " c o n t a c t " p o i n t have been d e p r e s s e d by t h e p o s i t i v e p r e s s u r e g e n e r a t e d i n the f l u x . At 0 . 3 s , t h e mould has t r a v e l l e d t o t h e bot tom of i t s s t r o k e when Vm=0, and o s c i l l a t i o n - g e n e r a t e d p r e s s u r e i n t h e f l u x has d i s a p p e a r e d 82 l e a v i n g t h e m e n i s c u s a g a i n w i t h an e q u i l i b r i u m shape p r e d i c t e d by E q . ( 4 - 3 6 ) . Beyond 0 . 3 s , the mould moves u p w a r d , g e n e r a t i n g a n e g a t i v e p r e s s u r e i n t h e f l u x and d r a w i n g t h e m e n i s c u s and c o n t a c t p o i n t a l s o u p w a r d . F i g u r e 4-41 shows t h e movement o f t h e c o n t a c t p o i n t of t h e m e n i s c u s w i t h the mould w a l l d u r i n g t h e mould o s c i l l a t i o n . I t i s seen t h a t t h e c o n t a c t p o i n t , w h i c h i s r e s p o n d i n g t o t h e mould v e l o c i t y , moves out of phase w i t h the mould d i s p l a c e m e n t by TX/2. M o r e o v e r t h e a m p l i t u d e of t h e c o n t a c t p o i n t movement i s g r e a t e r t h a n t h a t of t h e mould d i s p l a c e m e n t ; and a f t e r 0 . 3 s t h e c o n t a c t p o i n t r i s e s above i t s i n i t a l l e v e l . Thus a t t h i s t i m e , w h i c h c o r r e s p o n d s t o the end of t h e n e g a t i v e - s t r i p p e r i o d or j u s t beyond i t , t h e m o l t e n s t e e l s u r g e s t o w a r d t h e m o u l d w a l l ; and i f a t h i n r i g i d s k i n c o v e r s p a r t of t h e m e n i s c u s , o v e r f l o w may o c c u r . T h i s t i m i n g of t h e o v e r f l o w of the s t e e l i s i n agreement w i t h t h e o b s e r v a t i o n s made u s i n g a " m o u l d s i m u l a t o r " by Kawakami e t a l . 3 6 4 . 5 Mechanism Of O s c i l l a t i o n - M a r k F o r m a t i o n I n o r d e r t o f a c i l i t a t e t h e m a t h e m a t i c a l a n a l y s i s of f l u i d f l o w and m e n i s c u s shape i n t h e p r e c e d i n g s e c t i o n s , t h e m e n i s c u s has been assumed r e s p e c t i v e l y t o be c o v e r e d p a r t i a l l y w i t h a r i g i d s k i n or t o behave as a l i q u i d . The h e a t - f l o w a n a l y s i s has shown t h a t b o t h c a s e s a r e p o s s i b l e d e p e n d i n g on t h e s u p e r h e a t and l o c a l c o n v e c t i o n , F i g s . 4-27 - 4 - 3 4 . The p r e s e n c e of t h e m e n i s c u s s k i n and t h e g e n e r a t i o n of p r e s s u r e i n t h e f l u x c h a n n e l 83 a r e t h e b a s e s f o r t h e mechanism of o s c i l l a t i o n mark f o r m a t i o n . F i g s . 4-42 and 4~43 p r o v i d e a s c h e m a t i c r e p r e s e n t a t i o n o f t h e f o r m a t i o n o f t h e two t y p e s o f o s c i l l a t i o n marks, i . e . w i t h a n d w i t h o u t a d j a c e n t s u b s u r f a c e h o o k s r e s p e c t i v e l y . I n b o t h c a s e s t h e m e n i s c u s r e s p o n d s t o t h e mould o s c i l l a t i o n and f l u x p r e s s u r e i n t h e same manner. As d e s c r i b e d e a r l i e r , d u r r n g t h e n e g a t i v e - s t r i p t i m e ( S t a g e s 1-3), when t h e mould i s m oving downward more r a p i d l y t h a n t h e s t r a n d , t h e m e n i s c u s i s p u s h e d by t h e p o s i t i v e p r e s s u r e g e n e r a t e d i n t h e mould f l u x , away f r o m t h e mould w a l l . Then i n 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 ( S t a g e s 4 - 7 ) , t h e m e n i s c u s i s drawn back t o w a r d t h e m o u l d w a l l by t h e n e g a t i v e p r e s s u r e . I t i s most u n l i k e l y t h a t t h e p a r t i a l l y s o l i d i f i e d m e n i s c u s i s drawn back u n i f o r m l y , however, b e c a u s e t h e u p p e r p a r t o f t h e m e n i s c u s s k i n i s f a r t h e s t f r o m t h e c o o l i n g i n f l u e n c e o f t h e mould w a l l and t h e r e f o r e s h o u l d be t h e h o t t e s t and w e a k e s t . As a r e s u l t t h e u p p e r p a r t o f t h e s k i n i s e x p e c t e d t o be drawn back more by t h e n e g a t i v e f l u x p r e s s u r e and t h e i n e r t i a l f o r c e of t h e s u r g i n g l i q u i d s t e e l . The d i f f e r e n c e between t h e two t y p e s of o s c i l l a t i o n marks t h e n a r i s e s b e c a u s e of d i f f e r e n c e s i n t h e m e c h a n i c a l s t r e n g t h of t h e m e n i s c u s s k i n . In t h e c a s e o f o s c i l l a t i o n marks w i t h s u b s u r f a c e h o o k s , t h e s k i n i s r e l a t i v e l y s t r o n g , owing t o a g r e a t e r t h i c k n e s s (low s u p e r h e a t , s t a g n a n t l i q u i d ) a n d / o r low c a r b o n c o n t e n t . Thus t h e t o p o f t h e s k i n r e s i s t s b e i n g drawn back f u l l y t o w a r d t h e mould w a l l , and l i q u i d s t e e l o v e r f l o w s i t ( S t a g e 4, F i g . 4-42) t o form a s u b s u r f a c e hook. O'n t h e o t h e r hand, w i t h o s c i l l a t i o n marks h a v i n g no s u b s u r f a c e h o o k s , t h e s k i n i s weak and b e h a v e s 84 more l i k e a l i q u i d . Thus a t the b e g i n n i n g of p o s i t i v e s t r i p t h e t o p of t h e s k i n i s e a s i l y p u l l e d back w i t h the l i q u i d t o w a r d t h e mould so t h a t o v e r f l o w does not o c c u r (S tage 4 , F i g . 4 - 4 3 ) . T h i s mechanism i s c o n s i s t e n t w i t h r e s u l t s f rom t h e m e t a l l u r g i c a l s t u d y of o s c i l l a t i o n marks r e p o r t e d i n a p r e v i o u s s e c t i o n . T u r n i n g f i r s t t o t h e s u b s u r f a c e h o o k s , t h e y a r e s i g n i f i c a n t l y l o n g e r , F i g . 4 - 9 , t h a n was p r e d i c t e d by t h e h e a t -f l o w m o d e l , F i g . 4 - 2 7 . T h i s i s e x p l a i n e d by t h e f a c t t h a t t h e m e n i s c u s s k i n i s pushed t o w a r d t h e s t e e l by t h e f l u x p r e s s u r e , F i g . 4 - 4 2 , b e f o r e o v e r f l o w o c c u r s . The hooks i n 0 . 0 9 % - c a r b o n s l a b s form a s m a l l e r a n g l e w i t h the s u r f a c e t h a n hooks i n 0.26%-c a r b o n s l a b s , F i g . 4 - 9 , p r o b a b l y because the m e c h a n i c a l s t r e n g t h of t h e l o w e r c a r b o n m e n i s c u s s k i n i s g r e a t e r " 1 and more r e s i s t a n t t o t h e f l u x p r e s s u r e s . Even t h e n i t i s d i f f i c u l t t o r a t i o n a l i z e t h e p r e s e n c e of hooks i n the 0 .26%-carbon s l a b s t h a t a r e n e a r l y p e r p e n d i c u l a r t o t h e s l a b s u r f a c e , u n l e s s t h e c o m b i n a t i o n of f l u x p r e s s u r e and o v e r f l o w can d e f o r m them t o t h i s e x t e n t . S l a b s t h a t do not e x h i b i t s u b s u r f a c e hooks c h a r a c t e r i s t i c a l l y c o n t a i n e d i n e r t gas b l o w h o l e s . T h i s i n d i c a t e s t h a t c o n v e c t i o n was c r e a t e d c l o s e t o t h e m e n i s c u s by r i s i n g gas b u b b l e s w h i c h , as shown i n F i g . 4 - 3 3 , r e d u c e t h e t h i c k n e s s and s t r e n g t h of t h e m e n i s c u s s k i n so t h a t i t can be drawn back t o w a r d t h e mould w a l l w i t h o u t o v e r f l o w a t t h e b e g i n n i n g o f p o s i t i v e s t r i p . 85 A n o t h e r f i n d i n g i s t h a t the d e p t h of o s c i l l a t i o n marks w i t h s u b s u r f a c e hooks i s g r e a t e r i n 0 .09%-carbon s l a b s t h a n i n s l a b s c o n t a i n i n g 0.26% c a r b o n . A g a i n t h i s may be r e l a t e d t o t h e g r e a t e r s t r e n g t h e x p e c t e d f o r t h e l o w e r c a r b o n m e n i s c u s s k i n . A t t h e b e g i n n i n g of t h e p o s i t i v e - s t r i p t i m e , when n e g a t i v e f l u x p r e s s u r e s a r e d e v e l o p i n g , t h e e n t i r e m e n i s c u s s k i n i n t h e 0.26%-c a r b o n s l a b may be s u f f i c i e n t l y weak t o be drawn back t o w a r d t h e mould w a l l . In c o n t r a s t , t h e m e n i s c u s s k i n i n t h e 0 . 1 % - c a r b o n s l a b may have s u f f i c i e n t s t r e n g t h t o r e s i s t t h e n e g a t i v e f l u x p r e s s u r e e x c e p t near t h e t o p w h i c h bends p a r t i a l l y back t o w a r d t h e mould b e f o r e o v e r f l o w o c c u r s . That the d e p t h of o s c i l l a t i o n marks w i t h o u t s u b s u r f a c e hooks does not depend on t h e c a r b o n c o n t e n t of t h e s t e e l , F i g . 4 - 1 9 , can be e x p l a i n e d by a s i m i l a r argument based on m e c h a n i c a l p r o p e r t i e s . In t h i s c a s e , owing t o a h i g h s u p e r h e a t or enhanced c o n v e c t i o n , the m e n i s c u s s k i n i s s u f f i c i e n t l y t h i n and weak t h a t i t does no t d e v e l o p t h e m e c h a n i c a l r i g i d i t y of a s o l i d but behaves more l i k e a l i q u i d ; t h u s i t r e s p o n d s i n the same manner t o f l u x p r e s s u r e i r r e s p e c t i v e of t h e c a r b o n c o n t e n t of t h e s t e e l . The f i n d i n g t h a t t h e d e p t h of o s c i l l a t i o n marks w i t h s u b s u r f a c e hooks i n a l u m i n u m - k i l l e d s t e e l i s s l i g h t l y g r e a t e r t h a n i n s i l i c o n - k i l l e d s t e e l s may be due t o an i n c r e a s e i n mould f l u x v i s c o s i t y as a l u m i n a , f l o a t i n g out of t h e s t e e l , i s a b s o r b e d by t h e f l u x . The i n c r e a s e i n s l a g v i s c o s i t y i n c r e a s e s t h e p r e s s u r e g e n e r a t e d i n t h e mould f l u x , F i g . 4 - 2 0 , w h i c h i n t u r n s h o u l d i n c r e a s e t h e d e f o r m a t i o n of t h e m e n i s c u s d u r i n g n e g a t i v e s t r i p , and t h e d e p t h of o s c i l l a t i o n m a r k s . 86 The e l e c t r o m a g n e t i c a l l y d r i v e n f l o w d e c r e a s e s t h e d e p t h o f o s c i l l a t i o n marks, F i g . 4-22. I t was m e n t i o n e d e a r l i e r t h a t t h e s t e e l c o n v e c t i o n s u p p r e s s e s m e n i s c u s s o l i d i f i c a t i o n . F u r t h e r m o r e a t e m p e r a t u r e i n c r e a s e a t t h e m e n i s c u s due t o e l e c t r o m a g n e t i c s t i r r i n g was o b s e r v e d i n t h i s e x p e r i m e n t 8 8 a s a r e s u l t of m i x i n g o f s t e e l i n t h e m o u l d . T h i s s h o u l d i n c r e a s e t h e t e m p e r a t u r e i n t h e mould f l u x l a y e r w h i c h i n t u r n d e c r e a s e s t h e v i s c o s i t y o f mould f l u x and t h e f l u x p r e s s u r e and t h e r e b y r e d u c e 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 . 4.6 M e n i s c u s M o d e l P r e d i c t i o n s The f i n a l s t a g e i n t h i s s t u d y has been t o combine t h e m a t h e m a t i c a l a n a l y s e s of h e a t f l o w and f l u x p r e s s u r e i n a f i r s t -g e n e r a t i o n m e n i s c u s model and t o c a l c u l a t e a p p r o x i m a t e l y t h e e f f e c t o f t h e f o l l o w i n g v a r i a b l e s on t h e o s c i l l a t i o n mark f o r m a t i o n : s t r o k e l e n g t h , o s c i l l a t i o n f r e q u e n c y , n e g a t i v e s t r i p t i m e , mould f l u x v i s c o s i t y and m e n i s c u s l e v e l v a r i a t i o n . In t h e m o d e l , t h e h e a t f l o w a n a l y s i s i s a p p l i e d f i r s t t o p r e d i c t t h e t e m p e r a t u r e o f t h e mould f l u x a d j a c e n t t o t h e mould w a l l d u r i n g 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 o s c i l l a t i o n c y c l e . The a v e r a g e t e m p e r a t u r e o f t h e f l u x w i t h i n 5 0 0 i u m of t h e mould w a l l , a t y p i c a l d e p t h o f o s c i l l a t i o n mark, and o v e r a h e i g h t a t t h e m e n i s c u s e q u i v a l e n t t o t h e o s c i l l a t i o n s t r o k e l e n g t h t h e n i s c a l c u l a t e d s i n c e t h i s f l u x w i l l e n t e r t h e f l u x c h a n n e l on t h e s u c c e e d i n g d o w n s t r o k e of t h e mould. The v i s c o s i t y o f t h e f l u x i s computed f r o m t h e e m p i r i c a l r e l a t i o n s h i p 1 6 87 log y f = 0.578X10* ( T I 2 ? 3 ) - 2.979 (4-41) Next t h e f l u i d f l o w a n a l y s i s i s a p p l i e d t o p r e d i c t t h e p r e s s u r e g e n e r a t e d i n t h e mould f l u x , a s s u m i n g t h a t t h e l e n g t h of t h e f l u x c h a n n e l i s e q u a l t o t h e p i t c h of o s c i l l a t i o n marks and t h a t t h e t o p and b o t t o m w i d t h a r e 0 . 3 5 and 0 .05 mm r e s p e c t i v e l y . The t o p c h a n n e l w i d t h of 0.35mm i s i n the range of t h e d e p t h of o s c i l l a t i o n m a r k s , w h i l e t h e b o t t o m w i d t h has been e s t i m a t e d f rom t h e minimum t h i c k n e s s of mould f l u x f i l m c a l c u l a t e d f rom mould f l u x c o n s u m p t i o n . The t o t a l f l u x p r e s s u r e f o r c e , i n t e g r a t e d o v e r the c h a n n e l l e n g t h , has been computed a t t h e maximum downward v e l o c i t y of t h e m o u l d ; t h i s has been used as a measure of t h e d e f o r m a t i o n of t h e p a r t i a l l y s o l i d i f i e d m e n i s c u s , and of t h e d e p t h of o s c i l l a t i o n m a r k s . F i g . 4-44 shows t h e e f f e c t o f c h a n g i n g m e n i s c u s l e v e l , e x p r e s s e d i n terms of t h e p i t c h of t h e o s c i l l a t i o n m a r k s , on t h e t o t a l f o r c e g e n e r a t e d i n two mould f l u x e s h a v i n g d i f f e r e n t v i s c o s i t i e s . A sudden r i s e i n m e n i s c u s l e v e l l e n g t h e n s t h e p i t c h of t h e o s c i l l a t i o n marks and a t t h e same t i m e has t h e e f f e c t of r a i s i n g the mould v e l o c i t y . T h i s r e l a t i o n s h i p can be e x p r e s s e d by t h e f o l l o w i n g e q u a t i o n : (4-42) Thus as t h e p i t c h of the o s c i l l a t i o n marks i n c r e a s e s , t h e t o t a l f o r c e a c t i n g on t h e p a r t i a l l y s o l i d i f i e d m e n i s c u s i n c r e a s e s and t h e d e p t h of o s c i l l a t i o n marks a l s o s h o u l d i n c r e a s e . T h i s e f f e c t has been seen i n F i g s . 4-17 and 4-18. T h i s phenomenon 88 has been r e p o r t e d t o be a p r o b l e m when c a s t i n g w i t h h i g h o s c i l l a t i o n f r e q u e n c i e s . 6 1 A l s o i n F i g . 4-44, r a i s i n g t h e f l u x v i s c o s i t y i s p r e d i c t e d t o i n c r e a s e t h e t o t a l p r e s s u r e f o r c e , a s was s u g g e s t e d e a r l i e r w i t h r e s p e c t t o t h e c a s t i n g of a l u m i n u m - v s s i l i c o n - k i l l e d s t e e l s . F i g u r e 4-45 shows t h e e f f e c t of o s c i l l a t i o n s t r o k e on t h e t o t a l f l u x p r e s s u r e f o r c e f o r two o s c i l l a t i o n f r e q u e n c i e s . In b o t h c a s e s t h e t o t a l f o r c e i n c r e a s e s a l m o s t l i n e a r l y w i t h i n c r e a s i n g s t r o k e l e n g t h w h i c h i s i n r e a s o n a b l e a g r e e m e n t w i t h t h e m e a s u r e d r e l a t i o n s h i p , F i g . 4-21, and w i t h t h e r e p o r t e d r e l a t i o n s h i p by Emi e t a l . , 9 F i g . 2-3. T h i s i n f l u e n c e i s c a u s e d by t h e i n c r e a s e d mould s p e e d as t h e s t r o k e l e n g t h i s i n c r e a s e d a t c o n s t a n t f r e q u e n c y . F i g u r e 4-46 shows t h e e f f e c t o f o s c i l l a t i o n f r e q u e n c y on t h e f l u x p r e s s u r e f o r c e f o r s e v e r a l d i f f e r e n t s t r o k e l e n g t h s . W i t h l o n g e r s t r o k e s , r a i s i n g t h e o s c i l l a t i o n f r e q u e n c y m a r k e d l y r e d u c e s t h e t o t a l p r e s s u r e f o r c e ; but when s h o r t o s c i l l a t i o n s t r o k e s a r e e m ployed, f r e q u e n c y has o n l y a m i n o r e f f e c t . The same t e n d e n c y , w i t h r e s p e c t t o t h e d e p t h o f t h e o s c i l l a t i o n marks, has been r e p o r t e d by Kuwano e t a l . 2 1 f o r t h e c a s t i n g o f s t a i n l e s s s t e e l and by o t h e r s 8 2 2 w o r k i n g w i t h h i g h o s c i l l a t i o n f r e q u e n c i e s . V a r y i n g t h e f r e q u e n c y a f f e c t s t h e t o t a l p r e s s u r e f o r c e by c h a n g i n g b o t h t h e p i t c h of t h e o s c i l l a t i o n marks and t h e mould v e l o c i t y . At low f r e q u e n c i e s , t h e h i g h t o t a l p r e s s u r e f o r c e r e s u l t s f r o m t h e l o n g p i t c h of t h e o s c i l l a t i o n marks and t h e c o r r e s p o n d i n g l y l o n g f l u x c h a n n e l . However, b e c a u s e t h e 89 v e l o c i t y of t h e mould i s a l s o l o w , a low t o t a l p r e s s u r e f o r c e can be a t t a i n e d a t h i g h c a s t i n g s p e e d s , where V M -v i s s m a l l , d e p e n d i n g on t h e s t r o k e l e n g t h o f t h e m o u l d . Such an e f f e c t i s shown i n F i g . 4 - 4 7 . The n e g a t i v e - s t r i p t i m e , as d e f i n e d by E q . ( 2 - 3 ) , has been c a l c u l a t e d f o r a l l s e v e n t e e n c o n d i t i o n s i n d i c a t e d a b o v e ; and i t s e f f e c t on t h e t o t a l p r e s s u r e f o r c e g e n e r a t e d i n t h e f l u x c h a n n e l , a s s u m i n g a c a s t i n g speed o f 1.0 m / m i n , i s shown i n F i g . 4 - 4 8 . A s i n g l e c o r r e l a t i o n t h u s i s f o u n d , i n w h i c h w i t h o n l y m i n o r v a r i a t i o n s , the t o t a l p r e s s u r e f o r c e i n c r e a s e s w i t h n e g a t i v e - s t r i p t i m e . The same r e l a t i o n s h i p , between o s c i l l a t i o n mark d e p t h and t N , has been r e p o r t e d by M c P h e r s o n e t a l . 2 " H a s h i o e t a l . 2 3 and M i z o g u c h i e t a l . 2 " , see F i g . 2 - 5 . A s s u m i n g t h a t t h e f u l c r u m of b e n d i n g d e f o r m a t i o n of t h e m e n i s c u s i s a t t h e o u t l e t o f the f l u x c h a n n e l , the b e n d i n g moment can be c a l c u l a t e d as f o l l o w s : \= tf P ( x ) ( x - l f ) d x (4-43) The r e l a t i o n between the n e g a t i v e s t r i p t i m e and t h e c a l c u l a t e d b e n d i n g moment i s shown i n F i g . 4 - 4 9 , w h i c h shows a s i m i l a r t e n d e n c y t o t h a t i n F i g . 4 - 4 8 . However the t o t a l p r e s s u r e f o r c e seems t o be a more s u i t a b l e i n d e x o f t h e mark d e p t h , because the r i g i d i t y of t h e i n i t i a l l y s o l i d i f i e d s h e l l a t t h e m e n i s c u s i s not u n d e r s t o o d c o m p l e t e l y y e t . F i n a l l y a s e r i e s o f p l o t s of t o t a l p r e s s u r e f o r c e vs n e g a t i v e - s t r i p t i m e has been c a l c u l a t e d f o r d i f f e r e n t c a s t i n g speeds and a r e shown i n F i g . 4 - 5 0 . The 90 s o l i d l i n e s r e v e a l the e f f e c t of c a s t i n g speed on the t o t a l p r e s s u r e f o r c e . At low f r e q u e n c i e s and w i t h s h o r t s t r o k e s , t h e f o r c e d e c r e a s e s w i t h i n c r e a s i n g c a s t i n g s p e e d : but w i t h h i g h e r f r e q u e n c i e s and l o n g e r s t r o k e s i t i n c r e a s e s w i t h c a s t i n g s p e e d . I t s h o u l d be n o t e d t h a t i n t h i s c a l c u l a t i o n t h e e f f e c t of t h e c a s t i n g speed on the t e m p e r a t u r e d i s t r i b u t i o n i n t h e m e n i s c u s r e g i o n was not t a k e n i n t o a c c o u n t , owing t o a p a u c i t y of d a t a . I n g e n e r a l i n c r e a s i n g c a s t i n g speed i n c r e a s e s t h e m e n i s c u s t e m p e r a t u r e w h i c h r e d u c e s t h e v i s c o s i t y of mould f l u x . T h i s t e n d s t o r educe the d i f f e r e n c e s between t h e r e l a t i o n s h i p s f o r v a r i o u s c a s t i n g speeds , shown i n F i g . 4 - 5 0 . These p r e d i c t i o n s o b v i o u s l y have v a l u e o n l y i n r e v e a l i n g t r e n d s i n t h e d e p t h of o s c i l l a t i o n marks as a f u n c t i o n of o s c i l l a t i o n v a r i a b l e s . However t h e knowledge g a i n e d s h o u l d be u s e f u l i n t h e s e l e c t i o n of mould c o n d i t i o n s t h a t m i n i m i z e t h e d e p t h of o s c i l l a t i o n marks so t h a t h i g h s u r f a c e q u a l i t y can be a c h i e v e d . 91 T a b l e I I - C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n of t h e S l a b Samples f rom Company A C Mn S i P S A l C h e m i c a l 0, .08 0 .58 0 .20 0 .007 0, .009 0 .001 C o m p o s i t i o n - 0 , .26 - 0 . 8 9 - 0 . 2 4 - 0 . 0 0 9 - 0 , .019 - 0 . 0 3 3 T e m p e r a t u r e ( i n t u n d i s h ) : 1529 - 1 5 4 9 ° C C a s t i n g s p e e d : 0 .89 - 1.25 m/min Immers ion n o z z l e : b i f u r c a t e d , 2 5 ° down C a s t i n g M o u l d s i z e : 1829 - 1930mm x 178 - 203mm C o n d i t i o n s M o u l d o s c i l l a t i o n : 12.7mm s t r o k e , 40 - 90cpm f r e q u e n c y V i s c o s i t y of mould f l u x : 5 . 45 p o i s e a t 1 2 0 0 ° C 2 . 6 5 p o i s e a t 1 3 0 0 ° C Type of m a c h i n e : V e r t i c a l t y p e w i t h u n b e n d i n g (9mR) 92 T a b l e I I I - C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n S l a b Samples f rom Company B of t h e C Mn S i P S A l N C h e m i c a l C o m p o s i t i o n 0 .07 0 .25 0 .03 0 . 0 0 6 0 .012 0 .042 - 0 . 1 0 - 0 . 4 1 - 0 . 0 6 - 0 . 0 0 7 - 0 . 0 1 6 - 0 . 0 5 6 0 .0047 - 0 . 0 1 6 4 Tempera ture ( i n t u n d i s h ) : 1529 - 1 5 4 9 ° C C a s t i n g s p e e d : 1.37 - 1.57 m/min Immers ion n o z z l e : b i f u r c a t e d , 7 . 5 ° up C a s t i n g M o u l d s i z e : 985 - 1035mm x 180mm C o n d i t i o n s M o u l d o s c i l l a t i o n : 11mm s t r o k e , 95cpm f r e q u e n c y V i s c o s i t y of mould f l u x : not a v a i l a b l e ( s o f t e n i n g p o i n t 1 0 6 5 ° C , m e l t i n g p o i n t 1 1 4 0 ° C ) Type of m a c h i n e : V e r t i c a l t y p e w i t h u n b e n d i n g (9mR) 93 T a b l e IV - C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n of t h e S l a b Samples from Company C C Mn S i P S A l N C h e m i c a l C o m p o s i t i o n 0 .05 - 0 . 0 7 0 .26 - 0 . 3 3 0.021 - 0 . 0 2 6 0 . 0 0 3 - 0 . 0 0 5 0 .009 - 0 . 0 1 3 0 . 0 4 6 - 0 . 0 6 3 0 .0050 - 0 . 0 0 8 0 T e m p e r a t u r e ( i n t u n d i s h ) : 1546 - 1 5 5 5 ° C C a s t i n g s p e e d : 0 . 8 5 - 0 .91 m/min Immers ion n o z z l e : b i f u r c a t e d , 15 ° up C a s t i n g M o u l d s i z e : 1480 - 1880mm x 240mm C o n d i t i o n s M o u l d o s c i l l a t i o n : 5 .8 - 15mm s t r o k e , 60 - 95cpm f r e q u e n c y V i s c o s i t y of mould f l u x : t y p e l / 1.0 p o i s e a t 1 2 5 0 ° C 0 .6 p o i s e a t 1 3 0 0 ° C t y p e 2 / 4 . 9 p o i s e a t 1 2 5 0 ° C 2 .9 p o i s e a t 1 3 0 0 ° C Type of m a c h i n e : V e r t i c a l t y p e w i t h u n b e n d i n g (10.5mR) 94 T a b l e V - C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n of t h e S l a b Samples f rom Company D C Mn S i P S A l C h e m i c a l C o m p o s i t i o n 0 .043 0 . 1 9 0 .013 0 . 0 0 9 0.011 - 0 . 0 8 0 - 1 . 0 5 - 0 . 2 0 - 0 . 0 1 5 - 0 . 0 1 4 -0 .043 0 .070 Tempera ture ( i n t u n d i s h ) : 1541 - 1 5 5 4 ° C ( i n mould) : 1515 - 1 5 2 3 ° C C a s t i n g s p e e d : 0 .97 - 1.52 m/min Immers ion n o z z l e : 1) m u l t i - p o r t p r a c t i c e C a s t i n g C o n d i t i o n s 2) b i f u r c a t e d , 2 5 ° down M o u l d s i z e : 889 - 1346mm x 235mm M o u l d o s c i l l a t i o n : 19.8mm s t r o k e , 35 - 65cpm f r e q u e n c y V i s c o s i t y of mould f l u x : 60 - 65 c p a t 1 3 0 0 ° C Type of m a c h i n e : V e r t i c a l t y p e w i t h u n b e n d i n g (13mR) * S c h e m a t i c v i e w of the i m m e r s i o n n o z z l e f o r t h e m u l t i - p o r t p r a c t i c e i s i n F i g . 4 - 1 . 95 T a b l e VI - C a s t i n g C o n d i t i o n s and C h e m i c a l C o m p o s i t i o n of t h e S l a b Samples f rom Company E C Mn S i P S A l C h e m i c a l C o m p o s i t i o n 0 .04 0 .14 0 . 0 2 0 .020 0 .014 0.001 T e m p e r a t u r e ( i n t u n d i s h ) : 1 5 7 2 ° C C a s t i n g s p e e d : 0 .70 m/min Immers ion n o z z l e : b i f u r c a t e d , 4 5 ° down C a s t i n g M o u l d s i z e : 1600mm x 250mm C o n d i t i o n s M o u l d o s c i l l a t i o n : 7.0mm s t r o k e , lOOcpm f r e q u e n c y V i s c o s i t y o f mould f l u x : not a v a i l a b l e Type of m a c h i n e : V e r t i c a l t y p e w i t h u n b e n d i n g ( l 0 . 5 m R ) S t i r r i n g f o r c e ( f l o w v e l o c i t y o f s t e e l ) :0 - 1m/s * S c h e m a t i c f i e w of t h e e l e c t r o m a g n e t i c s t i r r e r i n t h e mould i s i n F i g . 4-2 96 T a b l e V I I - T h e r m a l P r o p e r t i e s and C o n d i t i o n s f o r C a l c u l a t i o n of M o u l d Heat F l u x S p e c i f i c hea t of w a t e r " : 4 . 18 J / g ° C V i s c o s i t y of w a t e r " : 0 . 8 3 x 1 0 " 2 P D e n s i t y o f w a t e r " : 0 . 9 9 5 g / c m 3 T h e r m a l c o n d u c t i v i t y of w a t e r " : 6 . 1 5 x lO~ 3 W/cm°C Water t e m p e r a t u r e a t t h e i n l e t of wate r c h a n n e l 1 0 0 : 2 1 ° C Water t e m p e r a t u r e a t t h e o u t l e t of w a t e r c h a n n e l 1 0 0 : 3 2 ° C F l o w v e l o c i t y of w a t e r 1 0 0 : 500cm/s H y d r a u l i c d i a m e t e r of w a t e r c h a n n e l 1 0 0 : 1cm Water c h a n n e l gap w i d t h 1 0 0 : 1.5cm T h e r m a l c o n d u c t i v i t y o f mould w a l l 1 8 : 3 .25 W/cm°C T h i c k n e s s of mould w a l l 1 8 : 4 . 8 cm L e n g t h of mould w a l l 1 8 : 80 cm D i s t a n c e of mould f l u x s u r f a c e from t h e mould t o p 1 8 : 8 cm D i s t a n c e of m e n i s c u s f rom t h e mould t o p 1 8 : 12 cm 97 T a b l e V I I I - A s s u m e d S t e e l C o m p o s i t i o n a n d T h e r m o p h y s i c a l P r o p e r t i e s f o r C a l c u l a t i o n s o f T e m p e r a t u r e D i s t r i b u t i o n a t t h e M e n i s c u s C Mn S i P S A l C h e m i c a l 0 . 1 6 C o m p o s i t i o n (%) 0 . 8 0 0 . 15 0 . 0 2 0 . 0 2 0 . 0 5 S t e e l l i q u i d u s : 1 5 1 7 ° C S t e e l s o l i d u s : 1 4 5 6 ° C S t e e l s u p e r h e a t : 5 ° C S t e e l s p e c i f i c h e a t : 0 . 6 7 J / g ° C S t e e l t h e r m a l c o n d u c t i v i t y 1 8 : 0 . 4 6 5 W / c m ° C S t e e l d e n s i t y : 7 . 2 g / c m I n i t i a l f l u x t e m p e r a t u r e 1 2 : 1 5 0 0 ° C F l u x s p e c i f i c h e a t : 1 .26 J / g ° C F l u x t h e r m a l c o n d u c t i v i t y 1 8 : 0 . 0 2 3 W / c m ° C F l u x d e n s i t y 1 8 : 2 . 8 g / c m 3 C a l c u l a t e d f r o m f o l l o w i n g e q u a t i o n s : 1 0 1 T = 1 5 3 6 - { 7 8 [ % C ] + 7 . 6 [ % S i ] + 4 . 9 [ % M n ] + 3 4 . 4 [ % P ] l l c l + 38 [%S] + 4 . 7 [ % C u j + 3 . 1 [ % N i ] + 1 . 3 [ % C r ] + 3 . 6 [ % A 1 ] } T = 1 5 3 6 - { 4 1 5 . 5 [ % C ] + l 2 . 3 [ % S i ] + 6 . 8 [ % M n ] + 1 2 4 . 5 [ % P ] s o 1 + 1 8 3 . 9 [ % S ] + 4 . 3 [ % N i ] + 1 . 4 [ % C r ] + 4.1 [ % A 1 3 } 98 F i g . 4-1 Immersion n o z z l e f o r t h e m u l t i - p o r t p r a c t i c e (Company D ) . F i g . 4-2 S c h e m a t i c view o f t h e e l e c t r o m a g n e t i c s t i r r e r i n t h e m o u l d 8 8 (Company E ) . 100 F i g . 4-3 Appearance of o s c i l l a t i o n marks on the narrow face of a s l a b (Company A ) . 101 E E tn D E c o o CO O o DL 18 16 14 12 10 H 8 t: 4 0 • A . 1 Avg Standard deviation Measurement position • A Centre of the narrow face • O Corner of the narrow face 0.7 A ° - A 1 1 1 0.9 I.I Casting speed (m/min) 1.3 F i g . 4-4 E f f e c t of c a s t i n g s p e e d on t h e p i t c h o f o s c i l l a t i o n marks (Company A ) . 102 F i g . 4-5 Appearance of o s c i l l a t i o n of a s l a b (Company C ) . marks on the narrow face 103 F i g . 4-6 F r e q u e n c y of mould o s c i l l a t i o n i n e a c h c a s t i n g o p e r a t i o n . (a) F i g . 4 - 7 Appearance of o s c i l l a t i o n marks on the narrow face of s l a b ca s t w i t h (a) B i f u r c a t e d n o z z l e and (b) M u l t i - p o r t p r a c t i c e (Company D ) . 105 F i g . 4-8 A p p e a r a n c e of o s c i l l a t i o n m a r k s on t h e n a r r o w f a c e o f s l a b c a s t (a) w i t h o u t m o u l d EMS a n d (b) w i t h m o u l d EMS (Company E ) . 106 F i g . 4-9 S u b s u r f a c e s t r u c t u r e near o s c i l l a t i o n marks e x h i b i t i n g "hooks" i n s t e e l s l a b s c o n t a i n i n g (a) 0.09%C and (b) 0.26%C (Company A ) . (x6) 107 F i g . 4-10 Subsur face s t r u c t u r e near o s c i l l a t i o n marks wi thout "hooks" i n s t e e l s l ab s c o n t a i n i n g (a) 0.08% C and (b) 0.26% C (Company A ) . (x6) 108 F i g . 4-11 Change o f s e c o n d a r y d e n d r i t e arm s p a c i n g a l o n g t h e c a s t i n g d i r e c t i o n C=0.09% (Company A ) . 109 F i g . 4-12 S u b s t a n c e a t t h e end of a s u b s u r f a c e hook (a) x 94 (b) x 2000 (Company B) . F i g . 4-13 X - r a y s p e c t r o g r a p h of t h e s u b s t a n c e n e a r t h e hook (Company B ) . 111 F i g . 4-14 Subsurface s t r u c t u r e near o s c i l l a t i o n marks in s t e e l s l a b s from Company C. (x6) 112 F i g . 4-15 Subsurface s t r u c t u r e near o s c i l l a t i o n marks i n s t e e l s l ab s ca s t by (a) b i f u r i c a t e d n o z z l e and (b) m u l t i -port p r a c t i c e (Company D ) . (x6.5) 113 F i g . 4-16 S u b s u r f a c e s t r u c t u r e n e a r o s c i l l a t i o n m a r k s i n s t e e l s l a b s c a s t ( a ) w i t h o u t EMS-M and (b) w i t h EMS-M (Company E ) . ( x 6 ) 1000 500 0 1 1 Subsurface Hooks • o Present • Absent 0 0 0 8 % • 0 . on o 8 8 ^ 1 J I I I 10 15 20 25 Pitch of Oscillation Marks (mm) F i g . 4-17 R e l a t i o n s h i p between d e p t h and p i t c h of o s c i l l a t i o n marks i n 0.08%-Carbon s l a b s (Company A ) . F i g . 4-18 R e l a t i o n s h i p between d e p t h and p i t c h o f o s c i l l a t i o n marks on 0.26%-Carbon s l a b s (Company A ) . T 1 1 1 r Subsurface Hooks o Present 0 0 0 0 5 010 015 0-20 0-25 Carbon Content of Slabs (%) 0-30 F i g . 4-19 I n f l u e n c e of c a r b o n c o n t e n t of t h e s l a b s on t h e d e p t h of o s c i l l a t i o n marks (Company A ) . 1500 1000 -500 10 I 5 20 Pitch of oscillation marks (mm) E f f e c t of A l on the shape of o s c i l l a t i o n marks (Company A ) . N o t e : a l l marks are h o o k - l i k e s t r u c t u r e s . 118 tn JC o E c o o tn o Q. Q 2 5 0 0 _ 2 0 0 0 E I 500 ° 1000 500 0. 0.10.2 03 T l r 035 0.4 f » 64 cpm vs= 0.91m/ min U f S Q.6P (at 1300°C) 1 oo-1 10 15 Stroke (mm) 20 F i g . 4-21 E f f e c t of o s c i l l a t i o n stroke on the depth of o s c i l l a t i o n Marks (Company C). 1 19 Flow velocity of molten steel (m/s) F i g . 4-22 E f f e c t of mould EMS on t h e d e p t h of o s c i l l a t i o n marks (Company E ) . 120 0 Yn Typel Type 2 Type3 • — • — • Type 4 Cooling water Steel Solidified shell Type 7 Type 8 Type9 F i g . 4-23 P h y s i c a l s y s t e m f o r t h e c o m p u t a t i o n o f h e a t t r a n s f e r i n t h e mould. 121 E E CO U CO ' c CP E o u c o 100 Distance From Cold Face (mm) 0 20 40 0 -20 •g -40 - 6 0 x T T 'Measured temperature (°C) - 8 0 ' Heat Flux (kW/mz) 1000 2000 3000 T T F i g . 4-24 T e m p e r a t u r e d i s t r i b u t i o n i n t h e mould w a l l and t h e he a t f l u x d i s t r i b u t i o n n e a r t h e m e n i s c u s ( C l o s e d c i r c l e s a r e v a l u e s of mould t e m p e r a t u r e r e p o r t e d by Nakato e t a l 1 8 ) . 122 F i g . 4-25 Geometry and n o d a l s y s t e m employed t o p r e d i c t t e m p e r a t u r e d i s t r i b u t i o n s i n t h e mould f l u x and s t e e l a t the m e n i s c u s . 123 ( s t ! r t ) f Read input data I Coord ina te system Set output time I F i r s t element I Ident i fy type of node I Ca lcu l a t e heat f lux i n each type of node I Ca lcu l a t e temperature i n each node I Ident i fy state of n o d e ( l i q u i d , s o l i d , o r mushy?) F i g . 4-26 Flow c h a r t f o r the c a l c u l a t i o n of th e temperature d i s t r i b u t i o n i n s t e e l and mould f l u x ( i ) . 124 F i g . 4-26 Flow c h a r t f o r t h e c a l c u l a t i o n o f t h e t e m p e r a t u r e d i s t r i b u t i o n i n s t e e l and mould f l u x ( i i ) . 125 Distance from mould wall (mm) F i g . 4-27 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e of 0.3s. S u p e r h e a t of s t e e l i s 5°C. 126 -4I11J u I I 1 0 2 4 6 8 10 Distance from mould wall (mm) F i g . 4-28 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e meni scus a f t e r a t i m e of 0 . 6 s . S u p e r h e a t of s t e e l i s 5 ° C . 1 27 Distance from mould wall (mm) F i g . 4-29 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x a nd f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a t i m e of 0.3s. S u p e r h e a t o f s t e e l i s 0°C. 128 0 2 4 6 8 10 Distance from mould wall (mm) F i g . 4-30 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n the s t e e l a t t h e m e n i s c u s a f t e r a t i m e of 0 . 6 s . Superhea t of s t e e l i s 0 ° C . 1 29 Distance from mould wall (mm) F i g . 4-31 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n the mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t the m e n i s c u s a f t e r a t ime of 0 . 3 s . S u p e r h e a t of s t e e l i s 2 0 ° C . 130 F i g . 4-32 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n the s t e e l a t the meni scus a f t e r a t i m e of 0 . 6 s . Superhea t of s t e e l i s 2 0 ° C . 131 F i g . 4-33 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n the s t e e l a t t h e meni scus a f t e r a t i m e of 0 . 3 s . C o n v e c t i o n i n m o l t e n s t e e l i n c o r p o r a t e d by a r t i f i c i a l l y r a i s i n g l i q u i d c o n d u c t i v i t y f o u r f o l d . 132 F i g . 4-34 P r e d i c t e d t e m p e r a t u r e d i s t r i b u t i o n i n t h e mould f l u x and f r a c t i o n s o l i d i f i e d i n t h e s t e e l a t t h e m e n i s c u s a f t e r a time o f 0.6s. C o n v e c t i o n i n m o l t e n s t e e l i n c o r p o r a t e d by a r t i f i c i a l l y r a i s i n g l i q u i d c o n d u c t i v i t y f o u r f o l d . 1 33 F i g . 4-35 Geometry of mould f l u x c h a n n e l a t t h e m e n i s c u s . 134 I O O I — 8 0 o Q_ Q_ - 2 0 - 4 0 - 6 0 — i 1 downward motion s = 8mm f = I00 cpm vs = l 0 m/min ^ f = 50p / i . f = IOp upward motion _L 0 2 4 6 8 10 Distance Down Mould Wall (mm) F i g . 4-36 P r e d i c t e d a x i a l p r e s s u r e p r o f i l e s i n t h e f l u x c h a n n e l near t h e m e n i s c u s . F l u x c h a n n e l d i m e n s i o n s assumed t o b e : h =0.35mm, h =0.05mm, 1 =lOmm. 135 Distance From Mould Wall (mm) 0 0 I 0 0 2 0 0 3 0 " — i 1 — i 1 — i — i — i F i g . 4-37 P r e d i c t e d v e l o c i t y d i s t r i b u t i o n s i n t h e f l u x c h a n n e l n e a r t h e m e n i s c u s a t t i m e of maximum downward v e l o c i t y o f t h e mould. C o n d i t i o n s as f o r F i g . 4-36 and assumed f l u x v i s c o s i t y i s 5P. 1 3 6 1.0 0.8 CM 0.6 E o \ Q) C T 3 0.4 tn | 0.2 "tn O CO 0.2 0.4 1 ju f = 5 0 P ju } = 10 P i j i . 1 1 1 0 0.2 0.4 0.6 0.8 Distance down the shell (cm) 1.0 F i g . 4-38 S h e a r s t r e s s d i s t r i b u t i o n a l o n g t h e s h e l l F i g . 4-39 Geometry o f t w o - d i m e n s i o n a l m e n i s c u s s y s t e m . F i g . 4-40 P r e d i c t e d change of m e n i s c u s shape w i t h t i m e r e s u l t i n g from s i n u s o i d a l mould o s c i l l a t i o n (s=8mm, f= l00cpm, a=l200 d y n e / c m , p =7.2 g / c m 3 , P j=2 .8 g / c m 3 ) . ? -CO 139 F i g . 4-41 Movement of " c o n t a c t p o i n t " of m e n i s c u s w i t h mould w a l l d u r i n g mould o s c i l l a t i o n . See c a p t i o n t o F i g . 4-40 f o r o s c i l l a t i o n c o n d i t i o n s . 140 F i g . 4-42 S c h e m a t i c r e p r e s e n t a t i o n of t h e f o r m a t i o n of an o s c i l l a t i o n mark w i t h s u b s u r f a c e h o o k . 141 F i g . 4-43 S c h e m a t i c r e p r e s e n t a t i o n of t h e f o r m a t i o n of an o s c i l l a t i o n mark w i t h o u t s u b s u r f a c e h o o k . F i g . 4-44 I n f l u e n c e of m e n i s c u s l e v e l v a r i a t i o n (shown as c h a n g i n g p i t c h of o s c i l l a t i o n marks) on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . 143 0 8 F i g . 4-45 I n f l u e n c e o f o s c i l l a t i o n s t r o k e on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v s = 1m/min. 4-46 I n f l u e n c e of o s c i l l a t i o n f r e q u e n c y on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v s = 1 m / m i n . 145 £ 2 0 Oscillation Frequency (cpm) F i g . 4-47 I n f l u e n c e of o s c i l l a t i o n f r e q u e n c y on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v s=1.2, 1.5 m/min. 1 46 I-5 F i g . 4-48 I n f l u e n c e of n e g a t i v e - s t r i p t i m e on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l (numbers i n p a r a n t h e s e s i n d i c a t e o s c i l l a t i o n f r e q u e n c i e s ) , v =1m/min. 147 0.81 E o i a> c >» c a> E o E c c OD 0.7 0.6 ^ Q51 X 0.4 0.3 0.2 0.1 / Q / f = IOOcpm (5^f=l50cpm d P = ° ^ 2 0 0 c p m |  0.1 0.2 0.3 0.4 Negative strip time (s) F i g . 4 - 4 9 I n f l u e n c e o f n e g a t i v e - s t r i p t i m e on b e n d i n g moment due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l s , v =1m/min. 148 2 0 F i g . 4-50 I n f l u e n c e of n e g a t i v e - s t r i p t i m e on t o t a l f o r c e due t o p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l . v =0.8, 1.0, 1.2, 1.5 m/min. 1 49 5. THE EFFECT OF OSCILLATION MARKS ON THE SURFACE QUALITY OF SLABS 5.1 I n t r o d u c t i o n T r a n s v e r s e c r a c k s and p o s i t i v e s e g r e g a t i o n o f s o l u t e e l e m e n t s a r e commonly e n c o u n t e r e d s u r f a c e d e f e c t s a s s o c i a t e d w i t h o s c i l l a t i o n m arks. The d e f e c t s a r e u s u a l l y o b s e r v e d n e a r t h e b o t t o m o f t h e o s c i l l a t i o n marks. Improvement i n t h e s u r f a c e q u a l i t y o f s l a b s t h e r e f o r e i s d e p e n d e n t on t h e c o n t r o l o f o s c i l l a t i o n marks w h i c h i n t u r n a r e a f f e c t e d by mould o p e r a t i o n . F o r example i t has been r e p o r t e d t h a t t h e d e f e c t s a r e r e d u c e d i n f r e q u e n c y and s e v e r i t y by d e c r e a s i n g 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 . 6 8 6 T r a n s v e r s e c r a c k s a r e one of t h e most common s u r f a c e d e f e c t s on c o n t i n u o s l y c a s t s l a b s and t h u s have been t h e s u b j e c t o f numerous s t u d i e s . Most p r o p o s e d mechanisms o f c r a c k i n i t i a t i o n i n v o l v e s e c o n d a r y c o o l i n g , a s summarized i n C h a p t e r 2, v i z f i n e s u r f a c e c r a c k s c a u s e d by t h e r m a l s t r e s s , 8 " and a u s t e n i t i c i n t e r g r a n u l a r e m b r i t t l e m e n t a s s o c i a t e d w i t h A r 3 t r a n s f o r m a t i o n 8 0 and w i t h p r e c i p i t a t i o n o f n i t r i d e 8 1 a n d / o r c a r b o n i t r i d e . 8 2 8 3 I t i s b e l i e v e d t h a t t h e s e e v e n t s a r e t h e n f o l l o w e d by t h e f o r m a t i o n o f c r a c k s a t t h e s t r a i g h t e n i n g p o i n t o f t h e c a s t m achine where t h e o s c i l l a t i o n marks s e r v e a s a n o t c h . 7 9 However, a s p r e s e n t e d i n t h e p r e v i o u s c h a p t e r , t h e b o t t o m of t h e o s c i l l a t i o n marks i s n o t s h a r p , and f u r t h e r m o r e 1 50 t h e s u r f a c e of s l a b s a r e u s u a l l y a l m o s t f l a t t e n e d by s u p p o r t r o l l s . Thus t h e f a c t t h a t t r a n s v e r s e c r a c k s a r e f o r m e d a l o n g t h e b o t t o m of o s c i l l a t i o n marks c a n n o t be e x p l a i n e d r e a s o n a b l y by t h e p r o p o s e d mechanisms. R e g a r d i n g p o s i t i v e s e g r e g a t i o n , P and N i have been r e p o r t e d t o s e g r e g a t e i n t h e v i c i n i t y of t h e b o t t o m of o s c i l l a t i o n marks i n p l a i n c a r b o n s t e e l s l a b s 2 8 2 9 and i n s t a i n l e s s s t e e l s l a b s , 2 7 r e s p e c t i v e l y . T h i s mechanism of p o s i t i v e s e g r e g a t i o n has been e x p l a i n e d o n l y i n a q u a l i t a t i v e manner as r e s u l t i n g f r o m s q u e e z i n g out o f i n t e r d e n d r i t i c s o l u t e - r i c h l i q u i d when t h e t o p o f t h e s h e l l i s d e f o r m e d d u r i n g mould o s c i l l a t i o n . 2 7 2 9 Such an i m p r e c i s e mechanism i s h a r d l y s u r p r i s i n g b e c a u s e t h e mechanism o f o s c i l l a t i o n - m a r k f o r m a t i o n h i t h e r t o had n o t been u n d e r s t o o d f u n d a m e n t a l l y . The p u r p o s e o f t h e p r e s e n t work i s t o c l a r i f y t h e e f f e c t o f o s c i l l a t i o n marks on t h e f o r m a t i o n of t r a n s v e r s e c r a c k s and a l s o o f p o s i t i v e s e g r e g a t i o n by m e t a l l o g r a p h i c i n v e s t i g a t i o n of t h e s e d e f e c t s , and by t h e o r e t i c a l a p p r o a c h e s u s i n g h e a t f l o w a n a l y s i s a n d a p p l y i n g t h e mechansim o f o s c i l l a t i o n mark f o r m a t i o n p r o p o s e d i n t h e p r e v i o u s c h a p t e r . 151 5.2 M e t a l l u r g i c a l S t u d y Of T r a n s v e r s e C r a c k s 5.2.1 C a s t i n g C o n d i t i o n s The c a s t i n g c o n d i t i o n s and t h e c o m p o s i t i o n s o f s l a b s a m p l e s f r o m Company B and C f o r t h e m e t a l l o g r a p h i c s t u d y o f t r a n s v e r s e c r a c k s a r e shown i n T a b l e IX. The t y p e s o f s t e e l s l a b i n v e s t i g a t e d f o r t h i s d e f e c t a r e p l a i n , low C, A l - k i l l e d s t e e l w i t h r e l a t i v e l y h i g h N c o n t e n t . F o r t h e p u r p o s e o f i n v e s t i g a t i n g t h e r e l a t i o n between t h e f o r m a t i o n o f t r a n s v e r s e c r a c k s and t h e o s c i l l a t i o n marks, s l a b samples f r o m Company B w i t h d i s t i n c t l y p e r i o d i c o s c i l l a t i o n marks were m a i n l y e x a m i n e d . S m a l l s a m p l e s were c u t out f r o m t h e narrow s i d e o f t h e s l a b s , where o s c i l l a t i o n marks had n o t been d e f o r m e d by t h e s u p p o r t r o l l s . 5.2.2 A p p e a r a n c e Of T r a n s v e r s e C r a c k s The o x i d i z e d s u r f a c e l a y e r on t h e narrow s i d e o f t h e s l a b s was f i r s t removed by s a n d b l a s t i n g t o r e v e a l t h e e x i s t e n c e o f t r a n s v e r s e c r a c k s . In g e n e r a l f i n e t r a n s v e r s e c r a c k s a r e h a r d t o d e t e c t on t h e s u r f a c e o f a s - c a s t s l a b s , w h i c h makes t r a n s v e r s e c r a c k s much more t r o u b l e s o m e i n a c a s t i n g o p e r a t i o n . F i g . 5-1 shows t h e e f f e c t o f N on t h e t r a n s v e r s e c r a c k f o r m a t i o n . Thus t h e c r a c k s were o b s e r v e d on t h e s u r f a c e o f s l a b s w h i c h have more t h a n 60 ppm o f N. T h i s c o n t e n t o f N f o r t h e c r a c k f o r m a t i o n i s r o u g h l y c o n s i s t e n t w i t h t h e i n d u s t r i a l d a t a r e p o r t e d by Mukai e t a l . 7 9 152 As m e n t i o n e d i n the p r e v i o u s c h a p t e r , o s c i l l a t i o n marks on t h e s l a b s of Company C a r e d i s o r d e r e d because of t h e l a r g e m e n i s c u s l e v e l v a r i a t i o n . T h e r e f o r e s l a b s samples f rom Company C were c o n s i d e r e d t o be u n s u i t a b l e f o r the p u r p o s e of t h i s i n v e s t i g a t i o n . H e r e a f t e r , i n v e s t i g a t i o n r e s u l t s of s l a b samples f rom Company B a r e d e s c r i b e d . F i g s . 5-2 and 5-3 show t h e a p p e a r a n c e of t r a n s v e r s e c r a c k s formed on t h e narrow s i d e o f t h e s l a b s . F i n e t r a n s v e r s e c r a c k s were o b s e r v e d not o n l y a t t h e b o t t o m but a l s o near t h e t o p of t h e o s c i l l a t i o n m a r k s . H o w e v e r , rows of r e l a t i v e l y l a r g e t r a n s v e r s e c r a c k s a l o n g t h e b o t t o m of o s c i l l a t i o n m a r k s , on t h e l o o s e ( i n s i d e r a d i u s ) s i d e of t h e s l a b s a re more c h a r a c t e r i s t i c . Due t o t h e t e n s i l e s t r a i n g e n e r a t e d on t h e l o o s e s i d e of t h e s l a b s a t the unbend ing p o i n t , l a r g e t r a n s v e r s e c r a c k s a r e f r e q u e n t l y o b s e r v e d on t h i s s i d e o f t h e s l a b s . 5 . 2 . 3 S u b s u r f a c e S t r u c t u r e I n The V i c i n i t y Of T r a n s v e r s e  C r a c k s S m a l l samples were c u t l o n g i t u d i n a l l y f rom t h e s l a b p e r p e n d i c u l a r t o the o s c i l l a t i o n m a r k s , i n t h e same way as d e s c r i b e d e a r l i e r f o r t h e m e t a l l o g r a p h i c s t u d y of o s c i l l a t i o n m a r k s . The l o n g i t u d i n a l c r o s s s e c t i o n s of t h e samples were m a c h i n e d f l a t , p o l i s h e d and e t c h e d w i t h P i c r a l , N i t a l and O b e r h o f f e r ' s r e a g e n t s t o i n s p e c t t h e s u b s u r f a c e s t r u c t u r e i n t h e v i c i n i t y of t r a n s v e r s e c r a c k s . F i g . 5-4 shows t h e d e n d r i t i c s t r u c t u r e of a l o n g i t u d i n a l c r o s s s e c t i o n f rom t h e s l a b a p p e a r i n g i n F i g . 5 - 2 . I t c l e a r l y i n d i c a t e s t h a t t h e t r a n s v e r s e 1 53 c r a c k s form a l o n g t h e bo t tom of o s c i l l a t i o n m a r k s . The a u s t e n i t i c s t r u c t u r e d e v e l o p e d i n t h e s u b s u r f a c e r e g i o n , and t h e c r a c k s p r o p a g a t e d a l o n g the a u s t e n i t e g r a i n b o u n d a r i e s . H o w e v e r , some i n t e r g r a n u l a r c r a c k s , w h i c h d i d not p r o p a g a t e t o t h e s l a b s u r f a c e , a l s o were o b s e r v e d i n t h e s u b s u r f a c e . F i g . 5-5 shows a N i t a l - e t c h e d c r o s s s e c t i o n of a n o t h e r sample c u t f rom t h e s l a b i n F i g . 5 - 2 , and p r o e u t e c t o i d f e r r i t e can be seen a l o n g t h e a u s t e n i t e g r a i n b o u n d a r i e s . As H . S u z u k i has r e p o r t e d , t r a n s v e r s e c r a c k s appear t o p r o p a g a t e a l o n g t h e g r a i n - b o u n d a r y p r o e u t e c t o i d f e r r i t e . The d i f f e r e n c e i n a u s t e n i t e g r a i n s i z e i s not c l e a r between t h e t o p and bo t tom of o s c i l l a t i o n marks i n F i g . 5 - 5 ; but i t i s e v i d e n t f rom F i g s . 5 -6 (a ) and (b) w h i c h show t h e m a g n i f i e d s o l i d i f i c a t i o n s t r u c t u r e o f F i g . 5-5 near t h e t o p and t h e b o t t o m r e s p e c t i v e l y of an o s c i l l a t i o n mark . There i s a s i g n i f i c a n t d i f f e r e n c e i n g r a i n s i z e a t t h e s e l o c a t i o n s ; t h e f e r r i t e - p e a r l i t e s t r u c t u r e i s much c o a r s e r a t t h e b o t t o m t h a n a t t h e t o p of the o s c i l l a t i o n mark . T h i s f i n d i n g i s 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 w h i c h c o n t r o l s the l o c a t i o n of t r a n s v e r s e c r a c k s because a l o n g t h e b o t t o m of o s c i l l a t i o n marks t r a n s v e r s e c r a c k s a r e l i k e l y t o p r o p a g a t e i n t h e b r i t t l e zone c o n s i s t i n g of t h e c o a r s e r s t r u c t u r e . T h i s phenomenon c a n n o t be e x p l a i n e d by t h e c o o l i n g r a t e i n t h e s p r a y zone but must be r e l a t e d t o e v e n t s i n t h e mould because i n a d d i t i o n , as was shown i n t h e p r e v i o u s c h a p t e r . , t h e r e i s a l a r g e d i f f e r e n c e i n t h e d e n d r i t e arm s p a c i n g between t h e t o p and t h e b o t t o m of o s c i l l a t i o n m a r k s . The gap c r e a t e d by each o s c i l l a t i o n mark a p p a r e n t l y i n c r e a s e s t h e 1 54 t h e r m a l r e s i s t a n c e f o r h e a t e x t r a c t i o n from t h e s l a b s u r f a c e w h i c h c a u s e s l o c a l l y h i g h t e m p e r a t u r e and r e s u l t s i n a n o n u n i f o r m s t r u c t u r e i n t h e m o u l d . At t h e same t i m e t h e c o o l i n g r a t e i s r e d u c e d l e a d i n g t o a c o a r s e r d e n d r i t i c s t r u c t u r e . T h i s a rgument t h u s l i n k s t h e c o a r s e d e n d r i t e s t r u c t u r e t o t h e c o a r s e a u s t e n i t e s t r u c t u r e , and c o n s e q u e n t l y t o t h e c o a r s e f e r r i t e -p e a r l i t e s t r u c t u r e , from w h i c h i t c a n be i n f e r r e d t h a t t h e n o n u n i f o r m c o o l i n g i n t h e mould r e g i o n i s r e p o n s i b l e f o r t h e n o n u n i f o r m s u b s u r f a c e s t r u c t u r e . T h i s phenomenon i s p u r s u e d f u r t h e r i n a l a t e r s e c t i o n w i t h t h e a i d o f a h e a t t r a n s f e r a n a l y s i s . N e x t , t h e p r i m a r y s o l d i f i c a t i o n s t r u c t u r e n e a r t h e t r a n s v e r s e c r a c k s was e x a m i n e d i n d e t a i l . F i g . 5-7 shows t h e d e n d r i t i c s t r u c t u r e n e a r t h e b o t t o m o f an o s c i l l a t i o n mark w h i c h d o e s n o t e x h i b i t a s u b s u r f a c e hook. D e n d r i t e s d e v e l o p n o r m a l t o t h e s l a b s u r f a c e , and a t t h e b o t t o m o f t h e marks, f i n e c r a c k s , w h i c h a r e n o t open t o t h e s u r f a c e , a r e o b s e r v e d between t h e d e n d r i t e s . T h e s e f i n e c r a c k s a r e i n t e r d e n d r i t i c , a s w i l l be p r o v e n l a t e r , and c l e a r l y i n d i c a t e t h a t t h e y were f o r m e d i n t h e m ould r e g i o n . F o r m a t i o n of t h e s e i n t e r d e n d r i t i c f i n e c r a c k s i s e x p l a i n e d as f o l l o w s ; t h e s o l i d i f y i n g s h e l l i n t h e mould r e g i o n i s s u b j e c t e d t o v a r i o u s k i n d s o f s t r e s s a n d / o r s t r a i n , v i z i ) m ould f r i c t i o n f o r c e may o p e r a t e on t h e s h e l l s u r f a c e when t h e l u b r i c a t i o n by mould f l u x i s i n s u f f i c i e n t , o r i i ) s h e l l s h r i n k a g e due t o 6 - 7 t r a n s f o r m a t i o n may o c c u r n o n u n i f o r m l y i n t h e a p p r o x i m a t e c a r b o n r a n g e 0.10% t o 0.15%. T h e s e s t r e s s e s and s t r a i n s c o n c e n t r a t e a t t h e b o t t o m o f o s c i l l a t i o n marks, where 155 t h e s h e l l i s t h i n n e r , owing t o t h e low h e a t e x t r a c t i o n , and h o t t e a r s r e s u l t . A n o t h e r t y p e of f i n e c r a c k was a l s o f o u n d a t t h e b o t t o m o f o s c i l l a t i o n marks w h i c h e x h i b i t e d s u b s u r f a c e h o o k s , F i g . 5-8. T h e s e f i n e c r a c k s have been g e n e r a t e d a l o n g t h e hook where p o s i t i v e l y s e g r e g a t e d P gave r i s e t o a b r i t t l e s t r u c t u r e . T a n a k a e t a l 2 9 have a l s o r e p o r t e d t h i s t y p e o f f i n e c r a c k a l o n g t h e hook o f t h e o s c i l l a t i o n marks a t s l a b c o r n e r s . Thus as d e s c r i b e d a b o v e , t r a n s v e r s e c r a c k s may be i n i t i a t e d * i n t h e m o u l d : 1) between d e n d r i t e s o r i n r e g i o n s o f p o s i t i v e s e g r e g a t i o n o f t h e bottom of o s c i l l a t i o n marks, 2) i n r e g i o n s o f p o s i t i v e s e g r e g a t i o n i n t h e v i c i n i t y o f o s c i l l a t i o n marks, and 3) i n a c o a r s e s o l i d i f i c a t i o n s t r u c t u r e . T h e s e s u b s u r f a c e c h a r a c t e r i s t i c s a l s o p r o v i d e good s i t e s f o r t r a n s v e r s e c r a c k f o r m a t i o n a l o n g t h e o s c i l l a t i o n marks i n t h e s e c o n d a r y c o o l i n g zone a n d / o r a t t h e u n b e n d i n g p o i n t o f t h e s t r a n d . 5.2.4 S u r f a c e Of T r a n s v e r s e C r a c k s The s u r f a c e o f t h e c r a c k s was e x a m i n e d by b r e a k i n g open a c r a c k - c o n t a i n i n g s t e e l s e c t i o n a t - 1 9 6 ° C a f t e r i m m e r s i o n i n l i q u i d n i t r o g e n . The c r a c k c o n t a i n i n g sample was c u t out f r o m t h e narrow s i d e of t h e s l a b s and was c h a r a c t e r i z e d by t h r e e d i f f e r e n t t y p e s o f c r a c k s . T h e s e w i l l be r e f e r r e d t o as C r a c k s [ 1 ] , [ 2 ] , a n d [3] as shown i n F i g . 5-9. C r a c k [1] f o r m s p a r t i a l l y a l o n g t h e b o t t o m o f o s c i l l a t i o n marks but g r a d u a l l y d e v i a t e s f r o m t h e b ottom; C r a c k [2] forms a l o n g t h e mark: C r a c k 1 56 [3] d o e s n o t open t o t h e s l a b s u r f a c e but can be o b s e r v e d i n t h e c r o s s s e c t i o n . The f r a c t u r e d s a m p l e s were immersed i n a s o l u t i o n o f 3ml H C l , 4ml 2 B u t y n e - 1 , 4 - D i o l and 50ml d i s t i l l e d w a t e r i n an u l t r a s o n i c c l e a n e r t o remove t h e o x i d e f i l m f r o m t h e c r a c k s u r f a c e . 1 0 2 W h i l e r e m o v i n g t h e o x i d e , t h e s o l u t i o n d o e s no t s e r i o u s l y a l t e r t h e o r i g i n a l s t r u c t u r e . The c r a c k s u r f a c e t h e n was o b s e r v e d w i t h a s c a n n i n g - e l e c t r o n m i c r o s c o p e . F i g . 5-10 shows t h e s u r f a c e of C r a c k [ 1 ] . The smooth s u r f a c e o f t h i s c r a c k c l e a r l y r e v e a l s t h e a n g u l a r a p p e a r a n c e of a u s t e n i t e g r a i n b o u n d a r i e s . The s u r f a c e i n t h e l o w e r p a r t o f F i g . 5-10 i s c h a r a c t e r i s t i c o f t h e l o w - t e m p e r a t u r e f a c e t e d , b r i t t l e f r a c t u r e and can be d i s t i n g u i s h e d f r o m t h e o r i g i n a l c r a c k e d s u r f a c e . F i g . 5-11 shows t h e s u r f a c e of C r a c k [ 2 ] . T h i s c r a c k s u r f a c e a l s o e x h i b i t s m o s t l y a u s t e n i t e g r a i n b o u n d a r i e s a s i n F i g . 5 -1 1 ( a ) ; however t h e p a r t i a l l y u n d u l a t i n g s u r f a c e i n F i g . 5-11(b) i n d i c a t e s t h e c r a c k i s i n t e r d e n d r i t i c i n some l o c a t i o n s . F i g . 5-12 shows t h e s u r f a c e of C r a c k [ 3 ] . The u p p e r u n d u l a t i n g s u r f a c e i s e v i d e n t l y i n t e r d e n d r i t i c . The l o w e r c r a c k s u r f a c e shows r e l a t i v e l y f l a t f a c e s w h i c h i n d i c a t e s a u s t e n i t e g r a i n b o u n d a r i e s . T h i s t y p e o f c r a c k s c o r r e s p o n d s t o t h e c r a c k o b s e r v e d on t h e c r o s s s e c t i o n o f t h e s l a b shown i n F i g . 5-7. The c r a c k p r o p a g a t i o n seen may o c c u r not o n l y t o w a r d t h e i n s i d e o f s l a b s b u t a l s o a l o n g t h e b o t t o m o f o s c i l l a t i o n marks where t h e p o s i t i v e s e g r e g a t i o n and c o a r s e g r a i n s c a u s e a b r i t t l e s t r u c t u r e . I n t e r d e n d r i t i c f i n e c r a c k s which a c t a s i n i t i a t i o n s i t e s a r e n o t n e c e s s a r i l y l o c a t e d c o n t i n u o u s l y a l o n g t h e b o t t o m o f o s c i l l a t i o n marks. 1 57 F i g . 5-13 shows a n o t h e r c r a c k s u r f a c e i n a sample f r o m Company C. A low m a g n i f i c a t i o n m i c r o g r a p h , F i g . 5 - 1 3 ( a ) , shows t h e c r a c k s u r f a c e c o n s i s t s m o s t l y of a u s t e n i t e g r a i n b o u n d a r i e s . In t h i s c a s e a d h e r e n t m a t e r i a l was o b s e r v e d on t h e c r a c k s u r f a c e b e f o r e c l e a n i n g . T h e r e f o r e t h e s u r f a c e was s u b j e c t e d t o X - r a y a n a l y s i s u s i n g a SEM t o d e t e r m i n e t h e c h e m i c a l c o m p o s i t i o n o f t h e m a t e r i a l . P o t a s s i u m d e t e c t e d i n t h e X - r a y s p e c t r o g r a p h c l e a r l y i n d i c a t e d t h a t t h e a d h e r e n t m a t e r i a l was mould f l u x . I t s h o u l d be n o t e d t h a t t h e mould f l u x u s e d i n t h i s e x p e r m i e n t d o e s n o t c o n t a i n s o d i u m . S u b s e q u e n t l y , t h i s c r a c k s u r f a c e was f u r t h e r e x a m i n e d w i t h a SEM a t r e l a t i v e l y h i g h m a g n i f i c a t i o n and a r e g i o n c o n t a i n i n g s m a l l i n t e r d e n d r i t i c c r a c k s was f o u n d c l o s e t o t h e s u r f a c e , F i g . 5 - 1 3 ( b ) . T h i s o b s e r v a t i o n s u g g e s t s t h a t t h e c r a c k s were p o s s i b l y i n i t i a t e d by i n t e r d e n d r i t i c c r a c k s w h i c h must have opened t o t h e s l a b s u r f a c e i n t h e m o u l d . T h i s c r a c k was t h e n e xpanded and p r o p a g a t e d i n t h e s u b - m o u l d r e g i o n a t t h e a u s t e n i t e g r a i n b o u n d a r i e s t o form t r a n s v e r s e c r a c k s a l o n g t h e o s c i l l a t i o n m arks. 5.2.5 The E f f e c t Of O s c i l l a t i o n - m a r k s Shape On L o c a l S h e l l  T h i c k n e s s . F i g . 5-15 shows t h e r e l a t i o n s h i p between t h e d e p t h of o s c i l l a t i o n marks and t h e n o n u n i f o r m i t y of t h e s h e l l t h i c k n e s s w h i c h was e s t i m a t e d from t h e w h i t e band c a u s e d by t h e i n p u t s t r e a m f r o m t h e i m m e r s i o n n o z z l e . Samples e x a m i n e d were t a k e n f r o m a 0.10%C s l a b from Company B. I t i s s e e n t h a t d e e p o s c i l l a t i o n marks g i v e r i s e t o a l o c a l l y t h i n s h e l l . T h i s 158 r e l a t i o n s h i p i s e x a m i n e d t h e o r e t i c a l l y u s i n g 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 i n a l a t e r s e c t i o n . I t may a l s o be n o t e d t h a t c r a c k s a r e a s s o c i a t e d w i t h t h e d e e p e s t o s c i l l a t i o n marks; no c r a c k s or o n l y f i n e s u b s u r f a c e c r a c k s were f o u n d w i t h s h a l l o w o s c i l l a t i o n m arks. I t i s w e l l known t h a t t h e s h e l l s h r i n k a g e due t o 6-7 t r a n s f o r m a t i o n p r e v e n t s u n i f o r m h e a t e x t r a c t i o n i n t h e mould w h i c h c a u s e s l o c a l l y r e d u c e d s h e l l t h i c k n e s s . The p r e s e n t work shows t h a t t h e o s c i l l a t i o n marks c r e a t e a l a r g e l o c a l t h e r m a l r e s i s t a n c e between mould w a l l and s h e l l s u r f a c e a l s o t o c a u s e n o n u n i f o r m i t y o f t h e s h e l l p r o f i l e . 5.2.6 Summary Of M e t a l l u r g i c a l I n v e s t i g a t i o n Of T r a n s v e r s e  C r a c k s From t h i s m e t a l l u r g i c a l i n v e s t i g a t i o n , t h e f o r m a t i o n mechanism o f t r a n s v e r s e c r a c k s a l o n g o s c i l l a t i o n marks c a n be e x p l a i n e d as f o l l o w s : 1) F i r s t l y t h e l a r g e gap between t h e s o l i d i f i e d s h e l l and t h e mould w a l l a t t h e b o t t o m of 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 s h e a t e x t r a c t i o n and t h e r a t e of s o l i d i f i c a t i o n . 2) S e c o n d l y f r i c t i o n f o r c e between t h e s o l i d i f i e d s h e l l and t h e mould w a l l v i a t h e mould f l u x , c o u p l e d w i t h s t r a i n due t o volume s h r i n k a g e c a u s e d by 6-7 t r a n s f o r m a t i o n i n t h e ra n g e from 0.10-0.15% c a r b o n c o n t e n t i n s t e e l , c o n c e n t r a t e a t t h e t h i n n e r p a r t o f t h e s h e l l , namely a t t h e b o t t o m o f o s c i l l a t i o n marks, t o form i n t e r d e n d r i t i c c r a c k s . 3) F i n a l l y below t h e mould t h e s e f i n e c r a c k s p r o p a g a t e b o t h 159 t o w a r d t h e i n s i d e of t h e s l a b and a l o n g t h e bo t tom of o s c i l l a t i o n marks where t h e s t r u c t u r e i s most b r i t t l e owing t o p o s i t i v e s e g r e g a t i o n of P and c o a r s e g r a i n s . E m b r i t t l e m e n t of a u s t e n i t e g r a i n b o u n d a r i e s i s due t o t h e p r e c i p i t a t i o n of b o t h a p r o e u t e c t o i d f e r r i t e f i l m and A l N . 5 . 3 M e t a l l u r g i c a l S t u d i e s Of P o s i t i v e S e g r e g a t i o n Near O s c i l l a t i o n Marks 5 .3 .1 C a s t i n g C o n d i t i o n s A m e t a l l o g r a p h i c i n v e s t i g a t i o n was made i n t o t h e p o s i t i v e s e g r e g a t i o n a t t h e bot tom of o s c i l l a t i o n marks b o t h i n p l a i n l o w - C , A l - k i l l e d s t e e l s l a b s and i n s t a i n l e s s s t e e l s l a b s . Samples f o r t h e p l a i n c a r b o n s l a b s a r e f rom Companies A and B; t h e s t a i n l e s s s t e e l samples a r e f rom Company F . S m a l l s amples were c u t f rom t h e narrow s i d e of t h e s l a b s i n t h e same way as t h o s e f o r t h e t r a n s v e r s e - c r a c k s t u d y . The c a s t i n g c o n d i t i o n s and c h e m i c a l c o m p o s i t i o n of s ample s examined a r e shown i n T a b l e X . The e t c h i n g r e a g e n t s used f o r t h e p l a i n - c a r b o n s t e e l s amples i s t h e same as t h o s e used f o r t h e m e t a l l u r g i c a l i n v e s t i g a t i o n o f t r a n s v e r s e c r a c k s . A s o l u t i o n of F e C l 3 ( 5 g ) + C u C l 2 ( 5 g ) + H C 1 ( l 0 0 m l ) + C 2 H 5 O H ( 8 0 m l ) + H 2 0 ( 3 0 0 m l ) 1 0 3 was used f o r t h e e t c h i n g of t h e s t a i n l e s s s t e e l s a m p l e s . 1 60 5.3.2 M e t a l l o g r a p h i c C l a s s i f i c a t i o n Of P o s i t i v e S e g r e g a t i o n In g e n e r a l , p o s i t i v e s e g r e g a t i o n o f s o l u t e e l e m e n t s , e s p e c i a l l y P, i s o b s e r v e d n e a r t h e b o t t o m o f o s c i l l a t i o n m a r k s . As shown i n C h a p t e r 2, t h e r e g i o n o f p o s i t i v e s e g r e g a t i o n i n t h e s u b s u r f a c e s t r u c t u r e c o n t a i n i n g h o o k s i s a l o n g t h e hook o r between the- hook and t h e s l a b s u r f a c e . When h o o k s a r e a b s e n t f r o m t h e s u b s u r f a c e s t r u c t u r e , t h e s e g r e g a t i o n i s f o u n d o u t s i d e of t h e b o t t o m o f t h e o s c i l l a t i o n m arks. In t h i s s e c t i o n t h e c h a r a c t e r i s t i c s o f p o s i t i v e s e g r e g a t i o n a r e c l a s s i f i e d i n a s s o c i a t i o n w i t h t h e v a r i o u s t y p e s o f s u b s u r f a c e s t r u c t u r e o b s e r v e d n e a r t h e o s c i l l a t i o n m arks. T y p i c a l e t c h e d s t r u c t u r e s n e a r t h e b o t t o m o f o s c i l l a t i o n marks a r e shown i n F i g . 5-16 - 5-20. A,B, and C i n d i c a t e t h e use of P i c r a l , O b e r h o f f e r , and N i t a l e t c h e s r e s p e c t i v e l y . The b l a c k r e g i o n i n A, c o r r e s p o n d i n g t o t h e w h i t e r e g i o n i n B, c l e a r l y i n d i c a t e s p h o s p h o r u s s e g r e g a t i o n . In C b l a c k or g r a y a r e a s i n d i c a t e p e a r l i t e g r a i n s , w h i l e t h e w h i t e a r e a s i n d i c a t e p r o e u t e c t o i d f e r r i t e g r a i n s . F i g . 5-16 and 5-17 show t h e p o s i t i v e s e g r e g a t i o n f o r m e d a t t h e b o t t o m o f o s c i l l a t i o n marks w h i c h have a d j a c e n t s u b s u r f a c e h o o k s . Sample B1, F i g . 5-16, and sample B2, F i g . 5-17, a r e f r o m Company A and Company B r e s p e c t i v e l y . The s a m p l e s e x h i b i t two common c h a r a c t e r i s t i c s : d e e p o s c i l l a t i o n marks and a hook o r i e n t e d a t a s h a l l o w a n g l e t o t h e s u r f a c e . B o t h o b s e r v a t i o n s c a n be a t t r i b u t e d t o t h e low c a r b o n c o n t e n t (C^O.10%) a s was see n i n an e a r l i e r c h a p t e r . A l a r g e r e g i o n o f s e g r e g a t i o n e x i s t s between t h e hook and t h e s u r f a c e of t h e s l a b . A n o t h e r 161 i n t e r e s t i n g c h a r a c t e r i s t i c was f o u n d i n t h e s u b s u r f a c e s t r u c t u r e i n F i g . 5 - 1 7 C . The t o p h a l f o f t h e hook d i s a p p e a r s t h e n r e a p p e a r s , and i t l o o k s as i f one o r more c h a n n e l s c r o s s t h e hook. The e x i s t e n c e of t h e s e c h a n n e l s a c r o s s t h e hook s u g g e s t s t h a t s t e e l ( i n t e r d e n r i t i c l i q u i d ) f l o w s from t h e i n s i d e t o t h e o u t s i d e o f t h e s h e l l . The f o r c e c a u s i n g t h e f l o w i s l i k e l y t o be t h e n e g a t i v e p r e s s u r e g e n e r a t e d i n t h e f l u x c h a n n e l d u r i n g 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 mould o s c i l l a t i o n c y c l e . T h i s w i l l be d i s c u s s e d i n a l a t e r s e c t i o n . Sample B3 was c u t from t h e same s l a b as Sample B1 and i t t o o e x h i b i t s hooks i n t h e s u b s u r f a c e s t r u c t u r e . However t h e h o o k s a r e l o c a t e d c l o s e r t o t h e t o p of t h e o s c i l l a t i o n marks r a t h e r t h a n a t t h e b o t t o m . A l t h o u g h t h e hook i s c l e a r l y d e l i n e a t e d i n F i g . 5-18A, i t i s n o t d i s t i n c t i n F i g s . 5-18B and 5-18C. O n l y a weak s e g r e g a t i o n l i n e s i m i l a r t o an i n t e r d e n d r i t i c s e g r e g a t i o n i s o b s e r v e d . The m a j o r d i f f e r e n c e between Sample B1 and Sample B3 i s t h e l o c a t i o n o f t h e hook and t h e r e f o r e of t h e o v e r f l o w of l i q u i d s t e e l . T h i s d i f f e r e n c e c a n be e x p l a i n e d b a s e d on t h e mechanism of o s c i l l a t i o n mark f o r m a t i o n p r o p o s e d i n t h e p r e v i o u s c h a p t e r . I n t h e c a s e o f Sample B1 t h e m e n i s c u s s h e l l i s more r i g i d ( t h i c k e r ) and r e s i s t s b e i n g p u l l e d back t o w a r d t h e m o u l d w a l l by t h e n e g a t i v e p r e s s u r e i n t h e mould f l u x and o v e r f l o w o c c u r s e a r l y i n t h e p o s i t i v e -s t r i p p e r i o d . However i n t h e c a s e o f Sample B3, t h e s h e l l i s drawn back by t h e n e g a t i v e p r e s s u r e b e c a u s e o f r e l a t i v e l y l o w e r r i g i d i t y o f t h e m e n i s c u s s h e l l ( l i k e l y t h i n n e r ) b e f o r e m o l t e n s t e e l o v e r f l o w s t h e hook. C o n s e q u e n t l y t h e hook i s c l o s e r t o 1 62 t h e t o p o f t h e o s c i l l a t i o n mark i n Sample B3 owing t o t h i s d e l a y of o v e r f l o w . S p e c i f i c a l l y , the o v e r f l o w r e g i o n i n Sample B1 i s f a r t h e r f rom t h e mould w a l l t h a n i n Sample B 3 , w h i c h i m p l i e s t h a t t h e t h e r m a l r e s i s t a n c e between mould w a l l and t h e b o t t o m of o s c i l l a t i o n marks of Sample B3 i s l e s s t h a n t h a t of Sample B 1 . T h i s s u g g e s t s t h a t the heat t r a n s f e r may have a s t r o n g e f f e c t on t h e p o s i t i v e s e g r e g a t i o n . F i g 5-19 shows the s u b s u r f a c e s t r u c t u r e of Sample B 4 . T h i s has a h i g h e r c a r b o n c o n t e n t (C=0.26%, Company A ) , and t h e r e f o r e t h e d e p t h of o s c i l l a t i o n marks i s s h a l l o w e r . A l s o , as seen e a r l i e r t h e hook forms a l a r g e a n g l e w i t h t h e s u r f a c e as compared w i t h l o w e r c a r b o n s t e e l s . However , t h e l o c a t i o n of t h e p o s i t i v e s e g r e g a t i o n i s a l m o s t t h e same as t h a t o f Sample B 3 , namely i n t h e form of a weak s e g r e g a t i o n l i n e a l o n g t h e h o o k . I n a d d i t i o n t o the d i s c o n t i n u i t y of t h e p r i m a r y s o l i d i f i c a t i o n s t r u c t u r e c h a r a c t e r i z e d by t h e h o o k , a n o t h e r d i s c o n t i n u t y i s o b s e r v e d i n t h e s e c o n d a r y s o l i d i f i c a t i o n s t r u c t u r e , v i z t h e p r o e u t e c t o i d f e r r i t e p r e c i p i t a t e d a l o n g t h e a u s t e n i t e g r a i n b o u n d a r i e s . Note t h a t t h e g r a i n - b o u n d a r y f e r r i t e was h a r d l y o b s e r v e d i n t h e sample w i t h C ^ 0.20%, because a t l o w e r c a r b o n c o n t e n t s t h e f e r r i t e p r e c i p i t a t e s more e a s i l y w i t h i n t h e a u s t e n i t e g r a i n . F i g . 5-20 shows the p o s i t i v e s e g r e g a t i o n a t t h e b o t t o m of an o s c i l l a t i o n mark i n Sample B5 w h i c h does no t have a d j a c e n t s u b s u r f a c e h o o k s . The c a s t i n g c o n d i t i o n s of Sample B5 a r e t h e 163 same as t h o s e o f Sample B4. The s o l i d i f i e d l a y e r a d j a c e n t t o t h e b o t t o m o f t h e o s c i l l a t i o n marks c a n be d i s t i n g u i s h e d from t h e s t r u c t u r e b e n e a t h , F i g . 5-20, a n d e x h i b i t s a h i g h p h o s p h o r u s c o n t e n t . B e c a u s e o v e r f l o w a t t h e m e n i s c u s does n o t g i v e r i s e t o t h i s t y p e of s t r u c t u r e , i t i s r e a s o n a b l e t o i n f e r t h a t t h e s e g r e g a t i o n o c c u r s by p e n e t r a t i o n o f i n t e r d e n d r i t i c l i q u i d t h r o u g h t h e s e m i - s o l i d i f i e d s h e l l , a s p r o p o s e d i n t h e c a s e of Sample B2. F i g . 5-21 shows t h e d e p e n d e n c e of t h e t h i c k n e s s o f t h i s s e g r e g a t i o n l a y e r on 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 . Thus t h e t h i c k n e s s o f t h e l a y e r i n c r e a s e s w i t h i n c r e a s i n g d e p t h o f o s c i l l a t i o n m arks. As p r o p o s e d b e f o r e , t h e n e g a t i v e p r e s s u r e g e n e r a t e d i n t h e mould f l u x d u r i n g t h e upward m o t i o n o f t h e mould i s a l s o c o n s i d e r e d t o be a s t r o n g d r i v i n g f o r c e f o r t h i s t y p e o f s e g r e g a t i o n . The r e l a t i o n s h i p shown i n F i g . 5-21 i s t o be d i s c u s s e d l a t e r . F i g . 5-22 shows t h e p o s i t i v e s e g r e g a t i o n o b s e r v e d a t t h e b o t t o m o f o s c i l l a t i o n marks i n t h e s t a i n l e s s s t e e l s l a b s . T h i s t y p e o f s e g r e g a t i o n i s s i m i l a r t o t h a t seen i n Sample B5, w h i c h i s c h a r a c t e r i z e d by t h e a b s e n c e o f s u b s u r f a c e h o o k s . As shown i n t h e n e x t s e c t i o n , N i i s c h i e f l y s e g r e g a t e d i n t h e w h i t e r e g i o n , t h e mechanism o f w h i c h i s t h o u g h t t o be t h e same a s t h a t i n Sample B5. 164 5 . 3 . 3 M i c r o a n a l y s i s Of P o s i t i v e S e g r e g a t i o n By CMA S e g r e g a t i o n of e l e m e n t s i n t h e s u b s u r f a c e a r e a of o s c i l l a t i o n marks have been i n v e s t i g a t e d w i t h a CMA (Computer a i d e d X - r a y M i c r o A n a l y z e r ) . 8 7 The p r i n c i p l e o f measurement of CMA i s s i m i l a r t o t h a t of EPMA. However CMA i s c a p a b l e o f c h a r a c t e r i z i n g a r e l a t i v e l y l a r g e a r e a of sample (max. 90x90 mm) w i t h h i g h l y c o m p u t e r i z e d f u n c t i o n s , w h i l e o r d i n a r y EPMA can d e t e r m i n e a s m a l l a r e a (max. 0 . 2 5 x 0 . 2 5 m m ) . I n t h i s s t u d y , t h e a n a l y s i s a r e a i s 3x3mm; t h e beam d i a m e t e r i s 10 um. F i g u r e s 5-23 t o 5-25 show t h e d i s t r i b u t i o n of Mn and P i n t h e s u b s u r f a c e a r e a of t h e o s c i l l a t i o n m a r k s . The r e s p e c t i v e s u b s u r f a c e s t r u c t u r e s a r e shown i n F i g . 5 - 1 7 , 5 - 1 8 , and 5 - 2 0 . P o s i t i v e s e g r e g a t i o n s of Mn and P a r e c l e a r l y o b s e r v e d a t t h e end of t h e o v e r f l o w r e g i o n i n F i g . 5-23 (Sample B 2 ) , w h i l e o n l y a weak s e g r e g a t i o n of P i s d e t e c t e d a l o n g t h e hook i n F i g . 5-24 (Sample B 3 ) . F i g . 5-25 r e v e a l s the s e g r e g a t i o n of b o t h e l e m e n t s a l o n g t h e p e r i p h e r y of t h e b o t t o m of t h e o s c i l l a t i o n mark (Sample B 5 ) . F i g . 5-26 shows t h e s e g r e g a t i o n of N i and P i n t h e o v e r f l o w r e g i o n a t t h e b o t t o m of t h e o s c i l l a t i o n m a r k s . The r e s p e c t i v e s u b s u r f a c e s t r u c t u r e i s shown i n F i g . 5-22 (Sample B6) . F i g . 5 -27 (A) and (B) show t h e c o n t o u r maps of s e g r e g a t i o n i n t h e a r e a s w h i c h a r e c o n s i d e r e d i n F i g . 5-23 and 5-25 r e s p e c t i v e l y . The h i g h e s t s e g r e g a t i o n r a t i o s o f M n O . 4 ) and P ( 3 . 0 ) a p p r o x i m a t e l y c o r r e s p o n d t o t h e r e c i p r o c a l numbers of e q u i l i b r i u m p a r t i t i o n r a t i o s of Mn ( 0 . 7 3 ) 1 0 9 and P ( 0 . 2 8 ) , 1 0 9 1 65 r e s p e c t i v e l y . 5 . 3 . 4 P o s i t i v e S e g r e g a t i o n Caused By O v e r f l o w At The S l a b  C o r n e r The i n v e s t i g a t i o n of p o s i t i v e s e g r e g a t i o n a t t h e bo t tom of o s c i l l a t i o n marks s u g g e s t s t h a t heat t r a n s f e r p l a y s an i m p o r t a n t r o l e i n t h e s e g r e g a t i o n mechani sm. However i t i s not easy t o d i s c u s s t h e heat t r a n s f e r e f f e c t i n d e p e n d e n t l y of i n t e r d e n d r i t i c f l o w w h i c h i s e v i d e n t f rom the m e t a l l o g r a p h i c i n v e s t i g a t i o n of l o n g i t u d i n a l c r o s s s e c t i o n s of s a m p l e s . To c l a r i f y t h e e f f e c t of hea t t r a n s f e r on p o s i t i v e s e g r e g a t i o n , samples i n t h e v i c i n i t y of t h e s l a b c o r n e r were examined m e t a l l o g r a p h i c a l l y . In g e n e r a l , p e r i o d i c a l o v e r f l o w of m o l t e n s t e e l i s o b s e r v e d a t t h e s l a b c o r n e r , but t h e s t e e l o v e r f l o w s not o n l y i n the c a s t i n g d i r e c t i o n but a l s o l a t e r a l l y i n t o t h e b o t t o m of o s c i l l a t i o n m a r k s , F i g . 5 - 2 8 . F i g . 5-29 shows the l o n g i t u d i n a l c r o s s s e c t i o n c l o s e t o t h e c o r n e r of the sample e t c h e d by P i c r a l seen i n F i g . 5 - 2 8 . The c a s t i n g c o n d i t i o n s of t h i s sample a r e t h e same as t h o s e of Sample B 5 . The d e p t h of the o s c i l l a t i o n mark , w h i c h does not i n c l u d e t h e t h i c k n e s s of t h e s o l i d i f i e d l a y e r on t h e b o t t o m , t e n d s t o i n c r e a s e t o w a r d the c o r n e r of the s l a b . The s h e l l a t the bo t tom of the o s c i l l a t i o n mark i s d i s c o n t i n u o u s i n S e c t i o n s #4 and #5, a l t h o u g h S e c t i o n s #1 - #3 show t h e t y p i c a l s u b s u r f a c e s t r u c t u r e s i n the absence of h o o k s . 1 66 W i t h r e s p e c t t o p o s i t i v e s e g r e g a t i o n , no s e g r e g a t i o n c o u l d be f o u n d i n S e c t i o n #4 and #5 w h i c h a r e l o c a t e d c l o s e s t t o t h e s l a b c o r n e r . However S e c t i o n #3 w h i c h i s t h e end o f t h e o v e r f l o w t o t h e b o t t o m of t h e o s c i l l a t i o n mark u n a m b i g u o u s l y i n d i c a t e s some s e g r e g a t i o n . To c h e c k t h e r e s u l t , a t r a n s v e r s e c r o s s s e c t i o n o f a s l a b c o r n e r a t t h e b o t t o m o f o s c i l l a t i o n marks was t a k e n from t h e same t y p e o f s a m p l e s as Sample B5. F i g . 5-31(a) and (b) show t h e d e n d r i t i c s t r u c t u r e e t c h e d by O b e r h o f f e r ' s r e a g e n t i n t h e l o n g i t u d i n a l c r o s s s e c t i o n and i n t h e t r a n s v e r s e s e c t i o n r e s p e c t i v e l y , see F i g . 5-30. F i g . 5-3 1 ( a ) shows t h a t t h e o s c i l l a t i o n mark i s not a c c o m p a n i e d by a s u b s u r f a c e hook, w h i l e F i g . 5-31(b) i n d i c a t e s t h a t s t e e l o v e r f l o w s a l o n g t h e b o t t o m of an o s c i l l a t i o n mark f r o m t h e s l a b c o r n e r . P h o s p h o r u s s e g r e g a t i o n i s c l e a r l y o b s e r v e d a t t h e end of t h e o v e r f l o w r e g i o n , a l t h o u g h i t i s not o b s e r v e d i n t h e l o n g i t u d i n a l c r o s s s e c t i o n . I t i s u n l i k e l y t h a t t h e s e g r e g a t i o n i s due s o l e l y t o i n t e r d e n d r i t i c f l o w t h r o u g h t h e s e m i - s o l i d s h e l l b e c a u s e t h i s would r e q u i r e a r e l a t i v e l y u n i f o r m s e g r e g a t i o n a l o n g t h e bottom o f t h e o s c i l l a t i o n mark. T h a t t h e p o s i t i v e s e g r e g a t i o n was g e n e r a t e d a t t h e end of t h e o v e r f l o w i s p r o b a b l y a h e a t f l o w e f f e c t . The t i p o f t h e o v e r f l o w c l o s e t o t h e b o t t o m of t h e o s c i l l a t i o n mark i s s e p a r a t e d from t h e mould w a l l by a l a r g e r gap t h a n o v e r f l o w c l o s e r t o t h e c o r n e r w h i c h f i l l s more of t h e o s c i l l a t i o n mark. The r e d u c e d h e a t t r a n s f e r a t t h e t i p o f t h e o v e r f l o w w i l l r e d u c e t h e s o l i d i f i c a t i o n r a t e and a l l o w more t i m e f o r s e g r e g a t i o n . T h i s w i l l be d i s c u s s e d t h e o r e t i c a l l y i n t h e n e x t s e c t i o n . 167 5.3.5 Summary Of M e t a l l u r g i c a l I n v e s t i g a t i o n s Of P o s i t i v e  S e g r e g a t i o n P o s i t i v e s e g r e g a t i o n was f o u n d i n t h e s u b s u r f a c e s t r u c t u r e a d j a c e n t t o o s c i l l a t i o n marks b o t h i n t h e p r e s e n c e and i n t h e a b s e n c e o f h o o k s . P and Mn were d e t e c t e d i n t h e s e g r e g a t i o n zone i n t h e p l a i n c a r b o n s t e e l s l a b s . In t h e c a s e o f s t a i n l e s s s t e e l s l a b s t h e s e g r e g a t e d e l e m e n t s a r e m a i n l y N i and P. The t y p e s of p o s i t i v e s e g r e g a t i o n o b s e r v e d i n t h i s i n v e s t i g a t i o n c a n be c l a s s i f i e d i n t o f o u r c a t e g o r i e s as shown i n F i g . 5-32. S e g r e g a t i o n t y p e 1 i s o b s e r v e d a t t h e b o t t o m o f d e e p o s c i l l a t i o n marks where o v e r f l o w e n d s . In t y p e 2 t h 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 an o v e r f l o w w h i c h i s c l o s e r t o t h e t o p of. t h e o s c i l l a t i o n m arks. The s e g r e g a t i o n i s weaker, a p p e a r i n g t o be i n t e r d e n d r i t i c i n o r i g i n and f o l l o w i n g t h e hook. In t y p e 3 t h e d e p t h o f t h e o s c i l l a t i o n mark i s s h a l l o w and s e g r e g a t i o n i s a l m o s t t h e same as i n t y p e 2. Type 4 i s o b s e r v e d a t t h e b o t t o m o f o s c i l l a t i o n marks w h i c h a r e n o t a c c o m p a n i e d by a s u b s u r f a c e hook. The t h i c k n e s s of t h e s e g r e g a t i o n l a y e r i n c r e a s e s w i t h i n c r e a s i n g d e p t h o f o s c i l l a t i o n m a r k s . T r a c e s o f f l o w t h r o u g h t h e s e m i - s o l i d i f i e d s h e l l was d e t e c t e d i n t h i s i n v e s t i g a t i o n . T h i s s u p p o r t s t h e r e p o r t e d mechanism s c h e m a t i c a l l y e x p l a i n e d by Tanaka e t a l . 2 9 I n d e e d , t h e s e g r e g a t i o n t y p e 4 can be e x p l a i n e d o n l y by t h i s mechanism. The 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 o f s a m p l e s f r o m t h e s l a b c o r n e r i n d i c a t e s t h a t a nonuniform' s o l i d i f i c a t i o n r a t e due t o t h e shape of o s c i l l a t i o n marks p l a y s a d o m i n a n t r o l e i n r e g a r d t o 1 68 s e g r e g a t i o n t y p e 1. 5.4 Heat T r a n s f e r A n a l y s i s Of S o l i d i f i c a t i o n In The M o u l d In The V i c i n i t y Of O s c i l l a t i o n M arks 5.4.1 O b j e c t i v e s And D e s c r i p t i o n Of The P h y s i c a l S ystem The m e t a l l u r g i c a l i n v e s t i g a t i o n d e s c r i b e d i n p r e v i o u s s e c t i o n s c l e a r l y i n d i c a t e s t h e s t r o n g e f f e c t o f n o n u n i f o r m h e a t t r a n s f e r , c a u s e d by t h e shape o f o s c i l l a t i o n marks, on t r a n s v e r s e c r a c k f o r m a t i o n and p o s i t i v e s e g r e g a t i o n . T h e r e f o r e a t w o - d i m e n s i o n a l h e a t f l o w a n a l y s i s i n t h e mould, t a k i n g a c c o u n t of t h e shape o f o s c i l l a t i o n marks, was u n d e r t a k e n t o d e t e r m i n e t h e f o l l o w i n g : (1) T e m p e r a t u r e d i s t r i b u t i o n b o t h i n t h e s t e e l and i n t h e mould f l u x i n t h e mark d e p r e s s i o n . ( 2 ) T e m p e r a t u r e h y s t e r e s i s o f e a c h r e g i o n of t h e s l a b s u r f a c e i n t h e mould. (3) I n f l u e n c e of t h e shape o f o s c i l l a t i o n marks on t h e n o n u n i f o r m i t y o f t h e s h e l l t h i c k n e s s . The p h y s i c a l s y s t e m f o r t h e m a t h e m a t i c a l m o d e l l i n g i s shown i n F i g . 5-33. I t i s assumed t h a t t h e o s c i l l a t i o n mark i s f o r m e d j u s t a t t h e m e n i s c u s , but o n c e f o r m e d , t h e mark r e t a i n s i t s shape i n t h e mould. The l e n g t h o f t h e s y s t e m i n t h e c a s t i n g d i r e c t i o n i s e q u a l t o t h e p i t c h o f t h e o s c i l l a t i o n m a r k s . The w i d t h of t h e s y s t e m i s t a k e n t o be a h a l f w i d t h of t h e mould. 169 T h i s s y s t e m moves down a t the c a s t i n g speed from the m e n i s c u s t o t h e o u t l e t of t h e m o u l d . T h e r e i s a mould f l u x l a y e r w h i c h a c t s as a t h e r m a l r e s i s t a n c e between t h e mould w a l l and t h e s o l i d i f i e d s h e l l . The mould f l u x i n t h e mark d e p r e s s i o n i s a d d i t i o n a l l y t a k e n i n t o a c c o u n t as a t h e r m a l r e s i s t a n c e . 5 . 4 . 2 M a t h e m a t i c a l M o d e l i n g -F i g . 5-34 shows the c o o r d i n a t e of t h e sy s tem under s t u d y . The x and y d i r e c t i o n s a r e the c a s t i n g d i r e c t i o n and t h e mould w i d t h d i r e c t i o n r e s p e c t i v e l y . The f o l l o w i n g a s s u m p t i o n s have been made i n t h e model f o r m u l a t i o n ; [ i ] . The shape of o s c i l l a t i o n mark i s s y m m e t r i c a t x=0 and can be r e p r e s e n t e d by a q u a d r a t i c e q u a t i o n a t O^x^x* , and beyond by a s t r a i g h t l i n e p a r a l l e l t o t h e mould w a l l f o r s i m p l i c i t y . Thus the p i t c h and the d e p t h of o s c i l l a t i o n mark a r e 2 x , and y , r e s p e c t i v e l y . [ i i ] . From the symmetry of t h e o s c i l l a t i o n mark , t h e r e i s no a x i a l h e a t f l o w a t x=0. [ i i i ] . The d e p r e s s i o n c r e a t e d by t h e o s c i l l a t i o n mark i s f i l l e d w i t h , mould f l u x i n t h e m o u l d . The t h e r m a l r e s i s t a n c e a t t h e i n t e r f a c e between t h e mould f l u x and s l a b s u r f a c e i s n e g l i g i b l e . [ i v ] . Heat i s e x t r a c t e d from t h e p l a n e y=0. The measured h e a t f l u x 1 8 i s used as t h e boundary c o n d i t i o n a t t h i s p l a n e . As t h e s h e l l moves down a t the c a s t i n g s p e e d , t h i s boundary c o n d i t i o n a l s o changes w i t h d r a w i n g t i m e f rom t h e m e n i s c u s t o t h e o u t l e t of t h e m o u l d . 170 The g o v e r n i n g h e a t c o n d u c t i o n e q u a t i o n s f o r t h e s t e e l and t h e mould f l u x a r e t w o - d i m e n s i o n a l and u n s t e a d y s t a t e . 2 2 3T X 3 T 3 T —s = § _ . s s . , 5 _ 1 . s ps 3x 3y 3T. X , 3 2 T 3 2 T F F ( T + — T ) ( 5" 2 ) 3t p,C v 2 , 2 f pf 3x 3y The i n i t i a l and b o u n d a r y c o n d i t i o n s a r e a s f o l l o w s ; t = 0, 0 <_ x <_ x* , 0 <_ y <. f (x) ; T f = T f ± (5-3) t = 0, 0 <_ x < x* , y > f (x) ; T g = T g i 3T t >_ 0, x = 0, y x < y < y 2 ; - X g = 0 (5-10) * 3T t >_ 0, y = y 2 , 0 < x £ x 2 ; - X g — ^ = 0 (5-11) (5-4) t = 0, x n < x < x 9 , y > 0 ; T c = T g . (5-5) * ' 3 T f t >_ 0, 0 < ^ x £ x 1 , y = 0 ; - X f — = q (x) (5-6) t r i 0 » 0 i x i x i > y = f ( x ) » 3T 3T f - X - r ^ = - X . = h , (T - T j (5-7) s 3r f 3r s-f s f * 9 T f t > 0, x = 0, 0 < y i y i ; - X f — = 0 (5-8) * * 3T t >_ 0, x]_ < x <_ x 2 , y = 0 ; - X g - ~ = q (x) (5-9) 171 A d d i t i o n a l l y f o r t h e s i m p l i f i c a t i o n of t h e h e a t f l o w c a l c u l a t i o n i t i s assumed t h a t t h e l o n g i t u d i n a l h e a t f l o w i s n e g l i g i b l e compared w i t h t h e l a t e r a l h e a t f l o w a t x = ± x 2 , r e p r e s e n t e d a s E q . ( 5 - 1 2 ) . * * 5 T t > 0, x = ±x 0 < y < y • - X = 0 (5-12) 2 - - 2 s 3y W i t h r e s p e c t t o t h e mould h e a t - f l u x d i s t r i b u t i o n i n t h e c a s t i n g d i r e c t i o n , t h e measured r e s u l t s by N a k a t o e t a l , 1 8 have been m o d i f i e d , e s p e c i a l l y i n t h e v i c i n i t y o f t h e m e n i s c u s , by t h e t w o - d i m e n s i o n a l mould h e a t - t r a n s f e r model d e s c r i b e d i n a p r e v i o u s c h a p t e r . The mould h e a t f l u x d i s t r i b u t i o n i s e x p r e s s e d by t h e f o l l o w i n g p o l y n o m i a l s a s a f u n c t i o n o f t i m e i n t h e mould u n d e r t h e c o n d i t i o n of 1) c a s t i n g s p e e d i s 1 m/min and 2) l e n g t h of t h e mould i s 0.8m. 0 <t £ 12 ; q*(t) = 251.04 - 23.341t + 1.9618t2 - 0.066233t3 (5-13) * 2 12 £ t £ 48 ; q (t) = 251.04 - 16.899t + 0.8250t 3 -4 4 (5-14) - 0.018468t + 1.5616x10 t K ° ' However t h i s h e a t f l u x d i s t r i b u t i o n d o e s n o t a p p l y a t o s c i l l a t i o n m arks. T h e r e f o r e a n o t h e r a s s u m p t i o n was made f o r t h e c o m p u t a t i o n o f s o l i d i f i c a t i o n w h i c h i s a f f e c t e d by t h e shape of t h e o s c i l l a t i o n marks. The h e a t f l u x a t t h e d e p r e s s i o n i s assumed t o d e c r e a s e i n p r o p o r t i o n t o t h e t h i c k n e s s o f t h e l a y e r o f m ould f l u x i n t h e d e p r e s s i o n . The t h i c k n e s s o f mould f l u x , w h i c h e f f e c t i v e l y r e d u c e s t h e h e a t f l u x by h a l f , i s d e f i n e d a s Y H . Ba s e d on t h i s a s s u m p t i o n , t h e l o c a l d i s t r i b u t i o n o f h e a t 1 72 f l u x i s e x p r e s s e d by t h e f o l l o w i n g e q u a t i o n s . (5 -15) q. (x ) = (1 - f ( x ) / 2 y ) q (t.) r n o X (5 -16) E q u a t i o n s (5-1) and (5-2) were s o l v e d , s u b j e c t t o t h e i n i t i a l a n d b o u n d a r y c o n d i t i o n s , E q s . (5-3) t o ( 5 - 1 6 ) , u s i n g t h e e x p l i c i t f i n i t e - d i f f e r e n c e method. T r i a n g u l a r volume e l e m e n t s have been e m p l o y e d t o s i m u l a t e t h e c u r v a t u r e o f t h e o s c i l l a t i o n m a r ks, as was t h e c a s e i n t h e c o m p u t a t i o n o f m e n s i s c u s h e a t t r a n s f e r i n C h a p t e r 4. The r e l e a s e o f l a t e n t h e a t o f s o l i d i f i c a t i o n has been i n c o r p o r a t e d by a d j u s t i n g t h e s p e c i f i c h e a t between t h e s o l i d u s and l i q u i d u s t e m p e r a t u r e . C a l c u l a t i o n s were p e r f o r m e d f o r a p l a t e g r a d e s t e e l u s i n g t h e h e a t f l u x p r o f i l e p r e s e n t e d i n E q s . (5-13) t o ( 5 - 1 6 ) . The c o m p o s i t i o n o f s t e e l and d a t a employed i n t h e c o m p u t a t i o n a r e g i v e n i n T a b l e V I I I . N ote t h a t t h e s u p e r h e a t of s t e e l i s t a k e n t o be 5°C, t h e i n i t i a l t e m p e r a t u r e o f t h e mould f l u x i s . l 5 0 0 ° C , 1 2 and i s 0.05 cm. 5.4.3 C a l c u l a t e d R e s u l t s 5.4.3.1 T e m p e r a t u r e D i s t r i b u t i o n In M o u l d F l u x And S t e e l F i g s . 5-35 and 36 show t h e t e m p e r a t u r e d i s t r i b u t i o n b o t h i n t h e s t e e l and t h e mould f l u x i n t h e o s c i l l a t i o n mark d e p r e s s i o n a f t e r 10s and 48s r e s p e c t i v e l y . T h o s e two r e s u l t s c o r r e s p o n d t o l o c a t i o n s 16.7 and 80cm b e n e a t h t h e m e n i s c u s f o r 173 t h e c a s t i n g s p e e d of 1.0m/min. In t h i s c a l c u l a t i o n t h e p i t c h (1) and t h e d e p t h (d) of t h e o s c i l l a t i o n mark a r e 1.0cm and 0.1cm r e s p e c t i v e l y ; The r a t i o of t h e l e n g t h o f t h e d e p r e s s i o n t o t h e p i t c h of o s c i l l a t i o n mark, v i z x*, /x* i s 0.5. In F i g . 5-35 (10s a f t e r t h e m e n i s c u s ) t h e mould f l u x r e m a i n s l i q u i d e x c e p t f o r a s o l i d i f i e d l a y e r c l o s e t o t h e mould w a l l . The m e l t i n g p o i n t o f mould f l u x a p p l i e d i n t h i s c o m p u t a t i o n i s e s t i m a t e d t o be a b o u t 1100°C. A n o n u n i f o r m i t y o f t e m p e r a t u r e d i s t r i b t u i o n i s o b s e r v e d i n t h e s t e e l w h i c h r e s u l t s f r o m t h e p r e s e n c e of t h e mould f l u x i n t h e o s c i l l a t i o n - m a r k d e p r e s s i o n . A t e m p e r a t u r e d i f f e r e n c e of a b o u t 150°C was c a l c u l a t e d between t h e t o p ( l e s s t h a n 1200°C) and t h e b o t t o m ( a b o u t 1350°C) o f t h e o s c i l l a t i o n mark. Even a f t e r 48s, l i q u i d mould f l u x s t i l l e x i s t s a t t h e b o t t o m o f t h e o s c i l l a t i o n mark. Thus t h e b o t t o m of t h e o s c i l l a t i o n mark i s a l w a y s i n c o n t a c t w i t h l i q u i d m ould f l u x so t h a t i f a c r a c k opens up a t t h e b o t t o m o f t h e o s c i l l a t i o n mark anywhere i n s i d e t h e mould, t h e l i q u i d m ould f l u x e a s i l y p e n e t r a t e s i n t o t h e c r a c k . T h i s f i n d i n g a g r e e s w e l l w i t h t h e o b s e r v a t i o n r e p o r t e d e a r l i e r , i e t h e mould f l u x was d e t e c t e d on t h e d e n d r i t i c c r a c k s u r f a c e , F i g . 5-14. A t t h e end o f t h e mould, t h e t e m p e r a t u r e d i f f e r e n c e between t h e t o p and t h e b o t t o m of t h e o s c i l l a t i o n mark i s a b o u t 100°C, a b o u t h a l f a s l a r g e a s th e d i f f e r e n c e j u s t below t h e m e n i s c u s . T h i s r e s u l t , however, i s o n l y v a l i d p r o v i d e d t h a t t h e o s c i l l a t i o n mark d e p r e s s i o n i s f i l l e d w i t h mould f l u x ; b u t t h i s may n o t be t h e c a s e l o w e r i n th e m o uld. T h e r e , mould f l u x may f l o w downward t o f o r m an a i r gap a n d / o r may s o l i d i f y and a g a i n f o r m t h e a i r gap t o i n c r e a s e 174 the t h e r m a l r e s i s t a n c e a t the s h e l l - f l u x i n t e r f a c e . In t h i s event hea t e x t r a c t i o n may not p r o c e e d c o m p l e t e l y by c o n d u c t i o n a l o n e but by b o t h c o n d u c t i o n and r a d i a t i o n . C o n s e q u e n t l y the n o n u n i f o r m i t y of t e m p e r a t u r e d i s t r i b u t i o n i n c r e a s e s f u r t h e r . However t h e c a l c u l a t e d s h e l l p r o f i l e i n t h e mould i s i n good agreement w i t h t h e measured p r o f i l e by N a k a t o e t a l . 1 8 u s i n g t r a c e r s . T h i s i n d i c a t e s t h e v a l i d i t y of t h e c a l c u l a t i o n d e s p i t e t h e numerous a s s u m p t i o n s . 5 . 4 . 3 . 2 The E f f e c t Of O s c i l l a t i o n - m a r k Shape On The  N o n u n i f o r m i t y Of S h e l l T h i c k n e s s Thus i t i s seen t h a t t h e mould f l u x a n d / o r t h e a i r gap i n t h e d e p r e s s i o n of the o s c i l l a t i o n marks s t r o n g l y a f f e c t s the t e m p e r a t u r e d i s t r i b u t i o n i n t h e s u b s u r f a c e of t h e a d j a c e n t s t e e l . T h i s i m p l i e s t h a t t h e r e i s a l s o a n o n u n i f o r m i t y i n t h e s h e l l t h i c k n e s s . To s t u d y t h i s e f f e c t f u r t h e r , more c a l c u l a t i o n s were c a r r i e d out t o e s t i m a t e t h e e f f e c t of o s c i l l a t i o n - m a r k shape on the n o n u n i f o r m i t y of s h e l l t h i c k n e s s i n t h e absence of an a i r gap . C a l c u l a t e d s h e l l p r o f i l e s ( f r a c t i o n s o l i d = l ) a d j a c e n t t o v a r i o u s shapes of o s c i l l a t i o n marks a f t e r 5 , 10, 20 , and 30s a r e shown i n F i g s . 5-37 t o 5-40 r e s p e c t i v e l y . The r a t i o x , / x 2 was h e l d c o n s t a n t a t 0 . 5 . Owing t o symmetry of t h e s y s t e m , o n l y the l o w e r p a r t of t h e s h e l l p r o f i l e s i s shown. F i g . 5-37 shows the change of s h e l l p r o f i l e w i t h t i m e when 1=1.0cm and d = 0 . l 0 c m . A s l i g h t n o n u n i f o r m i t y i s found a t t h e s o l i d i f i c a t i o n f r o n t a f t e r 5 s , but a u n i f o r m l y t h i c k s h e l l i s t h e r e a f t e r soon r e c o v e r e d . Even though t h e d e p t h 1 75 i n c r e a s e s t o d = 0 . l 5 c m , i t does not much a f f e c t t h e n o n u n i f o r m i t y of t h e s o l i d i f i c a t i o n f r o n t a t 1=1.Ocm, see F i g . 5 - 3 8 . In t h e s e c a s e s , t h e d e p t h of o s c i l l a t i o n marks i s much l a r g e r t h a n t h e l o c a l l y r e d u c e d t h i c k n e s s of the s o l i d s h e l l . F i g . 5-39 r e p r e s e n t s t h e s h e l l p r o f i l e s when the o s c i l l a t i o n - m a r k p i t c h i s g r e a t e r , 1=2.Ocm, and d = 0 . l 0 c m . The r e d u c t i o n i n s h e l l t h i c k n e s s i n t h e v i c i n i t y of t h e bot tom of o s c i l l a t i o n mark c l e a r l y i n c r e a s e s as compared w i t h the s h o r t p i t c h (1=1.Ocm). A u n i f o r m l y t h i c k s h e l l c a n n o t be a t t a i n e d even a f t e r 3 0 s . In t h e c a s e of 1=2.Ocm, t h e l o c a l l y t h i n s h e l l i s e x a c e r b a t e d w i t h an i n c r e a s e i n t h e d e p t h of o s c i l l a t i o n m a r k s , F i g . 5 - 4 0 . Next t h e r a t i o x * / x * 2 , see F i g . 5 -34 , was changed from 0 . 3 t o 0 . 7 , where t h e p i t c h and the d e p t h of marks were kep t c o n s t a n t (1=2.Ocm, d = 0 . l 0 c m ) . C o m p a r i s o n of F i g . 5-41 ( x * / x * = 0 . 7 ) and F i g . 5-42 (x*/x*2 = 0 .3 ) c l e a r l y i n d i c a t e s t h a t i n c r e a s i n g t h e f l a t p a r t of the s u r f a c e s i g n i f i c a n t l y i m p r o v e s t h e u n i f o r m i t y of s h e l l t h i c k n e s s . These c a l c u l a t e d r e s u l t s r e v e a l t h a t b o t h t h e p i t c h and t h e d e p t h of o s c i l l a t i o n marks a r e i m p o r t a n t f a c t o r s w h i c h i n f l u e n c e t h e n o n u n i f o r m i t y of t h e s h e l l t h i c k n e s s , see F i g . 5 - 4 3 . P a r t i c u l a r l y , i t s h o u l d be n o t e d t h a t the h e a t c o n d u c t i o n i n s t e e l i n t h e c a s t i n g d i r e c t i o n p l a y s an i m p o r t a n t r o l e i n t h e u n i f o r m i t y of s h e l l p r o f i l e . As m e n t i o n e d i n C h a p t e r 4 t h e shape of o s c i l l a t i o n marks i s a f f e c t e d by t h e c a r b o n c o n t e n t of t h e s t e e l . The o s c i l l a t i o n marks on s l a b s h a v i n g 0.10% c a r b o n a r e deep and c u r v e d , w h i l e they a r e r e l a t i v e l y f l a t and s h a l l o w 176 on h i g h e r c a r b o n s l a b s ; t h e s e were e x p l a i n e d by t h e c a r b o n d e p e n d e n t - d e f o r m a b i l i t y o f t h e s h e l l . Such c h a r a c t e r i s t i c s h a p e s o f t h e o s c i l l a t i o n mark s u g g e s t s t h a t t h e v a r i a t i o n i n s h e l l t h i c k n e s s i n c a s t i n g of a b o u t 0.10% c a r b o n s l a b s i s more s e v e r e . 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 shape of o s c i l l a t i o n mark i s mould r e c i p r o c a t i o n f r e q u e n c y . H i g h - f r e q u e n c y mould o s c i l l a t i o n h as been r e p o r t e d t o d r a s t i c a l l y i m p r o v e t h e s u r f a c e q u a l i t y of s l a b s , e s p e c i a l l y w i t h r e s p e c t t o t r a n s v e r s e c r a c k s . The d e p t h o f o s c i l l a t i o n marks i s r e d u c e d by h i g h e r f r e q u e n c y o f mould o s c i l l a t i o n , f o r w h i c h a mechanism was p r o p o s e d i n C h a p t e r 4. F u r t h e r m o r e t h i s h i g h e r f r e q u e n c y d e c r e a s e s t h e p i t c h o f o s c i l l a t i o n marks. Heat f l o w a n a l y s i s p r e d i c t s t h a t b o t h o f t h e s e phenomena y i e l d a more u n i f o r m l y t h i c k s h e l l . As m e n t i o n e d i n a p r e v i o u s s e c t i o n , t h e t r a n s v e r s e c r a c k s m i g h t have been i n i t i a t e d i n t h e mould where t h e s t r e s s a n d / o r s t r a i n a r e c o n c e n t r a t e d a t t h e t h i n n e r p a r t of t h e s h e l l , v i z i n t h e v i c i n i t y of t h e bottom of o s c i l l a t i o n marks. I t i s t h e r e f o r e i n f e r r e d t h a t t h e f o r m a t i o n of t r a n s v e r s e c r a c k s i s r e d u c e d by m a king t h e s h e l l t h i c k n e s s u n i f o r m w i t h i n c r e a s i n g t h e o s c i l l a t i o n f r e q u e n c y . 5.4.3.3 C o o l i n g Rate D i s t r i b u t i o n Near The O s c i l l a t i o n Marks I t has been seen t h a t o s c i l l a t i o n marks i n t h e mould c a u s e a n o n u n i f o r m t e m p e r a t u r e d i s t r i b u t i o n on t h e s l a b s u r f a c e and c o n s e q u e n t l y s t r o n g l y a f f e c t t h e s h e l l p r o f i l e . T h i s f a c t i n d i c a t e s t h a t t h e r e m i g h t be l a r g e d i f f e r e n c e s i n t h e c o o l i n g 177 r a t e between t h e t o p and the b o t t o m of o s c i l l a t i o n m a r k s . F i g . 5-44 shows t h i s t o be t h e ca se f o r o s c i l l a t i o n marks of d i f f e r e n t s h a p e . As d e s c r i b e d p r e v i o u s l y , not o n l y r e d u c i n g t h e o s c i l l a t i o n - m a r k d e p t h but a l s o d e c r e a s i n g t h e mark p i t c h r e d u c e s t h e s e t e m p e r a t u r e d i f f e r e n c e s . F i g . 5-45 shows t h e change of c o o l i n g r a t e w i t h t i m e when 1=1.Ocm and d = 0 . l 0 c m . The c o o l i n g r a t e s a t b o t h l o c a t i o n s a f t e r 5s a r e a l m o s t same, w h i l e t h e y a r e v e r y d i f f e r e n t b e f o r e 5 s . I t i s w e l l known t h a t t h e c o o l i n g r a t e a f f e c t s the p r i m a r y s o l i d i f i c a t i o n s t r u c t u r e , e s p e c i a l l y t h e s e c o n d a r y d e n d r i t e arm s p a c i n g . The r e l a t i o n s h i p between t h e s e v a r i a b l e s as measured by S u z u k i 1 0 " i s shown i n F i g . 5 - 4 6 . The average v a l u e of t h e s e c o n d a r y d e n d r i t e arm s p a c i n g measured i n t h e p r e s e n t w o r k , d e s c r i b e d i n C h a p t e r 4 , a r e a l s o g i v e n i n t h i s f i g u r e . C o o l i n g r a t e s f rom t h e d a t a of S u z u k i a r e c o n s i s t e n t w i t h t h e c a l c u l a t e d v a l u e s . The s h e l l g r o w t h r a t e s have a l s o been c a l c u l a t e d t o a s s e s s t h e e f f e c t of t h e shape of the o s c i l l a t i o n mark . F i g . 5-47 shows t h e change of the s h e l l t h i c k n e s s w i t h t i m e b o t h a t t h e b o t t o m and a t t h e t o p of an o s c i l l a t i o n mark, when 1=1.Ocm and d = 0 . l 0 c m . The s h e l l t h i c k n e s s does no t show t h e g e n e r a l l i n e a r r e l a t i o n s h i p w i t h t 0 ' 5 but r a t h e r i s l i n e a r w i t h t 0 ' 7 5 as r e p o r t e d by Kumai e t a l . 1 0 5 u s i n g a r a d i o i s o t o p e i n t h e m o u l d . A l a r g e d i f f e r e n c e of s o l i d i f i c a t i o n r a t e i s not f o u n d a t t h e two l o c a t i o n s but t h e r e i s a d i f f e r e n c e i n t h e t i m e a t w h i c h s o l i d i f i c a t i o n commences. The t i m e d e l a y f o r s o l i d i f i c a t i o n a t t h e t o p of o s c i l l a t i o n marks i s due m a i n l y t o t h e s u p e r h e a t o f t h e s t e e l ; a t t h e b o t t o m of t h e mark t h e 178 t h e r m a l r e s i s t a n c e of the mould f l u x l a y e r f u r t h e r c o n t r i b u t e s t o make t h e t i m e d e l a y l o n g e r . The change of s h e l l t h i c k n e s s w i t h t i m e i s shown f o r an o s c i l l a t i o n mark h a v i n g 1=2.Ocm and d = 0 . l O c m , F i g . 5 - 4 8 , and 1=2.Ocm and d=0.15cm, F i g . 5 - 4 9 . Thus t h e l o n g e r t h e mark p i t c h a n d / o r t h e deeper t h e mark d e p t h , t h e g r e a t e r t h e d i f f e r e n c e i n s o l i d i f i c a t i o n s t a r t t i m e between t h e t o p and t h e bot tom of t h e o s c i l l a t i o n m a r k s . These f i n d i n g s h e l p t o e x p l a i n s e g r e g a t i o n o b s e r v e d near s u b s u r f a c e hooks r e p o r t e d e a r l i e r . The o v e r f l o w r e g i o n on t h e hook a t t h e b o t t o m of t h e o s c i l l a t i o n mark i s e x p e c t e d t o have a c e r t a i n t i m e d e l a y b e f o r e s o l i d i f i c a t i o n commences compared t o t h e t o p of t h e m a r k . The s o l u t e e l e m e n t s a r e t h u s c o n c e n t r a t e d a t t h e f i n a l s o l i d i f i c a t i o n p o i n t , namely a t the b o t t o m of t h e o s c i l l a t i o n mar k . 5 . 5 D i s c u s s i o n On The F o r m a t i o n Of P o s i t i v e S e g r e g a t i o n A l a r g e a r e a of p o s i t i v e s e g r e g a t i o n i s f o u n d a d j a c e n t t o o s c i l l a t i o n marks h a v i n g s u b s u r f a c e hooks when t h e f o l l o w i n g t h r e e c o n d i t i o n s a r e s a t i s f i e d : a) The a n g l e o f t h e hook t o t h e s l a b s u r f a c e i s not l a r g e and a narrow and l o n g o v e r f l o w r e g i o n i s formed o v e r the hook , b) t h e o v e r f l o w r e g i o n i s l o c a t e d a t t h e b o t t o m of o s c i l l a t i o n m a r k s , and c ) t h e o s c i l l a t i o n mark i s d e e p . The c o n c e n t r a t e d l i q u i d s t e e l s e g r e g a t e s a t t h e end of t h e o v e r f l o w , s i n c e i t s o l i d i f i e s l a t e r t h a n t h e r e m a i n d e r of t h e r e g i o n between t h e hook and s t e e l - f l u x i n t e r f a c e . T h i s mechanism i s s u p p o r t e d by t h e p o s i t i v e s e g r e g a t i o n f o u n d a t t h e 179 end of t h e o v e r f l o w from t h e s l a b c o r n e r a l o n g t h e b o t t o m o f t h e o s c i l l a t i o n mark, F i g . 5-31. F u r t h e r m o r e , t h e mechanism i s c o n s i s t e n t w i t h t h e r e s u l t s o f t h e h e a t t r a n s f e r m o d e l . On t h e o t h e r hand t h e p o s i t i v e - s e g r e g a t i o n l a y e r on t h e b o t t o m o f t h e o s c i l l a t i o n marks h a v i n g no s u b s u r f a c e h o o k s c a n n o t be e x p l a i n e d by s u c h a h e a t t r a n s f e r e f f e c t b e c a u s e t h e r e i s no o v e r f l o w . In t h i s c a s e , i t has been p o s t u l a t e d t h a t t h e weak s h e l l c a u s e d by h i g h s u p e r h e a t o r h i g h c a r b o n c o n t e n t e t c . i s drawn back t o t h e m o u l d w a l l by t h e n e g a t i v e p r e s s u r e g e n e r a t e d i n t h e mould f l u x c h a n n e l d u r i n g t h e upward m o t i o n o f t h e m o u l d . The t o p of t h e s h e l l i s i n t h e s e m i - s o l i d s t a t e and hence i t i s c o n c e i v a b l e t h a t i n t e r d e n d r i t i c l i q u i d c o u l d be drawn o u t t o t h e s u r f a c e by t h e n e g a t i v e p r e s s u r e . To i n v e s t i g a t e t h i s p o s s i b i l i t y , t h e n e g a t i v e p r e s s u r e f o r c e g e n e r a t e d i n t h e f l u x c h a n n e l d u r i n g upward mould m o t i o n was e s t i m a t e d by c o m b i n i n g m e n i s c u s h e a t t r a n s f e r w i t h l u b r i c a t i o n t h e o r y a s d e s c r i b e d i n a p r e v i o u s c h a p t e r . C o m p u t a t i o n s were c a r r i e d o u t f o r t h e s e v e n t e e n d i f f e r e n t o s c i l l a t i o n c o n d i t i o n s . The c a l c u l a t e d p r e s s u r e d i s t r i b u t i o n P ( x ) was c o n v e r t e d t o a v e r a g e p r e s s u r e P by t h e f o l l o w i n g e q u a t i o n . l l Pa = / r P(x) dx (5-17) \ 0 The r e s u l t s a r e shown i n F i g . 5-50. Thus t h e a v e r a g e n e g a t i v e p r e s s u r e 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 . As 180 suggested t h i s may cause i n t e r d e n r i t i c f low, the v e l o c i t y of which can be e s t i m a t e d from D ' a r c y ' s L a w : 1 0 6 k v T = (P + AP ) (5-18) L T a s y s L c g L In t h i s case the s t a t i c p r e s s u r e d i f f e r e n c e A P s i s a lmost n e g l i g i b l e compared w i t h the magnitude of Pp. Assuming t h a t the f r a c t i o n l i q u i d , g L , i s a p p r o x i m a t e l y 0.5 because the top of the s h e l l i s s e m i - s o l i d s t a t e at the meni scus , the p e r m e a b i l i t y K can be e s t i m a t e d to be about I 0 ~ 8 c m 2 based on P iwonka ' s d a t a 1 0 7 shown i n F i g . 5-51. The l e n g t h of c h a n n e l L c can be regarded as the s h e l l t h i c k n e s s ( g L = 0 . 5 ) . T h e r e f o r e L c i s rough ly e s t i m a t e d to be 0.05cm from the c a l c u l a t e d r e s u l t s of the meniscus heat t r a n s f e r . With the v i s c o s i t y of mol ten s t e e l at 7 c P , 1 5 the f low v e l o c i t y V"T i s computed to be a p p r o x i m a t e l y 0 .6cm/s , which suggests tha t the i n t e r d e n d r i t i c l i q u i d can move outward to the s h e l l su r f ace w i t h i n a p e r i o d of one mould o s c i l l a t i o n . T h i s mechanism i s c o n s i s t e n t w i t h observed data i n i n d u s t r i a l p l a n t s , tha t the f requency and the t h i c k n e s s of sur f ace s e g r e g a t i o n i n c r e a s e 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 ime , 2 7 and a l s o w i t h the i n v e s t i g a t e d r e s u l t shown i n F i g . 5-21. A l s o i n the case of the o s c i l l a t i o n marks hav ing subsur face hooks , t h i s i n t e r d e n d r i t i c flow l i k e l y takes p l a c e i n a d d i t i o n to the heat t r a n s f e r e f f e c t , as shown i n F i g . 5-17. Tanaka et a l . 2 9 have r e p o r t e d t h a t p o s i t i v e s e g r e g a t i o n of phosphorus was observed a long the o u t s i d e of the hook, w h i l e n e g a t i v e s e g r e g a t i o n e x i s t s below the p o s i t i v e s e g r e g a t i o n l i n e . T h i s 181 r e s u l t i s e x p l a i n e d by t h e mechanism p r o p o s e d h e r e , a l t h o u g h T a n a k a e t a l assumed t h a t t h e i n t e r d e n d r i t i c f l o w o c c u r s d u r i n g downward m o t i o n of t h e mould. The i n t e r d e n d r i t i c f l o w s h o u l d be more s i g n i f i c a n t i n t h e c a s e of o s c i l l a t i o n marks h a v i n g no s u b s u r f a c e h o o k s , s i n c e t h e r i g i d i t y o f t h e s h e l l i s l i k e l y t o be l e s s t h a n t h a t of t h e s h e l l when hooks f o r m due t o o v e r f l o w a t t h e m e n i s c u s . T a b l e IX - C a s t i n g C o n d i t i o n s of Samples f o r M e t a l l u r g i c a l Study of T r a n s v e r s e C r a c k s Chem i ca Co m p o s i t i o n (%) C a s t i n g C o n d i t i o n s Samp Comp c Mn Si P s Al N Tc C O Vs S (mm) f(cpm) Type of Type of Nozz1e Mou 1 d N o . (m/min) F l u x A1 B 0. 10 0.41 0.06 0.007 0.013 0.042 0. 164 1538 1 .57 1 1 95 7 . 5' up b A2 B " " " II " " " 154 1 1 . 37 " 11 " b A3 c 0.07 0.33 0.03 0.003 0.012 0.062 0.0080 1547 0.91 5 . 8 64 1 5' up C1 Tc : C a s t i n g temperature of s t e e l i n t u n d i s h Vs : C a s t i n g speed S S t r o k e of mould o s c i l l a t i o n f : Frequency of mould o s c i l l a t i o n T a b l e X - C a s t i n g C o n d i t i o n s of Samples f o r M e t a l l u r g i c a l Study of P o s i t i v e S e g r e g a t i o n Chemica Compos i t i on (%) Cast i ng Cond i t i on Samp Comp C Mn Si P S A t Cr Ni N T c C C) Vs S(mm) f(cpm) Type of Type of No. (m/m i n) Nozz1e Mou 1 d F 1 ux B1 A 0.09 0, 79 0.24 0.008 0.014 0.033 - _ 0 0080 1538 0.94 12.7 44 25'dwn a B2 B 0. 10 0.41 0.06 0.007 0.013 0.042 - - 0 0164 1538 1 . 57 1 1 95 7.5'dwn b B3 A 0.09 0. 79 0.24 0.008 0.014 0.033 - - 0 0080 1541 0.91 12.7 43 25'dwn a B4 A 0. 26 0. 76 0.22 " 0.012 0.001 - - 0 0070 1538 0.89 " 42 " a B5 A II " " " " - - " 1532 1 .02 " 48 " a B6 F 0. 10 1 .03 0.67 0.033 0.004 - 16.82 7 . 64 0 0200 1491 1 .00 12.7 60 15' up f Tc C a s t i n g temperature of s t e e l i.n.tundish Vs C a s t i n g speed S S t r o k e of mould o s c i l l a t i o n f Frequency of mould o s c i l l a t i o n 184 700 600 E 3 500 \ \ \ \ \ \ o C o m p a n y B C o m p a n y C N o C r a c k s O A F i n e C r a c k s 3 A T r a n s v e r s a C r a c k s • \ \ K= [ % A i l y [ % N j y A \ \ \ \ \ \ \ s s 400 \ \ \ K a t l l O O ° C v \ II30°C ^ II50°C 3 0 0 40 80 120 N (ppm) 160 F i g . 5-1 E f f e c t of N c o n t e n t on t h e c r a c k s . N o t e : e q u i l i b r i u m c o n s t a n t D a r k e n e t a l . 1 0 8 f o r m a t i o n of t r a n s v e r s e K = [ % A l ] r [ % N ] r by L.S. 185 F i g . 5-2 A p p e a r a n c e of t r a n s v e r s e c r a c k s ( i ) (Company B ) . F i g . 5-3 Appearance of t r a n s v e r s e c r a c k s ( i i ) (Company B). A T r a n s v e r s e c r a c k s f o r m e d a t t h e b o t t o m o f o s c i l l a t i o n m a r k s ; n i t a l e t c h i n g (Company B ) . (x3 .6 ) F i g . 5-6 F e r r i t e - p e a r l i t e s t r u c t u r e (a) a t t h e t o p a n d (b) a t t h e b o t t o m of an o s c i l l a t i o n m a r k ; n i t a l e t c h i n g (Company B ) . ( x 1 4 6 ) _ F i g . 5-7 Interdendritic cracks at the bottom of o s c i l l a t i o n marks; p i c r a l etching (Company B ) . (a):x6.5, (b):x32. KD O F i g . 5-8 S m a l l c r a c k o b s e r v e d a l o n g and n e a r t h e s u b s u r f a c e hook; p i c r a l e t c h i n g (Company B ) . ( a ) : x 6 . 5 , ( b ) : x 4 3 . [1] [ 2 ] [3] F i g . 5-9 T r a n s v e r s e c r a c k s on the s u r f a c e of a s l a b sample from Company B. ( x l . 1 8 ) 194 F i g . 5-13 Sur f ace of t r a n s v e r s e crack from Company C . ( a ) : x 4 0 , (b ) : x200 . Fe s Ka, ] Ca Ka, * •*. • • • .* ,\ •*• • . • • . . • Fe • Ka„ «*. 2 ..•.'V/ X - r a y s p e c t r o g r a p h o f a d h e r e n t m a t e r i a l s u r f a c e of t r a n s v e r s e c r a c k , Company C. Df =7 38mm Surface White Band O Large Crack # with Subsurface Crack X O C = O I O % 0 500 1000 Depth of Oscillation Marks (/im) 1500 g. 5-15 R e l a t i o n s h i p between t h e d e p t h o f o s c i l l a t i o n marks and t h e n o n u n i f o r m i t y of t h e s h e l l t h i c k n e s s (Company B ) . F i g . 5 16 S u b s u r f a c e s t r u c t u r e n e a r p o s i t i v e s e g r e g a t i o n (Sample B 1 ) . (x38.7) 201 F i g . 5-18 S u b s u r f a c e s t r u c t u r e near a h o o k ; 0 0 . 1 0 % (Sample B 3 ) . U 3 8 . 7 ) 202 F i g . 5 - 1 9 S u b s u r f a c e s t r u c t u r e n e a r a hook; C-0.26% (Sample B 4 ) . U 3 8 . 7 ) F i g . 5-20 S u b s u r f a c e s t r u c t u r e i n t h e v i c i n i t y of p o s i t i v e s e g r e g a t i o n l a y e r w i t h h o o k s a b s e n t (Sample B 5 ) . ( x 3 8 . 7 ) O CO 204 Depth of Oscillation Marks (ftm) 5-21 R e l a t i o n s h i p between t h e d e p t h of o s c i l l a t i o n marks and t h e t h i c k n e s s of s e g r e g a t i o n l a y e r (Company A ) . 205 F i g . 5-22 P o s i t i v e s e g r e g a t i o n at the bottom of o s c i l l a t i o n mark i n s t a i n l e s s s t e e l s l a b (Sample B 6 ) . (x6) F i g . 5-24 S e g r e g a t i o n of Mn and P i n t h e s u b s u r f a c e a r e a o f t h e o s c i l l a t i o n mark d e t e r m i n e d by CMA (Sample B 3 ) . o —I p 0 1.0 (mm) I I 1 S e g r e g a t i o n of Mn a n d P i n t h e s u b s u r f a c e a r e a o f t h e o s c i l l a t i o n mark d e t e r m i n e d by CMA (Sample B 5 ) . ro o CO F i g . 5-26 S e g r e g a t i o n o f N i and P i n t h e s u b s u r f a c e a r e a o f t h e o s c i l l a t i o n mark d e t e r m i n e d by CMA (Sample B 6 ) . ro o 2 1 0 F i g . 5-27 Contour map of the s e g r e g a t i o n r a t i o s , R, for (1) Mn and (2) P in (A) Sample B7 and (B) Sample B5. Mn: Red/R=1.4 P: Red/R=3.0 Blue/R=1.2 Blue/R=1.2 Green/R=1.1 Green/R=1.1 21 1 Narrow Face F i g . 5-28 Appearance of over f low at the s l a b c o r n e r and the l o c a t i o n of c r o s s s e c t i o n for m e t a l l o g r a p h i c i n s p e c t i o n , Company A . 2 1 2 F i g . 5-29 Subsur face s t r u c t u r e in each l o n g i t u d i n a l c r o s s s e c t i o n shown i n F i g . 5-28; p i c r a l e t c h i n g (Company A ) . (x5 .4) 213 Longitudinal cross section H o r i z o n t a l cross section F i g . 5-30 S l a b c o r n e r sample f o r the i n v e s t i g a t i o n i n t o t h e s t r u c t u r e i n t h e h o r i z o n t a l c r o s s s e c t i o n (Company A ) . (x2) 214 (b) F i g . 5-31 S u b s u r f a c e s t r u c t u r e (a) i n l o n g i t u d i n a l c r o s s s e c t i o n , and ( b ) i n h o r i z o n t a l c r o s s s e c t i o n of the sample shown i n F i g . 5-30; O b e r h o f f e r e t c h (Company A ) . ( a ) : x 6 . 5 , ( b ) : x 6 . 2 1 5 Types of Oscillation Mark Types of Positive Segregation Hooks in subsurface structure Type I large segregation area Type 2 weak segregation line Type 3 weak segregation line Hooks absent in subsur face structure Type 4 segregation layer F i g . 5-32 C l a s s i f i c a t i o n of p o s i t i v e s e g r e g a t i o n . F i g . 5-33 P h y s i c a l s y s t e m f o r m a t h e m a t i c a l model of h e a t f l o w i n t h e v i c i n i t y o f t h e o s c i l l a t i o n mark. 217 Xj : half length of mark depression x* : half pitch of oscillation marks y( -. depth of oscillation mark y* . half width of the mould F i g . 5 - 3 4 C o o r d i n a t e of t h e c o n t r o l l e d s y s t e m ( s y m m e t r i c s y s t e m ) . 218 Distance From Mould Wall (mm) F i g . 5-35 T e m p e r a t u r e d i s t r i b u t i o n i n mould f l u x and s t e e l a f t e r 10s . 219 Distance From Mould Wall (mm) F i g . 5-36 Tempera ture d i s t r i b u t i o n i n mould f l u x and s t e e l a f t e r 4 8 s . 220 Distance From Mould Wall (mm) 0 2 4 6 8 10 12 14 ^ o 4 a 2 5s 10s 20s 30s F i g . 5-37 Change of s h e l l p r o f i l e w i t h t i m e . (1=1.Ocm, d = 0 . ! 0 c m , x * / x * = 0 . 5 ) Distance From Mould Wall (mm) 0 2 4 6 8 10 12 14 11 n 1 1 — i n 1 1 (UULU) UM( ice Dc Wall 2 Distar Mould 4 5s 10s 20s 30s F i g . 5-38 Change of s h e l l p r o f i l e w i t h t i m e . (1=1.Ocm, d=0.15cm, x , / x * 2 = 0 . 5 ) F i g . 5-39 Change of s h e l l p r o f i l e w i t h t i m e . (1 = 2 . Ocm, d= 0 . l 0 c m , x t / x * = 0 . 5 ) 222 F i g . 5-40 Change of s h e l l p r o f i l e w i t h t i m e . (1 = 2 .Ocm, d = 0 . l 5 c m , x?/x * = 0 . 5 ) 223 F i g . 5-41 Change of s h e l l p r o f i l e w i t h t i m e . '(1 = 2. Ocm, d=0.lOcm, x * , A * s 0 . 3 ) 224 Distance From Mould Wall (mm) 0 2 4 6 8 10 12 14 30s F i g . 5-42 Change of s h e l l p r o f i l e w i t h t i m e . (1 = 2 . Ocm, d = 0 . l 0 c m , x V x * = 0 . 7 ) 225 10 F i g . 5-43 E f f e c t of t h e shape of o s c i l l a t i n mark on t h e n o n u n i f o r m i t y of s h e l l t h i c k n e s s a f t e r 10s. ( x * A * = 0 . 5 ) 2 2 6 F i g . 5 - 4 4 Change of t e m p e r a t u r e w i t h t i m e a t t h e b o t t o m and a t t h e t o p of o s c i l l a t i o n m a r k s . 227 200 F i g . 5-45 Change of c o o l i n g r a t e w i t h t i m e a t t h e t o p and a t t h e b o t t o m of o s c i l l a t i o n mark . (1=2.Ocm, d=0. lOcm) 228 F i g . 5-46 R e l a t i o n s h i p between c o o l i n g r a t e a n d t h e s e c o n d a r y d e n d r i t e arm s p a c i n g by A. S u z u k i e t a l . 1 0 " N o t e : M e a s u r e d a v e r a g e v a l u e s of s e c o n d a r y arm s p a c i n g a t t h e t o p and a t t h e b o t t o m o f o s c i l l a t i o n mark a r e i n d i c a t e d . 229 Time (s) 0 5 10 2 0 3 0 I i 1 1 1— F i g . 5 - 4 7 R e l a t i o n s h i p between s h e l l t h i c k n e s s and t i m e . ( 1 = 1 .Ocm, d= 0 . 1Ocm) 230 Time (s) 0 5 10 2 0 3 0 I — i 1 1 \— F i g . 5-48 R e l a t i o n s h i p between s h e l l t h i c k n e s s and t i m e . (1=2.Ocm, d=0.lOcm) 231 F i g . 5-49 R e l a t i o n s h i p between s h e l l t h i c k n e s s and t i m e . (1=2.Ocm, d=0 . l5cm) 232 F i g . 5-50 R e l a t i o n s h i p between Pa and n e g a t i v e - s t r i p t i m e . 233 F i g . 5-51 R e l a t i o n s h i p between p e r m e a b i l i t y and f r a c t i o n l i q u i d f r o m Piwonka e t a l . 1 0 7 234 6. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 6.1 C o n c l u s i o n s I n i t i a l s o l i d i f i c a t i o n phenomena i n t h e c o n t i n u o u s c a s t i n g s l a b mould have been s t u d i e d by m e t a l l u r g i c a l l y e x a m i n i n g s l a b s a mples and by u n d e r t a k i n g a s e r i e s o f t h e o r e t i c a l a n a l y s e s . F i r s t l y an i n v e s t i g a t i o n has been c o n d u c t e d t o e l u c i d a t e t h e mechanism of o s c i l l a t i o n - m a r k f o r m a t i o n . The m e t a l l u r g i c a l i n v e s t i g a t i o n has r e v e a l e d t h e f o l l o w i n g : [ i ] As r e p o r t e d by p r e v i o u s w o r k e r s , o s c i l l a t i o n marks were f o u n d w i t h and w i t h o u t h o o k s i n t h e a d j a c e n t s u b s u r f a c e s t r u c t u r e . [ i i ] Hooks i n t h e s u b s u r f a c e s t r u c t u r e o f 0 . 0 9 % - c a r b o n s l a b s form a s m a l l e r a n g l e w i t h t h e s u r f a c e t h a n i n 0.26%-c a r b o n s l a b s . [ i i i ] S l a b s h a v i n g o s c i l l a t i o n marks w i t h o u t s u b s u r f a c e h o o k s c h a r a c t e r i s t i c a l l y a l s o c o n t a i n e d i n e r t - g a s b l o w h o l e s , w h i c h s u g g e s t t h a t f l u i d f l o w of s t e e l c a u s e d by a r g o n gas i n j e c t i o n t h r o u g h t h e submerged n o z z l e had i n f l u e n c e d 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 . In o t h e r c a s e s , t h e m u l t i - p o r t p r a c t i c e and e l e c t r o m a g n e t i c s t i r r i n g i n t h e mould s u p p r e s s e d m e n i s c u s s o l i d i f i c a t i o n due t o r a p i d c o n v e c t i o n i n t h e m e n i s c u s r e g i o n . [ i v ] The d e p t h of o s c i l l a t i o n marks w i t h s u b s u r f a c e hooks i s 235 g r e a t e r i n l o w - c a r b o n (0.09%) t h a n i n med ium-carbon (0.26%) s l a b s . O s c i l l a t i o n marks w i t h o u t hooks do not show a c a r b o n dependence of d e p t h . In l o w - c a r b o n s l a b s t h e d e p t h of o s c i l l a t i o n marks w i t h s u b s u r f a c e hooks i s s l i g h t l y g r e a t e r when the s t e e l i s k i l l e d w i t h a l u m i n u m t h a n when k i l l e d w i t h s i l i c o n . The t h e o r e t i c a l a n a l y s e s c o n t r i b u t e d new knowledge on m e n i s c u s phenomena. [ i ] P r e d i c t i o n of t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e m e n i s c u s r e g i o n , u s i n g e x p e r i m e n t a l h e a t f l u x e s , showed t h a t p a r t i a l 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 , t o form a t h i n r i g i d s k i n , depends s t r o n g l y on b o t h l o c a l c o n v e c t i o n and s u p e r h e a t . A l o w - t e m p e r a t u r e r e g i o n was a l s o p r e d i c t e d i n t h e mould f l u x a d j a c e n t t o the mould w a l l . [ i i ] A f l u i d - f l o w a n a l y s i s has shown t h a t , owing t o t h e shape of the f l u x c h a n n e l between t h e m e n i s c u s and mould w a l l , p r e s s u r e i s g e n e r a t e d i n t h e f l u x by t h e mould o s c i l l a t i o n . The p r e s s u r e i s p o s i t i v e when t h e mould i s moving downward f a s t e r t h a n t h e s t r a n d ( n e g a t i v e s t r i p ) and i s n e g a t i v e d u r i n g t h e p o s i t i v e -s t r i p p e r i o d . The p r e s s u r e i s much l a r g e r t h a n t h e s h e a r s t r e s s a c t i n g i n the f l u x c h a n n e l . [ i i i ] C a l c u l a t i o n of t h e m e n i s c u s shape ( i n t h e absence of a r i g i d s k i n ) has i n d i c a t e d t h a t t h e m e n i s c u s i s pushed away from t h e mould w a 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 when p o s i t i v e p r e s s u r e i s g e n e r a t e d i n t h e f l u x 236 and i s drawn back t o w a r d t h e mould w a l l by n e g a t i v e f l u x p r e s s u r e d u r i n g p o s i t i v e s t r i p . O v e r f l o w t h u s may o c c u r a t 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 . The mechanism of o s c i l l a t i o n m a r k - f o r m a t i o n i s b a s e d upon t h e g e n e r a t i o n of p r e s s u r e i n t h e f l u x c h a n n e l and t h e p r e s e n c e o f a r i g i d o r s e m i - r i g i d s k i n a t t h e m e n i s c u s . I f t h e s k i n i s r i g i d , o v e r f l o w a t t h e commencement o f p o s i t i v e - s t r i p c a u s e s a s u b s u r f a c e hook t o form; whereas i f t h e s k i n i s s e m i - r i g i d i t moves w i t h t h e m e n i s c u s t o w a r d t h e mould, and o v e r f l o w d o e s n o t o c c u r and hooks do n o t fo r m . T h i s mechanism and a r e s u l t i n g f i r s t - g e n e r a t i o n m e n i s c u s model c a n e x p l a i n o b s e r v a t i o n s o f o s c i l l a t i o n marks made i n t h i s work and i n o t h e r s t u d i e s . S e c o n d l y , e x t e n s i v e s t u d i e s have been made on t h e mechanism of t r a n s v e r s e c r a c k f o r m a t i o n and p o s i t i v e s e g r e g a t i o n a t t h e s u r f a c e o f s l a b s , w h i c h a r e c l o s e l y t i e d t o o s c i l l a t i o n - m a r k f o r m a t i o n . [ i ] A l a r g e a r e a o f . p o s i t i v e s e g r e g a t i o n was f o u n d a d j a c e n t t o deep o s c i l l a t i o n marks h a v i n g s u b s u r f a c e h o o k s when a narrow and l o n g o v e r f l o w r e g i o n f o r m e d o v e r t h e h o o k s . [ i i ] A t w o - d i m e n s i o n a l , h e a t - f l o w a n a l y s i s , t a k i n g i n t o a c c o u n t t h e shape o f o s c i l l a t i o n marks, p r e d i c t s a l o c a l d e l a y o f c o o l i n g a t t h e b o t t o m o f o s c i l l a t i o n m arks. I f t h e end o f t h e o v e r f l o w r e g i o n i s l o c a t e d a t t h e b o t t o m o f t h e o s c i l l a t i o n marks, p o s i t i v e s e g r e g a t i o n forms a t t h e p o i n t where s o l i d i f i c a t i o n i s d e l a y e d . 2 3 7 [ i i i ] A n o t h e r t y p e of p o s i t i v e s e g r e g a t i o n was f o u n d i n a l a y e r on t h e b o t t o m of o s c i l l a t i o n marks h a v i n g no s u b s u r f a c e h o o k s . The t h i c k n e s s o f t h e s e g r e g a t i o n l a y e r i n c r e a s e s w i t h i n c r e a s i n g d e p t h of o s c i l l a t i o n m a r k s . T h i s e v e n t c a n n o t be e x p l a i n e d by t h e h e a t -t r a n s f e r m o d e l, but i n s t e a d c a n be r a t i o n a l i z e d by a p e n e t r a t i o n m o d el, i n w h i c h i n t e r d e n d r i t i c , e n r i c h e d l i q u i d i s drawn o u t by t h e n e g a t i v e p r e s s u r e g e n e r a t e d i n t h e mould f l u x c h a n n e l d u r i n g t h e upward m o t i o n of t h e m ould. In b o t h t y p e s of p o s i t i v e s e g r e g a t i o n , t h e s e g r e g a t i o n o f p h o s p h o r u s was c h a r a c t e r i s t i c a l l y d e t e c t e d . [ i v ] As r e p o r t e d i n p r e v i o u s s t u d i e s , t r a n s v e r s e c r a c k s f o r m e d a l o n g t h e b o t t o m of o s c i l l a t i o n marks and a r e a f f e c t e d l a r g e l y by t h e A l and N c o n t e n t i n s t e e l . [ v ] In t h e s u b s u r f a c e s t r u c t u r e o f s l a b s , t r a n s v e r s e c r a c k s were o b s e r v e d a l o n g t h e a u s t e n i t e g r a i n b o u n d a r y , 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 , and i n t h e r e g i o n o f p o s i t i v e s e g r e g a t i o n n e a r t h e b o t t o m o f t h e o s c i l l a t i o n m a rks. However a l m o s t a l l o f t h e t r a n s v e r s e c r a c k s u r f a c e s e x h i b i t e d a p a r t i a l l y i n t e r d e n d r i t i c s u r f a c e i n t h e v i c i n i t y of t h e s l a b s u r f a c e . Some of t h e c r a c k s c o n t a i n e d mould f l u x . [ v i ] N o n u n i f o r m i t y o f t h e s h e l l p r o f i l e i n t h e mould was d e t e c t e d from t h e w h i t e band c a u s e d by t h e s t r e a m of s t e e l f r o m t h e submerged n o z z l e . The s h e l l was t h i n n e s t a d j a c e n t t o d e e p o s c i l l a t i o n marks. 238 T r a n s v e r s e c r a c k s form a t t h i s t h i n n e s t p a r t of t h e s h e l l . [ v i i ] A h e a t f l o w a n a l y s i s has r e v e a l e d t h a t t h e shape o f o s c i l l a t i o n marks, v i z . not o n l y t h e d e p t h but a l s o t h e p i t c h , i s l a r g e l y r e s p o n s i b l e f o r t h e n o n u n i f o r m i t y o f t h e s h e l l p r o f i l e . T h e r e f o r e i n c r e a s i n g t h e f r e q u e n c y of t h e mould o s c i l l a t i o n e f f e c t i v e l y i m p r o v e s t h e u n i f o r m i t y of t h e s h e l l t h i c k n e s s , a n d c o n s e q u e n t l y r e d u c e s t h e s i t e s of t r a n s v e r s e c r a c k s . The r e s u l t s of t h e s e s t u d i e s o f f e r new i n f o r m a t i o n f o r t h e b e t t e r u n d e r s t a n d i n g of i n i t i a l s o l i d i f i c a t i o n phenomena, e s p e c i a l l y i n t h e v i c i n i t y of t h e m e n i s c u s o f t h e c o n t i n u o u s -c a s t i n g s l a b mould, where s e v e r a l i n t e r r e l a t e d phenomena t a k e p l a c e s i m u l t a n e o u s l y . 6.2 S u g g e s t i o n s F o r F u t u r e Work In t h e p r e s e n t work a m a t h e m a t i c a l model d e s c r i b i n g o s c i l l a t i o n - m a r k f o r m a t i o n has h e l p e d t o e x p l a i n , t h e m e t a l l o g r a p h i c r e s u l t s o b t a i n e d as w e l l as p r e v i o u s l y r e p o r t e d i n d u s t r i a l d a t a . However i t i s a f i r s t - g e n e r a t i o n model w h i c h i s b a s e d on many a s s u m p t i o n s . In t h e f u t u r e an e x t e n s i v e programme i s r e q u i r e d t o enhance t h e a b i l i t y o f t h e model t o s i m u l a t e i n i t i a l s o l i d i f i c a t i o n phenomena more a c c u r a t e l y . T h i s w o u l d i n c l u d e t h e f o l l o w i n g : [A] Most i m p o r t a n t i s t h e e x p e r i m e n t a l measurement of h e a t f l u x i n t h e v i c i n i t y of t h e m e n i s c u s i n c l u d i n g t h e 239 mould f l u x z o n e . T h i s measurement c o u l d be u n d e r t a k e n u s i n g an a r r a y of t h e r m o c o u p l e s imbedded i n t o t h e m o u l d w a l l . M e a s u r e d t e m p e r a t u r e s would be c o n v e r t e d t o a h e a t - f l u x d i s t r i b u t i o n by t w o - d i m e n s i o n a l , h e a t - f l o w a n a l y s i s . The e f f e c t s o f t h e c a s t i n g c o n d i t i o n s u c h a s t h e s u p e r h e a t , c a s t i n g s p e e d , t y p e o f s t e e l , t y p e o f mould f l u x , o s c i l l a t i o n s t r o k e , and o s c i l l a t i o n f r e q u e n c y on t h e amount o f h e a t f l u x w o u l d be e x a m i n e d . S i m u l t a n e o u s l y t h e s u b s u r f a c e s t r u c t u r e o f s l a b s and t h e c o n s u m p t i o n r a t e o f mould f l u x w o u l d be i n v e s t i g a t e d . The d e t e c t i o n of t h e m e n i s c u s s h e l l p r o f i l e u s i n g t r a c e r s s u c h as s u l p h u r o r r a d i o a c t i v e A u 1 9 8 a r e a l s o q u i t e i m p o r t a n t from t h e s t a n d p o i n t o f d e t e r m i n i n g t h e r i g i d i t y o f t h e s h e l l . S e c o n d l y , an e x t e n s i o n o f t h e p r e s e n t work would be t o d e v e l o p a t h e o r e t i c a l a n a l y s i s of m o u l d - f l u x c o n s u m p t i o n i n w h i c h h e a t and f l u i d f l o w s h o u l d be c o u p l e d and s o l v e d s i m u l t a n e o u s l y . T h i s s u b j e c t i s o f g r e a t i n t e r e s t i n t h e s t e e l i n d u s t r y w h i c h i s s t r i v i n g t o c o n t i n u o u s l y c a s t s l a b s w i t h h i g h s u r f a c e q u a l i t y a t h i g h s p e e d . The a n a l y s i s c o u l d be b a s e d on t h e p r o p o s e d model o f o s c i l l a t i o n mark f o r m a t i o n . The measurement of mould f r i c t i o n f o r c e a l s o w o u l d be u s e f u l f o r t h i s t h e o r e t i c a l a n a l y s i s . 240 BIBLIOGRAPHY 1. N.A. McPherson and S. H e n d e r s o n : I r o n m a k i n g a n d S t e e l m a k i n g , 1983, v o l . 10, no. 6, pp. 259-268. 2. P r o c . o f 2nd P r o c e s s T e c h n o l o g y C o n f . on C o n t i n u o u s C a s t i n g o f S t e e l , C h i c a g o , ISS-AIME, 1981, v o l . 2 . 3. S p e c i a l I s s u e on C o n t i n u o u s C a s t i n g o f S t e e l ( I I ) , T e t s u -t o - H a g a n e ' , 1 9 8 1 , v o l . 6 7 , no.8. 4. Y. Takemura, S. M i z o g u c h i , 0. T s u b a k i h a r a , T. Kuwabara, and M. S a i t o : N i p p o n S t e e l T e c h . R e p o r t , 1983, no. 21, pp. 189-201. 5. T. Kohno, T. Shima, T. Kuwabara, S. M i z o g u c h i , T. Yamamoto, H. M i s u m i , and S. T s u n e o k a : T e t s u - t o - H a g a n e ' , 1981, v o l . 86, no. 13, p p l 1764-1772. 6. N.A. McP h e r s o n and R.E. M e r c e r : I r o n m a k i n g and S t e e l m a k i n g , 1980, v o l 7, pp. 167-179. 7. Y. N u r i , T. O h a s h i , N. M i y a s a k a , K. Shima, and Y. U c h i d a : T e t s u - t o - H a g a n e ' , 1979, v o l . 65, p. S701. 8. T. O k a z a k i , H. Tomono, T. O z a k i , and Y. Aka b a n e : T e t s u - t o - H a g a n e ' , 1982, v o l . 68, no. 11, p. S929. 9. T.Emi, H. N a k a t o , Y. I i d a , K. Emoto, R. T a c h i b a n a , T. I m a i , and H.> Bada: P r o c . 6 1 s t NOH-BOSC, 1978, pp. 350-361 . 10. P.V. R i b o u d and M. L a r r e c q : P r o c . 62nd NOH-BOSC, 1979, pp. 78-92. 11. R. G r a y and H. M a r s t o n : P r o c . 62nd NOH-BOSC, 1979, pp. 93-102. 12. T. S a k u r a y a , T. Emi, T. I m a i , K. Emoto, and M. Kodama: T e t s u - t o - H a g a n e ' , 1981, v o l . 67, no. 8, pp. 1220-1228. 13. T. Nakano, M. F u j i , K. Nagano, S. M i z o g u c h i , T. Yamamoto, and K. A s a n o : T e t s u - t o - H a g a n e ' , 1981, v o l . 67, no. 8, pp. 1210-1219. 14. M.D. L a n y i and C . J . R o s a : P r o c . 2nd P r o c e s s T e c h . C o n f . , 1981, pp. 133-140. 15. A Hand Book on P r o p e r t y of L i q u i d I r o n and S l a g , The I r o n a n d S t e e l I n s t i t u e o f J a p a n , 1971. 241 16. K . S o r i m a c h i , H . Yamakawa, M . Kuga , H . S h i t a k a , and M . S a i g u s a : i n P r o c . of M o d e l i n g o f C a s t i n g and W e l d i n g P r o c e s s , E n g i n e e r i n g F o u n d a t i o n , New Y o r k , 1983, p p . 195-198. 17. A . N i s h i w a k i and K. O g i n o : 140th C o m m i t t e e , The J apan S o c i e t y f o r t h e P r o m o t i o n of S c i e n c e , D e c . 1979. 18. H . N a k a t o and I . M u c h i : T e t s u - t o - H a g a n e ' , 1980, v o l . 66 , p p . 3 3 - 4 2 . 19. K . O g i n o , A . N i s h i w a k i - , and K . Yamamoto: T e t s u - t o -H a g a n e ' , 1979, n o . 65 , p . S 6 8 3 . 2 0 . W . C . K . Boemer and A . G . R a p e r : J I S I , 1970, v o l . 208 , p p . 1 8 - 2 7 . 2 1 . T . Kuwano, N . S h i g e m a t s u , F . H o s h i , and H . O g i w a r a : I r o n m a k i n g and S t e e l m a k i n g , 1983, v o l . 10, p p . 7 5 - 8 1 . 2 2 . H . O k a , Y . E d a , T . K o s h i k a w a , H . N a k a t o , T . N o z a k i , and Y . H a t o : T e t s u - t o - H a g a n e ' , 1983, v o l . 69 , p . S 9 3 2 . 2 3 . M . H a s h i o , T . Watanabe , T . Yamamoto, M . Marukawa , and M . K a w a s a k i : T e t s u - t o - H a g a n e ' , 1982, v o l . 68 , p . S 9 8 1 . 24 . S . M i z o g u c h i , H . M i s u m i , and S . T a n a k a : i n p r e p r i n t of J apan - US J o i n t Seminar on S o l i d i f i c a t i n o P r o c e s s i n g , J u n e , 1983, B o s t o n . 2 5 . N . A . M c P h e r s o n , A . W . . H a r d i e , and G . P a t r i c : ISS T r a n s a c t i o n s , 1983, v o l . 3 , p p . 2 1 - 3 6 . 2 6 . I . Saucedo , J . B e e c h , and G . J . D a v i s : P r o c . 6 t h I n t l . Vacuum M e t a l l u r g y C o n f . , 1979, p p . 8 8 5 - 9 0 4 . 2 7 . H . T a k e u c h i , S . M a t s u m u r a , and Y . I k e g a r a : T e t s u - t o -H a g a n e ' , 1983, v o l . 69 , n o . 16, p p . 1 9 9 5 - 2 0 0 1 . 2 8 . S . T a n a k a , H . M i s u m i , S . M i z o g u c h i , and H . H o r i g u c h i : T e t s u - t o - H a g a n e ' , 1 9 8 1 , v o l . 6 7 , p . S 1 7 2 . 2 9 . S . T a n a k a , H . M i s u m i , H . K i b e , T . O h t a , and S. M i z o g u c h i : T e t s u - t o - H a g a n e ' , 1981 , v o l . 6 7 , p . S 8 5 2 . 3 0 . R. S a t o : P r o c . 62nd NOH-BOSC, I S S - A I M E , 1979, p p . 48-6 7 . 3 1 . J . Savage and W . H . P r i t c h a r d : J . I r o n S t e e l I n s t . , 1954, v o l . 178, p p . 2 6 9 - 2 7 7 . 242 32. H. Tomono, H. Ackermann, W. K u r z , and W. Heinemann: i n C a s t i n g of S m a l l S e c t i o n , TMS-AIME, W a r r e n d a l e , PA., 1982, pp. 55-73. 33. T. A r a k i and Y. S u g i t a n i : T e t s u - t o - H a g a n e ' , 1973, v o l . 59, pp. A17-A20. 34. R. S c h o e f f m a n n : I r o n and S t e e l E n g r . , 1972, v o l . 49, pp. 25-36. 35. H.P J u n g , K . J . Kremer, H. S p i t z e r , H. Voge, and R. H e n t r i c h : S t a h l u. E i s e n , 1984, v o l . 104, no. 4, pp. 197-204. 36. K. Kawakami, T. K i t a g a w a , H. M i z u k a m i , H. U c h i b o r i , S. M i y a h a r a , M. S u z u k i , and Y. S h i r a t a n i : T e t s u - t o - H a g a n e ' , 1981, v o l . 67, no. 8, pp. 1190-1199. 37. H. T a k e u c h i , S. Matsumura, R. H i d a k a , Y. Nagano, and Y. S u z u k i : T e t s u - t o - H a g a n e ' , 1983, v o l . 69, no. 2, pp. 248-253. 38. I . Saucedo, J . B e e c h , and G . J . D a v i e s : M e t a l T e c h . , 1982, v o l . 9 , pp. 282-291. 39. H. N a k a t o , Y. Habu, T. Emi, K. S o r i m a c h i , T. K o s h i k a w a , and H. K o j i m a : T e t s u - t o - H a g a n e ' , 1981, v o l . 67, p. S908. 40. R. A l b e r n y , A. L e c l e r c q , D. Amaury, and M. L a h o u s s e : Rev. Met., 1976, v o l . 73, pp. 545-557. 41. M. W o l f : T r a n s . I S I J , 1980, v o l . 20, pp. 710-717. 42. H. N a k a t o , Y. Habu, T. Emi, K. K i n o s h i t a , Y. Tomura, N. Ueda, and T. I m a i : T e t s u - t o - H a g a n e ' , 1976, v o l . 62, p. S506. 43. T. S a e k i , S. O h g u c h i , S. M i z o g u c h i , T. Yamamoto, H. M i s u m i , and S. T s u n e o k a : T e t s u - t o - H a g a n e ' , 1982, v o l . 68, pp. 1773-1781. 44. S.N. S h i n g h and K.E. B l a z e k : J . M e t a l s , 1974, v o l . 26, pp. 17-27. 45. K. K i n o s h i t a , T. Emi, and M. K a s a i : T e t s u - t o - H a g a n e ' , 1979, v o l . 65, no. 14, pp. 2022-2031. 46. I.V. S a m a r a s e k e r a and J.K. Brimacombe: Can. Met. Q u a r t . , 1979, v o l . 18, pp. 251-266. 243 47. T. Nakano, K. Koyama, S. N a k a m o r i , H. M i s u m i , and T. N a i t o : T e t s u - t o - H a g a n e ' , 1983, v o l . 69, no. 12, p. S1036. 48. H. N a k a t o , M. Ozawa, K. K i n o s h i t a , Y. Habu, and T. Emi: T e t s u - t o - H a g a n e ' , 1981, v o l . 67, no. 8, pp. 1200-1 209. 49. H. Yamakawa, J . I k e d a , T. N i s h i y a , and S. Ando: T e t s u -t o - H a g a n e ' , 1983, v o l . 69, no. 4, p. S164. 50. M. W o l f : T e t s u - t o - H a g a n e ' , 1981, v o l . 67, p. S904. 51. K. Hamagami, H. Bada, M. Enomoto, M. Kuga, and S. Ohmiya: T e t s u - t o - H a g a n e ' , 1983, v o l . 69, no. 4, p. S162. 52. H. Bada, K. Mamagami, M. Kuga, and M. Enomoto: T e t s u -t o - H a g a n e ' , 1983, v o l . 69, no. 14, p. S103 1 . 53. H. M i z u k a m i , M. Komatsu, T. K i t a g a w a , K. Kawakami, H. U c h i b o r i , and H. M i y a n o : T e t s u - t o - H a g a n e ' , 1983, v o l . 69, no. 14, p. S1032. 54. K. T o k i w a , F. K a t a o k a , S. T s u n e o k a , Y. N a k a m o r i , and Y. F u j i g a k e : T e t s u - t o - H a g a n e ' , 1983, v o l . 69, no. 14, p. S1033. 55. Y. M i y a w a k i , M. Hanmyo, S. U c h i d a , T. T e r a o k a , Y. S h i r a t a n i , and Y. I s h i d a : T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 4, p. S143. 56. Y. M i y a w a k i , M. Hanmyo, S. U c h i d a , T. M o r i , Y. S h i r a t a n i , and Y. I s h i d a : T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 4, p. S144. 57. Y. Nagano, K. Koyama, T. Nakano, T. M u k a i , T. Komai, and S. Kaneko: T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 4, p. S145. 58. Y. Mimura, F. Yamaguchi, T. T a k a h a s h i , N. O g i b a y a s h i , H. Yamaguchi, and K. Koyama: T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 4, p. S146. 59. K. Koyama, Y. Nagano, T. Nakano, H. Y a m a g u c h i , N. O g i b a y a s h i , and Y. Mimura: T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 4, p. S147. 60. T. Koyama, M. Yamaguchi, H. S a k a i , A. Yamagami, and C. Matsumura: T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 4, p. S1 48. 244 6 1 . H . N a k a t o , T . N o z a k i , Y . Habu , H . O k a , N . U e d a , and H . B a d a : T e t s u - t o - H a g a n e ' , 1984, v o l . 7 0 , n o . 4 , p . S149 . 62 . K . T o k i w a , F . K a t a o k a , S. T s u n e o k a , Y . N a k a m o r i , and Y . F u j i g a k e : T e t s u - t o - H a g a n e ' , 1984, v o l . 70 , n o . 4 , p . S150 . 6 3 . H . M i z u k a m i , T . K i t a g a w a , and K . Kawakami : T e t s u - t o -H a g a n e ' , 1984, v o l . 70 , n o . 4, p . S 1 5 1 . 64 . Y . N a k a m o r i , 0 . I c h i k o , Y . M i m u r a , Y . E d a , and M . O h t a : T e t s u - t o - H a g a n e ' , 1984, v o l . 7 0 , n o . 4 , p . S152 . 65 . Y . N a k a m o r i , O. I c h i k o , T . W a s h i t a n i , M . O h t a , Y . M i m u r a : T e t s u - t o - H a g a n e ' , 1984, v o l . 7 0 , n o . 4 , p . S153 . 66 . H . M i z u k a m i , M . Komatsu , T . K i t a g a w a , K . Kawakami , H . U c h i b o r i , and H . M i y a n o : T e t s u - t o - H a g a n e ' , 1983, v o l . 69 , n o . 14, p . S1032 . 67 . K . T o k i w a , T . O k a z a k i , T . S h i r a i , Y . N a k a m o r i , and Y . F u i g a k e : T e t s u - t o - H a g a n e ' , 1983, v o l . 6 9 , n o . 12, p.. S1034. 68 . Y . K o m a t s u , Y . U c h i d a , Y . S h i r a t a n i , S . M i y a h a r a , M . S u z u k i , J . F u k u m i , T . D o i h a r a , and 0 . Nomura : T e t s u - t o -H a g a n e ' , 1982, v o l . 68 , p . S928 . 69 . G . J . W . K o r : i n P r o c . of 2nd P r o c e s s T e c h n o l o g y C o n f . on C o n t i n u o u s C a s t i n g of S t e e l , C h c a g o , I S S - A I M E , 1981, v o l . 2 , p p . 124-132 . 70 . Y . N u r i and T . O h a s h i : T e t s u - t o - H a g a n e ' , 1979, v o l . 65 , p . S702 . 7 1 . Y . N u r i and T . O h a s h i : T e t s u - t o - H a g a n e ' , 1979, v o l . 65 , p . S703 . 72 . M . K o m a t s u , T . K i t a g a w a , and K . Kawakami : T e t s u - t o -H a g a n e ' , 1982, v o l . 68 , p . S927 . 7 3 . T . K i m u r a , T . O z a k i , Y . Akabane , M . Nakamura , and Y . S h i r a i s h i : T e t s u - t o - H a g a n e ' , 1984, v o l . 70 , p . S154 . 74 . Y . N a k a m o r i , 0 . I c h i k o , K . T o k i w a , and F . K a t a o k a : T e t s u - t o - H a g a n e ' , 1983, v o l . 69 , p . S1035 . 7 5 . S . O h m i y a , H . N a k a t o , Y . Habu , T . E m i , K . Hamagami, H . Bada , and Y . F u k u h a r a : T e t s u - t o - H a g a n e ' , 1982, v o l . 68 , p . S926 . 245 76. M. H a s h i o , M. K a w a s a k i , T. Watanabe, Y. O h t a n i , and J . Murayama: T e t s u - t o - H a g a n e ' , 1980, v o l . 66, p. s757. 77. C. O f f e r m a n , C. D a c k e r , and C. E n s t r o m : S c a n d i n a v i a n J . o f M e t a l l u r g y , 1981, v o l . 10, pp. 115-119. 78. L. S c h m i d t and A. J o s e f s s o n : S c a n d i n a v i a n J . of M e t a l l u r g y , 1974, v o l . 3, pp. 193-199. 79. T. M u k a i , N. O g i b a y a s h i , R. T s u j i n o , T. N a i t o , H. S u z u k i , Y. Abe, and S. N a g a t a : T e t s u - t o - H a g a n e ' , 1982, v o l . 68, pp. A161 - 164. 80. H. S u z u k i , S. N i s h i m u r a , J . Imamura, and Y. Nakamura: T e t s u - t o - H a g a n e ' , 1981, v o l . 67, no. 8, pp. 1180-1189. 81. Y. I i d a , K. M o r i w a k i , N. Ueda, and Y. Habu: T e t s u - t o -Hagane', 1973, v o l . 59, p. S89. 82. Y. M i y a s h i t a , M. S u z u k i , K. T a g u c h i , S. U c h i d a , H. S a t o , and M. Yamamura: N i p p o n Kokan T e c h . R e p o r t , 1982, no. 36, pp. 1-10. 83. J . Fukumi, Y. M i y a w a k i , M. Hanmyo, M. I s h i k a w a , and Y. I s h i d a : T e t s u - t o - H a g a n e ' , 1982, v o l . 68, p. S985. 84. T. S u g i t a n i : i n p r e p r i n t o f CONCAST S l a b S e m i n a r , 1973, Z u r i c h . 85. M. Yamaki e t a l . : T e t s u - t o - H a g a n e ' , 1974, v o l . 60, p. S455. 86. H. T a k e u c h i , S. Matsumura, T. Y a n a i , and Y. I k e h a r a : T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 7, pp. 687-693. 87. Y. Fukuda, S. M i z o g u c h i , T. M a t s u m i y a , H. Hamada, T. M i y a z a k i , and R. S a s a k i : T e t s u - t o - H a g a n e ' , 1984, v o l . 70, no. 4, p. S282. 88. E . T a k e u c h i , H. F u j i i , T. O h a s h i , H. Tanno, S. Tokao, I . F u r u g a k i , and H. K i t a m u r a : T e t s u - t o - H a g a n e ' , 1983, v o l . 69, no. 14, pp. 1615-1622. 89. T. Nakano, M. F u g i , T. K i s h i , K. Koyama,. T. Komai, and T. N a i t o : T e t s u - t o - H a g a n e ' , 1983, v o l . 69, S163. 90. W. H. McAdam: "Heat T r a n s m i s s i o n " , 3ed., 1954, New Y o r k , M c G r a w - H i l l . 91. R. H i g b i e : T r a n s . Am. I n s t . Chem. Eng., 1935, v o l . 31, pp. 365-389. 246 92. J . S z e k e l y and N . J . T h e m e l i s : "Rate Phenomena i n P r o c e s s M e t a l l u r g y " , W i l e y - I n t e r s c i e n c e , 1971, pp. 427-431. 93. B. C a r n a h a n , H.A. L u t h e r , and J.O. W i l k e s : " A p p l i e d N u m e r i c a l Method", W i l e y , New Y o r k , 1969, pp. 432-433. 94. T. M a t s u m i y a , T. S a e k i , J . T a n a k a , and T. A r i y o s h i : T e t s u - t o - H a g a n e ' , 1982, v o l . 68, p. 1782-1791. 95. M. I s h i g u r o , K. Kawakami, M. I t o , and S. M i y o s h i : T e t s u - t o - H a g a n e ' , 1974, v o l . 60, p. S885. 96. J . H a r r i s : " R h e o l o g y and Non-Newtonian F l o w " , Longmans,1977, pp. 280-289. 97. J . J . B i d e r m a n : " P h y s i c a l S u r f a c e s " , A c a d e m i c P r e s s , 1970, p.12. 98. E. M a t i j e v i c : " S u r f a c e and C o l l o i d S c i e n c e " , W i l e y -I n t e r s c i e n c e , 1969, v o l . 1, p.81. 99. F. K r e i t h and W.Z. B l a c k : " B a s i c Heat T r a n s f e r " , 1980, New Y o r k , H a r p e r & Row. 100. P r i v a t e C o m m u n i c a t i o n , N i p p o n S t e e l C o r p o r a t i o n . 101. K. Kawakami, T. K i t a g a w a , K. Murakami, Y. M i y a s h i t a , Y. T s u c h i d a , and K. Kawawa: N i p p o n Kokan T e c h . R e p o r t , 1983, no. 93, pp. 149-163. 102. J.K. Brimacombe, F. W e i n b e r g , and E.B. H a w b o l t : Met. T r a n s . B, 1979, v o l 10B, pp. 279-292. 103. H. T a k e u c h i , S. Matsumura, Y. I k e h a r a , T. Komano, and T. Y a n a i : T e t s u - t o - H a g a n e ' , 1983, v o l . 69, no. 1, pp. 73-79. 104. A. S u z u k i and Y. Nagaoka: J . J a p a n I n s t . M e t a l s , 1969, v o l . 33, pp. 658. 105. H. Kumai, K. Asano, T. O h a s h i , E. Nomura, and H. F u j i i : T e t s u - t o - H a g a n e ' , 1974, v o l . 60, no. 7, pp. 894-914. 106. M.C. F l e m i n g s : " S o l i d i f i c a t i o n P r o c e s s i n g " , 1974, New Y o r k , M c G r a w - H i l l , pp. 234-239. 107. T.S. Piwonka and M.C. F l e m i n g s : T r a n s a c t i o n s o f t h e M e t a l l u r g i c a l S o c i e t y o f AIME, 1966, v o l . 236, pp. 1157-1165. 247 108. L.S. D a r k e n , R.W. G u r r y : " P h y s i c a l C h e m i s t r y of M e t a l s " , 1953, New Y o r k , M c G r a w - H i l l . 109. W.A. F i s c h e r and H. F r y e : A r c h . E i s e n h " t t e n w . , 1970, v o l . 41, pp.293. 248 APPENDIX I NODAL EQUATIONS FOR THE HEAT FLOW CALCULATION IN THE MOULD WALL Nodal equations for each type of node are as follows: < t y p e 1 > ( T1.0 " T0>0 ) + ( T ( M " T ° . Q ) + 5w_ ( V T 0 Q ) = 0 (A-l-1) Ax z Ay z A y ^ w u ' u < type 2 > ( T 0 , n - l ~ 2 T0,n + T0,n+1> + ( T l , n ~ W _ Q 2Ay z Ax z < t y p e 3 > ( T 0 , n - l ^ T0,n> + ( T l , n ~ W + m Q Ay Z Ax z A y ^ < t y p e 4 > (Tm-1.0 " 2 V P + Tm+1,0> + <Tm,l " Tm.0> + _K_ 2 Ax 2 Ay 2 Ay>^ * ( Tw " Tm,0) = 0 (A-l-4) < type 5 > ( T m - l . " " 2 Tm,n + Tm+l.n> + (Tm,n-1 ~ 2 Tm.n + Tm.n+1> m Q Ax 2 Ay 2 (A-1-5) < type 6 > ( V l , " " 2 Tm,n + Tm+l,n> + (Tm,n-1 ' Tm.n> + ^ o ^ ) = Q 2 Ax 2 Ay 2 AyX^ (A-l-6) < t y p e 7 > ( V l , 0 2 " xm,0^ + V ^ , l ~ X^,0 ; W Ax" Ay- A y ^ <Tw " Tm,0) - 0 (A-1-7) 249 < type 8 > m ' n 1 5^3 m.n+l' + m-l,n m,n' = Q ( A _ ] _ 8 ) 2Ay z Ax z < t y p e 9 > ( T*,"-1 : T°»,n) + ( V - l , n ~ Tm,n> + M f l = Q (A-^g) Ay z Ax z A y ^ These simultaneous equations are rewritten i n a matrix system, which can be solved with the Gauss-Siedel method under the given conditions. 250 APPENDIX II NODAL EQUATIONS FOR THE TEMPERATURE DISTRIBUTION IN THE MENISCUS REGION Node Q 2 04 a 0 %<x> Q30 Q40 c b 0 Q20 Q30 Q40 o •H c 0 Q20 Q30 <Tf - T f i ) 00 cu d Q10 q 0< x> Q30 Q40 u e Q10 Q20 Q30 Q40 X f Q10 Q20 Q30 ( T f - T f ± ) Q40 3 iH g Q10 Q20 Q31 h Q10 Q20 Q32 Q40 i Q10 Q20 Q32 ( T f = T f i ) j Q10 Q20 Q31 Q41 k Q l l Q21 Q33 mm 1 - Q22 034 Q41 •H 0) m Q12 Q23 Q30 040 CU4-1 n Q12 Q20 Q30 Q40 CO o Q12 Q20 Q30 <Ts = T s i ) Q40 P Q13 Q23 Q30 X q Q10 Q31 Q41 3 i-l r Q10 Q33 -U  s Q13 Q23 Q30 Q40 t Q13 Q30 Q40 C o u Q10 Q20 Q30 040 •H 60 V Q10 Q20 Q30 <*s = n T s l ) Q40 0) w Q10 q 0< x> Q30 X Q10 %w 0 Q40 CU cu y Q10 Q20 0 Q40 CO z Q10 Q20 0 <Ts - T s i > Nodal equations for each type of node are as follows: Q10 = A y ( Tm-l.n m tn / Ax. l , n m * V n " (A-2-1) Q l l = )(Ax m_ 1/2 + Ax 2{(Ay/6)ii + ( A x/3 + Ax. _ 1/2) 7} (A-2-2) 251 Ay (I*; 1"* - T n) Q12 = m ^ ^ (A-2-3) ^ m - l + _ J _ + , n 8 - r 2 V , n Q13 = ( V l . n + ( V l . n " V n > ( * » m / 2 + A x ^ / 3 ) _ 2_ 2{(Ay/6)* + ('Axni_1/3) + ( A x m / 2 ) 2 non - Nil.n-1 + \ n , n ^ T m , n - l ~ T m . n ^ , A 0 cv Q20 = » 2 Ay ^— (A-2-5) 021 = 1 5 A y A x m (V.n-1 + V . n ) < V n - l ' Tn.n> ( A . 2 _ 6 ) 25 Ay2^ + Ax 2 Q 2 2 = 2 A y A x m ( T m ^ - T m < n ) / / A x £ + Ay*  / A ^ 2 + Ay Z/12 4 ^ x + /Ax m 2 + A y 2 / 1 2 V > n + l / h 8 _ f Q23 - 1 5 f r * W ° - l + V t n ) ( T m , n - l ^ m < n ) ( A _ 2 _ g ) 25 Ay* + Ax*, Q30 - W m + l . n - Tm.n> ^ m + l +  2 X m + l , n 2 V , n (A-2-9) 252 03i = *»<V»l.n + V n X V u . n ~ + ^ r i - l / 3 > (A-2-10) 2 {(Ay/62 + ( A x ^ j/3 + Ax m/2) 2} Q32 = A , ^ ( T m + l t n - T m i n ) ^ (A-2-11) ^ + 1 + 1 , J*m  2\n+l,n h s - f 2\n,n 2 Ay Ax,, ( T ^ ^ 1 - T_ „ ) / /Ax 2 + Ay 2 Q 3 3 = Y ^ M » N £ ^ S 1 (A-2-12) /AxJ + Ay 2/12V, n + /Ax m + A 7 2 / 1 2 ^ e l + 1 / h s - f Q 3 4 - < V n . n + W f ( T ^ . n - + Ax m/2) 2{(Ay/6) Z + (Ax m/3 + Q40 = ^ V . n + l + V . n ^ V n + l ~ T i , n ) (A-2-14) 0 4 1 = 1 5 A y A x m ( V n + 1 + V n > ( V n + l ~ V n ) ( A _ 2 _ 1 5 ) 25Ay 2 + Ax2, F i n a l l y temperature of element (m,n) a f t e r At i s , T' = T M „ + R . „ * At (A-2-16) m,n m,n CpAy*x^ v ' for the t r i a n g l e element (m,n) a f t e r At i s calculated by, T : ; n - c p ^ ? i ^ * (A-2-17) 253 APPENDIX III SHAPE OF MENISCUS Generally the geometry of the s t e e l droplet i n molten f l u x i s expressed with Eq.(A-3-l): a ( f - + f -) = AP (A-3-1) r l r2 The meniscus system can be expressed by the equation of meridional and c y l i n d r i c a l curvature r e s p e c t i v e l y . r7 - U <A"3"2> ^ - J L L f ~ i (A-3-3) In the case of the meniscus i n the continuous casting mould, however, T2 Is so large that the two dimensional external meniscus can be appl i c a b l e . Also following r e l a t i o n s can be obtained from F i g . 4-39. {(f - cos • (A-3-5) | f = - s i n <{> (A-3-6) The^pressure d i f f e r e n c e AP i s expressed by Eq.(A-3-7) when the f l u i d pressure P (x) e x i s t s i n the f l u x layer. AP = ( p s - p f)gx - P*(x) (A-3-7) From equations as above, . i „ • j * . <Ps - - ' * w C A . 3 . 8 ) 254 With the boundary condition, <t> = 0; x = 0 (A-3-9) , 4> , X (Po " Pf)gX ~ P * (X) J s i n dxitj) = J — = = —5 dx (A-3-10) ,. cos $ = 1 - ( P s ^ g P ^ 8 x 2 + & £ L (A-3-11) where R(x) = J q X P*(x)dx » J * {P(x) - p fgx}dx (A-3-12) From Eqs.(A-3-5),(6) and (11) dy_ 2a(p s - p f ) g x 2 - 4q{R(x) + 0}  [ p 8 - P f ) 2 g 2 x 4 - 4 ( P g - p f){R(x) + a}gx 2 + 4R(x){R(x) + 2 a } ] 1 / 2 (A-3-13) S t a t i c shape of the meniscus can be calculated a n a l y t i c a l l y by Eq. (A-3-13) when R(x) = 0. V a r i a t i o n of equation i s as follows: = - a 2 " , x 2 „ CA-3-14) :/2a 2 - x 2 where a i s a c a p i l l a r constant a 2 * g ( P 6 q - p f) ( A" 3- 1 5> Integration of Eq.(A-3-14) i s 2 2 / dy = / (- a " x -) dx (A-3-16) x/2a 2 - x 2 = ^ / d x - / 1 / 2 3 2 ; x 2 dx (A-3-17) x/2a 2 - x 2 255 a 2 ( - J L - m /2a 2 + / a 2 - x 2 ) _ ( / 2 a 2 _ x2 _ feZ^ /2a • In /2a + /2a - x x. + ^ (A-3-18) = - /2a 2 - x 2 + iSf- In ^  + /2a* - x i_ + ^ ( A _ 3 _ 1 9 ) i—o- A ,A„2 1 + cos = - / 2 a r cos £ + ->^f— In ^—A + C Q (A-3-20) s i n With the boundary condition, 4) = n/2; y = 0 (A-3-21) y = - /2a 2 cos 1 + In "*" + C ° l ^ + 0.3768 a (A-3-22) s i n = - /2a 2 - x 2 + - ^ f ^ In fi*7 + £2a2 ~ x Z + 0.3768 a (A-3-23) On the other hand change of meniscus shape by dynamic pressure i n the f l u x channel i s calculated i n the following way. ( i ) The dynamic pressure i s ca l c u l a t e d by the meniscus shape at t = t ^ , assuming that the shape i s expressed as the second order of a curve. ( i i ) Calculated pressure p r o f i l e i s modified to a simple form by Eq.(A-3-24), thus the meniscus shape at t = t^+1 i s calculated by Eq.(A-3-13). R(x)/R(x c, t ) = x 2 / x 2 t (A-3-24) c c i + l c , t i + l t i + 1 = t± + At (A-3-25) ict j obtained from Eq.(A-3-ll). x„ i s the "conta  point" of meniscus with the mould wall and i s c , t 1 + 1 256 i+1 2g R(x c t ) (A-3-26) Computation of ( i ) and ( i i ) are i t e r a t e d by the time step At for one cyc l e of the mould o s c i l l a t i o n . APPENDIX IV DYNAMIC PRESSURE IN THE MOULD FLUX CHANNEL The s o l u t i o n to Eqs. (4-26) and (4-27) for the pressure p r o f i l e and v e l o c i t y d i s t r i b u t i o n i n the mould f l u x channel has been obtained as follows. Equation (4-26) was integrated with respect to y, with l i m i t s set by B.C.'s (4-29) and (4-3 0), to obtain u x = <vm - V ^ 1 'ft ~ fe < £ + Pf«) i U " ft> CA-4-1) Then Eq.(A-4-l) was substituted into Eq.(4-27) which was integrated, and rearranged to give 3x _ + Pf« " p - °-R (A-4-2) Q R, the r e l a t i v e f l u x consumption r a t e , was e v a l u a t e d by r e a r r a n g i n g Eq. (A-4-2) and i n t e g r a t i n g from x = 0 to x = JLf ( F i g . 4-35). n - Pfg*f - ( p f - p i > + 6 ^ f ( v m - v s> £<*f> . „ Q R - 1 2 p f C ( l f ) (A-4-3) where e(A f) = / f .,dx (A-4-4) * 0 h/(x) Eqs.(A-4-3) and (A-4-5) were substituted i n t o Eq.(A-4-2) which was integrated again from 0 to x, giving P(x) = P i + P fgx + 6 u f ( v m - v s ) e(x) - { P fgA f - (F± - P f) + 6 u f ( v m - v g ) e(A f)> (A-4-6) where e(x) and C(x) are defined as in Eqs.(A-4-4) and (A-4-5). The r e l a t i v e v e l o c i t y d i s t r i b u t i o n was obtained by combining Eqs.(A-4-l) - (A-4-5) to y i e l d 258 u x " <vm " V * 1 " " z f c * - J z ) « ^ f ( v 1 B - v 8 ) } PjgJLf - ( P f - P ±) + 6 ^ 0 ^ - v g ) e(A f) } (A-4-7) The expressions f o r e(y) and C(y) depend on the function used to approximate the segment of the meniscus under consideration. I f a l i n e a r f u n ction i s adopted, eg. h(x) = ax + 6 (A-4-8) then « x > = - 75 t J W * 2 " 7^1 (A-4-10) {h(x)}^ ^ For the physical system i n F i g . 4-3 5 h f - h, a = —=-j (A-4-11) and p - h A (A-4-12) Sub s t i t u t i o n of Eqs.(A-4-9) - (A-4-12) into Eqs.(A-4-6) and (A-4-7) leads to Eqs.(A-4-9) and (A-4-10) r e s p e c t i v e l y . On the other hand, i f a quadratic function for the lower part of the meniscus region i s chosen, v i z . h(x) = ax 2 + bx + c (a,b,c const.) (A-4-13) the evaluation of e(x) and C(x) depends on the value of K = 4ac - b . If K > 0 / \ _ 1 /2ax + b _ b\ . 4a /arc tan /2ax + b\ _ arc tan b •> £<x> - T < h(x) c> + 7377 { ( R l / 2 > 7l7T} (A-4-14) 259 1 (2ax + b)(K + 6ah(x)) b(K + 6ac) «*> - ^ { rj-: ~2 > 2K* h^(x) c' o . 12a /arc tan ,2ax+bx arc tan b \ , U N + ^5/7 < ( K l / 2 > £I77> (A-4-15) but i f K < 0 K h ( x ) c (-K) (2ax + b + /=K)(b - ^=K) (A-4-16) C ( X ) = 1 {(2ax + bVK + 6ah(x)) _ b j O L + M d } 2K Z h Z ( x ) c 2 +  6alf., In {(2ax + b - /=K) (b + /=K) } (A-4-17) ( - K ) V * (2ax + b + /=IQ ( b -This refinement of the f l u i d flow c a l c u l a t i o n s has not been pursued further i n the present study. N 0 I 9 3 a 133J.S NI Xnid-1V3H *V3d S30 OOO N0i93a xnid a i n o w NI x m j xv3H d30 0 1331S JO 1V3H 1N3J.V1 SIIVH 0 i 33 i s a a i d i a n o s do A i i A i i o n a N O O ivwast-u sswva 0 xnid a a i d i a n o s dd Ai iAi ionaNoo ivwaam Sdwva 0 i33 i s a i n o n d o AiiAiisnaNOo ivwaam l s w v a 0 xnid a i n o n d o AiiAiisnaNOO ivwasm idwva 3 T 3 3 i s a s i d i a n o s do xiiSNsa s s n o a 0 xrnd c m d i a n o s d o AIISNSO Sdnoa 0 T331S a i n o n do Ai iSNsa isnoa 0 xrnd a i n o n d o AIISNSCJ idnoa 0 i33 i s a a i d i a n o s do IVSH o id i s sa s sso 3 i 33 i s a i n o n d o iV3H o id iosds ISO 3 xnid a 3 i d i a n o s do ±V3H o id iosas Sd3 3 xrnd a i n o n d o IVSH o i d i o s d s IdO 3 xnid ainOW ONV 1331S N33/U38 lN3ISIdd300 d3dSNVdi 1V3H H 3 1331S dO 3aniVd3dW31 ONIUnOd di 3 T 3 3 i s do 3aniva3dW3i s n a n o s SlOSi 3 T 3 3 i s do 3aniva3dW3i s n a i n o n SOI1J. 3 133J.S dO 3anXVa3dW31 1VI1INI i sn 3 x n d ainow do 3aniva3dW3i IVIJ.INI idn 3 d3±S 3WI1 i a 3 Nonoaaia x NI NOissa aaAano do Hi9N3i IX 3 NOIlDSaia Z NI N0I33a 1331S dO H19N31 e z 3 N o i i o s a i a z NI snosiNsw do NOiosa asAano do HIONSI zz 3 snosiN3w NO x n i d ainow d o SS3N*OIHI vz 3 N o i i o s a i a x NI snosiNsw d o N o i s s a i v i d NI SOON do asawnN sw 3 N o n o s a i a x NI snos iNsw do NOisaa 03Aano NI SOON d o aaawnN i\n 3 No i io sa i a z NI N0i93a i 33 i s NI aaoN d o aaawnN CN 3 N O i i o a a i a z NI SOOSINSW do Noi33a aaAano NI aaoN do asawnN SN 3 N o i i o s a i a z NI Noiosa x n i d NI SOON do asawnN l-N 3 N o n n a i a i s i a x r n d IVSH (N)O 3 anos 'AHSnw 'a inon '• 1331s d o 3 i v i s SN1 3 1331S dO 31V1S SWT 3 x m j a i n o w da AIISN30 dnoa 3 1331S dO A1ISN30 snoa 3 xrnd ainow do IVSH oidi33ds dO 3 1331S dO 1V3H 3IdI33dS so 3 1331S dO AilAIlOnaNOO "IVWa3Hl swva 3 xnid ainow JO A i i A i i o n a N o o ivwa3H± dwva 3 ( N 0 I i 0 3 a i a 9NIJ.SVO) NOIX03aia Z NI 300N dO 3ZIS z a 3 N o i i o s a i a x NI SOON d o 3 Z i s x a 3 xnid annow da 3aniva3dW3i MSN sdn 3 1331S dO 3aniVa3dW31 W3N s s n 3 xnid ainow d o 3aniva3dW3i d n 3 i33 i s JO aaniva3dW3i sn 3 *\ s i o a w A S do i s i i •J 3 3 **************************************************************** 3 eeeviva = SWVN 3 i i d v i v a 3 eeeiv3H : 3WVN s l i d wvasoad 3 3 ******************************************************************************3 ssaoN avinoNviai asanivad snssiNsw i v 3 wvasoad aadSNvai IVSH 30N3aaddia 3 i i N i d 3 i v i s AavsiSNn i v N o i S N 3 w i a - o « i o ******************************************************************************3 N0I93H S11DSIN3W 3HI NI Nonoiasraa 3"anivHa<iwai am aoi wvnooHd aanidwoo A XIWKddY 093 261 Q* ********************************************************************** ******* C DIMENSION Q****************************************************************************** c c DIMENSION U S ( 1 0 0 , 1 0 0 ) , U F ( 1 0 0 , 1 0 0 ) ,US2 (100 ,100 ) ,UF2 (100 ,100 ) ,DZ ( A100 ) ,RAMF(100 ,100 ) ,RAMS (100 ,100 ) ,CS (100 ,100 ) ,CF (100 ,100 ) ,ROUS (100 A, 100) ,ROUF( 1 0 0 , 1 0 0 ) , L M F ( 1 0 0 , 1 0 0 ) , L N F ( 1 0 0 , 1 0 0 ) , L M S ( 1 0 0 , 1 0 0 ) , L N S ( 1 0 0 A , 1 0 0 ) , 0 ( 1 0 0 ) C C Q****************************************************************************** C READ DATA Q****************************************************************************** c c READ(5,500) N1,N2,N3,M1,M2,Z1 ,Z2 ,Z3 ,X1,DT 500 F0RMAT (5 I 4 ,4F8 .4 ,F10 .e ) READ(5,501 ) U F I , U S I , T L I O F , .TLIOS, , TSOLF, .TSOLS, ,TP,H 501 F0RMAT(8F 10 .5) READ(5,502) CFL .CSL .ROUFL, , ROUSL, , RAMFL, ,RAMSL, ,HLATF,HLATS 502 F0RMAT(8F 10 .5) READ(5,503) CFS.CSS.ROUFS, , ROUSS, , RAMFS, ,RAMSS, ,OEF,OES 503 F0RMAT(8F 10 .5) c c Q****************************************************************************** C WRITE DATA c****************************************************************************** c c WRITE(6,2000) N1,N2,N3,M1,M2,Z1,Z2,Z3,X1,DT 2000 FORMAT(10H N1 = , I4/10H N2 = , I4/10H N3 = , 14 A/10H M1 = , I4/10H M2 = , I4/10H Z1 = . F 1 2 . 6 / A10H Z2 = , F 1 2 . 6 / 1 0 H Z3 = , F12 . 6 / 10H X1 = , F12.6 A/10H DT = , F 1 2 . 6 ) WRITE(6,2001) UF I ,US I . TL IOF ,TL IOS ,TSOLF ,TSOLS ,TP , H 2001 F0RMAT(10H UFI = , F12 .6 /10H USI = . F12.6/10H TLIOF = , AF12 .6 /10H TLIOS = , F12 .6 /10H TSOLF = , F12 . 6 / 10H TSOLS = , AF12.G/10H TP = . F 1 2 . 6 / 1 0 H H = , F12 . 6 ) WRITE(6 ,2002) CFL,CSL,ROUFL,ROUSL,RAMFL,RAMSL,HLATF,HLATS 2002 FORMAT(10H CFL = , F12 .6 /10H CSL = , F12 . 6 /10H ROUFL = , AF12 .6 /10H ROUSL = , F 1 2 . 6 / 1 0 H RAMFL = , F12.6/10H RAMSL = , A F 1 2 . 6 / 1 0 H HLATF = , F12.G/10H HLATS = , F12.6) WRITE(6,2003) CFS,CSS,ROUFS,ROUSS,RAMFS,RAMSS,OEF,QES 2003 FORMATOOH CFS = , F12 .6 /10H CSS = , F12.6/10H ROUFS = , AF12 .6 /10H ROUSS = , F 1 2 . 6 / 1 0 H RAMFS = , F12.6/10H RAMSS = . AF12 .6 /10H OEF = , F12.G/10H OES = , F12.G) C C Q****************************************************************************** C OUTPUT TIME SET C C END1=0.160 END 1A = END1+DT END2=0.162 END2A=END2+DT END3=0.164 END3A=END3+DT END4=0.165 END4A=END4+DT END5=0.168 END5A=END5+DT END6=0.169 END6A=END6+DT END7=0.173 END7A=END7+DT END8=0.179 END8A=END8+DT END9=0.181 END9A=END9+DT END10=0.192 END 10A = END10+DT END 11=0.203 END 11A=END11+DT END12 = 0. 2.16 END 12A = END12+DT END13=0.221 END13A=END13+DT END14=0.227 END14A=END14+DT END15=0.234 END 15A = END15+DT END16=0.238 END 16A = END16+DT END17=0.256 END 17A = END17+DT END 18=0.300 END 18A = END 18 + DT END19=0.339 END19A=END19+DT END20=0.348 END20A=END20+DT END21=0.358 END21A=END21+DT END22=0.362 END22A=END22+DT END23=0.374 END23A=END23+DT END24=0.375 END24A=END24+DT END25=0.378 END25A=END25+DT END26=0.395 END26A=END26+DT END27=0.402 END27A=END27+DT END28=0.422 END28A=END28+DT END29=0.432 END29A=END29+DT END30=0.466 END30A=END30+DT END31=0.494 END31A=END31+DT END32=0.542 END32A=END32+DT 263 END33=0.600 END33A=END33+DT END34=0.609 END34A=END34+DT END35=0.637 END35A=END35+DT END36=0.640 END36A=END36+DT END37=0.667 END37A=END37+DT END38=0.678 END38A=END38+DT END39=0.716 END39A=END39+DT END40=0.720 END40A=END40+DT END41=0.731 END41A=END41+DT END42=0.793 END42A=END42+DT END43=0.967 C C c****************************************************************************** C DIMENSION OF SYSTEM c c NZ=N1+N2+N3 MX=M1+M2 N11=N1+1 N21=N1+N2 N20=N21-1 N22=N21+1 M10=M1-1 M12=M1+1 DX=X1/FL0AT(M1) X2=DX*FL0AT(M2) T=0.0 IF(M1-N2) 8 0 0 , 8 0 1 , 8 0 0 800 WRITE(G,850) 850 FORMAT(1H0.20H M1=N2 NOT EQUAL ) GO TO 1500 801 CONTINUE DO 200 N=1,N1 200 DZ(N) = Z1/FL0AT(N1 ) DO 201 N=N11,N21 201 DZ (N )=SQRT(Z2* *2 /X1* (X1 -DX*FL0AT(N-N1-1 ) ) ) - SORT( Z2**2/X1 * (X 1-ADX*FL0AT(N -N1 ) ) ) DO 202 N=N22,NZ 202 DZ(N)=Z3/FL0AT(N3) C C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C PROFILE OF HEAT FLUX C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c c DO 701 N=1,N1 264 701 Q(N)=QEF DO 702 N=N11.N21 702 0(N) = -OES*OEF* ( (N2+1 ) * *2 ) / (N22*S0RT( -OES ) -N1*SORT( -QEF) A -N * ( SORT ( -QES ) - SORT ( -OEF ) ) )**2 DO 703 N=N22,NZ 703 0(N)=OES C C Q****** *********************************************************************** * C IN IT IAL CONDITION ETC. C****************************************************************************** C C DO 205 N=1,NZ DO 205 M=1,MX UF(M,N)=UFI US(M,N)=USI LMF(M,N)=0 LNF(M,N)=0 LMS(M,N)=0 LNS(M,N)=0 CF(M,N)=CFL CS(M,N)=CSL R0UF(M,N)=R0UFL R0US(M,N)=R0USL RAMF(M,N)=RAMFL 205 RAMS(M,N)=RAMSL CFF =HLATF/ (TL IOF-TSOLF)+CFL CSF=HLATS/(TL IOS-TSOLS)+CSL R0UFF=(R0UFL+R0UFS)/2.0 RAMFF = (RAMFL+RAMFS)/2.0 R0USF=(R0USL+R0USS)/2.0 RAMSF=(RAMSL+RAMSS)/2.0 GXF=RAMFL*DT/(CFL*R0UFL*DX**2) GXS=RAMSL*DT/(CSL*R0USL*DX**2) GZ1F=RAMFL*DT/ (CFL*R0UFL*DZ(1 ) * *2 ) GZ2F=RAMFL*DT/ (CFL*R0UFL*DZ(N11)**2 ) GZ2S=RAMSL*DT/(CSL*R0USL*DZ(N11)**2) GZ3=RAMSL*DT/(CSL*R0USL*DZ(N22)**2) I F ( G X F - 0 . 5 ) 5 6 0 , 5 7 0 , 5 7 0 560 I F ( G X S - 0 . 5 ) 5 6 1 , 5 7 0 , 5 7 0 561 I F ( G Z 1 F - 0 . 5 ) 5 6 2 , 5 7 0 , 5 7 0 562 I F ( G Z 2 F - 0 . 5 ) 5 6 3 , 5 7 0 . 5 7 0 563 I F ( G Z 2 S - 0 . 5 ) 5 6 4 , 5 7 0 , 5 7 0 564 I F ( G Z 3 - 0 . 5 ) 5 6 5 , 5 7 0 , 5 7 0 570 WRITE(6,575) 575 FORMAT(1H0,46HG IS MORE THAN 0 . 5 , CANNOT PERFORM CALCULATION) GO TO 1500 565 CONTINUE C C Q****************************************************************************** C TIME INCREASES Q****************************************************************************** C C T=T+DT C C Q******************************************************************************* 2 6 5 C HEAT FLUX CALCULATION IN EACH TYPE OF NODE C C 1 CONTINUE DO 999 N=1,NZ DO 999 M=1,MX IF(N-1) 10.10,15 10 IF(M-1) 12,12,11 12 CALL SUB 1(DX.DZ(N),DZ(N+1).RAMF(M,N).RAMF(M+1,N).RAMF(M,N+1),UF(M, AN+1),UF(M,N),UF(M+1,N),OF,0(N)) US(M,N)=0 GO TO 991 11 IF(M-MX) 13,14,14 13 CALL SUB2(DX,DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N),RAMF(M+1,N) , ARAMF(M,N+1),UF(M-1,N).UF(M,N).UF(M,N+1),UF(M+1,N),OF) US(M,N)=0 GO TO 991 14 CALL SUB3(DX,DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N),RAMF(M,N+1), AUF(M-1,N),UF(M,N),UF(M,N+1).UFI.OF) US(M,N)=0 GO TO 991 C C 15 IF(N-N1) 1G, 18,38 16 IF(M-1) 17, 17,19 17 CALL SUB4(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M+1,N).RAMF(M.N) ,RAMF(M,N-A1),RAMF(M,N+1),UF(M,N-1),UF(M,N),UF(M,N+1),UF(M+1,N),QF,Q( N)) US(M.N)=0 GO TO 991 19 IF(M-MX) 20,50,50 20 CALL SUB5(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N) , ARAMF(M+1,N),RAMF(M,N-1),RAMF(M,N+1).UF(M,N-1),UF(M,N),UF(M-1,N), AUF(M,N+1),UF(M+1,N),0F) US(M,N)=0 GO TO 991 50 CALL SUB6(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N) , RAMF (M , N-A1),RAMF(M,N+1),UF(M,N-1),UF(M,N),UF(M- 1 ,N) ,UF(M,N+1),UFI , OF) US(M,N)=0 GO TO 991 C c 18 IF(M-1) 51,51,52 51 CALL SUB4(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M+1,N),RAMF(M,N),RAMF(M,N-A1 ) , RAMF(M,N+1),UF(M,N-1),UF(M,N),UF(M.N+1),UF(M+1,N),OF,Q( N)) US(M,N)=0 GO TO 991 52 IF(M-M1) 25,21,22 25 CALL SUB5(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N) , ARAMF(M+1,N),RAMF(M,N-1),RAMF(M,N+1),UF(M,N-1),UF(M,N),UF (M-1,N), AUF(M,N+1),UF(M+1,N),0F) US(M,N)=0 GO TO 991 21 CALL SUB7(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M-1,N).RAMF(M,N),RAMF(M,N+ A1 ),RAMF(M+1,N),RAMF(M,N-1),UF(M,N-1),UF(M,N),UF(M-1,N),UF(M,N+1), AUF(M+1,N),QF) US(M,N)=0 GO TO 991 266 22 IF(M-MX) 23.24,24 23 CALL SUB8(DX.DZ(N-1).DZ(N),DZ(N+1),RAMF(M-1.N),RAMF(M,N),RAMF (M+1 , AN),RAMF(M,N-1),RAMS(M,N+1),UF(M,N-1),UF(M.N),UF(M-1,N),US(M.N+1), AUF(M+1,N),H,QF) US(M,N)=0 GO TO 991 24 CALL SUB9(DX ,DZ(N-1 ) ,DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N) , RAMF (M, N-A1),RAMS(M,N+1),UF(M,N-1),UF(M.N),UF(M-1,N),US(M.N+1),UFI ,H,OF) US(M,N)=0 GO TO 991 C C 38 IF(N-N11) 39,39,40 39 IF(M-1) 26,26,27 26 CALL SUB4(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M+1,N),RAMF(M,N),RAMF(M,N-A1),RAMF(M,N+1),UF(M,N-1),UF(M,N),UF(M,N+1),UF(M+1,N),OF,Q(N)) US(M,N)=0 GO TO 991 27 IF(M-M10) 30,31,32 • 30 CALL SUB5(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M.N), ARAMF(M+1,N),RAMF(M,N-1) ,RAMF(M.N+1),UF(M,N-1).UF(M,N),UF(M-1,N), AUF(M,N+1),UF(M+1,N),QF) US(M,N)=0 GO TO 991 31 CALL SUB10(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N),RAMF(M,N A+1),RAMF(M+1,N),RAMF(M,N-1),UF(M,N-1),UF(M,N),UF(M-1,N),UF(M,N+1), AUF(M+1,N),0F) US(M,N)=0 GO TO 991 32 IF(M-M12) 33,34,35 33 CALL SUB 1 1 (DX,DZ(N-1),DZ(N),RAMF(M,N-1),RAMF(M,N),RAMF(M-1 ,N) ,RAMS A(M,N),UF(M,N-1),UF(M,N),UF(M-1,N),US(M,N),H,OF) CALL SUB12(DX,DZ(N),DZ(N+1),RAMF(M,N),RAMS(M,N+1).RAMS(M.N), ARAMS(M+1.N),UF(M,N),US(M.N),US(M,N+1).US(M+1,N),H.OS) GO TO 990 34 CALL SUB13(DX,DZ(N-1),DZ(N),DZ(N+1).RAMF(M.N-1),RAMS(M-1,N).RAMS(M A.N),RAMS(M+1,N),RAMS(M,N+1),UF(M,N-1),US(M,N),US(M-1,N),US(M,N+1), AUS(M+1,N).H.QS) UF(M,N)=0 GO TO 992 35 IF(M-MX) 62,63,63 62 CALL SUB14(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M,N-1),RAMS(M-1,N),RAMS(M A.N),RAMS(M+1,N),RAMS(M,N+1),UF(M,N-1),US(M,N),US(M-1,N),US(M,N+1) , AUS(M+1,N),H,OS) UF(M,N)=0 GO TO 992 63 CALL SUB15(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M,N-1),RAMS(M-1,N),RAMS(M A.N) ,RAMS(M,N+1),UF(M,N-1).US(M.N),US(M-1,N),US(M,N+1),TP,H.QS) UF(M,N)=0 GO TO 992 C C 40 IF(N-N20) 69,70,80 69 MC1=M1-(N-N1 ) MC2=MC1+2 IF(M-1 ) 41,41 ,42 41 CALL SUB4(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M+1.N),RAMF(M,N),RAMF(M,N-A1),RAMF(M,N+1),UF(M,N-1),UF(M.N),UF(M.N+1).UF(M+1.N).OF,Q(N)) 267 US(M,N)=0 GO TO 991 42 IF(M-MC1) 43,44,45 43 CALL SUB5(DX,DZ(N-1),DZ(N),DZ(N+1),RAMF(M-1,N),RAMF(M,N), ARAMF(M+1,N),RAMF(M,N-1),RAMF(M,N+1),UF(M,N-1),UF (M,N),UF(M-1,N), AUF(M,N+1),UF(M+1,N),QF) US(M,N)=0 GO TO 991 44 CALL SUB10(DX,DZ(N-1),DZ(N) ,DZ(N+1),RAMF(M-1,N),RAMF(M.N),RAMF(M,N A+1 ) , RAMF(M+1,N),RAMF(M,N-1),UF(M,N-1),UF(M,N),UF(M-1.N).UF(M,N+1), AUF(M+1,N),QF ) US(M,N)=0 GO TO 991 45 IF(M-MC2) 46,47,48 46 CALL SUB 11(DX,DZ(N-1 ) ,DZ(N) ,RAMF(M,N-1),RAMF(M,N),RAMF(M-1,N),RAMS A(M,N),UF(M,N-1),UF(M.N) ,UF(M-1,N),US(M,N),H,QF) CALL SUB12(DX,DZ(N),DZ(N+1),RAMF(M,N),RAMS(M,N+1),RAMS(M,N), ARAMS(M+1,N).UF(M.N),US(M,N) ,US(M,N+1),US(M+1,N).H,QS) GO TO 990 47 CALL SUB16(DX,DZ(N-1 ) ,DZ(N) ,DZ(N+1),RAMS(M,N-1),RAMS(M,N), ARAMS(M-1,N),RAMS(M+1,N) ,RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1).US(M+1,N),0S) UF(M,N)=0 GO TO 992 48 IF(M-MX) 49,60,60 49 CALL SUB5(DX,DZ(N-1 ) ,DZ(N),DZ(N+1),RAMS(M-1,N),RAMS(M,N), ARAMS(M+1,N),RAMS(M,N-1),RAMS(M,N+1),US(M.N-1),US(M, N).US(M-1 , N) , AUS(M,N+1),US(M+1,N),0S) UF(M,N)=0 GO TO 992 60 CALL SUB23(DX,DZ(N-1 ) ,DZ(N),DZ(N+1),RAMS(M-1,N) , RAMS (M , N) , ARAMS(M,N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US (M-1 , N) , AUS(M,N+1).TP.OS) UF(M,N)=0 GO TO 992 C C 70 IF(M-1) 71,71,72 71 CALL SUB17(DX,DZ(N-1),DZ(N) ,DZ(N+1),RAMF(M,N+1),RAMF(M,N),RAMF(M+1 A , N) , RAMF(M,N-1), UF ( M, N-1 ) , UF (M, N) , UF (M, N+1 ) , UF (M+1 , N) , AQF.Q(N)) US(M,N)=0 GO TO 991 72 IF(M-3) 73,74,75 73 CALL SUB 11(DX,DZ(N-1),DZ(N),RAMF(M,N-1).RAMF(M,N),RAMF(M-1,N),RAMS A(M,N),UF(M,N-1),UF(M,N),UF(M-1,N),US(M.N),H,OF) CALL SUB12(DX,DZ(N),DZ(N+1 ),RAMF(M,N),RAMS(M,N+1),RAMS(M,N), ARAMS(M+1,N),UF(M,N),US(M,N),US(M,N+1),US(M+1 ,N) ,H.OS) GO TO 990 74 CALL SUB16(DX,DZ(N-1 ) ,DZ(N),DZ(N+1),RAMS(M,N-1),RAMS(M,N), ARAMS (M-1,N).RAMS(M+1,N),RAMS(M,N+1),US(M,N-1),US(M,N) ,US(M-1 ,N) , AUS(M,N+1),US(M+1,N),OS) UF(M,N)=0 GO TO 992 75 IF(M-MX) 76,77,77 76 CALL SUB5(DX,DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N),RAMS(M,N), ARAMS(M+1,N),RAMS(M,N-1 ) ,RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),US(M+1,N),0S) UF(M,N)=0 268 GO TO 992 77 CALL SUB23(DX,DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N),RAMS(M,N), ARAMS(M,N-1),RAMS(M,N+1),US(M.N-1),US(M,N),U5(M-1,N), AUS(M.N+1),TP,OS) UF(M,N)=0 GO TO 992 C Q****************************************************************************** c 80 IF(N-N22) 81,90,100 81 IF(M-2) 82,83,84 82 CALL SUB18(DX,DZ(N-1),DZ(N),RAMF(M,N-1),RAMF(M,N),RAMS(M,N), AUF(M.N-1),UF(M,N).US(M.N),H,OF,Q(N)) CALL SUB12(DX,DZ(N),DZ(N+1),RAMF(M,N),RAMS(M,N+1),RAMS(M,N), ARAMS(M+1,N),UF(M,N),US(M,N),US(M,N+1),US(M+1,N),H,OS) GO TO 990 83 CALL SUB1G(DX,DZ(N-1),DZ(N),DZ(N+1),RAMS(M,N-1),RAMS(M,N) , ARAMS(M-1,N).RAMS(M+1,N),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1.N), AUS(M.N+1).US(M+1,N),OS) UF(M,N)=0 GO TO 992 84 IF(M-MX) 85,86,86 85 CALL SUB5(DX,DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N),RAMS(M.N), ARAMS(M+1,N),RAMS(M,N-1 ) ,RAMS(M,N+1 ) ,US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),US(M+1,N),OS) UF(M,N)=0 GO TO 992 86 CALL SUB23(DX,DZ(N-1),DZ(N) ,DZ(N+1),RAMS(M-1,N),RAMS(M,N), ARAMS(M,N-1),RAMS(M,N+1 ) ,US(M,N-1),US(M,N),US(M-1 , N) , AUS(M,N+1),TP,OS) UF(M,N)=0 GO TO 992 C Q****************************************************************************** c 90 IF(M-1) 91.91,92 91 CALL SUB19(DX,DZ(N-1),DZ(N),DZ(N+1),RAMS(M,N-1),RAMS(M,N) , ARAMS(M+1,N),RAMS(M,N+1),US(M,N-1),US(M,N),US(M.N+1),US(M+ 1 , N) , A0S,0(N)) UF(M,N)=0 GO TO 992 92 IF(M-MX) 93,94,94 93 CALL SUB5(DX,DZ(N-1),DZ(N) ,DZ(N+1 ) ,RAMS(M-1,N),RAMS(M,N), ARAMS(M+1,N),RAMS(M,N-1),RAMS(M,N+1 ),US(M,N-1),US(M,N),US(M-1 , N) , AUS(M,N+1),US(M+1,N),QS) UF(M,N)=0 GO TO 992 94 CALL SUB23(DX,DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N),RAMS(M,N) , ARAMS(M,N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N) , AUS(M,N+1),TP,0S) UF(M,N)=0 GO TO 992 C Q****************************************************************************** c 100 IF(N-NZ) 101,110,110 101 IF(M-1) 102,102,103 102 CALL SUB24(DX,DZ(N-1 ) ,DZ(N) ,DZ(N+1 ) ,RAMS(M+1,N),RAMS(M,N),RAMS A(M,N-1 ) ,RAMS(M,N+1),US(M,N-1),US(M.N),US(M,N+1),US(M+1 ,N) .OS.O(N)) 269 UF(M.N)=0 GO TO 992 103 IF(M-MX) 104,105,105 104 CALL SUB5(DX,DZ(N-1),OZ(N),DZ(N+1),RAMS(M-1,N).RAMS(M.N), ARAMS(M+1,N),RAMS(M,N-1),RAMS(M,N+1),US(M.N-1),US(M,N),US(M-1,N), AUS(M,N+1),US(M+1,N),OS) UF(M,N)=0 GO TO 992 105 CALL SUB23(DX,DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N),RAMS(M,N), ARAMS(M,N-1),RAMS(M.N+1),US(M.N-1),US(M,N),US(M-1,N), AUS(M,N+1).TP,OS) UF(M,N)=0 GO TO 992 C Q************ ***************************************************** ************* c 110 IF(M-1) 11 1 , 111,120 1 11 CALL SUB20(DX,DZ(N-1),DZ(N).RAMS(M+1,N),RAMS(M,N),RAMS(M.N-1), AUS(M.N-1),US(M.N).US(M+1,N),OS,0(N)) UF(M,N)=0 GO TO 992 120 IF(M-MX) 121 , 122, 122 121 CALL SUB21(DX.DZ(N-1),DZ(N),RAMS(M-1,N),RAMS(M,N),RAMS(M+1,N), ARAMS(M,N-1),US(M,N-1),US(M,N),US(M-1,N),US(M+1,N).OS) UF(M,N)=0 GO TO 992 122 CALL SUB22(DX,DZ(N-1),DZ(N),RAMS(M-1,N),RAMS(M,N),RAMS(M,N-1), AUS(M,N-1),US(M,N),US(M-1,N).TP,OS) UF(M,N)=0 GO TO 992 C C Q****************************************************************************** C NEW TEMPERATURE CALCULATION Q****************************************************************************** c c 990 UF2(M,N)=UF(M,N)+2.O*DT*0F/(CF(M,N)*ROUF(M,N)*DX*DZ(N)) US2(M,N)=US(M,N)+2.O*DT*0S/(CS(M,N)*R0US(M,N)*DX*DZ(N)) GO TO 999 991 UF2(M.N)=UF(M,N)+DT*QF/(CF(M,N)*R0UF(M,N)*DX*DZ(N)) US2(M,N)=US(M,N) GO TO 999 992 US2(M.N)=US(M,N)+DT*0S/(CS(M,N)*ROUS(M,N)*DX*DZ(N)) UF2(M,N)=UF(M,N) GO TO 999 999 CONTINUE C C Q****************************************************************************** C TEMPERATURE CHANGE Q****************************************************************************** c c DO 150 N=1,NZ DO 150 M=1,MX UF(M,N)=UF2(M,N) 150 US(M,N)=US2(M,N) DO 175 N=1,N21 DO 175 M=1.MX IF(LNF(M,N)) 175,199,175 199 IF(UF(M,N).GT.TLIOF) GO TO 175 I F(LMF(M,N ) ) 173.174,173 174 UF(M,N)=TLIOF-CFL*(TLIOF-UF(M.N))/CFF CF(M,N)=CFF ROUF(M,N)=ROUFF RAMF(M,N)=RAMFF LMF(M,N)=1 173 IF(UF(M,N).GT.TSOLF) GO TO 175 ROUF(M.N)=ROUFS CF(M,N)=CFS RAMF(M,N)=RAMFS LNF(M,N)=1 175 CONTINUE DO 275 N=N11,NZ DO 275 M=1,MX IF(LNS(M,N)) 275,299,275 299 IF(US(M,N).GT.TLIOS) GO TO 275 IF(LMS(M,N)) 273,274,273 274 US(M,N)=TLIOS-CSL*(TLIOS-US(M.N))/CSF CS(M,N)=CSF ROUS(M,N)=ROUSF RAMS(M.N)=RAMSF LMS(M,N)=1 273 IF(US(M,N).GT.TSOLS) GO TO 275 ROUS(M,N)=ROUSS CS(M,N)=CSS RAMS(M,N)=RAMSS LNS(M,N)=1 275 CONTINUE C C Q* *************************************** * C OUTPUT Q***************************************** c c IF(T-END1 ) 565,1000,9001 9001 IF(T-END1A) 1000,9002,9002 9002 IF(T-END2) 565,1000,9003 9003 IF(T-END2A) 1000,9004,9004 9004 IF(T-END3) 565,1000,9005 9005 IF(T-END3A) 1000,9006,9006 9006 IF(T-END4) 565,1000,9007 9007 IF(T-END4A) 1000,9008,9008 9008 IF(T-END5 ) 565,1000,9009 9009 IF(T-END5A) 1000,9010,9010 9010 IF(T-END6) 565,1000.9011 9011 IF(T-END6A) 1000,9012.9012 9012 IF(T-END7) 565,1000,9013 9013 IF(T-END7A) 1000,9014,9014 9014 IF(T-END8) 565,1000,9015 9015 IF(T-END8A) 1000,9016,9016 9016 IF(T-END9) 565,1000,9017 9017 IF(T-END9A) 1000,9018,9018 9018 IF(T-ENDIO) 565,1000,9019 9019 IF(T-END10A) 1000,9020,9020 9020 IF(T-END11) 565,1000,9021 9021 IF(T-END11A) 1000,9022,9022 9022 IF(T-END12) 565,1000.9023 9023 IF( T- END12A 9024 IF( T- END 13) 9025 IF( T- END 13A 9026 IF( T- END 14) 9027 IF( T- END 14A 9028 IF( T- END 15) 9029 IF( T- END15A 9030 IF( T- END 16 ) 9031 IF( T- END16A 9032 IF( T- END 17) 9033 IF( T- END17A 9034 IF( T- END 18) 9035 IF( T- END18A 9036 IF( T- END 19 ) 9037 IF( T- END19A 9038 IF( T- END20) 9039 IFI T- END20A 9040 IF( T- END21) 9041 IFI T- END21A 9042 IFI T- END22) 9043 IF T- END22A 9044 IF T- END23) 9045 IF T- END23A 9046 IF T- END24) 9047 IF T- END24A 9048 IF T- END25) 9049 IF T- END25A 9050 IF T- END26) 9051 IF T- END26A 9052 IF T- END27) 9053 IF [T- END27A 9054 IF T- END28) 9055 IF [T- END28A 9056 IF T- END29) 9057 IF (T- END29A 9058 IF [T- END30) 9059 IF (T- END30A 9060 IF (T- END31) 9061 IF (T- END31A 9062 IF (T- END32) 9063 IF (T- END32A 9064 IF T- END33) 9065 IF (T- END33A 9066 IF T- END34) 9067 IF ;T- END34A 9068 IF (T- END35) 9069 IF :T- END35A 9070 IF (T- END36) 9071 IF (T- END36A 9072 IF (T- END37) 9073 IF (T- END37A 9074 IF (T--END38) 9075 IF (T- END38A 9076 IF ;T- END39) 9077 IF ;T- END39A 9078 IF (T- END40) 9079 IF (T--END40A 9080 IF (T-•END41) 9081 IF ( T-END41A 9082 IF (T--END42) 1000.9024,9024 565.1000,9025 1000,9026,9026 565,1000,9027 1000,9028,9028 565,1000,9029 1000,9030,9030 565.1000,9031 1000,9032,9032 565.1000,9033 1000,9034,9034 565,1000,9035 1000,9036,9036 565,1OOO,9037 1000,9038,9038 565,1000,9039 1000,9040,9040 565,1000,9041 1000,9042,9042 565,1000,9043 1000,9044,9044 565,1000,9045 1000,9046,9046 565,1000,9047 1000,9048,9048 565,1000,9049 ' 1000,9050,9050 565,1000,9051 > 1000,9052,9052 565,1000,9053 ' 1000,9054,9054 565,1000,9055 1000,9056,9056 565,1000,9057 i 1000,9058,9058 565,1000,9059 i 1000,9060,9060 565,1000,9061 i 1000,9062,9062 565,1000,9063 I 1000,9064,9064 565,1000,9065 l 1000,9066,9066 565,1000,9067 I 1000,9068,9068 565,1000.9069 I 1000.9070,9070 565,1000,9071 I 1000,9072,9072 565,1000,9073 I 1000,9074,9074 565,1000,9075 I 1000,9076,9076 565,1OOO.9077 I 1000,9078,9078 565,1000,9079 I 1000,9080,9080 565,1000,9081 ) 1000,9082,9082 565,1000,9083 9083 IF(T-END42A) 1000,9084,9084 9084 IF(T-END43) 565,1000.1000 10OO CONTINUE WRITE(6.300) T 300 FORMAT(10H TIME =,F10.7) WRITE(6,303) END 1,END2 , END3,END4,END5,END6,END7,END8 , END9 , END 10 303 FORMAT(4F10.7) WRITE(6,301) 301 FORMAT(5H U-1 ,5H U-2 ,5H U-3 ,5H U-4 ,5H U-5 ,5H U-6 ,5H U-7 A,5H U-8 ,5H U-9 ,5H U-10.5H U-11.5H U-12.5H U-13.5H U-14.5H U-15,5 AH U-16.5H U-17.5H U-18.5H U-19.5H U-20.5H U-21.5H U-22.5H U-23) DO 350 N=1,40 WRITE(6,3350) (UF(M,N),M=1,23) 3350 FORMAT(23F5.O) 350 CONTINUE WRITE(6,302) 302 F0RMAT(5H U-1 ,5H U-2 ,5H U-3 ,5H U-4 ,5H U-5 ,5H U-6 ,5H U-7 A.5H U-8 ,5H U-9 ,5H U-10.5H U-11.5H U-12.5H U-13.5H U-14.5H U-15,5 AH U-16.5H U-17.5H U-18.5H U-19.5H U-20.5H U-21.5H U-22.5H U-23) DO 351 J=1,40 WRITE(6,3510) (US(M,J),M=1,23) 3510 FORMAT(23F5.0) 351 CONTINUE IF(T-END43) 565,1500,1500 1500 STOP END C C C C SUBROUTINE C c SUBROUTINE SUB 1(DX,DZ1,DZ2,RAM 11,RAM21,RAM12,U12,U11,U21,0,010) RAM=(RAMI1+RAM21)/2.0 Q2=Q10*DZ1 Q3=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) 04=RAM*DZ1*(U21-U11)/DX 0=02+03+04 RETURN END SUBROUTINE SUB2(DX,DZ1,DZ2,RAM01,RAMI 1,RAM21,RAM12,U01,U11, AU12.U21,0) RAMI = (RAM01+RAM11)/2 .0 RAM2=(RAM11+RAM21)/2 .0 02=RAM1*DZ1*(U01-U11)/DX 03=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) Q4=RAM2*DZ1*(U21-U11)/DX 0=04+02+03 RETURN END SUBROUTINE SUB3(DX,DZ1,DZ2,RAM01,RAM 11.RAM12.U01,U11,U12,UFI,0) RAM=(RAM01+RAM11)/2.0 02=RAM*DZ1*(U01-U11)/DX 273 03=DX* (U12-U11) / (DZ2/ (2 .0*RAM12)+DZ l / (2 .0*RAM11) ) 04=2.0*RAM11*DZ1*(UFI -U11)/DX 0=02+03+04 RETURN END C C SUBROUTINE SUB4(DX,DZO,DZ1,DZ2,RAM21,RAM 11,RAM 10,RAM 12,U10,U11,U12 A , U 2 1 , 0 , 0 1 0 ) RAM=(RAM21+RAM11)/2.0 01=DX*(U10-U11)/ (DZ0/(2.0*RAM10)+DZ1/(2.0*RAMI 1)) Q2=Q10*DZ1 Q3=DX*(U12-U11)/ (DZ2/ (2.0*RAM12)+DZ1/ (2.0*RAM11) ) Q4=RAM*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB5(DX,DZO,DZ1,DZ2,RAM01,RAM 11,RAM21,RAM 10,RAM 12,U10,U A11 ,U01 .U12 .U21 ,0 ) RAM 1 = (RAM01 + RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 01=DX* (U10-U11) / (DZ0/ (2 .0*RAM10)+DZ l / (2 .0*RAM11) ) 02=RAM1*DZ1*(U01-U11)/DX 03=DX*(U12-U11) / (DZ2/ (2.0*RAM12)+DZ1/ (2.0*RAM11) ) Q4=RAM2*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB6(DX,DZO,DZ1,DZ2,RAM01,RAMI 1,RAM 10,RAM 12,U10,U11, AU01 ,U12 ,UF I , 0 ) RAM=(RAM01+RAM11)/2.0 Q1=DX*(U10-U11)/ (DZ0/ (2.0*RAM10)+DZ1/ (2.0*RAM11) ) Q2=RAM*DZ1*(U01-U11)/DX 03 = DX*(U12-U11)/ (DZ2/ (2.0*RAM12)+DZ1/ (2.0*RAMI 1)) 04=2.0*RAM11*DZ1*(UFI -U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB7(DX,DZO,DZ1.DZ2,RAM01,RAM 11,RAM12,RAM21,RAM 10, AU10.U11,U01,U12,U21,0 ) RAM 1 = (RAMO1 + RAM11)/2.0 RAM2=(RAM 12+RAM 1 1 ) / 2 . 0 RAM3 = ( RAM2 1+RAM1 D / 2 . 0 01=DX*(U10-U11) / (DZ0/ (2.0*RAM10)+DZ1/ (2.0*RAM11) ) 02=RAM1*DZ1*(U01-U11)/DX 03 = RAM2*DX*(U12-U11 ) * (DZ1 /2 .0+DZ2/3 .O ) / ( (DX /G .0 ) * *2+ (DZ2/3 .0+ A D Z 1 / 2 . 0 ) * * 2 ) 04=RAM3*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB8(DX,DZO,DZ1,DZ2,RAM01,RAMI 1,RAM21.RAM 10,RAMS,U10, 274 AU11,U01.U12.U21.H.O) RAM 1 = (RAM01+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 Q1=DX*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=RAM1*DZ1*(U01-U11)/DX 03 = DX*(U12-U11)/(DZ1/(2.0*RAM11 )+1.0/H+DZ2/(2.0*RAMS)) 04=RAM2*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB9(DX,DZO,DZ1.DZ2.RAM01,RAM 11,RAM 10.RAMS,U10,U11, AU01,U12,UFI,H,0) RAM=(RAM01+RAM11)/2.0 Q1=DX*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=RAM*DZ1*(U01-U11)/DX 03=DX*(U12-U11)/(DZ1/(2.0*RAM11)+1.0/H+DZ2/(2.0*RAMS)) 04=2.0*RAM11*DZ1*(UFI-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB10(DX,DZO,DZ1,DZ2,RAM01,RAM 11,RAM 12,RAM21 , RAM 10, AU10.U11,U01,U12,U21,Q) RAM 1 = (RAM01+RAM11)/2.0 RAM2 = (RAM 12+RAM 11)/2.0 RAM3=(RAM21+RAM11)/2.0 01=DX*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=RAM1*DZ1*(U01-U11)/DX 03 = RAM2*DX*(U12-U11 )*(DZ1/2 . 0+DZ2/3.0)/((DX/6.0)**2+ A(DZ2/3.0+DZ1/2.0)**2) 04 = 30.0*DX*DZ1*RAM3*(U21-U11)/(25.0*DX**2+DZ1 **2) 0=01+02+03+04 RETURN END C C SUBROUTINE SUB 11(DX.DZO.DZ1 ,RAM 10,RAM 11,RAM01,RAMS,U10,U1 1,U01, AUS11.H.Q) RAM 1 = (RAM10+RAM11)/2.0 RAM2=(RAM01+RAM11)/2.0 Q1=DX*RAM1*(U10-U11)*(DZO/2.0+DZ1/3.0)/((DX/6.O)**2+(DZ1/3.0+ ADZO/2.0) * *2) 02 = 30.0*DX*DZ1*RAM2*(U01-U11)/(25.0*DX**2+DZ1 **2 ) 03=(2.0*DX*DZ1*(US11-U11)/S0RT(DZ1**2+DX**2))/(SORT(DZ1 **2+DX**2)/ A( 12.0*RAM11)+1.O/H+S0RT(DZ1**2+DX**2)/(12.0*RAMS)) Q=Q1+Q2+Q3 RETURN END C C SUBROUTINE SUB12(DX,DZ1,DZ2,RAMF,RAM 12,RAM 11,RAM21 , AUF11,U11,U12,U21,H,0) RAM1=(RAM12+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 02 = 2.O*DX*DZ1*(UF11-U11)/SORT(DZ1 **2+DX**2)/(SORT(DZ1 **2 + DX* *2 ) / A( 12.0*RAMF)+1.O/H+S0RT(DZ1**2+DX**2)/(12.0*RAM11)) 03 = RAM1*DX*(U12-U11)*(DZ2/2.0+DZ1/3.0)/((DX/6.0)* *2+(DZ1/3.0+DZ2/ 275 A2.0) * *2 ) 04 = 30.0*DX*DZ1*RAM2*(U21-U11)/(25.0*DX**2+DZ1 **2) 0=02+03+04 RETURN END C C SUBROUTINE SUB13(DX,DZO.DZ1,DZ2,RAMF,RAM01,RAM11,RAM21,RAM 12, AUF10.U11,U01.U12.U21,H,0) RAMI=(RAM01+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 Q1=DX*(UF10-U11)/(DZ0/(2.0*RAMF)+1.O/H+DZ1/(2.0*RAM11)) 02=30.0*DX*DZ1*RAM1*(U01-U11)/(25.0*DX**2+DZ1**2) Q3=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) Q4=RAM2*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB 14(DX,DZO.DZ1,DZ2,RAMF,RAM01,RAMI 1.RAM21,RAM 12, AUF10.U11,U01.U12.U21,H,0) RAMI=(RAM01+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 Q1=DX*(UF10-U11)/(DZO/(2.0*RAMF)+1.O/H+DZ1/(2.0*RAM11)) 02=RAM1*DZ1*(U01-U11)/DX 03=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) Q4=RAM2*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB15(DX,DZO,DZ1,DZ2,RAMF,RAM01,RAMI 1,RAM 12, AUF10.U11,U01,U12,TP,H,0) RAM=(RAM01+RAM11)/2.0 Q1=DX*(UF10-U11)/(DZ0/(2.0*RAMF)+1.O/H+DZ1/(2.0*RAM11)) 02=RAM*DZ1*(U01-U11)/DX 03=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) 04=2.0*RAM11*DZ1*(TP-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB16(DX.DZO,DZ1,DZ2.RAM 10.RAMI 1,RAM01.RAM21,RAM 12 , AU10.U11,U01,U12,U21,0) RAMI=(RAM10+RAM11)/2.0 RAM2 = (RAM01+ RAM11)/2.0 RAM3=(RAM21+RAM11)/2.0 01=RAM1*DX*(U10-U11)*(DZ1/2.0+DZ0/3.0)/((DX/6.0)**2+ A(DZO/3.0+DZ1/2.0)**2) 02=30.0*DX*DZ1*RAM2*(U01-U11)/(25.0*DX**2+DZ1**2) 03=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) Q4=RAM3*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END C C 276 SUBROUTINE SUB 17(DX,DZO,DZ1.DZ2,RAM 12,RAMI 1,RAM21.RAM 10, AU10.U11.U12.U21,0.010) RAMI= (RAMI2+RAM11 )/2 .0 RAM2=(RAM21+RAM11)/2.0 Q1=DX*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) 02=010*DZ1 03=RAM1*DX*(U12-U11)*(DZ1/2.0+DZ2/3.O)/((DX/6.0)**2+ A(DZ2/3.0+DZ1/2.0)**2) 04 = 30.0*DX*DZ1*RAM2*(U21-U11)/(25.0*DX**2+DZ1 **2) 0=01+02+03+04 RETURN C C END SUBROUTINE SUB 18(DX,DZO,DZ1.RAM10,RAMI 1,RAMS,U10,U11,US11,H.Q, AQ10) RAM=(RAM 10+RAM 11)/2.0 Q1=DX*RAM*(U10-U11)*(DZ0/2.0+DZ1/3.0)/((DX/6.0)**2 A+(DZ1/3.0+DZ0/2.0)**2) Q2=010*DZ1 03=(2.0*DX*DZ1*(US11-U11)/SQRT(DZ1 **2+DX**2))/(SORT(DZ1**2+DX**2) A/(12.0*RAM11)+1.O/H+S0RT(DZ1**2+DX**2)/(12.0*RAMS)) 0=01+02+03 RETURN END C C SUBROUTINE SUB 19(OX,DZO,DZ1,DZ2,RAM 10,RAMI 1,RAM21,RAM 12, AU10.U11,U12,U21,0,010) RAM 1 = (RAM10+RAM11)/2.0 RAM2 = (RAM21+RAM11)/2.0 01=RAM1*DX*(U10-U11)*(DZ1/2.0+DZ0/3.0)/((DX/6.0)**2+(DZO/3.0+ ADZ1/2 .0) * *2) 02=010*DZ1 Q3=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) 04 = RAM2*DZ1*(U21-U1 1 )/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB20(DX,DZO,DZ1,RAM21,RAM 11.RAM 10,U10,U11,U21,0.010) RAM=(RAM21+RAM 11)/2.0 Q1=DX*(U10-U11)/(DZO/(2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=Q10*DZ1 Q4=RAM*DZ1*(U21-U11)/DX 0=01+02+04 RETURN END C C SUBROUTINE SUB21(DX.DZO,DZ1,RAM01,RAM 11.RAM21,RAM 10,U10,U11,U01, AU21,0) RAM 1 = (RAM01+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 Q1=DX*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) 02=RAM1*DZ1*(U01-U11)/DX 04=RAM2*DZ1*(U21-U11)/DX 0=01+02+04 RETURN 277 END C C SUBROUTINE SUB22(DX,DZO,DZ1,RAM01,RAM11,RAM10,U10,U11,U01,TP,Q) RAM=(RAM01+RAM11)/2.0 Q1=DX*(U10-U11)/(DZ0/(2.O*RAM1O)+0Z1/(2.O*RAM11)) 02=RAM*DZ1*(U01-U11)/DX 04=2.0*RAM11*DZ1*(TP-U11)/DX 0=01+02+04 RETURN END . C C SUBROUTINE SUB23(DX,DZO,DZ1,DZ2,RAM01,RAM11,RAM 10,RAM12,U10,U11, AU01,U12,TP,0) RAM=(RAM01+RAM11)/2.0 01=DX*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM1 1 )) 02=RAM*DZ1*(U01-U11)/DX 03=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) 04=2.0*RAM*DZ1*(TP-U11)/DX 0=01+02+03+04 RETURN END C C SUBROUTINE SUB24(DX,DZO,DZ1,DZ2,RAM21,RAM 11,RAM10,RAM 12 , U10, U11,U12, AU21,0,010) RAM=(RAM21+RAM11)/2.0 Q1=DX*(U10-U11)/(DZ0/(2.0*RAM 10)+DZ1/(2.0*RAM11)) Q2=Q1O*0Z1 03=DX*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) Q4=RAM*DZ1*(U21-U11)/DX 0=01+02+03+04 RETURN END 278 APPENDIX VI COMPUTER PROGRAM FOR THE CALCULATION OF FLUID PRESSURE IN THE FLUX CHANNEL Q* **************************************************************************** * C DYMANIC PRESSURE ACT ING ON THE SHELL C****************************************************************************** c C PROGRAM F I L E NAME : FL0W44 C DATA F I L E NAME : DATA33 C Q****************************************************************************** c C L I S T OF SYMBOLS C C P PRESSURE D I S T R I B U T I O N C RAM INTEGRATION OF P C ARAM INTEGRATED AVERAGE OF P C BM BENDING MOMENT C MU V I S C O S I T Y OF MOULD FLUX C XHI X-COORDINATE OF I N L E T OF FLUX CHANNEL C YHI Y-COORDINATE OF I N L E T OF F L U X CHANNEL C XHF X-COORDINATE OF O U T L E T OF FLUX CHANNEL C YHF Y-COORDINATE OF O U T L E T OF FLUX CHANNEL C RF DENSITY OF MOULD FLUX C RS DENS ITY OF S T E E L C VS V E L O C I T Y OF SLAB C VM VELOC ITY OF MOULD C 0 FLUX CONSUMPTION C OO R E L A T I V E FLUX-CONSUMPT ION C Q************************************************** **************************** C c I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) REAL MU DIMENSION P ( 3 0 0 0 ) , R A M ( 3 0 0 0 ) , B M ( 3 0 0 0 ) C C R E A D ( 5 , 1 0 0 ) X H I , Y L I , X H F , Y L F R E A D ( 5 , 1 1 0 ) R F . R S . M U , V S . V M 100 F 0 R M A T ( 4 F 1 0 . 7 ) 1 1 0 F O R M A T ( 5 F 1 0 . 5 ) C C W R I T E ( 6 , 2 0 0 ) X H I , Y L I , X H F , Y L F W R I T E ( 6 . 2 1 0 ) R F . R S . M U , V S . V M 2 0 0 FORMAT(1OH HI = , F 1 2 . G / 10H L I = , F 1 2 . 6 C / 10H HF = , F 1 2 . 6 / 10H LF = , F 1 2 . 6 ) 2 1 0 FORMAT(10H RF = , F 1 2 . 6 / 10H RS = . F 1 2 . 6 C / 10H MU = , F 1 2 . 6 / 10H VS = , F 1 2 . 6 C / 10H VM = , F 1 2 . 6 ) C C C****************************************************************************** C SHAPE OF FLUX CHANNEL : X=A*Y+B Q* **************************************************************************** * C C A = ( X H I - X H F ) / ( Y L I - Y L F ) B = ( X H F * Y L I - X H I * Y L F ) / ( Y L I - Y L F ) C C E P L F = - ( 1 / A ) * ( 1 / X H F - 1 / B ) ZELF=- (1 /2 . /A) * (1 / (XHF* *2 ) -1 / (B * *2 ) ) Q=((2*RF-RS)*980*YLF+6*MU*( VM+VS)*EPLF)/(12*MU*ZELF) WRITE(6,330) 0 330 FORMAT(10H 0 = , F12.6) 00=((2*RF-RS)*980*YLF+6*MU*(VM-VS)*EPLF)/(12*MU*ZELF) II=(YLF-YLI+0.001)*1000.0 DO 405 1=1,11 Y=FL0AT(I)/1OOO.0 G=A*Y+B EPY=-(1/A)*(1/G-1/B) ZEY=-1/ (2 *A) * (1 / (G* *2)-1 / (B* *2 ) ) P(I)=(2*RF-RS)*980*Y+6*MU*(VM-VS)*EPY-12*MU*QQ*ZEY 405 CONTINUE RAM(1)=0.0 BM(1)=0.0 DO 600 1=2,11 RAM(I)=RAM(I-1)+P(I)*0.0010 BM(I)=BM(1-1)+P(I)*0.0010*(11-1)/1000 600 CONTINUE WRITE(6,610) II 610 FORMAT(10H II = , 15) WRITE(6,620) RAM(II) 620 FORMAT(IOH RAM(II)- , F15.6) ARAM = RAM(II ) / (YLF-YLI) WRITE(6,630) ARAM 630 FORMAT(10H ARAM = , F15.6) WRITE(6,650) BM(II) 650 FORMAT(10H BM(II) = , F15.6) STOP END 280 APPENDIX VII COMPUTER PROGRAM FOR THE CALCULATION OF THE CHANGE OF MENISCUS SHAPE c ******************************************************** C CHANGE OF MENISCUS SHEPE C****************************************************************************** c C PROGRAM FILE NAME : MENISCUS10 C DATA FILE NAME : DATA33M C C******************************************************************* c c /-< LIST I DF SYMBOLS c p PRESSURE DISTRIBUTION c RAM INTEGRATION OF P c MU VISCOSITY OF MOULD FLUX c XHI COORDINATE OF INLET OF FLUX CHANNEL c YHI COORDINATE OF INLET OF FLUX CHANNEL c XHM COORDINATE OF MIDDLE POINT OF FLUX CHANNEL c YHM COORDINATE OF MIDDLE POINT OF FLUX CHANNEL c XHF COORDINATE OF OUTLET OF FLUX CHANNEL c YHF COORDINATE OF OUTLET OF FLUX CHANNEL c RF DENSITY OF MOULD FLUX c RS DENSITY OF STEEL c VS VELOCITY OF SLAB c VM VELOCITY OF MOULD c 0 FLUX CONSUMPTION c 00 RELATIVE FLUX-CONSUMPTION c SIGMA INTERFACIAL TENSION c CAP CAPILLARY CNSTANT c CAPMODMODIFI ED CAPILLARY CONSTANT c XX(I) CALCULATED SHAPE OF MENISCUS c c YY(I ) CALCULATED SHAPE OF MENISCUS Q****************************************************************************** C c IMPLICIT REAL*8(A-H,0-Z) REAL MU DIMENSION P(30CO),RAM(3000),XX(1000),YY( 1000) C C C****************************************************************************** c C READ AND WRITE DATA C £ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C c READ(5,100) XHI,YLI.XHM,YLM,XHF,YLF READ(5,110) RF,RS,MU,VS,VM 100 FORMAT(6F10.7) 110 F0RMAT(5F10.5) C C C*************************»**************************************************** c c WRITE(6,200) XHI,YLI , XHM, YLM WRITE(6,210) XHF,YLF,RF,RS WRITE(6,220) MU,VS,VM 200 FORMAT(10H HI = , F12.6/ 10H LI = , F12.6/ 10H HM = , 281 C F12.6/ 10H Ltf = .F12.6) 210 F0RMAT(10H HF = . F12.6/ 10H LF = , F12.6/ 10H RF C F12.6/ 10H RS = . F12.6) 220 F0RMATO0H MU = , F12.6/ 10H VS = . F12.G/ 10H VM = , C F12.6) C C C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c C APPLOXIMATED SHAPE OF MENISCUS X=A*Y**2.+B*Y+C C C c A=((XHI-XHM)*(YLI-YLF)-(XHI-XHF)*(YLI-YLM))/((YLI-YLM)*( YLI-YLF) C*(YLM-YLF ) ) B = ( (XHI-XHM)* (YLF* *2 . -YLI * *2 . ) + (XHI-XHF)* (YLI* *2 . -YLM**2 . ) ) / C( (YLI-YLM)*(YLI-YLF)*(YLM-YLF) ) C = XHI-(YLI*YLF*(YLF-YLI)*(XHI-XHM)-YLI*YLM*(YLM-YL I ) C*(XHI-XHF))/ ( (YLI-YLM)*(YLI-YLF)*(YLM-YLF)) C C ^ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 4 ; 4 ; 4 ; * * C c D=4*A*C-B**2 C C c c E=2*A*YLF+B F=A*YLF**2+B*YLF+C C C c c IF(D-O) 310,300,300 C C C C 3CO EPLF=(1/D)*(E/F-B/C)+4*A/D**1.5*(DATAN(E/D**0.5) C-DATAN(B/D**0.5)) ZELF=(1 /2 . /D* *2 . ) * (E * (D+6*A*F) /F * *2 . -B * (D+G*A*C) /C* *2 . ) C+12. *A* *2 . /D* *2 .5 * (DATAN(E/D**0 .5)-DATAN(B/D**0 .5) ) GO TO 320 C C c c 310 EPLF = (1 /D) * (E/F-B/C) + 2 *A/ ( -D) * *1 .5*DL0G((E-(-D)**0.5) C* (B+(-D)* *0 .5 ) / (E+(-D)* *0 .5 ) / (B-(-D)* *0 .5 ) ) ZELF=(1 /2 . /D* *2 . ) * (E * (D+6*A*F ) /F * *2 . -B * (D+6 . *A*C) /C* *2 . ) C+6 . *A* *2 . / ( -D) * *2 .5 *DL0G((E- ( -D) * *0 .5 ) * (B+(-D) * *0 .5 ) C / (E+(-D) * *0 .5 ) / (B- ( -D) * *0 .5 ) ) C c c 320 0=((2.*RF-RS)*980*YLF+6.*MU*(VM+VS)*EPLF)/(12.*MU*ZELF) C C c * * * * * * * * * * * * * * * * » * * * * * * * * * * * * « * * * * » * * * ^ ^ C c WRITE(6,330) 0 330 F0RMATO0H 0 = , F12.6) C C c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * • , * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * » * * . C c 0Q=((2.*RF-RS)*980*YLF+6.*MU*(VM-VS)*EPLF)/(12.*MU*ZEIF) C C c**»******************************************************************. c c IF(D-O) 410.400,400 C C c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * » * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , c c 400 II=(YLF+0.0001)*1000.0 DO 405 1=1 ,11 Y=FL0AT(I)/1000.0 G=2.*A*Y+B H=A*Y**2.+B*Y+C EPY=(1/D)*(G/H-B/C)+4.*A/D**1.5*(DATAN(G/D**0.5)-DATAN(B/D**0.5)) ZEY = ( 1/2 . / D * * 2 . )*(G*(D+6 . * A * H ) / H * * 2 . -B*(D+G'. * A * C ) / C * * 2 ) C+12. *A* *2 . /D* *2 .5 * (DATAN(G/D**0 .5)-DATAN(B/D**0 .5) ) C C c*********************************************************************, C c P(I)=(2.*RF-RS)*980*Y+6*MU*(VM-VS)*EPY-12.*MU*00*ZEY C C C c 405 CONTINUE GO TO 500 C C c * * * * * * * * * » * * * * * » * » * * * * * * * * * * * * * * * * * * * * * * * ^ ^ C c 410 II=(YLF+0.0001)*1000.0 DO 415 1=1 ,11 Y=FLOAT(I)/10O0.0 G=2.*A*Y+B H=A*Y**2.+B*Y+C 283 EPY=(1/D)* (G/H-B/C)+2. *A/(-D)* *1 .5*DL0G((G-(-D)* *0 .5) C* (B+(-D) * *0 .5 ) / (G+(-D) * *0 .5 ) / (B- ( -D) * *0 .5 ) ) ZEY = ( 1 / 2 . / D * * 2 . ) * (G* (D+6 . *A*H) /H* *2 . -B * (D+6 . *A*C) /C* *2 . ) C+6 . *A* *2 . / ( -D) * *2 .5 *DL0G((G- ( -D) * *0 .5 ) * (B+( -D) * *0 .5 ) C / (G+(-D) * *0 .5 ) / (B- ( -D) * *0 .5 ) ) C C c * * * * * * * * * * * * * * » * » * * * * * * * * * * * * * * * » * * * » * * * * * * * * C c P( I ) = (2.*RF-RS)*980*Y+6.*MU*(VM-VS)*EPY-12.*MU*QQ*ZEY C C c * * * * * * * * * » * * * * * * * * * * * * * * » * * * * * * * * * * * * * » * * * * * * * C c 415 CONTINUE 50O RAM(1)=0.0 DO 600 1=2,11 RAM(I)=RAM(I-1)+P(I)*0.0010 GOO CONTINUE WRITE(6,610) II G10 FORMAT(10H II = , 15) WRITE(6,615) ( I . I , P ( I ) , I, RAM(I), 1=1,11) 615 F O R M A T ( I 4 , 3 X , ' P ( ' , I 4 , ' ) = ' , F 16.6,3X, 'RAM(',14, ' ) = ' ,F1G.6) C C READ(5,1CO0) SIGMA,YMIN.H 1000 F0RMAT(3F8.5) CAP=2.*SIGMA/(RS-RF)/980. IF(RAM(11)-O. ) 1010,1020,1020 1010 YMAX=0.0O0 CAPM0D = 2.*SIGMA/((RS-RF)*980.-2.*RAM(II ) /YLF**2) GO TO 1030 1020 YMAX=(CAP*(1.+RAM(11)/SIGMA))**0.5 CAPM0D = 2.*SIGMA/((RS-RF)*980.-2.*RAM( 11)/YMAX* *2) 1030 WRITE(6, 1100) SIGMA,YMIN.H 11CO FORMAT( 10H SIGMA = .F12.6/ 10H YMIN = .F12.6/ IOH H = , CF12.6) WRITE(6,1200) CAP,YMAX,CAPMOD 1200 FORMAT( 10H CAP = ,F12.6/ 10H YMAX = .F12.6/ IOH CAPMOD = , CF12.6/ 1H ,6X, 1HX.15X, 1HY, 1H ) C C c * * » » » * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * + * C c YY(1)=YMIN N= 1 1300 XX(N)=-(2.*CAPM0D-YY(N)**2)**0.5 C+(CAPM0D/2.)**0.5*DLOG(((2.*CAPMOD)**0.5 C+(2.*CAPM0D-YY(N)**2)**0.5)/YY(N)) 284 C+0.3768*CAPM0D**0.5 N=N+1 YY(N)=YY(N-1)+H YF=(2.*CAPM0D)**0.5 IF(YF-YY(N)) 1400,1300,1300 1400 M=N-1 WRITE(G,1500) (XX(I) ,YY(I) ,I=1,M) 1500 F0RMAT( 1H , F10.6. F1G.G) STOP END 285 APPENDIX VIII COMPUTER PROGRAM FOR THE CALCULATION OF NONUNIFORM!TY OF SHELL PROFILE IN THE MOULD Q****************************************************************************** C 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 F I N I T E D I F F E R E N C E H E A T T R A N S F E R P R O G R A M C O F S O L I D I F I C A T I O N N E A R O S C I L L A T I O N M A R K S F E A T U R E D T R I A N G U L A R N O D E S Q* **************************************************************************** * c C P R O G R A M F I L E N A M E : TRANSVERSE2 C D A T A F I L E N A M E : DATA555 C Q****************************************************************************** C C L I S T O F S Y M B O L S C C U S T E M P E R A T U R E O F S T E E L C U F T E M P E R A T U R E O F M O U L D F L U X C US2 NEW T E M P E R A T U R E O F S T E E L C UF2 NEW T E M P E R A T U R E O F M O U L D F L U X C DX S I Z E O F N O D E I N X D I R E C T I O N C D Z S I Z E O F N O D E I N Z D I R E C T I O N ( C A S T I N G D I R E C T I O N ) C R A M F T H E R M A L C O N D U C T I V I T Y O F M O U L D F L U X C R A M S T H E R M A L C O N D U C T I V I T Y O F S T E E L C C S S P E C I F I C H E A T O F S T E E L C C F S P E C I F I C H E A T O F M O U L D F L U X C R O U S D E N S I T Y O F S T E E L C R O U F D E N S I T Y O F M O U L D F L U X C L M S S T A T E O F S T E E L C L N S S T A T E O F S T E E L ; L I Q U I D , M U S H Y , S O L I D C N1 N U M B E R O F N O D E I N C U R V E D R E G I O N O F O . M . I N Z D I R E C T I O N C N2 N U M B E R O F N O D E I N F L A T R E G I O N O F O . M . I N Z D I R E C T I O N C M1 N U M B E R O F N O D E O F F L U X L A Y E R I N X D I R E C T I O N C M2 N U M B E R O F N O D E O F S L A B W I D T H I N X D I R E C T I O N - M1 C Z1 L E N G T H O F C U R V E D R E G I O N OF O . M . I N Z D I R E C T I O N C X1 D E P T H O F O S C I L L A T I O N MARK C X2 W I D T H O F S L A B - X1 C D T T I M E S T E P C U F I I N I T I A L T E M P E R A T U R E O F MOULD F U X C U S I I N I T I A L T E M P E R A T U R E O F S T E E L C T L I O S L I Q U I D U S T E M P E R A T U R E O F S T E E L C T S O L S S O L I D U S T E M P E R A T U R E O F S T E E L C T P P O U R I N G T E M P E R A T U R E O F S T E E L C H H E A T T R A N S F E R C O E F F I S I E N T B E T W E E N S T E E L A N D M O U L D F L U X C C F L S P E C I F I C H E A T O F L I Q U I D F L U X C C F S S P E C I F I C H E A T O F S O L I D I F I E D F L U X C C S L S P E C I F I C H E A T O F L I Q U I D S T E E L C C S S S P E C I F I C H E A T O F S O L I D I F I E D S T E E L C R O U F L D E N S I T Y OF L I Q U I D F L U X C R O U S L D E N S I T Y O F L I Q U I D S T E E L C R O U F S D E N S I T Y O F S O L I D I F I E D F L U X C R O U S S D E N S I T Y O F S O L I D I F I E D S T E E L C R A M F L T H E R M A L C O N D U C T I V I T Y O F L I Q U I D F L U X C R A M S L T H E R M A L C O N D U C T I V I T Y O F L I Q U I D S T E E L C R A M F S T H E R M A L C O N D U C T I V I T Y O F S O L I D I F I E D F L U X C R A M S S T H E R M A L C O N D U C T I V I T Y O F S O L I D I F I E D S T E E L C H A L T S L A T E N T H E A T O F S T E E L C Q ( N ) H E A T F L U X D I S T R I B U T I O N A L O N G O S C I L L A T I O N MARK C Q01 H E A T F L U X D I S T R I B U T I O N DOWN MOULD W A L L C X H V T H I C K N E S S O F F L U X L A Y E R T H A T R E D U C E S H E A T F L U X A S H A L F C C Q****************************************************************************** C D I M E N S I O N 286 Q* **************************************************************************** * c c DIMENSION US(60.10),UF(GO.10),US2(60.10),UF2(G0, 10) ,DZ( A10).RAMF(60.10).RAMS(60,10),CS(60.10),CF(GO.10) ,R0US(60 A, 10),R0UF(60,10),LMF(60, 10),LNF(60,10),LMS(G0,10).LNS(60 A,10),Q(10).DX(60) C C c * * » * » * * * * * » * * * * » * » , * • » » * * * * * * • * * * * » . » » * * * » » * * * * * * * * * * * * * * . * * * » * * * * * * * » * * * * * * * » C READ DATA c»**«***************************************** C C READ(5,SOO) N1.N2.M1.M2.Z1,Z2.X1.X2.DT 500 FORMAT(4I4,4F8.4,F10.6) READ(5,501 ) UFI,USI,TLIOF,TLIOS,TSOLF,TSOLS,TP , H 501 F0RMAT(8F10.5) READ(5,502) CFL,CSL,ROUFL,ROUSL,RAMFL.RAMSL,HLATF , HLATS 502 FORMAT(8F10.5) READ(5.503) CFS,CSS.ROUFS.ROUSS,RAMFS,RAMSS 503 FORMAT(6F10.5) C c Q************************************************************************ ****** C WRITE DATA Q****************************************************************************** c c WRITE(6,2000) N1,N2,M1,M2,Z1,Z2,X1,X2,DT 2000 F0RMAT(10H N1 = , I4/10H N2 = , 14 A/10H M1 = . I4/10H M2 = , I4/10H Z1 = , F12.6/ A10H Z2 = , F12.6/10H X1 = , F12.6/10H X2 = . F12.6 A/10H DT = , F12.6) WRITE(6,2001) UFI.USI.TLIOF,TLIOS,TSOLF.TSOLS,TP, H 2001 FORMAT(10H UFI = , F12.6/10H USI = , F12.6/10H TLIOF = , AF12.6/10H TLIOS = , F12.6/10H TSOLF = . F12.6/10H TSOLS = , AF12.6/10H TP = , F12.6/10H H = , F12.6) WRITE(6,2002) CFL,CSL,ROUFL,ROUSL.RAMFL,RAMSL,HLATF,HLATS 2002 FORMAT(10H CFL = , F12.6/10H CSL = . F12.6/10H ROUFL = , AF12.6/10H ROUSL = , F12.6/10H RAMFL = , F12.6/10H RAMSL = , AF12.6/10H HLATF = , F12.6/10H HLATS = , F12.6) WRITE(6,2003) CFS.CSS,ROUFS,ROUSS,RAMFS,RAMSS 2003 FORMATOOH CFS = , F12.6/10H CSS = , F12.6/10H ROUFS = , AF12.6/10H ROUSS = , F12.6/10H RAMFS = . F12.6/10H RAMSS = . AF12.6) C C c * » * * * * * * * * * * * * * * * * * * * * « * * * * * * * * * * * * * * * * * * * * C OUTPUT TIME SET c * * * * * * * * * * * * * * * « * * * * * * « * * * * « * * * * * * * * * * * * * * C c END 1=0.30 END 1A = END1+DT END2=0.60 END2A=END2+DT END3=0.90 END3A=END3+DT END4=1.20 END4A=END4+DT END5=1.50 END5A=END5+DT ENDG=1.80 END6A=END6+DT END7=2.10 END7A=END7+DT END8=2.40 END8A=END8+DT END9=2.70 END9A=END9+DT END10=3.0 END10A=END10+OT END 11=4 .0 END11A=END11+DT END12=5.0 END 12A = END12+DT END13=6.0 END 13A = END13+DT END14=7.0 END14A=END14+DT END 15=8.0 END15A=END15+DT END16=9.0 END16A=END16+DT END17=10.0 END17A=END17+DT END18=1 1 .0 END 18A = END 18+DT END19=12.0 END 19A = END19+DT END20=13.0 END20A=END20+DT END21=14.0 END21A=END21+DT END22=15.0 END22A=END22+DT END23=16.0 END23A=END23+DT END24=17.0 END24A=END24+DT END25=18.0 END25A=END25+DT END2G=19.0 END26A=END26+DT END27=20.0 END27A=END27+DT END28=21.0 END28A=END28+DT END29=22.0 END29A=END29+DT END30=23.0 END30A=END30+DT END31=24.0 END31A=END31+DT END32=26.0 END32A=END32+DT END33=28.0 END33A=END33+DT 288 END34=30.0 END34A=END34+DT END35=32.0 END35A=END35+DT END36=34.O END3GA=END36+DT END37=3G-O END37A=END37+DT END38=38.0 END38A=END38+DT END39=40.0 END39A=END39+DT END40=42.0 END40A=END40+DT END41=44.0 END41A=END41+DT END42=46.0 END42A=END42+DT END43=48.0 C C C******************************************** C DIMENSION OF SYSTEM C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c c M21=M1+M2 N10=N1-1 N11=N1+1 N21=N1+N2 M10=M1-1 M11=M1+1 T=0.0 IF(M1-N1) 800,801,800 800 WRITE(6,850) 850 FORMAT(1H0.20H M1=N1 NOT EQUAL ) GO TO 1500 801 CONTINUE DO 200 M=1,M1 200 DX(M)=X1/FL0AT(M1) DO 202 N=1,N1 202 DZ(N)=SQRT(Z1**2/X1*(X1-DX(1)*FL0AT(N-1))) A-SQRT(Z1**2/X1*(X1-DX(1)*FLOAT(N))) DO 201 M=M1,M21 201 DX(M)=X2/FL0AT(M2) DO 203 N=N11,N21 203 DZ(N)=Z2/FL0AT(N2) C C C INITIAL CONDITION ETC. C C DO 205 N=1,N21 DO 205 M=1,M21 UF(M,N)=UFI US(M,N)=USI LMF(M,N)=0 2 8 9 LNF(M,N)=0 LMS(M,N)=0 LNS(M,N)=0 CF(M.N)=CFL CS(M,N)=CSL ROUF(M,N)=ROUFL R0US(M,N)=R0U5L RAMF(M.N)=RAMFL 205 RAMS(M,N)=RAMSL CFF=HLATF/(TLIQF-TSOLF)+CFL CSF=HLATS/(TLI0S-TS0LS)+CSL R0UFF=(R0UFL+R0UFS)/2.O RAMFF=(RAMFL+RAMFS)/2.0 R0USF=(R0USL+R0USS)/2.0 RAMSF=(RAMSL+RAMSS)/2.0 GXF=RAMFL*DT/(CFL*R0UFL*DX(1)**2) GXS=RAMSL*DT/(CSL*ROUSL*DX(1)**2) GZ1F=RAMFL*DT/(CFL*R0UFL*DZ(1)**2) GZ2F=RAMFL*DT/(CFL*R0UFL*DZ(N11)**2) GZ2S=RAMSL*DT/(CSL*R0USL*DZ(1)**2) GZ3=RAMSL*DT/(CSL*R0USL*DZ(N11)**2) IF(GXF-0.5) 560,570,570 560 IF(GXS-0.5) 561,570,570 561 IF(GZ1F-0.5) 563,570,570 563 IF(GZ2S-0.5) 564,570,570 564 IF(GZ3-0.5) 565,570,570 570 WRITE(6,575) 575 F0RMAT(1H0.46HG IS MORE THAN 0.5 , CANNOT PERFORM CALCULATION) GO TO 1500 565 CONTINUE GO TO 710 599 T=T+DT C C Q* ************************************************** C PROFILE OF HEAT FLUX (NOTE ; M1=N1=3) Q****************************************************************************** C c 710 CONTINUE XHV=0.05 DZZ1=DZ(1)*(XHV/(XHV+2.5*DX(1))) DZZ2=DZ(2)*(XHV/(XHV+1.5*DX(1))) DZZ3=DZ(3)*(XHV/(XHV+0.5*DX(1))) IF(T-12.0) 701,701,705 701 001 = -1 .0* (60 .0-5 . 5786*T+0.46889*T**2-0.01583*T**3) 002=001*(Z1+Z2)/(DZZ1+DZZ2+DZZ3+Z2) DO 702 N=1,N1 702 Q(N)=(XHV/(XHV+(3.5-N)*DX(1)))*002 DO 703 N=N11,N21 703 0(N)=002 GO TO 7 20 705 QQ1=-1.0*(60.0-4.0389*T+0.19717*T**2 A-4.4139E-03*T**3+3.5889E-05*T**4) 002 = 001 *(Z1 + Z2 )/(DZZ1+DZZ2+DZZ3+Z2) DO 706 N=1,N1 706 0(N)=(XHV/(XHV+(3.5-N)*DX(1)))*002 DO 707 N=N11,N21 707 0(N)=002 720 CONTINUE 290 c c Q******************************************************************************* C H E A T F L U X C A L C U L A T I O N I N E A C H T Y P E O F N O D E Q****************************************************************************** c c c 1 C O N T I N U E DO 9 9 9 N=1,N21 DO 9 9 9 M=1,M21 3 8 I F ( N - 1 ) 3 9 , 3 9 , 4 0 3 9 I F ( M - 1 ) 2 6 , 2 6 , 2 7 2 6 C A L L S U B 1 ( D X ( M ) , D Z ( N ) , D Z ( N + 1 ) , R A M F ( M + 1 , N ) . R A M F ( M . N ) , A R A M F ( M , N + 1 ) , U F ( M , N ) , U F ( M , N + 1 ) , U F ( M + 1 , N ) , O F , Q ( N ) ) U 5 ( M , N ) = 0 GO T O 9 9 1 2 7 I F ( M - M 1 ) 3 1 , 3 2 , 3 2 31 C A L L S U B 1 0 A ( D X ( M ) , D Z ( N ) , D Z ( N + 1 ) , R A M F ( M - 1 , N ) , R A M F ( M , N ) , R A M F ( M , N A + 1 ) , R A M F ( M + 1 , N ) , U F ( M , N ) , U F ( M - 1 , N ) , U F ( M , N + 1 ) , A U F ( M + 1 , N ) , 0 F ) U S ( M , N ) = 0 G O T O 9 9 1 3 2 I F ( M - M 1 1 ) 3 3 , 3 4 , 3 5 3 3 C A L L S U B 1 1 A ( D X ( M ) , D Z ( N ) , R A M F ( M , N ) , R A M F ( M - 1 , N ) , R A M S A ( M , N ) , U F ( M , N ) , U F ( M - 1 , N ) . U S ( M . N ) , H , O F ) C A L L S U B 1 2 A ( D X ( M ) , D X ( M + 1 ) , D Z ( N ) , D Z ( N + 1 ) . R A M F ( M . N ) , R A M S ( M , N + 1 ) , A R A M S ( M . N ) , R A M S ( M + 1 , N ) , U F ( M , N ) , U S ( M , N ) , U S ( M , N + 1 ) , U S ( M + 1 , N ) , H , O S ) GO T O 9 9 0 3 4 C A L L S U B 1 3 A ( D X ( M - 1 ) , D X ( M ) , D Z ( N ) , D Z ( N + 1 ) , R A M S ( M - 1 , N ) , R A M S ( M A . N ) , R A M S ( M + 1 , N ) , R A M S ( M , N + 1 ) , U S ( M , N ) , U S ( M - 1 , N ) , U S ( M , N + 1 ) , A U S ( M + 1 , N ) , H , O S ) U F ( M , N ) = 0 GO T O 9 9 2 3 5 I F ( M - M 2 1 ) 6 2 , 6 3 . 6 3 6 2 C A L L S U B 1 4 A ( D X ( M ) , D Z ( N ) , D Z ( N + 1 ) , R A M S ( M - 1 , N ) , R A M S ( M A . N ) . R A M S ( M + 1 , N ) , R A M S ( M , N + 1 ) . U S ( M . N ) . U S ( M - 1 , N ) , U S ( M , N + 1 ) , A U S ( M + 1 , N ) , H , O S ) U F ( M , N ) = 0 GO T O 9 9 2 6 3 C A L L S U B 1 5 A ( D X ( M ) , D Z ( N ) , D Z ( N + 1 ) , R A M S ( M - 1 , N ) , R A M S ( M A , N ) , R A M S ( M , N + 1 ) . U S ( M . N ) , U S ( M - 1 , N ) , U S ( M , N + 1 ) , T P , H , O S ) U F ( M . N ) = 0 G O T O 9 9 2 C Q* ***************************************************************************** c 4 0 I F ( N - N 1 0 ) 7 0 . 7 0 , 8 0 7 0 I F ( M - 1 ) 7 1 . 7 1 , 7 2 7 1 C A L L S U B 1 7 ( D X ( M ) , D Z ( N - 1 ) , D Z ( N ) , D Z ( N + 1 ) , R A M F ( M , N + 1 ) , R A M F ( M . N ) , A R A M F ( M + 1 , N ) . R A M F ( M . N - 1 ) , U F ( M , N - 1 ) , U F ( M , N ) , U F ( M , N + 1 ) , U F (M+ 1 , N ) , A Q F , Q ( N ) ) U S ( M , N ) = 0 G O T O 9 9 1 7 2 I F ( M - 3 ) 7 3 , 7 4 , 7 5 7 3 C A L L S U B 1 1 ( D X ( M ) , D Z ( N - 1 ) , D Z ( N ) , R A M F ( M . N - 1 ) , R A M F ( M . N ) , R A M F ( M - 1 , N ) , A R A M S ( M . N ) , U F ( M , N - 1 ) , U F ( M , N ) , U F ( M - 1 , N ) , U S ( M , N ) , H , Q F ) C A L L S U B 1 2 A ( D X ( M ) , D X ( M + 1 ) , D Z ( N ) , D Z ( N + 1 ) . R A M F ( M . N ) , A R A M S ( M , N + 1 ) . R A M S ( M . N ) , A R A M S ( M + 1 , N ) , U F ( M , N ) , U S ( M , N ) , U S ( M , N + 1 ) , U S ( M + 1 , N ) , H . Q S ) 291 GO TO 990 74 CALL SUB16A(DX(M),DX(M+1),DZ(N-1),DZ(N),DZ(N+1).RAMS(M.N-1), ARAMS(M,N), ARAMS(M-1,N).RAMS(M+1,N),RAMS(M,N+1).US(M.N-1),US(M,N),US(M-1,N), AUS(M,N+1).US(M+1,N),OS) UF(M,N)=0 GO TO 992 75 IF(M-M21) 76,77,77 76 CALL SUB5A(DX(M-1),DX(M),DX(M+1),DZ(N-1),DZ(N),DZ(N+1), ARAMS(M-1,N),RAMS(M.N), ARAMS(M+1,N),RAMS(M,N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),US(M+1,N).OS) UF(M,N)=0 GO TO 992 77 CALL SUB23A(DX(M),DZ(N-1),DZ(N).DZ(N+1),RAMS(M-1,N),RAMS(M,N). ARAMS(M.N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N) , AUS(M,N+1),TP,OS) UF(M,N)=0 GO TO 992 C Q****************************************************************************** C 80 IF(N-N11) 81,90,100 81 IF(M-2) 82,83,84 82 CALL SUB18(DX(M),DZ(N-1),DZ(N),RAMF(M,N-1),RAMF(M,N),RAMS(M,N), AUF(M.N-1),UF(M,N),US(M.N),H,QF,Q(N)) CALL SUB12A(DX(M),DX(M+1),DZ(N),DZ(N+1),RAMF(M.N).RAMS(M,N+1), ARAMS(M.N), ARAMS(M+1,N),UF(M,N).US(M.N),US(M,N+1),US(M+1,N),H,QS) GO TO 990 83 CALL SUB16A(DX(M),DX(M+1),DZ(N-1),DZ(N),DZ(N+1),RAMS(M,N-1), ARAMS(M,N), ARAMS(M-1,N),RAMS(M+1,N),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),US(M+1,N).OS) UF(M,N)=0 GO TO 992 84 IF(M-M21) 85,86,86 85 CALL SUB5A(DX(M-1),DX(M),DX(M+1),DZ(N-1),DZ(N),DZ(N+1), ARAMS(M-1,N),RAMS(M,N), ARAMS(M+1,N),RAMS(M,N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),US(M+1,N),0S) UF(M,N)=0 GO TO 992 86 CALL SUB23A(DX(M),DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N),RAMS(M,N), ARAMS(M.N-1),RAMS(M,N+1).US(M.N-1),US(M,N),US(M-1,N), AUS(M,N+1),TP,OS) UF(M,N)=0 GO TO 992 C Q****************************************************************************** c 90 IF(M-1 ) 91,91,92 91 CALL SUB19(DX(M),DZ(N-1),DZ(N),DZ(N+1),RAMS(M,N-1),RAMS(M,N), ARAMS(M+1,N),RAMS(M,N+1),US(M,N-1),US(M,N),US(M,N+1),US(M+1,N), AOS,0(N)) UF(M.N)=0 GO TO 992 92 IF(M-M21) 93,94,94 93 CALL SUB5A(DX(M-1),DX(M),DX(M+1),DZ(N-1),DZ(N),DZ(N+1), ARAMS(M-1, N) ,RAMS(M,N), 292 ARAMS(M+1,N),RAMS(M,N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1.N), AUS(M,N+1),US(M+1,N),0S) UF(M.N)=0 GO TO 992 94 CALL SUB23A(DX(M),DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N).RAMS(M.N) . ARAMS(M,N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),TP,QS) UF(M,N)=0 GO TO 992 C Q****************************************************************************** c 100 IF(N-N21) 101,110,110 101 IF(M-1) 102,102,103 102 CALL SUB24(DX(M),DZ(N-1),DZ(N),DZ(N+1),RAMS(M+1,N),RAMS(M,N),RAMS A(M,N-1),RAMS(M,N+1).US(M,N-1),US(M,N),US(M,N+1),US( M+1,N),OS,0( N)) UF(M,N)=0 GO TO 992 103 IF(M-M21) 104,105,105 104 CALL SUB5A(DX(M-1),DX(M),DX(M+1),DZ(N-1),DZ(N).DZ(N+1),RAMS(M-1,N) A.RAMS(M.N), ARAMS(M+1,N),RAMS(M,N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),US(M+1,N),0S) UF(M,N)=0 GO TO 992 105 CALL SUB23A(DX(M),DZ(N-1),DZ(N),DZ(N+1),RAMS(M-1,N).RAMS(M.N), ARAMS(M.N-1),RAMS(M,N+1),US(M,N-1),US(M,N),US(M-1,N), AUS(M,N+1),TP,OS) UF(M,N)=0 GO TO 992 C C****************************************************************************** C 110 IF(M-1 ) 1 11,111,120 1 1 1 CALL SUB20(DX(M),DZ(N-1),DZ(N),RAMS(M+1,N),RAMS(M,N),RAMS(M,N-1), AUS(M,N-1),US(M,N),US(M+1,N),QS,Q(N)) UF(M,N)=0 GO TO 992 120 IF(M-M21) 121,122,122 121 CALL SUB21A(DX(M-1),DX(M),DX(M+1),DZ(N-1),DZ(N),RAMS(M-1,N), ARAMS(M,N).RAMS(M+1,N), ARAMS(M,N-1).US(M.N-I).US(M.N),US(M-1,N),US(M+1,N).OS) UF(M.N)=0 GO TO 992 122 CALL SUB22A(DX(M),DZ(N-1 ) ,DZ(N).RAMS(M-1,N),RAMS(M.N),RAMS (M.N-1), AUS(M,N-1),US(M,N),US(M-1,N).TP,OS) UF(M,N)=0 GO TO 992 C C Q****************************************************************************** C NEW TEMPERATURE CALCULATION Q* ***************************************************************************** 990 UF2(M,N)=UF(M,N)+2.O*DT*OF/(CF(M,N)*R0UF(M,N)*DX(M)*DZ(N)) US2(M,N)=US(M,N) + 2.0*DT*OS/(CS(M,N)*ROUS(M,N)*DX (M)*DZ (N)) GO TO 999 991 UF2(M,N)=UF(M,N)+DT*0F/(CF(M,N)*R0UF(M,N)*DX(M)*DZ(N)) US2(M,N)=US(M,N) GO TO 999 992 US2(M,N)=US(M,N)+DT*0S/(CS(M,N)*R0US(M,N)*DX(M)*DZ(N)) 2 9 3 UF2(M,N)=UF(M,N) GO TO 999 999 CONTINUE C C 0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C TEMPERATURE CHANGE c c DO 150 N=1,N21 DO 150 M=1,M21 UF(M,N)=UF2(M,N) 150 US(M,N)=US2(M,N) DO 175 N=1,N1 DO 175 M=1,M21 IF(LNF(M, N ) ) 175,199,175 199 IF(UF(M,N).GT.TLIQF) GO TO 175 IF(LMF(M,N) ) 173,174,173 174 UF(M,N)=TLIQF-CFL*(TLIQF-UF(M,N))/CFF CF(M,N)=CFF ROUF(M,N)=ROUFF RAMF(M,N)=RAMFF LMF(M,N)=1 173 IF(UF(M,N).GT.TSOLF) GO TO 175 ROUF(M.N)=ROUFS CF(M,N)=CFS RAMF(M.N)=RAMFS LNF(M,N)=1 175 CONTINUE DO 275 N=1,N21 DO 275 M=1,M21 IF(LNS(M,N)) 275,299,275 299 IF(US(M,N).GT.TLIOS) GO TO 275 IF(LMS(M,N)) 273.274.273 274 US(M,N)=TLI0S-CSL*(TLI0S-US(M.N))/C5F CS(M,N)=CSF ROUS(M,N)=ROUSF RAMS(M,N)=RAMSF LMS(M,N)=1 273 IF(US(M,N).GT.TSOLS) GO TO 275 ROUS(M,N)=ROUSS CS(M,N)=CSS RAMS(M.N)=RAMSS LNS(M,N)=1 275 CONTINUE C C 0* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C OUTPUT ^ * * * * * * * * * * * * * * * * * * * * * * 4 c * * * * * * * A * 4 ; 4 : * * : t : * * * * * * * * I t r « * * 4 - * t * * * * * * * * * * * * * * * * * * * * * * * * * * C IF(T-END1) 599,1000,9001 9001 IF(T-END1A) 1000.9002,9002 9002 IF(T-END2) 599,1000,9003 9003 IF(T-END2A) 1000,9004,9004 9004 IF(T-END3) 599,1000,9005 9005 IF(T-END3A) 1000,9006,9006 9006 IF(T-END4) 599,1000,9007 9007 IF(T-END4A) 1000,9008,9008 9008 IF 9009 IF 9010 IF 9011 IF 9012 IF 9013 IF 9014 IF 9015 IF 9016 IF 9017 IF 9018 IF 9019 IF 9020 IF 9021 IF 9022 IF 9023 IF 9024 IF 9025 IF 9026 IF 9027 IF 9028 IF 9029 IF 9030 IF 9031 IF 9032 IF 9033 IF 9034 IF 9035 IF 9036 IF 9037 IF 9038 IF 9039 IF 9040 IF 9041 IF 9042 IF 9043 IF 9044 IF 9045 IF 9046 IF 9047 IF 9048 IF 9049 IF 9050 IF 9051 IF 9052 IF 9053 IF 9054 IF 9055 IF 9056 IF 9057 IF 9058 IF 9059 IF 9060 IF 9061 IF 9062 IF 9063 IF 9064 IF 9065 IF 9066 IF 9067 IF T-END5) 599.1000,9009 T-END5A) 1000,9010,9010 T-END6) 599,1000,9011 T-END6A) 1000,9012.9012 T-END7) 599,1000,9013 T-END7A) 1000.9014,9014 T-END8) 599,1000,9015 T-END8A) 1000.9016,9016 T-END9) 599,1000,9017 T-END9A) 1000.9018.9018 T-END 10) 599.1000,9019 T-END10A T-END11) T-END11A T-END12) T-END12A T-END13) T-END13A T-END14) T-END14A T-END15) T-END15A T-END16) T-END16A T-END17) T-END17A T-END18) END18A END 19 ) END19A END20) END20A END21) END21A END22) T-END22A -END23) -END23A -END24) -END24A -END25) T-END25A T-END26) T-END26A T-END27) T-END27A T-END28) T-END28A T-END29) T-END29A T-END30) T-END30A T-END31) T-END31A T-END32) T-END32A T-END33) T-END33A T-END34) T-END34A 1000,9020,9020 599,1000,9021 1000,9022,9022 599,1000,9023 1000,9024,9024 599,1000,9025 1000,9026,9026 599,1000,9027 1000,9028,9028 599.1000,9029 1000,9030,9030 599,1000,9031 1000,9032,9032 599,1000,9033 1000.9034,9034 599,1000,9035 1000,9036,9036 599,1000,9037 1000,9038,9038 599,1000,9039 1000,9040,9040 599,1000,904 1 1000.9042,9042 599,1000,9043 1000,9044,9044 599,1000,9045 1000,9046,9046 599,1000,9047 1000,9048,9048 599.1000,9049 1000,9050,9050 599,10O0.9051 1000,9052,9052 599,1000,9053 1000,9054,9054 599,1000,9055 1000,9056,9056 599,1000,9057 1000,9058,9058 599,1000,9059 1000,9060,9060 599,1000,9061 1000,9062,9062 599.1000,9063 1000,9064,9064 599,1000,9065 1000,9066,9066 599,1000,9067 1000,9068,9068 295 9068 IF(T-END35) 599,1OO0,9069 9069 IF(T-END35A) 1000.9070,9070 9070 IF(T-END36) 599,1000,9071 9071 IF(T-END36A ) 1000,9072.9072 9072 IF(T-END37) 599,1000,9073 9073 IF(T-END37A) 1000,9074,9074 9074 IF(T-END38) 599,1000,9075 9075 IF(T-END38A ) 1000,9076,9076 9076 IF(T-END39) 599,1000,9077 9077 IF(T-END39A ) 1000,9078,9078 9078 IF(T-END40) 599,1000,9079 9079 IF(T-END40A) 1000,9080,9080 9080 IF(T-END41) 599,1000,9081 9081 IF(T-END41A ) 1000,9082,9082 9082 IF(T-END42) 599,1000,9083 9083 IF(T-END42A ) 1000,9084,9084 9084 IF(T-END43) 599,1000,1000 1O00 CONTINUE WRITE(6,300) T ,0 (1 ) ,0 (2 ) .0 (3 ) ,0 (4 ) .0 (5 ) ,0 (6 ) 300 F0RMAT(10H TIME =,F10.7/10H 0(1) =,F10.5/10H 0(2) AF10.5/10H 0(3) =,F10.5/10H 0(4) =,F10.5/10H 0(5) AF10.5/10H 0(6) =.F10.5) WRITE(6,301) 301 FORMAT(5H U-1 ,5H U-2 ,5H U-3 ,5H U-4 ,5H U-5 ,5H U-6 ,5H U-7 A,5H U-8 ,5H U-9 ,5H U-10.5H U-11.5H U-12.5H U-13.5H U-14.5H U-15,5 AH U-16.5H U-17.5H U-18.5H U-19.5H U-20.5H U-21.5H U-22.5H U-23) DO 350 N=1,6 WRITE(6,3350) (UF(M,N).M=1.23) 3350 FORMAT(23F5.0) 350 CONTINUE WRITE(6,302) 302 FORMAT(5H U-1 ,5H U-2 ,5H U-3 ,5H U-4 ,5H U-5 .5H U-G ,5H U-7 A,5H U-8 ,5H U-9 ,5H U-10.5H U-11.5H U-12.5H U-13.5H U-14.5H U-15,5 AH U-16.5H U-17.5H U-18.5H U-19.5H U-20.5H U-21.5H U-22.5H U-23) DO 351 0=1,6 WRITE(6,3510) (US(M,J),M=1,23) 3510 FORMAT(23F5.O) 351 CONTINUE IF(T-END43) 599,1500,1500 1500 STOP END C C ' C Q****************************************************************************** C SUBROUTINE C****************************************************************************** C c SUBROUTINE SUB 1(DX1,DZ1,DZ2,RAM21,RAMI 1,RAM12,U11,U12,U21,0,010) RAM=(RAM 11+RAM21)/2.0 Q2=Q10*DZ1 03 = DX1*(U12-U1 1 )/(DZ2/(2.0*RAM12)+DZ1/ ( 2.0*RAM1 1 )) 04=RAM*DZ1*(U21-U11)/DX1 0=02+03+04 RETURN END C C SUBROUTINE SUB5A(DXO,DX1,DX2,DZ0,DZ1,DZ2,RAM01,RAM 11,RAM21,RAM 10, 296 ARAM12.U10.U A1 1 ,U01 .U12.U21 ,0) RAMI=(RAM01+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 Q1=DX1*(U10-U11)/(DZO/(2.0*RAM10)+DZ1/(2.0*RAM1 1 )) 02=RAM1*DZ1*(U01-U11)/(DX0/2+DX1/2) 03=DX1*(U12-U11)/(DZ2/(2.0*RAM12)+DZl/(2.0*RAM1 1 )) Q4=RAM2*DZ1*(U21-U11)/(DX1/2+DX2/2) 0=01+02+03+04 RETURN END C C SUBROUTINE SUB10A(DX1,DZ1,DZ2,RAM01 ,RAM11,RAM12 , RAM2 1 , AU11,U01.U12.U21,0) RAMI = (RAM01+RAM11 ) /2.0 RAM2=(RAM12 + RAM1 1 ) /2.0 RAM3=(RAM21+RAM11)/2.0 02=RAM1*DZ1*(U01-U11)/DX1 03=RAM2*DX1*(U12-U11)*(DZ1/2.0+DZ2/3.0)/((DX1/6.0)**2+ A(DZ2/3.0+DZ1/2.0)**2) 04 = 30.0*DX1*DZ1*RAM3*(U21-U1 1 )/(25 .0*DX1**2+DZ1**2) 0=02+03+04 RETURN END C C SUBROUTINE SUB 11(0X1,DZ0,DZ1.RAM10,RAM11,RAM01,RAMS,U10,U11.U01, AUS11,H.O) RAMI = (RAMI0+RAM1 1 )/2.O RAM2=(RAM01 + RAM1 1 )/2.0 01=DX1*RAM1*(U10-U11)*(DZ0/2.0+DZ 1/3.0)/((DX1/6.0)**2+(DZ1/3.0+ ADZO/2.0)* *2) 02 = 30.0*DX1*DZ1*RAM2*(U01-U11)/(25.0*DX1**2+DZ 1 **2) 03=(2.0*DX1*DZ1*(US11-U11 )/SQRT(DZ 1 **2+DX1**2))/( SORT (DZ 1 **2 + ADX1**2)/ A(12.0*RAM11)+1.0/H+SORT(DZ1**2+DX1**2)/(12.0*RAMS)) 0=01+02+03 RETURN END C C SUBROUTINE SUB 11A(DX1,DZ1,RAM 11,RAM01.RAMS,U1 1 ,U01 , AUS11,H,0) RAM2=(RAM01+RAM11 )/2.0 02 = 30.0*DX1*DZ1*RAM2*(U01-U11 )/(25 . 0*DX1**2+DZ1**2) 03 = (2.0*DX1*DZ1*(US11-U11)/SQRT(DZ 1 **2+DX1**2))/( SORT (DZ1 **2 + ADX1**2)/ A( 12.0*RAM11) + 1.0/H+SORT(DZ1**2+DX1**2)/(12.0*RAMS)) 0=02+03 RETURN END C C SUBROUTINE SUB12A(DX1,DX2,DZ1,DZ2,RAMF,RAM 12,RAMI 1 , RAM21 , AUF11,U11,U12,U2 1,H,0) RAMI = (RAM12 + RAM1 1 )/2.0 RAM2=(RAM21+RAM1 1 )/2.0 02 = 2.0*DX1*DZ1*(UF11-U11)/SORT(DZ 1 **2+DX1 **2)/(SORT (DZ1 * *2 + ADX1**2)/(12.0*RAMF)+1.0/H+SQRT(DZ 1 * *2+DX1 * *2 ) / ( 12 .0*RAMI 1 )) 297 Q3=RAM1*DX1* (U12-U11) * (DZ2/2 .O+DZ1/3 .0 ) / ( (DX1/6 .0 ) * *2+(DZ1/3 .0 A+DZ2/2 .0 ) * *2 ) Q4 = (DZ1*(DX1/3+DX2/2) / ( (DX1/3+DX2/2)* *2+(DZ1/6)* * 2 ) ) A*RAM2*(U21-U11) 0=02+03+04 RETURN END C C SUBROUTINE SUB 13A(DX0,DX1,DZ1,DZ2,RAM01,RAMI 1,RAM21,RAM 12, AU11,UO1,U12,U21,H,0) RAM 1 = (RAM01 + RAM11)/2 .0 RAM2=(RAM21+RAM11)/2.0 02=(DZ1* (DX0/3+DX1/2) / ( (DX0/3+DX1/2) * *2+(DZ1/6) * *2 ) ) A*RAM1*(U01-U11) 03=DX1*(U12-U11)/ (DZ2/ (2.0*RAM12)+DZ1/(2.0*RAM11)) Q4=RAM2*DZ1*(U21-U11)/DX1 0=02+03+04 RETURN END C C SUBROUTINE SUB 14A(DX1.DZ1,DZ2,RAM01.RAM 11,RAM21,RAM 12, AU11,U01,U12,U21,H,0 ) RAM 1 = (RAMO1+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 02=RAM1*DZ1*(U01-U11)/DX1 03=DX1*(U12-U11)/ (DZ2/ (2.0*RAM12)+DZ1/(2.0*RAM11)) Q4=RAM2*DZ1*(U21-U11)/DX1 0=02+03+04 RETURN END C c SUBROUTINE SUB15A(DX1,DZ1,DZ2,RAM01,RAM 11,RAM 12, AU11,U01,U12,TP .H,0 ) RAM=(RAMO1+RAM11)/2.0 02=RAM*DZ1*(U01-U11)/DX1 Q3=DX1*(U12-U11)/ (DZ2/(2.0*RAM12)+DZ1/(2,0*RAM11)) 0=02+03 RETURN END C C SUBROUTINE SUB16A(DX1,DX2,DZ0,DZ1.DZ2,RAM10,RAMI 1.RAM01,RAM21 , ARAM12.U10.U11,U01,U12,U21,0) RAM 1 = (RAM 10+RAM11)/2.0 RAM2=(RAM01+RAM11)/2.0 RAM3=(RAM21+RAM11 )/2 .0 01=RAM1*DX1* (U10-U11)* (DZ1/2.0+DZ0/3.0 ) / ( (DX1/6.0 ) * *2+ A (DZ0 /3 .0+DZ1/2 .0 ) * *2 ) 02 = 30.0*DX1*DZ1*RAM2*(U01-U11)/ (25.0*DX1**2+DZ1 * *2 ) 0 3 = D X 1 « ( U 1 2 - U 1 1 ) / ( D Z 2 / ( 2 . 0 * R A M 1 2 ) + D Z l / ( 2 , 0 * R A M 1 1 ) ) Q4 = RAM3*DZ1*(U21-U11)/(DX1/2+DX2/2 ) 0=01+02+03+04 RETURN END C C 298 SUBROUTINE SUB17(DX1,DZ0,DZ1,DZ2,RAM12,RAM 11,RAM21,RAM 10, AU10 .U11 ,U12 ,U21 ,0 ,010 ) RAMI=(RAM12+RAM11)/2.0 RAM2=(RAM21+RAM11)/2.0 01=DX1* (U10-U11) / (DZ0/ (2 .0*RAM10)+DZ l / (2 .0*RAM11) ) Q2=Q10*DZ1 Q3=RAM1*DX1* (U12-U11)* (DZ1/2.0+DZ2/3.O) / ( (DX1/6.O)**2+ A ( D Z 2 / 3 . 0 + D Z 1 / 2 . 0 ) * * 2 ) 04 = 30.0*DX1*DZ1*RAM2*(U21-U11) / (25.0*DX1 **2+DZ1 * *2) 0=01+02+03+04 RETURN C C END SUBROUTINE SUB 18(DX1,DZO,DZ1.RAM10,RAMI 1,RAMS,U10,U11,US11,H,0. A010) RAM=(RAM10+RAM11)/2.0 Q1=DX1*RAM*(U10-U11)* (DZO/2.O+DZ1/3.O) / ( (DX1/6.0 ) * *2 A+ (DZ1/3 .0+DZ0/2 .0 ) * *2 ) Q2=Q10*DZ1 Q3=(2 .0*DX1*DZ1*(US11-U11)/SQRT(DZ1**2+DX1 * * 2 ) ) / ( SORT(DZ1**2+ ADX1**2)/(12.0*RAM11)+1.0/H+SQRT(DZ1 * *2+DX1 * * 2 ) / ( 1 2 . 0 * R A M S ) ) 0=01+02+03 RETURN END C C SUBROUTINE SUB19(DX1,DZ0,DZ1,DZ2,RAM10,RAM 11,RAM21,RAMI 2, AU10 .U11 ,U12 ,U21 ,0 ,010 ) RAM 1 = (RAM 10+RAMI 1 ) / 2 . 0 RAM2=(RAM21+RAM11)/2.0 Q1=RAM1*DX1*(U10-U11 ) * (DZ1 /2 .0+DZO/3 .O ) / ( (DX1 /6 .O ) * *2+(DZ0/3.0+ A D Z 1 / 2 . 0 ) * * 2 ) 02=Q10*DZ1 03=DX1*(U12-U11 )/ (DZ2/(2.0*RAM12)+DZ1/(2.0*RAM1 1 )) 04=RAM2*DZ1*(U21-U11)/DX1 0=01+02+03+04 RETURN END C C SUBROUTINE SUB20(DX1,DZO,DZ1,RAM21,RAM 11,RAM10,U10,U11,U21,0,010) RAM=(RAM21+RAM11)/2.0 Q1=DX1*(U10-U11)/ (DZ0/ (2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=Q10*DZ1 Q4=RAM*DZ1*(U21-U11)/DX1 0=01+02+04 RETURN END C C SUBROUTINE SUB21A(DXO,DX1,DX2,DZ0,DZ1,RAM01,RAM 11,RAM21,RAM 10,U10, AU11,U01,U21,0 ) RAM 1 = (RAM01+RAM11)/2.O RAM2=(RAM21+RAM11)/2.0 01=DX1*(U10-U11)/ (DZ0/(2.0*RAM10)+DZ1/(2.0*RAM1 1 )) Q2=RAM1*DZ1*(U01-U11)/(DX0/2+DX1/2) 04=RAM2*DZ1*(U21-U11)/(DX1/2+DX2/2) 0=01+02+04 RETURN 299 END C C SUBROUTINE SUB22A(DX1,DZO,DZ1,RAM01,RAM 11,RAM 10,U10,U11,U01,TP,0) RAM=(RAM01+RAM11 )/2.0 Q1=DX1*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=RAM*DZ1*(U01-U11)/DX1 Q=Q1+Q2 RETURN END C C SUBROUTINE SUB23A(DX1,DZ0,DZ1,DZ2,RAM01.RAMI 1,RAM 10,RAM 12,U10,U11, AU01,U12,TP,0) RAM=(RAM01+RAM11)/2.0 Q1=DX1*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=RAM*DZ1*(U01-U11)/DX1 03=DX1*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) 0=01+02+03 RETURN END C C SUBROUTINE SUB24(DX1,DZO,DZ 1 .DZ2,RAM21,RAMI 1,RAM 10,RAM 12,U10.U11. AU12.U21,0,010) RAM=(RAM21+RAM11)/2.0 01=DX1*(U10-U11)/(DZ0/(2.0*RAM10)+DZ1/(2.0*RAM11)) Q2=Q10*DZ1 03=DX1*(U12-U11)/(DZ2/(2.0*RAM12)+DZ1/(2.0*RAM11)) 04=RAM*DZ1*(U21-U11)/DX1 0=01+02+03+04 RETURN END 300 F i g . A1 T y p i c a l nodes and t h e i r number of s u b r o u t i n e s i n computer program f o r t h e c a l c u l a t i o n o f t e m p e r a t u r e d i s t r i b u t i o n a t t h e m e n i s c u s 301 It A I 2 A 1 I 0 A / l 3 A . I 4 A . I 5 A 1 7 1 1 / / l 2 A I 6 A f l 2 A I 6 A 1 9 A 2 3 A 2 4 2 0 . 21 A _ 2 2 A F i g . A2 T y p i c a l nodes and t h e i r number of subrou t i n e s i n computer program f o r the c a l c u l a t i o n of nonuniformity of s h e l l p r o f i l e i n the mould. 

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