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Thermal distortion of continuous casting moulds Samarasekera, Indira Vasanti 1980

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THERMAL DISTORTION OF CONTINUOUS CASTING MOULDS by INDIRA VASANTI SAMARASEKERA .Sc. (Hons), Universi ty of Sri Lanka, Peradeniya, 1 M.S. Universi ty o f C a l i f o r n i a , Davis, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Metal lurgical Engineering We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1980 (c) I.V. Samarasekera, 1980 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at 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 , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e 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 the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that 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 not 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 . Department o f | vLgTALL UR &-i CA L I^N 6-f rv&Sgj The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT H e a t - f l o w models based on the f i n i t e d i f f e r e n c e and f i n i t e e l emen t methods have been deve l oped to c a l c u l a t e the s t e a d y - s t a t e t he rma l f i e l d i n b i l l e t and s l a b mou l d s . A t r a n s i e n t h e a t - f l o w a n a l y s i s wh i ch employs an i m p l i c i t a l t e r n a -t i n g - d i r e c t i o n f i n i t e - d i f f e r e n c e method has been f o r m u l a t e d e x p r e s s l y to i n v e s t i g a t e the e f f e c t o f s u r f a c e b o i l i n g on b i l -l e t mould t e m p e r a t u r e s . The models were v a l i d a t e d by compar i ng p r e d i c t i o n s w i t h e x p e r i m e n t a l measurements r e p o r t e d i n the l i t -e r a t u r e . The the rma l d i s t o r t i o n o f b i l l e t moulds have been com-pu ted u s i n g a t h r e e - d i m e n s i o n a l e l a s t o - p l a s t i c f i n i t e - e l e m e n t mode l . I t has been shown t h a t the rma l e x pan s i o n i s the dom i -nant component o f the t o t a l d i s t o r t i o n w h i l e the c o n t r i b u t i o n due to b end i ng a s s o c i a t e d w i t h the t h r ough t h i c k n e s s g r a d i e n t i s r e l a t i v e l y s m a l l . The d i s t o r t e d mould e x h i b i t s an ou tward bu l g e i n the men i s cus r e g i o n , wh i ch has a maximum v a l u e i n the ' 0 . 1 " 0 . 3 mm r a n g e , a p p r o x i m a t e l y 6 cms be low the m e n i s c u s . The peak d i s t o r t i o n i s bounded above and be low by r e g i o n s o f nega -t i v e and p o s i t i v e t a p e r r e s p e c t i v e l y . Measurements o f w a l l movement i n an o p e r a t i n g b i l l e t c a s t e r compare f a v o u r a b l y w i t h the p r e d i c t e d d i s t o r t i o n s . The p e r i o d i c movement o f the mould w a l l o b s e r v ed i n the men i scus r e g i o n has been a t t r i b u t e d to n u c l e a t e b o i l i n g i n the c o o l i n g c h a n n e l . i i A s y s t e m a t i c a n a l y s i s o f the e f f e c t o f o p e r a t i n g v a r i -a b l e s on w a l l t empe r a t u r e s has r e v e a l e d t h a t owing to the t h i c k e r w a l l s , peak s l a b mould t empe r a t u r e s a r e h i g h e r t h an i n b i l l e t mou l d s . I t i s t h e r e f o r e recommended t h a t m a t e r i a l s w i t h enhanced s o f t e n i n g r e s i s t a n c e be employed i n s l a b mou l d s . In c o n t r a s t to t h i s s o f t e n i n g i s u n l i k e l y i n b i l l e t moulds but however owing to h i g h e r c o l d - f a c e t empe r a t u r e s b o i l i n g may o c c u r a t s t a n d a r d f l o w - r a t e s ; wa t e r q u a l i t y must t h e r e f o r e be adequate to a v e r t d e p o s i t i o n o f s c a l e and the a t t e n d a n t h i g h -w a l l t empe r a t u r e s and e x c e s s i v e d i s t o r t i o n . A compa r i s on o f the t he rma l d i s t o r t i o n p r o f i l e s c a l c u l a t e d f o r s i x i n d u s t r i a l b i l l e t moulds has shown t h a t mould c o n s t r a i n t s and w a l l t h i c k -ness have a p r o f ound i n f l u e n c e on permanent d i s t o r t i o n . Lower -i n g the men i s cus l e v e l o r m o d i f y i n g the c o n s t r a i n t s so as to m i n i m i z e the r e s t r a i n t s to t he rma l e x p a n s i o n t end to r educe the p l a s t i c s t r a i n and c o u l d be b e n e f i c i a l f rom the s t a n d p o i n t o f mould l i f e . A mechanism has been p roposed r e l a t i n g l o n g i t u d i n a l c o r -ne r c r a c k i n g and r h o m b o i d i t y i n b i l l e t s to an i n t e r m i t t e n t b o i l i n g c y c l e i n the mou ld . I t has been demons t r a t ed t h a t i f b o i l i n g on two a d j a c e n t f a c e s o f the mould i s o u t - o f - p h a s e w i t h b o i l i n g on the o t h e r two , unev°en c o o l i n g and r h o m b o i d i t y wou ld r e s u l t . Th i s can l e a d to c o r n e r c r a c k s by h o t - t e a r i n g i n i n t e r d e n d r i t i c r e g i o n s near the s o l i d i f i c a t i o n f r o n t . i i i TABLE OF CONTENTS Page. A b s t r a c t i i T ab l e o f Con t en t s i v L i s t o f Tab l e s x L i s t o f F i g u r e s x i i i A cknow ledgemen t s . x x i i i Chap t e r 1.0 INTRODUCTION . .' 1 2 .0 REVIEW OF PAST WORK AND OBJECTIVES OF ' RESEARCH PROJECT 8 2.1 Mechanism o f H e a t - T r a n s f e r and Gap Fo rma t i on 8 2 .2 L o n g i t u d i n a l H e a t - F l u x P r o f i l e s 12 2 .2 .1 The I n f l u e n c e o f Mou ld d e s i g n and Geometry on Heat F low 17 . 2 . 2 . 2 I n f l u e n c e o f O p e r a t i n g V a r i a b l e s . on Heat T r a n s f e r 22 2 . 2 . 3 I n f l u e n c e o f S t e e l C o m p o s i t i o n on Heat T r a n s f e r 2 8 2 .3 Ma t hema t i c a l M o d e l l i n g o f Heat F low i n Moul ds 3 0 2.4 D i s t o r t i o n o f Moulds 2.5 I n f l u e n c e o f Mould C o n d i t i o n s on Q u a l i t y 38 2 .5 .1 Mould R e l a t e d S u r f a c e D e f e c t s and C racks i n C o n t i n u o u s l y Ca s t S l a b s . 3 9 2 . 5 . 2 Mould R e l a t e d Q u a l i t y P rob lems i n B i l l e t s and Blooms 41 i v Chapter Page 2.6 Objec t ives and Plan of Research P r o j e c t . . 44 3.0 THERMAL FIELDS IN CONTINUOUS CASTING MOULDS . . 47 3.1 Mathematical Model l ing of Heat Flow in Moul ds 47 3.1.1 C h a r a c t e r i z a t i o n of the Hot-Face Heat-F lux P r o f i l e 48 3.1 .2 Heat -Trans fe r C o e f f i c i e n t at the Cool ing-Water I n t e r f a c e 48 3 .1 .3 Development o f Heat-Flow Models . . 60 3 .1 .3 .1 Assumptions 60 3 .1 .3 .2 Heat-Flow Equations and Boundary Cond i t ions 64 3.1.4 Numerical S o l u t i o n Techniques . . . . 67 3 .1 .4 .1 C a l c u l a t i o n o f the Steady State Temperature Distr ibution i n Moulds by the F i n i t e -D i f fe rence Method 67 3 .1 .4 .2 C a l c u l a t i o n of the Steady State Temperature D i s t r i -bu t i on in Moulds by the F in i t e -E lemen t Method . . . . 83 3 .1 .4 .3 An I m p l i c i t F i n i t e - D i f -ference Technique f o r C a l c u l a t i n g the T r a n s i e n t Temperature D i s t r i b u t i o n in B i l l e t Moulds . . . 98 3.1.5 V a l i d a t i o n o f Heat-Flow Models by Comparison w i t h Experimental Data. 100 3.2 P red ic ted Thermal F ie lds in Slab and B i l l e t Moulds 105 v Chapter Page 4.0 MOULD DISTORTION 122 4.1 I n t r o d u c t i o n . 1 2 2 4.2 Mechanical Proper ty Data 1 2 4 4.3 Mould Cons t ra in ts V. 1 2 5 4.4 The Development o f Mathematical Models . , to Ca lcu la te Mould D i s t o r t i o n 1 2 6 4 . 4 . 1 Assumptions 1 2 7 4.4 .2 E l a s t i c D e f l e c t i o n of the Mould From the P la te Bending Theory . . . . I 2 8 4 . 4 . 3 C a l c u l a t i o n of Average Mould Wall Movement Due to Thermal Expansion of a Transverse S l i c e I 3 8 4 .4 .4 Three-dimensional El a s t o - p l a s t i c Stress Ana lys is of the Mould ™5 4 . 4 . 4 . 1 Mathematical D e s c r i p t i o n of P l a s t i c Flow ' 4 7 4 . 4 . 4 . 2 Numerical S o l u t i o n Techniques 4 . 4 . 4 . 3 A Mathematical Model f o r Mould D i s t o r t i o n in Three Dimensions . . . . 1 5 2 4 . 4 . 4 . 4 Pred ic ted Mould Wall D i s t o r t i o n 1 5 5 4.5 Experimental Measurements o f Mould D i s t o r t i o n 4 .5 .1 S e l e c t i o n o f I n s t r u m e n t a t i o n 1 6 3 4.5 .2 C a l i b r a t i o n 1 6 6 4 .5 .3 D e t a i l s of Experimental Set-up . . . 1 6 6 4 .5 .4 Experimental T r i a l s 1 6 9 4.5 .5 Results of Mould D i s t o r t i o n Measurement v i Chap t e r Page 4 .6 Compar i son o f Model P r e d i c t i o n s w i t h E x p e r i m e n t a l Measurements, and A n a l y s i s o f R e s u l t s 183 4 . 7 Summary . . . . . . . . . . . . . . . . 199 5.0 EFFECT OF OPERATING VARIABLES ON THE THERMAL RESPONSE OF SLAB AND BILLET MOULDS 204 5.1 I n t r o d u c t i o n . . . . 204 5.2 S e l e c t i o n o f S u i t a b l e Ho t - F a c e H e a t - F l u x P r o f i l e s f o r V a r i o u s O p e r a t i n g C o n d i t i o n s 205 5 .3 Thermal F i e l d s i n S l ab and B i l l e t Moulds 208 5 .3 .1 I n f l u e n c e - o f Mould Des i gn Pa rame te r s 210 5 . 3 . 2 The I n f l u e n c e o f C o o l i n g - W a t e r V a r i a b l e s on the Thermal F i e l d . . . 216 5 . 3 . 3 The I n f l u e n c e o f C a s t i n g Speed on the Thermal F i e l d 225 5 . 3 . 4 The I n f l u e n c e of Carbon Con t en t o f the S t e e l on the Thermal F i e l d i n Moulds 2 3 0 5 . 3 . 5 A Summary o f the I n f l u e n c e o f O p e r a t i n g V a r i a b l e s on the Thermal F i e l d i n B i l l e t Moulds 2 3 1 5.4 Mould Wal l S o f t e n i n g . . 2 3 4 5.5 Case S t u d i e s o f I n d u s t r i a l B i l l e t Mould O p e r a t i o n s 2 3 9 6 .0 THE INFLUENCE OF MOULD DISTORTION ON BILLET QUALITY • 2 4 7 6.1 I n t r o d u c t i o n 2 4 7 6.2 The R e l a t i o n s h i p between N u c l e a t e B o i l i n g , Uneven C o o l i n g and R h o m b o i d i t y . . 2 5 2 v i i Chap t e r Page 6 .3 Mechanism f o r the Fo rma t i on o f L o n g i t u d i n a l Co rne r C r a ck s 269 6 .4 E f f e c t o f C o o l i n g - w a t e r V e l o c i t y on Rhombo id i t y and L o n g i t u d i n a l Co rne r C r a c k i n g 276 6 .5 The I n f l u e n c e o f S t e e l C o m p o s i t i o n on L o n g i t u d i n a l Co rne r C r a c k i n g 286 6 .6 The Use o f S o f t - c o o l i n g P r a c t i c e s to C o n t r o l Rhombo i d i t y and L o n g i t u d i n a l Co rne r C r a c k i n g 290 6 .7 The I n f l u e n c e o f S e c t i o n S i z e on Rhombo id i t y and L o n g i t u d i n a l Co rne r C r a c k i n g 292 6 .8 The I n f l u e n c e o f O the r F a c t o r s on L o n g i t u d i n a l Co rne r C r a c k i n g . . . . . . . . . . . . 3 0 2 6.9 Summary 3 0 4 7.0 ' SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 3 0 8 7.1 Summary 3 0 8 7.2 C o n c l u s i o n s 3 1 0 7.3 Recommendat ions f o r F u t u r e Work 3 ^ 4 SYMBOLS 3 1 5 REFERENCES 3 2 2 APPENDICES 3 3 1 Append i x I C o n s t r u c t i o n o f the Fo r ced C o n v e c t i o n B o i l i n g Curve f o r a G i ven Se t o f Mould Water C o n d i t i o n s 332 I I E q u a t i o n s f o r R a d i a t i v e Heat Exchange Between Mould W a l l s and Me ta l S u r f a c e f o r a B i l l e t Mould 3 3 7 v i i i Append i x Page t I I I F i n i t e - D i f f e r e n c e Equa t i o n s f o r S t e a d y -S t a t e Heat Flow i n the Mould 343 IV The rmophy s i c a l P r o p e r t i e s o f Copper 352 V D e r i v a t i o n o f the F i n i t e - E l e m e n t E q u a t i o n s f o r S t e a d y - S t a t e H e a t - F l o w f o r Two Types o f E l ement s 356 > VI E q u a t i o n s f o r T r a n s i e n t H e a t - F l o w Based on the A l t e r n a t i n g D i r e c t i o n I m p l i c i t F i n i t e - D i f f e r e n c e Method 366 V I I Me chan i c a l P r o p e r t y Data f o r a V a r i e t y o f Moul d M a t e r i a l s 379 V I I I D e r i v a t i o n o f E q u i v a l e n t F o r c e s and Moments f rom a Thermal G r a d i e n t f o r a P l a t e Bend ing E lement . 384 IX D e r i v a t i o n o f the F i n i t e - E l e m e n t E q u a t i o n s to C a l c u l a t e the E l a s t i c De f o rma t i o n o f a Body due to a Thermal F i e l d 394 X Sample C a l c u l a t i o n o f the Gap Wid th i n a Con t i nuou s C a s t i n g B i l l e t Mou ld 410 XI Sou r ce L i s t i n g o f Computer P rogram to C a l c u l a t e the S t e a d y - S t a t e Thermal F i e l d u s i n g a F i n i t e - D i f f e r e n c e M e t h o d . . . 414 XI I Source L i s t i n g o f Computer P rogram to C a l c u l a t e the S t e a d y - S t a t e Thermal F i e l d U s i ng the F i n i t e - E l e m e n t Method . . . 4 2 4 X I I I Source L i s t i n g o f Computer P rogram to C a l c u l a t e the T r a n s i e n t Thermal Response Us i ng an I m p l i c i t A l t e r n a t i n g D i r e c t i o n F i n i t e - D i f f e r e n c e Method 4 3 3 XIV Source L i s t i n g o f Computer P rogram to C a l c u l a t e the De f o rma t i o n o f a Body due to Thermal Load i ng 4 4 5 i x LIST OF TABLES Tab l e Page I Adve rse S l ab Mould C o n d i t i o n s and A s s o c i a t e d De f e c t s 40 I I Adve r se B i l l e t Mould C o n d i t i o n s and A s s o c i a t e d De f e c t s . 42 I I I Water Q u a l i t y Data ,. 59 IV C o n d i t i o n s f o r B i l l e t Moulds based on I n d u s t r i a l P r a c t i c e 76 V Mou l d -Wa l l Tempera tu res w i t h and w i t h o u t the I n c o r p o r a t i o n o f Tempera tu re dependent Thermal C o n d u c t i v i t y . . . 79 VI A Compar i son o f the Mou l d -Wa l l Tempera tu res P r e d i c t e d by the F i n i t e -D i f f e r e n c e Method f o r Two D i f f e r e n t Mesh S i z e s 80 V I I Tempera tu res P r e d i c t e d f o r a Copper P l a t e w i t h S imp le Boundary C o n d i t i o n s . . . . 82 V I I I A Compar i son o f the Mou l d -Wa l l Tempera tu res P r e d i c t e d by the F i n i t e -E lement Method f o r Two D i f f e r e n t Mesh S i z e s 94 IX A Compar i son o f the Mou l d -Wa l l Tempera tu res P r e d i c t e d by the F i n i t e -D i f f e r e n c e and F i n i t e - E l e m e n t Methods . . . . 95 X Mou l d -Wa l l Tempera tu res P r e d i c t e d by the Q u a d r a t i c Tempera tu re E lement Model 96 XI A Compar i son of the S t e a d y - S t a t e Mou l d -Wa l l Tempera tu res P r e d i c t e d by the F i n i t e -D i f f e r e n c e and I m p l i c i t A l t e r n a t i n g D i r e c t i o n F i n i t e - D i f f e r e n c e Methods 101 X I I Mould D e t a i l s and O p e r a t i n g C o n d i t i o n s f o r S l a b C a s t i n g w i t h Low-heat F l u x P r a c t i c e . 117 x o Tab le ; Page X I I I Mould d e t a i l s and O p e r a t i n g C o n d i t i o n s f o r S l a b C a s t i n g w i t h H i g h - h e a t F l u x P r a c t i c e 118' XIV E f f e c t i v e S t r e s s e s a t S e l e c t e d L o c a t i o n s i n the Mould 139 XV B i l l e t Mould C o n d i t i o n s f o r P r e l i m i n a r y D i s t o r t i o n C a l c u l a t i o n s 156 XVI A Compar i son o f S t r e s s e s a t I d e n t i c a l L o c a t i o n s i n the Mould f o r Two D i f f e r e n t Mesh C o n f i g u r a t i o n s 161 XVII C h a r a c t e r i s t i c s o f LVDT's S e l e c t e d I 6 5 XV I I I Mould D imens ions A f t e r Re fo rm ing O p e r a t i ons 1 6 7 XIX D e t a i l s o f E x p e r i m e n t a l T r i a l s 170 XX I n d i c a t o r Read ings a t the B e g i n n i n g and End o f a Thermal C y c l e 180 XXI R e s u l t s o f Mould Wa l l Movement From Data o f T r an sdu ce r 3 182 XXII R e s u l t s o f Mould Wa l l Movement From Data o f T r a n s du c e r 2 I 8 4 XX I I I Des ign D e t a i l s o f I n s t r umen t ed Mou ld I 8 5 XXIV O p e r a t i n g C o n d i t i o n s Du r i ng E x p e r i m e n t a l T r i a l s I 8 6 XXV A Compar i son o f Measured and P r e d i c t e d Va l ues o f Mould D i s t o r t i o n I 9 0 XXVI Va lues o f C a s t i n g V a r i a b l e s f o r B i l l e t Moulds 2 0 9 XXVII Va lues o f C a s t i n g V a r i a b l e s f o r S l a b Moulds 209 XXVI I I C a l c u l a t e d Peak Tempera tu res i n t he Mould Wal 1 2 3 6 xi Tab l e Page XXIX S t a n d a r d O p e r a t i n g and Des ign C o n d i t i o n s f o r S i x B i l l e t Moulds . 241 XXX R e s u l t s o f Hea t - F l ow and D i s t o r t i o n C a l c u l a t i o n s 2 4 3 XXXI D e t a i l s o f C a s t i n g P a r a m e t e r s ' o f Mould J . . 251 XXXII Reduced H e a t - F l u x Owing to a Change i n Gap S i z e f o r H igh Carbon S t e e l s a t H igh Wa t e r - F l ow Rates 257 XXX I I I Tempera tu re D i s t r i b u t i o n A c r o s s the T h i c k n e s s o f the Wa l l a t the Upper and Lower L i m i t o f B o i l i n g 2 6 2 XXXIV Reduced H e a t - F l u x Owing to a Change i n Gap S i z e f o r H igh Carbon S t e e l s a t Low Wa t e r - F l ow Rates 2 8 ^ XXXV Reduced H e a t - F l u x Owing to a Change i n Gap S i z e f o r a Th in Wa l l Mould 2 9 8 x i i LIST OF FIGURES F i g u r e Page 1 S chema t i c Diagram o f the Con t i nuous C a s t i n g P r o c e s s f o r S t e e l 2 2 Three Main Types o f M o u l d s 1 3 0 ' 1 3 1 . . .4 3 S chema t i c View o f a T r a n s v e r s e S e c t i o n o f a B i l l e t Showing A i r Gap F o r m a t i o n 9 11 4 B i l l e t Mould Heat F l u x as a F u n c t i o n o f D i s t a n c e f rom the Top o f the Mould21 13 5 A Double Peak i n a B i l l e t Mould H e a t -F l u x P r o f i l e 2 1 15 6 V a r i a t i o n o f H e a t - T r a n s f e r C o e f f i c i e n t Down the M o u l d 4 16 7 S p e c i f i c Heat E x t r a c t i o n as a F u n c t i o n o f D i s t a n c e f rom the T O D o f the Mould f o r D i f f e r e n t L u b r i c a n t s " 27 8 Average B i l l e t Mould H e a t - F l u x as a F u n c t i o n o f Carbon Conten f21 29 9 D imens i ona l Changes i n a B i l l e t Mou ld A f t e r S e v e r a l H e a t s 5 5 33 10 R e v e r s a l o f Taper i n a Bloom M o u l d 5 7 34 11 Thermal Load Map f o r a S l ab Mould P l a t e 6 0 36 9 7 R 9 12 H y s t e r e s i s E f f e c t i n the B o i l i n g Curve °^ 13 A S chema t i c Diagram I l l u s t r a t i n g the Con-s t r u c t i o n o f a B o i l i n g Curve f o r Sub -c o o l e d W a t e r 9 8 5 5 14 Fo r c ed C o n v e c t i o n B o i 1 i n g Curves f o r Sub-c o o l e d Water f o r Three D i f f e r e n t Water V e l o c i t i e s . 56 15 Fo r c ed C o n v e c t i o n B o i l i n g Curves f o r Sub-c o o l e d Water f o r Three D i f f e r e n t Degrees c 7 o f Subcoo l i ng 5 ' x i i i F i g u r e Page 16 A S chema t i c D iagram o f t he L o n g i t u d i n a l M i d - P l a n e Through a Mould Wa l l 61 17 S chema t i c D iagram o f a Two -D imens i ona l F i n i t e - D i f f e r e n c e Mesh 69 18 Flow Cha r t o f the F i n i t e - D i f f e r e n c e H e a t -Flow Program to Compute the S t e a d y - S t a t e Thermal F i e l d i n Moulds '72 19 H e a t - F l u x P l o t t e d A g a i n s t T i m e - i n - t h e -Mould (Dwe l l Time) f o r a B i l l e t Mou ld f o r S t e e l s w i t h D i f f e r e n t Carbon Con t en t s 77 20 A S chema t i c D iagram o f a Two -D imens i ona l F i n i t e - E l e m e n t Mesh 8 5 21 Flow Cha r t o f the F i n i t e - E l e m e n t H e a t -F low Program to Compute the S t e a d y - S t a t e Thermal F i e l d i n Moulds 8 9 22 C o l d - F a c e H e a t - F l u x P r o f i l e i n a B i l l e t 2 1 Mould F i t t e d to E x p e r i m e n t a l Measurements by T r i a l and E r r o r • 1 0 3 23 A Compar i son Between P r e d i c t e d and Measured Wa l l Tempera tu res i n a B i l l e t Mould 1 0 4 24 L o n g i t u d i n a l Tempera tu re P r o f i l e i n a S l ab M o u l d ' ' ' F i t t e d to E x p e r i m e n t a l Mea su r e -ments by T r i a l and E r r o r 105 25 P r e d i c t e d S t e a d y - S t a t e I so the rms i n the Wa l l o f a B i l l e t Mould Under S t a n d a r d C o n d i t i o n s f o r a C o o l i n g - W a t e r V e l o c i t y o f 1 0 m/sec 1 0 7 26 Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould Under S t anda r d Con -d i t i o n s f o r a C o o l i n g - W a t e r V e l o c i t y o f 10 m/sec 1 1 0 27. Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould Showing Upper and Lower Tempe ra tu re L i m i t s Du r i ng B o i l i n g a t the Mou ld /Wate r I n t e r f a c e f o r a Water V e l o c i t y o f 7 m/sec 111 xiv F i g u r e Page 28 P r e d i c t e d I so the rms i n the Wa l l o f a B i l l e t Mould f o r the Upper Tempe ra tu re L i m i t o f the B o i l i n g C y c l e 112 29 P r e d i c t e d I s o t he rms i n the Wa l l o f a B i l l e t Mould f o r the Lower Tempera tu re L i m i t o f the B o i l i n g C y c l e . . . . 1 1 3 30 P r e d i c t e d S t e a d y - S t a t e I so the rms i n the Wa l l o f a S l ab Mould w i t h Low H e a t - F l u x P r a c t i c e ^ 5 31 P r e d i c t e d S t e a d y - S t a t e I so the rms i n the Wa l l o f a S l ab Mould w i t h H igh H e a t - F l u x P r a c t i c e . . 1 1 6 32 D i s t o r t i o n o f the H e a t - F l u x P r o f i l e as i t T r a v e r s e s the Wa l l o f a S l ab Mould w i t h H igh H e a t - F l u x P r a c t i c e 1 1 9 33 H e a t - F l u x P l o t t e d A g a i n s t T i m e - i n - t h e Mou ld (Dwe l l Time) f o r a S l ab Mou ld w i t h H i gh and Low H e a t - F l u x P r a c t i c e ' 2 ' 34 D i s c r e t i z a t i o n o f the Mou ld Cont inuum i n the P l ane o f a B i l l e t Mould Wa l l f o r a F i n i t e - E l e m e n t P l a t e A n a l y s i s 1 3 2 35 E l a s t i c D e f l e c t i o n o f a B i l l e t Mou ld Wa l l Due to the S t e a d y - S t a t e Thermal G r a d i e n t s C a l c u l a t e d f rom a F i n i t e - E l e m e n t P l a t e M o d e l . . 1 3 5 36 P r e d i c t e d E l a s t i c D e f l e c t i o n s A c r o s s the Ho t -Face o f a B i l l e t Mou ld Wal l a t D i f f e r e n t D i s t a n c e s (d) f rom the Top o f the Mou ld . . . . . . . 1 3 6 37 D i s c r e t i z a t i o n o f a F ou r t h o f a T r a n s v e r s e S e c t i o n o f a B i l l e t Mould f o r a P l a n e -S t r e s s F i n i t e - E l e m e n t A n a l y s i s 1 4 1 38 F l o w - C h a r t o f the Computer Program t o Compute the Thermal D i s t o r t i o n o f a T r a n s -v e r s e S l i c e o f t he Mould Based on a P l a n e -S t r e s s F i n i t e - E l e m e n t Model 1 4 3 xv F i gure Page 39 P r e d i c t e d E l a s t i c D e f l e c t i o n P r o f i l e A c r o s s the Ho t - Fa ce o f a B i l l e t Mou ld Wa l l ~30 mm f rom the Top o f the Mou ld C a l c u l a t e d f rom a P l ane S t r e s s F i n i t e -E lement Model "146 40 A S chema t i c D iagram o f the Three Dimen-s i o n a l F i n i t e - E l e m e n t Mesh f o r the M i d -Face to Co rne r Reg ion o f a B i l l e t Mou ld 1 5 3 41 P r e d i c t e d S t e a d y - S t a t e I so the rms i n the Wa l l o f a B i l l e t Mould Based on Savage and P r i t c h a r d ' s H e a t - F l u x Curve 157 42 43 44 45 46 47 48 49 A Compar i son Between B i l l e t Mou ld D i s -t o r t i o n s . P r e d i c t e d by the T h r e e -D imens i ona l E l a s t o - P l a s t i c F i n i t e -E lement A n a l y s i s f o r T w o - D i f f e r e n t Mesh . C o n f i g u r a t i ons 158 P r e d i c t e d D i s t o r t i o n s A c r o s s the H o t - F a c e o f a B i l l e t Mould a t D i f f e r e n t D i s t a n c e s (d) f rom the Top o f the Mould 1 6 0 A S chema t i c Diagram o f the L o n g i t u d i n a l P l a n e Through the I n s t r umen t ed B i l l e t Mou ld Assemb ly 1 6 8 T y p i c a l Examples o f the S i g n a l s Gene r a t ed by T r a n s d u c e r s 1, 2 and 3 Wh i l e M o n i t o r i n g Mould Wa l l Movement Du r i n g Heat No. D05946 T y p i c a l Examples o f by T r a n s du c e r s 1, 2 Mould Wa l l Movement the S i g n a l s and 3 Wh i l e Du r i ng Heat Gene r a t ed Moni t o r i ng No. D05947 T y p i c a l Examples o f the S i g n a l s Gene ra t ed by T r a n s du c e r s 1, 2 a n d ' 3 Wh i l e M o n i t o r i n g Mou ld Wa l l Movement Du r i n g Heat No. E06591 T y p i c a l Examples o f the S i g n a l s Gene r a t ed by T r a n s du c e r s 1, 2 and 3 Wh i l e M o n i t o r i n g Mould Wa l l Movement Du r i ng Heat No. D05951 T y p i c a l Examples o f the S i g n a l s Gene ra t ed by T r an sdu ce r s 1, 2 and 3 Wh i l e M o n i t o r i n g Mould Wa l l Movement Du r i ng Heat No. E06592 1 72 173 17.4 175 176 xvi F i g u r e Page 50 T y p i c a l E x a m p l e s o f t h e S i g n a l s G e n e r a t e d by T r a n s d u c e r s 1 , 2 and 3 W h i l e M o n i t o r i n g M o u l d W a l l Movement D u r i n g H e a t No . D05952 . . . 177 51 A C o m p a r i s o n o f t h e D i s p l a c e m e n t S i g n a l A s s o c i a t e d w i t h T r a n s d u c e r 2 P o s i t i o n e d 65 mm B e l o w t h e M e n i s c u s w i t h T e m p e r a t u r e F l u c t u a -t i o n s M e a s u r e d i n a M e t a l l i c S u r f a c e D u r i n g N u c l e a t e B o i l i n g 178 52 P r e d i c t e d H o t - and C o l d - F a c e T e m p e r a t u r e P r o f i l e s f o r t h e I n s t r u m e n t e d B i l l e t M o u l d S h o w i n g t h e U p p e r and Lower T e m p e r a t u r e L i m i t s D u r i n g B o i l i n g 188 53 P r e d i c t e d D i s t o r t i o n P r o f i l e a t t h e M i d - S i d e o f t h e I n s t r u m e n t e d M o u l d Down t h e H o t - F a c e . . 1 8 9 54 A C o m p a r i s o n o f t h e I n t e r n a l D i m e n s i o n s o f t h e I n s t r u m e n t e d M o u l d A f t e r t h e E x p e r i m e n t a l T r i a l s w i t h t h e O r i g i n a l D i m e n s i o n s as S p e c i f i e d by t h e M a n u f a c t u r e r 1 9 3 55 A C o m p a r i s o n o f t h e P r e d i c t e d D i s t o r t i o n B e t w e e n t h e M i d - S i d e and t h e C o r n e r o f t h e I n s t r u m e n t e d B i l l e t M o u l d a t t h e H o t - F a c e as a F u n c t i o n o f D i s t a n c e f r o m t h e Top A f t e r a S i n g l e T h e r m a l C y c l e 1 9 5 56 P r e d i c t e d E f f e c t i v e S t r e s s C o n t o u r s i n a T r a n s v e r s e S e c t i o n o f t h e I n s t r u m e n t e d M o u l d a t ( i ) 3 6 . 7 5 mm and ( i i ) 5 7 . 7 5 mm B e l o w t h e Top o f t h e M o u l d 1 9 7 57 P r e d i c t e d E f f e c t i v e S t r e s s C o n t o u r s i n a T r a n s v e r s e S e c t i o n o f t h e I n s t r u m e n t e d M o u l d a t ( i ) 9 9 . 7 5 mm and ( i i ) 5 3 0 . 2 5 mm B e l o w t h e Top o f t h e M o u l d 1 9 8 58 P r e d i c t e d E f f e c t i v e S t r e s s C o n t o u r s i n Two L o n g i t u d i n a l P l a n e s T h r o u g h t h e W a l l o f t h e I n s t r u m e n t e d M o u l d ( i ) 5 . 5 5 mm f r o m t h e M i d - F a c e and ( i i ) P l a n e T h r o u g h t h e C e n t r e of. t h e Two C o r n e r E l e m e n t s ^ 0 0 x v i i F i gure 59 T h r e e - D i m e n s i o n a l V i s u a l i z a t i o n o f the D i s t o r t e d Mould 60 R e s i s t a n c e s to Hea t - F l ow Between the S t e e l S u r f a c e and the Mould C o o l i n g Water f o r the B i l l e t Mould and S l a b Moulds w i t h Low and H igh H e a t - F l u x P r a c t i c e s 61 I n f l u e n c e o f Mould Wa l l T h i c k n e s s on the Ho t -and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould 62 I n f l u e n c e o f Mould Wa l l T h i c k n e s s on the Ho t -and C o l d - F a c e Tempera tu re P r o f i l e s o f a S l a b Mou ld w i t h H igh H e a t - F l u x P r a c t i c e 63 I n f l u e n c e o f Mould Thermal C o n d u c t i v i t y on t he Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould 64 I n f l u e n c e o f Mould Thermal C o n d u c t i v i t y on the Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a S l a b Mould w i t h Low H e a t - F l u x P r a c t i c e . . 65 I n f l u e n c e o f C o o l i n g - W a t e r V e l o c i t y on the Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mou ld 66 I n f l u e n c e o f C o o l i n g - W a t e r V e l o c i t y on the Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a S l ab Mould w i t h Low H e a t - F l u x P r a c t i c e . . . . . 67 I n f l u e n c e o f S c a l e F o rma t i o n on t he Ho t -and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould 68 I n f l u e n c e o f Water Channe l E x i t P r e s s u r e on the Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould 69 I n f l u e n c e o f Water I n l e t Tempera tu re on the Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould 70 The E f f e c t o f R e v e r s i n g the D i r e c t i o n o f Water F low on the Ho t - and C o l d - F a c e Tempera tu res o f a B i l l e t Mould xvi i i F i g u r e Page 71 I n f l u e n c e o f C a s t i n g Speed on the Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould 227 72 I n f l u e n c e o f C a s t i n g Speed on Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a S l a b Mould w i t h Low H e a t - F l u x P r a c t i c e 228 73 I n f l u e n c e o f C a s t i n g Speed on Ho t - and Co ld -Face Tempera tu re P r o f i l e s o f a S l a b Mould w i t h H igh H e a t - F l u x P r a c t i c e 229 74 I n f l u e n c e o f Carbon Con ten t o f the S t e e l on Hot- and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mou ld 232 75 H a l f S o f t e n i n g Tempera tu res f o r B i l l e t Mould M a t e r i a l s 5 5 5 ' 1 d i 238 76 A Compar i son o f the P r e d i c t e d Thermal D i s t o r t i o n P r o f i l e s A l ong the Cen t e r o f the W a l l s a t the Ho t - F a c e f o r S i x I n -d u s t r i a l B i l l e t Moulds 240 77 A L o n g i t u d i n a l Co r ne r C rack i n a B i l l e t 1 3 2 . . . 249 78 P r e d i c t e d Ho t - and Co l d - F a c e Tempera tu re P r o f i l e s f o r Mould J a t H igh Wa t e r - F l ow Rates (wa te r v e l o c i t y = 8.16 m/sec) 254 79 Thermal D i s t o r t i o n P r o f i l e s o f Mou ld J A l ong the Cen t r e o f the Wa l l a t the H o t - F a c e a t D i f -f e r e n t S tages o f the I n t e r m i t t e n t B o i l i n g C y c l e 255 80 A Compar i son of the T ime -Ave raged H e a t - F l u x P r o f i l e as a F u n c t i o n o f T i m e - i n - t h e - M o u l d (Dwe l l Time) and the Reduced H e a t - F l u x P r o f i l e a t the Upper Tempera tu re L i m i t o f the I n t e r m i t t e n t B o i l i n g C y c l e 258 81 The Ho t - and Co l d - F a c e Tempera tu re P r o f i l e s o f Mould J A s s o c i a t e d w i t h the Reduced Hea t -F l u x P r o f i l e Owing to I n t e r m i t t e n t B o i l i n g . . . 259 x ix F i gure 82 D i s c r e t i z a t i o n o f a T r a n s v e r s e S e c t i o n o f the Mou ld Cont inuum to Compute the D i s t o r t i o n Due to Non Un i f o rm Tempera tu res i n t he Mould by a P l a n e - S t r e s s F i n i t e - E l e m e n t A n a y l s i s 2 83 A S chema t i c Diagram o f a 130x130 mm B i l l e t Mould C o n f i g u r a t i o n s f o r ( i ) Two O p p o s i t e Faces o f the Mould a t the Upper Tempe ra tu re ' . L i m i t o f the I n t e r m i t t e n t B o i l i n g C y c l e and the Other Two a t the Lower Tempera tu re L i m i t ( i i ) B o i l i n g on Two A d j a c e n t F a c e s , No B o i l -i n g on the Othe r Two ( i i i ) B o i l i n g On ly on One Face ( i v ) B o i l i n g on Three Faces 84 A S chema t i c D iagram o f B i l l e t and Mould Con-f i g u r a t i o n s With and W i t hou t Uneven C o o l i n g . ( i ) No Rhombo id i t y ( i i ) Rhombo i d i t y o f S t r a n d and Mould Showing an I n c r e a s e i n Co r ne r Gaps a t Obtuse Ang l e Co rne r s 85 A S chema t i c D iagram I l l u s t r a t i n g F a c t o r s t h a t Cause T e n s i l e S t r e s s e s Near the S o l i d i f i c a t i o n F r o n t ( i ) F e r r o s t a t i c P r e s s u r e Over the Un-s u p p o r t e d Co rne r and S h r i n k a g e o f the S h e l l ( i i ) the I n c r e a s e i n Ang l e a t Obtuse Ang l e Co r ne r s ( i i i ) R ehea t i n g o f the S h e l l a t Obtuse Ang l e Co rne r s Due to the I n c r e a s e i n Co r ne r Gap S i z e 86 Pho tog raph of a T r a n s v e r s e S e c t i o n o f a B i l l e t Showing a L o n g i t u d i n a l Co rne r C rack i n the S h e l l 1 3 3  87 P r e d i c t e d B i l l e t S u r f a c e Tempera tu res and S h e l l T h i c k n e s s as a F u n c t i o n o f T i m e - i n -t h e - M o u l d (Dwe l l T i m e ) 8 2  88 P r e d i c t e d Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s f o r Mould J a t Low Wa te r - F l ow R a t e s . . 89 Thermal D i s t o r t i o n P r o f i l e s o f Mou ld J A l o n g the Cen t r e o f the Wa l l a t the Ho t - F a c e a t the I n c e p t i o n o f B o i l i n g f o r Low Wa t e r - F l ow Rates xx F i gure . Page 90 A Compar i son o f the T ime -Ave raged H e a t - F l u x P r o f i l e as a F u n c t i o n o f T i m e - i n - t h e - M o u l d (Dwe l l Time) and the Reduced H e a t - F l u x P r o -f i l e A s s o c i a t e d w i t h the Tempera tu re Ove r -shoo t a t the I n c e p t i o n o f B o i l i n g 282 91 A C o o l i n g - W a t e r V e l o c i t y / W a t e r E x i t P r e s s u r e Diagram Demar ca t i ng the B o i l i n g / N o - B o i l i n g Reg ions f o r B i l l e t Mould O p e r a t i o n when C a s t i n g H igh Carbon Grades o f S t e e l 285 92 Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a B i l l e t Mould f o r a Low Carbon (0.1%) S t e e l . . . 287 93 The I n f l u e n c e o f S t e e l C o m p o s i t i o n on the Thermal D i s t o r t i o n o f Mould J a t t he C e n t r e o f the Wa l l as a F u n c t i o n o f D i s t a n c e Down the Mould 94 A C o o l i n g - W a t e r V e l o c i t y / W a t e r E x i t P r e s s u r e Diagram Demarca t i ng the B o i l i n g / N o - B o i l i n g Reg ions f o r B i l l e t Mould O p e r a t i o n When C a s t i n g Low-Carbon Grades o f S t e e l 97 Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s P r e -d i c t e d f o r a B i l l e t Mould w i t h A Wa l l T h i c k -ness o f 5.875 mm w i t h a l l O the r C o n d i t i o n s C o r r e s p o n d i n g to Mould J 98 Thermal D i s t o r t i o n P r o f i l e s o f the Th i n Mould A l ong the Cen t re o f t he Wa l l a t the Ho t - F a c e at the I n c e p t i o n o f B o i l i n g . . . . 288 289 95 Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s P r e d i c t e d f o r a S o f t - C o o l e d B i l l e t Mould 291 96 A S chema t i c D iagram o f a 181x181 mm2 B i l l e t Mould C o n f i g u r a t i o n Showing the R e l a t i v e Movement o f the Co rne r s Due to B o i l i n g on Two A d j a c e n t Faces o f the Mould N o - B o i l i n g on the O the r Two 293 295 296 99 A Compar i son o f the Reduced H e a t - F l u x P r o f i l e s P l o t t e d as a F u n c t i o n o f T i m e - i n - t h e - M o u l d (Dwe l l Time) f o r B i l l e t Moulds o f Two D i f -f e r e n t Wa l l T h i c k n e s s e s xx i F i gure Page TOO Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f a Th i n Wa l l Mou ld A s s o c i a t e d w i t h the Reduced H e a t - F l u x P r o f i l e Owing to I n t e r m i t t e n t B o i l i n g 300 2 101 A S chema t i c D iagram of a 111x111 mm B i l l e t Mould C o n f i g u r a t i o n Showing the R e l a t i v e Move-ment o f the Co rne r s Due to B o i l i n g on Two A d j a c e n t Faces o f the Mou l d , N o - B o i l i n g on the Othe r Two 301 xxi i ACKNOWLEDGEMENTS The au t ho r i s d eep l y g r a t e f u l t o P r o f . J . K . Br imacombe f o r h i s i n v a l u a b l e gu idance and i n s p i r a t i o n t h r o u g h o u t the c ou r s e o f t h i s r e s e a r c h . S i n c e r e t h a n k s a re e x t ended to P r o f . D.L . Anderson f o r the numerous s t i m u l a t i n g d i s c u s s i o n s and f o r h i s e x p e r t a d v i c e . The c o o p e r a t i o n and a s s i s t a n c e o f Mr . R. Pugh and the O p e r a t i n g P e r s onne l a t the S t e e l Company o f Canada , Edmonton w o r k s , i s g r a t e f u l l y a c know l edged . The au t ho r a l s o w i s h e s to e x p r e s s her a p p r e c i a t i o n to Mr. B .N . W a l k e r , Mr . P. Wenman, Mr. H. Tump, Mr. E. K l a s s e n and Mr. P. M u s i l f o r t h e i r t e c h n i -c a l a s s i s t a n c e a t v a r i o u s s t a ge s o f the p r o j e c t . D i s c u s s i o n s w i t h g r adua t e s t u d e n t s and o t h e r members o f the f a c u l t y have been h e l p f u l and a re acknowledged w i t h t h a n k s . The G radua te F e l l o w s h i p awarded by the U n i v e r s i t y o f B r i t i s h Co lumb ia p r o v i d e d f i n a n c i a l s u p p o r t d u r i n g the r e s e a r c h f o r wh i ch the a u t ho r i s t r u l y g r a t e f u l . xxi i i 1.0 INTRODUCTION A c t i v e deve lopment o f the c o n t i n u o u s c a s t i n g p r o c e s s f o r s t e e l o ve r the pas t two decades has l e d to i t s w i d e -sp r ead a d o p t i o n i n the s t e e l i n d u s t r y . I t s a c c ep t a n c e stems f rom the f a c t t h a t s e m i - f i n i s h e d p r o du c t s o f good q u a l i t y can be d i r e c t l y p roduced bo th e c o n o m i c a l l y and a t h i g h p r o -d u c t i v i t i e s . T h i s has made c o n t i n u o u s c a s t i n g more a t t r a c -t i v e than the c o n v e n t i o n a l s t a t i c c a s t i n g r o u t e f o r the p r o d u c t i o n o f b i l l e t s , blooms and s l a b s used i n the manufac -t u r e o f bar p r o d u c t s , s t r u c t u r a l s e c t i o n s , s t r i p and p l a t e . In i t s s i m p l e s t form the 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 n v o l v e s the p o u r i n g o f mo l t en s t e e l f rom a t u n d i s h i n t o a r e c i p r o c a t i n g wa te r c o o l e d copper mould f rom wh i ch the s t e e l s t r a n d i s c o n t i n u o u s l y w i t hd rawn ( F i g . 1 ) . The emerg ing s e c t i o n has o n l y a t h i n s h e l l s u p p o r t i n g the mo l t en co re so t h a t f u r t h e r c o o l i n g i s r e q u i r e d be low the mou l d . T h i s i s e f f e c t e d by a s e r i e s o f wa t e r s p r a y s wh i ch imp inge d i r e c t l y be low the s t r a n d . For the c a s t i n g o f s l a b s , c l u s t e r s o f r o l l s benea th the mould g u i d e , s u p p o r t and w i t hd r aw the s t r a n d to f a c i l i t a t e smooth passage o f the s e c t i o n t h rough the c a s t e r . Lower i n the machine where a s o l i d i f i c a t i o n i s comp le te the s t r a n d i s c u t t r a n s v e r s e l y 1 2 Input Stream From Tundish F i g . 1 S chema t i c Diagram o f the Con t i nuou s C a s t i n g P r o c e s s f o r S t e e l . 3 i n t o d e s i r e d l e n g t h s . C o n s i d e r a b l e r e s e a r c h has been devo t ed to i m p r o v i n g a l l a s p e c t s o f the c o n t i n u o u s c a s t i n g p r o c e s s to b r i n g i t to i t s p r e s e n t l e v e l o f s o p h i s t i c a t i o n . However t he t a s k i s by no means c o m p l e t e . One a rea t h a t r e q u i r e s f u r t h e r i n v e s t i g a t i o n i s the mould because o f i t s p r o f ound i n -f l u e n c e on q u a l i t y and o p e r a t i n g p r o b l e m s . Th i s i s t he s u b j e c t o f the p r e s e n t t h e s i s . S t r a i g h t and cu r ved moulds o f t h r e e b a s i c d e s i g n s a re i n use t o d a y . F i g . 2 i l l u s t r a t e s the s a l i e n t f e a t u r e s o f each t y p e . The b l o c k mould i s g e n e r a l l y used f o r t he c a s t i n g o f beam b l a n k s o r blooms and i s c h a r a c t e r i z e d by a t h i c k w a l l t h rough wh ich v e r t i c a l c o o l i n g wa t e r c h a n n e l s have been d r i l l e d . P l a t e moulds a re employed f o r t he c a s t i n g o f s l a b s and are compr i s ed o f f o u r coppe r p l a t e s b o l t e d to s t e e l b a c k i n g p l a t e s assemb led t o form the mould c a v i t y . V e r t i c a l c o o l i n g - w a t e r c hanne l s a r e mach ined e i t h e r i n the coppe r p l a t e s o r the s t e e l b a c k i n g p l a t e s . The moulds used f o r c a s t i n g b i l l e t s have t h i n n e r w a l l s t han e i t h e r the p l a t e o r b l o c k mou l d s . In t h i s case an a n n u l a r c o o l i n g wa t e r channe l i s formed between t he mould tube and a s t e e l l i n e r o f l a r g e r d i m e n s i o n s . A l l t h r e e t ypes o f moulds a re c o o l e d by f o r c e d c i r c u l a t i o n o f wa t e r t h r ough the ( i ) S o l i d B l o c k Mould ( i i ) P l a t e Mould ( i i i ) Tube Mould 130 131 F i g . 2 Three Main Types o f Moulds. U ' , J 1 5 c o o l i n g c hanne l s a t v e l o c i t i e s o f 6~12 m/sec , c o u n t e r c u r -r e n t to the c a s t i n g d i r e c t i o n . A more r e c e n t i n n o v a t i o n i s the m u l t i s t a g e mould wh i ch has a s h o r t e r c o n v e n t i o n a l l y c o o l e d s e c t i o n f o l l o w e d by a sp r ay c o o l e d second s t a g e t h a t a l s o p r o v i d e s s u p p o r t f o r the s t r a n d w i t h s p r i n g l o a ded p l a t e s . The moulds a re n o r m a l l y p r o v i d e d w i t h a t a p e r to compensate f o r s h r i n k a g e o f t he s t e e l and t o improve hea t t r a n s f e r by a l l o w i n g c l o s e r c o n t a c t between the s t r a n d and the mou l d . In o p e r a t i o n moulds are o s c i l l a t e d w i t h a f i x e d s t r o k e and f r e quen c y wh i ch depend on the type o f c a s t e r and the c a s t i n g s p e e d . N ega t i v e s t r i p , i n wh i ch the mould moves f a s t e r than the s t r a n d i s employed to p r e v e n t the s t r a n d f rom s t i c k i n g to the mou l d . L u b r i c a t i o n i s a l s o p r o v i d e d f o r t he same p u r p o s e . In b i l l e t moulds v e g e t a b l e or s y n t h e t i c o i l s a r e used w h i l s t mould powders a r e u t i l i z e d ,in s l a b c a s t e r s . The l a t t e r mode o f l u b r i c a t i o n a l s o p r o -t e c t s the s t e e l f rom r e o x i d a t i o n , r educes hea t l o s s and f l u x e s a l u m i n a t e s . Submerged n o z z l e s a re employed w i t h powder l u b r i c a t i o n and c r e a t e the d e s i r e d f l u i d f l o w c o n -d i t i o n s i n the mou ld . F l u i d f l o w i n the mould may a l s o be c o n t r o l l e d by e l e c t r o m a g n e t i c s t i r r i n g w i t h the i n t e n t be i ng to improve c a s t s t r u c t u r e o f the a s - c a s t p r o d u c t . C o n t i n u e d e f f o r t s a re be i ng d i r e c t e d toward a b e t t e r u n d e r s t a n d i n g o f heat t r a n s f e r and s o l i d i f i c a t i o n i n the mou l d . The o b j e c t i v e i s to s o l v e m o u l d - r e l a t e d q u a l i t y and o p e r a t i n g p rob lems such as l o n g i t u d i n a l c o r n e r c r a c k s and b r e a k - o u t s . I t i s b e l i e v e d t h a t t he se p rob lems i n many cases may be due t o the r e sponse o f the mould to the hea t f l o w , wh i ch i s a f f e c t e d by o p e r a t i n g c o n d i t i o n s , mould de -s i g n and mould p r o p e r t i e s . T u r n i n g to the b e h a v i o u r o f the mould d u r i n g o p e r a -t i o n , i t i s e v i d e n t t h a t the mould w a l l i s s u b j e c t t o d i f -f e r e n t i a l t he rma l e xpan s i o n due to the t he rma l g r a d i e n t s s e t up, wh i ch c oup l e d w i t h the r e s t r a i n t s imposed by the mould s u p p o r t s y s t e m , g ene r a t e s s t r e s s e s i n the c o p p e r . S t r e s s e s o f magn i tude s u f f i c i e n t to cause y i e l d i n g u s u a l l y r e s u l t s i n the men i s cus r e g i o n where the l a r g e s t t empe r a t u r e g r a d i e n t s and h i g h e s t t empe r a t u r e s o b t a i n . The e n s u i n g p l a s t i c f l o w g i v e s r i s e to permanent d i s t o r t i o n o f the mou l d . Mould d i s t o r t i o n c oup l ed w i t h wear has been found t o a d v e r s e l y a f f e c t s u r f a c e q u a l i t y and b r e a k - o u t f r e q u e n c y . S i n c e the the rma l f i e l d o b t a i n e d i n a mould i s a f f e c t e d by o p e r a t i n g v a r i a b l e s , mould d i s t o r t i o n and r e l a t e d q u a l i t y p rob lems c o u l d be l inked to adve r s e o p e r a t i n g c o n d i t i o n s . In p r a c t i c e many s t e p s have been t a ken to improve the r e s i s t a n c e o f the mould to the rma l d i s t o r t i o n . Two examples a re work h a r d e n i n g , and u t i l i z a t i o n o f coppe r a l l o y s o f s u p e r i o r s t r e n g t h and s o f t e n i n g r e s i s t a n c e f o r the manu-f a c t u r e o f mou l d s . In a d d i t i o n to t hese c o r r e c t i v e measures moulds a re p e r i o d i c a l l y examined f o r permanent shape change s . The i n t e r n a l d imens i on s o f the mould c a v i t y a re measured down the l e n g t h o f the mould and i f d i s t o r t i o n above s p e c i f i e d l i m i t s i s d e t e c t e d the mould i s e i t h e r r e c o n d i t i o n e d o r d i s c a r d e d . D e s p i t e t he se measures mould d i s t o r t i o n c o n t i n u e s to rema in a p r o b l e m , a f f e c t i n g p r o du c t q u a l i t y and l i m i t i n g mould l i f e . 2.0 REVIEW OF PAST WORK AND OBJECTIVES OF RESEARCH PROJECT ' In the c o n t i n u o u s c a s t i n g o f s t e e l , hea t f l o w and s o l i d i f i c a t i o n i n the mould a re a s p e c t s o f t he p r o c e s s t h a t have r e c e i v e d c o n s i d e r a b l e a t t e n t i o n due to t h e i r i n t i m a t e l i n k w i t h p r o du c t q u a l i t y . A l t h ough e ven t s i n the mould a re no t w e l l u n d e r s t o o d , t h e r e i s a w e a l t h o f i n f o r m a t i o n i n the l i t e r a t u r e t h a t sheds l i g h t on i m p o r t a n t a s p e c t s o f mould b e h a v i o u r . Th i s c h a p t e r a t t emp t s to draw t o g e t h e r a v a i l a b l e i n f o r m a t i o n on the s u b j e c t . T op i c s r e v i ewed i n c l u d e mechanism o f hea t t r a n s f e r to the mould and measurement o f the h e a t -t r a n s f e r r a t e as a f u n c t i o n o f t i m e . The i n f l u e n c e o f mould d e s i g n , geome t r y , o p e r a t i n g v a r i a b l e s and s t e e l c o m p o s i t i o n on the h e a t - f l u x d i s t r i b u t i o n i s a l s o e xam ined . Ma t h ema t i c a l models have been f omu l a t ed to s i m u l a t e hea t f l o w i n moulds and t he se a re d i s c u s s e d . A v a i l a b l e i n f o r m a t i o n on the s u b j e c t o f mould d i s t o r t i o n , and i t s r e l a t i o n to p r o d u c t q u a l i t y i s p r e s e n t e d . In c o n c l u s i o n the o b j e c t i v e s o f the r e s e a r c h are o u t l i n e d . 2.1 Mechanism o f H e a t - T r a n s f e r and Gap Fo rma t i o n Heat f rom the s t r a n d s u r f a c e i s t r a n s f e r r e d to the mould c o o l i n g wa t e r by a sequence of s t e p s i n s e r i e s . These 8 9 c o n s i s t o f c o n d u c t i o n and r a d i a t i o n a c r o s s an a i r gap s e p a r a t i n g the mould and the s t r a n d , c o n d u c t i o n t h r ough the mould w a l l i t s e l f and c o n v e c t i o n at the m o u l d / c o o l i n g w a t e r i n t e r f a c e . The r e l a t i v e impo r t an ce or each s t e p e x p r e s s e d i n terms o f r e s i s t a n c e to hea t f l o w has been c a l c u l a t e d by 4 Watanab le e t a l . u s i n g mould w a l l t empe r a t u r e measu re -ments . They found t h a t the a i r gap c o n s t i t u t e s the l a r g e s t r e s i s t a n c e b e i n g 84% o f the t o t a l , w h i l e the mould w a l l and m o u l d / c o o l i n g wa t e r i n t e r f a c e a c coun t ed f o r o n l y 2 and 14% r e s p e c t i v e l y . C l e a r l y t h e r e f o r e the p a t t e r n o f hea t r e -moval i n the mould i s l a r g e l y dependent on the dynamics o f gap f o r m a t i o n . F a c t o r s t h a t a l t e r the s i z e and n a t u r e o f the gap have a s t r o n g i n f l u e n c e on hea t f l o w to the mou l d . The e v o l u t i o n o f the gap depends on the a b i l i t y o f the s o l i d i f y i n g s h e l l to w i t h s t a n d the f o r c e s due to f e r -5 r o s t a t i c p r e s s u r e . Because v a r i a b l e s such as t empe r a t u r e and s t e e l c o m p o s i t i o n a f f e c t the s t r e n g t h o f the s o l i d s h e l l i n a complex w a y , 6 gap f o r m a t i o n i s s t i l l p o o r l y u n d e r s t o o d . V iewed i n s i m p l e terms i t has been p roposed t h a t gaps a re p r e s e n t o n l y i n the l owe r r e g i o n o f the mould w h i l e nea r the men i s cus good c o n t a c t e x i s t s between the s o l i d i f y i n g 2 3 7 s h e l l and the mou l d . ' ' In between t he se r e g i o n s a zone 1 8 o f i n t e r m i t t e n t c o n t a c t p r e v a i l s . ' V a r i a t i o n s i n gap s i z e a l s o o c cu r i n the t r a n s v e r s e d i r e c t i o n . The gap 10 i n i t i a t e s a t the c o r n e r s wh i ch a re c o l d e s t and may s p r e a d 9-11 a c r o s s the f a c e , as s c h e m a t i c a l l y shown i n F i g . 3 . Ma t hema t i c a l models f o r m u l a t e d i n the p a s t have not been 7 12 13 1 a b l e to c a l c u l a t e t r a n s v e r s e v a r i a t i o n s i n gap s i z e . ' » , ^» ' 5 Two n o t a b l e e x c e p t i o n s are the work o f G r i l l e t a l . and 1 5 F r e d r i k s s o n and T h e g e r s t r o m . The f o rmer based on a two-d i m e n s i o n a l ma thema t i c a l model have p r e d i c t e d t h a t i n s l a b s , gaps form a t a f i n i t e d i s t a n c e below the men i s cus and a re a maximum a t the c o r n e r s . They a l s o found t h a t a l a r g e p o r t i o n o f the w ide f a c e i s i n c o n t a c t w i t h the mould due t o f e r r o s t a t i c p r e s s u r e . F r e d r i k s s o n and T h e g e r s t r o m 1 s c a l c u -l a t i o n s f o r s l a b mou ld s , show t h a t the gap i s a lmos t non -e x i s t e n t a l o ng the m i d - f a c e o f the wide p l a t e . For b i l l e t moulds gap s i z e v a r i a t i o n s are seen to be m a i n l y i n the c o r n e r r e g i o n s . Th i s i n f o r m a t i o n s ugge s t s t h a t f o r any t r a n s v e r s e s l i c e , n o n - u n i f o r m hea t f l o w i s p r edom inan t i n the c o r n e r r e g i o n s . Measurements o f gaps have been made on i n s t r u m e n t e d m o u l d s ^ and w i d t h s o f 0.6~12 mm and 0 .2~0.8 mm have been r e p o r t e d f o r the wide and narrow f a c e s o f the s l a b r e -s p e c t i v e l y . For the case o f o i l l u b r i c a t i o n gas samp les f rom the gap have shown t h a t hydrogen i s the ma jo r c o n -s t i t u e n t and i s r e l a t e d to the d e c o m p o s i t i o n o f t he l u b r i -can t .^ S i n c e the the rma l c o n d u c t i v i t y o f the gas i s h i gh 11 + MOLD -SOLID STEEL_ LIQUID S T E E L lb AIR GAP-F i g . 3 S chema t i c View o f a T r a n s v e r s e S e c t i o n o f a B i l l e t Showing A i r g Gap Fo rma t i on . i t s i g n i f i c a n t l y i n c r e a s e s hea t t r a n s f e r a c r o s s s m a l l gaps (up to 1 mm) where c o n d u c t i o n i s the dominan t mode o f 18 1 hea t t r a n s f e r . Ak imenko and S k v o r t s o v o b s e r v e d t h a t the c o m p o s i t i o n o f hydrogen i n the gas i s a maximum as the l u b r i c a n t i s f i r s t i n t r o d u c e d (45-55%) but the a ve r age c o m p o s i t i o n d u r i n g c a s t i n g i s c l o s e r to 10-20%. Model 1 9 s t u d i e s based on H i l l ' s i n t e g r a l p r o f i l e t e c h n i q u e r e v e a l t h a t a l t h o u g h i n j e c t i o n o f hydrogen i n t o the gap i n c r e a s e s the h e a t - t r a n s f e r r a t e s i t a l s o has u n d e s i r a b l e e f f e c t s such as h i g h e r mould w a l l t empe r a t u r e s a t t he m e n i s c u s . T h i s may g i v e r i s e to h i g h e r s t r e s s e s and d i s t o r t i o n . 2.2 L o n g i t u d i n a l H e a t - F l u x P r o f i l e s L o n g i t u d i n a l h e a t - t r a n s f e r p r o f i l e s have been 4 20 o b t a i n e d by measu r i ng mould w a l l t e m p e r a t u r e s ' o r c o o l i n g wa t e r t empe r a t u r e s a t the i n l e t and o u t l e t o f 21 22 h o r i z o n t a l passages mach ined i n a mou l d . ' The p r o f i l e s , a t y p i c a l example o f wh i ch can be seen i n F i g . 4 , e x h i b i t a c h a r a c t e r i s t i c o f maximum hea t f l u x a few c e n t i m e t e r s 21 be low the m e n i s c u s . S ingh and B l a z e k have p r opo sed t h a t the low h e a t - t r a n s f e r r a t e s at the men i s cus i t s e l f a r e caused by the s e p a r a t i o n o f the s h e l l f rom the w a l l due to s h r i n k a g e o r s u r f a c e t e n s i o n f o r c e s . F u r t h e r down the mould the f e r r o s t a t i c p r e s s u r e causes the s h e l l t o make c o n t a c t w i t h the mould a g a i n , t h e r eb y i n c r e a s i n g t he hea t 0 5 10 15 20 25 30 35 D i s t a n c e f r o m t o p o f m o l d ( c m ) F i g . 4 B i l l e t Mould H e a t - F l u x as a F u n c t i o n o f D i s t a n c e 21 f rom the Top o f the Mou ld . CO 14 t r a n s f e r . Be low t h i s p o i n t the heat f l u x p r o g r e s s i v e l y d rops o f f as the s o l i d s h e l l t h i c k e n s and the heat f l o w 21 r e s i s t a n c e c o r r e s p o n d i n g l y i n c r e a s e s . S ingh and B l a z e k a l s o found t h a t a doub l e peak p e r s i s t e d i n t h e i r measu re -ments as shown i n F i g . 5 wh ich has a l s o been r e p o r t e d by 2 3 Ru s s i a n w o r k e r s . As t h e r e was no f e a s i b l e m e c h a n i s t i c , e x p l a n a t i o n f o r the dua l peak i t was a t t r i b u t e d to d i s t o r -t i o n due t o the p re sence o f the c o o l i n g wa te r c h a n n e l s . A l l measurements o f heat f l u x i n the p a s t have been made e i t h e r a t the c o l d f a c e o r some s e c t i o n in-bet.ween the hot and c o l d f a c e s . The s i m i l a r i t y o f the hea t f l u x p r o f i l e a t the ho t f a c e to the measured p r o f i l e wou ld depend on whe the r the heat f l o w a c r o s s the mould i s u n i - d i m e n s i o n a l . No a t t emp t s have been made to i n v e s t i g a t e the e x t e n t o f t w o - d i m e n s i o n a l hea t f l o w , i f any , i n the mou l d . U s i ng hea t f l o w da ta many i n v e s t i g a t o r s have a t t emp t ed to p r e d i c t l o n g i t u d i n a l h e a t - t r a n s f e r c o e f f i c i e n t p r o f i l e s . 4 Fo r example Watanabe e t a l . o b t a i n e d the p r o f i l e shown i n 24 F i g . 6 . H i l l s has p roposed t h a t the heat t r a n s f e r c o -e f f i c i e n t h m , d e c r e a se s a c c o r d i n g to the f o l l o w i n g e q u a t i o n : h m = h Q ( 1 - n z ) . . . ( 1 ) where h i s the v a l u e o f h a t the men i s cus and z i s the o m depth be low the m e n i s c u s . Akimenko and S k v o r t s o v 1 7 have r e p o r t e d t h a t f o r a 1.5 m l o ng mould the upper t h i r d i s most e f f e c t i v e f rom a heat t r a n s f e r p o i n t o f v iew s i n c e the ± i i i i 0 5 10 15 20 25 30 35 4 0 Distance from top of mold (cm) F i g . 5 A Double Peak i n a B i l l e t Mould H e a t - F l u x P r o f i l e . 2 1 16 F i g . 6 V a r i a t i o n o f H e a t - T r a n s f e r C o e f f i c i e n t 4 Down the Mou l d . 17 s l a b s u r f a c e t empe r a t u r e was ob se r ved to be a lmos t i n -v a r i a n t i n the l owe r r e g i o n . Mould d e s i g n , geome t r y , o p e r a t i n g v a r i a b l e s and s t e e l c h e m i s t r y have been found to have an i n f l u e n c e on the r a t e o f heat f l o w , and the subsequen t s e c t i o n s a t t emp t to draw t o g e t h e r a l l t he a v a i l a b l e i n f o r m a t i o n on the s u b j e c t . 2 . 2 . 1 The I n f l u e n c e o f Mould Des ign and Geometry  on Heat F l ow . (a) Mould Tape r : In the case o f b i l l e t s i t has been ob se r ved t h a t mould t a p e r imp roves h e a t -t r a n s f e r 1 0 ' 2 5 and a l s o d e c r e a s e s the s u r f a c e t empe r a t u r e at t h e mould e x i t , 1 7 p r e sumab l y because o f the r e d u c t i o n o f the gap w i d t h o ve r the l owe r r e g i o n o f the mou l d . S i n c e th\i.s g.a,p i s m a i n l y due to s t e e l s h r i n k a g e , e s t i m a t e s o f the r e q u i r e d t a p e r can be made. 5 G r i l l e t a l . have i n v e s t i g a t e d the e f f e c t s o f d i f f e r e n t end p l a t e t a p e r s i n a s l a b mould by means o f a ma thema t i c a l m o d e l . T h e i r r e s u l t s demons t r a t e the adve r s e e f f e c t s o f e x c e s s i v e t a p e r . I f t h i s c o n d i t i o n o b t a i n s , 18 the r e d u c t i o n o f w i d t h owing to t a p e r i n the l owe r r e g i o n o f the mould i s l a r g e r t han t h a t caused by the rma l s h r i n k a g e ; thus the nar row f a ce o f the s l a b i s compressed by the mould wh i ch causes e x c e s s i v e e n d - p l a t e wear nea r the mould e x i t . I t a l s o enhances c o o l i n g and s h r i n k a g e of t h i s f a c e , wh i ch i n c r e a s e s the gap on the w ide f a c e . The r e s u l t a n t \ drop i n l o c a l hea t e x t r a c t i o n causes a ho t s po t to form o f f the c o r n e r s on the w ide f a c e , and t h i s can g i v e r i s e to o f f - c o r n e r l o n g i t u d i n a l c r a c k s o r b r e a k o u t s . The 2 fi e x p e r i m e n t a l r e s u l t o f S a i t o and Ta te i n -d i c a t e the u n d e s i r a b i 1 i t y o f e x c e s s i v e e n d -p l a t e t a p e r i n s l a b mou l d s . The l a t t e r a u t h o r s have recommended t h a t the t a p e r be s e l e c t e d a c c o r d i n g to c a s t i n g speed and grade o f s t e e l b e i n g c a s t , s i n c e t he se two f a c t o r s cause c o n s i d e r a b l e v a r i a t i o n i n hea t f l u x . A t p r e s e n t however t h e r e i s no d e f i n i t e r u l e t h a t can be a p p l i e d f o r a p a r t i c u l a r s l a b s i z e under a g i v e n s e t o f c a s t i n g c o n d i t i o n s . 27 Hememann i n v e s t i g a t e d the e f f e c t o f mould shape on hea t f l o w . H i s r e s u l t s show 19 a 50% i n c r e a s e i n heat e x t r a c t i o n f o r a p a r a b o l i c mould shape . Based on t h e s e r e s u l t s a u n i v e r s a l tube mould d e s i g n has been a r r i v e d a t wh ich i s m u l t i - t a p e r e d . (b) I n t e r n a l Work ing S u r f a c e o f the Mou l d : Many Ru s s i an moulds have been d e s i g n e d w i t h d i r e c t i o n a l c o r r u g a t i o n and t a p e r e d 2 8 r i b s . I t has been sugge s t ed t h a t t h i s d e s i g n reduces the f o r m a t i o n o f a i r gaps between the s t r a n d and mould i n t e r f a c e t h e r e b y 29 29 i m p r o v i n g hea t t r a n s f e r . Perminov e t a l . c l a i m t h a t the 8~9% i n c r e a s e i n e f f e c t i v e p e r i m e t e r caused by the c o r r u g a t i o n s i n -c r e a s e s the ave rage hea t f l u x by 8%. The r e d u c t i o n i n t r a n s v e r s e d i s p l a c e m e n t o b s e r v ed w i t h t h i s t ype o f mould a p p a r e n t l y e n su r e s more u n i f o r m c o o l i n g and a r e d u c t i o n i n s t e e l e x i t t empe r a t u r e s o f 50 K. ( c ) Mould D i s t o r t i o n : Very l i t t l e i n f o r m a t i o n i s a v a i l a b l e a t i p r e s e n t on the i n f l u e n c e o f mould d i s t o r t i o n 30 on hea t t r a n s f e r . H a l l i d a y has r e p o r t e d t h a t an i n c r e a s e i n heat t r a n s f e r c o u l d be 20 o b t a i n e d by u s i n g f l e x i n g t u b u l a r coppe r moulds as compared to normal b l o c k moulds o r t u b u l a r mou l d s . He sugge s t ed t h a t the e l a s t i c b e h a v i o u r o f the moulds i s as i m p o r t a n t as the e l a s t o - p l a s t i c b e h a v i o u r o f the s o l i d i -f y i n g s h e l l . T h i s a spe c t o f mould b e h a v i o u r c o n s i d e r a b l y c o m p l i c a t e s any a t t emp t to de t e rm ine t h e o r e t i c a l l y 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 the mould and the s h e l l on the b a s i s o f gap f o r m a t i o n . (d) Work ing L e n g t h : 20 21 From F i g . 4 and o t h e r s t u d i e s ' i t i s appa r en t t h a t more than 50% o f the hea t i s e x t r a c t e d i n the upper h a l f o f the mou l d . Thus the l owe r h a l f f u n c t i o n s l a r g e l y as a s u p p o r t to the s o l i d s h e l l . Fo r t h i s r ea son t h e r e seems l i t t l e j u s t i f i c a t i o n f o r i n -c r e a s i n g mould l e n g t h s beyond t h e i r p r e s e n t v a l u e o f 60~70 cms u n l e s s s i g n i f i c a n t i n c r e a s e s i n c a s t i n g speed a re c o n t e m p l a t e d . In t h i s 31 c o n n e c t i o n Br imacombe has s ugge s t ed t h a t the mould l e n g t h s h ou l d be i n c r e a s e d p r o -p o r t i o n a t e l y to the i n c r e a s e i n c a s t i n g s p e e d , i f the s h e l l t h i c k n e s s a t the mould e x i t i s to be kept c o n s t a n t . 21 Fu tu re deve lopments i n t h i s a r ea un -d o u b t e d l y w i l l be i n the d i r e c t i o n o f s h o r t e r mou l d s , not o n l y f o r r e a sons o f hea t t r a n s f e r but because s h o r t moulds o f f e r advan tages o f r educed c o s t and i n c r e a s e d s e r v i c e l i f e . There i s p r o b a b l y a l i m i t , however t o the , p o s s i b l e r e d u c t i o n i n l e n g t h because o f the i n c r e a s e d danger o f b r e a k - o u t s . The use o f s p r ay c o o l e d sub -mou ld s u p p o r t a s s e m b l i e s 3 2 such as i n the ' m u l t i - s t a g e ' mou ld may h e l p to overcome t h i s c h r o n i c p r o b l e m . (e) Mould m a t e r i a l s : For opt imum pe r f o rmance the mould m a t e r i a l has to r e t a i n s t r e n g t h a t the ave r age o p e r a t i n g t e m p e r a t u r e s , be i n e r t to mo l t en s t e e l , and above a l l , have good h e a t - t r a n s f e r c h a r a c t e r i s t i c s . D e o x i d i z e d coppe r was one o f the e a r l i e s t c h o i c e s , and c o n t i n u e s to be u s ed , owing to i t s h i gh t he rma l c o n d u c t i v i t y . However because o f i t s low y i e l d s t r e n g t h , t he rma l s t r e s s e s p roduced a t o p e r a t i n g t empe r a t u r e s are s u f f i c i e n t to cause p e r -manent d i s t o r t i o n and r educed mould l i f e . Thus a s e r i e s o f h i gh y i e l d - s t r e n g t h coppe r 22 33 a l l o y s have been deve l oped wh i ch have a w o r k i n g l i f e s e v e r a l t imes t h a t o f d e o x i d i z e d coppe r as w e l l as improved wear r e s i s t a n c e . S i n c e a l l o y i n g d e c r e a s e s the t he rma l c o n -d u c t i v i t y , the c h o i c e o f a l l o y a l s o d.epends on i t s the rma l p r o p e r t i e s . 34 S o v i e t s c i e n t i s t s examined the p o s -s i b i l i t y o f u s i n g s t e e l as a mould m a t e r i a l . From a heat t r a n s f e r p o i n t o f v i e w , i t a p -pea red t h a t by u s i n g t h i n n e r s t e e l w a l l s i t wou ld be p o s s i b l e to o b t a i n heat f l u x e s comparab le i n magn i tude to t hose a t t a i n e d i n r e g u l a r copper mou l d s . However c o n t a c t o f the b i l l e t w i t h a s t e e l mould had a d e l e t e r -i o u s e f f e c t on p r o du c t q u a l i t y . 2 . 2 . 2 I n f l u e n c e o f O p e r a t i n g V a r i a b l e s on Hea t - T r a n s f e r (a) C a s t i n g s p e e d . Many i n v e s t i g a t o r s have ob se r ved an 2 21 i n c r e a s e i n heat f l u x ' and mould w a l l t empe r a t u r e s ' w i t h an i n c r e a s e i n c a s t i n g s p e e d . The a v e r a g e , o v e r a l l h e a t - t r a n s f e r 23 c o e f f i c i e n t has a l s o been no t ed to i n c r e a s e 4 17 34 w i t h c a s t i n g s p e e d . ' ' A l t h o u g h a d e -f i n i t e r e l a t i o n has no t been f o r m u l a t e d , i t has been sugge s t ed t h a t the i n c r e a s e i s 37 l i n e a r . Br imacombe e t a l . have d emon s t r a t e d t h a t the mean h e a t - t r a n s f e r c o e f f i c i e n t can be r e l a t e d to dwe l l t ime as f o l l o w s f o r d i f f e r e n t mou l d s . h" = 1.696 - 0.0162 t . . . ( 2 ) m m 2 where h m (kW/m °C) i s the mean h e a t - t r a n s f e r co-e f f i c i e n t i n the mould and t i s the dwe l l m t ime i n s e e s . The a u t h o r s have a l s o shown t h a t a t i m e - a v e r a g e d form o f Savage and 51 P r i t c h a r d s r e l a t i o n , q = 2680 - 221 .9 vT^ . . . ( 3 ) a d e q u a t e l y d e s c r i b e s the v a r i a t i o n o f t he o v e r a l l hea t f l u x q Q (kW/m ) w i t h dwe l l t i m e t ( s e e s ) f o r 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 . m D e s p i t e the i n c r e a s e i n h e a t - t r a n s f e r r a t e w i t h c a s t i n g s peed , i t i s i m p o r t a n t t o 24 no te t h a t the s p e c i f i c hea t e x t r a c t i o n ( J / k g ) de c r ea se s r e s u l t i n g i n a net d e c r e a s e i n s h e l l t h i c k n e s s a t a g i v e n p o s i t i o n be low the men i scus . The magn i tude of t empe r a t u r e f l u c t u a t i o n s i n mould p l a t e s has been found to i n c r e a s e 35 w i t h c a s t i n g s p e e d . Mo r eove r , mould powders wh i ch f l ow uneven l y between the s h e l l and the mould g i v e r i s e , t o t empe r a t u r e f l u c t u a t i o n s wh i ch i n c r e a s e ma r ked l y w i t h h i g h e r c a s t i n g speeds . (b) P o u r i n g t e m p e r a t u r e : I n c r e a s e i n c a s t i n g t empe r a t u r e w i t h i n r e a s o n a b l e l i m i t s has a n e g l i g i b l e e f f e c t on 2 25 1 21 hea t t r a n s f e r . ' S i ngh and B l a z e k found t h a t f o r t h r e e c a s t i n g t empe r a t u r e s i n the range 1577 to 1622°C, the h e a t - t r a n s f e r p r o f i l e s rema ined the same. The m i d - s e c t i o n s h e l l t h i c k n e s s was a l s o the same f o r the t h r e e c a s e s . However e x a m i n a t i o n o f the t r a n s v e r s e s e c t i o n o f b r e a k - o u t s h e l l s r e v e a l e d t h a t w i t h i n c r e a s i n g s u p e r h e a t , the s h e l l t h i c k n e s s a t the c o r n e r s d e c r e a s e d . From a s t r i c t l y 25 h e a t - t r a n s f e r p o i n t o f v i e w , the r eason f o r t h i s i s no t c l e a r . ( c ) C o o l i n g Water V e l o c i t y and Tempe r a t u r e : E x p e r i m e n t a l s t u d i e s r e v e a l t h a t i n -c r e a s e s i n c o o l i n g wa t e r v e l o c i t y above 6m/sec do not improve the o v e r a l l hea t t r a n s f e r t o 1 25 34 the mou ld . ' ' T h i s f i n d i n g c o n f i r m s t h a t the h e a t - t r a n s f e r c o e f f i c i e n t a t the mou l d / wa t e r i n t e r f a c e c o n t r i b u t e s a s m a l l f r a c t i o n to the t o t a l r e s i s t a n c e to hea t f l o w and m ino r v a r i a t i o n s i n the f o rme r do not a f f e c t the l a t t e r . C o o l i n g wa te r t empe r a t u r e f l u c t u a -t i o n s between 20°C and 60°C have no t been ob se r ved to have any e f f e c t on the hea t f l o w . (d) Mould L u b r i c a n t s : Of the f a c t o r s t h a t do i n f l u e n c e the hea t f l ow to the mou l d , l u b r i c a n t t ype i s one o f the most i m p o r t a n t . O i l l u b r i c a n t s commonly used i n b i l l e t mou l d s , p a r t i a l l y breakdown a t h i gh t empe r a t u r e to p roduce a hydrogen 9 38 r i c h atmosphere i n the gap. * Mould powders employed to l u b r i c a t e s l a b moulds do not de -compose i n t h i s manner but s i m p l y me l t and 26 wet the s t e e l . Due to the h i g h c o n d u c t i v i t y o f h y d r o g e n , o i l l u b r i c a t i o n o f t e n r e s u l t s i n h i g h e r heat f l u x e s i n the upper mould compared 3 8 to powders . In s l a b c a s t e r s the r a t e o f hea t f l o w to the mould depends on t he t ype o f mould powder and f l u i d - f l o w c o n d i t i o n s i n the mou l d , the l a t t e r b e i n g a f f e c t e d by n o z z l e 39 t y p e . A r e l a t i v e l y "1 ow h e a t - f 1 ux" p r o f i 1 e may o b t a i n w i t h the use o f a h i g h - m e l t i n g powder and a b i - f u r c a t e d n o z z l e i n wh i ch the n o z z l e s t reams a re d i r e c t e d ' towards the na r row f a c e s o f the mou ld . The " h i g h h e a t - f l u x " p r o f i l e may be found w i t h a l o w - m e l t i n g powder and a m u l t i - p o r t n o z z l e , p a r t i c u l a r l y i f t he n o z z l e s t r eam washes a g a i n s t the b road f a c e o f the s l a b . F i g . 7 i s a c ompa r i s on o f t he h e a t - f l u x p r o f i l e s o b t a i n e d f o r t h r e e d i f f e r e n t l u b r i -22 c a t i o n p r a c t i c e s . The i n c r e a s e i n hea t f l u x nea r the bot tom o f the mould f o r a l l t h r e e cases has been a t t r i b u t e d to p e n e t r a t i o n o f wa t e r and i t s e n s u i n g d e c o m p o s i t i o n to p roduce a hydrogen r i c h a tmosphe r e . S i n c e the l o w -m e l t i n g powder wets the s t r a n d more e f f e c t i v e l y , the e f f e c t i s r e d u c e d . The t ype o f powder used a l s o seems to have an i n f l u e n c e on the e x t e n t 27 F i g . 7 S p e c i f i c Heat E x t r a c t i o n as a F u n c t i o n o f D i s t a n c e f rom the Top o f the Mould f o r D i f f e r e n t L u b r i -9 c a n t s . o f t empe r a t u r e f l u c t u a t i o n s i n the coppe r p l a t e s o f a s l a b mou ld , some c a u s i n g l a r g e r f l u c t u a t i o n s than o t h e r s . 2 . 2 . 3 I n f l u e n c e o f S t e e l C ompo s i t i o n on Heat  Trans f e r Heat t r a n s f e r i n the mould v a r i e s w i t h the 40 c o m p o s i t i o n o f the s t e e l b e i n g c a s t . An i n v e s t i g a t i o n on a bench s c a l e c a s t e r has r e v e a l e d t h a t the hea t f l u x i s a minimum when c a s t i n g 0.10% ca rbon s t e e l s as seen i n F i g . 8 . Th i s e f f e c t was r e p o r t e d to be more p ronounced f o r the 2 7 case o f the p a r a b o l i c mould shape . The r eason f o r the low h e a t - t r a n s f e r r a t e s has been e x p l a i n e d by G r i l l and B r i m a -41 combe. The p roposed mechanism i s based on the l a r g e s h r i n k a g e a s s o c i a t e d w i t h the 8-y t r a n s f o r m a t i o n i n the s o l i d s t a t e 0.1% carbon s t e e l s . In t h i s c o n n e c t i o n the i n t e r i o r and e x t e r i o r s u r f a c e s o f b r e a k - o u t s h e l l s have been examined and found to have a r i p p l e d appea rance due to 40 the phase t r a n s f o r m a t i o n . A second s t udy unde r t a ken by 21 S ingh and B l a z e k c o n f i r m e d t h e i r p r e v i o u s o b s e r v a t i o n s and shed new l i g h t on the i n f l u e n c e o f s t e e l c o m p o s i t i o n . The ca rbon c o n t e n t was found to have a n e g l i g i b l e i n f l u e n c e on the shape o f the h e a t - f l u x p r o f i l e but the peak heat 21 f l u x e s were l e a s t f o r 0.10% carbon s t e e l s ; . O the r e l emen t s were a l s o ob s e r v ed to have an e f f e c t on mould hea t t r a n s f e r . CM, ^ 2 0 0 0 1800 CD +— a K_ i _ o> <+-CO c o £ 1600 0) . c o E 1400 > < 1200 I i V 0.4 Casting speed = - 21.2mm / sec 1 1 0.8 1.2 Steel carbon content (%) 1.6 F i g . 8 Average B i l l e t Mould H e a t - F l u x as a F u n c t i o n o f Carbon C o n t e n t . For 0.10% ca rbon s t e e l s , 0.30% s u l p h u r i n c r e a s e d the mould hea t f l u x by 25%, w h i l e 1% n i c k e l caused a 11% i n c r e a s e . In c o n t r a s t to t h i s the same l e v e l o f s u l p h u r i n 0.70% 40 ca rbon s t e e l s d e c r e a s ed the hea t f l u x by 5%. 2.3 Ma t hema t i c a l M o d e l l i n g o f Heat Flow i n Moulds S e v e r a l s t u d i e s have been c o n d u c t e d 3 ' ^ > 4 2 - 5 0 t Q m a t h e m a t i c a l l y model the heat f l ow i n the mould e x p r e s s l y f o r c a l c u l a t i n g the s u r f a c e t empe r a t u r e s and s h e l l t h i c k n e s s o f the s t r a n d . The boundary c o n d i t i o n a t the m o u l d / s t e e l i n t e r f a c e has been mode l l e d as a h e a t - t r a n s f e r c o e f f i c i e n t deduced f rom measurements o r c a l c u l a t e d by v a r i o u s t e c h n i -„ 4 6 , 5 0 , 4 2 . . ques. The e q u a t i o n , q . = 2680 - 335 / t . . . (4) r e l a t i n g the i n s t a n t a n e o u s hea t f l u x q . (kW/m ) to t ime t ( s e e s ) , was i n i t i a l l y o b t a i n e d by Savage and P r i t c h a r d ^ 1 f o r a w a t e r - c o o l e d s t a t i c mou l d . A l t h ough i t p r e d i c t s ave rage hea t f l u x e s i n c o n t i n u o u s c a s t i n g moulds r e a s o n a b l y w e l l , f rom the p r e c e d i n g d i s c u s s i o n i t i s e v i d e n t t h a t i t may not be a p p l i c a b l e f o r d i f f e r e n t s t e e l c o m p o s i t i o n s and l u b r i c a t i o n p r a c t i c e s . The t empe r a t u r e f i e l d f o r the s t r a n d has been c a l c u l a t e d i n the pa s t by many w o r k e r s , but i t was o n l y r e c e n t l y t h a t papers d e s c r i b i n g the the rma l r e sponse o f t he mould to the hea t f l ow from the s t r a n d began to appear i n the l i t e r a t u r e . 5 2 , 5 3 , 5 4 53 In a s t udy on hea t f l o w i n s l a b mou ld s , the h e a t -f l u x d i s t r i b u t i o n i n a t r a n s v e r s e p l ane has been c a l c u l a t e d f rom t empe r a t u r e measurements . The the rma l f i e l d was com-puted u s i n g a t w o - d i m e n s i o n a l h e a t - f l u x mode l . R e s u l t s o f t h i s i n v e s t i g a t i o n have l e d to the r e d e s i g n o f mould c o o l i n g - w a t e r c hanne l s i n the c o r n e r r e g i o n i n an a t t emp t to reduce the l o c a l t empe r a t u r e g r a d i e n t s . A r e d u c t i o n i n c o r n e r gap s i z e was a n t i c i p a t e d w i t h the new d e s i g n wh i c h wou l d d e c r e a s e t he s e v e r i t y o f c o r n e r c r a c k s and b r e a k -54 o u t s . A more r e c e n t work d e s c r i b e s the i n f l u e n c e o f d i f f e r e n t mould powders and c a s t i n g speeds on the ave rage hea t f l u x i n the mou ld . The the rma l f i e l d i n a two-d i m e n s i o n a l p l ane has a l s o been computed . 2 .4 D i s t o r t i o n o f Moulds Thermal s t r e s s e s gene r a t ed d u r i n g o p e r a t i o n a re g e n e r a l l y known to cause permanent d i s t o r t i o n i n mou l d s . Th i s i s e x e m p l i f i e d i n a s t udy conduc ted by Br imacombe e t 55 a l . i n which, c o l d measurements o f a number o f used b i l -l e t moulds f rom d i f f e r e n t companies were made. The e x t e n t 32 o f d i s t o r t i o n was found to v a r y f rom mould to mould i n d i c a t i n g a dependence on e i t h e r o p e r a t i n g c o n d i t i o n s , mould d e s i g n , m a t e r i a l p r o p e r t i e s , s e r v i c e l i f e o r a c o m b i n a t i o n o f t he se f a c t o r s . The d i s t o r t i o n was g e n e r a l l y i n the form o f outward b u l g i n g i n the upper mould r e g i o n w i t h a peak s e v e r a l c e n t i m e t e r s be low the m e n i s c u s . T h i s r e s u l t s i n a r e g i o n o f n e g a t i v e t a p e r j u s t be low the s t e e l l e v e l . A n e g a t i v e t a p e r e i t h e r due to t he rma l d i s t o r t i o n o r wear was a l s o ob se r ved nea r the bot tom o f some mou l d s . 55 F i g . 9 p r e s e n t s a t y p i c a l r e s u l t . Mould d i s t o r t i o n i n the fo rm o f a n e g a t i v e t a p e r e f f e c t j u s t be low the men i s cus has a l s o been r e p o r t e d by A n d r e w . 5 6 A ke t a and U s h i j u m a 1 0 have a t t r i b u t e d the s t eady de c r ea se i n hea t t r a n s f e r ob se r v ed i n b i l l e t moulds to the p r o g r e s s i v e d i s t o r t i o n o f the mou l d . The e f f e c t o f t h i s shape change on q u a l i t y w i l l be d i s c u s s e d 1 a t e r . D imens i ona l changes i n the men i s cus r e g i o n o f b loom moulds were found to be o p p o s i t e to t h a t o b s e r v e d i n b i l l e t mou l d s . Inward bowing o f the mould w a l l was found to o c c u r 50 to 75 mm below the m e n i s c u s , 5 ^ ' 5 ^ ( F i g . 1 0 ) . In some moulds w i t h p r o l o n g e d s e r v i c e p r o g r e s s i v e r e v e r s a l o f t a p e r 10 57 59 was found to o c cu r ' ' wh i ch i n t u r n l e d to an i n c r e a s e d s e v e r i t y o f l o n g i t u d i n a l c r a c k i n g . M o n i t o r i n g d i m e n s i o n a l changes o f mould w a l l s has 33 Distance Between Opposite F a c e s (in.) 7 0 5 0 7 1 0 0 7 1 5 0 7 0 5 0 7 1 0 0 7 1 5 0 I 1 Straight Wall • 100 | 2 0 0 2 O 3 0 0 Q. £ 4 0 0 E o i t 5 0 0 a> o c o 5 6 0 0 7 0 0 8 0 0 h_ B682 _| Constraints T 1 C u r v e d Wal l CA ^ -OD O A • O A • O A D o A n a n CD CD ao QD OO a a c n a • CA • O A • OA Meniscus • OA T o • _ i _ 0CX\ O A D o A n o A n o/n o n O o z n a n QQ Q D c n o n Ch • Ch OA — I t • 0 2 4 6 _ 8 ~ T3 10 3 o 12 5 14 Q. o 16 *~ i8 i 2 0 o 2 2 o </> 2 4 b 2 6 2 8 3 0 179 180 181 179 180 181 182 Distance Between Opposite F a c e s (mm) F i g . 9 Dimensional Changes in a B i l l e t Mould A f t e r Several H e a t s . 5 5 r a r e l y been a t t e m p t e d ; the q u e s t i o n a r i s e s as to the a c -c u r a c y o f measurement fo r sma l l movements o f the copper i n a h i g h t empe r a t u r e env i r onmen t t h a t i s s u b j e c t to a c on -27 s i d e r a b l e amount o f v i b r a t i o n and n o i s e . Hememann has e xpe r imen t ed w i t h a s e r i e s o f i n d u c t i v e t r a n s d u c e r s and r e p o r t e d t h a t f o r b i l l e t moulds shape changes were ve r y s m a l l . P u h r i n g e r and H o l l e i s have measured s t ud f o r c e s i n s l a b moulds u s i n g s t r a i n gauges . A f i n i t e - e l e m e n t model was a l s o deve l oped to o b t a i n a t h r e e - d i m e n s i o n a l t he rma l l o a d map shown i n F i g . 1 1 . The c a l c u l a t e d and measured the rma l l o ad s were found to be i n good ag reemen t . These r e s u l t s were then used to compute the d i s t o r t i o n o f the coppe r p l a t e . The r e s u l t i n g p r o f i l e e x h i b i t e d an ou tward bu l g e i n the men iscus r e g i o n o f the coppe r p l a t e . E l e c t r i c a l ana l ogues and p h o t o - e l a s t i c m o d e l l i n g t e c h n i q u e s have a l s o been used to d e t e rm i ne the t empe r a t u r e and s t r e s s d i s t r i b u t i o n i n c o r r u g a t e d coppe r m o u l d s . 6 1 The r e s u l t s o f t h i s s t udy i n d i c a t e t h a t the s t r e s s e s i n the mould exceed the y i e l d - p o i n t o f copper a t many l o c a t i o n s . In o r d e r to reduce the s t r e s s e s they recommend the use o f mould powders to d e c r e a s e heat e x t r a c t i o n r a t e s . The l i f e o f a mould c o u l d be l i m i t e d by the f o rma -t i o n o f f a t i g u e c r a c k s i n the men i scus r e g i o n , r e s u l t i n g f rom c y c l i c s t r e s s e s g ene r a t ed by mould o s c i l l a t i o n . 11 Thermal Load Map f o r a S l ab Mould P l a t e . 6 0 Y o s h i h a r a e t a l . t h rough a s e r i e s o f c a l c u l a t i o n s have c shown t h a t the the rma l s t r e s s e s at the men i scus can be r educed by i n c r e a s i n g the c o o l i n g - w a t e r t e m p e r a t u r e . H i g h e r wa t e r t empe r a t u r e s reduce the g r a d i e n t a c r o s s the men i s cus t h e r e b y l o w e r i n g the the rma l s t r e s s e s . For opt imum pe r f o rmance and l ong s e r v i c e l i f e , m a t e r i a l s e l e c t i o n i s an a s p e c t o f pr ime i m p o r t a n c e . D e o x i d i z e d copper wh i ch was one o f the e a r l i e s t c h o i c e s , due to i t s h i gh the rma l c o n d u c t i v i t y , i s s t i l l w i d e l y u s e d , but i n r e c e n t y e a r s h i gh s t r e n g t h coppe r a l l o y s and s u r f a c e ha rdened moulds have become p o p u l a r due to the improved r e s i s t a n c e to permanent d i s t o r t i o n and wea r . E l e c t r o -formed mou ld s , wh i ch c o n s i s t o f a p r e c i p i t a t i o n hardened coppe r body l i n e d w i t h a .3 mm l a y e r o f n i c k e l have w i t h -s t o od 500~600 h e a t s / 1 i n i n g . Chromium p l a t e d moulds a re p o p u l a r due to t h e i r wear r e s i s t a n t p r o p e r t i e s , and have s u c c e s s f u l l y been used to a v e r t hot s h o r t n e s s o f the s t r a n d 64 by coppe r a b r a s i o n . A s e r i e s o f h i gh s t r e n g t h coppe r a l l o y s t h a t c o u l d be e xpec t ed to have a l i f e s e v e r a l t ime s t h a t o f d e o x i d i z e d copper have been deve l oped and a re b e i n g 33 t e s t e d i n p l a n t . A r e c e n t s u r v ey o f mould m a t e r i a l s i n d i c a t e s t h a t 70% o f c o n t i n u o u s c a s t i n g moulds a re made 6 5 o f coppe r a l l o y e d w i t h 0.1% s i l v e r . S i l v e r has been known to i n c r e a s e the r e c r y s t a l 1 i z a t i o n t empe r a t u r e o f c o p p e r . In the case o f commerc i a l g rade coppe r the r e -c r y s t a l l i z a t i o n k i n e t i c s appea r to be i n f l u e n c e d by t he many r e s i d u a l e l ements t h a t a re p r e s e n t , and the a d d i t i o n £ 55 o f sma l l q u a n t i t i e s o f s i l v e r does not have much e f f e c t . Hardness measurements o f s e v e r a l b i l l e t moulds t h a t had been i n s e r v i c e f o r v a r y i n g l e n g t h s o f t ime were found t o be u n i f o r m , i n d i c a t i n g t h a t l i t t l e o r no s o f t e n i n g had 55 t aken p l a c e . Th i s s ugge s t s t h a t mould l i v e s c o u l d be improved by go i ng to h i g h e r s t r e n g t h a l l o y s r a t h e r than by me re l y chang i ng the r e c r y s t a l 1 i z a t i o n k i n e t i c s . 2.5 I n f l u e n c e o f Mould C o n d i t i o n s on Q u a l i t y Many t ypes o f c r a c k s found i n c o n t i n u o u s l y c a s t s e c t i o n s can be t r a c e d to c o n d i t i o n s i n the mou l d . 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 i n bo th the m i d - f a c e and c o r n e r r e g i o n s a re good e xamp l e s . The mould i s p a r t i c u l a r l y s u s c e p t i b l e to the g e n e r a t i o n o f s u r f a c e c r a c k s because o f the h i gh t empe r a t u r e of the s u r f a c e o f the s t r a n d nea r the m e n i s c u s . Moreover i t has been found t ha t , a t t empe r a -t u r e s 50~100°C be low the s o l i d u s , s t e e l has ve r y low d u c t i l i t y and s t r e n g t h , and i s i n c a p a b l e o f w i t h s t a n d i n g t e n s i l e s t r a i n s i n e x ce s s o f 0 . 2 - 0 . 3 % . 6 6 Br imacombe and S o r i m a c h i 6 have examined the d i f f e r e n t t ypes o f s u r f a c e and i n t e r n a l c r a c k s t h a t can a r i s e and have i d e n t i f i e d the main f a c t o r s c o n t r i b u t i n g to each d e f e c t . 39 2 .5 .1 Mould R e l a t e d S u r f a c e D e f e c t s and C r a c k s  i n C o n t i n u o u s l y Cas t S l a b s Q u a l i t y p rob lems i n s l a b s t h a t a re due to imp rope r mould c o n d i t i o n s a re p r e d o m i n e n t l y s u r f a c e c r a c k s . Tab l e I i s a summary o f u n f a v o u r a b l e mould c o n d i t i o n s and the r e l a t e d d e f e c t . In g ene r a l s u r f a c e c r a c k i n g i n s l a b moulds f a l l i n t o t h r e e c a t e g o r i e s . L o n g i t u d i n a l m i d - f a c e c r a c k s , l o n g i t u d i n a l o f f - c o r n e r o r edge c r a c k s and t r a n s v e r s e c r a c k s . A mechanism f o r the f o r m a t i o n o f the f i r s t t ype o f c r a c k s has been p roposed by Br imacombe e t a l . , 6 ^ e x p l a i n i n g the i n f l u e n c e o f e n d - p l a t e t a p e r , sub -mou ld s u p p o r t and o t h e r o p e r a t i n g f a c t o r s , t o g e t h e r w i t h s u g g e s t i o n s f o r r e -du c i ng t h e i r o c c u r r e n c e . O f f - c o r n e r edge c r a c k s seem to be a s s o c i a t e d w i t h b u l g i n g o f the s l a b edge due to i n s u f f i c i e n t s u p p o r t . 6 8 T r a n s v e r s e c r a c k s t h a t appear i n o s c i l l a t i o n m a r k s 1 1 0 have been l i n k e d to the p r e c i p i t a t i o n A1N 6 and the low d u c t i l i t y o f s t e e l i n the 700~900°C t empe r a t u r e r a n g e . A l t h ough the c r a c k s o r i g i n a t e i n the sub -mou ld r e g i o n t h e i r i n c i d e n c e has been shown to d e c r e a s e , by i n c r e a s i n g the r a t e o f o s c i l l a t i o n . Other p r o c e s s changes t h a t have s u c c e s s f u l l y r educed t r a n s v e r s e c r a c k i n g i n c l u d e c o n t r o l o f s o l u b l e a l um i n i um i n the s t e e l as w e l l as c o n t r o l o f the p a t t e r n and f l o w r a t e o f s p r a y - w a t e r . 1 1 0 40 Tab le I Adverse Slab Mould C o n d i t i o n s and A s s o c i a t e d D e f e c t s . MOULD CONDITION DEFECT Mould d i s t o r t i o n , wear and and loss of taper. Increases long i tud ina l m i d - f a c e 6 ' 6 9 and edge c r a c k s . 6 ^ I n s u f f i c i e n t taper on the Enhances long i tud ina l o f f - c o r n e r c r a c k s . 2 6 end p la te . I n s u f f i c i e n t r o l l e r apron support. Enhances long i tud ina l mid-face c r a c k s . 6 9 ' 7 0 Poor alignment between mould and support system. pc CO Enhances edge 5 cracks and long i tud ina l mid-face c r a c k s . 7 0 Uneven cool ing in the mould. Results in long i tud ina l surface cracks.6 .69,70 Improper mould powder usage. Surface cracking and i n c l u s i o n s . 7 1 Improper mould o s c i l l a t i o n condi t ions. Transverse cracks at o s c i l l a t i o n i . 73,74 marks near slab corners. Pick up of copper from * ' 73,75,76 Star cracking on surface. surface of mould. 4 1 2 . 5 . 2 Mould R e l a t e d Q u a l i t y Problems i n B i l l e t s and Blooms A l t h o u g h s u r f a c e and i n t e r n a l q u a l i t y o f b i l l e t s and blooms i s a s u b j e c t t h a t has been w i d e l y d i s -cussed i n t he l i t e r a t u r e , t he p rob lem has n o t been s o l v e d due t o i n c o m p l e t e u n d e r s t a n d i n g o f how t h e s e c r a c k s o r i g i -n a t e . Many o f t he c r a c k s o c c u r r i n g i n b i l l e t s and blooms have been r e l a t e d t o adve rse mould c o n d i t i o n s and Tab le I I i s a summary o f t h e i n f o r m a t i o n i n t he l i t e r a t u r e . The two most p r e v a l e n t fo rms o f s u r f a c e and i n t e r n a l c r a c k s a re t h e l o n g i t u d i n a l c o r n e r c r a c k and t h e o f f - c o r n e r i n t e r n a l c r a c k . S e v e r a l i n v e s t i g a t o r s i n t h e f i e l d have e x p r e s s e d the v iew t h a t l o n g i t u d i n a l c o r n e r c r a c k s a re l i n k e d t o t h e rhombo id c o n d i t i o n o f t he b i l l e t . 2 9 ' 7 7 ' 8 0 , 8 1 S t u d i e s have r e v e a l e d t h a t when the r h o m b o i d i t y i s reduced t h r o u g h c o r r e c t i v e measures f o r adverse mould c o n d i t i o n s , t h e r e i s 29 77 80 81 a marked decrease i n the s e v e r i t y o f the c r a c k i n g . » " » u » 0 1 T h i s t y p e o f c r a c k a l s o o c c u r s i n t h e absence o f r h o m b o i d i t y due t o i m p r o p e r c o r n e r r a d i u s , i n b i l l e t m o u l d s , 1 0 ' 1 1 e x c e s s i v e c o r n e r gaps i n bloom m o u l d s , 5 8 and mould d i s -t o r t i o n , w e a r and l o s s o f t a p e r . 2 9 ' 5 7 ' 5 8 ' 7 1 However when the two phenomena do o c c u r t o g e t h e r t hey t e n d t o appear a t 29 77 the o b t u s e a n g l e d c o r n e r s o f the rhomboid b i l l e t . ' I t i s a l m o s t c e r t a i n t h a t c o r n e r c r a c k s f o r m i n the m o u l d . 42 Tab l e I I Adve r se B i l l e t Mould C o n d i t i o n s and A s s o c i a t e d D e f e c t s . MOULD CONDITION DEFECT Mould d i s t o r t i o n , wear and loss of taper. Increases long i tud ina l corner cracks in blooms57>58>77 a n c j 77 29 b i l l e t s and rhomboidity. Excessive corner gaps. Leads to longi tud ina l corner cracks in b looms 5 8 and often 78 resu l ts in break-outs. Improper corner rad ius . Enhances longi tudina l o f f -2,79 ' , 7 9 corner ' and corner cracks. Uneven mould coo l ing . Causes r h o m b o i d i t y , 2 9 ' 7 7 , 8 0 ' 8 1 29 of f -corner internal cracks and longi tud ina l corner : cracks. 77,80,81 Increasing serv ice l i f e . 29 77 Causes rhomboidity ' and longi tudina l corner cracking in : b i l l e t s . 2 9 ' 7 7 Inadequate mould l ub r i ca t i on Transverse cracking in b i l l e t s . and improper mould o s c i l l a t i o n . 6 O f f - c o r n e r i n t e r n a l c r a c k s wh ich i s the s u b j e c t o f a r e c e n t 82 s t u d y are b e l i e v e d to o c c u r i n the l o w e r mould o r upper s p r a y z o n e s . The c r a c k s a re a t t r i b u t e d t o b u l g i n g o f t h e b i l l e t i n t h e mould and appear t o be enhanced by r e v e r s a l o f t a p e r i n the l o w e r m o u l d . R e t u r n i n g to the s u b j e c t o f l o n g i t u d i n a l c o r n e r c r a c k s , uneven c o o l i n g seems u n d o u b t e d l y one o f t h e most 29 77 80 81 i m p o r t a n t f a c t o r s l e a d i n g t o t h e i r o c c u r r e n c e . ' ' ' Mould d i s t o r t i o n , wear and poor a l i g n m e n t between t h e mould and sub -mou ld assembly have been known to a g g r a v a t e 29 8 0 8 1 77 r h o m b o i d i t y ' ' and c o r n e r c r a c k i n g . S t e e l s c o n t a i n -i n g carbon i n t he 0 .18 - 0.25% range have been i d e n t i f i e d as b e i n g most s e n s i t i v e t o the f o r m a t i o n o f c o r n e r c r a c k s 77 83 and r h o m b o i d i t y ' w h i l e o t h e r s c l a i m t h a t t h e p r o b l e m i n -80 c r e a s e s i n s e v e r i t y w i t h i n c r e a s i n g c a r b o n . R h o m b o i d i t y has been r e p o r t e d to be a more f r e q u e n t p r o b l e m when c a s t i n g s m a l l e r s e c t i o n s . 7 7 O p e r a t i n g f a c t o r s such as h i g h p o u r i n g t e m p e r a t u r e s and h igh c a s t i n g speeds have been f o u n d t o 2 10 enhance c o r n e r c r a c k s . ' I n p r a c t i c e c o r r e c t i v e measures have been a d o p t e d to reduce t h e i n c i d e n c e o f 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 , wh ich have been o n l y p a r t i a l l y s u c c e s s f u l i n c e r t a i n c a s e s . 44 Proper a d j u s t m e n t o f f o o t r o l l e r s and ma in tenance o f a l i g n -ment between the mould and sub-mou ld assembly i s a measure t h a t has met w i t h some s u c c e s s , a l t h o u g h i n c e r t a i n cases 80 the p rob lem was no t t o t a l l y e l i m i n a t e d . A t t he Funabash i o o S t e e l Works i n Japan a s o f t - c o o l i n g p r a c t i c e has .proved to be a s u c c e s s f u l method o f r e d u c i n g r h o m b o i d i t y and c r a c k s . In s o f t - c o o l i n g moulds s e r r a t i o n s were made on the o u t s i d e 29 s u r f a c e o f the mould t u b e . Perminov e t a l . have a d v o c a t e d the use o f moulds w i t h i n t e r n a l l y c o r r u g a t e d s u r f a c e s w i t h wh ich t h e y obse rved a r e d u c t i o n in r h o m b o i d i t y and c r a c k i n g . A t e c h n i q u e c u r r e n t l y used to c o n t r o l c o r n e r c r a c k s i n h i g h -carbon grades i s the r e d u c t i o n o f w a t e r f l o w w i t h i n c r e a s e d 84 88 c a r b o n . " T h i s has been s u c c e s s f u l e x c e p t where heavy 84 d e p o s i t s o f s c a l e were formed due t o poor w a t e r q u a l i t y , wh ich s u b s e q u e n t l y l e d t o e x c e s s i v e mould d i s t o r t i o n , and a marked i n c r e a s e i n t he s e v e r i t y o f c o r n e r c r a c k i n g . I t has been r e p o r t e d t h a t w i t h t h i s method o f q u a l i t y c o n t r o l moulds have t o f r e q u e n t l y be r e c o n d i t i o n e d o r r e p l a c e d a f t e r a r e l a t i v e l y sma l l number o f h e a t s , o t h e r w i s e 85 d i s t o r t i o n and c o r n e r c r a c k s b e g i n t o i n c r e a s e . 2 .6 O b j e c t i v e s and Plan o f Research P r o j e c t The p r e c e d i n g r e v i e w o f the l i t e r a t u r e c l e a r l y i n d i c a t e s a need f o r a b e t t e r u n d e r s t a n d i n g o f t he r e l a t i o n s h i p between mould o p e r a t i n g v a r i a b l e s and the 45 t he rma l d i s t o r t i o n o f mou l d s , and a l s o the i n f l u e n c e o f the l a t t e r on p r o d u c t q u a l i t y . Thus the f i r s t two o b j e c t i v e s o f t h i s p r o j e c t have been to c a l c u l a t e the the rma l f i e l d i n the mould f o r a v a r i e t y o f c a s t i n g c o n d i t i o n s and s t e e l c o m p o s i t i o n s , and based on peak mould t empe r a t u r e s p r e d i c t e d , t o i d e n t i f y t he pa r ame te r s t h a t c o u l d l e a d to s o f t e n i n g , l o s s o f s t r e n g t h o f the mould and d i s t o r t i o n . As was shown i n the p r e v i o u s s e c t i o n s mould d i s t o r t i o n l i m i t s s e r v i c e l i f e , a n d has been l i n k e d to a v a r i e t y o f c a s t i n g d e f e c t s . Hence the t h i r d o b j e c t i v e o f t h i s s t udy has been to c a l c u l a t e the d i s t o r t i o n o f the mould i n r e sponse to the the rma l f i e l d wh i ch o b t a i n s under d i f f e r e n t o p e r a t i n g c o n d i t i o n s . The e f f e c t o f mou ld c o n s t r a i n t s , s e c t i on:-s i ze and o t h e r d e s i g n pa r ame te r s on d i s t o r t i o n c o u l d then be e v a l u a t e d . I n s t r u m e n t a t i o n o f an o p e r a t i n g mould f o r t h e pu rpose o f m o n i t o r i n g mould w a l l movement o ve r s e v e r a l h e a t s was p l a n n e d , f i r s t l y to i n v e s t i g a t e the f e a s i b i l i t y o f a c c u r a t e l y measu r i ng d i s t o r t i o n , and s e c o n d l y to c o r r e l a t e shape changes measured w i t h c a l c u l a t e d v a l u e s . In t he p a s t , mechanisms have been p roposed f o r mou l d -r e l a t e d s u r f a c e d e f e c t s i n s l a b s , and the r ea sons f o r the 46 adverse i n f l u e n c e o f c e r t a i n mould and o p e r a t i n g p a r a m e t e r s have been i d e n t i f i e d , wh ich makes t h e p rob lem more amenable to s o l u t i o n . In b i l l e t moulds no s a t i s f a c t o r y e x p l a n a t i o n s have been p roposed f o r the mechanism o f c o r n e r c r a c k i n g and r h o m b o i d i t y , or f o r the p a r t i a l success o f c e r t a i n . r e m e d i a l measures . Thus the f i n a l o b j e c t i v e o f t h i s s t u d y was t o d e v e l o p a mechanism to e x p l a i n the o c c u r r e n c e o f r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k s on the b a s i s o f h e a t f l o w and d i s t o r t i o n c a l c u l a t i o n s . 3.0 THERMAL FIELDS IN CONTINUOUS CASTING MOULDS 3.1 Ma t hema t i c a l M o d e l l i n g o f Heat Flow i n Moulds The the rma l f i e l d o f a c o n t i n u o u s c a s t i n g mould i s a f u n c t i o n o f the r a t e o f heat f l ow to the mould f rom the s o l i d i f y i n g s t r a n d , the e f f i c i e n c y o f hea t e x t r a c t i o n o f the m o u l d - c o o l i n g w a t e r , and the t h e r m o p h y s i c a l p r o p e r t i e s o f the mou l d . The su c ce s s o f m a t h e m a t i c a l l y m o d e l l i n g t h i s hea t e x t r a c t i o n p r o c e s s i s dependent on the a c c u r a c y w i t h wh i ch the above pa ramete r s can be de t e rm ined and the ade -quacy o f the e q u a t i o n s chosen t o d e s c r i b e the phenomena. T h i s c h a p t e r commences w i t h a d i s c u s s i o n o f the h o t -and c o l d - f a c e boundary c o n d i t i o n s and t h e i r n u m e r i c a l c h a r a c t e r i z a t i o n . Th i s i s f o l l o w e d by an a c coun t o f the deve lopment o f the heat f l o w model and the g o v e r n i n g e q u a t i o n s . Numer i ca l s o l u t i o n t e c h n i q u e s a re a l s o d e s c r i b e d . The l a s t s e c t i o n i s devo ted to v a l i d a t i o n o f the ma t hema t i c a l models and a p r e s e n t a t i o n o f the t he rma l f i e l d s c a l c u l a t e d f o r s l a b and b i l l e t moulds under s t a n d a r d o p e r a t i n g c o n d i -t i o n s . 47 48 3 .1 .1 C h a r a c t e r i z a t i o n o f the Ho t - F a c e H e a t - F l u x  P r o f i l e From the p r e c e d i n g c h a p t e r d e s c r i b i n g p a s t work i n t h i s f i e l d , i t i s e v i d e n t t h a t the r a t e o f h ea t f l o w to the mould i s dependent on the g rade o f s t e e l b e i n g c a s t , mould l u b r i c a t i o n , mould d e s i g n f a c t o r s such as t a p e r , and a l s o the n a t u r e o f the a i r gap t h a t f o r m s . C l e a r l y the hea t f l o w to the mould canno t be c h a r a c t e r i z e d by a s i n g l e d i s t r i b u t i o n p a t t e r n . However i t i s p o s s i b l e to d e t e rm i n e s u i t a b l e h e a t - f l u x p r o f i l e s , t o c o v e r d i f -f e r e n t o p e r a t i n g p r a c t i c e s and g rades o f s t e e l f rom e x -p e r i m e n t a l measurements . These measurements a re g e n e r a l l y ( i n the form o f l o n g i t u d i n a l h e a t - f l u x p r o f i l e s measured by 21 22 s e c t i o n a l hea t b a l a n c e s o f the c o o l i n g w a t e r , ' o r t empe r a t u r e da ta f rom s e t s o f t h e rmocoup l e s l o c a t e d on p a r a l -89 90 l e i p l a n e s i n a l o n g i t u d i n a l s l i c e . ' 3 . 1 . 2 H e a t - T r a n s f e r C o e f f i c i e n t a t the C o o l i n g - Water I n t e r f a c e Heat exchange between the coppe r mould and the c o o l i n g wa t e r i s by f o r c e d c o n v e c t i o n . C o o l i n g wa t e r i s c i r c u l a t e d a t v e l o c i t i e s i n the 6~12 m/sec range c o u n t e r c u r r e n t t o the c a s t i n g d i r e c t i o n , t h r ough a s e r i e s o f v e r t i c a l s l o t s mach ined i n e i t h e r the coppe r p l a t e o r s t e e l b a c k i n g p l a t e i n the case o f s l a b mou l d s , o r t h r ough the a n n u l a r channe l formed between the coppe r tube and s t a i n -l e s s s t e e l j a c k e t i n the b i l l e t mould a s s emb l y . E m p i r i c a l c o r r e l a t i o n s have been p roposed t o mathe-m a t i c a l l y q u a n t i f y the h e a t - t r a n s f e r c o e f f i c i e n t t h a t o b t a i n s a t a hea ted s u r f a c e c o o l e d by a f l u i d f l o w i n g p a s t i t . For f l u i d f l o w i n g under t u r b u l e n t c o n d i t i o n s i n a duc t where h^ c (kW/m' 1) i s the f o r c e d - c o n v e c t i o n h e a t - t r a n s f e r c o e f f i c i e n t . T h i s r e l a t i o n has been found to be v a l i d f o r P r a n d t l numbers r a n g i n g f rom 0.7 to 120, Reyno lds numbers o f 10 ,000 to 120 ,000 and when l e n g t h to h y d r a u l i c d i a m e t e r p r o p e r t i e s o f the f l u i d a re e v a l u a t e d a t the b u l k t empe r a -t u r e o f the l i q u i d . T h i s e m p i r i c a l c o r r e l a t i o n was d e r i v e d by f i t t i n g a mean cu r ve t o a l a r g e c o l l e c t i o n o f e x p e r i m e n -92 t a l d a t a under v a r y i n g c o n d i t i o n s ; Bernado and E i a n , e x p e r i m e n t i n g w i t h wa t e r i n a hea ted tube found t h a t the h e a t - t r a n s f e r c o e f f i c i e n t s c o r r e l a t e d w i t h a maximum d e v i a -t i o n o f 20%. o f n o n - c i r c u l a r c r o s s - s e c t i o n , one such r e l a t i o n i s (5) r a t i o f o r t he channe l i s g r e a t e r than 60 . 91 In Eq . (5) the The h i gh degree o f s u b c o o l i n g o f the wa t e r does no t p r e c l u d e the p o s s i b i l i t y o f s u r f a c e b o i l i n g . Bubb l e s a r e n u c l e a t e d i n the s upe r hea t ed l a y e r a d j a c e n t to t he coppe r i n l o c a l r e g i o n s o f h i gh t e m p e r a t u r e . These bubb l e s can de tach f rom the i n t e r f a c e and r i s e t h rough the s u p e r -hea t ed l a y e r . On r e a c h i n g the main s t r eam they condense s i n c e f l u i d t empe r a t u r e s are w e l l be low s a t u r a t i o n , and t h e r e i s no ne t g e n e r a t i o n o f v apou r . The a g i t a t i o n caused by bubb l e mot ion and c o l l a p s e , r e s u l t s i n m i x i n g i n the " s t a g n a n t " s upe r h ea t ed l a y e r a d j a c e n t to t he coppe r and s e t s up a r e c i r c u l a t i o n pa th f o r c o l d e r f l u i d f rom t he main s t r e am to the i n t e r f a c e . T h i s phenomena l o c a l l y e n -hances the hea t t r a n s f e r f rom the mould to the c o o l i n g 93 w a t e r . 94 B e r g l e s e t a l . have deve l oped a q u a n t i t a t i v e p r o -cedu re f o r d e t e r m i n i n g the p o i n t o f i n c i p i e n t b o i l i n g f o r wa t e r ove r a p r e s s u r e range 225 to 13790 k P a . The hea t f l u x a t the i n c e p t i o n o f b o i l i n g i s g i v e n by the f o l l o w i n g e x p r e s s i o n . <in = 5.281 x ID" 3 P 1 ' 1 5 6 [1.8(T -T ) ] 2 - 4 0 / P ° " ° 2 3 4 . . . ( 6 ) L w sat For such a g e n e r a l i z e d e x p r e s s i o n to be v a l i d f o r a s p e c i f i c f l u i d / s u r f a c e c o m b i n a t i o n , a wide range o f c a v i t y s i z e s f o r y 4 y ID n u c l e a t i o n must e x i s t . ' F u r t he rmo re t h e s e s i t e s a r e 95 a c t i v e o n l y i f t hey c o n t a i n en t r apped g a s . Many com-m e r c i a l l y f i n i s h e d s u r f a c e s appear to s a t i s f y t he se c o n -94 d i t i o n s as B e r g l e s e t a l . have found good agreement b e -tween t h e i r r e s u l t s and e x p e r i m e n t a l measurements f o r s e v -e r a l cases . For s u r f a c e s b o i l i n g ove r a l ong p e r i o d o f t i m e , i t has been ob se r ved t h a t the s u r f a c e i s g r a d u a l l y d e -gas sed c a u s i n g a d e c r e a s e i n the number o f a c t i v e n u c l e i w i t h the r e s u l t t h a t l a r g e r s u p e r h e a t s are r e q u i r e d t o 96 i n i t i a t e and s u s t a i n n u c l e a t e b o i l i n g . C o r t y and 9 7 Fous t have r e p o r t e d t h a t the b o i l i n g cu r ve c o u l d e x h i b i t a h y s t e r e s i s e f f e c t as s c h e m a t i c a l l y shown i n F i g . 12, the e x t e n t o f wh i ch i s dependen t , on the pa s t h i s t o r y and n a t u r e o f t he s u r f a c e . They ob s e r v ed t h a t i f t he b o i l i n g sys tem i s r e p e t i t i v e l y c y c l e d t h rough p e r i o d s o f r a p i d b o i l i n g f o l l o w e d by q u i e s c e n t p e r i o d s , i n wh i ch a l l t he bubb l e s are a l l o w e d to d i e o u t , and the s u r f a c e i s m a i n -t a i n e d bubb l e f r e e f o r 10 to 15 m i n u t e s , the h y s t e r e s i s e f f e c t shown i n F i g . 12 i s o b t a i n e d . Nucl e'ate b o i l i n g i s then s upp r e s s ed u n t i l s u r f a c e t empe r a t u r e s a r e as h i g h as 28°C above the s a t u r a t i o n t e m p e r a t u r e , a l t h o u g h normal e x ce s s t empe r a t u r e s f o r the i n c e p t i o n o f n u c l e a t e b o i l i n g a re a p p r o x i m a t e l y 12 to 14°C. They have a t t r i b u t e d t h i s A T F i g . 12 H y s t e r e s i s E f f e c t i n the 97 B o i l i n g Cu r v e . phenomena t o the d e a c t i v a t i o n o f n u c l e a t i n g c a v i t i e s due t o ou twa rd d i f f u s i o n o f vapour f rom t h e s e s i t e s . C l e a r l y c o n d i t i o n s d e s c r i b e d above c o u l d e a s i l y be o b t a i n e d a t t he c o o l i n g - w a t e r i n t e r f a c e o f a coppe r mould as the l a t t e r i s s u b j e c t e d to p e r i o d i c h e a t i n g and c o o l i n g c y c l e s f rom campaign to c ampa ign . Under such c o n d i t i o n s an i n c u b a t i o n s upe r hea t o f 10~15 o C s h o u l d be added to t he t empe r a t u r e o f i n c i p i e n t b o i l i n g c a l c u l a t e d f rom E q . ( 6 ) t o o b t a i n a r e a l i s t i c e s t i m a t e o f the i n t e r f a c e s u p e r h e a t r e q u i r e d to i n i t i a t e b o i l i n g . Once n u c l e a t e b o i l i n g i s w e l l e s t a b l i s h e d the l o c a l h e a t - t r a n s f e r c o e f f i c i e n t can be computed f rom c o r r e l a t i o n s a r r i v e d a t u s i n g e x p e r i m e n t a l da t a on poo l b o i l i n g . I t i s i m p o r t a n t to note t h a t the geometry o f t he h e a t i n g s u r f a c e and v e l o c i t y o f the main s t r eam o f f l u i d have no a p p r e c i -a b l e e f f e c t on l o c a l b o i l i n g h e a t - t r a n s f e r c o e f f i c i e n t s , as n u c l e a t e b o i l i n g and i t s i n f l u e n c e i s r e s t r i c t e d t o t he 9 1 9 9 f l u i d l a y e r s a d j a c e n t t o the s u r f a c e . ' The s u p e r h e a t o f the s u r f a c e , i t s p r o p e r t i e s and f l u i d p r e s s u r e a r e t h e most i m p o r t a n t f a c t o r s g o v e r n i n g the b o i l i n g h e a t - t r a n s f e r c o e f f i c i e n t . Based on a l a r g e c o l l e c t i o n o f e x p e r i m e n t a l 9 8 da ta Rohsenow has d e r i v e d the f o l l o w i n g c o r r e l a t i o n , 54 W V T s a t > = C s f H fg V y A g g ( p ^ - p v ) n 0.33 . . . ( 7 ) where the c o e f f i c i e n t s C ^ and s depend on the f l u i d s u r f a c e c o m b i n a t i o n and p r o p e r t i e s 0 .013 and 1.0 r e s p e c t i v e l y p r o p e r t i . For w a t e r / c o p p e r and s a re 98 From E q . ' s ( 5 ) , (6) and (7) f o r c e d - c o n v e c t i o n b o i l i n g c u r v e s f o r a g i v en p r e s s u r e can be c a l c u l a t e d f o r d i f f e r e n t v e l o c i t i e s and degrees o f s u b c o o l i n g a c c o r d i n g to a p r o c e d u r e 9 8 sugge s t ed by Rohsenow. F i g . 13 s c h e m a t i c a l l y r e p r e s e n t s the recommended t e c h n i q u e f o r c o n s t r u c t i n g the c o o l i n g cu r ve f o r a g i v e n s e t o f f l u i d p r o p e r t i e s . The i n t e r p o l a t i o n e q u a t i o n g i v en be low i s u t i l i z e d to c a l c u l a t e v a l u e s o f b o i l i n g h e a t - f l u x f o r t h e t r a n s i t i o n r e g i o n between t he i n c i p i e n t b o i l i n g p o i n t on the f o r c e d c o n v e c t i o n c u r v e and the f u l l y d e ve l oped poo l b o i l i n g c u r v e . ' t r = q fc 1 + % 1 ' in 1 f c 0.5 • •(8) A sample c a l c u l a t i o n f o r a g i v en s e t o f c o n d i t i o n s commonly o b t a i n e d i n o p e r a t i n g b i l l e t c a s t e r s i s p r e s e n t e d i n Append i x I . F i g . ' s 14 and 15 show the r e s u l t s o b t a i n e d f o r d i f f e r e n t wa t e r v e l o c i t i e s and degrees o f s u b c o o l i n g , / E q ( 7 ) | E q / ( 6 ) / i / E q { 7 N c. Eq(5)" ^ ^ 7  V. Wfc / | / (q) b t i o 3 1 7 — X 1 / 1/ a <o I 1 (Tw - T s a t ) °C cn cn F i g . 13 A Schemat i c Diagram I l l u s t r a t i n g the C o n s t r u c t i o n o f a B o i l i n g 9 8 Curve f o r Subcoo l ed Wate r . F i g .14 100 Forced Conve c t i o n B o i l i n g Curves f o r Subcoo l ed Water f o r Three D i f f e r e n t Water V e l o c i t i e s . Ch < T - T s a t ) 0 C F i g . 1 5 Fo r ced C o n v e c t i o n B o i l i n g Curves f o r Subcoo l ed Water for . Three - D i f f e r e n t Degrees o f S u b c o o l i n g . 58 a t a p r e s s u r e o f 241 .3 k P a , wh i ch i s a t y p i c a l v a l u e o f mould wa t e r p r e s s u r e near the e x i t o f the c o o l i n g w a t e r channe l i n the mould a s s emb l y . N u c l e a t e b o i l i n g a t the m o u l d / c o o l i n g wa t e r i n t e r -f a c e f a v o u r s the d e p o s i t i o n o f s c a l e i f wa t e r q u a l i t y i s p o o r . F o u l i n g has a l s o been found to enhance the h y s t e r e s i s e f f e c t d i s c u s s e d e a r l i e r . 9 6 ' 1 0 0 F o rma t i o n o f s c a l e on the c o l d f a c e o f the mould can be a s e r i o u s p rob l em because i t i n t r o d u c e s a d d i t i o n a l t he rma l r e s i s t a n c e and l o c a l l y r educes the h e a t - t r a n s f e r c o e f f i c i e n t a t t h e mould c o o l i n g - w a t e r i n t e r f a c e . There a re no e m p i r i c a l c o r r e l a t i o n s f o r e s t i m a t i n g the l o c a l h e a t - t r a n s f e r c o -e f f i c i e n t due to d e p o s i t i o n o f s c a l e as i t i s dependent on the t ype o f s c a l e f o rmed , t he t empe r a t u r e and v e l o c i t y o f the c o o l i n g w a t e r , the l o c a l mould w a l l t e m p e r a t u r e , the t y pe o f w a t e r t r e a t m e n t and the t ime the mould i s i n s e r v i c e . Tab l e I I I 1 0 1 i s a sample o f l o c a l h e a t - t r a n s f e r c o e f f i c i e n t s o b t a i n e d under a range o f i n d u s t r i a l c o n d i t i o n s . In c a l c u l a t i n g h e a t - t r a n s f e r c o e f f i c i e n t s f o r t h e m o u l d / c o o l i n g - w a t e r i n t e r f a c e c o n s i d e r a t i o n s h o u l d be g i v e n to a l l t he f a c t o r s d i s c u s s e d above . C l e a r l y the h e a t -e x t r a c t i o n p r o c e s s a t t h i s boundary i s c o m p l i c a t e d , and can be l i n k e d to s e v e r a l f a c t o r s some o f w h i c h , such as s c a l e f o r m a t i o n canno t be e a s i l y q u a n t i f i e d as a f u n c t i o n o f mould Tab l e I I I Water Qua! i t y Data . Water Flow C o n d i t i o n s Tempera tu re o f H e a t i n g Medium <116°C Tempera tu re o f Water < 52°C Water V e l o c i t y > 1 m/sec Type o f Water Heat T r a n s f e r C o e f f i c i en t kW/m 2 o C Di s t i l l e d wa te r 11 .36 T r e a t e d B o i l e r Feed wa te r 11 .36 C i t y and R i v e r wa te r 5.67 Hard wa te r 0 .97 60 l i f e . D e s p i t e t h e s e u n c e r t a i n t i e s , the b e s t p o s s i b l e e s t i -mate was a r r i v e d a t based on o p e r a t i n g c o n d i t i o n s and t he a v a i l a b l e e m p i r i c a l c o r r e l a t i o n s . 3 . 1 . 3 Deve lopment o f Hea t - F l ow Models Owing to the c o m p l e x i t y o f the h e a t - t r a n s f e r p r o c e s s e s t a k i n g p l a c e a t each boundary o f the coppe r mould c e r t a i n s i m p l i f y i n g a s sump t i on s a re n e c e s s a r y i n t he mathe -m a t i c a l f o r m u l a t i o n . The adequacy o f the h e a t - f l o w model depends u n q u e s t i o n a b l y on the v a l i d i t y o f the a s s u m p t i o n s and t h e s e a re examined i n d e t a i l i n the subsequen t s e c t i o n . T h i s i s f o l l o w e d by the h e a t - f l o w e q u a t i o n s and a d e s c r i p -t i o n o f the s o l u t i o n p r o c e d u r e . 3 . 1 . 3 . 1 As sumpt i ons The h e a t - f l o w model has been f o r m u -l a t e d f o r a l o n g i t u d i n a l p l ane p e r p e n d i c u l a r to t he f a c e o f the mould w a l l a t the m i d - s e c t i o n assuming t h a t t he h e a t -f l o w i n the t r a n s v e r s e d i r e c t i o n i s n e g l i g i b l e . F i g . 16 i s a s c h e m a t i c v iew o f t he p l a n e o f i n t e r e s t t h r ough the mould w a l l , c o o l i n g - w a t e r channe l and w a t e r - j a c k e t / b a c k i n g p l a t e . Th i s c o n f i g u r a t i o n i s r e p r e s e n t a t i v e of a b i l l e t mould or a slab.mould.with a rectangular c o o l i n g , c h a n n e l , a l t h o u g h s l a b moulds more commonly have v e r t i c a l s l o t s mach ined i n the 61 Z = 0 X=0 X=XM X z = zF Backing plate/ mould jacket Z = Z M i y . Meniscus Molten steel i Cooling water F i g . 16 A S chema t i c D iagram M i d - P l a n e Through a o f the L o n g i t u d i n a l Mould W a l l . 62 coppe r f o r c o o l i n g p u r p o s e s . In the l a t t e r case the model wou ld r o u g h l y a p p l y to the r e g i o n between the s l o t s and t he hot f a c e o f t he mou l d . M o d e l l i n g the the rma l f i e l d i n a l l t h r e e d imens i on s wou ld r e q u i r e t empe ra t u r e measurements on an e x t e n s i v e l y i n s t r u m e n t e d mou l d . Th i s was not c o n s i d e r e d w o r t h w h i l e as a v a i l a b l e i n f o r m a t i o n i n d i c a t e s t h a t the v a r i a -t i o n i n hea t f l o w i n a t r a n s v e r s e p l a ne i s o f s i g n i f i c a n c e a t the c o r n e r s where n o n - u n i f o r m gaps e x i s t , w h i l e m i d - f a c e t empe r a t u r e s are more u n i f o r m . A c rude c a l c u l a t i o n based on F o u r i e r s law f o r a 140 x 140 mm b i l l e t mou l d , assuming a m i d - f a c e to c o r n e r t empe r a t u r e v a r i a t i o n o f 25°C g i v e s a 2 t r a n s v e r s e h e a t - f l u x o f 139 kW/m . T h i s i s a p p r o x i m a t e l y an o r d e r o f magn i tude s m a l l e r than the ave rage h e a t - f l u x o f 2 1730 kW/m f l o w i n g t h r ough the t h i c k n e s s , c a l c u l a t e d f rom Eq . ( 4 ) . T h e r e f o r e a r e a s o n a b l y a c c u r a t e e s t i m a t e o f t he t he rma l f i e l d o f the mould can be o b t a i n e d f rom a two-d i m e n s i o n a l model f o r the l o n g i t u d i n a l m i d - p l a n e o f the mould w a l l . T h i s a s sump t i on a l s o r e s u l t s i n a s i g n i f i c a n t s a v i n g i n comput i ng c o s t s . The e f f e c t s o f mould r e c i p r o c a t i o n on the t empe r a t u r e f i e l d has been n e g l e c t e d and t i m e - a v e r a g e d v a l u e s o f measured hea t f l u x e s and wa t e r t empe r a t u r e s a re used as d a t a . Du r i ng o p e r a t i o n the t empe r a t u r e f i e l d wou ld o s c i l l a t e as the mould r e c i p r o c a t e s , w i t h the g r e a t e s t t empe r a t u r e v a r i a t i o n b e i n g seen c l o s e to the m e n i s c u s , where the heat f l u x i s g r e a t e s t . 63 M o d e l l i n g o f the t he rma l c y c l i n g e f f e c t s wou ld r e q u i r e da t a f rom a f i n e g r i d o f t h e rmocoup l e s i n the men i s cu s r e g i o n t o g e t h e r w i t h a r e c o r d o f the e x a c t l o c a t i o n o f t h e men i s cus t h r o u g h o u t the d u r a t i o n o f the c a s t . Th i s i n f o r m a t i o n c oup l e d w i t h a dynamic model c o u l d g i v e an a c c u r a t e p i c t u r e o f the t he rma l r e sponse o f t he mould p r o v i d e d t he t i m e s t e p s s e l e c t e d a r e s i g n i f i c a n t l y s m a l l e r than t he p e r i o d o f o s c i l l a t i o n o f the mou l d . From the s t a n d p o i n t o f t h e o b j e c -t i v e s o f t h i s s t u d y , wh i ch was to c a l c u l a t e t he change i n shape o f t he mould i n r e sponse to the t he rma l f i e l d , a com-p u t a t i o n o f t h i s n a t u r e wou ld not l e a d to a d i s t o r t i o n p r o -f i l e s i g n i f i c a n t l y d i f f e r e n t f rom t h a t c a l c u l a t e d n e g l e c t i n g o s c i l l a t i o n . The r ea son i s t h a t the s t r o k e o f t he o s c i l -l a t i o n wh i ch i s i n the 10 to 20 mm r a n g e , i s two t o t h r e e o r d e r s o f magn i tude s m a l l e r than the l e n g t h o f t h e mould (700 mm) and t h e r e f o r e t empe r a t u r e v a r i a t i o n s due t o o s c i l -l a t i o n wou ld o n l y r e s u l t i n l o c a l i z e d changes i n d i s t o r t i o n wh i ch wou ld no t a l t e r the o v e r a l l d i s t o r t i o n p r o f i l e s i g n i -f i c a n t l y . O the r a s sump t i on s i n c l u d e p l u g f l o w i n t he c o o l i n g -wa te r c h a n n e l , n e g l i g i b l e hea t t r a n s f e r between the c o o l i n g wa t e r and b a c k i n g p l a t e and an e x t e n s i o n o f the c o o l i n g c i r c u i t to the top and bot tom ends o f the mou l d . In c u r r e n t d e s i g n , the r e g i o n s o f the mould t h a t e x t end beyond the 64 f o r c e d - c o n v e c t i o n c o o l i n g path a c coun t f o r l e s s than 10% o f t he t o t a l l e n g t h and a re r e g i o n s o f low hea t f l o w . T h e r e -f o r e the e r r o r s i n v o l v e d i n e x t e n d i n g the c o o l i n g c i r c u i t c o u l d be e xpe c t ed to be s m a l l . The top and bot tom s u r f a c e s o f the mould a re t r e a t e d as a d i a b a t i c b ounda r i e s and the on l y mode o f hea t t r a n s f e r to t he mould above the s t e e l l e v e l i s assumed to be r a d i a n t exchange w i t h the mo l t en p o o l . 3 . 1 . 3 . 2 Hea t - F l ow Equa t i o n and Bounda ry - C o n d i t i o n s The g o v e r n i n g e q u a t i o n f o r hea t c o n d u c t i o n i n the mould w a l l i n two - d imen s i o n s i s , ^ { k " f } + t { k ™ l j = p ^ f - ( 9 ) A s e c t i o n a l hea t b a l a n c e f o r the c o o l i n g wa te r based on the p l u g - f l o w a s sump t i on i s as f o l l o w s ; p w ^w^w p^w 3"V - h ( z , t ) [ T ( o , z , t ) az w T , ( z , t ) ] = o • (10) 65 The r e l e v a n t boundary c o n d i t i o n s a r e , ( i ) c o l d f a ce o f the mould x - o o<z«Z t>o m -k 3T = h w ( z , t ) [ T ( o , z , t ) - T w ( z , t ) ] • . . . ( 1 1 ) = m w w ( i i ) top and bot tom o f mould w a l l o^x-sX z = o , z = Z m t>o m m -k m i l = o . . . ( 1 2 ) 8x ( i i i ) h o t - f a c e be low the men i s cus x = X m , Z.<z«Z t>o m t m k m I I = q_ ( z ) . . . ( 1 3 ) 8x where q g ( z ) i s the t i m e - a v e r a g e d h e a t - f l u x p r o f i l e a t the hot f a c e . ( i v ) h o t - f a c e above the men i s cus x = X o<z<Z r t>o m f •k H = h ( z , t ) [ T ( X m , z , t ) - T ] . . . ( 1 4 ) m a x a m a where h i s a "pseudo" r a d i a n t h e a t - t r a n s f e r c o e f f i c i e n t 66 c a l c u l a t e d f rom a s e r i e s o f e q u a t i o n s d e s c r i b i n g the r a d i a n t heat exchange between the mould w a l l s , and the l i q u i d s t e e l s u r f a c e i n the mou l d . Append ix I I p r e s e n t s sample c a l c u l a -t i o n s o f h f o r c o n t i n u o u s c a s t i n g mou ld s , a (v) i n l e t t empe r a t u r e o f wa t e r z = Z m T = r w w (15) An a l t e r n a t i v e f o r m u l a t i o n f o r the t i m e - i n d e p e n d e n t v e r s i o n o f E q . ' s . (9) t h r ough (15) i s the v a r i a t i o n a l p r i n c i p l e . , 102 g i v e n b e l o w . J(T) = •ill " s T d s h b + ? 2 k 3T  c , k 3T m — + m 3x 1 h r 2 w 3Z dxdz TT w dS cb (16) The f u n c t i o n T ( x , z ) t h a t m i n i m i z e s the f u n c t i o n a l J ( T ) a l s o s a t i s f i e s the g o v e r n i n g hea t c o n d u c t i o n e q u a t i o n s and boundary c o n d i t i o n s , wh i ch are the E u l e r - L a g r a n g e c o n d i t i o n s f o r J ( T ) . The n o n - l i n e a r r a d i a t i o n boundary c o n d i t i o n , the s p a t i a l v a r i a t i o n o f h e a t - f l u x a l o ng the h o t - f a c e , l o c a l v a r i a t i o n s i n h e a t - t r a n s f e r c o e f f i c i e n t due to b o i l i n g a t the c o l d - f a c e , and the t empe r a t u r e dependence o f t he t he rmo-p h y s i c a l p r o p e r t i e s p r e c l u d e s the use o f an a n a l y t i c a l 67 t e c h n i q u e f o r s o l v i n g . t h e e q u a t i o n s . Numer i c a l methods o f s o l u t i o n f o r p rob lems o f t h i s t ype a re e x t e n s i v e l y d e s c r i b e d i n the l i t e r a t u r e , and t h e r e f o r e the subsequen t s e c t i o n s w i l l be r e s t r i c t e d to a b r i e f d e s c r i p t i o n o f the methods adop t ed i n t h i s s t u d y . 3 . 1 . 4 Numer i ca l S o l u t i o n Techn i ques C a l c u l a t i o n o f the t he rma l f i e l d i n s l a b and b i l l e t moulds has been d i v i d e d i n t o two p a r t s . In t he f i r s t , n u m e r i c a l s o l u t i o n s f o r the s t e a d y - s t a t e p r ob l em have been o b t a i n e d under c o n d i t i o n s where the peak t empe r a t u r e a t the c o l d - f a c e o f the mould was be low the i n c i p i e n t b o i l i n g p o i n t . In the second p a r t a n ume r i c a l scheme has been adop ted t o c a l c u l a t e the t r a n s i e n t r e sponse o f the b i l l e t mould to n u c l e a t e b o i l i n g . The r ea son f o r l i m i t i n g t he s t u d y o f t he i n f l u e n c e o f n u c l e a t e b o i l i n g to b i l l e t moulds o n l y , w i l l become appa r en t when the p r e d i c t e d t empe r a t u r e d i s t r i b u t i o n s f o r s l a b and b i l l e t moulds a re p r e s e n t e d . 3 . 1 . 4 . 1 C a l c u l a t i o n o f S t e a d y - s t a t e Tempera tu re  D i s t r i b u t i o n ^ Moulds by the Fini te^  Difference Method The s t e a d y - s t a t e h e a t - c o n d u c t i o n e q u a t i o n wh ich i s the t i m e - i n d e p e n d e n t v e r s i o n o f Eq . (9) i s an e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n . The f i n i t e -d i f f e r e n c e method has been w i d e l y used to o b t a i n n u m e r i c a l s o l u t i o n s f o r t h i s c l a s s o f e q u a t i o n s . 1 0 3 ' 1 0 4 In the f i n i t e - d i f f e r e n c e scheme, the mould c on t i nuum i s r e p r e s e n t e d by a s e r i e s o f nodes formed a t the i n t e r -s e c t i o n o f the l i n e s o f a squa re or r e c t a n g u l a r g r i d . The s o l i d l i n e s o f F i g . 17 show such an a r rangement f o r the l o n g i t u d i n a l m i d - p l a n e o f the mould d e s c r i b e d e a r l i e r . The f i n i t e - d i f f e r e n c e e q u a t i o n s f o r the sys tem a re o b t a i n e d f rom the h e a t - f l o w e q u a t i o n s , by r e p l a c i n g the d i f f e r e n t i a l o p e r -a t o r s w i t h a p p r o p r i a t e d i f f e r e n c e q u o t i e n t s . T h i s c o u l d be a c c o m p l i s h e d by a T a y l o r s e r i e s a p p r o x i m a t i o n o f t he d i f -f e r e n t i a l o p e r a t o r s o r by a p p l y i n g a heat b a l a n c e t o an e l emen t a l volume o f m a t e r i a l s u r r o u n d i n g the node . The e l emen t s a re formed i n t h i s case by an i n t e r p e n e t r a t i n g l a t t i c e i d e n t i c a l to the noda l g r i d shown as d o t t e d l i n e s i n F i g . 17 . For an i n t e r i o r node, f o r e xamp l e , t he f o l l o w i n g e q u a t i o n i s o b t a i n e d assuming t h a t the t he rma l c o n d u c t i v i t y i s i n dependen t o f t e m p e r a t u r e . T, - 2T. h i + T i + l , j + T i , j - 1 - 2T. + T i,j+1 ( d z ) 2 (dx ) 2 + 0 (dx 2) + 0 (dz 2 ) = 0 . . . (17) 69 E o IT) I I N ! ( i - 1 f A k -i) ! a ,J> !<U; - ! ) ! _ _L i 1 i j ) ! ' * i J e » t i — • — < • — • — i >< — - H I H H dx = 238mm F i g . 17 S chema t i c D iagram o f a Two-D imens i ona l F i n i t e - D i f f e r e n c e Mesh. 70 2 2 The o r d e r terms 0 (dx ) , 0 (dz ) r e p r e s e n t the e r r o r i n -v o l v e d i n t h i s a p p r o x i m a t i o n , and can be seen to d e c r e a s e as s m a l l e r and s m a l l e r v a l u e s a re chosen f o r dx and d z , wh i ch r e p r e s e n t the node s p a c i n g i n the X- and Z- d i r e c t i o n s . There i s a p r a c t i c a l l i m i t however to the p r o g r e s s i v e de -c r e a s e i n mesh s i z e as the c o s t e s c a l a t e s w i t h the number o f nodes w h i l e the improvement i n a c c u r a c y a f t e r a p a r t i -c u l a r mesh s i z e i s r eached becomes ve r y s m a l l . A p p l i c a t i o n o f the d i f f e r e n c e e q u a t i o n s to a l l t he nodes r e s u l t s i n a s e r i e s o f s i m u l t a n e o u s e q u a t i o n s o f the f o l l o w i n g f o rm ; [A] {T} = {B} . . . ( 1 8 ) The c o e f f i c i e n t m a t r i x , [ A ] , i s a d i a g o n a l l y dominan t s p a r s e m a t r i x , the maximum number o f n on - z e r o e l emen t s per row b e i n g f i v e f o r a t w o - d i m e n s i o n a l p r o b l e m . A c o n s i d e r a b l e s a v i n g i n computer s t o r a g e c o s t s can be r e a l i z e d i f o n l y t he n o n - z e r o e l emen t s a re s t o r e d as a u n i - d i m e n s i o n a l a r r a y , t o g e t h e r w i t h a v e c t o r c o n t a i n i n g the mesh t o p o l o g y d a t a . The G a u s s - S i e d e l i t e r a t i v e t e c h n i q u e has been w i d e l y used f o r s o l v 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 as i t i s c o m p a t i b l e 103 104 w i t h the above -men t i oned s t o r a g e scheme. ' In t h i s method , a s o l u t i o n i s o b t a i n e d i t e r a t i v e l y , s t a r t i n g w i t h an i n i t i a l a p p r o x i m a t i o n o f the s o l u t i o n v e c t o r . I t has been found t h a t the speed o f c ove rgence can be s i g n i f i c a n t l y 71 10 5 improved by u s i n g an o v e r - r e l a x a t i o n t e c h n i q u e . T h i s i s a c c o m p l i s h e d by f o r c i n g the r e s i d u a l s c a l c u l a t e d f o r each e q u a t i o n , i n each s u c c e s s i v e a p p r o x i m a t i o n o f the s o l u t i o n , to exceed z e r o , by an amount depend ing on the r e l a x a t i o n pa ramete r c h o s e n , u n l i k e the pure G a u s s - S i e d e l method where the r e s i d u a l s a re s e t equa l to z e r o . An o v e r - r e l a x a t i o n , method has t h e r e f o r e been s e l e c t e d to s o l v e the e q u a t i o n s g o v e r n i n g hea t f l o w i n the mou l d . Append i x I I I l i s t s the f i n i t e - d i f f e r e n c e e q u a t i o n s f o r the n i n e d i f f e r e n t t ypes o f nodes e n c oun t e r e d i n the c o n f i g u r a t i o n r e p r e s e n t i n g the m i d - p l a n e o f t he mould w a l l . The t empe r a t u r e dependence o f t he rma l c o n d u c t i v i t y has been i n c o r p o r a t e d i n t hese e q u a t i o n s . Under t h e s e c i r c u m s t a n c e s a r e p e t i t i t v e s o l u t i o n p r o cedu r e has been adop ted i n wh i ch t he t empe r a t 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 i n the p r e v i o u s i t e r a t i o n i s used to compute the new v a l u e s o f t he rma l c o n d u c t i v i t y a t each node . The p r o cedu r e i s c o n t i n u e d u n t i l t he d i f f e r e n c e i n the t he rma l f i e l d between s u c c e s s i v e a p p r o x i m a t i o n s i s s m a l l . The c o o l i n g - w a t e r t empe r a t u r e d i s t r i b u t i o n , i n i t i a l l y assumed t o va ry l i n e a r l y between i n l e t and o u t l e t t e m p e r a t u r e s , and the r a d i a t i v e h e a t - t r a n s f e r r a t e above the men i s cus a re a l s o r e c a l c u l a t e d a t each i t e r a t i o n u n t i l c onve rgence i s a t t a i n e d . The f l ow c h a r t i n F i g . 18 t r a c e s the p rogram l o g i c and d e s c r i b e s the s t ep s i n v o l v e d i n the s o l u t i o n p r o c e d u r e . 72 READ INPUT DATA t PRINT INPUT DATA I START ITERATION COUNT I = 1,2,. . .L IDENTIFY NODE LOCATION I COMPUTE NON ZERO TERMS FROM FINITE DIFFERENCE EQUATIONS F i g . 18 Flow Cha r t o f the F i n i t e - D i f f e r e n c e Hea t - F l ow Program to Compute the S t e a d y - S t a t e Thermal F i e l d i n Mou l d s , ( c o n t . ) 73 ASSEMBLE MASTER COEFFICIENT VECTOR ASSEMBLE TOPOLOGY VECTOR SOLVE BY GSSOR COMPUTE NEW WATER TEMPERATURES BY HEAT BALANCE F i g . 18 F low Cha r t o f the F i n i t e - D i f f e r e n c e H e a t - F l o w Program to Compute the S t e a d y - S t a t e Thermal F i e l d i n M o u l d s . ( c o n t . ) 74 F i g . 18 F low Cha r t o f the F i n i t e - D i f f e r e n c e Hea t - F l ow Program to Compute the S t e a d y - S t a t e Thermal F i e l d i n Mou l d s . 75 Inpu t da ta f o r the model i s based on t y p i c a l i n -d u s t r i a l p r a c t i c e and Tab l e IV p r e s e n t s i n f o r m a t i o n p e r -t a i n i n g to b i 1 1 e t • mou l d s . C o o l i n g - w a t e r v e l o c i t i e s n o r -m a l l y employed i n b i l l e t moulds a re i n t he 6~12 m/sec r a nge , w i t h 7 m/sec be i ng more t y p i c a l . F o r the pu rpose o f t h i s c a l c u l a t i o n a v e l o c i t y o f 10 m/sec was s e l e c t e d t o , ensu re t h a t the peak t empe r a t u r e a t the c o l d - f a c e i s be low the c r i t i c a l v a l u e a t wh i ch n u c l e a t e bo i l im i g wou ld i n i t i a t e i n the channe l . Shou l d b o i l i n g o c c u r , then t h e l o c a l h e a t -t r a n s f e r c o e f f i c i e n t s va r y as a f u n c t i o n o f lo .ca l w a l l t e m p e r a t u r e ; such a p r o c e s s cannot be s i m u l a t e d by a s t e a d y -s t a t e model and a t r a n s i e n t model wou ld be r e q u i r e d . How-eve r under n o - b o i l i n g c o n d i t i o n s a s t e a d y - s t a t e model i s more e conom i ca l and hence the reason f o r i t s d e v e l o p m e n t . The f o r c e d - c o n v e c t i o n h e a t - t r a n s f e r c o e f f i c i e n t c o r r e s p o n d -i n g to a wa t e r v e l o c i t y o f 10 m/sec was c a l c u l a t e d f rom Eq . (5) f o r the s t e a d y - s t a t e mode l . At the h o t - f a c e , a t i m e - a v e r a g e d l o n g i t u d i n a l h e a t -f l u x p r o f i l e f o r a h i gh ca rbon grade was used as i n p u t d a t a . Th i s was c a l c u l a t e d f rom h e a t - f l u x measurements a t t he c o l d -f a c e i n an e x p e r i m e n t a l b i l l e t c a s t e r , and the d e t a i l s o f the c a l c u l a t i o n a re d e s c r i b e d i n a s ub sequen t s e c t i o n . F i g . 19 shows the v a r i a t i o n o f h e a t - f l u x w i t h d w e l l t ime o b t a i n e d f rom such c a l c u l a t i o n s f o r two d i f f e r e n t g r ade s o f s t e e l . 76 Tab l e IV C o n d i t i o n s f o r B i l l e t Moulds Based on I n d u s t r i a l P r a c t i c e Var iable Mould Length 700 mm Wall Thickness 9.525 nm Water In le t Temperature 30°C Water Temperature Rise 8.5°C Water Ve loc i ty 10.0 m/sec Section Size 130mm x 130 mm Casting Speed 2.03 m/min Mould Free board 100.0 mm Water Channel Gap 5.0 mm Direct ion of Water flow up Mould Material copper Lubr icat ion o i l Mould water pressure (ex i t ) 241.3 kPa 77 4000 3000 CM E X 3 ^ 2000 o CO X 1000! J 1 1 1 1 1 1 1 1 1 1 1 ) — \ \ \ \ \ \ \ \ \ Carbon content 0.9% 0.1 % \ \ \ \ \ \ \ \ - \ \ \ \\ \ \ \ \ \ \ > \ \ \ \ N — ~N. \ V \ V \ . S «^s^  \ V. — — • I I I ! 1 1 1 1 1 1 1 8 12 16 Dwell time (s) 20 2 4 F i g . 19 H e a t - F l u x P l o t t e d A g a i n s t T i m e - i n - t h e - M o u l d (Dwe l l Time) f o r a B i l l e t Mould f o r S t e e l s w i t h D i f f e r e n t Carbon C o n t e n t s . 78 The t h e r m o p h y s i c a l p r o p e r t i e s ove r a range o f t empe r a t u r e a re p r e s e n t e d i n Append i x IV . S i n c e the v a r i a t i o n o f t he rma l c o n d u c t i v i t y o f c oppe r w i t h t empe r a t u r e i s seen to be q u i t e sma l l i t s e f f e c t on mould w a l l t empe r a t u r e s was i n v e s t i g a t e d , t o check whe the r t h i s r e f i n e m e n t o f the program was r e a l l y n e c e s s a r y . T a b l e V compares the c a l c u l a t e d t empe r a t u r e s w i t h and w i t h o u t t empe r a t u r e dependent the rma l c o n d u c t i v i t y , a t i d e n t i c a l l o c a t i o n s i n t he copper mould f o r a 705 node , (5 x 141) mesh. I t i s e v i d e n t t h a t the maximum d i f f e r e n c e i n computed t empe r a t u r e s a t the m e n i s c u s , where the v a r i a t i o n can be e x -pe c t ed to be the g r e a t e s t , i s 2°C, w h i l e the i n c r e a s e i n , computer t ime was 25%. From a p r a c t i c a l s t a n d p o i n t t h e s e sma l l d i f f e r e n c e s i n t empe r a t u r e do not j u s t i f y t he e x t r a compu t i ng c o s t s . A l l subsequen t c a l c u l a t i o n s have been pe r f o rmed n e g l e c t i n g the t empe r a t u r e dependence o f t h e rma l conduct!* v i t y . To t e s t f o r u n i f o rm conve rgence o f the n u m e r i c a l s o l u t i o n , the t empe r a t u r e s c a l c u l a t e d f o r two d i f f e r e n t mesh s i z e s were compared . The r e s u l t s a re p r e s e n t e d i n Tab l e V I . In the c o a r s e r mesh, wh ich c o n s i s t s o f 213 nodes , t he d i s -t ance between nodes i n bo th the X- and Z- c o - o r d i n a t e 79 Tab l e V Mould Wa l l Tempera tu res W i th and W i thou t t he I n c o r p o r a t i o n o f Tempera tu re Dependent  Thermal C o n d u c t i v i t y Distance Down the Mould (mm) Mould Wall Temperatures -°C Hot- Face Cold- Face k=k(T) k=0.3889 k=k(T) k=0.3889 0.0 39.57 39.47 39.04 38.98 50.0 40.75 40.50 39.71 39.83 100.0 162.31 163.48 100.24 100.04 140.0 223.87 225.46 139.71 139.48 160.0 222.53 224.04 139.12 138.84 180.0 219.57 221.04 137.41 137.12 200.0 215.88 217.33 135.27 134.99 220.0 211 .59 213.01 132.76 132.51 260.0 199.71 201.05 125.95 125.72 300.0 181 .69 182.89 115.83 115.62 340.0 160.94 161.95 104.20 104.02 380.0 141.89 142.75 93.49 93.33 420.0 127.27 128.04 85.20 85.08 460.0 117.41 118.04 79.54 79.37 500.0 108.48 109.02 74.37 74.22 540.0 99.62 100.10 69.28 69.13 580.0 91.55 91.97 64.61 64.48 620.0 84.86 85.25 60.69 60.58 660.0 78.85 79.19 57.17 57.06 700.0 74.93 75.22 54.94 54.82 80 Tab l e VI A Compar i son o f the Mou l d -Wa l l Tempe ra tu r e s P r e d i c t e d by the F i n i t e - D i f f e r e n c e Method f o r Two D i f f e r e n t Mesh S i z e s Distance Down the Mould (mm) Mould Wall Temperatures °C Hot- Face Cold- Face : 705; Nodes .213 Nodes 705 Nodes 213, Nodes 0 39.47 39.55 38.98 39.01 50.0 40:50 40.92 39.83 39.78 100.0 163.48 175.52 100.04 110.13 140.0 225.46 224.87 139.48 140.18 160.0 224.04 223.10 138.84 139.28 180.0 221.04 220.06 137.12 137.52 200.0 217.33 216.36 134.99 135.88 220.0 213.01 212.06 132.51 132.89 260.0 201.05 200.17 125.72 126.08 300.0 182.89 182.11 115.62 115.94 340.0 161.95 161.27 104.02 104.29 380.0 142.75 142.18 93.33 93.56 420.0 128.04 127.55 . 85.08 85.29 460.0 118.04 117.60 79.37 79.55 500.0 109.02 108.63 74.22 74.38 540.0 100.10 99.75 . 69.13 69.27 580.0 91 .97 91 .67 64.48 64.60 620.0 85.25 84.97 60.58 60.69 660.0 79.19 78.94 57.06 57.16 700.0 75.22 74.94 54.82 54.89 81 d i r e c t i o n s i s t w i c e t h a t i n the f i n e r mesh wh i ch c o n s i s t s o f 705 nodes . I t i s a ppa r en t f rom Tab l e VI t h a t the d i f f e r e n c e i n t empe r a t u r e between the two g r i d s a t i d e n t i c a l l o c a t i o n s i s g e n e r a l l y l e s s than 1°C eve rywhere e x c ep t i n the men i s cus r e g i o n . Here t he d i f f e r e n c e stems f rom the f a c t t h a t i n a p p l y i n g a heat b a l a n ce to an e l emen t a l volume s u r r o u n d i n g the node a t the m e n i s c u s , w i t h the c o a r s e r mesh, t he peak heat f l u x i s a p p l i e d to a l a r g e r e l emen t a l volume than w i t h the f i n e r mesh, wh i ch r e s u l t s i n h i g h e r t e m p e r a t u r e s f o r the f o r m e r . O the r than t h i s d i s c r e p a n c y , the conve rgence e v e r y -where e l s e i s good . A check on the a c c u r a c y o f the n u m e r i c a l s o l u t i o n was c a r r i e d ou t by e xam in i ng the c a l c u l a t e d t e m p e r a t u r e s f o r a s i m p l e case o f a 4 .7625 mm t h i c k coppe r w a l l , 700 mm l o n g , 2 r e c e i v i n g a c o n s t a n t heat f l u x o f 2000 kW/m ove r the l e n g t h o f one f a c e , w h i l e l o s i n g heat to a l a r g e mass o f c o o l a n t a t a f i x e d t empe r a t u r e o f 30°C ove r the l e n g t h o f the o p p o s i t e f a c e ; the i n t e r f a c e heat t r a n s f e r c o e f f i c i e n t i s assumed to 2 be 34 .48 kW/m °C. Tab l e V I I i s a sample o f the t e m p e r a t u r e s computed a t a few l o c a t i o n s i n the copper w a l l . C l e a r l y the the rma l f i e l d i s u n i d i m e n s i o n a l as e x p e c t e d . A p p l y i n g a hea t b a l a n c e a t the c o o l i n g s u r f a c e , q u i = 34.48 (88-30.0) = 1999.84 kW/m2 . . . (19) 82 Tab 1 e V I I Tempera tu res P r e d i c t e d f o r a Copper P l a t e w i t h S imp le Boundary C o n d i t i o n s Distance from the Mould Wal l Temperatures °C Top of the Distance from the Cold-Face (d) Plate (mm) d = 0 d = 2.3812 mm d = 4.7625 mm 0 88.00 100.24 112.48 100 88.00 100.24 112.48 200 88.00 100.24 112.48 300 88.00 100.24 112.48 400 88.00 100.24 112.48 500 88.00 100.24 112.48 600 88.00 100.24 112.48 700 88.00 100:24 112.48 83 2 wh i ch compares w e l l w i t h the i n p u t heat f l u x o f 2000 kW/m . A p p l i c a t i o n o f F o u r i e r s Law to the s y s t e m , q w = k c dT = 0.3889 (112.48 - ,80.0) dx 4.7625 x 10" 3 = 1999.007 kW/m2 . . . (20) wh i ch a l s o compares w e l l w i t h the i n p u t hea t f l u x . When the o i n p u t hea t f l u x was s e t equa l to z e r o , the t e m p e r a t u r e s c a l -c u l a t e d eve rywhere were found to be equa l to 30°C wh i ch was the t empe r a t u r e o f the c o o l a n t . These s i m p l e c a l c u l a t i o n s a re a check on the i n t e r n a l c o n s i s t e n c y o f the f i n i t e -d i f f e r e n c e e q u a t i o n s f o r m u l a t e d f o r the mould and a l s o i l -l u s t r a t e the a c c u r a c y o f the nume r i c a l s o l u t i o n p r o c e d u r e . 3 . 1 . 4 . 2 C a l c u l a t i o n o f the S t e a d y - S t a t e Tempera tu re D i s t r i b u t i o n i n Moulds  by the F i n i t e - E l e m e n t Me thod . The f i n i t e - e l e m e n t me thod , wh i ch was o r i g i n a l l y d e ve l oped to compute s t r e s s e s i n s t r u c t u r a l mem-b e r s , has been ex tended to many o t h e r a r ea s one o f wh i ch i s to c a l c u l a t e the t empe r a t u r e f i e l d i n a h e a t - f l o w p r o b l e m . S i n c e the s t r e s s e s t h a t a r i s e i n many e n g i n e e r i n g components are o f t e n the r e s u l t o f t he rma l g r a d i e n t s the e x t e n s i o n i s a l o g i c a l one . 84 Many t e c h n i q u e s have been de ve l oped t o d e r i v e f i n i t e e l ement e q u a t i o n s f rom the p h y s i c a l laws g o v e r n i n g t he be -i n ? h a v i o u r o f a s y s t e m . In t h i s s t udy the a p p r o p r i a t e e q u a t i o n s have been o b t a i n e d f rom the v a r i a t i o n a l p r i n c i p l e f o r the hea t f l o w prob lem p r e s e n t e d e a r l i e r , ( E q . ( 1 6 ) ) . The p r e l i m i n a r y s t ep i n a f i n i t e - e l e m e n t a n a l y s i s i s t o d i s c r e t i z e the con t i nuum i n t o a s e r i e s o f e l e m e n t s . A scheme s e l e c t e d f o r the copper mould can be seen i n F i g . 20 . The e l emen t s shown i n t h i s F i g . a re t h r e e node l i n e a r tem-p e r a t u r e t r i a n g u l a r e l e m e n t s . The t empe r a t u r e w i t h i n each e l ement i s assumed t o v a r y l i n e a r l y and i s d e s c r i b e d by t he f o l l o w i n g r e l a t i o n . 1 0 2 T (x,z) = z N. (x,z) T. = [N] {T} e . . . (21) e i=l 1 1 Here T ( x , z ) d e s c r i b e s the t empe r a t u r e v a r i a t i o n w i t h i n an e l e m e n t , T / r e p r e s e n t s the d i s c r e t e noda l t e m p e r a t u r e s wh i ch a re s o u g h t , and N . ( x , z ) a re the a p p r o p r i a t e i n t e r p o l a t i o n f u n c t i o n s chosen to d e s c r i b e the assumed t e m p e r a t u r e v a r i a -t i o n w i t h i n an e l e m e n t . n , wh i ch r e p r e s e n t s the number o f nodes i s t h r e e f o r a l i n e a r t empe r a t u r e t r i a n g u l a r e l e m e n t . A q u a d r a t i c t empe r a t u r e e l ement wh i ch i s a l o g i c a l e x t e n s i o n o f t h i s c oncep t i s a s i x node t r i a n g u l a r e l e m e n t , and the a p p r o p r i a t e i n t e r p o l a t i o n f u n c t i o n s are second o r d e r p o l y -n o m i a l s . D e t a i l s o f t he se two t ypes o f e l emen t s can be h H dx = 238 mm F i g . 20 A S chema t i c D iagram o f a Twp-D imens i ona l F i n i t e - E l e m e n t Mesh 86 found in Appendix V. Having descr ibed the temperature d i s t r i -bu t i on w i t h i n each element as a f u n c t i o n of d i s c r e t e nodal tempera tures , the next step is to fo rmu la te a se r ies o f equat ions based on the phys ica l laws governing the system in terms o f the unknown nodal tempera tu res , which could then be solved f o r the l a t t e r . Since the temperature d i s t r i b u t i o n f u n c t i o n T ( x , z ) , which minimizes the f u n c t i o n a l J(T) of Eq. (16) a lso s a t i s f i e s the heat f low Eq . ' s (9) through ( 1 4 ) , the element equat ions are obta ined by s e t t i n g the d e r i v a t i v e s o f the f u n c t i o n a l J ( T g ) w i t h respect to the d i s c r e t e nodal temperatures to ze ro , f o r each element. This i s e q u i v a l e n t to . . . ( 2 2 ) For any node i , in any e lement , one such equat ion i s , e 9Z 3N. dxdz + h T N.) dS W W 1 cb = 0 . . . ( 2 3 ) In m a t r i x n o t a t i o n t h i s can be r e w r i t t e n as [ K t ]6 {T} 6 + [ K h ]e {T} e = { K t w }6 - {q} G .(24) where K t ( i , j ) q ( i ) K t w ^ k 3N. 3N. + k 3N. 3N. m i _ ;i m v j 3x 3x 3Z 3Z ^ ( L J ) . / h w N . N . d s c b 5cb / q N i dS hb >hb h T N. dS . w w i cb 'hb dxdz 87 (25) .(26) (27) ..(28) A s s e m b l i n g t he se e q u a t i o n s f o r a l l the e l emen t s g i v e s a s e r i e s o f s i m u l t a n e o u s e q u a t i o n s o f the form [ K ] :m {Q} ...(29) [ K ] , the mas te r t he rma l s t i f f n e s s m a t r i x i s a symmet r i c banded m a t r i x , the b and -w i d t h b e i n g dependent on the maximum d i f -f e r e n c e between node numbers i n any e l e m e n t . In o r d e r to r e -duce computer s t o r a g e c o s t s , a node number ing scheme was s e l e c t e d to m i n i m i z e the b a n d - w i d t h , and o n l y the l owe r h a l f band o f the m a t r i x i n c l u d i n g the d i a g o n a l was s t o r e d . The r e s u l t i n g e q u a t i o n s were s o l v e d by a Cho l e s k y method 103,105 The r e l e v a n t f i n i t e - e l e m e n t e q u a t i o n s f o r the l i n e a r and q u a d r a t i c t empe r a t u r e e l emen t s are p r e s e n t e d i n Append i x V. The i n t e r p o l a t i o n p o l y n o m i a l s s e l e c t e d f o r bo th t h e s e cases s a t i s f y c o m p a t i b i l i t y and comp l e t enes s r e q u i r e m e n t s wh i ch en su re s mono ton i c conve rgence o f the s o l u t i o n w i t h mesh r e f i n e m e n t . 1 0 2 Emery and C a r s o n 1 0 6 have p o i n t e d ou t t h a t the l i n e a r t empe r a t u r e f i n i t e - e l e m e n t a p p r o x i m a t i o n i s e q u i v a l e n t to the f i n i t e d i f f e r e n c e method as the a s semb led c o e f f i c i e n t m a t r i c e s a re most o f t e n i d e n t i c a l . However the f i n i t e e l ement method can b e t t e r app rox ima t e c o n d i t i o n s a t the bounda r y . For example i n the p r e s e n t a n a l y s i s , i n f o r m u l a t i n g the f i n i t e - e l e m e n t e q u a t i o n s f o r the l i n e a r t empe r a t u r e e l emen t , the heat f l u x , and c o o l i n g - w a t e r t empe r a t u r e s have been assumed to v a r y l i n e a r l y between boundary nodes . In c o n t r a s t to t h i s , i n a p p l y i n g the h e a t -b a l a n c e p r i n c i p l e to e l emen t a l volumes i n the f i n i t e -d i f f e r e n c e method each o f the a f o r e - m e n t i o n e d pa rame te r s were assumed c o n s t a n t ove r t he e l emen t . In the q u a d r a t i c t empe r a t u r e model a q u a d r a t i c v a r i a t i o n was assumed between boundary nodes . F i g . 21 i s a f l ow c h a r t o f the computer program f o r the f i n i t e e l ement method. The o p e r a t i n g c o n d i t i o n s o f a b i l l e t c a s t e r p r e s e n t e d i n Tab l e IV were used as i n p u t da ta to the m o d e l . 89 GENERATE MESH TOPOLOGY NODE COORDINATES BOUNDARY CODE F i g . 21 Flow Cha r t o f the F i n i t e - E l e m e n t Hea t - F l ow Program to Compute the S t e a d y - S t a t e Thermal F i e l d i n Moulds . ( c o n t . ) 9 0 COMPUTE INFLUENCE MATRIX FOR CONVECTION COMPUTE EFFECTIVE STIFFNESS MATRIX COMPUTE CONVECTIVE 'LOAD ' VECTOR COMPUTE HEAT-FLUX 'LOAD' VECTOR F i g . 21 Flow C h a r t o f t he F i n i t e - E l e m e n t Hea t -Flow Program to Compute the S t e a d y -S t a t e Thermal F i e l d s i n Mou lds , ( c o n t . ) 91 COMPUTE EFFECTIVE THERMAL LOAD VECTOR ASSEMBLE MASTER COEFFICIENT STIFFNESS VECTOR CONTAINING LOWER HALF BAND ASSEMBLE MASTER THERMAL LOAD VECTOR SOLVE EQUATIONS BY CHOLESKY ROUTINE F i g . 21 Flow Chart of the F in i t e -E lemen t Heat-Flow Program to Compute the Steady-State Thermal F ie lds in Moulds, ( c o n t . ) 92 O COMPUTE NEW WATER TEMPERATURES BY HEAT BALANCE ^ N ^ F i g . 21 F low Cha r t o f the F i n i t e - E l e m e n t Hea t -Flow Program to Compute the S t e a d y -S t a t e Thermal F i e l d s i n Mou l d s . 93 As a p r e l i m i n a r y s t e p , the conve rgence o f the l i n e a r -t empe r a t u r e f i n i t e - e l e m e n t model was t e s t e d by compar ing the t empe r a t u r e s c a l c u l a t e d a t i d e n t i c a l l o c a t i o n s f o r a 213 node , ( ( 3 x 7 1 ) m e s h ) , and 705 node, ( (5x141)mesh) s y s t e m . The maximum d i f f e r e n c e i n p r e d i c t e d w a l l t empe r a t u r e s between the two ca se s i s g e n e r a l l y o f the o r d e r o f 1.0°C e x c ep t a t t he men i s cus where a g r e a t e r d i f f e r e n c e i s ob se r ved f o r the same r e a s on s d i s c u s s e d f o r the f i n i t e - d i f f e r e n c e scheme. T h e r e f o r e i t can be assumed t h a t t he s o l u t i o n conve rges u n i f o r m l y . Compar ing the r e s u l t s w i t h the t e m p e r a t u r e d i s t r i b u t i o n p r e d i c t e d by t he f i n i t e d i f f e r e n c e method ( Tab l e IX) i t i s a ppa r en t t h a t the maximum d i f f e r e n c e i n t empe r a t u r e s i s o f the o r d e r o f 2°C. I t i s e v i d e n t t h a t the d i f f e r e n c e i n the r e s u l t s ' w i t h the improved a p p r o x i m a t i o n to the boundary f l u x e s and wa te r t empe r a t u r e s i s n e g l i g i b 1 y s m a l l . To e v a l u a t e the improvement o b t a i n a b l e by a s s u m i n g a q u a d r a t i c tem-p e r a t u r e v a r i a b l e wi t h i n an e l ement, the r e s u l t s o f a 705 node q u a d r a t i c t empe r a t u r e e l emen t model were s c r u t i n i z e d . A sample o f the p r e d i c t e d t empe r a t u r e d i s t r i b u t i o n o v e r 100 mm o f the mould l e n g t h a r e shown i n Tab l e X. I t i s a p p a r e n t t h a t w a l l tem-p e r a t u r e v a r i a t i o n down the mould d i s p l a y s l o c a l i r r e g u l a r -i t i e s , the e f f e c t b e i n g a maximum a t the c o o l i n g - w a t e r i n t e r f a c e . A l t h o u g h the q u a d r a t i c e l ement model i s a h i g h e r o r d e r a p p r o x i m a t i o n o f the v a r i a b l e , t han the l i n e a r e l ement model and g e n e r a l l y c o u l d be e x p e c t e d to be more a c c u r a t e than the l a t t e r , the type of i n a c c u r a c y o b s e r v e d , wh i ch i s i n h e r e n t l y p e c u l i a r to t he q u a d r a t i c 94 Tab l e V I I I A Compar i son o f the Mou l d -Wa l l Tempera tu res  P r e d i c t e d by the F i n i t e - E l e m e n t Method f o r Two D i f f e r e n t Mesh S i z e s Di s t a n c e Down the Moul d (mm) Mould Wa l l Tempe ra tu re s °C Hot--Face C o l d - Face 705 Nodes 213 Nodes 705 Nodes 213 Nodes 0 38.54 38.55 38 .53 38 .53 50 40 .04 40 .32 39 .55 39.70 100 160 .82 171.37 100 .68 108 .75 140 226 .60 225 .48 140 .43 140 .80 160 224 .98 223.44 139 .58 139 .62 180 221.77 220.15 137 .65 137 .63 200 217 .85 216 .25 135 .32 135 .28 220 213 .33 211 .75 132 .63 132.59 260 200 .98 199.50 125 .47 125 .45 300 182.51 181 .21 115 .07 115.05 340 161 .34 160 .23 103 .25 103 .23 380 142.01 141 .07 92 . 45 92 .44 420 127 .25 126 .45 84 . 17 84 .17 460 117 .27 116 .53 78 .49 78 .48 500 108 .33 107.66 73 .42 73 .42 540 99 .52 98 .93 68 .45 68 .45 580 91 .54 91 .03 63 .96 63 .96 620 85 .00 84 .54 60 .25 60 .26 660 79 .1 7 78 .75 56 .97 56 .96 700 75.40 75 .04 54 .92 54 .90 95 Tab 1e IX A Compar i son o f the Mou l d -Wa l l Tempera tu res P r e d i c t e d by the F i n i t e - D i f f e r e n c e and  F i n i t e - E l e m e n t Methods (5x141 Mesh) Distance Down the Mould (mm) Mould Wall Temperatures °C Hot- Face Cold -Face F.D F.E F.D F.E 0 39.47 38.54 38.98 38.53 50 40.50 40.04 39.83 39.55 100 163.48 160.82 100.04 100.68 140 225.46 226.60 139.48 140.43 160 224.04 224.98 138.84 139.58 180 221.04 221.79 137.12 137.65 200 217.33 217.85 134.99 135.32 220 213.01 213.33 132.51 132.63 260 201.05 200.98 125.72 125.47 300 182.89 182.51 115.62 115.07 340 161 .95 161.34 104.02 103.25 380 142.75 142.01 93.33 92.45 420 128.04 127.25 85.08 84.17 460 1.18.04 117.27 79.37 78.49 500 109.02 108.33 74.22 73.42 540 100.10 99.52 69.13 68.45 580 91.97 91.54 64.48 63.96 620 85.25 85.00 60.58 60.25 660 79.19 79.17 57.06 56.97 700 75.22 75.40 54.82 54.92 96 Tab l e X Mou l d -Wa l l Tempera tu res P r e d i c t e d by the Q u a d r a t i c Tempera tu re E lement Model Distance Mould Wa l l Tempera tu res a t D i f f e r e n t D i s t a n c e s Down the f rom the Ho t - F a ce Mould (mm) d = 0.0 (mm) d= 2.38 (mm) d= 4.76 < (mm) d= 7.1437 (mm) d = 9.525 (mm) 100 154.42 136.39 120.42 106.00 91.75 105 181.60 160.40 141.29 123.93 108.26 110 197.07 175.14 154.50 134.88 114.66 115 206.15 184.27 163.24 143.09 124.17 120 211.62 189.63 168.11 146.40 124.52 125 214.84 192.97 171.49 150.46 130.32 130 216.49 194.59 172.85 150.63 127.98 135 217.30 195.57 174.02 152.76 132.22 140 217.43 195.69 173.98 151.67 128.81 145 217.34 195.75 174.28 153.02 132.41 150 216.90 195.31 173.69 151 .44 128.60 155 216.44 195.00 173.65 152.48 131.94 160 215.75 194.31 172.83 150.70 127.97 165 215.12 193.84 172.63 151 .59 131.17 170 214.31 193.03 171 .70 149.72 127.14 175 213.57 192.45 171 .40 150.53 130.26 180 212.66 191 .55 170.40 148.60 126.20 185 211.82 190.89 170.03 149.33 129.24 190 210.83 189.93 168.97 147.37 125.17 195 209.95 189.92 168.56 148.05 128.15 200 208.91 188.21 167.45 146.06 124.08 t empe r a t u r e e lement model wou ld a r i s e i f a h e a t - t r a n s f e r c o e f f i c i e n t boundary c o n d i t i o n had to be s i m u l a t e d . T h i s stems f rom the f a c t t h a t the c o n t r i b u t i o n t o the o v e r a l l s t i f f n e s s m a t r i x o f the m i d s i d e nodes a r i s i n g f rom [K^] i s f o u r t imes g r e a t e r than the c o n t r i b u t i o n o f the c o r n e r node s . Owing to t h i s i r r e g u l a r i t y i n p r e d i c t e d w a l l t empe ra tu re s , i t i s d i f f i c u l t to a s se s s the a c c u r a c y o f t he q u a d r a t i c t empe r a t u r e e l ement model and i t was t h e r e f o r e d i s c a r d e d . Wi th r e g a r d to s t o r a g e and e x e c u t i o n c p s t s , t h e r e was h a r d l y any d i f f e r e n c e between the f i n i t e - d i f f e r e n c e and f i n i t e - e l e m e n t models f o r a (5 x 1 4 1 ) , 705 node , mesh . T h i s i s a c o i n c i d e n c e f o r t h i s p a r t i c u l a r c o n f i g u r a t i o n s i n c e the b and -w i d t h o f the f i n i t e - e l e m e n t model was equa l t o the number o f n on - z e r o e l emen t s per row i n the f i n i t e d i f f e r e n c e method . Hence f o r t h i s s t udy e i t h e r model can be used w i t h t he same a c c u r a c y and comparab le c o s t . In g e n e r a l however , i f t h e r e a re a g r e a t e r number o f e l emen t s i n the t h r o u g h -t h i c k n e s s d i r e c t i o n o f the mou ld , or a t h i c k e r mould were b e i n g s i m u l a t e d m a i n t a i n i n g the e lement s i z e the same as b e f o r e , then the b and -w i d t h o f the as semb led c o e f f i c i e n t m a t r i x o f the f i n i t e - e l e m e n t e q u a t i o n s wou ld be g r e a t e r than the b and -w i d t h o f the f i n i t e - d i f f e r e n c e c o e f f i c i e n t m a t r i x wh ich wou ld s t i l l be f i v e ; under t he se c i r c u m s t a n c e s t he f i n i t e - d i f f e r e n c e method wou ld have a d e f i n i t e edge ove r the f i n i t e - e l e m e n t method i n terms o f comput ing c o s t s . 98 3 . 1 . 4 . 3 An Imp l i c i t F i n i t e D i f f e r e n c e Te chn i que f o r C a l c u l a t i n g the T r a n s i e n t Tempe ra tu re ' D i s t r i b u t i o n i n B i l l e t M o u l d s . A t r a n s i e n t t w o - d i m e n s i o n a l h e a t - f l o w model has been deve l oped e x p r e s s l y to compute the t h e rma l r e s pon se o f a b i l l e t mould to i n t e r m i t t e n t and c o n t i n u o u s s u r f a c e b o i l i n g i n the c o o l i n g - w a t e r c h a n n e l . As w i l l be shown, b o i l i n g i s a key f a c t o r i n m o u l d - r e l a t e d q u a l i t y p r o b -lems f o r t he c a s t i n g o f b i l l e t s . The h e a t - f l u x p r o f i l e em-p l o y e d f o r t h i s a n a l y s i s was the same as t h a t used t o c a l -c u l a t e the s t e a d y - s t a t e r e sponse o f a b i l l e t c a s t e r s i n c e the e f f e c t s o f mould o s c i l l a t i o n were n e g l e c t e d . Of t he a v a i l a b l e n u m e r i c a l t e c h n i q u e s , 1 0 3 - 1 0 5 the i m p l i c i t a l t e r n a t i n g - d i r e c t i o n f i n i t e d i f f e r e n c e method d e s c r i b e d by Peaceman and R a c h f o r d , 1 0 7 and D o u g l a s 1 0 8 ' 1 0 9 was s e l e c t e d f o r the f o l l o w i n g r e a s o n s . The c o e f f i c i e n t m a t r i x o f the f i n i t e - d i f f e r e n c e e q u a t i o n s wh i c h i s t r i -d i a g o n a l , does not r e q u i r e l a r g e a l l o c a t i o n s o f memory and i s amenable to a r a p i d s o l u t i o n . In t h i s method two d i f -f e r e n c e e q u a t i o n s are used i n s u c c e s s i o n each f o r h a l f the t o t a l t ime s t e p , d t . The f i r s t e q u a t i o n i s assumed i m p l i c i t i n the x - d i r e c t i o n , wh ich f o r the case o f t he mould i s t he t h r ough t h i c k n e s s d i r e c t i o n , and v a l u e s computed o ve r t h i s 99 h a l f t ime s t ep are used i n the second e q u a t i o n wh i c h i s assumed i m p l i c i t i n the z - d i r e c t i o n , where the l a t t e r c o r -r e sponds t o the a x i a l d i r e c t i o n o f t h e . c a s t i n g . Based on t h i s method , the a p p r o p r i a t e f i n i t e - d i f f e r e n c e e q u a t i o n s have been d e r i v e d f o r the l o n g i t u d i n a l m i d - p l a n e o f the mould w a l l and a re p r e s e n t e d i n Append i x V I . T h i s method o f s o l u t i o n has been found to be u n c o n d i t i o n a l l y s t a b l e f o r any v a l u e o f the t ime s t e p d t , as w e l l as noda l s p a c i n g s dx and dz i n t he two p e r p e n d i c u l a r d i r e c t i o n s , and i s n o t r e s t r i c -t i v e w i t h r e s p e c t to t h e i r s e l e c t i o n as compared w i t h e x p l i c i t 103 me thods . The s o l u t i o n conve rges w i t h a d i s c r e t i z a t i o n 2 2 e r r o r o f 0 ( d t + dx ) and t h e r e f o r e a s m a l l e r t i m e s t e p and f i n e r node s p a c i n g wou ld y i e l d b e t t e r a c c u r a c y . However the c o s t i n c r e a s e s w i t h f i n e r d i s c r e t i z a t i o n o f t h e con t i nuum and t h e r e f o r e a compromise between c o s t and a c c u r a c y must be r eached i n s e l e c t i n g a s u i t a b l e t ime s t e p . Emery and C a r s o n 1 0 6 have shown t h a t an i m p l i c i t f i n i t e - d i f f e r e n c e t e c h n i q u e i s more a t t r a c t i v e than a f i n i t e - e l e m e n t method f o r t r a n s i e n t p rob lems s i n c e both e x e c u t i o n t ime and s t o r a g e r e q u i r e m e n t s may be an o r d e r o f magn i tude l e s s . The o v e r a l l program l o g i c f o r the t r a n s i e n t model i s s i m i l a r to t h a t o f the s t e ad y s t a t e f i n i t e - d i f f e r e n c e mode l , e x c e p t t h a t i t e r a t i o n s i n the f l o w - c h a r t o f F i g . 18 100 a re f o r the advance o f t i m e ; t h e r e f o r e a s e p a r a t e f l o w -c h a r t w i l l no t be p r e s e n t e d . To check the a c c u r a c y o f the model t h e program was f i r s t employed to compute the t empe r a t u r e d i s t r i b u t i o n f o r the b i l l e t mould c o n d i t i o n s g i v en i n Tab l e I V . I t was found t h a t a s t e a d y - s t a t e was r eached i n 5 .0 sees , the t empe r a t u r e d i f f e r e n c e between s u c c e s s i v e i t e r a t i o n s a t i d e n t i c a l l o c a t i o n s r e a c h i n g 0 . 2 ° C . These r e s u l t s a r e compared i n Tab l e XI w i t h the s t eady s t a t e s o l u t i o n t o t he p rob l em o b t a i n e d by the f i n i t e - d i f f e r e n c e method and the maximum d e v i a t i o n was found to be 3%. The s m a l l t e m p e r a t u r e dependence o f t he rma l c o n d u c t i v i t y was no t i n c o r p o r a t e d i n the m o d e l . 3 . 1 . 5 V a l i d a t i o n o f Heat F low Mode l s by  Compar i son w i t h E x p e r i m e n t a l Data An i m p o r t a n t s t a ge i n the deve l opmen t o f a ma t hema t i c a l model i s the compar i son of model p r e d i c t i o n s w i t h i n d u s t r i a l measurements , p r i o r to i t s use as a p r e d i c -t i v e t o o l . S i n c e the the rma l r e sponse o f b i l l e t moulds as p r e d i c t e d by the f i n i t e - d i f f e r e n c e and f i n i t e - e l e m e n t models d e ve l oped i n the p r e c e d i n g s e c t i o n , a re i n agreement w i t h one a n o t h e r , c ompa r i s on o f the r e s u l t s o f one model w i t h e x p e r i m e n t a l da ta i s s u f f i c i e n t to p rove t he v a l i d i t y o f 101 Table XI A Comparison of the Steady-State Mould-Wall Temperatures Pred ic ted by the F i n i t e - D i f f e r e n c e  and I m p l i c i t A l t e r n a t i n g D i r e c t i o n F i n i t e -D i f fe rence Methods Distance Down the Mould (mm) Mould Wall Temperatures °C Hot- Face Cold-Face F.D. I.A.D. F.D. F.D. I.A.D. F.D. 0 39.47 38.49 38.98 38.40 50 40.50 38.79 39.83 38.69 100 163.48 160.76 100.04 100.14 140 225.46 221.26 139.48 141.60 160 224.04 220.11 138.84 139.85 180 221.04 218.19 137.12 137.75 200 217.33 212.46 134.99 135.40 220 213.01 209.99 132.51 132.76 260 201.05 198.72 125.72 129.63 300 182.89 179.24 115.62 115.31 340 161 .95 158.21 104.02 103.38 380 142.75 138.98 93.33 92.50 420 128.04 125.11 85.08 84.14 460 118.04 114.85 79.37 78.53 500 109.02 106.47 74.22 73.48 540 100.10 97.17 69.13 68.50 580 91.97 88.66 64.48 63.98 620 85.25 81.54 60.58 60.28 660 79.19 76.01 57.06 56.95 700 75.22 71.71 54.82 54.25 102 a l l t he models . The p r o cedu r e adop ted here to v a l i d a t e the t r a n s i e n t h e a t - f l o w mode l , was to u t i l i z e e x p e r i m e n t a l l y measured t i m e - a v e r a g e d h e a t - f l u x and c o o l i n g - w a t e r d a t a as i n p u t to the model and compare the measured t e m p e r a t u r e s a t a l o n g i -t u d i n a l s e c t i o n to p r e d i c t e d v a l u e s . The e x p e r i m e n t a l d a t a 21 o f S i ngh and B l a z e k f o r a 0.9% ca rbon s t e e l was s e l e c t e d f o r t h i s p u r p o s e . They measured the hea t f l u x a t the c o l d -f a c e by p e r f o r m i n g hea t b a l a n c e s on the c o o l i n g - w a t e r i n h o r i z o n t a l c o o l i n g c hanne l s o f an e x p e r i m e n t a l b i l l e t c a s t e r . S i n c e the ma thema t i c a l model r e q u i r e s a h o t - f a c e h e a t - f l u x p r o f i l e as i n p u t d a t a , i n t h i s p a r t i c u l a r c a s e , the measured c o l d - f a c e h e a t - f l u x p r o f i l e t o g e t h e r w i t h t he c o o l i n g - w a t e r da ta were used to back c a l c u l a t e the h o t - f a c e h e a t - f l u x p r o f i l e by a t r i a l and e r r o r p r o c e d u r e . When the c o r r e c t h o t - f a c e h e a t - f l u x p r o f i l e was f i n a l l y o b t a i n e d the c a l c u -l a t e d and measured c o l d - f a c e h e a t - f l u x p r o f i l e s matched ve r y w e l l as can be seen i n F i g . 2 2 . The a c c u r a c y o f the model was then e v a l u a t e d by compar i ng t e m p e r a t u r e s measured 3.175 mm from the h o t - f a c e to the p r e d i c t e d , v a l u e s . The r e s u l t s a re shown i n F i g . 23 and i t i s e v i d e n t t h a t t h e r e i s good ag reemen t . The h o t - f a c e h e a t - f l u x p r o f i l e o b t a i n e d by the V) 5 Meniscus - 4 0 0 • c o cu 5 0 0 O C o 6 0 0 o o Experimental measurements Model fitted 1000 2 0 0 0 Heat flux (kW/m 2 ) 3 0 0 0 4 0 0 0 F i g . 22 Cold-Face Heat-Flux P r o f i l e in a B i l l e t Mould F i t t e d to 21 Experimental Measurements by T r i a l and E r r o r . o oo 2 0 0 o E a> SZ c 4 0 0 5 o T3 O C o 6 0 0 h -1 o o Measured temperature Model prediction 1 1 1 1 4 0 160 8 0 120 Temperature (°C) F i g . 2 3 A Compar ison Between P r e d i c t e d and Measured Wa l l Tempera tu re i n a B i l l e t Mou ld . 105 p r o c edu r e d e s c r i b e d above i s shown i n F i g . 19 and was the b a s i s o f the c a l c u l a t i o n s p r e s e n t e d i n the p r e c e d i n g s e c t i o n s . The h o t - f a c e h e a t - f l u x p r o f i l e f o r a 0.1% ca rbon s t e e l was o b t a i n e d by a s i m i l a r p r o c edu r e and i s a l s o p r e s e n t e d i n F i g . 19 . For s l a b moulds the h o t - f a c e boundary c o n d i t i o n s f o r the h i gh and l o w - m e l t i n g powder l u b r i c a t i o n p r a c t i c e s were a l s o o b t a i n e d by back c a l c u l a t i o n f rom e x p e r i m e n t a l l y measured t empe r a t u r e s a l o n g a l o n g i t u d i n a l s e c t i o n . 1 1 1 Tab l e X I I p r e s e n t s the o p e r a t i n g c o n d i t i o n s f o r a s l a b c a s t e r w i t h low h e a t - f l u x p r a c t i c e . F i g . 24 shows the match o b t a i n e d between measured and c a l c u l a t e d mould w a l l t em-p e r a t u r e s 6 .35 mm f rom the h o t - f a c e when t he t r i a l and e r r o r p r o c edu r e t o o b t a i n h o t - f a c e h e a t - f l u x p r o f i l e was c o m p l e t e d . Tab l e X I I I p r e s e n t s the o p e r a t i n g c o n d i t i o n s f o r a s l a b c a s t e r w i t h h i g h - h e a t f l u x p r a c t i c e i n wh i ch mould w a l l t empe r a t u r e measurements were c a r r i e d o u t . The h o t - f a c e h e a t - f l u x p r o f i l e s c a l c u l a t e d as d e s c r i b e d , a re shown i n F i g . 33 as a f u n c t i o n o f t i m e . 3.2 P r e d i c t e d Thermal F i e l d s i n S l ab and B i l l e t Moulds Hav i ng a s c e r t a i n e d t h a t deve l oped i n s e c t i o n 3 . 1 . 4 a re the ma t hema t i c a l mode ls c apab l e o f a c c u r a t e l y 106 Temperature (°C) Fig.24 Long i tud ina l Temperature P r o f i l e in a Slab M o u l d 1 1 1 F i t t e d to Experimental Measurements by T r i a l and E r r o r . 107 F i g . 2 5 P r e d i c t e d S t e a d y - S t a t e I so the rms i n the Wa l l o f a B i l l e t Mould Under S t a nda r d C o n d i t i o n s f o r a C o o l i n g - W a t e r V e l o c i t y o f 10 m/sec . 108 p r e d i c t i n g the t empe r a t u r e f i e l d i n a mou l d , the model was then used to compare the the rma l r e sponse o f s l a b and b i l l e t moulds under t y p i c a l i n d u s t r i a l ' c o n d i t i o n s . The t empe r a t u r e Mso the rms f o r a b i l l e t mould w a l l under c o n d i t i o n s g i v e n i n Tab l e IV a re shown i n F i g . 25 . The v a l u e s o f o p e r a t i n g pa rame te r s s e l e c t e d a re based on t y p i c a l i n d u s t r i a l p r a c t i c e . A c o o l i n g - w a t e r v e l o c i t y o f 10 m/sec has been employed i n t h i s a n a l y s i s to en su r e t h a t b o i l i n g does not o c c u r i n the wa t e r c h a n n e l . T h i s i s an i m p o r t a n t c o n d i t i o n as w i l l be shown l a t e r when the i n -f l u e n c e o f b o i l i n g on b i l l e t q u a l i t y i s d i s c u s s e d . The the rma l f i e l d o b t a i n e d , ( F i g . 2 5 ) , i s based on a s t e a d y -s t a t e h e a t - f l o w mode l , and the peak c o l d - f a c e t e m p e r a t u r e o f 139°C i s be low the i n c i p i e n t b o i l i n g p o i n t , f o r s u r f a c e b o i l i n g i n the c h a n n e l . C l e a r l y the peak t empe r a t u r e i n t he mou ld , wh i ch has a v a l u e o f 225°C , does no t o c c u r a t the men i s cus bu t a p p r o x i m a t e l y 2.5~3 cms be low i t . F u r t he rmo re i t i s e v i d e n t t h a t t w o - d i m e n s i o n a l h e a t - f l o w i s s i g n i f i c a n t i n the upper r e g i o n s o f the mou l d . The r ea son o f c ou r s e i s t h a t the exposed s e c t i o n s o f the mould above the men i s cus a re s u b s t a n t i a l l y c o l d e r than the p o r t i o n i n c o n t a c t w i t h the s t e e l , r e s u l t i n g i n l a r g e l o n g i t u d i n a l g r a d i e n t s f o r h e a t - f l o w . In the l owe r r e g i o n s o f the mould however t he h e a t - f l o w i s a p p r o x i m a t e l y u n i - d i m e n s i o n a l and the i s o t h e r m s 109 a re n e a r l y p a r a l l e l to the mould w a l l . F i g . 26 i s a p l o t o f h o t - and c o l d - f a c e t empe r a t u r e s as a f u n c t i o n o f d i s t a n c e down the mou l d . The i n f l u e n c e o f n u c l e a t e b o i l i n g on the t e m p e r a t u r e i s o t h e r m s was i n v e s t i g a t e d f o r the mould c o n d i t i o n s o f t h e p r e c e d i n g example e x c ep t w i t h a l owe r wa t e r v e l o c i t y o f 7.0 m/sec , wh i ch i s a v a l u e t h a t i s more commonly e n c o u n t e r e d i n i n d u s t r i a l b i l l e t ma ch i n e s . Th i s c ompu t a t i o n was c a r r i e d out u s i n g the t r a n s i e n t h e a t - f l o w mode l . The the rma l r e s pon se o f t he mould w a l l took the fo rm o f a s t e a d y o s c i l l a t i o n between an appa r en t "upper t empe r a t u r e f i e l d " and " l o w e r t empe r a t u r e f i e l d " . F i g . 27 shows the h o t - and c o l d - f a c e t empe r a t u r e p r o f i l e s p r e d i c t e d a t the upper and l owe r l i m i t s . A s a l i e n t f e a t u r e o f the c o l d - f a c e t empe r a t u r e p r o f i l e a t the " l o w e r - l i m i t " i s t h a t i t has two peaks i n s t e a d o f o n e . T h i s i s s i m i l a r to the t empe r a t u r e d i s t r i b u t i o n measured by S i ngh and B l a z e k i n the e x p e r i m e n t a l b i l l e t c a s t e r p r e -s en t ed i n F i g . 2 3 . They a l s o r e p o r t e d t h a t a doub l e peak p e r s i s t e d i n t h e i r c o l d - f a c e h e a t - f l u x p r o f i l e measurements i n the same s t u d y . Th i s e v i d en c e s u g g e s t s t h a t the two peaks i n the h e a t - f l u x p r o f i l e a re due to n u c l e a t e b o i l i n g i n the c o o l i n g wa te r c h a n n e l . The i m p l i c a t i o n s o f t h i s even t on mould pe r f o rmance w i l l be d e f e r r e d u n t i l the c o r -r e s p o n d i n g d i s t o r t i o n p r o f i l e s a re p r e s e n t e d . F i g . ' s 28 T e m p e r a t u r e (°C) 0 5 0 1 0 0 1 5 0 2 0 0 26 Hot - and Co l d - F a ce Temperature P r o f i l e s o f a B i l l e t . Mould Under S t anda rd C o n d i t i o n s f o r a C o o l i n g - W a t e r V e l o c i t y o f 10 m/sec . T e m p e r a t u r e (°C) 0 5 0 1 0 0 150 2 0 0 F i g . 27 Hot - and C o l d - F a c e Temperature P r o f i l e s o f a B i l l e t Mould Showing Upper and Lower Tempera tu re L i m i t s Dur ing B o i l i n g a t the Mou ld /Wate r I n t e r -f a ce f o r a Water V e l o c i t y o f 7 m/sec. 112 5mm 5 mm 105 170 Meniscus 26 5 h 535 F i g . 2 8 P r e d i c t e d I so the rms i n the Wa l l o f a B i l l e t Mould f o r the Upper Tempera tu re L i m i t o f the B o i l i n g C y c l e . 113 I 5 mm | -r r 5 mm 100 170 265H 5'30r-Meniscus 80 100 L J Fig.29 Pred ic ted Isotherms in the Wall of a B i l l e t Mould f o r the Lower Temperature L i m i t of the B o i l i n g Cyc le . 114 and 29 show the i s o t h e rms f o r the upper and l owe r l i m i t o f t e m p e r a t u r e . The peak h o t - f a c e t empe r a t u r e f o r the two cases were 227°C and 217°C w h i l s t the c o l d - f a c e peak tem-p e r a t u r e s were 157°C and 141°C , r e s p e c t i v e l y . F i g . ' s 30 and 31 show the the rma l f i e l d i n a s l a b mould under the s t a n d a r d c o n d i t i o n s p r e s e n t e d i n T ab l e s XI I and X I I I f o r " l ow" and " h i g h " h e a t - f l u x mould p r a c t i c e s r e s p e c t i v e l y . As e x p e c t e d , the p r e d i c t e d t empe r a t u r e s w i t h a h i gh h e a t - f l u x p r a c t i c e a re h i g h e r than w i t h a low h e a t -f l u x p r a c t i c e . In both cases the peak h o t - f a c e t empe r a t u r e r can be seen to o c c u r a few c e n t i m e t e r s be low the m e n i s c u s . The peak c o l d - f a c e t empe r a t u r e s a re we l 1 . b e ! ow t h a t r e q u i r e d to i n i t i a t e n u c l e a t e b o i l i n g . Hence wa t e r q u a l i t y and i t s e f f e c t on s c a l e d e p o s i t i o n d u r i n g b o i l i n g i s no t as c r u c i a l f o r the o p e r a t i o n o f a s l a b mould as i t i s f o r b i l l e t mou l d s . Owing to t he e x t e n s i v e t w o - d i m e n s i o n a l hea t f l o w i n the upper r e g i o n s o f t he mould t h e r e i s a c o n s i d e r a b l e q u a n t i t y o f hea t f l o w i n g up as w e l l as a c r o s s the mould w a l l to the c o o l i n g w a t e r . T h e r e f o r e the h e a t - f l u x d i s t r i b u t i o n measured a t the c o l d - f a c e w i l l be d i f f e r e n t f rom t h a t a t the h o t - f a c e a l t h o u g h the i n t e g r a t e d ave rage w i l l be the same. Th i s i s c l e a r l y i l l u s t r a t e d i n F i g . 32 wh i ch i s a p l o t o f l o n g i t u d i n a l h e a t - f l u x p r o f i l e s a t d i f f e r e n t s e c t i o n s o f a F i g . 3 0 P r e d i c t e d S t e a d y - S t a t e I so the rms i n the Wa l l o f a S l a b Mould w i t h Low H e a t - F l u x P r a c t i c e . 1 1 6 F i g . 3 1 P r e d i c t e d S t e a d y - S t a t e I so the rms i n the Wa l l o f a S l ab Mould w i t h H igh H e a t - F l u x P r a c t i c e . 117 Tab l e XII Mould D e t a i l s and O p e r a t i n g C o n d i t i o n s f o r S l ab C a s t i n g Wi th Low-heat F l u x P r a c t i c e Mould and C a s t i n g Pa rame te r s Section Size 1270x200 mm2 Mould Length 730 mm Wall Thickness 39 mm Casting Speed 0.8 m/min Water Ve loc i ty 7.9 m/sec Water Temperature Rise 8.5°C Lubr icat ion mould powder E f fec t i ve Cooling-Water Heat-Transfer Coef f i c ient 23.5 kW/m2oC 118 Tab l e X I I I Mould D e t a i l s and O p e r a t i n g C o n d i t i o n s f o r  S l ab C a s t i n g Wi th H igh -hea t F l u x P r a c t i c e Mould and C a s t i n g Pa rame te r s Section Size 1540x210 mm2 Mould Length 700 mm Wall Thickness 35 mm Casting Speed 1.3. m/mi n Water Ve loc i ty 7.0 m/sec Water Temperature Rise 8.2°C Lubr icat ion E f fec t i ve Cooling-Water Heat-transfer Coef f i c ien t powder/mul t i po r t nozzle 21.12 kW/m2°C 8 7 6 5 4 3 2 1 H E E O E cu SZ c o X J CD O c D £ 200 100 150 1 No X(mm) 1 2.5 2 7.5 3 12.5 4 17.5 5 22.5 6 27.5 7 32J5 8 37.5 1 0 5 CF 0 CF I 2 3 4 0 Through mould heat flux XI0 3(kW/m 2) 10 15 20 25 30 35 40 HF Z = 185.4 mm J I I 1 1 L 10 20 30 Z =288.4 mm J I L J I L c T3 3 o c CO a c X 0 -I 40 H.F I 10 20 30 40 Distance from the cold face (mm) oi 04 - I 5 C 3 ro F i g . 32 D i s t o r t i o n o f the H e a t - F l u x P r o f i l e as i t T r a v e r s e s the Wa l l o f a S l ab Mould w i t h High H e a t - F l u x P r a c t i c e . 1 2 0 s l a b mou l d . The peak o f the h e a t - f l u x p r o f i l e t h a t the c o o l i n g - w a t e r ' s e e s ' i s s e v e r a l c e n t i m e t e r s be low the men i s -cus a l t h o u g h the i n com ing h o t - f a c e h e a t - f l u x p r o f i l e has a. peak a t the m e n i s c u s . T h i s has i m p o r t a n t i m p l i c a t i o n s f rom the s t a n d p o i n t o f mould i n s t r u m e n t a t i o n f o r the measurement o f h e a t - f l u x d i s t r i b u t i o n i n both b i l l e t and s l a b mou l d s . The measured p r o f i l e a t any s e c t i o n p a r a l l e l t o the h o t -f a c e w i l l be d i s t o r t e d i n compa r i s on w i t h i n c om ing h e a t - f l u x p r o f i l e ow ing to t w o - d i m e n s i o n a l hea t f l o w and f o r b i l l e t moulds due to b o i l i n g . Hence, to o b t a i n an a c c u r a t e e s t i -mate o f the h e a t - f l u x d i s t r i b u t i o n a t the h o t - f a c e , measured mould w a l l t empe r a t u r e s have to be a n a l y z e d u s i n g a mathe-m a t i c a l model to back c a l c u l a t e the h e a t - f l u x a t the h o t -f a c e , as has been done i n t h i s a n a l y s i s . F i g . ' s 19 and 33 show the r e s u l t s o b t a i n e d i n t h i s way and i t i s c l e a r t h a t i n a l l cases the peak h e a t - f l u x i s a t t he m e n i s c u s . 121 4 4 0 0 4000p-I I 3600ff-I 320o|j-_ 2800 CM E 2400 X 3 3 2 0 0 0 CD X 1 6 0 0 K -200 8 0 0 r — 4 0 0 Casting operation Billet (Savage a Pritchard) Slab ; high heat flux practice Slab ; low heat flux practice 1 1 1 F i g . 3 3 10 H e a t - F I u x T i m e ) f o r P r a c t i c e . 20 50 60 30 40 Time ( s ) P l o t t e d A g a i n s t T i m e - i n - t h e - M o u l d ( D w e l l a S l a b M o u l d w i t h H i g h a n d Low H e a t - F l u x 4 .0 MOULD DISTORTION 4 .1 I n t r o d u c t i o n The c o n t i n u o u s c a s t i n g mould i s r e p e a t e d l y hea ted and c o o l e d t h r o ughou t i t s s e r v i c e l i f e . Du r i ng each h e a t , the mould i s s u b j e c t e d to a w i d e l y v a r y i n g t empe r a t u r e d i s t r i -b u t i o n as i s e v i d e n t f rom F i g . ' s 2 5 , 28-31 , o f the p r e c e d i n g c h a p t e r . The the rma l f i e l d s o b t a i n e d a re ' q u a s i ' s t e a d y -s t a t e d i s t r i b u t i o n s ; supe r imposed on t h e s e a re t empe r a t u r e f l u c t u a t i o n s caused by f a c t o r s such as l o c a l i z e d v a r i a t i o n s i n gap w i d t h a t the m o u l d - s t e e l i n t e r f a c e , the growth and detachment o f bubb l e s d u r i n g n u c l e a t e b o i l i n g a t the c o l d -f a c e , uns teady meta l l e v e l , and c y c l i c v a r i a t i o n s i n t empe r a -t u r e caused by mould o s c i l l a t i o n . The n o n - u n i f o r m t empe r a t u r e d i s t r i b u t i o n i n the mould g i v e s r i s e to d i f f e r e n t i a l e x pan s i o n o f the coppe r l e a d i n g to a d i s t o r t i o n o f the shape o f the mould c a v i t y , wh ich may va r y w i t h t ime due to the f a c t o r s men t i oned above . The d i f -f e r e n t i a l t he rma l e xpan s i on o f the coppe r r e s u l t s i n t he rma l s t r e s s e s o f magn i tude s u f f i c i e n t to cause y i e l d i n g i n r e g i o n s o f h i gh the rma l g r a d i e n t s , such as near the m e n i s c u s , p a r t i -c u l a r l y because the y i e l d s t r e s s i s l o c a l l y r educed due to the h i g h e r t e m p e r a t u r e s . The e n s u i n g p l a s t i c f l o w r e s u l t s i n permanent d i s t o r t i o n . The c o n s t r a i n t s e x e r t e d by the 122 mould s u p p o r t sys tem and the g e o m e t r i c c o n f i g u r a t i o n o f the mould i t s e l f , r e s t r a i n the f r e e e x p a n s i o n o f the c o p p e r , wh i ch i n du ce s a d d i t i o n a l s t r e s s e s t h a t can c o n t r i b u t e t o y i e l d i n g . The a c t u a l magn i tude o f the d i s t o r t i o n i s depen -dent on the imposed the rma l f i e l d , the m a t e r i a l p r o p e r t i e s , t h e geometry o f the mould and the mould c o n s t r a i n t s . From the p r e c e d i n g d i s c u s s i o n i t i s c l e a r t h a t t he d i s t o r t i o n o f the mould i s a r e s u l t o f t he i n t e r a c t i o n b e -tween the rma l and r e s t r a i n i n g f o r c e s , and i t i s d i f f i c u l t to d i s c e r n wh i ch f a c t o r s have the most s i g n i f i c a n t i n f l u e n c e . Th i s c h a p t e r d e s c r i b e s the sequence of c a l c u l a t i o n s t h a t were pe r f o rmed to o b t a i n a b e t t e r u n d e r s t a n d i n g o f the r e -sponse o f the mould to the the rma l l o a d s . The p l a t e -a n a l y s i s , wh i ch was f o l l o w e d by a p l a n e - s t r e s s a n a l y s i s o f a • t r an s ve r s e s l i c e o f the mould a t a s e l e c t e d d e p t h , p r o v i d e d a means f o r compar ing the r e l a t i v e magn i tudes o f the v a r i o u s components o f the t o t a l d i s t o r t i o n , and s e r v e d as a r a t i o n a l b a s i s f o r the s e l e c t i o n o f a t h r e e - d i m e n s i o n a l e l a s t o -p l a s t i c s t r e s s a n a l y s i s f o r the f i n a l d i s t o r t i o n c a l c u l a t i o n s . S i n c e one o f the o b j e c t i v e s o f t h i s s t udy was to i n v e s t i g a t e the r e l a t i o n s h i p between mould d i s t o r t i o n and q u a l i t y i n the c a s t i n g o f b i l l e t s , the s u b j e c t o f s l a b mould d i s t o r t i o n has no t been a d d r e s s e d . The t h e o r e t i c a l a n a l y s i s o f b i l l e t mould d i s t o r t i o n p r e s e n t e d i n t h i s c h a p t e r i s f o l l o w e d by a 124 d e s c r i p t i o n o f e x p e r i m e n t a l measurements o f mould w a l l movement i n an o p e r a t i n g b i l l e t c a s t e r . These measurements have been compared w i t h the p r e d i c t e d v a l u e s o f mould d i s -t o r t i o n to check the a c c u r a c y o f the c a l c u l a t i o n s . E f f e c -t i v e s t r e s s c o n t o u r s are p r e s e n t e d f o r d i f f e r e n t t r a n s v e r s e and l o n g i t u d i n a l s e c t i o n s i n the mould w a l l t o o b t a i n some i n s i g h t as to wh i ch r e g i o n s o f the mould a re most s u s c e p t i b l e to y i e l d i n g and p l a s t i c f l o w . 4 . 2 Me chan i c a l P r o p e r t y Data The a c c u r a c y o f d i s t o r t i o n and s t r e s s c a l c u l a t i o n s , f o r the c o n t i n u o u s c a s t i n g mould i s dependent on the a c c u r a c y o f the mechan i c a l p r o p e r t y d a t a . In the p a s t a l t h o u g h s t u d i e s have been c onduc t ed to o b t a i n such da ta f o r c o p p e r , i t was o n l y r e c e n t l y t h a t the mechan i c a l b e h a v i o u r o f mould m a t e r i a l s pe r s e , wh i ch are a l l o y s o f copper c o n t a i n i n g phosphorous o r s i l v e r , o r commerc i a l g rade coppe r wh ich c o n t a i n s m a l l q u a n t i -55 t i e s of a v a r i e t y o f t r a c e e l e m e n t s , became a v a i l a b l e . F u r t h e r -more ve r y l i t t l e was known about the mechan i c a l s t a t e o f the mould m a t e r i a l i n terms o f p e r c e n t work h a r d e n i n g . 125 A r e c e n t s t ud y conduc t ed a t the U n i v e r s i t y of B r i t i s h 55 Co lumb ia by Br imacombe e t a l . p r o v i d e s i n f o r m a t i o n for a v a r i e t y of mould m a t e r i a l s based on mechan i c a l p r o p e r t y measurements on samples t aken f rom d i f f e r e n t moulds bo th new and used.. Append i x VII p r e s e n t s the da t a e x t r a c t e d f rom t h a t work wh i ch were used i n the p r e s e n t s t u d y . 4.3 Mould C o n s t r a i n t s The d i s t o r t i o n t h a t o c c u r s i n a b i l l e t mould w i l l be s t r o n g l y i n f l u e n c e d by the method of mould c o n s t r a i n t s i n c e the t ype of s u p p o r t of the mould tube w i t h i n the mould a s -sembly wou ld de t e rm ine the e x t e n t of l o c a l i z e d r e s t r a i n t to t he rma l e x p a n s i o n and the magn i tude of t he s t r e s s e s t h a t r e s u l t s . 55 From the s tudy c onduc t ed by Br imacombe e t a l . on b i l l e t moulds f rom s e v e r a l i n d u s t r i e s , i t appea r s t h a t t h e r e are t h r e e common t ypes o f mould s u p p o r t s y s t e m s . In the f i r s t t y p e , h o r i z o n t a l s l o t s a re mach ined a p p r o x i m a t e l y 12 mm f rom the top o f the mould on the o u t s i d e o f a l l f o u r f a c e s . A s p l i t p l a t e f i t s i n t o the s l o t s , and i s b o l t e d a round the p e r i p h e r y to a s uppo r t p l a t e t h a t i s an i n t e g r a l , p a r t o f the mould h o u s i n g . In the second t y p e , s l o t s are mach ined on two o p p o s i t e f a c e s on l y and s p l i t p l a t e s f i t i n t o t he se s l o t s as i n the p r e v i o u s c a s e . In the t h i r d t y p e , the mould i s s u ppo r t e d at the top and bot tom between two p l a t e s w i t h g a s k e t s s e a l i n g the ma t i ng s u r f a c e s . The e f f e c t o f t he se t ypes o f c o n s t r a i n t s on mould d i s t o r t i o n has been c a l c u l a t e d and w i l l be r e p o r t e d l a t e r . 4.4 The Deve lopment o f Ma t hema t i c a l Mode ls to  C a l c u l a t e Mould D i s t o r t i o n In f o r m u l a t i n g a ma thema t i c a l model to c a l c u l a t e d i s t o r t i o n , a l l t h r e e d imens i on s have to be c o n s i d e r e d , i n c o n t r a s t to the h e a t - f l o w a n a l y s i s where t he t h i r d d imen s i o n was n e g l e c t e d due to the n e g l i g i b l y s m a l l e f f e c t s o f t r a n s -v e r s e heat f l o w . The b i l l e t mould b e i n g a square tube o f r e l a t i v e l y sma l l c r o s s - s e c t i o n , has an i n h e r e n t r i g i d i t y impa r t e d to i t by v i r t u e o f i t s s hape . Thus c o r n e r s can be e x p e c t e d to e x e r t a r e s t r a i n i n g e f f e c t on the movement o f the m i d - f a c e . Th i s i n t r o d u c e s c o n s i d e r a b l e c o m p l e x i t y t o the p rob l em and can r e s u l t i n p r o h i b i t i v e l y h i gh compu t i ng c o s t s . P r i o r to embark ing on a t h r e e - d i m e n s i o n a l model a p p l i c a t i o n o f t he t h e o r y o f p l a t e s to the mould was i n -v e s t i g a t e d as i t o f f e r s an a l t e r n a t i v e s o l u t i o n to the mould d i s t o r t i o n p r ob l em , p r o v i d e d the main c o n t r i b u t i o n to d i s -t o r t i o n i s due to b e n d i n g . In o r d e r to check t h i s , a p l a t e a n a l y s i s was f i r s t a p p l i e d to the mould and the e l a s t i c d e f l e c t i o n s were c a l c u l a t e d . The r e s u l t s were compared w i t h 1 2 7 mould w a l l movement due to t he rma l e xpan s i o n c a l c u l a t e d f o r a t r a n s v e r s e s l i c e i n two - d imen s i o n s assuming p l ane s t r e s s . The l a t t e r a n a l y s i s i s an i d e a l i z a t i o n , s i n c e the p l a n e s t r e s s a s sump t i on i s v i b l a t e d . n e a r s u p p o r t s and a l s o i n r e g i o n s o f r a p i d l y c hang i ng t empe r a t u r e due to the i n f l u e n c e o f a d j a c e n t l a y e r s wh i ch expand by d i f f e r e n t amounts . How-eve r the i n t e n t was to o b t a i n an e s t i m a t e o f the t he rma l e x pan s i o n i n r e g i o n s where the t empe r a t u r e v a r i a t i o n i s no t s t e e p . The r e s u l t s o f t he se a n a l y s e s c l e a r l y i n d i c a t e d a need f o r a t h r e e - d i m e n s i o n a l mould d i s t o r t i o n .mode l . These t o p i c s a re d e a l t w i t h i n the subsequen t s e c t i o n s . 4 . 4 . 1 A s sumpt i ons As i n the h e a t - f l o w a n a l y s i s , l o c a l i z e d changes i n d i s t o r t i o n e i t h e r due to c y c l i c v a r i a t i o n s i n ^tempera-t u r e caused by mould o s c i l l a t i o n o r o t h e r random even t s such as imp rope r meta l l e v e l c o n t r o l were no t c a l c u l a t e d . I t i s a n t i c i p a t e d t h a t the o v e r a l l d i s t o r t i o n p r o f i l e c a l c u -l a t e d f rom s t e a d y - s t a t e t empe r a t u r e s i s a r e a s o n a b l y a c -c u r a t e " a v e r a g e " p r e d i c t i o n and t h a t p e r t u r b a t i o n s i n d i s -t o r t i o n caused by t empe r a t u r e f l u c t u a t i o n s w i l l not mod i f y the o v e r a l l shape o f t h i s " a v e r a g e " p r o f i l e . S i n c e the s u b j e c t o f mould d i s t o r t i o n and i t s r e l a t i o n s h i p to q u a l i t y i s i n i t s i n f a n c y , t h e r e i s much to be g a i n ed f rom a comp le te a n a l y s i s o f the ave rage d i s t o r t i o n f o r d i f f e r e n t o p e r a t i n g c o n d i t i o n s . T h e r e f o r e a t r a n s i e n t model i n t h r e e d i m e n s i o n s , b e s i d e s be i ng p r o h i b i t i v e l y c o s t l y , i s p r e m a t u r e . The i n f l u e n c e o f n u c l e a t e b o i l i n g on mould d i s t o r t i o n was a n a l y z e d by c a l c u l a t i n g the d i s t o r t i o n p r o f i l e s c o r -r e s p o n d i n g to the upper and l owe r l i m i t s o f t empe r a t u r e ' a t t a i n e d d u r i n g the t empe r a t u r e c y c l e i nduced by t h e b o i l i n g phenomena. Because permanent d i s t o r t i o n has been ob s e r v ed i n mou l d s , bo th an e l a s t i c and e l a s t o - p l a s t i c a n a l y s i s was c a r r i e d o u t . The e f f e c t s o f c r eep were n e g l e c t e d s i n c e a t the peak t empe r a t u r e s reached by the c o p p e r , ~20Q°C, the c r eep s t r a i n f o r a mould l i f e o f 500 hours i s o f t h e o r d e r o f 0.05% at a s t r e s s l e v e l o f 140 MPa. i C 4 . 4 . 2 E l a s t i c D e f l e c t i o n o f the Mou ld f rom the  P l a t e Bend ing Theory The c l a s s i c a l t h e o r y o f p l a t e s p r o v i d e s a means f o r c a l c u l a t i n g the d e f l e c t i o n s , s t r e s s e s and s t r a i n s due to bend i ng o f a s t r u c t u r e s u b j e c t e d to t r a n s v e r s e l o a d -i n g , p r o v i d e d the th rough t h i c k n e s s i s s u b s t a n t i a l l y s m a l l e r than the o t h e r two d i m e n s i o n s . In a p l a t e a n a l y s i s i t i s assumed t h a t the v a r i a t i o n s o f s t r a i n s and s t r e s s e s a l o ng l i n e s normal to the p l ane o f the p l a t e a re l i n e a r , wh i ch i s 129 a good a p p r o x i m a t i o n i f the p l a t e i s t h i n and the d e f l e c t i o n s a re s m a l l . The obv i ou s advan tage o f t h i s a n a l y s i s i s t h a t a t h r e e - d i m e n s i o n a l p rob l em i s r educed to a t w o - d i m e n s i o n a l one w i t h o u t a s i g n i f i c a n t l o s s i n a c c u r a c y p r o v i d e d the assump-t i o n s rema in v a l i d . T h i s wou ld r e s u l t i n a c o n s i d e r a b l e s a v i n g i n comput ing c o s t s . The t r a d i t i o n a l a p p l i c a t i o n o f the t h e o r y o f p l a t e s has been to p rob lems i n wh i ch the s t r u c t u r e i s s u b j e c t e d to t r a n s v e r s e l o a d s . To e x t end t h i s a n a l y s i s to compute the e l a s t i c bend ing o f the mould w a l l s u b j e c t e d to a v a r y i n g t empe r a t u r e g r a d i e n t , an e q u i v a l e n t ' t h e r m a l l o a d and moment' d i s t r i b u t i o n i s computed and t r e a t e d as e x t e r n a l l o a d s . I t i s assumed t h a t the mould w a l l i s f i x e d a g a i n s t r o t a t i o n a l o n g the two edges by symmetry . The top edge o f the mould .which i s s l o t t e d to a c cep t a s u p p o r t p l a t e i s assumed to be s i m p l y s u p p o r t e d , w h i l e the bot tom edge i s f r e e . The f i n i t e - e l e m e n t method i s emminen t l y s u i t e d f o r s o l v i n g p rob lems w i t h complex l o a d i n g c o n d i t i o n s and s e v e r a l d i f f e r e n t t ypes o f e l ements have been de ve l oped f o r p l a t e s . 1 1 The v a r i a t i o n a l p r i n c i p l e f o r the p l a t e bend i ng p rob l em i s 102 g i v en be l ow ; J(w) = Ed 3 24(1-v2). 9 W I + 2v 8 w 3 w + 2" ( 2 2 3x ) ^ 3y A 1 3 0 A.}2+ 2 ( l -v ) /AJ 2 3y3x j dA 3y' _. E, F.W. 1=1 1 1 1=1 n * .z, M. -3W ay or 3W 3X W wds . . . (30) In E q . ( 3 0 ) J(w) i s the f u n c t i o n a l , w the t r a n s v e r s e d i s -p l a cemen t f u n c t i o n , x and y a re the c a r t e s i a n c o - o r d i n a t e d i r e c t i o n s i n t he p l ane o f the p l a t e ; f o r t h i s a n a l y s i s t he z - a x i s i s assumed to be the t h r ough t h i c k n e s s . d i r e c t i o n o f * the p l a t e . W* i s the s p e c i f i e d s u r f a c e l o a d i n g , F^  a r e * s p e c i f i e d c o n c e n t r a t e d l o ad s i n the z - d i r e c t i o n and NL a r e s p e c i f i e d c o n c e n t r a t e d moments. S i n c e the f u n c t i o n a l J ( w ) , c o n t a i n s second o r d e r d e r i v a t i v e s o f the d i s p l a c e m e n t f u n c t i o n w, f o r comp l e t enes s w i t h i n an e l e m e n t , the second d e r i v a t i v e o f w must be c o n t i n u o u s , w h i l e the s l o p e s must be c o n t i n u o u s a c r o s s i n t e r - e l e m e n t b o u n d a r i e s f o r com-p a t i b i l i t y . I d e a l l y , f o r u n i f o r m c o n v e r g e n c e , t h e se two c o n d i t i o n s must be met . However i n the case o f the p l a t e bend i ng e l ement due to the d i f f i c u l t y o f o b t a i n i n g com-p a t i b l e d i s p l a c e m e n t f u n c t i o n s , e l emen t s based on non-113 c on f o rm i ng shape f u n c t i o n s have been d e v e l o p e d . I t has been demons t r a t ed t h a t f o r the 4 node 12 degree o f f r eedom r e c t a n g u l a r e l e m e n t , wh i ch i s one such n o n - c o n f o r m i n g e l e m e n t , the a c c u r a c y o f t he s o l u t i o n and the r a t e o f c onve rgence i s g o o d . 1 1 4 In the t r e a t m e n t o f the mould w a l l as a p l a t e , the r e c t a n g u l a r e l ement d e s c r i b e d above has been used and the mould con t i nuum has been d i s c r e t i z e d i n t o a ne twork o f such e l emen t s as shown i n F i g . 34 . Each node , a t the v e r t i c e s o f an e l emen t , has t h r e e deg rees o f f r e e d o m , and t h e d i s -p l a cemen t w i t h i n an e lement i s e x p r e s s e d as a f u n c t i o n o f the noda l degrees o f f r eedom; w = [N] {§} . . . (31 ) where r w . aw ay 1 8w 3x ....'(32) The a p p r o p r i a t e shape f u n c t i o n s f o r t h i s e l emen t a r e p r e -s en t ed i n Append i x V I I I . F o l l o w i n g the t r a d i t i o n a l f i n i t e -e l emen t p r o c e d u r e , e q u a t i o n s o f the f o r m , [K] {s} = <P> (33) can be f o r m u l a t e d f o r the e n t i r e sys tem where [K ] i s an as semb led mas te r s t i f f n e s s m a t r i x and {P} i s an e q u i v a l e n t mas te r l o a d v e c t o r . The s t i f f n e s s m a t r i x f o r t h i s p a r t i -c u l a r e l emen t has been d e r i v e d 113 and i s a v a i 1 a b l e i n the 132 433 435 437 439 441 4 2 4 4 I 5 406 397 388 432 423 414 405 396 E E CO m 5; II >. I Y, 55 4 6 37 28 19 10 63 54 45 36 27 1 8 2 8 dx = 16.25 mm F i g . 34 D i s c r e t i z a t i o n o f the Mould Cont inuum i n the P l a ne o f a B i l l e t Mould Wal l f o r a F i n i t e - E l e m e n t P l a t e A n a l y s i s . 133 l i t e r a t u r e . The e q u i v a l e n t l o a d v e c t o r f o r each e l emen t i s e e g i v e n as {F> , where {F^} f o r node i i s {F.>-rF W . ) Fwx . l Fwy / J ....(34) Fw^ i s a l a t e r a l l o a d and Fwx.. and Fwy • a r e c o u p l e s . The " t h e rma l l o a d v e c t o r " f o r each e lement can be c a l c u l a t e d f r om t he t empe r a t u r e g r a d i e n t a c r o s s the e l emen t by the p r i n c i p l e s o f v i r t u a l wo r k . T y p i c a l l y , Fw. 6w . = / /(M 6c + M 6c ) dxdy i AJ x x y y ' (35) where 6c x and Sc^ are v i r t u a l c u r v a t u r e s caused by the v i r t u a l d i sp l a cement•Sw^ a t node i . M x and a r e the ' t h e r m a l moments' wh i ch p roduce a c u r v a t u r e e q u i v a l e n t to t h a t caused by t he t empe r a t u r e g r a d i e n t . The magn i tude o f the moments can t h e r e f o r e be c a l c u l a t e d f rom the c u r v a t u r e due to the g r a d i e n t and i s g i v en by the f o l l o w i n g e x p r e s s i o n : M x = M y ad'E AT 12( l-v) ..(36) a i s the c o e f f i c i e n t o f t he rma l e x p a n s i o n , d i s the w a l l t h i c k n e s s , E the modulus o f e l a s t i c i t y and AT i s the 19 t empe r a t u r e g r a d i e n t The noda l c o u p l e s Fwx^, and Fwy^ can be computed f rom M and M i n a s i m i l a r manner . D e t a i l s x y 1 3 4 o f t he se c a l c u l a t i o n s are p r e s e n t e d i n Append i x V I I I . The t he rma l g r a d i e n t s c a l c u l a t e d f rom the t he rma l f i e l d p r e d i c t e d by the s t e a d y - s t a t e h e a t - f l o w model , and shown i n F i g . 26 were a p p l i e d down the l e n g t h o f mould w a l l t r e a t e d he re as a p l a t e ( t h e g r a d i e n t s were assumed to be u n i f o r m a c r o s s the f a c e ) . A p l a t e - b e n d i n g computer program deve l oped by the Depar tment o f C i v i l E n g i n e e r i n g a t the U n i v e r s i t y o f B r i t i s h Co lumb ia was m o d i f i e d to i n -c o r p o r a t e t he i n p u t o f t he rma l g r a d i e n t s i n s t e a d o f l o a d s f rom wh i ch e f f e c t i v e l o ad v e c t o r s were computed i n t e r n a l l y . The e l a s t i c d e f l e c t i o n o f the p l a t e computed f o r two d i f f e r e n t mesh s i z e s i s shown i n F i g . 35 . A l t h o u g h the f l e x i b i l i t y o f the sys tem improves w i t h the f i n e r g r i d , t he e l a s t i c d e f l e c t i o n a t the men i scus computed f o r the c o a r s e r g r i d (125 nodes) i s l a r g e r than f o r the f i n e r g r i d (441 n o d e s ) . T h i s i s due to the d i s c r e t i z a t i o n o f the t e m p e r a t u r e g r a d i e n t p r o f i l e , wh ich l e ad s to l a r g e r d i f f e r e n c e i n g r a d i e n t between a d j a c e n t e l ements i n the c o a r s e r g r i d than i n the f i n e r g r i d . The d e f l e c t i o n p r o f i l e a c r o s s the f a c e o f t he mould a t d i f f e r e n t t r a n s v e r s e l e v e l s i s shown i n F i g . 36 . Due to bend ing a c r o s s the f a c e the m i d - f a c e moves towards the s t r a n d ove r most o f the l e n g t h the main e x c e p t i o n s b e i n g i n the men i s cus r e g i o n , and a t the bot tom o f the mou l d . 135 E E o E c * o o c a CO b 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 - 3 0 F i g .35 - 2 0 - 1 0 0 D e f l e c t i o n a w a y f r o m s t r a n d X I 0 J ( m m ) E l a s t i c D e f l e c t i o n o f a B i l l e t Mould Wa l l Due to the S t e a d y - S t a t e Thermal G r a d i e n t s C a l c u l a t e d f rom a F i n i t e - E l e m e n t P l a t e Mode l . 136 F i g . 36 P r e d i c t e d E l a s t i c D e f l e c t i o n s A c r o s s the Ho t - F a c e o f a B i l l e t Mould Wa l l a t D i f f e r e n t D i s t a n c e s (d) f rom the Top o f the Mou l d . 137 A f u r t h e r r e f i n e m e n t o f the mesh was not c a r r i e d out s i n c e i t i s c l e a r t h a t the e l a s t i c d e f l e c t i o n o f t he mould w a l l due to bend ing i s o f the o r d e r o f - . 0 3 mm wh i ch i s s m a l l ; i t i s t h e r e f o r e l i k e l y t h a t the the rma l e x p a n s i o n e f f e c t s wou ld o v e r r i d e the bend ing e f f e c t s . B e f o r e p r o c e e d i n g to a t r a n s v e r s e s l i c e a n a l y s i s i t i s i n s t r u c t i v e to compare the e f f e c t i v e s t r e s s e s i n the mould c a l c u l a t e d f rom the p l a t e a n a l y s i s , w i t h the y i e l d s t r e s s o f the m a t e r i a l . The s t r e s s e s a t a node can be c a l c u l a t e d f rom the moments M , M , M wh i ch were computed x y xy w i t h i n the p rog ram. 3M • (37) xy 3M d< 3M xy . . . ( 3 8 ) . . . ( 3 9 ) The p r i n c i p a l s t r e s s e s can be o b t a i n e d f rom the f o l l o w i n g r e l a t i o n s o, = a + a + 1 x y 2 o_ = a + o 2 _x y_ 2 a X a y 2 2 -xy 0.5 2 - a -X a y 2 4. 2 1 xy 0.5 2 . . ( 4 0 ) (41) A p p l y i n g Von M i ses y i e l d c r i t e r i o n , f o r y i e l d i n g (a-j _ + 0^ + 0^ ^ 2 a 2 . . . (42) T h e r e f o r e i f t h i s c r i t e r i a i s s a t i s f i e d anywhere i n the mould p l a s t i c f l ow wou ld r e s u l t . Tab l e XIV p r e s e n t s the r e s u l t s o f c a l c u l a t i o n s based on E q . ' s (37) - (42) f o r s e v e r a l l o c a t i o n s i n the mould where the moments were found to be h i g h . C l e a r l y the s t r e s s e s a t the men i s cus and a l s o a t the bot tom o f the mould c o u l d r e s u l t i n y i e l d i n g o f the c o p p e r . An e l a s t o - p l a s t i c model i s t h e r e f o r e n e c e s s a r y to c a l c u l a t e the magn i tude of the p l a s t i c d i s t o r t i o n . 4 . 4 . 3 C a l c u l a t i o n o f Average Mould Wa l l Movement Due to Thermal E xpans i on o f a T r a n s v e r s e S l i c e A t r a n s v e r s e s l i c e o f the mould a t any dep th w i l l a t t emp t to expand due to the i n c r e a s e i n the ave rage l o c a l t empe r a t u r e o f the s l i c e r e l a t i v e to room t e m p e r a t u r e . The e x p a n s i o n i s a i d e d or r e s t r a i n e d depend ing on the magn i -tude o f the ave rage e xpan s i o n s o f s l i c e s above and benea th i t . At a p o s i t i o n a p p r o x i m a t e l y 30 mm be low the men i s cus the peak t empe r a t u r e o c c u r s and i t i s a n t i c i p a t e d t h a t s i n c e t he a x i a l t empe r a t u r e g r a d i e n t i s l o c a l l y z e r o , the l o c a l bend ing s t r e s s e s i n the a x i a l d i r e c t i o n w i l l be s m a l l . Thus a p l ane s t r e s s a s sump t i on has been made. A t w o - d i m e n s i o n a l p l a n e - s t r e s s model was d e ve l oped to c a l c u l a t e the ave rage Tab l e XIV E f f e c t i v e S t r e s s e s a t S e l e c t e d L o c a t i o n s i n the Mould Location Stresses Pr inc ipa l Stresses E f f . Stress Y ie ld Stress * (mm) * (mm) 0 X MPa a y MPa MPa °1 MPa °2 MPa ' a e MPa % MPa 102.22 0 -82.83 -155.59 - 3.15 -155.73 -82.69 134.9 172.9 102.22 16.25 -81.26 -163.78 -28.98 -172.94 -72.10 150.47 172.9 102.22 32.50 -101.73 -154.76 -36.54 -173.39 -83.10 150.23 172.9 102.22 48.75 - 81.5 -101.73 -34.96 -128.01 -55.22 111.22 172.9 102.22 65.00 +209.4 + 69.20 +18.27 211.74 66,86 181.40 172.9 700.00 65.00 +325.0 +107.40 +34.02 330.20 82.81 29 7.. 61 187.3 distance from the top of the mould distance from center! ine of face 140 the rma l e x pan s i o n o f the s l i c e a t t h i s l o c a t i o n . T h i s model has been used i n a subsequen t p a r t o f the work t o s i m u l a t e n o n - u n i f o r m h e a t i n g o f the mou l d , t o c a l c u l a t e r e s u l t a n t d i s t o r t i o n . Due to symmetry o n l y a q u a r t e r o f the c r o s s - s e c t i o n o f the mould was c o n s i d e r e d . The con t i nuum was d i s c r e t i z e d i n t o a s e r i e s o f t r i a n g u l a r e l emen t s as shown i n F i g . 37 . The p a r t i c u l a r node number ing scheme s e l e c t e d m i n i m i z e s the band w i d t h o f the assemb led c o e f f i c i e n t m a t r i x and t h e r e f o r e r e s u l t s i n the minimum computer s t o r a g e c o s t s f o r t h i s c o n -f i g u r a t i o n . A q u a d r a t i c v a r i a t i o n o f d i s p l a c e m e n t was assumed w i t h i n each e l ement wh i ch ensu re s t h a t both t he c omp l e t ene s s and c o m p a t i b i l i t y r e q u i r e m e n t s are s a t i s f i e d , s i n c e the f u n c t i o n a l c o n t a i n s o n l y f i r s t o r d e r d e r i v a t i v e s . The f i n i t e -.e lement e q u a t i o n s f o r the mould w a l l i n m a t r i x fo rm i s as f o l l o w s f /[V 2 N ] T [ D ] [V 2N] dA{ S ; } 6 + / / [ V 2 N ] T [ D ] {a}{M} {T} e dA A A - f f [N]T{F> dA - JyT>2N]T[D] {a} To dA A A f[N]T{n2}[D] {a}{M}{T}edS + f [N] T{n 2> [D]{a> To dS a dS = 0 . . . (43) / [ N ] T .37 D i s c r e t i z a t i o n o f a Fou r t h o f a T r a n s v e r s e S e c t i o n o f a B i l l e t Mould f o r a P l a n e - S t r e s s F i n i t e - E l e m e n t A n a l y s i s . The d e r i v a t i o n o f t h i s e q u a t i o n f rom the c oup l e d t he rmo-e l a s t i c e q u a t i o n s i s g i v e n i n Append ix IX . The r e l e v e n t s t i f f n e s s m a t r i c e s and e q u i v a l e n t t he rma l l o a d s f o r an e l emen t a re a l s o p r e s e n t e d i n Append ix IX . The r e s u l t i n g e l emen t e q u a t i o n s can be assemb led to y i e l d a s e t o.f s i m u l t a n e o u s e q u a t i o n s o f the form [K] {s} = {P} . . . (44) where [K ] i s a mas te r s t i f f n e s s m a t r i x and {P} i s a mas t e r l o a d v e c t o r . The r e s u l t i n g f i n i t e - e l e m e n t e q u a t i o n s were 10 3 s o l v e d by a Cho l e s k y scheme. F i g . 38 wh i ch i s a f l o w ° c h a r t , t r a c e s the s t ep s i n v o l v e d i n the c o m p u t a t i o n . To check the a c c u r a c y o f the mode l , a u n i f o r m mould t empe r a t u r e o f 130°C was s i m u l a t e d f o r an ambient t empe r a t u r e o f 30°C . The ou tward e x p a n s i o n o f the h o t - f a c e was u n i f o r m eve r ywhe re .and equa l to 0 .1119 mm. Th i s compares w e l l w i t h the e x p e c t e d r e s u l t c a l c u l a t e d be l ow . 6£ = oa AT . . . (45) 2 = 1 .73 x 10" 5 x 130.0 x 100 2 = 0.11245 mm From the r e s u l t s o f the s t e a d y - s t a t e h e a t - f l o w model shown i n F i g . 26 , the peak t empe r a t u r e a t the h o t - f a c e and the c o r r e s p o n d i n g c o l d - f a c e t empe r a t u r e s a re 225 and 140°C r e s p e c t i v e l y . Assuming t h a t the t empe r a t u r e i s u n i f o r m 143 GENERATE MESH TOPOLOGY NODE COORDINATES BOUNDARY CONDITION COMPUTE ELEMENT STIFFNESS MATRIX COMPUTE THERMAL LOAD VECTOR 0 F i g . 3 8 F l o w - C h a r t of the Computer Program to Compute the Thermal D i s t o r t i o n o f a T r a n s v e r s e S l i c e o f the Mould Based on a P l a n e - S t r e s s F i n i t e -E lement Mode l , ( c o n t . ) ASSEMBLE MASTER STIFFNESS MATRIX ASSEMBLE MASTER LOAD VECTOR APPLY BOUNDARY CONDITIONS SOLVE EQUATIONS BY CHOLESKY METHOD PRINT RESULTS F i g . 38 F l o w - C h a r t o f the Computer P rogram to Compute the Thermal D i s t o r t i o n o f a T r a n s v e r s e S l i c e o f the Mould Based on a P l a n e - S t r e s s F i n i t e -El ement Mode l . 145 a c r o s s the f a ce o f the mou ld , t h r o ughou t the t h i c k n e s s , the a p p r o p r i a t e d i s p l a c e m e n t s were c a l c u l a t e d f o r a q u a r t e r o f the mou l d . The t r a n s v e r s e d e f l e c t i o n a l o ng the h o t - f a c e i s shown i n F i g . 39 . The the rma l e x pan s i o n causes the h o t - f a c e to move away f rom the s t e e l by 0 .1666 mm at the m i d 7 f a c e and 0 .170 mm at the c o r n e r s . Th i s i s a p p r o x i m a t e l y an o r d e r . o f magn i tude l a r g e r than the maximum e l a s t i c d e f l e c t i o n due to b e n d i n g . From these two a n a l y s e s i t i s a ppa r en t .that t he t he rma l e x pan s i o n due to the ave rage l o c a l t empe r a t u r e i s s i g n i f i c a n t l y l a r g e r than the d i s t o r t i o n due t o b e n d i n g . Fu r t he rmo re the p l a t e a n a l y s i s r e v e a l e d t h a t the s t r e s s e s i n the coppe r c o u l d cause l o c a l i z e d y i e l d i n g . In the l i g h t o f t he se r e s u l t s , a t h r e e - d i m e n s i o n a l e l a s t o - p l a s t i c a n a l y s i s i s u ndoub t ed l y •the most r e l i a b l e t e c h n i q u e f o r c a l c u l a t i n g mould d i s t o r t i o n due to the t empe r a t u r e f i e l d . 4 . 4 . 4 T h r e e - P i mens i ona l E l a s t o - P l a s t i c S t r e s s  A n a l y s i s o f the Mould The e l a s t o - p l a s t i c b e h a v i o u r - o f the mould i s dependent on the c h a r a c t e r i s t i c s o f the s t r e s s - s t r a i n c u r v e o f t h e mould m a t e r i a l and the t empe r a t u r e dependence o f the y i e l d s t r e s s and w o r k - h a r d e n i n g p a r a m e t e r s . . These f a c t o r s undoub t ed l y i n f l u e n c e the e x t e n t o f the p l a s t i c zone and the co -*> co -o -s — i . -5 o -—i CD 3 — * CL CD — j . <-+ O c+ -a ~~ fD — • O OL 3 — i m n> CL — ' i co to «-+ -s — ' CD — ' o to to i o CO CD -n o -+> — i . — • 3 ro — ' • 3 o r+ r+ ro i. 1 03 O m CD . — i j CD O - a 3 -5 CD O 3 r+ -t) 3" i. n> — i CD O -2 Cu CD D> CD o i _ i . -s • CO o o to sr' to to <-t-O 3" CU CD .—J o _r c o — i <-»• cu i <-+ -n CD 0) CL o CD O - h CU Deflection away from the strand (mm ) T 9H 147 t o t a l p l a s t i c s t r a i n . Owing t o t he c o m p l e x i t y o f t he p l a s t i c b e h a v i o u r o f m a t e r i a l s , ma thema t i c a l t r e a t m e n t o f the phenomena i s based on s e v e r a l i d e a l i z a t i o n s . T h i s t o p i c has been e x t e n s i v e l y d i s c u s s e d i n the l i t e r a t u r e and t h e r e f o r e o n l y a b r i e f d e s c r i p t i o n o f the e q u a t i o n s f o r p l a s t i c f l o w p e r t a i n i n g t o t h i s s tudy a re p r e s e n t e d i n the subsequen t s e c t i o n . 4 . 4 . 4 . 1 Ma t hema t i c a l D e s c r i p t i o n o f P l a s t i c Flow The e l a s t o - p l a s t i c b e h a v i o u r o f a m a t e r i a l i s c h a r a c t e r i z e d by two fundamenta l r e l a t i o n s o ve r and above t h a t r e q u i r e d f o r e l a s t i c i t y a l o n e . The f i r s t , c a l l e d the y i e l d c r i t e r i a , d e t e rm i ne s whe the r the s t r e s s c o n d i t i o n i s s u f f i c i e n t to cause y i e l d , i . e . go f r om an e l a s t i c to a p l a s t i c c o n d i t i o n . The second u s u a l l y c a l l e d the f l o w l a w , r e l a t e s i n c r e m e n t a l s t r e s s e s and s t r a i n s i n the y i e l d e d c o n d i t i o n . The t o t a l s t r a i n i n a m a t e r i a l unde rgo i ng e l a s t o - p l a s t i c d e f o r m a t i o n i s composed o f two components . de. . = de?. + d e P. . . . (46) e D where d e . . and de^ . a re the i n c r e m e n t a l e l a s t i c and p l a s t i c components o f the t o t a l s t r a i n v e c t o r d e . . ' . The y i e l d s u r f a c e d e s c r i b e d by the e q u a t i o n F = 1 a."., a . ' . - T 2 Ce?. T) = 0 . . . (47) 2 was deve l oped by Von M i ses and has been f ound to d e s c r i b e 11 5 the y i e l d i n g o f me t a l s f a i r l y w e l l . x i s the y i e l d s s t r e s s i n s i m p l e shea r and a . ' , i s the d e v i a t o r i c s t r e s s v e c t o r and T the t e m p e r a t u r e . x g i s assumed to be o f the form T s = T o ( e V ) - T l ( T ) = a o • • ' ( 4 8 ) / T where a Q i s the y i e l d s t r e s s i n t e n s i o n and 1? i s the e f f e c t i v e p l a s t i c s t r a i n g i v e n by the f o l l o w i n g r e l a t i o n = /2 ep . e P. 3 The a s s o c i a t e d f l o w r u l e wh i ch r e l a t e s the i n c r e m e n t a l s t r e s s e s and s t r a i n s d u r i n g p l a s t i c f l ow i s , de9.. = dx 3F . . . (50) s i n c e Eq . (47) i s o f the form F (o:., T) = 0 . . . (51) i t can be r e w r i t t e n as f o l l o w s ! ! _ d ° i J + ! L _ d e i j + 8 F d T = 0 - - - ( S 2 ) f rom E q . ' s (50) and (52) dx can be o b t a i n e d 149 dx = - 9 F do-:. +.3F dT . . . (53) 9 F 9 F From E q . ' s ( 4 6 ) - (53) and the e q u a t i o n s o f e l a s t i c i t y the f o l l o w i n g i n c r e m e n t a l s t r e s s - s t r a i n r e l a t i o n s can be d e r i v e d d * i j = 2 G d e i j + d e k k 6 i j " ^ w d T 6 i j l-2v ( l -2v) " ° 1 j ! k £ > M * 2 6 Q ° i 0 d T ' • • ( 5 4 ) where S = 2 o 2 H_l + 1 . . . (55) 3 3G and Q = BcF_ . . . (56) G/3 H' = 9^_ . . . (57) 9iP a and a re the e f f e c t i v e s t r e s s and s t r a i n and H i s the s l o p e o f the a', e * 3 c u r ve a t any p o i n t . " - / f i d "ii •••(58) 3 i n Eq . (56) i s g i v e n by the e x p r e s s i o n 150 3-^ . . . (59) 3 T ~ and i s assumed c o n s t a n t a t each t empe r a t u r e s t e p . In m a t r i x fo rm Eq . (54) r educes to {da} = [ D P ] {de}- 2G{a'}& dT ....(60) S where [ D p ] c o r r e s pond s to the e l a s t o - p l a s t i c m a t e r i a l m a t r i x Eq . (60) i s the g o v e r n i n g e q u a t i o n f o r the p l a s t i c b e h a v i o u r o f the body . Th i s e q u a t i o n t o g e t h e r w i t h the ap -p r o p r i a t e boundary c o n d i t i o n s i s r e p e t i t i v e l y s o l v e d by n u m e r i c a l methods f o r sma l l i n c r emen t s o f t he t o t a l l o a d , u n t i l the f u l l l o a d i s a p p l i e d . The p l a s t i c i t y m a t r i x D p , Q, and S depend on the mechan i c a l s t a t e o f t he m a t e r i a l and have to be recomputed a t each s t e p , n e c e s s i t a t i n g an i n -c r emen t a l s o l u t i o n t e c h n i q u e . 4 . 4 . 4 . 2 Numer i ca l S o l u t i o n Te chn i que The f i n i t e - e l e m e n t method has been w i d e l y a p p l i e d to the s o l u t i o n o f n o n - l i n e a r m a t e r i a l p r o b l e m s , and the e q u a t i o n s l i s t e d i n the p r e c e d i n g s e c t i o n d e s c r i b e a t y p i c a l e xamp le . S e v e r a l f o r m u l a t i o n s have appea red i n t he l i t e r a t u r e , 1 1 7 o f wh i ch the i n i t i a l s t r e s s 1 1 7 " 1 1 9 app roach has been the b a s i s o f t h i s a n a l y s i s . 151 In t h i s method , an i n c r e m e n t o f t h e t o t a l l o a d i s a p p l i e d to the con t inuum and the e l a s t i c s t r e s s e s and s t r a i n s a r e computed . A l l the e l a s t i c v a l u e s a re then s c a l e d up t o i n du ce the e l emen t w i t h the maximum e f f e c t i v e s t r e s s to y i e l d . The s t i f f n e s s m a t r i x o f the who le sys tem i s m o d i f i e d , i n c o r p o r a t i n g the p l a s t i c i t y m a t r i x o f the y i e l d e d e l e m e n t , w h i l e the o t h e r e l ement s t i f f n e s s e s rema in unchanged . Ano t he r i n c r emen t o f the l o a d i s a p p l i e d and c o r r e s p o n d i n g i n c r emen t s o f s t r e s s and s t r a i n a re c a l c u l a t e d . I f d u r i n g an i n c r emen t more e l ements r e a ch the y i e l d s t a t e the s t r e s s e s w i l l n o t be i n e q u i l i b r i u m w i t h t he l o a d and so a r e d i s t r i -b u t i o n o f s t r e s s e s i s p e r f o r m e d . Th i s p r o c edu r e i s c a r r i e d out u n t i l the l o a d v e c t o r c o r r e s p o n d i n g to the s t r e s s e s i s l e s s than some p r e d e t e r m i n e d f r a c t i o n o f the i n c r e m e n t a l l o a d v e c t o r . For sys tems where t he rma l l o a d i n g i s p r e s e n t , as i n the coppe r mou l d , the y i e l d s t r e s s f o r each e l emen t i s m o d i f i e d a t each i n c r e m e n t a l s t ep o f l o a d i n g to i n c o r p o r a t e the e f f e c t s o f the new e l emen t t empe r a t u r e on the y i e l d s t r e s s , and the s t r e s s r e d i s t r i b u t i o n t e c h n i q u e d e s c r i b e d above i s c a r r i e d ou t u n t i l c onve rgence i s r e a c h e d . A t t h i s p o i n t a no t h e r i n c r emen t o f the l o a d i s a p p l i e d and the p r o -cess i s r e p e a t e d u n t i l the f u l l l o a d has been a p p l i e d . 1 5 2 4 . 4 . 4 . 3 A Ma t hema t i c a l Model f o r Mou ld  D i s t o r t i o n i n Th ree D imens i ons A computer p r og r am , ELAS65 , d e v e l o p e d by the Computer S t r u c t u r a l A n a l y s i s Group o f Duke U n i v e r s i t y , based on the p r i n c i p l e s d e s c r i b e d i n the p r e c e d i n g s e c t i o n s , h a s b e e n u s e d f o r t h e a n a l y s i s . Assuming symmetry , o n l y an e i g h t h o f t he mould f rom the m i d - f a c e to the c o r n e r was a n a l y z e d . The c on t i n uum was d i s -c r e t i z e d i n t o a s e r i e s o f h e xahed r a l e l emen t s as shown i n F i g . 4 0 . Owing to symmet ry , the v - componen t o f d i s p l a c e m e n t o f the nodes i n the l o n g i t u d i n a l m i d - p l a n e ABCD i s equa l to z e r o . By s i m i l a r r e a s o n i n g , t h e r e s h o u l d be no r o t a t i o n o f the c o r n e r p l a ne EFGH. Th i s c o n d i t i o n i s s i m u l a t e d by c o n -s t r a i n i n g the nodes l y i n g i n the c o r n e r p l a n e to move w i t h i n the p l a n e and have no component o f d i s p l a c e m e n t p e r p e n d i c u l a r to t h i s p l a n e . T h i s i s a c c o m p l i s h e d by s t i p u l a t i n g t h a t the u and v components o f d i s p l a c e m e n t o f ea ch node be equa l t o one a n o t h e r , when f o r m u l a t i n g the bounda ry c o n d i t i o n s . The f i r s t t ype o f mould c o n s t r a i n t d e s c r i b e d i n s e c t i o n 4 . 3 was s i m u l a t e d f o r the p r e l i m i n a r y c a l c u l a t i o n s . In a c c o r dan ce w i t h the t ype o f c o n s t r a i n t e x p e c t e d , the u - d i s p l a c e m e n t o f a l l the nodes a l o ng KL and MN i n F i g . 40 were a s s i g n e d ze ro v a l u e s . Here KM c o r r e s p o n d s to t he F i g . 4 0 A S chema t i c Diagram o f the Three D i m e n s i o n a l F i n i t e - E l e m e n t Mesh f o r the M i d - F a ce to Co rne r Reg ion of a B i l l e t Mou l d . 154 t h i c k n e s s o f the s u p p o r t p l a t e . In a d d i t i o n to t h e s e c o n -s t r a i n t s , the w-component o f d i s p l a c e m e n t , o f nodes a l o n g MN are s e t equa l to z e r o . Th i s l a s t c o n d i t i o n a pp r o x ima t e s the p h y s i c a l c o n d i t i o n and ensu re s t h a t a l l r i g i d body t r a n s l a -t i o n a l and r o t a t i o n a l modes a re p r e v e n t e d . The program i n t e r n a l l y d i v i d e s each h e x ahed r a l e l emen t i n t o two t e t r a h e d r a l e l e m e n t s . The t e m p e r a t u r e i s assumed c o n s t a n t w i t h i n each hexahed r a l e l emen t . The t empe r a t u r e dependence o f the y i e l d s t r e s s was i n c o r p o r a t e d i n t o the c a l c u l a t i o n s bu t the w o r k - h a r d e n i n g pa r ame t e r s had t o be assumed t empe r a t u r e i n d e p e n d e n t . Other a s s ump t i o n s i n c l u d e i s o t r o p y o f m a t e r i a l p r o p e r t i e s , and a t ime i ndependen t s t r e s s - s t r a i n c u r v e . The conve rgence c r i t e r i a wh i ch i s equa l to the r a t i o o f the l o a d v e c t o r c o r r e s p o n d i n g to the s t r e s s e s i n the e l ements a f t e r a r e d i s t r i b u t i o n p r o c e d u r e to the i n c r e m e n t a l l o a d v e c t o r was s e t e qua l to 0 .25 and o n l y one e l ement was p e r m i t t e d to y i e l d p e r i t e r a t i o n . A s upp l emen t a r y program was f o r m u l a t e d to g ene r a t e a u t o m a t i c a l l y t he f i n i t e - e l e m e n t mesh and r e l a t e d mesh t o p o l o g y da ta f o r the mould c o n t i n u u m . The a p p r o p r i a t e d i s p l a c e m e n t boundary c o n d i t i o n s were f o r m u l a t e d by an i ndependen t s ub - p r og r am , and the da ta w e r e s t o r e d on d i s k f i l e s f o r r e p e a t e d u se . 155 4 . 4 . 4 . 4 P r e d i c t e d Mould Wa l l D i s t o r t i o n The p r e l i m i n a r y c a l c u l a t i o n s were p e r -formed f o r a s t a n d a r d b i l l e t mould w i t h t he c o n d i t i o n s g i v e n i n Tab l e XV. The s t e a d y - s t a t e t empe r a t u r e d i s t r i b u t i o n was i n d e p e n d e n t l y c a l c u l a t e d by the h e a t - f l o w model d e v e l o p e d e a r l i e r . The t empe r a t u r e i s o t h e rms f o r t h i s p a r t i c u l a r case a r e p r e s e n t e d i n F i g . 4 1 . In the f i r s t i d e a l i z a t i o n o f the mould c on t i nuum a 3x7x71 mesh was s e l e c t e d . The t o t a l number o f nodes amounted to 1491 and the number o f e l ements e q u a l l e d 840 . D i s c r e t i z e d c u m u l a t i v e p l a s t i c s t r e s s - s t r a i n cu r ve s f o r t y p i c a l i n d u s t r i a l coppe r moulds a re p r e s e n t e d i n Append i x V I I and the da t a o f Company G were used i n t h i s c a l c u l a t i o n . F i g . 42 shows the c a l c u l a t e d d i s t o r t i o n p r o f i l e f o r the case i n v e s t i g a t e d . The cu r ve i s c h a r a c t e r i z e d by s e v e r a l d i s t i n c t f e a t u r e s . The f i r s t i s the peak d i s t o r t i o n wh i ch i n t h i s case has a v a l u e o f 0 .13 mm and i s l o c a t e d 65 mm be low the m e n i s c u s . The peak d i s t o r t i o n c a l c u l a t e d here i s o f the same o r d e r o f magn i tude as the t he rma l e x p a n s i o n o f the t r a n s v e r s e s l i c e i n the men i scus r e g i o n shown i n F i g . 39 . T h i s c l e a r l y i n d i c a t e s the dominance o f the d i f f e r e n t i a l t he rma l e x pan s i o n o f the mould w a l l on t he o v e r a l l d i s t o r t i o n p r o f i l e . The r e g i o n o f the mould above t he peak has a > 156 Tab l e XV B i l l e t Mould C o n d i t i o n s f o r P r e l i m i n a r y D i s t o r t i o n C a l c u l a t i o n s V a r i a b l e Mould Length 700 mm Wal l T h i c k n e s s 10 mm S e c t i o n S i z e 111 mm C a s t i n g Speed 2 .25 m/mi n Water I n l e t Tempera tu re 30 .0°C Water Tempera tu re r i s e 8 . 5 0 C Water V e l o c i t y 7 . 6 m/sec Water Channe l Gap 5.0 mm D i r e c t i o n o f Water f l o w up Mould Water P r e s s u r e ( e x i t ) 241 .3 k Pa Mould M a t e r i a l coppe r H o t - f a c e Heat f l u x Q = 2 6 8 0 - 3 3 5 / t kW/m 2 (Savage & P r i t c h a r d ) Mou ld Free board 85 .0 mm 157 F i g . 4 1 P r e d i c t e d S t e a d y - S t a t e I s o t h e r m s i n t he Wal l o f a B i l l e t Mould Based on Savage and P r i t c h a r d ' s H e a t - F l u x Curve . 0 0.02 0.04 0.06 0.08 0.10 0.12 Distortion (m m ) .42 A Compar i son Between B i l l e t Mould D i s t o r t i o n s P r e d i c t e d by the Th r e e -D imen s i o na l E l a s t o - P l a s t i c F i n i t e - E l e m e n t A n a l y s i s f o r T w o - D i f f e r e n t Mesh C o n f i g u r a t i o n s . 159 n e g a t i v e t a p e r o f 1..64% /m. Below the peak the mould a t t a i n s a p o s i t i v e t a p e r o f 0 .25%/m, due to t h e l o n g i t u d i n a l t empe r a -t u r e g r a d i e n t down the mou l d . F i g . 43 shows the d i s t o r t i o n a c r o s s the f a c e a t s e v e r a l t r a n s v e r s e s e c t i o n s . The c o r n e r s a re seen to move f u r t h e r away f rom the s t e e l than the m i d -f a c e ove r most o f the mould l e n g t h e x c e p t a t the bot tom where due to the edge moment the r e v e r s e i s seen to o c c u r . To t e s t the conve rgence o f the m o d e l , mesh r e f i n e m e n t was c o n s i d e r e d . Due to l i m i t a t i o n s i n compute r s t o r a g e space and c o s t , o n l y 100 mm o f the mesh i n the men i s cu s r e g i o n was r e f i n e d , by d o u b l i n g the number o f nodes i n each d i r e c t i o n . Th i s r e g i o n was chosen because the h i g h e s t t empe r a t u r e s and l a r g e s t g r a d i e n t s o c cu r h e r e . The mould l e n g t h had to be r educed f rom 700 mm to 650 mm by o m i t t i n g 50 mm a t the b o t -tom, s i n c e w i t h the mesh r e f i n e m e n t p r o c e d u r e , the s t o r a g e exceeded the a v a i l a b l e memory on the c ompu t e r . The r e s u l t s o f t h i s a n a l y s i s have been p l o t t e d i n F i g . 42 f o r c ompa r i s on w i t h the p r e v i o u s c a l c u l a t i o n s ; i t i s e v i d e n t t h a t the a g r e e -ment i n the p r e d i c t e d v a l u e s o f d i s t o r t i o n between the c o a r s e and r e f i n e d mesh a re good . Tab l e XVI i s a compa r i s on o f the s t r e s s e s a t i d e n t i c a l l o c a t i o n s f o r the two d i f f e r e n t mesh c o n f i g u r a t i o n s . A l t h o u g h the t r e n d i s the same f o r both c a s e s , t h e c o a r s e r mesh tends £ £ C a o SZ £ o o 5 o c o o T J <U O O O X 0.12 0.1 I d =150 m m 1 1 i i i 0 . 0 7 5 10 2 0 3 0 4 0 5 0 Distance from the mid-plane (mm) 6 0 F i g . 4 3 P r e d i c t e d D i s t o r t i o n s A c r o s s the Ho t - F a c e o f a B i l l e t Mould a t D i f f e r e n t D i s t a n c e s (d) f rom the Top o f the Mou l d . Tab le XVI A Compar ison o f S t r e s s e s a t I d e n t i c a l L o c a t i o n s i n the Mould f o r Two D i f f e r e n t Mesh C o n f i g u r a t i o n s . * do Stress X = 2 .38 mm X = 7.14 mm x = 2. 38 mm x = 7 .14 mm Component y = 5 56 mm y = 5 56 mm y = 54.57 mm y = 58.15 mm mm (MPa) Coarse Fine Coarse Fine Coarse Fine Coarse Fine Mesh Mesh Mesh Mesh Mesh Mesh Mesh Mesh °yy -127.8 -116.0 44.6 30.6 -186.7 -122.4 - 39.6 - 28.5 °xx - 5.4 - 1.8 - 5.2 - 1.2 -194.6 -153.0 - 4.3 - 1.9 n o 0 zz -104.2 - 97.9 84.2 67.3 -149.2 -116.0 90.8 95.76 T xy 0.4 0.5 0.4 .48 - 77.6 - 56.6 - 66.6 - 63.96 T yz - 0.9 - 1.0 - 0.9 - 1.3 0.7 1.8 - 0.6 - 1.6 T xz - 2,3 - 1.8 - 1.6 - 1.3 0.2 2.0 - 0.7 1.8 0 yy - 23.3 - 20.1 45.3 30.9 - 56.9 - 48.1 - 43.9 - 42.3 °xx 15.6 20.6 1.02 1.0 - 74.4 - 62.2 - 11.7 - 17.8 85 °zz - 45.5 - 40.5 38.6 51.0 - 64.4 - 50.5 15.6 38.3 T xy 0.01 0.1 0.08 .36 - 56.9 - 43.9 - 47.6 - 46.2 T yz - 4.3 - 3.8 - 2.2 - 3.9 - 8.8 - 10.5 - 15.9 -.11.5 xz - 2.5 - 5.6 - 7.7 - 7.2 - 2.2 - 15.2 - 14.8 - 15.6 0 yy 30.6 26.2 53.3 50.0 0.2 - 5.3 - 33.9 - 30.0 0 XX 15.8 7.2 6.2 2.6 - 18.7 - 7.8 - 7.4 - 6.2 75 °zz - 6.5 0.3 21.7 20.6 - 27.0 - 15.3 - 22.5 - 22.1 T xy - 0.1 0.7 - 0.09 - 0.3 - 41.7 - 28.0 - 32.4 - 24.0 V - 2.9 - 4.9 - 2.2 - 4.8 - 7.0 - 7.6 - 12.2 - 1.1 T XZ - 5.5 - 4.5 - 5.4 - 3.4 - 13.8 - 5.8 - 8.4 - 0.2 en * *-d Q - distance from the top of the mould 162 to p r e d i c t h i g h e r s t r e s s e s than the r e f i n e d mesh. T h i s c o u l d be due to t he f a c t t h a t w i t h the r e f i n e d mesh the i n c r e a s e d f l e x i b i l i t y due to the l a r g e r number o f e l emen t s wou ld d e -c r e a s e t he the rma l moments wh i ch wou ld l e a d to l owe r s t r e s s e s . 4.5 E x p e r i m e n t a l Measurements o f Mould D i s t o r t i o n In the p r e c e d i n g s e c t i o n , p r e d i c t i o n s o f mould d i s -t o r t i o n have been based on the " q u a s i " s t e a d y - s t a t 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 g n o r i n g the e f f e c t s o f mould o s c i l l a t i o n and o t h e r f a c t o r s t h a t c o u l d l e a d to random f l u c t u a t i o n s i n t empe r a t u r e and d i s t o r t i o n . The v a l i d i t y o f the model can o n l y be a s c e r t a i n e d by compar i ng the r e s u l t s p r e d i c t e d , w i t h v a l u e s o f mould d i s t o r t i o n measured d u r i n g o p e r a t i o n . F u r t h e r -more, i n s i t u measurements on an o p e r a t i n g b i l l e t c a s t e r c o u l d p r o v i d e i n f o r m a t i o n on the i n t e r a c t i o n between mould d i s t o r t i o n and gap d ynam i c s , and the i n f l u e n c e o f n u c l e a t e b o i l i n g on mould w a l l movement. The s u c c e s s o f measu r i ng mould w a l l movement i n an env i r onmen t i n wh i ch the v i b r a t i o n and n o i s e l e v e l s a r e h i g h , depends undoub t ed l y on the ruggedness and s e n s i t i v i t y o f the measu r i ng d e v i c e . The subsequen t s e c t i o n d e a l s w i t h the s e l e c t i o n o f i n s t r u m e n t a t i o n , d e t a i l s o f c a l i b r a t i o n , i n -s t a l l a t i o n and measurement . 163 4 .5 .1 S e l e c t i o n of I ns t rumen ta t i on Obviously the h o t - f a c e o f the mould i s i n -access ib le and t h e r e f o r e a l l measurements would have to be c a r r i e d out at the co ld f a c e . The re fo re , the device se-l e c t e d , should be capable of measuring wa l l movements in the 0 ~ 0.1 mm range w i th s i g n i f i c a n t accuracy, in the presence of r a p i d l y f l o w i n g coo l i ng wa te r , in which p u l s a t i o n s could occur as a r e s u l t of surges in the water supply as w e l l as due to growth and co l l apse o f bubbles i f b o i l i n g takes p lace . Based on these requ i rements , L inear Va r iab le D i f -f e r e n t i a l Transformers b e t t e r known as LVDT displacement t ransducers were se lec ted LVDT's have been used in the past f o r a wide v a r i e t y of a p p l i c a t i o n s under adverse e n v i r o n -mental c o n d i t i o n s and have proved to be h i g h l y r e s i s t a n t to v i b r a t i o n and e l e c t r i c a l i n t e r f e r e n c e . The LVDT's are p ro tec ted from damage by a rugged metal housing and the c o i l s are epoxy encapsulated and magne t i ca l l y s h i e l d e d . In a t y p i c a l LVDT the sensing probe i s a non-magnetic rod a x i a l l y p o s i t i o n e d in a c o - a x i a l c o i l assembly. In o p e r a t i o n the t i p o f the probe is con t inuous ly in con tac t w i t h the moving t e s t p iece . The output vo l tage s igna l generated in the c o i l assembly is d i r e c t l y p r o p o r t i o n a l to the d i s p l a c e -ment of the t ransducer probe. 164 An e x p e r i m e n t a l program was p l anned i n c o o p e r a t i o n w i t h the S t e e l Company o f Canada , Edmonton Works , to i n -s t r umen t a b i l l e t mould w i t h LVDT's and to m o n i t o r i t s d i s t o r t i o n d u r i n g s e v e r a l hea t s i n s u c c e s s i o n . The c h a r a c t e r -i s t i c s o f the two t ypes o f LVDT's s e l e c t e d a r e p r e s e n t e d i n Tab l e X V I I . O the r f a c t o r s t h a t were g i v e n due c o n s i d e r a t i o n i n the s e l e c t i o n p r o ce s s were the s i z e o f the t r a n s d u c e r s , owing to space r e s t r i c t i o n s i n the mould h ou s i n g a s s e m b l y , and the range of t r a v e l . A d d i t i o n a l d e s i g n f e a t u r e s i n -c l u d e a s p r i n g l o aded probe to ensu re c o n t i n u o u s c o n t a c t w i t h the mou l d , and a r e p l a c e a b l e t i p . The t r a n s d u c e r s were s p e c i a l l y wa t e r p r o o f e d by the m a n u f a c t u r e r t o ensu re s t a b l e o p e r a t i o n i n the mould h o u s i n g . A D a y t r o n i c 3230 s i g n a l c o n d i t i o n e r s u p p l i e d r e g u l a t e d ac e x c i t a t i o n to the t r a n s d u c e r and demodu la ted the r e s u l t i n g s i g n a l . The s i g n a l c o n d i t i o n e r was equ i pped w i t h a d i g i t a l i n d i c a t o r and an ana l og o u t p u t f a c i l i t y . The l a t t e r was c o n n e c t e d to a c h a r t r e c o r d e r f o r c o n t i n u o u s r e c o r d i n g o f t h e s i g n a l . S i n c e t h r e e t r a n s d u c e r s were to be used s i m u l t a n e o u s l y , to m o n i t o r the d i s t o r t i o n a t t h r e e l o c a t i o n s i n the mou l d , an e l e c t r i c a l s w i t c h i n g mechanism was d e s i g n ed to e n a b l e s e l e c t i v e m o n i t o r i n g o f the ou t pu t s i g n a l u s i n g a s i n g l e s i g n a l c o n d i t i o n e r . 165 T a b l e XVII C h a r a c t e r i s t i c s o f LVDT's S e l e c t e d Transducer Type Linear Range Sens i t i v i t y mv/v/.0254 mm Ambient Temperature Range DS100Z ±1.27 mm 4.0 -65 + 200 F DS200A ±2.54 mm 3.0 -65 + 200 F 166 4 . 5 . 2 C a l i b r a t i o n P r i o r to i n s t a l l a t i o n , a l l t h r e e t r a n s d u c e r s were c a l i b r a t e d a g a i n s t a depth m i c r o m e t e r . The sys tem was c a l i b r a t e d to p roduce a ± 500 .0 u n i t d i g i t a l r e adou t f o r ± 0 .635 mm t r a n s d u c e r c o r e d i s p l a c e m e n t . The two t ypes o f t r a n s d u c e r s were o p p o s i t e i n p o l a r i t y , the DS100A g i v i n g a + 500 .0 r e adou t f o r 0 .635 mm e x t e n s i o n o f t he probe and the DS200A LVDT g i v i n g a + 500 .0 r e adou t f o r 0 .635 mm com-p r e s s i o n o f the p r o be . 4 . 5 . 3 D e t a i l s o f E x p e r i m e n t a l S e t - u p The mould t h a t was to be m o n i t o r e d had a s i n g l e t a p e r , and had been r e f o rmed by an e x p l o s i v e f o rm i ng p r o c e s s . Tab l e XV I I I p r e s e n t s the d imens i on s o f the mould a f t e r the r e f o r m i n g o p e r a t i o n as s p e c i f i e d by the m a n u f a c t u r e r . Three t r a n s d u c e r s were mounted on the s t a i n l e s s " s t e e l j a c k e t s u r r o u n d i n g the mould by means o f s l e e v e s t h a t c o u l d be t h r e a d e d i n t o ho l e s d r i l l e d f o r the pu rpose a t l o c a t i o n s shown i n F i g . 44 . With the mould i n p l a c e , the a x i a l p o s i -t i o n o f each t r a n s d u c e r i n the s t a i n l e s s s t e e l j a c k e t was a d j u s t e d such t h a t the s p r i n g l o aded probe i n c o n t a c t w i t h the mould w a l l was a t t he c e n t r e o f i t s t r a v e l . T h i s c o r -r e sponded to a z e r o v o l t a g e ou t pu t and a n u l l r e a d i n g on the i n d i c a t o r . Once the ad j u s tmen t s were comp l e t e the 167 Tab l e XV I I I Mould D imens ions A f t e r Re fo rm ing O p e r a t i o n s Pa ramete r (mm) S t a n d a r d D e v i a t i o n Mould Length 735.00 Average Top D imens ion 110 .92 .0254 (mm) Average Bot tom D imens ion 110 .46 .0254 (mm) 168 50.8 m m Meniscus Mould 1 Water jacket 2 Transducer 3 Feed through 4 Mould housing 5 Cover plate 6 F ig.44 A S chema t i c D iagram o f the L o n g i t u d i n a l P l a n e Through the I n s t r umen t ed B i l l e t Mould A s s emb l y . 169 t ransducers were locked in p o s i t i o n . During assembly i t ' was found t h a t the probes were d isp laced from t h e i r i n i t i a l s e t t i n g , due to assembly o p e r a t i o n s . I t was f o r t u i t o u s t h a t reasonably la rge ranges of t r a v e l had been se lec ted as o therwise the i n i t i a l s e t t i n g could have very e a s i l y gone complete ly o f f - s c a l e . The e l e c t r i c a l leads were brought out of the mould assembly through a hole i n the s i d e - w a l l o f the mould housing by means of a threaded leak proof feed-th ro ugh. 4 . 5 . 4 Experimental T r i a l s The inst rumented mould was i n s t a l l e d in a three s t rand b i l l e t c a s t i n g machine at S t e l c o ' s Edmonton Works. 2 The e n t i r e t r i a l which invo lved the c a s t i n g of 111 x 111 mm b i l l e t s of va ry ing grades, l a s t e d approx imate ly 60 hours . Of the 28 heats c a s t , the inst rumented mould was in the c a s t i n g machine f o r 21 heats . Table XIX l i s t s the heat numbers in c h r o n o l o g i c a l order i n d i c a t i n g those f o r which mould move-ment was recorded . The d i g i t a l readout corresponding to each t ransducer was recorded before s t a r t - u p . During opera-t i o n , the movement of the mould wal l at each l o c a t i o n was moni tored at r e g u l a r i n t e r v a l s , by s w i t c h i n g from one t ransducer to another . A f t e r the heat was complete ly c a s t , reco rd ing of the s igna l was cont inued f o r a per iod o f 15 to 20 m inu tes , u n t i l the s igna ls on a l l th ree t ransducers were steady and i n v a r i a n t . Tab le XIX D e t a i l s o f E xpe r imen t a l T r i a l s Thermal Cycle NO. Heat No. Start Time Finish Time Steel Composition Comments C Mn P S S1 NI Cr 1 D05946 E06586 10.06 11.36 11.36 13.15 .3 .29 .74 .65 .028 .019 .029 .035 .21 .22 .12 .12 .16 .15 Signals monitored Sequence cast 2 005947 14.02 16.20 .29 .45 .024 .033 .20 .13 .87 Signals monitored 3 £06587 17.49 17.51 .30 .53 .024 .025 .22 .14 .89 Break out. Mould assembly removed Signals monitored Sequence cast 4 E06591 005951 E06592 9.42 11.13 12.42 11.13 12.42 14.15 .30 .31 .30 .62 .67 .69 .015 .021 .017 .032 .042 .032 .21 .21 .20 .10 .10 .17 .14 .17 .19 5 005952 15.46 17.15 .59 .95 .018 .028 .24 .18 .96 Signals monitored 6 E06593 18.05 19.30 .57 .83 .012 .024 .23 .08 .76 Signals monitored 7 D05953 E06594 20.20 21.45 21.45 23.15 .60 .56 .83 .52 .008 .004 .024 .026 .22 .25 .09 .09 .76 .74 Sequence cast Not recorded 8 005954 E06595 23.50 01.20 01.20 02.50 .68 .63 .77 .77 .023 .010 .028 .026 .17 .19 .10 .10 .19 .15 Sequence cast Not recorded 9 005955 E06596 03.27 04.52 04.52 06.25 .53 .52 .71 .73 .015 .008 .026 .021 .22 .20 .12 .12 .17 .20 Sequence cast Not recorded 10 • D05956 E06597 DO 595 7 07.04 08.25 09.56 08.25 09.56 11.55 .51 .27 .30 .89 .52 .71 .028 .012 .007, .028 .033 .024 .19 .22 .22 .15 .28 .16 .24 .14 .11 Sequence cast Signals recorded 11 E06598 D05958 E06599 12.32 14.00 15.35 14.00 15.35 17.30 .31 .30 .50 .67 .74 .89 .019 .023 .029 .030 .038 .041 .21 .19 .19 .15 .12 .14 .15 .13 .04 Sequence cast Digital readout recorded o 171 4 . 5 . 5 R e s u l t s o f Mould D i s t o r t i o n Measurement The c h a r a c t e r i s t i c s o f the s i g n a l s o f each t r a n s d u c e r were found to be w i d e l y d i f f e r e n t f rom one a n o t h e r . T y p i c a l samples o f the t r a n s d u c e r s i g n a l s f o r a number o f hea t s a r e p r e s e n t e d i n F i g . ' s 4 5 - 5 0 . T r a n s d u c e r s , wh i c h was p o s i t i o n e d 100 mm above the bot tom o f the mould e x h i b i t e d ve r y l i t t l e t o t a l f l u c t u a t i o n . The d i s p l a c e m e n t m o n i t o r e d by t r a n s d u c e r 2, p o s i t i o n e d a p p r o x i m a t e l y 65 mm below t he men i s cus e x h i b i t s a p e r i o d i c v a r i a t i o n w i t h t ime as can be seen i n F i g . ' s 4 5 - 5 0 . The h i gh f r e q u e n c y peaks on the c u r v e s wh i ch c o r r e s p o n d to mould o s c i l l a t i o n , a re s u p e r -imposed on a c u r v e hav i ng a much l a r g e r p e r i o d . T h i s l owe r f r e q u e n c y o s c i l l a t i o n i s t hough t t o be due to n u c l e a t e b o i l i n g a t the m o u l d / c o o l i n g wa te r i n t e r f a c e i n the men i s cus r e g i o n . The b a s i s f o r t h i s d e d u c t i o n i s the r ema r k ab l e s i m i l a r i t y between the d i s p l a c e m e n t s i g n a l o f t r a n s d u c e r 2 and the measured s u r f a c e t empe r a t u r e v a r i a t i o n s o f a s t e e l 120 sample d u r i n g poo l b o i l i n g o f wa te r ( F i g . 5 1 ) . Such a c ompa r i s on o f a d i s p l a c e m e n t s i g n a l w i t h a t e r m p e r a t u r e s i g n a l i s p o s s i b l e because the l o c a l i z e d v a r i a t i o n i n d i s -p l a cemen t o f the mould w a l l a t the l o c a t i o n o f t r a n s d u c e r 2 i s a r e f l e c t i o n o f the l o c a l t empe r a t u r e f l u c t u a t i o n s a t t h a t p o i n t . A h e a t - f l o w a n a l y s i s o f t he i n s t r u m e n t e d b i l l e t mould was pe r f o rmed and the r e s u l t s i n d i c a t e d t h a t f o r the g i v e n c o o l i n g - w a t e r c o n d i t i o n s n u c l e a t e b o i l i n g wou ld F i g . 4 5 T y p i c a l Examples o f by T r a n s du c e r s 1 , 2 Mou ld Wa l l Movement the S i g n a l s and 3 Wh i l e Dur ing Heat Gene r a t ed M o n i t o r i ng No. D05946. 173 F i g . 4 6 T y p i c a l Examples o f t h e S i g n a l s G e n e r a t e d by T r a n s d u c e r s 1,2 and 3 W h i l e M o n i t o r i n g Mould Wal l Movement D u r i n g Heat No. D05947. F i g . 4 7 T y p i c a l Examples o f by T r an sduce r s 1 , 2 Mould Wa l l Movement the S i g n a l s and 3 Wh i l e Dur ing Heat Gene ra ted Mon i t o r i ng No. E06591 . 175 F i g . 4 8 T y p i c a l Examples o f by T r a n s d u c e r s 1 , 2 Mould Wal1 Movement the S i g n a l s and 3 Wh i l e Du r i ng Heat Gene r a t e d M o n i t o r i n g No. D05951 . 176 F i g . 4 9 T y p i c a l Examples of by T r an sduce r s 1 , 2 Mould Wa l l Movement the S i g n a l s and 3 Wh i l e Dur ing Heat Gene r a t ed M o n i t o r i ng No. E06592 . 177 F ig .5 0 T y p i c a l Examples o f by T r an sdu ce r s 1 , 2 Mould Wal l Movement the S i g n a l s and 3 Wh i l e Du r i ng Heat Gene r a t ed Moni t o r i ng No. D05952. 178 3 Z5tU . fe» ( i ) Tempera tu re F l u c t u a t i o n s Measured Du r i n g B o i l i n g on a S t e e l P l a t e . ( i i ) S i g n a l o f T r a n s d u c e r No. 2 Du r i ng Heat No. D05957 F i g . 5 1 A Compar i son o f the D i s p l a c emen t S i g n a l A s s o c i a t e d w i t h T r a n s d u c e r 2 P o s i t i o n e d 65 mm Below the Men i s cu s w i t h Tempera tu re F l u c t u a t i o n s Measured i n a M e t a l l i c S u r f a c e Du r i ng N u c l e a t e B o i l i n g . 179 p r o b a b l y o c c u r i n the c o o l i n g channe l wh i ch a l s o s u b s t a n t i -a t e s the above c l a i m . These r e s u l t s a re p r e s e n t e d l a t e r . R e t u r n i n g to the s u b j e c t o f t r a n s d u c e r s i g n a l c h a r a c t e r -i s t i c s , t r a n s d u c e r 1, wh i ch was l o c a t e d 333 mm be low the men i s cus was l e s s c o n s i s t e n t than t r a n s d u c e r s 2 o r 3 . I t was ob s e r v ed t h a t f o r the f i r s t t h r e e hea t s the s i g n a l was e x t r e m e l y s t a b l e and e x h i b i t e d ve r y l i t t l e l o c a l f l u c t u a t i o n . Du r i ng the f o u r t h heat a b reak out o c c u r r e d f rom the s t r a n d , and the mould assemb ly was removed f rom the mach i ne , c l e a n e d , and r e i n s t a l l e d . Subsequent to t h i s e v e n t , the s i g n a l o f t r a n s d u c e r 1 was e x t r e m e l y e r r a t i c . E x am i n a t i o n o f the t r a n s d u c e r s i g n a l s b e f o r e and a f t e r each i n d i v i d u a l h e a t i n g and c o o l i n g c y c l e p r o v i d e f u r t h e r i n s i g h t on the c o n s i s t e n c y o f t he r e s u l t s o b t a i n e d . T ab l e XX summar i zes t h i s i n f o r m a -t i o n . I t i s e v i d e n t t h a t p r i o r to t he b reak ou t the r e adou t o f a l l t h r e e t r a n s d u c e r s a t t he s t a r t and c o n c l u s i o n of a t he rma l c y c l e were not e q u a l . Th i s d i f f e r e n c e c o u l d be e i t h e r due t o permanent d i s t o r t i o n o f the mould w a l l o r t r a n s l a t i o n o f the mould w i t h i n the s u p p o r t s due to the shock o f t he rma l l o a d i n g . Subsequent to the b reak o u t , t r a n s d u c e r s 2 and 3 d i s p l a y e d r ema rkab l e c o n s i s t e n c y i n the v a l u e o f the r e c o r d e d s i g n a l s a t the b e g i n n i n g and end o f a t he rma l c y c l e . I t i s e v i d e n t t h a t the d i f f e r e n c e i s l e s s than 3.0 u n i t s o f the i n d i c a t o r s c a l e , wh i ch c o r r e s p o n d s to an e x c e e d i n g l y sma l l d i s p l a c e m e n t o f 0 .003 mm. The Tab l e XX I n d i c a t o r Read ings at the B e g i n n i n g and End of a Thermal C y c l e Thermal Cycle No. Heat Numbers Indicator Transc • Readings ucer 3 Indicator Transc • Readings ucer 2 Indicator Transc Readings ucer 1 I n i t i a l Final I n i t i a l F inal I n i t i a l Final 1 D05946 E065586 -67.8 -120.0 +190.7 +105.0 -102.2 -170.0 2 D05947 -120.0 -170.0 +105.0 +128.0 -170.0 -220.0 3 E06587 (Breakout) -170.0 -180.0 Not Recorded Not Recorded 4 E06591 D05951 E06592 -172.7 -171.2 +140.0 +141.1 - 69.7 - 17.0 5 D05952 -171.2 -170.2 +142.7 +139.2 - 12.7 - .4.60 6 E06593 -171.0 -173.4 +139.2 Not Recorded - 14.0 Not Recorded 10 E06597 D05957 Transducer Disconnected Loose Cable Not Recorded +161.0 Not Recorded + 11.70 11 E06598 D05958 E06599 +161.0 +166.5 + 11.40 + 18.80 181 r e s u l t s o f t r a n s d u c e r 1 d i d not d i s p l a y t h i s l e v e l o f c o n -s i s t e n c y . I t i s p o s s i b l e t h a t t h i s t r a n s d u c e r was not w e l l s e c u r e d to the s t a i n l e s s s t e e l h o u s i n g , due to t o l e r a n c e s i n the l o c k i n g d e v i c e wh i ch wou ld p e r m i t s m a l l l a t e r a l movements o f the t r a n s d u c e r r e s u l t i n g i n e r r a t i c p e r f o r m a n c e . Due t o t h i s u n c e r t a i n t y no d e f i n i t e c o n c l u s i o n s can be drawn f rom the r e s u l t s o f t r a n s d u c e r 1. In the l i g h t o f t h i s a n a l y s i s , the d i f f e r e n c e i n the i n s t a n t a n e o u s , and i n i t i a l o r f i n a l p o s i t i o n s o f t r a n s d u c e r s 2 and 3 ( f o r the the rma l c y c l e s a f t e r the b reak ou t ) i s u n q u e s t i o n a b l y a measure o f l o c a l mould w a l l movement r a t h e r than random t r a n s l a t i o n o f t he mould w i t h i n the s u p p o r t s . T ab l e XXI p r e s e n t s the mean mould w a l l d i s p l a c e m e n t , 100 mm f rom the bo t tom o f the mou l d , c a l c u l a t e d f rom the mean p o s i t i o n o f t r a n s d u c e r 3 f o r t he rma l c y c l e s 4 , 5 and 6 . The 95% c o n f i d e n c e l i m i t s are ve r y c l o s e to t he mean v a l u e , i n d i c a t i n g t h a t the s i g n a l e x h i b i t e d ve r y l i t t l e f l u c t u a t i o n t h r o u g h o u t the h e a t s . The d i s p l a c e m e n t o f t r a n s d u c e r 2 on the o t h e r hand had a low f r e q u e n c y v a r i a t i o n d u r i n g the hea t p e r i o d wh i ch was a t t r i b u t e d to b o i l i n g i n the c o o l i n g - w a t e r c h a n n e l s . S i n c e the d i f f e r e n c e between the maxima and min ima of the d i s p l a c e m e n t s i g n a l was s i g n i f i c a n t , a v e r a g i n g the i n d i c a t o r r e a d i n g s wou ld not be m e a n i n g f u l . For t h i s case the peak Tab le XXI R e s u l t s o f Mould Wa l l Movement From Data o f T r a n s d u c e r 3 Thermal Cycle No. Heat No. Indicator Results Average Mould Wall Movement 95% Confidence Limits Direct ion . of Movement I n i t i a l Final Average 95% Confidence Limits 4 E06591 DO 59 51 E06592 -172.7 -171.2 -114.43 -115.38 -113.48 0.0740 .0728 .0752 Away From Steel 5 D05952 -171.2 -170.2 -119.12 -122.36 -115.88 0.0648 .0606 .0689 Away From Steel 6 E06593 -171.0 -173.4 -128.77 -130.53 -127.01 .0536 .051 .055 Away From Steel CO ro 183 d i s t o r t i o n d u r i n g the low f r e quen c y d i s p l a c e m e n t c y c l e was c a l c u l a t e d f rom the ave rage v a l u e o f the maxima o f the c u r v e . T ab l e XXII p r e s e n t s the r e s u l t s o b t a i n e d f o r t h r e e t he rma l c y c l e s . 4 .6 Compar i son o f Model P r e d i c t i o n s w i t h E x p e r i m e n t a l Measurements and A n a l y s i s o f R e s u l t s Compar i son o f the r e s u l t s o f a model w i t h e x p e r i m e n t a l measurements i s an i m p o r t a n t a s pe c t o f p r o c e s s m o d e l l i n g . I f the model p r e d i c t i o n s compare f a v o u r a b l y w i t h measured v a l u e s then the model can be used w i t h c o n f i d e n c e as a p r e d i c t i v e t o o l to e v a l u a t e the i n f l u e n c e o f a v a r i e t y o f p r o c e s s v a r i -a b l e s on the pa ramete r t h a t i s o f i n t e r e s t . For t h e - m o u l d , the v a l i d i t y o f the t h r e e - d i m e n s i o n a l e l a s t o - p l a s t i c model f o r p r e d i c t i n g d i s t o r t i o n i s e v a l u a t e d by compar i ng the c a l c u l a t e d v a l u e s o f d i s t o r t i o n a t the two l o c a t i o n s c o r -r e s p o n d i n g to the p o s i t i o n s o f t r a n s d u c e r 2 and 3 w i t h t he e x p e r i m e n t a l l y measured ave rage v a l u e s p r e s e n t e d i n T a b l e s XXI and X X I I . D e t a i l s o f t h i s a n a l y s i s a r e d i s c u s s e d i n t h i s s e c t i o n . , The d e s i g n d e t a i l s o f the. i n s t r u m e n t e d mould and o p e r a t i n g c o n d i t i o n s d u r i n g the t r i a l s a r e p r e s e n t e d i n T ab l e s X X I I I and XXIV. A t r a n s i e n t h e a t - f l o w a n a l y s i s was c ondu c t e d f o r the mould based on the h i g h ca rbon h e a t - f l u x Tab l e XXII R e s u l t s o f Mould Wal l Movement From Data Of T r an sdu ce r 2 Thermal Cycle Heat No. Indicator Readings Maximum Mould Movement Direct ion of No. I n i t i a l Final Peak 951 Confidence Limits Mean 95% Conf. Limits Movement E06591 +76.74 .0803 Away from 4 D05951 E06592 +140.0 +141.1 +79.79 +82.83 .0765 .0726 Steel +46.26 0.1224 Away from 5 D05952 +142.7 +139.2 +56.52 +66.77 .1094 0.0963 Steel E06598 +84.86 .1036 Away from 11 D05958 E06599 +161.0 +166.5 +92.0 +99.13 .0946 .0855 Steel 185 Tab l e XX I I I Des ign D e t a i l s o f I n s t r umen t ed Mould Mould Pa ramete r s Mould 1ength 735 mm Wal l t h i c k n e s s 9 .525 mm Water channe l gap 4 .50 mm Mould c o n s t r a i n t s Type I M a t e r i a l (assumed) DLP120 coppe r Mould t a p e r .6% S e c t i o n s i z e 111 mm 186 Tab l e XXIV O p e r a t i n g C o n d i t i o n s Du r i ng E x p e r i m e n t a l T r i a l s C a s t i n g Pa ramete r s C a s t i n g speed 3.048 m/min Moul d f r e e boa rd 52.0 mm Mould o s c i 1 1 a t i o n 90 cpm Mould wa t e r v e l o c i t y 8.16 m/sec Water p r e s s u r e near channe l e x i t 243.7 kPa Water i n l e t t empe r a t u r e 30.0°C Water t empe r a t u r e r i s e 8.5°C 187 p r o f i l e g i v e n i n F i g . 19 . I t was found t h a t f o r t h e g i v e n c o n d i t i o n s , n u c l e a t e b o i l i n g wou ld o c c u r a t the mou l d / c o o l i n g - w a t e r i n t e r f a c e . As was d i s c u s s e d e a r l i e r , t he c h a r a c t e r i s t i c s o f the s i g n a l o f t r a n s d u c e r 2 s u g g e s t t h a t b o i l i n g c o u l d be o c c u r r i n g i n the men i s cus r e g i o n . . F i g . 52 shows a p l o t o f the h o t - and c o l d - f a c e t e m p e r a t u r e p r o f i l e s f o r the mould as c a l c u l a t e d f rom the t r a n s i e n t a n a l y s i s a t the i n c e p t i o n o f b o i l i n g . The upper and l owe r t e m p e r a t u r e d i s t r i b u t i o n s o f the h o t - and c o l d - f a c e s shown i n F i g . 52 a re due to the b o i l i n g phenomena. The d i s t o r t i o n p r o f i l e f o r t he " uppe r " t empe r a t u r e f i e l d has been c a l c u l a t e d u s i n g the t h r e e - d i m e n s i o n a l model and i s shown i n F i g . 5 3 . The mould p r o p e r t i e s g i v e n i n T ab l e V I I - I o f Append i x V I I were assumed. Tab l e XXV p r e s e n t s a c ompa r i s on o f t h e measured " a v e r a g e " , and c a l c u l a t e d d i s t o r t i o n s o f the mould a t i d e n t i c a l l o c a t i o n s . N o t w i t h s t a n d i n g the use o f t h e h e a t -21 f l u x p r o f i l e d e r i v e d f rom S i ngh and B l a z e k s s t u d y to c a l c u l a t e the the rma l f i e l d f o r the i n s t r u m e n t e d m o u l d , the p r e d i c t i o n s a r e good . C l e a r l y the agreement between t h e measured and c a l c u l a t e d v a l u e s i s good a t the 635 mm l e v e l wh i ch c o r r e s pond s to the l o c a t i o n o f t r a n s d u c e r 3 . However a t the 114 mm l e v e l wh i ch i s 63 . 5 mm be low the m e n i s c u s , t h e r e i s a g r e a t e r d i s c r e p a n c y between the measured and p r e d i c t e d v a l u e s . Th i s may be due to a c o m b i n a t i o n o f many f a c t o r s . The t empe r a t u r e o s c i l l a t i o n s i n the men i s cu s r e g i o n due to b o i l i n g causes the mould w a l l t o move 0 _ 100 E E — 200 O I 300 SZ CL O CD SZ 400 500 E 2 600 CD O o 700 «/> Q 800 900 Meniscus 1 Hotface Temperature profi les Upper limit due to boiling L o w e r limit due to boi l ing 0 50 100 150 Temperature (°C) 200 F i g . 5 2 250 P r e d i c t e d Ho t - and C o l d - F a c e Temperature P r o f i l e s f o r the In s t r umen ted B i l l e t Mould Showing the Upper and Lower Tempera tu re L i m i t s Dur ing B o i l i n g . CO CO 189 F i g .53. P r e d i c t e d D i s t o r t i o n P r o f i l e a t the M i d - S i d e o f the I n s t r umen ted Moul d Down the Ho t - F a c e . Tab le XXV A Compar i son of Measured and P r e d i c t e d Va l ue s o f Mould D i s t o r t i o n 190 Distance Below the Top of the Mould (mm) Predicted D is tor t ion (mm) Average D is to r t ion (mm) Thermal Cycle No.'s 4 5 6 11 635.0 0.0620 .0740 .0648 .0536 -114.3 0.162 .0765* .1094* - .0946* * Average Va l ue o f Peak D i s t o r t i o n 191 i n t e r m i t t e n t l y away f rom the s t e e l . S i n c e the s t r a n d does not n e c e s s a r i l y f o l l o w the mould i n s t a n t a n e o u s l y , the d imens i on s o f the gap a t the i n t e r f a c e c o n t i n u o u s l y c hange s . I t has been e s t a b l i s h e d t h a t the t he rma l r e s i s t a n c e o f the gap i s the r a t e c o n t r o l l i n g s t ep i n the hea t e x t r a c t i o n p r o -c e s s ; t h e r e f o r e a v a r i a t i o n i n gap s i z e wou ld a l t e r the . r a t e o f h e a t - t r a n s f e r to the mou l d . The r e s u l t i n g t he rma l f i e l d i n t he mould changes a c c o r d i n g l y , wh i ch has an i n -f l u e n c e on the b o i l i n g c y c l e . Th i s i n t u r n l e a d s t o v a r i a -t i o n s i n t he b o i l i n g h e a t - t r a n s f e r c o e f f i c i e n t wh i ch then a l t e r s the t he rma l f i e l d and the d i s t o r t i o n . Due to t h i s complex i n t e r p l a y between mould d i s t o r t i o n , gap d y n a m i c s , and n u c l e a t e b o i l i n g , the p r e d i c t i o n s o f the model based on " t i m e - a v e r a g e d " h e a t - f l u x da ta cannot be e x pe c t e d t o p e r -f e c t l y match the measured d i s t o r t i o n . Ano t he r f a c t o r t h a t c o u l d a l s o be r e s p o n s i b l e f o r the d i s c r e p a n c y a re t he c o n -s t r a i n t s due to the s p a c i n g s c r e w s . These s c r e w s , e i g h t i n t o t a l per mould f a c e , a re used to c e n t r e the m o u l d . i n t he s t a i n l e s s s t e e l wa te r j a c k e t . Once the mould i s c e n -t r a l l y p o s i t i o n e d and b o l t e d to the h o u s i n g , the s c rews a r e t u r n e d back a c oup l e o f t u r n s to a l l o w f o r t he rma l ex -p a n s i o n . A f t e r the t r i a l s sc rew marks were o b s e r v ed i n the men i s cus r e g i o n on the c o l d - f a c e o f the mou ld , i n d i c a t i n g t h a t the a l l o w a n c e f o r e xpan s i o n was i n s u f f i c i e n t and the sc rews p r o b a b l y o f f e r e d some degree o f r e s t r a i n t to the the rma l e x p a n s i o n . Th i s may be p a r t l y r e s p o n s i b l e f o r the low v a l u e s o f d i s t o r t i o n measured i n c ompa r i s on t o the c a l c u l a t e d v a l u e s . D e s p i t e t he se s h o r t c o m i n g s , t h e t h r e e -d i m e n s i o n a l d i s t o r t i o n model has paved the way t oward a b e t t e r u n d e r s t a n d i n g o f q u a l i t y p rob lems i n b i l l e t moulds and the r e l a t i o n s h i p to o p e r a t i n g p a r a m e t e r s . These t o p i c s a r e d i s c u s s e d i n c h a p t e r 6 . 0 . Once the t r i a l s were comp le te the i n t e r n a l d i m e n s i o n s o f the i n s t r u m e n t e d mould were measured to d e t e c t permanent shape change s . F i g . 54 i s a t r a v e r s e down the mould o f the d i s t a n c e between the i n s t r u m e n t e d f a c e and t he o p p o s i t e f a c e . There i s e v i d e n c e o f permanent ou tward d i s t o r t i o n i n the men i scus r e g i o n . The o r i g i n a l d imen s i on s and t a p e r as s p e c i f i e d by the m a n u f a c t u r e r a re a l s o shown i n F i g . 54 . Compar ing the o r i g i n a l mould shape w i t h the f i n a l measured s hape , i t appears t h a t ove r most o f the mould l e n g t h the mould w a l l has moved i n by a p p r o x i m a t e l y 0 .4 mm. Th i s i s h i g h l y imp r obab l e and i t i s more l i k e l y t h a t the o r i g i n a l measurements o f the mould have an e r r o r o f a p p r o x i m a t e l y 0.4 mm ove r the l e n g t h o f the mou ld . I f a u n i f o r m c o r r e c -t i o n o f 0.4 mm were a p p l i e d the o r i g i n a l t a p e r wou ld c o -i n c i d e w i t h t he f i n a l shape ove r the l o w e r r e g i o n and the peak permanent d i s t o r t i o n ob se r ved i n the men i s cus r e g i o n wou ld amount to a p p r o x i m a t e l y 0 .425 mm/face . A s i n g l e t he rma l c y c l e was s i m u l a t e d by the model f o r 1 9 3 F i g . 54 A Compar i son o f the I n t e r n a l D imens i ons o f the I n s t r umen t ed Mould A f t e r the E x p e r i m e n t a l T r i a l s w i t h the O r i g i n a l D imens ions as S p e c i f i e d by the M a n u f a c t u r e r . 194 the i n s t r u m e n t e d mould and the r e s u l t s a re shown i n F i g . 55 . The permanent d i s t o r t i o n p r o f i l e p r e d i c t e d a t the end of a s i n g l e c y c l e , i s ve r y s i m i l a r i n shape to the measured one , e x c e p t t h a t t he model p r e d i c t i o n f o r a s i n g l e c y c l e i s much s m a l l e r than the t o t a l permanent d i s t o r t i o n measured a f t e r 28 h e a t s . However the peak d i s t o r t i o n i s p r e d i c t e d to o c c u r 98 mm below the top o f the mould wh i ch i s a l m o s t i d e n t i -c a l t o the l o c a t i o n o f the measured peak . In F i g . 55 i s a l s o p l o t t e d the p r e d i c t e d permanent d i s t o r t i o n o f the c o r n e r r e g i o n a f t e r a s i n g l e the rma l c y c l e . I t i s e v i d e n t t h a t the permanent d i s t o r t i o n a t the c o r n e r s i s much g r e a t e r t han a t the m i d - f a c e . Th i s s ugge s t s t h a t w i t h i n c r e a s i n g mou ld l i f e the c o r n e r s move f u r t h e r away f rom the s t e e l than the m i d -f a c e . Th i s i s because d u r i n g , a h e a t , p l a s t i c d i s t o r t i o n i n any t r a n s v e r s e s e c t i o n i s a lways i n i t i a t e d a t the c o r n e r s . E v i dence o f t h i s can be o b t a i n e d f rom the r e s u l t s o f the model where the c u m u l a t i v e p l a s t i c s t r a i n can be seen to be h i g h e s t a t the c o r n e r s . The e f f e c t i v e s t r e s s i s an i n v a r i a n t q u a n t i t y c a l c u -l a t e d f rom the f o l l o w i n g e x p r e s s i o n . + (a - a )2 + zz xx' 2 xy + 6x xz 2 + 6T 2 yz . . . ( 6 1 ) 4 0 0 — 5 0 0 -6 0 0 -700 — 2 6 10 14 18 Permanent distortion XI0 3 (mm) .55 A Compar i son of t he P r e d i c t e d D i s t o r t i o n Between the M i d - S i d e and the Co rne r o f t he I n s t r umen t ed B i l l e t Mould a t the Ho t - F a c e as a F u n c t i o n o f D i s t a n c e f rom the Top A f t e r a S i n g l e Thermal C y c l e . 196 A c c o r d i n g to Von M i ses C r i t e r i o n , p l a s t i c f l o w o c c u r s when t he e f f e c t i v e s t r e s s i s equa l to the u n i - a x i a l y i e l d s t r e s s i n t e n s i o n . For the mould F i g . 1 s 56 and 57 are e f f e c t i v e s t r e s s c o n t o u r s a t s e v e r a l t r a n s v e r s e s e c t i o n s i n the mou l d . These p l o t s s e r ve t o i l l u s t r a t e the h i gh s t r e s s l e v e l s i n the men i s cus r e g i o n ( F i g . 5 6 ( i i ) and F i g . 5 7 ( i ) ) . T h i s i n -d i c a t e s t h a t permanent d i s t o r t i o n w i l l be a maximum i n the men i s cus r e g i o n . Exam in i ng , the d i s t r i b u t i o n o f e f f e c t i v e s t r e s s a t e i t h e r t r a n s v e r s e s e c t i o n ( F i g . ' s 5 6 ( i i ) o r 5 7 ( i ) ) i t i s e v i d e n t t h a t the e f f e c t i v e s t r e s s i s a maximum near the c o r n e r a t the c o l d - f a c e r a t h e r than a t the c o r n e r on the h o t -f a c e . Fu r t he rmore the s t r e s s e s near the c o r n e r a t the h o t -f a c e a re even l owe r than the e f f e c t i v e s t r e s s e s near t he c e n t r e o f a s i d e . Th i s r e s u l t , a l t h o u g h s u r p r i s i n g a t f i r s t , can be e x p l a i n e d by e xam in i ng Tab l e XVI wh ich l i s t s the i n -d i v i d u a l component s t r e s s e s f o r a b i l l e t mould i n the men i s cus r e g i o n . I t i s e v i d e n t t h a t the a x x » a y y a n d a z z components o f s t r e s s e s a re o f the same o r d e r o f magn i tude i n r e g i o n s c l o s e t o t he c o r n e r o f the h o t - f a c e ( x = 2.38 mm, y = 54 .57 mm). T h i s e f f e c t i v e l y i s a t r i a x i a l s t a t e o f s t r e s s w i t h the r e -s u l t t h a t the e f f e c t i v e s t r e s s and p o t e n t i a l to y i e l d i s r e -duced . Everywhere e l s e the s t r e s s component a wh i ch i s the X X d i r e c t s t r e s s i n the t h r o u g h - t h i c k n e s s d i r e c t i o n i s s ub -s t a n t i a l l y l e s s than a o r a , w i t h the r e s u l t t h a t the yy zz e f f e c t i v e s t r e s s i s h i g h e r . T h i s i s most p ronounced f o r the f i r s t case examined i n Tab l e XVI wh ich c o r r e s p o n d s to a 1 9 7 • o.o E E 0) o D o xz 40.0 50.0 0> SZ E o 0> o c D b '0.0 50.0 Distance from the midface (mm) F ig . 56 Predicted E f f ec t i ve Stress Contours in a Transverse Section of the Instrumented Mould at ( i ) 36.75 mm and ( i i ) 57.75 mm Below the Top of the Mould. 198 o O . O E E CD O D 0.0 40 .0 50.0 SZ E o 0) <-> c D JO Q o O . O 0.0 10.0 _1 20.0 30.0 _ j 40.0 _ J 10.0 20.0 30.0 40.0 50.0 _ j 45 30 5560 65 50.0 60.0 Distance from the midface(mm) F i g . 57 P r e d i c t e d E f f e c t i v e S t r e s s Con tou r s i n a T r a n s v e r s e S e c t i o n o f the I n s t r umen t ed Mould a t ( i ) 99 .75 mm and ( i i ) 530.25 mm Below the Top o f the Mou l d . 199 depth o f 110 .0 mm be low the top of the mou ld . J u s t above the men i s cus t h e r e i s a marked dec rease in s t r e s s l e v e l s o b v i o u s l y due to the r educed t empe r a t u r e s ( F i g . 5 6 ( i ) ) . Lower i n the mou l d , where y i e l d i n g does not o c c u r , i t can be seen f rom F i g . 5 7 ( i i ) t h a t the e f f e c t i v e s t r e s s i s a maximum nea r t he c o r n e r a d j a c e n t t o the h o t - f a c e . F i g . 58 shows e f f e c t i v e s t r e s s c o n t o u r s i n two l o n g i t u d i n a l p l a n e s i n the men i s cus r e g i o n . F i g . 5 8 ( i ) r e p r e s e n t s a l o n g i t u d i n a l s l i c e 5.65 mm from the c e n t r e o f a s i d e , w h i l e F i g . . 5 8 ( i i ) i s a l o n g i t u d i n a l s l i c e t h rough the m i d - p o i n t s o f the two e l emen t s a d j a c e n t to the l o n g i t u d i n a l c o r n e r p l a n e . I t i s e v i d e n t t h a t the e f f e c t i v e s t r e s s i s h i g h e r a t the h o t - f a c e as com-pa red w i t h the c o l d e x c ep t near the c o r n e r s where the t r i a x i a l s t a t e o f s t r e s s reduces the e f f e c t i v e s t r e s s . F i g . 59 i s a t h r e e - d i m e n s i o n a l s u r f a c e v i s u a l i z a t i o n o f the d i s t o r t e d shape o f the mou l d , f rom the m i d - f a c e t o the c o r n e r , w i t h the d i s t o r t i o n m a g n i f i e d r e l a t i v e to the l a t e r a l d imens i on s o f the mou l d . 4 . 7 Summary The t he rma l d i s t o r t i o n o f b i l l e t moulds have been a n a l y z e d to o b t a i n a b e t t e r u n d e r s t a n d i n g o f the phenomena. I t was shown u s i n g two p r e l i m i n a r y a n a l y s e s t h a t the e l a s t i c d e f l e c t i o n due to bend ing a r i s i n g f rom the t h rough t h i c k n e s s 5 . 0 10. ft a O . O 5 . 0 Distance from hotface (mm) F i g . 58 P r e d i c t e d E f f e c t i v e S t r e s s Con tou r s i n Two L o n g i t u d i n a l P l anes Through the Wa l l o f the I n s t r umen t ed Mould ( i ) 5.55 mm f rom the M i d -Face and ( i i ) P l ane Through the Cen t r e o f the Two Co rne r E l emen t s . F i g . 5 9 T h r e e - D i m e n s i o n a l D i s t o r t e d M o u l d . V i s u a l i z a t i o n o f t h e 202 g r a d i e n t i s sma l l compared to the the rma l e x p a n s i o n o f a t r a n s v e r s e s l i c e . The s t r e s s e s c a l c u l a t e d f rom the f i r s t a n a l y s i s r e v e a l e d t h a t they exceeded the y i e l d s t r e s s o f the m a t e r i a l a t s e v e r a l l o c a t i o n s . Based on t h e s e r e s u l t s a t h r e e - d i m e n s i o n a l e l a s t o - p l a s t i c f i n i t e - e l e m e n t a n a l y s i s ' was chosen f o r the f i n a l d i s t o r t i o n c a l c u l a t i o n s . The d i s t o r t e d shape c a l c u l a t e d u s i n g t he t h r e e -d i m e n s i o n a l a n a l y s i s e x h i b i t e d a maximum o f a p p r o x i m a t e l y 0 . 13 mm, a t a l o c a t i o n 65 mm below the m e n i s c u s , and i s bounded above and below by r e g i o n s o f n e g a t i v e and p o s i t i v e t a p e r . The c o r n e r s moved f u r t h e r away f rom the s t r a n d e v e r y -where e x c e p t nea r the bot tom where due to t he edge moment the r e v e r s e o c c u r s . Mould w a l l movement was mon i t o r e d i n 'an o p e r a t i n g b i l -l e t c a s t i n g machine u s i n g L i n e a r V a r i a b l e D i s p l a c e m e n t T r a n s -duce r s f o r s e v e r a l h e a t s . The p r e d i c t e d v a l u e s o f d i s t o r t i o n compare f a v o u r a b l y w i t h measured v a l u e s . Mou ld w a l l movement i n the men i scus r e g i o n e x h i b i t e d a p e r i o d i c v a r i a t i o n wh i ch has been h y p o t h e s i z e d as be i ng due to n u c l e a t e b o i l i n g . E x am i n a t i o n o f the e f f e c t i v e s t r e s s e s i n the mould i n d i c a t e the permanent d i s t o r t i o n w i l l be a maximum i n the men i s cus r e g i o n and y i e l d i n g i n i t i a t e s nea r the c o r n e r s 203 a d j a c e n t to the c o l d - f a c e . The permanent d i s t o r t i o n c a l c u -l a t e d a f t e r a s i n g l e c y c l e r e v e a l s t h a t the c o r n e r s c e r t a i n l y y i e l d more than the m i d - f a c e and move f u r t h e r away f rom the s t e e l . 5.0 EFFECT OF OPERATING VARIABLES ON THE THERMAL RESPONSE OF SLAB AND BILLET MOULDS 5.1 I n t r o d u c t i o n Many o f the d e f e c t s i n a s - c a s t p r o d u c t s have been l i n k e d to m o u l d - d e s i g n pa rame te r s and o p e r a t i n g v a r i a b l e s . To d e t e rm i n e the r e l a t i o n s h i p between t h e s e q u a l i t y p rob lems and t he rma l d i s t o r t i o n o f the mou l d , i t i s i n s t r u c t i v e f i r s t to a n a l y z e the i n f l u e n c e o f p r o c e s s v a r i a b l e s on mould p e r -f o rmance . In t h i s c h a p t e r a s y s t e m a t i c h e a t - f l o w a n a l y s i s o f the e f f e c t o f the d e s i g n and o p e r a t i n g v a r i a b l e s on the t he rma l f i e l d i n s l a b and b i l l e t moulds i s i n i t i a l l y p r e s e n t e d . The r e s u l t s o f the h e a t - f l o w a n a l y s i s a re t hen u t i l i z e d to o b t a i n some i n s i g h t i n t o the mould c o n d i t i o n s t h a t c o u l d l e a d to a l o s s o f s t r e n g t h o f the mould w a l l t h r ough s o f t e n i n g . Gu ide l i n e s f o r m a t e r i a l s e l e c t i o n based on t h i s a n a l y s i s a r e d i s c u s s e d . A combined s t r e s s and h e a t - f l o w a n a l y s i s f i n a l l y has been u t i l i z e d to i d e n t i f y the v a r i a b l e s t h a t a re l i k e l y t o e x a c e r b a t e d i s t o r t i o n i n b i l l e t mou ld s . Case s t u d i e s o f s e v e r a l i n d u s t r i a l o p e r a t i o n s a re u n d e r t a k e n to e x e m p l i f y the dua l e f f e c t o f o p e r a t i n g and d e s i g n pa r ame t e r s on the t he rma l d i s t o r t i o n o f t hese mou l d s . The r e s u l t s o f t h i s a n a l y s i s s e r v e as a b a s i s f o r i d e n t i f y i n g s a l i e n t d e s i g n 204 205 and o p e r a t i n g f e a t u r e s t h a t ma rked l y i n f l u e n c e mould d i s t o r t i o n . 5.2 S e l e c t i o n o f S u i t a b l e H o t - f a c e , H e a t - f l u x P r o f i l e s The f i r s t s t e p i n t h i s a n a l y s i s was t o i d e n t i f y the p r o c e s s v a r i a b l e s t h a t have a p r o f ound i n f l u e n c e on the hea t e x t r a c t i o n r a t e . S e l e c t i o n o f s u i t a b l e h e a t - f l u x p r o -f i l e s f o r the h o t - f a c e boundary were then made on the b a s i s o f t h i s know ledge . The o p e r a t i n g v a r i a b l e s examined i n t h i s phase o f the s t udy can t h e r e f o r e be c l a s s i f i e d i n t o two c a t e g o r i e s de -pend ing on whe the r o r not they s i g n i f i c a n t l y a f f e c t the h e a t - f l u x p r o f i l e a t the h o t - f a c e . V a r i a b l e s wh i ch do no t a l t e r the h e a t - f l u x a p p r e c i a b l y i n c l u d e : V a r i a t i o n o f any o f t he se pa rame te r s a l t e r s the r e s i s t a n c e to h e a t - f l o w e i t h e r i n the mould o r a t the m o u l d / c o o l i n g wa te r i n t e r f a c e . No rma l l y t hese r e s i s t a n c e s a re much s m a l l e r f o r V a r i o u s O p e r a t i n g C o n d i t i o n s (1) ( i i ) ( i i i ) ( i v ) (v) ( v i ) ( v i i ) m o u l d - w a l l t h i c k n e s s ; mould the rma l c o n d u c t i v i t y ; c o o l i n g - w a t e r v e l o c i t y ; i n l e t t empe r a t u r e o f c o o l i n g w a t e r ; d i r e c t i o n o f c o o l i n g - w a t e r f l o w ; wa t e r q u a l i t y and s c a l e f o r m a t i o n ; mould wa te r e x i t p r e s s u r e 206 than the r e s i s t a n c e o f the mould s t e e l gap , and t h e r e f o r e i n g e n e r a l do not s i g n i f i c a n t l y a f f e c t the h e a t - f l u x p r o f i l e . To i l l u s t r a t e t h i s p o i n t , F i g . 60 shows the magn i tude o f the i n d i v i d u a l r e s i s t a n c e s as a f u n c t i o n o f d i s t a n c e be low the men i s cus f o r the s t a n d a r d b i l l e t mould and s l a b mould w i t h h i gh and low h e a t - f l u x p r a c t i c e s . Over most o f the mould l e n g t h the combined mould and mou l d /wa te r r e s i s t a n c e s c o n -s t i t u t e l e s s than 20% o f the t o t a l . For t h i s a n a l y s i s i t was t h e r e f o r e assumed t h a t the i n f l u e n c e o f t h e s e pa r ame te r s on the h e a t - e x t r a c t i o n r a t e i s s m a l l . However , i t i s im -p o r t a n t to note t h a t i f v a r i a t i o n s i n any p a r ame t e r a re accompan ied by l a r g e changes i n the rma l d i s t o r t i o n wh i ch s i g n i f i c a n t l y a l t e r the gap s i z e then the hea t e x t r a c t i o n r a t e w i l l change a c c o r d i n g l y . Th i s i s d i s c u s s e d i n the s u b -sequen t c h a p t e r . As a p r e l i m i n a r y s t ep towards e v a l u a t i n g the i n f l u e n c e o f the above -men t i oned v a r i a b l e s on t he t he rma l f i e l d i n the mou l d , the p r o f i l e s d e s c r i b e d e a r l i e r ( F i g . 19 and F i g . 33) f o r each o f the s t a n d a r d b i l l e t and s l a b cases have been used unchanged . In the second c a t e g o r y , v a r i a b l e s wh i ch do a l t e r the h e a t - f l u x p r o f i l e i n c l u d e : ( i ) l u b r i c a t i o n and p o u r i n g p r a c t i c e ; ( i i ) ca rbon c o n t e n t o f the s t e e l ; ( i i i ) c a s t i n g speed 207 100 2 80 o -I r S 60 40 2.4 2.0 1.6 1.2 0.8k-0.4 W Slob-(High heat flux practice) Slab-(Low heat flux practice) Billet Gap-(Low heat flux practice) Gap-(High heat flux practice) Gap-(Billet) •» — Slab mould wall Billet mould wall • • — Cooling water interface r o — o 100 200 Meniscus 300 400 500 Distance down the mould (mm) 600 F i g . 6 0 R e s i s t a n c e s to Hea t - F l ow Between the S t e e l S u r f a c e and the Mou ld C o o l i n g - W a t e r f o r the B i l l e t Mould and S l a b Moulds w i t h Low and High H e a t - F l u x P r a c t i c e s . The d i f f e r e n t h e a t - f l u x p r o f i l e s a r i s i n g f rom d i f f e r i n g l u b r i c a t i o n and p o u r i n g p r a c t i c e s have a l r e a d y been d i s c u s s e d e x t e n s i v e l y f o r b i l l e t and s l a b c a s t i n g ( s e c t i o n 3 . 2 ) . To examine the e f f e c t o f ca rbon c o n t e n t on m o u l d - w a l l t e m p e r a -t u r e s i n b i l l e t s , the h e a t - f l u x p r o f i l e s d e r i v e d f rom the 21 da t a o f S i ngh and B l a z e k have been used ( F i g . 1 9 ) . F i n a l l y the i n f l u e n c e o f c a s t i n g speed has been s t u d i e d by assuming t h a t f o r a l l speeds the l o c a l h e a t - f l u x i s u n i q u e l y d e t e rm i ned by the l o c a l r e s i d e n c e t ime i n the mould (= z / V c ) . 5 .3 Thermal F i e l d s i n S l ab and B i l l e t Moulds The the rma l f i e l d s i n moulds were computed f o r a v a r i e t y o f o p e r a t i n g c o n d i t i o n s . A l i s t o f the v a r i a b l e s examined and t h e i r s t a n d a r d v a l u e s t o g e t h e r w i t h an upper and l owe r l i m i t f o r each pa rame te r a re g i v e n i n T ab l e s XXVI and X X V I I . The two v a l u e s s e l e c t e d on e i t h e r s i d e o f the s t a n d a r d span the range n o r m a l l y found i n i n d u s t r y . To s i m p l i f y the p r e s e n t a t i o n o f the computed t he rma l f i e l d f o r each s e t o f v a r i a b l e s , o n l y the t empe r a t u r e p r o -f i l e s a t the h o t - and c o l d - f a c e s a re p r e s e n t e d . The h o t - f a c e t empe r a t u r e p r o f i l e i s i m p o r t a n t because i t d e l i n e a t e s the r e g i o n o f the mould wh i ch i s most s u s c e p t i b l e to s o f t e n i n g and the rma l d i s t o r t i o n . The c o l d f a c e t empe r a t u r e p r o f i l e , on the o t h e r hand when examined i n r e l a t i o n to the i n c i p i e n t 209 Tab l e XXVI Va lues o f C a s t i n g V a r i a b l e s f o r B i l l e t Moulds Casting or Mould Parameter Lower Value / Standard Upper Value Water i n l e t temperature (°C) 15.0 30.0 45.0 Water ve loc i t y (ms~ )^ 5.0 8.0 11.0 Water ex i t pressure (KPa) - 241.3 378.9 Direct ion of water flow - up down Mould wall thickness 5.875 9.525 12.7 Mould wall material copper a l loy copper Mould thermal conduct iv i ty 0.3256 0.3889 Casting speed m/min~^ 1.5 2.03 3.0 % carbon in stee l 0.1% 0.2 - 0.7 Tab l e XXVII Va l u e s o f C a s t i n g V a r i a b l e s f o r S l ab Moulds Casting or Mould Parameter Lower Standard Upper Value Value Water i n l e t temperature 15.0 30.0 45.0 Water ve loc i t y 4.0 7.0 11.0 Mould wal l thickness 25.0 40.0 55.0 Mould wall material copper a l l oy copper Mould thermal conduct iv i ty 0.3256 0.3889 Casting speed 0.6 0.9 1.2 b o i l i n g t empe r a t u r e a t mould wa t e r p r e s s u r e s , g i v e s an i n d i c a -t i o n o f the p o s s i b i l i t y o f n u c l e a t e b o i l i n g a t the mou l d / c o o l i n g - w a t e r i n t e r f a c e . 5 .3 .1 The I n f l u e n c e o f Mould Des ign Pa rame te r s Mould w a l l t h i c k n e s s i s an i m p o r t a n t d e s i g n pa r ame te r s i n c e i t d e t e rm ine s the a b i l i t y o f the mould w a l l to r e s i s t d i s t o r t i o n . F i g . ' s 6 1 , 62 show the c a l c u l a t e d t empe r a t u r e p r o f i 1 es f or a b i 1 1 e t and a slab mould f o r d i f f e r e n t w a l l t h i c k n e s s e s . Thus i t i s seen t h a t f o r b i l l e t mou l d s , ow ing to the h i g h e r c o l d - f a c e t empe r a t u r e s t h a t o b t a i n com-pa red to s l a b mou ld s , n u c l e a t e b o i l i n g t a k e s p l a c e , and as d i s c u s s e d e a r l i e r , i n i t i a t e s a t empe r a t u r e o s c i 1 1 a t i o n i n the mould w a l l . The upper and l ower l i m i t s o f the t empe r a -t u r e o s c i l l a t i o n a t the i n i t i a t i o n o f b o i l i n g a re p r e s e n t e d where a p p l i c a b l e , i n F i g . 61 . In gene r a l the change i n c o l d -f a c e t empe r a t u r e s i s q u i t e sma l l o ve r the range o f t h i c k -nesses e xam ined . However, f o r b i l l e t mou l d s , f o r the 12 .70 mm t h i c k w a l l , the s l i g h t d e c r ea se i n c o l d - f a c e t e m p e r a t u r e s i s s u f f i c i e n t to supp re s s b o i l i n g as i s e v i d e n t f rom F i g . 61 . The h o t - f a c e t empe ra t u r e s on the o t h e r hand i n c r e a s e w i t h i n c r e a s i n g w a l l t h i c k n e s s . In go i ng f rom the s t a n d a r d w a l l t h i c k n e s s to a h i g h e r v a l u e the i n c r e a s e i n h o t - f a c e t empe r a -t u r e i s g r e a t e r f o r the s l a b mould w i t h h i g h h e a t - f l u x p r a c t i c e ( 5 3 ° C ) , w h i l e f o r the s l a b mould w i t h a low h e a t -f l u x p r a c t i c e and f o r a b i l l e t mould the i n c r e a s e s a r e 33°C, z Temperature (°C) 275 1 - J 5 Mould wall thickness 25 mm 40 mm . 55 mm 10 15 20 25 500 700 Temperature (°F ) F i g . 6 2 I n f l u e n c e o f Mould Wal l T h i c kne s s on the Ho t - and C o l d - F a c e Tempera ture P r o f i l e s o f a S lab Mould w i t h H igh H e a t - F l u x P r a c t i c e 213 and 11°C r e s p e c t i v e l y . Fo r a l l cases t h e r e f o r e t h i c k e r moulds are l i k e l y to have h i g h e r s t r e s s e s , owing to the i n c r e a s e d g r a d i e n t s a c r o s s the t h i c k n e s s . T h i s combined w i t h the l owe r y i e l d s t r e s s e s may r e s u l t i n h i g h e r permanent d e f o r m a t i o n . Compar ing s l a b and b i l l e t mould t e m p e r a t u r e s i t i s e v i d e n t t h a t the r i s k o f s o f t e n i n g i s h i g h e r i n s l a b moulds due to the h i g h e r t e m p e r a t u r e . Th i s w i l l be d i s c u s s e d i n g r e a t e r d e t a i l when the i n f l u e n c e o f mould t e m p e r a t u r e s on s o f t e n i n g and r e c r y s t a l 1 i z a t i o n i s e x am ined . H i g h - s t r e n g t h a l l o y s o f copper have been d e v e l o p ed f o r moulds e x p r e s s l y to r e s i s t d i s t o r t i o n . Compared to pure c o p p e r , t h e se a l l o y s have a l owe r t h e rma l c o n d u c t i v i t y . The i n f l u e n c e o f the d e c r e a s ed t he rma l c o n d u c t i v i t y on t he t he rma l f i e l d i n moulds has been i n v e s t i g a t e d f o r one such a l l o y . F i g . ' s 63 ,64 show the r e s u l t s f o r a b i l l e t and .a s l a b mou ld . Changes i n the rma l c o n d u c t i v i t y have a g r e a t e r e f f e c t on h o t - f a c e t e m p e r a t u r e s , w h i l e the c o l d - f a c e t empe r a -t u r e s rema in v i r t u a l l y unchanged . The i n c r e a s e i n peak t empe r a t u r e a t the h o t - f a c e was found to be 14°C f o r the b i l l e t mou l d , and 36°C and 58°C f o r the s l a b mould w i t h low and h i gh h e a t - f l u x p r a c t i c e r e s p e c t i v e l y . C l e a r l y t h e r e f o r e t he b e n e f i t s o f r e s o r t i n g to an e x pen s i v e h i g h - s t r e n g t h a l l o y can o n l y be r e a l i z e d i f the s o f t e n i n g r e s i s t a n c e and s t r e n g t h o f the a l l o y a t the h i g h e r o p e r a t i n g t empe r a t u r e s a re s u p e r i o r to t h a t o f the u n a l l o y e d commerc ia l copper a t the l owe r 0 5 0 100 1 5 0 2 0 0 2 5 0 T e m p e r a t u r e ( ° C ) g.63 I n f l u e n c e o f Mould Thermal C o n d u c t ! v i t y . on the Ho t - and . C o l d - F a c e Temperature P r o f i l e s o f a B i l l e t Mou l d . 75 175 Temperature (°C ) 275 375 475 e E Z3 o E 0> SZ c ? o •o CO o c o 100 200 300 400 t; 500 tob 600 700 \ i i l l I I i I Meniscus — ) —^ ""^ ^ ' / X / \ / 7 — — / / 1 \ I 1 / /' / Cold / / Hot ( face / j face — Thermal conductivity of mould / / 7 / /  1 \  1 Copper 0.39kW/m°C _ Copper alloy 0.32 " — \ / 1 / / / / / f \ / 1 — — i / / // - — 1 I 1 / I 1 1 1 1 1 1 1 100 300 900 F i g 500 700 Temperature (°F) 64 I n f l u e n c e o f Mould Thermal C o n d u c t i v i t y on the Ho t - and C o l d - F a c e Temperature P r o f i l e s o f a S l ab Mould w i t h Low H e a t - F l u x P r a c t i c e . 216 t e m p e r a t u r e . 5.3.2 The I n f l u e n c e o f C o o l i n g - w a t e r V a r i a b l e s  on the Thermal F i e l d C o n s i d e r a b l e a t t e n t i o n has been focus ' sed on the mould c o o l i n g - w a t e r as i t i s known to be a key f a c t o r i n a s u c c e s s f u l c a s t i n g o p e r a t i o n . Water v e l o c i t y i s an i m p o r t a n t pa rame te r because o f i t s e f f e c t on the h e a t -t r a n s f e r c o e f f i c i e n t a t the c o o l i n g - w a t e r i n t e r f a c e (Eq. (5)). F i g . 65 shows the i n f l u e n c e o f t h r e e d i f f e r e n t wa t e r v e l o c i t i e s on b i l l e t mould t e m p e r a t u r e s . Changes i n wa t e r v e l o c i t y can be seen to a f f e c t both the h o t - and c o l d - f a c e t e m p e r a t u r e s , w h i l e i t s i n f l u e n c e on the the rma l g r a d i e n t s s e t up i n t he mould i s ve r y s l i g h t . In the case o f b i l l e t mou l d s , f o r the two v e l o c i t i e s at the l owe r end o f t h e range (5 and 8 m/sec) b o i l i n g i n i t i a t e s i n the c h a n n e l . As was shown e a r l i e r , ( F i g . 1 4 ) , f o r a l owe r wa t e r v e l o c i t y the i n c e p t i o n o f b o i l i n g i s a t a l owe r w a l l s upe r hea t wh i ch r e s u l t s i n l ower peak h o t - f a c e t empe r a t u r e s as. can be seen f rom F i g . 65. I n c r e a s i n g the wa te r v e l o c i t y to 11 m/sec r e s u l t s i n a s u f f i c i e n t r e d u c t i o n o f c o l d - f a c e t empe r a t u r e s to s upp r e s s b o i l i n g c o m p l e t e l y . H o t - f a c e t e m p e r a t u r e s a re c o r r e s p o n d i n g l y r e d u c e d . Thus i t i s e v i d e n t t h a t o v e r the range o f wa te r v e l o c i t i e s n o r m a l l y e n c oun t e r e d i n b i l l e t mou l d s , t he mould wa te r c o u l d be i n a b o i l i n g or n o - b o i l i n g mode depend ing on the v a l u e o f the wa te r v e l o c i t y . Th i s i s an i m p o r t a n t f i n d i n g s i n c e i t may be l i n k e d to l o n g i t u d i n a l Temperature (C°) F i g . 6 5 I n f l u e n c e o f C o o l i n g - W a t e r V e l o c i t y on the Ho t - and C o l d - F a c e Temperature P r o f i l e s o f a B i l l e t Mou l d . 218 c o r n e r c r a c k i n g and r h o m b o i d i t y p rob lems ob s e r v ed i n s e v e r a l b i l l e t c a s t i n g o p e r a t i o n s ; i t may a l s o e x p l a i n the r e d u c t i o n i n s e v e r i t y o f the q u a l i t y p rob lem a c h i e v e d , t h r ough p r o p e r 85-88 c o n t r o l o f wa t e r f l o w r a t e s . ~ Th i s t o p i c i s d i s c u s s e d i n d e t a i l i n the subsequen t c h a p t e r . The e f f e c t o f v a r y i n g wa te r v e l o c i t y i n s l a b moulds on the o t h e r hand me re l y r a i s e s or l owe r s h o t - a n d c o l d - f a c e t e m p e r a t u r e s a l i k e as i s e v i d e n t f rom the r e s u l t s f o r a l ow h e a t - f l u x p r a c t i c e , shown i n F i g . 6 6 . Owing to the t h i c k e r w a l 1 , c o l d - f a c e t empe r a t u r e s a re be low the i n c i p i e n t b o i l i n g p o i n t e x c e p t w i t h the l o w - m e l t i n g powder p r a c t i c e where due t o the h i g h e r h e a t - f l u x b o i l i n g i n i t i a t e s i n the c h a n n e l . On a f i n a l n o t e , f o r s l a b moulds i n g e n e r a l , wa t e r q u a l i t y i s not as c r u c i a l as i n the case o f b i l l e t mou l d s , where f o r . c e r t a i n wa t e r v e l o c i t i e s b o i l i n g and s c a l e d e p o s i t i o n c o u l d o c c u r . S i n c e under s t a n d a r d o p e r a t i n g c o n d i t i o n s n u c l e a t e b o i l i n g i s l e s s l i k e l y i n s l a b mou ld s , the a d v e r s e e f f e c t s o f poor wa t e r q u a l i t y and the r e s u l t i n g s c a l e on mould t empe r a t u r e s has been i n v e s t i g a t e d on l y f o r a b i l l e t mou l d . S c a l e f o r m a t i o n has been s i m u l a t e d by a p p l y i n g a r educed h e a t - t r a n s f e r c o e f f i c i e n t o f 11.4 kW/m °C a t the c o o l i n g -wa t e r i n t e r f a c e a t l o c a t i o n s where the mould w a l l t e m p e r a t u r e s Mould temperature (°C) 100 200 300 400 500 600 700 800 900 Mould temperature (°F) F i g . 6 6 I n f l u e n c e o f C o o l i n g - W a t e r V e l o c i t y on the Ho t - and C o l d - F a c e Temperature P r o f i l e s o f a S l ab Mould w i t h Low H e a t - F l u x P r a c t i c e . ro i—> to 220 exceeds the i n c i p i e n t b o i l i n g p o i n t . Th i s v a l u e o f 2 11.4 kW/m °C was o b t a i n e d f rom i n d u s t r i a l t e s t da t a p r e -s e n t e d e a r l i e r i n Tab l e I I I . 1 0 1 In c a r r y i n g ou t t h i s c ompu t a t i o n w i t h the t r a n s i e n t mode l , i t was found t h a t the r e g i o n o f the c o l d - f a c e , w i t h t empe r a t u r e s above the i n -c i p i e n t b o i l i n g - p o i n t , c o n t i n u e d to expand w i t h t he advance o f t ime u n t i l f i n a l l y a " s t e a d y - s t a t e " s i t u a t i o n was r e a c h e d , w i t h the d i f f e r e n c e between mould t e m p e r a t u r e s a t s u c c e s s i v e i t e r a t i o n s r e a c h i n g 0 . 25 °C . The h o t - and c o l d -f a c e t empe r a t u r e p r o f i l e s f o r t h i s p a r t i c u l a r . c o n d i t i o n a re shown i n F i g . 67 . E x am i n a t i o n o f the c o l d - f a c e t e m p e r a t u r e s r e v e a l s t h a t s c a l e i s formed f rom the men i s cus r e g i o n to a depth o f ~ 200 mm be low the m e n i s c u s . The peak h o t - and c o l d - f a c e t empe r a t u r e s o f the mould were found to be 371 and 304°C r e s p e c t i v e l y wh i ch i s a s u b s t a n t i a l i n c r e a s e ove r •the peak v a l u e s o f 227°C and 156°C c a l c u l a t e d f o r t h e s t a n d a r d c o n d i t i o n . Over the r e g i o n o f s c a l e f o r m a t i o n the t empe r a t u r e g r a d i e n t i n the d i r e c t i o n o f the l e n g t h o f the mould can be seen to be e x t r e m e l y h i g h . Th i s c o u p l e d w i t h the l o c a l l y h igh t empe r a t u r e s c o u l d l e a d to p l a s t i c d i s -t o r t i o n o f the mou ld . E v i d ence o f t h i s has been o b s e r v e d i n i n d u s t r y where e x c e s s i v e mould d i s t o r t i o n i s v e r y o f t e n 84 accompan ied by d e p o s i t s o f s c a l e on the c o l d f a c e . S i n c e s c a l e f o r m a t i o n has such a d e l e t e r i o u s e f f e c t on the t he rma l f i e l d i n the mou ld , i t i s i m p o r t a n t t h a t w a t e r q u a l i t y be a d e q u a t e l y m a i n t a i n e d . 222 Ano the r method of a v e r t i n g e x c e s s i v e d e p o s i t i o n o f s c a l e i s to r a i s e the i n c i p i e n t b o i l i n g p o i n t f o r the sys tem by r a i s i n g mould wa te r e x i t p r e s s u r e s . No rma l l y a back p r e s s u r e o f 140 kPa (gauge) i s m a i n t a i n e d to ensu re t h a t the wa t e r channe l rema ins f i l l e d . The e f f e c t o f r a i s i n g the sys tem p r e s s u r e to 275 kPa (gauge) i s shown i n F i g . 68 . I t i s e v i d e n t t h a t n u c l e a t e b o i l i n g i s s u pp r e s s ed w i t h the i n c r e a s e i n p r e s s u r e bu t w a l l t empe r a t u r e s t end to be s l i g h t l y h i g h e r . Th i s i s undoub t ed l y due to the f a c t t h a t t he a t t e n d a n t l o c a l b o i l i n g h e a t - t r a n s f e r c o e f f i c i e n t s a re h i g h e r than the f o r c e d c o n v e c t i o n h e a t - t r a n s f e r c o e f f i c i e n t p r o v i d e d no s c a l e i s f o rmed . T u r n i n g to the e f f e c t o f wa te r i n l e t t empe r a t u r e on the the rma l f i e l d , the t empe r a t u r e p r o f i l e s i n b i l l e t moulds f o r wa te r i n l e t t empe r a t u r e s o f 15 , 30 and 45°C a re p r e s e n t e d i n F i g . 69 . I t i s e v i d e n t t h a t t h e r e i s n e g l i g i b l e change i n the t empe r a t u r e s i n the men i scus r e g i o n , w i t h the g r e a t e s t d i f f e r e n c e o c c u r r i n g a t the top and bo t tom o f the mou l d . The t empe r a t u r e o f the mould at the bot tom i s i n c r e a s e d f rom 56°C to 79°C f o r a change i n wa t e r i n l e t t empe r a t u r e f rom 15°C to 45°C . The e f f e c t o f wa t e r t empe r a t u r e on s l a b moulds i s v e r y s i m i l a r , e x c ep t t h a t b o i l i n g does not o c c u r and t h e r e -f o r e t h e r e i s no t empe r a t u r e o s c i l l a t i o n . In c u r r e n t p r a c t i c e , c o o l i n g wa t e r f l o w s i n the . 6 8 I n f l u e n c e o f Water Channel E x i t P r e s s u r e on the Ho t - and C o l d -Face Tempera tu re P r o f i l e s o f a B i l l e t M o u l d . Temperature (°C) F i g . 6 9 I n f l u e n c e o f Water I n l e t Tempera ture on the Ho t - and C o l d - F a c e Temperature P r o f i l e s o f a B i l l e t Mou l d . ro ro channe l c o un t e r c u r r e n t to the c a s t i n g d i r e c t i o n . Th i s t o g e t h e r w i t h the back p r e s s u r e a re methods employed t o en su r e t h a t the mou l d -wa te r channe l rema ins f i l l e d . Re-v e r s i n g the d i r e c t i o n o f f l ow i n the c o o l i n g c hanne l s o f b i l l e t moulds has a s l i g h t but b e n e f i c i a l e f f e c t as shown i n F i g . 70 p r o v i d e d the channe l rema ins f u l l . For the s t a n d a r d c a s e , s i n c e peak c o l d - f a c e t empe r a t u r e s were j u s t above t h a t r e q u i r e d to i n i t i a t e b o i l i n g , the s l i g h t r e -d u c t i o n a s s o c i a t e d w i t h the r e v e r s a l o f f l o w was s u f f i c i e n t to s upp r e s s b o i l i n g i n the c h a n n e l . 5 . 3 . 3 The I n f l u e n c e o f C a s t i n g Speed on the  Thermal F i e l d The e f f e c t o f c a s t i n g speed on the h o t - and c o l d - f a c e t empe r a t u r e s i n bo th b i l l e t and s l a b moulds i s shown i n F i g . 1 s 71 - 73 . In a l l t h r e e c a s e s , i n c r e a s i n g the c a s t i n g speed g e n e r a l l y r a i s e s the mould t e m p e r a t u r e s w i t h the g r e a t e s t e f f e c t b e i ng seen i n the l owe r r e g i o n o f the mou l d . For examp le , f o r the s l a b mould w i t h a h i g h - h e a t f l u x p r a c t i c e , a t w o - f o l d i n c r e a s e i n c a s t i n g speed r a i s e s the h o t - f a c e t empe r a t u r e f rom 100°C to 188°C a t the mould b o t t o m . In the case o f b i l l e t moulds i n c r e a s i n g the c a s t i n g speed ex tends the r e g i o n ove r wh ich n u c l e a t e b o i l i n g o c c u r s so t h a t a l a r g e r p o r t i o n o f the mould undergoes t e m p e r a t u r e c y c l i n g . The e f f e c t o f i n c r e a s i n g the c a s t i n g speed on peak 0 50 100 150 200 250 T e m p e r a t u r e (°C) F i g . 7 0 Trie ^Effect o f R e v e r s i n g the D i r e c t i o n o f Water F low on the Hot - and Co l d - F a c e Tempera tures o f a B i l l e t M o u l d . 1 0 0 E 2 0 0 CO CL O 3 0 0 cu £ 4 0 0 E o S 5 0 0 1 c o 6 0 0 7 0 0 M e n i s c u s face C a s t i n g s p e e d ( m m / m i n ) . 1 5 0 0 2 0 3 0 3 0 0 0 F i g . 7 1 1 0 0 1 5 0 T e m p e r a t u r e (°C) I n f l u e n c e o f C a s t i n g Speed on the Ho t - and C o l d - F a c e Temperature P r o f i l e s o f a B i l l e t Mou l d . 2 0 0 2 5 0 ro ro Distance down the mould (mm ) C O ro - a >—i - s 3 o - h - h —• ->• c — ' fD fD 3 O O fD O - h O - h Ol o CO B> — i (/> CD c+ c r - J . 3 2 t o . O c c o —• T 3 D_ fD fD £ CL. i. <-+ O 3 " 3 O s fD Cu r + l - n E -a n -5 Co cu n O fD o I O O O fD fD 3 X J fD S Di tz -i fD 51 3 CD CD o -n CD 3 T3 CD C CD o Distance down the mould (in ) 823 230 t empe r a t u r e s at. the h o t - and c o l d - f a c e i s q u i t e s m a l l . C l e a r l y t h e r e f o r e the ma jo r i n f l u e n c e o f c a s t i n g speed i s to a l t e r the l o n g i t u d i n a l t he rma l g r a d i e n t s i n the mou l d . 5 . 3 . 4 The I n f l u e n c e o f Carbon Con t en t o f the S t e e l  on the Thermal F i e l d i n B i l l e t Moulds The ca rbon c o n t e n t o f the s t e e l has been ob s e r v ed to have a p r o f ound i n f l u e n c e on the magn i tude o f 21 the h e a t - e x t r a c t i o n r a t e s i n the mou ld . S i ngh and B l a z e k e x p e r i m e n t i n g w i t h d i f f e r e n t g rades i n b i l l e t mou l d s , o b s e r v ed t h a t a l t h o u g h the shape o f the h e a t - f l u x p r o f i l e i s v i r t u a l l y u n a f f e c t e d by the s t e e l c o m p o s i t i o n , the magn i -tude o f the peak , and ave rage h e a t - f l u x e s do change w i t h the ca rbon c o n t e n t . As was seen e a r l i e r ( F i g . 8) t h e i r s t u d i e s r e v e a l e d t h a t the peak h e a t - f l u x i s a minimum f o r s t e e l s c o n t a i n i n g 0.1% c a r b o n , and f o r ca rbon l e v e l s g r e a t e r than a p p r o x i m a t e l y 0.2% t h e r e i s l i t t l e v a r i a t i o n i n the h e a t - f l u x . For b i l l e t moulds the h e a t - f l u x p r o f i l e s p r e -s en t ed i n F i g . 19 f o r 0.1% ca rbon and 0.9% ca rbon have been used to compare the mould w a l l t empe r a t u r e s t h a t wou ld o b t a i n when c a s t i n g low and h i gh ca rbon (>0.2%) g rades o f s t e e l . The h e a t - f l u x p r o f i l e s shown i n F i g . 19 were o b t a i n e d by a n a l y s i n g the da t a o f S ingh and B l a z e k as d e s c r i b e d i n s e c t i o n 3 . 1 . 5 . Owing to the p a u c i t y o f h e a t - f l u x da ta f o r s l a b moulds d u r i n g the c a s t i n g o f d i f f e r e n t g rades o f s t e e l , an i n v e s t i g a t i o n o f the i n f l u e n c e o f s t e e l c o m p o s i t i o n on s l a b mould t empe r a t u r e s c o u l d not be c a r r i e d o u t . The the rma l f i e l d i n a b i l l e t mould was c a l c u l a t e d f o r h i gh and low ca rbon s t e e l g rades and the h o t - and c o l d -f a c e t empe r a t u r e p r o f i l e s a re shown i n F i g . 74 . The mou ld -w a l l t empe r a t u r e s a s s o c i a t e d w i t h l o w - c a r b o n g rades can be seen to be ma r ked l y l owe r than those o b t a i n e d w i t h the h i g h e r c a r bon s t e e l s ; the peak h o t - f a c e t e m p e r a t u r e i s i n -c r e a s e d f rom 179°C to 227°C. Of g r e a t e r s i g n i f i c a n c e i s the absence o f n u c l e a t e b o i l i n g when c a s t i n g low ca r bon g rades o f s t e e l a t , s t a n d a r d wa te r f l o w r a t e s and mould wa t e r p r e s s u r e s . The r e s u l t s o f t h i s a n a l y s i s s u g g e s t t h a t t he i n c r e a s e i n s e v e r i t y o f l o n g i t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y ob se r ve s w i t h h i gh ca rbon g rades o f s t e e l , 85 87 ,(>0.4%), may be l i n k e d to the b o i l i n g phenomena; p a r t i c u l a r l y s i n c e i t i s h i g h l y . u n l i k e l y t h a t the c r a c k i n g t endency i s due to poor d u c t i l i t y o r a r e d u c t i o n i n s t r e n g t h s i n c e on the c o n t r a r y , i t has been ob s e r v ed t h a t t he s t r a i n to f r a c t u r e i s a minimum f o r s t e e l s c o n t a i n i n g 0 .17 to 0.20% c a r b o n . T h i s t o p i c i s d i s c u s s e d i n g r e a t e r d e t a i l i n the subsequen t c h a p t e r on q u a l i t y . 5.3.5 A Summary o f the I n f l u e n c e o f O p e r a t i n g V a r i a b l e s on the Thermal F i e l d s i n Mou lds Summar i z i ng the r e s u l t s o f t h i s a n a l y s i s , i t zzz has been shown t h a t a de c r ea se i n t he rma l c o n d u c t i v i t y o f the mou l d , and an i n c r e a s e i n w a l l t h i c k n e s s r a i s e s the t empe r a t u r e s a t the h o t - f a c e w h i l e c o l d - f a c e t e m p e r a t u r e s rema in v i r t u a l l y unchanged . Of the c o o l i n g - w a t e r p a r a m e t e r s , wa t e r v e l o c i t y has a p r o f ound i n f l u e n c e on w a l l t empe r a t u r e s i n bo th s l a b and b i l l e t mou l d s . I t has been shown t h a t i n s l a b mou l d s , owing to the t h i c k e r w a l l s wh i ch l e a d s to l ower c o l d - f a c e t e m p e r a t u r e s , b o i l i n g would r a r e l y o c c u r . In c o n t r a s t to t h i s , i n b i l l e t moulds b o i l i n g c o u l d i n i t i a t e a t s t a n d a r d o p e r a t i n g c o n d i t i o n s i n the w a t e r c h a n n e l ; i n t h i s case p r o p e r s e l e c t i o n o f wa t e r v e l o c i t y i s n e c e s -s a r y to s upp r e s s b o i l i n g i n the c h a n n e l . I f wa t e r q u a l i t y i s poor b o i l i n g may l e a d to s c a l e f o r m a t i o n wh i ch i s a s s o c i a -t e d w i t h a marked r e d u c t i o n i n t he h e a t - t r a n s f e r c o e f f i c i e n t . T h i s has been shown to r e s u l t i n a s u b s t a n t i a l i n c r e a s e i n w a l l t e m p e r a t u r e s , and l o n g i t u d i n a l t he rma l g r a d i e n t s , wh ich may l e a d to e x c e s s i v e mould d i s t o r t i o n . H i g h e r wa t e r p r e s -s u r e s can be employed i n the mould to e f f e c t i v e l y s upp r e s s b o i l i n g . R e v e r s i n g the d i r e c t i o n o f wa te r f l o w has a s l i g h t bu t b e n e f i c i a l e f f e c t on the the rma l f i e l d i n b i l l e t mou ld s . Chang ing the c a s t i n g speed has the g r e a t e s t e f f e c t on the t e m p e r a t u r e s a t t h e bot tom o f the mou l d , w h i l e the i n f l u e n c e on the t empe r a t u r e s a t the men i s cus i s much s m a l l e r . In the e ven t o f b o i l i n g , wh i ch i nduce s the rma l c y c l i n g o f the mould w a l l , r a i s i n g the c a s t i n g speed tends to e x t end the r e g i o n ove r wh i ch the rma l c y c l i n g o c c u r s . F i n a l l y f o r b i l l e t mou lds , 234 i t has been demons t r a t ed t h a t when c a s t i n g low ca r bon g rades o f s t e e l (0.1%) mould w a l l t empe r a t u r e s a re ma r ked l y r educed and b o i l i n g i s u n l i k e l y to o c cu r i n the c o o l i n g c h a n n e l s . 5.4 Mould Wa l l S o f t e n i n g The t empe r a t u r e f i e l d t h a t a r i s e s i n a c o n t i n u o u s c a s t i n g mould has two e f f e c t s on the mould w a l l . F i r s t , the mould a t t emp t s to expand o r c o n t r a c t l o c a l l y i n r e spon se to d i f f e r e n t i a l t empe r a t u r e changes , and thus g ene r a t e s t r a i n s and second the s t r e n g t h and d i s t o r t i o n r e s i s t a n c e o f t h e mould i s r educed as the t empe r a t u r e o f the mould w a l l i n c r e a s e s . 55 R e c e n t l y a s t udy was conduc t ed by Br imacombe e t a l . to d e t e rm i ne the mechan i c a l p r o p e r t i e s and i n f l u e n c e o f t empe r a t u r e on the r e c r y s t a l 1 i z a t i o n k i n e t i c s and s t r e n g t h o f a v a r i e t y o f b i l l e t mould m a t e r i a l s . Ha rdness measu re -ments o f the a s - r e c e i v e d mould m a t e r i a l s has i n d i c a t e d t h a t moulds have some r e s i d u a l c o l d work , p r o b a b l y r e l a t e d to the f a b r i c a t i o n p r o c e s s , wh i ch was found to be e q u i v a l e n t t o 55 a p e r c e n t r e d u c t i o n o f 7 ~ 14%. I t was a l s o o b s e r v ed t h a t i f mould w a l l t empe r a t u r e s a re below 300°C. the re , wo'ul d be ve r y 1 i t t l e s o f t e n i n g d u r i n g the l i f e o f t he mou l d . The r e s i s t a n c e to s o f t e n i n g i n t he se mould m a t e r i a l s i s b e l i e v e d to be due to the p r e sence o f r e s i d u a l e l emen t s such as Ag and P wh i ch r e t a r d t he r e c r y s t a l 1 i z a t i o n k i n e t i c s . Upon e xam in i ng 235 the peak t empe r a t u r e s computed f o r each o f the v a r i o u s ca ses examined i n t he p r e c e d i n g s e c t i o n ( T a b l e X X V I I I ) , i t i s e v i -den t t h a t f o r a l l c a s e s , w i t h the e x c e p t i o n o f s c a l e f o r m a -t i o n , b i l l e t mould t empe r a t u r e s a re be low 3 0 0 a C . In p r a c t i c e i t has been ob se r ved t h a t s c a l e f o r m a t i o n l e a d s to b a d l y d i s t o r t e d moulds and t h e r e f o r e wa te r q u a l i t y i s m a i n t a i n e d to a v e r t t h i s e v e n t . The e v i d en c e o f s o f t e n i n g . i n mould m a t e r i a l s s ugge s t s t h a t i n b i l l e t mou l d s , the d i s t o r t i o n o b s e r v ed i s due to the t h e r m a l l y g e n e r a t e d s t r a i n s , w i t h the d e c r e a s e i n s t r e n g t h due to s o f t e n i ng at the high t e m p e r a t u r e s in the men i s cus r e g i o n p r o b a b l y h a v i ng o n l y a s m a l l e f f e c t on the o v e r a l l d i s t o r t i o n . Based on t h i s i n f o r m a t i o n i t appea r s t h a t f o r b i l l e t moulds f rom the s t a n d p o i n t o f imp r o v i n g the d i s t o r t i o n r e s i s t a n c e , i t wou ld be more e f f e c t i v e to r e s o r t t o wo rk -h a r d e n i n g , or. to use h igh s t r e n g t h a l l o y s i n the manu f a c t u r e o f moulds r a t h e r than a l l o y i n g the coppe r w i t h e l emen t s such as Ag wh i ch may o n l y i n f l u e n c e the r e c r y s t a l 1 i z a t i o n k i n e t i c s . For s l a b - m o u l d s t h e r e are no p u b l i s h e d da t a on the chem i ca l c o m p o s i t i o n s , mechan i c a l p r o p e r t i e s , and r e c r y s t a l -l i z a t i o n k i n e t i c s , o f a s - r e c e i v e d mould m a t e r i a l s . T h e r e f o r e computed peak t empe r a t u r e s f o r the d i f f e r e n t o p e r a t i n g c o n d i -t i o n s p r e s e n t e d i n Tab l e XXVI I I have been compared w i t h a v a i l -a b l e i n f o r m a t i o n on the s o f t e n i n g o f c oppe r and v a r i o u s a l l o y s 121 o f c o p p e r . F i g . 75 i s the da ta o f D e s t i t o and P l o v a n i c h Table XXVII I Ca lcu la ted Peak Temperatures in the Mould Wall (°C) Conditions . . B i l le t Slab low heat f lux Slab high heat f lux Standard 227 260 383 Water in let temperature - low 226 256 378 - high 229 263 388 Water velocity - low 215 285 425 - high 204 247 361 Downward water flow 225 - -Sealing 371 - -Wall thickness - small 206 226 325 - large 228 293 436 Copper alloy 241 296 438 Casting speed - low 227 261 378 - high 228 272 410 0.2 - 0.7% C 227 0.1% C 177 . -ro Co 237 55 as p r e s e n t e d by Brimacombe e t a l . The l a t t e r a u t h o r s ' measurements on b i l l e t mould m a t e r i a l s have a l s o been p l o t t e d f o r compa r i son and are seen grouped t o g e t h e r on the f a r r i g h t o f F i g . 75 . Exam in ing the peak t empe r a t u r e s i n s l a b moulds w i t h low h e a t - f l u x p r a c t i c e ( T ab l e X X V I I I ) , i t appea r s t h a t mould t empe r a t u r e s c o u l d r i s e to 300°C i n c e r t a i n c a s e s . In 1 21 r e l a t i o n to D e s t i t o and . P I o v a n i c h ' s r e s u l t s i t appea r s t h a t e l e c t r o l y t i c t o u g h - p i t c h , o x y g e n - f r e e , and d e o x i d i z e d h i g h - p h o s p h o r o u s coppe r s a re u n s u i t a b l e f o r t h i s a p p l i c a t i o n . The s i l v e r b e a r i n g coppe r s seem b e t t e r s u i t e d owing t o the l o n g e r s o f t e n i n g t imes a t a g i v e n t e m p e r a t u r e . The p o i n t s to the ext reme r i g h t o f F i g . 75 c o r r e s p o n d t o the most r e c e n t da t a on b i l l e t mould m a t e r i a l s , wh i ch c o n t a i n e i t h e r Ag o r an a l l o y o f Ag and P, and t r a c e s o f o t h e r r e s i d u a l e l e m e n t s . These m a t e r i a l s appear to have b e t t e r s o f t e n i n g c h a r a c t e r -i s t i c s than measured i n r e g u l a r g r a d e s . I f s l a b moulds a re o f the same m a t e r i a l as b i l l e t mou l d s , then w i t h the low h e a t - f l u x p r a c t i c e l i t t l e s o f t e n i n g c o u l d be e x pe c t e d to o c c u r i n the mou l d . Here too i t wou ld be b e n e f i c i a l t o use the moulds i n a wo r k -ha rdened s t a t e f rom t he s t a n d p o i n t o f d i s t o r t i o n r e s i s t a n c e . H igh s t r e n g t h a l l o y s wou ld o f f e r the same b e n e f i t s w i t h o u t the a t t e n d a n t r i s k o f s o f t e n i n g . For s l a b moulds w i t h h i gh h e a t - f l u x l u b r i c a t i o n p r a c t i c e , owing to the h i gh t empe ra t u r e s t h a t are o b t a i n e d i n the mou l d , a l l o y s o f coppe r w i t h b e t t e r s o f t e n i n g c h a r a c t e r i s t i c s may have to be u t i l i z e d to a v e r t e x c e s s i v e d i s t o r t i o n due to Time (sec) 3 - 6 X I 0 2 3 - 6 X I 0 3 3 - 6 X I 0 4 3 - 6 X I 0 9 1 j — I 1 O CDA 110 E T P @ " 102 O F • " 1 1 4 0 0 3 4 % A g • " 1 1 6 0 0 8 5 % " 3 0 0 10 T i m e (hr) 100 F i g . 75 H a l f S o f t e n i n g T e m p e r a t u r e s f o r B i l l e t M o u l d M a t e r i a l s . 5 5 ' ^ 2 1 oo 00 239 the l o s s o f s t r e n g t h . 5.5 Case S t u d i e s o f I n d u s t r i a l B i l l e t Mou ld O p e r a t i o n s Owing to the h i g h c o s t o f c a l c u l a t i n g mould d i s t o r t i o n u s i n g the t h r e e - d i m e n s i o n a l e l a s t o - p l a s t i c m o d e l , a' s y s t e -m a t i c a n a l y s i s o f the i n f l u e n c e o f p r o c e s s v a r i a b l e s on d i s t o r t i o n was uneconomic . I n s t e a d , i n f o r m a t i o n on the com-b i n e d e f f e c t o f s e v e r a l o p e r a t i n g v a r i a b l e s on mould d i s -t o r t i o n was g l e aned f rom case s t u d i e s o f s i x o p e r a t i n g b i l l e t mou l d s . , Tab l e XXIX p r e s e n t s the s t a n d a r d o p e r a t i n g and d e s i g n c o n d i t i o n s o f the s i x b i l l e t mou ld s . Based on t he se c o n d i t i o n s the the rma l f i e l d was computed f o r each c a s e . These r e s u l t s c o up l e d w i t h mechan i c a l p r o p e r t y da ta i n Append i x V I I were u t i l i z e d to c a l c u l a t e the d i s t o r t i o n o f each mou l d . F i g . 76 shows the c a l c u l a t e d d i s t o r t i o n p r o -f i l e s a l o ng the c e n t r e o f a s i d e , a t the h o t - f a c e f o r each o f the s i x mou l d s . The d i s t a n c e o f the men i s cus be low the top o f the mould f o r each o p e r a t i o n i s a l s o i n d i c a t e d i n the same F i g . A l l the p r o f i l e s e x h i b i t the same c h a r a c t e r -i s t i c s . A r e g i o n o f n e g a t i v e t a p e r ex tends f rom the top o f the mould to a l o c a t i o n be low the m e n i s c u s , a t wh i ch peak d i s t o r t i o n o c c u r s . Th i s i s f o l l o w e d by a r e g i o n o f p o s i t i v e t a p e r due to the l o n g i t u d i n a l t he rma l g r a d i e n t i n the mou l d . 240 Distor t ion ( m m ) . 76 Compar i son o f the P r e d i c t e d Thermal D i s t o r t i o n P r o f i l e s A l ong the Cen t re o f the W a l l s a t the Ho t - F a c e f o r S i x I n d u s t r i a l B i l l e t M o u l d s . t a b l e XXIX Standard Operat ing and Design Condi t ions f o r Six B i l l e t Moulds Va r iab le Des ignat ion f o r Each Mould Opera t ion A B G H 0 K Mould length (mm) 700.0 812.8 757.0 735.0 812.0 803.28 Wall thickness (mm) 14.14 13.00 5.875 9.875 9.525 12.70 Section size (mm) 184x184 181x181 139x139 111x111 131x131 133x133 Mould material DHP 122 DHP 122 DHP 120 + Ag DHP 120 + Ag DHP 120 + Ag DHP 120 + Ag Water in le t temperature (°C) 30.0 30.0 24.0 30.0 30.0 22.5 Temperature r ise (°C) 5.0 4.5 8.0 8.5 8.5 7.7 Water velocity (m/sec) 7.87 7.00 10.85 8.16 8.15 10.71 Computed value of h w 29.8 25.52 39.68 28.90 28.84 36.77 Pressure at mould ex i t (kPa) 241.3 241.3 241 .3 241.3 241.3 241.3 Casting speed (m/min) 1.27 1.52 2.41 3.05 2.03 2.21 Mould free board (mm) 60.0 66.6 150.00 50.8 100.0 101.6 Mould support Type 3 Type 2 Type 1 Type 1 Type 2 Type 1 4=» 242 The i m p o r t a n t r e s u l t s o f the h e a t - f l o w and d i s t o r t i o n a n a l y s i s a re p r e s e n t e d i n Tab l e XXX. E x am i n a t i o n o f the r e s u l t s r e v e a l t h a t mould w a l l t h i c k n e s s and s e c t i o n s i z e have a p r o f ound i n f l u e n c e on the peak d i s t o r t i o n . Cases H and J are a lmos t i d e n t i c a l i n a l l r e s p e c t s e x c ep t f o r the d i f f e r e n c e i n s e c t i o n s i z e , H h a v i n g a s m a l l e r s e c t i o n . C l e a r l y t h e r e f o r e the h i g h e r d i s t o r t i o n o f mould J ( 0 . 188 mm) ove r and above t h a t c a l c u l a t e d f o r mould H (0~.166 mm) i s due to the e f f e c t o f s e c t i o n s i z e . The i n f l u e n c e o f w a l l t h i c k n e s s on the peak d i s t o r t i o n stems f r om the h i g h e r h o t - f a c e t e m p e r a t u r e s . S i n c e the d i s t o r t e d shape o f the mould i s due to the d i f f e r e n t i a l t he rma l e x -p a n s i o n r e s u l t i n g f rom the l o n g i t u d i n a l t h e rma l g r a d i e n t down the l e n g t h o f the mou ld , h i g h e r ave rage w a l l t empe r a -t u r e s wou ld l e a d to g r e a t e r average e x p a n s i o n s . The com-b i n e d e f f e c t o f l a r g e s e c t i o n s and t h i c k w a l l s i s r e f l e c t e d i n the h i gh v a l u e s o f d i s t o r t i o n c a l c u l a t e d f o r moulds A and B ( 0 . 2 2 4 , 0 .236 mm). The e f f e c t i v e n e s s o f h i g h e r wa te r v e l o c i t i e s i n r e -du c i n g the peak d i s t o r t i o n i n t h i c k e r mou l d s , t h r ough a r e -d u c t i o n i n w a l l t e m p e r a t u r e s , i s seen i n t he r e s u l t s f o r mould K. He r e , the peak d i s t o r t i o n c a l c u l a t e d was 0 .168 mm wh i ch i s comparab le to the peak d i s t o r t i o n o f 0 .166 mm Table XXX Resul ts o f Heat Flow and D i s t o r t i o n C a l c u l a t i o n s Parameter Des ignat ion f o r Each Mould Opera t ion A B G H J K Peak hot-face temperature(°C) 236.0 234.0 187.0 229.0 230.0 215.0 Peak cold-face temperature(°C) 149.0 151.0 133.0 155.0 155.0 125.0 Peak temperature difference(°C) 87.0 83.0 54.0 74.0 75.0 70.0 Peak distor t ion (mm) 0.224 0.238 0.198 0.166 0.188 0.168 Maximum plastic strain 8.1xl0" 4 10.9xl0" 4 l .OxlO" 4 6.9x l0" 4 5.6xl0" 4 3.3xl0" 4 Location of peak distor t ion below meniscus (mm) 80.0 100.0 85.0 90.0 80.0 90.0 Negative taper above peak (%/m) 1.14 2.06 1.31 2.12 1.75 1.27 • Posi t i ve taper below peak (%/m) 0.375 0.353 0.38 0.33 0.386 0.383 Location of maximum plast ic strain belowmeniscus (mm) 30.0 14.6 12.30 13.0 16.1 24.6 Cooling-water condition no boi l ing boi l ing no boi l ing boi l ing boi l ing no boi l ing 244 o b t a i n e d i n mould H wh ich has a t h i n n e r w a l l and s m a l l e r s e c t i o n . The n e g a t i v e t a p e r t h a t i s seen to o c c u r j u s t be low the men i s cus i s dependent on the peak d i s t o r t i o n , as w e l l as on the mould c o n s t r a i n t s , and the depth o f the mould f r e e -boa rd r e l a t i v e to the c o n s t r a i n t s . Mould A i s s u p p o r t e d a t the top and bot tom between two cove r p l a t e s , where as a l l the o t h e r moulds are s l o t t e d a p p r o x i m a t e l y 1.2 ~ 2.5 cms f rom the top to a c c ep t s p l i t s u p p o r t p l a t e s . The l a t e r a l r e s t r a i n t to t he rma l e x p a n s i o n i n the s u p p o r t r e g i o n c o u l d be e xpe c t ed to be h i g h e r w i t h the 1 a t t e r t ype o f s u p p o r t . T h i s wou ld l e a d to a l a r g e r s l o p e i n the d i s t o r t i o n p r o f i l e above the peak as w e l l as i n c r e a s e d permanent d i s t o r t i o n . A compa r i s on o f the v a l u e o f n e g a t i v e t a p e r and p l a s t i c s t r a i n o f mould A w i t h t hose c a l c u l a t e d f o r mould B c l e a r l y i l l u s t r a t e s t h i s p o i n t . R a i s i n g the men i s cus l e v e l c l o s e r to the s u p p o r t a l s o i n c r e a s e s the n e g a t i v e t a p e r as a r e s u l t o f the h i g h e r s l o p e i n the d i s t o r t i o n . p r o f i 1 e i n t he r e g i o n . T h i s i s e v i d e n t f rom the v a l u e s o f n e g a t i v e t a p e r c a l c u l a t e d f o r moulds H and K. These two moulds have a lmos t i d e n t i c a l v a l u e s o f peak d i s t o r t i o n but d i f f e r e n t men i s cus l e v e l s . The low n e g a t i v e t a p e r o f mould K (1.27%/m) as compared to t h a t c a l c u l a t e d f o r mould H (2.12%/m) i s c l e a r l y due to the l owe r men i s cus l e v e l , i n mould K. An added b e n e f i t o f o p e r a t i n g w i t h a l ower s t e e l l e v e l w i t h r e s p e c t to t he depth a t wh i ch the mould i s c o n s t r a i n e d , i s the r educed p l a s t i c s t r a i n . A s i m i l a r o b s e r v a t i o n can be made by compar i ng the r e s u l t s o b t a i n e d f o r mould G w i t h J-. In mould G the men i s cus l e v e l i s 150 mm below the top o f the mou l d . T h e r e -f o r e the r e s t r a i n t to the rma l e xpan s i o n i n the men i s cus r e g i o n i s l o w , wh i ch r e s u l t s i n a peak d i s t o r t i o n o f h i g h e r magn i tude than c a l c u l a t e d f o r mould J a l t h o u g h the l a t t e r has a s i m i l a r s e c t i o n s i z e and h i g h e r ave rage w a l l t e m p e r a - : t u r e s . The low r e s t r a i n t to e x p a n s i o n i n mould G c o u p l e d w i t h the l owe r mould w a l l t empe r a t u r e s r e s u l t s i n a low v a l u e o f peak p l a s t i c s t r a i n ( 1 . 0 x 10 ) wh i ch i s f i v e t imes l e s s than the maximum p l a s t i c s t r a i n i n mould J . The p o s s i b l e i n f l u e n c e o f e x c e s s i v e n e g a t i v e t a p e r on b i l l e t q u a l i t y w i l l be d i s c u s s e d i n t he subsequen t c h a p t e r . One f i n a l o b s e r v a t i o n can be made about the c h a r a c t e r -i s t i c s o f the d i s t o r t i o n p r o f i l e . The p o s i t i v e t a p e r o f a p p r o x i m a t e l y 0 .3 ~ 0.4%/m ob se r v ed i n a l l c a ses r e p r e s e n t s a 40 ~ 45% i n c r e a s e ove r and above the v a l u e o f 0.9%/m c u r -r e n t l y employed i n b i l l e t moulds to compensate f o r s h r i n k a g e o f the s t r a n d owing to c o o l i n g . Th i s dynamic p o s i t i v e t a p e r t h a t i s due to t he l o n g i t u d i n a l t he rma l g r a d i e n t down the l e n g t h o f the mould may be p a r t l y r e s p o n s i b l e f o r wear i n b i l l e t mo u l d s . S e v e r a l i m p o r t a n t p o i n t s can be drawn f rom t h i s 2 4 6 c ompa r a t i v e s t udy o f h e a t - f l o w and d i s t o r t i o n i n b i l l e t mou l d s . In g e n e r a l , a r e d u c t i o n i n pe rmanent d i s t o r t i o n may be a c h i e v e d by r e d u c i n g the s t r e s s l eveT i n the h i gh s t r e s s r e g i o n s . In the case o f the b i l l e t raroulds i t appears t h a t t h i s may be a c c o m p l i s h e d by r e d u c i n g t h e l a t e r a l r e -s t r a i n t to the rma l e x pan s i o n by o p e r a t i n g w i t h l ower men i scus l e v e l s ( p r o v i d e d mould l e n g t h s a re a p p r o p r i a t e l y i n c r e a s e d to reduce the r i s k o f b r e a k - o u t s owing to t h e t h i n n e r s h e l l a t mould e x i t ) . Ano t he r a l t e r n a t i v e i s to m o d i f y the method o f mould c o n s t r a i n t s . A r e d u c t i o n i n a v e r age w a l l t empe ra -t u r e s t h a t wou ld r e s u l t f rom emp l oy i ng h i g h e r wa te r v e l o c i -t i e s i s b e n e f i c i a l f rom the s t a n d p o i n t o f r e d u c i n g peak d i s t o r t i o n , n e g a t i v e t a p e r and p l a s t i c s t r a i i m . E x c e s s i v e permanent d i s t o r t i o n i s u n d e s i r a b l e s i n c e i t l i m i t s mould l i f e and has been a s s o c i a t e d w i t h an i n c r e a s e i n l o n g i t u d i n a l 2 9 7 7 c o r n e r c r a c k i n g . ' Mould pe r f o rmance and! . i t s r e l a t i o n to q u a l i t y i s the s u b j e c t o f the f o l l o w i n g c h a p t e r . 6 .0 THE INFLUENCE OF MOULD DISTORTION ON BILLET QUALITY 6 .1 I n t r o d u c t i on A s i g n i f i c a n t outcome of the h e a t - f l o w and d i s t o r t i o n a n a l y s i s , as w e l l as the e x p e r i m e n t a l measurements i n an o p e r a t i n g b i 1 1 e t c a s t e r , i s t h a t under s t a n d a r d c o n d i t i o n s s u r f a c e b o i l i n g may o c cu r i n the c o o l i n g c h a n n e l . I t has been shown t h a t d u r i n g b o i l i n g mould t e m p e r a t u r e s i n the men i s cu s r e g i o n c y c l e between an upper and a l owe r v a l u e . From the p r e c e d i n g c h a p t e r i t was seen t h a t i n c r e a s i n g the c o o l i n g -wa t e r v e l o c i t y i s an e f f e c t i v e way o f s u p p r e s s i n g b o i l i n g w h i l e f o r low ca rbon g rades o f s t e e l (~0.1%), owing t o the low h e a t - t r a n s f e r r a t e s to the mou l d , b o i l i n g i s h i g h l y un -l i k e l y . These two o b s e r v a t i o n s a re i m p o r t a n t s i n c e t h e y l i n k l o n g i t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y i n b i l l e t moulds to t he b o i l i n g phenomena. For i n s t a n c e , i t has been ob se r ved t h a t c o r n e r c r a c k i n g and r h o m b o i d i t y i n c r e a s e 80 i n s e v e r i t y w i t h the ca rbon c o n t e n t o f the s t e e l and c u r -r e n t l y when c a s t i n g h i gh ca rbon g rades o f s t e e l (>0 .4%) , w a t e r - f l o w r a t e s a r e d r a s t i c a l l y r educed w i t h a marked r e -84 — 88 d u c t i o n or an e l i m i n a t i o n o f the p r o b l e m . ~ I f w a t e r q u a l i t y i s poor t h i s method o f q u a l i t y c o n t r o l p r o v e s un -s a t i s f a c t o r y as heavy d e p o s i t s are formed on the c o o l i n g -wa t e r f a c e wh ich i s accompan ied by e x c e s s i v e mould d i s t o r t i o n 247 84 and an i n c r e a s e i n the s e v e r i t y o f c o r n e r c r a c k i n g . These o b s e r v a t i o n s s t i m u l a t e d t h i n k i n g a l o ng the l i n e s t h a t c o r n e r c r a c k i n g and r h o m b o i d i t y are p r o b a b l y r e l a t e d to the b o i l i n g e v e n t , and l e d to the i n v e s t i g a t i o n d e s c r i b e d i n t h i s c h a p t e r . F o c u s s i n g a t t e n t i o n on the c h a r a c t e r i s t i c s o f the c o r n e r c r a c k , i t i s c l e a r t h a t they o r i g i n a t e i n the mould as they a re commonly obse r ved on the s u r f a c e o f the b i l l e t 2 l e a v i n g the mou l d . F i g . 77 i s a t y p i c a l example o f a l o n g i t u d i n a l c o r n e r c r a c k . These c r a c k s are g e n e r a l l y o f the o r d e r o f 1.6 mm deep but c r a c k s o f depth 6~7 mm have been 122 ob s e r v ed under u n f a v o u r a b l e c o n d i t i o n s . When l o n g i -t u d i n a l c o r n e r c r a c k s o c cu r c o n c u r r e n t l y w i t h r h o m b o i d i t y , t hey appea r more f r e q u e n t l y a t d i a g o n a l l y o p p o s i t e ob t u s e .angle c o r n e r s . 2 9 2 3 With the s t r i n g e n t q u a l i t y r e q u i r e m e n t s f o r s p e c i a l ba r q u a l i t y and o t h e r g rades o f s t e e l f o r s p e c i f i c a p p l i c a -t i o n s , t h e r e i s a g r e a t e r need to p r e v e n t the o c c u r r e n c e o f t he se d e f e c t s . Be i ng s u r f a c e d e f e c t s , the c r a c k s u r f a c e s o x i d i z e due to exposu re to the a tmosphere and do no t r ewe l d d u r i n g r o l l i n g , but appear i n the f i n a l p r o d u c t . T h e r e f o r e they have to be removed by hand g r i n d i n g or s c a r f i n g , to meet q u a l i t y r e q u i r e m e n t s . 249 F i g . 77 A L o n g i t u d i n a l Co rne r Crack i n a B i l l e t . 250 Before techniques can be p resc r ibed to e l i m i n a t e these d e f e c t s , a b e t t e r understanding of how they occur must be e s t a b l i s h e d . I t i s be l ieved t h a t l o n g i t u d i n a l corner c rack ing and rhomboid i ty are c l o s e l y r e l a t e d and uneven c o o l i n g has been i d e n t i f i e d as a pr imary f a c t o r lead ing to t h i s q u a l i t y p r o b l e m . 2 9 , 7 7 , 8 0 ' 8 1 Other f a c t o r s t h a t have been observed to have an i n f l u e n c e are mould 29 77 d i s t o r t i o n , mould wear and loss of t a p e r , ' and poor 29 80 8 al ignment between the mould and the submould assembly. ' ' The s e v e r i t y o f the problem has been known to increase when c a s t i n g smal le r s e c t i o n s 7 7 and has been found to increase w i t h mould l i f e . 2 9 ' 7 7 In t h i s chapter a mechanism has been proposed f o r the fo rma t ion of l o n g i t u d i n a l corner cracks and r h o m b o i d i t y , based on the b o i l i n g phenomena. Since uneven coo l i ng has been i d e n t i f i e d as a pr imary f a c t o r the p r e l i m i n a r y step has been to determine the r e l a t i o n s h i p between uneven c o o l i n g and b o i l i n g , and how t h i s could lead to r h o m b o i d i t y . R a t i o n a l i z a t i o n o f the observat ions of o ther workers coupled w i th the r e s u l t s of the above ana lys i s has helped e l u c i d a t e the mechanism o f . c r a c k f o r m a t i o n . The observed i n f l u e n c e of ope ra t i ng v a r i a b l e s on corner c rack ing and rhombo id i t y has a lso been eva luated in terms of the proposed mechanism. Based on the r e s u l t s of t h i s ana lys i s a technique Tab le XXXI D e t a i l s o f C a s t i n g Pa ramete r s o f Mould J V a r i a b l e Mould Length (mm) 812 .8 . Wa l l Thi ckness (mm) 9 .525 S e c t i o n S i z e (mm 2) 131x131 C a s t i n g Speed (m/mi n) 2 .03 . W a t e r - i n l e t Tempera ture (°C) 30 .0 Water Tempera tu re R i s e (°C) • 8 .5 Water V e l o c i t y (m/sec) 8 .15 Water channe l Gap (mm) 5.0 D i r e c t i o n o f Water f l o w up Mould Water P r e s s u r e ( e x i t ) (kPa) 241 .3 Mould M a t e r i a l DLP120+Ag Mould Free board (mm) 100 .0 252 f o r the s e l e c t i o n o f the c r i t i c a l o p e r a t i n g v a r i a b l e s to a v e r t c r a c k i n g has been s u g g e s t e d . 6 .2 The R e l a t i o n s h i p between N u c l e a t e B o i l i n g , Uneven C o o l i n g , and Rhombo id i t y Uneven c o o l i n g has been i d e n t i f i e d as the p r ima r y f a c t o r r e s p o n s i b l e f o r r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r 29 77 80 81 c r a c k i n g . ' ' ' S i n c e i t i s t hough t t h a t b o i l i n g i s l i n k e d to t he se d e f e c t s , t h i s i n v e s t i g a t i o n was c a r r i e d out to d e t e r m i n e the i n f l u e n c e o f n u c l e a t e b o i l i n g on the r e l a -t i v e c o o l i n g o f the f o u r f a c e s o f the b i l l e t . The mould d e s i g n a t e d as J i n the p r e v i o u s c h a p t e r was s e l e c t e d f o r the s t u d y . The r ea son f o r the s e l e c t i o n o f t h i s p a r t i c u l a r mou ld , i s t h a t when c a s t i n g h i g h ca rbon g rades o f s t e e l (>0.4%), f o r the c o n d i t i o n s o f mould J , w a t e r - f l o w r a t e s have to be d r a s t i c a l l y r educed to p r e v e n t the o c c u r r e n c e o f l o n g i t u d i n a l c o r n e r c r a c k s and r h o m b o i d i t y . T h e r e f o r e , i f i ndeed t h e r e i s a l i n k between w a t e r - f l o w r a t e s , b o i l i n g , uneven c o o l i n g , r h o m b o i d i t y and c o r n e r c r a c k i n g , then a h e a t - f l o w and d i s t o r t i o n a n a l y s i s o f t h i s p a r t i c u l a r mould may p r o v i d e some i n f o r m a t i o n on how t he se f a c t o r s are l i n k e d and h e l p e l u c i d a t e t he mechanism o f f o r m a t i o n o f t hese d e f e c t s . The d e t a i l s o f the mould and o p e r a t i n g c o n d i t i o n s a re g i v e n i n Tab l e XXX I . A h e a t - f l o w 253 a n a l y s i s o f the mould a t the wa te r f l o w - r a t e s , f o r a h i gh ca rbon s t e e l (>0.4%) was c a r r i e d out u t i l i z i n g the t r a n s i e n t h e a t - f l o w m o d e l . An " i n c u b a t i o n " e x ce s s t e m p e r a t u r e o f 15 .0°C was added to the c a l c u l a t e d b o i l i n g i n c e p t i o n t empe r a -t u r e ( Eq . 5 ) to s i m u l a t e the h y s t e r e s i s e f f e c t on the b o i l i n g c u r v e . I t was f ound t h a t b o i l i n g was i n i t i a t e d i n t he men i s cus r e g i o n , and the upper and l owe r l i m i t t e m p e r a t u r e p r o f i l e s o f the h o t - and c o l d - f a c e s o f the mould d u r i n g the b o i l i n g c y c l e a re shown i n F i g . 78 . The d i s t o r t i o n p r o f i l e c o r r e s p o n d i n g to each t empe r a t u r e d i s t r i b u t i o n i s shown i n F i g . 79 . The t h i r d cu r ve o f F i g . 79 i s the d i s t o r t i o n c a l c u l a t e d f o r a r educed t empe r a t u r e p r o f i l e d i s c u s s e d b e l o w . I t has been e s t a b l i s h e d t h a t the h ea t t r a n s f e r r e d f rom the s t e e l s t r a n d to the mould i s h i g h l y dependent upon 4 the w i d t h o f t h e a i r gap a t the s t r a n d / m o u l d i n t e r f a c e . I f changes i n mould d i s t o r t i o n due to the t e m p e r a t u r e o s c i l -l a t i o n s caused by b o i l i n g s i g n i f i c a n t l y a l t e r the gap w i d t h , t hen t h i s c o u l d have a p r o f ound i n f l u e n c e on the l o c a l hea t e x t r a c t i o n r a t e s . To a s s e s s the impo r t an ce o f t h i s e f f e c t , gap w i d t h s as a f u n c t i o n o f d i s t a n c e be low the men i s cus 1 24 were computed f rom a b i l l e t s u r f a c e t e m p e r a t u r e p r o f i l e and mould w a l l t empe r a t u r e s i n F i g . 78 . The l o c a l i z e d changes i n d i s t o r t i o n a s s o c i a t e d w i t h the ou tward movement o f the w a l l d u r i n g the t empe r a t u r e o v e r s h o o t i n the men i s cus E E ZJ o E CD SZ 0 100 200 300 o 4 0 0 CL O <u 500 SZ E 2 600 cu | 700 to b 800 900 0 Temperature profiles Upper temp limit Lower temp limit 1 1 J L 50 100 150 Temperature (°C) 200 250 78 P r e d i c t e d Ho t - and Co l d - F a c e Tempera tu re P r o f i l e s f o r Mould J a t H igh Wate r -F l ow Rates (wa te r v e l o c i t y = 8 .16 m / s e c ) . 255 E E 100 2 0 0 300 | 400[ 5 0 0 c o | 600f o 7 0 0 8 0 0 h -M e n i s c u s Distortion prof i les Upper temperature limit at incept ion of bo i l ing L o w e r temperature limit at inception of bo i l ing R e d u c e d heat f lux owing to bo i l ing cyc le 900' i 1 1 0.04 0.08 0.12 0.16 Distortion (mm) 0.20 F i g .79 Thermal D i s t o r t i o n P r o f i l e s o f Mould J A l ong the Cen t r e o f the Wal l a t the Ho t - F a c e a t D i f f e r e n t S tages o f the I n t e r m i t t e n t B o i l i n g C y c l e . 256 r e g i o n a t the i n i t i a t i o n o f b o i l i n g were o b t a i n e d f rom F i g . 79 . C a l c u l a t i o n s , g i v e n i n Append i x X, to de t e rm i ne the change i n h e a t - t r a n s f e r r a t e due to the i n c r e a s e d gap show t h a t a r e d u c t i o n o f up to 18% i s p o s s i b l e . Tab l e XXXII p r e s e n t s r e s u l t s o f the c a l c u l a t i o n s a t d i f f e r e n t p o s i t i o n s a l o n g the mou l d . The reduced h e a t - f l u x p r o f i l e i s compared w i t h the t i m e - a v e r a g e d hea t f l u x i n F i g . 8 0 . The r educed h e a t - f l u x p r o f i l e has two peaks i n s t e a d of one ; a phenomena 21 measured i n b i l l e t c a s t e r s by S i ngh and B l a z e k as w e l l as 2 3 Aseev e t a l . Based on the reduced h e a t - f l u x p r o f i l e a new t empe r a t u r e f i e l d was c a l c u l a t e d f o r the mould and i s shown i n F i g . 8 1 . The peak h o t - and c o l d - f a c e t e m p e r a t u r e s were f ound to be 198°C and 137°C. The l a t t e r i s w e l l be low t he i n c i p i e n t b o i l i n g p o i n t f o r the g i v e n f l o w c o n d i t i o n s wh i ch wou ld cause b o i l i n g to c e a s e . The l owe r t e m p e r a t u r e s i n the men i s cus r e g i o n would cause the mould w a l l t o move back towards t he s t r a n d . Th i s i s the t h i r d p r o f i l e i n F i g . 79 . T h i s causes a r e d u c t i o n i n gap w i d t h , the h e a t - e x t r a c t i o n r a t e s b e g i n to r i s e a g a i n , the t empe r a t u r e o f the mould w a l l i n c r e a s e s and b o i l i n g i s r e i n i t i a t e d i n the men i s cu s r e g i o n . As bubb l e s n u c l e a t e , grow and de tach f rom the mould s u r f a c e , the boundary l a y e r i s a g i t a t e d , l e a d i n g to enhanced h e a t -t r a n s f e r c o e f f i c i e n t s . Th i s l e a d s to a r e d u c t i o n i n mould w a l l t empe r a t u r e s and b o i l i n g i s a ga i n e x t i n g u i s h e d and the h o t - f a c e l o c a l l y moves back towards the s t r a n d . A b o i l i n g / n o - b o i l i n g c y c l e o f t h i s n a t u r e may o c c u r r e p e t i t i v e l y Table XXXII Reduced H e a t - f l u x Owing to a Change in Gap Size f o r High Carbon S tee ls  a t High Water F low- ra tes ^men t Tm Ts *ag * g 3 0 q q red Aq/q (mm) (sees) (°C) (°C) (mm) (mm) (kW/nf) (kW/m2) % 4.49 0.13 202.8 1506.0 0.011 0.0699 3690 3109 15.74 11.61 0.343 220.4 1504.0 0.011 0.0697 3645 3026 15.80 23.22 0.685 220.5 1500.0 0.011 0.0704 3594 3032 15.63 34.83 1.020 219.1 1496.9 0.012 0.0713 3545 2948 16.84 46.44 1.371 215.4 1492.8 0.013 0.0720 3510 2876 18.06 58.05 1.714 210.7 1488.8 0.013 0.0728 3476 2855 17.86 69.66 2.056 202.8 1467.2 0.013 0.0727 3440 2824 17.90 81.27 2.399 190.9 1440.1 0.013 0.0728 3397 2790 17.86 92.88 2.742 176.8 1413.1 0.012 0.0729 3354 2801 16.48 104.49 3.048 162.9 1385.3 0.010 0.0732 3306 2854 13.67 116.10 3.428 149.7 1358.7 0.008 0.0737 3246 2893 10.87 127.71 3.770 138.0 1331.3 0.007 0.0741 3186 2884 9.46 139.92 4.131 1 2 9 . 3 1304.2 0 .005 0 . 0748 3108 2900 6.69 150.93 4.456 122.5 1276.4 0.004 0.0760 3005 2846 5.29 162.54 4.799 116.5 1249.4 0.003 0.0772 2902 2789 8.89 174.15 5.142 110.6 1246.6 0.002 0.0807 2786 2716 2.61 185.76 5.485 104.8 1243.7 0.001 0.0848 2658 2626 1.20 g Q - or ig ina l gap width Ag - change in gap width ro 258 0 8 16 Dwell time ( s ) F i g . 80 A Compar i son of the T ime-Ave raged H e a t - F l u x P r o f i l e as a F u n c t i o n o f T i m e - i n - t h e - M o u l d (Dwe l l T ime) and the Reduced H e a t - F l u x P r o f i l e a t the Upper Tempera-t u r e L i m i t o f the I n t e r m i t t e n t B o i l i n g C y c l e . Meniscus 9 0 0 l I I I I I" I I I [ 1 0 50 100 150 '200 250 Temperature (°C) 81 The Ho t - and C o l d - F a c e Tempera tu re P r o f i l e s o f Mou ld J A s s o c i a t e d w i t h the Reduced H e a t - F l u x P r o f i l e Owing to I n t e r m i t t e n t B o i l i n g . 260 t h r o ughou t the h e a t , i t s p e r i o d depend ing on the " t he rma l r e sponse t i m e " o f the s t r a n d / m o u l d s y s t em . The e x p e r i m e n t a l measurements o f mould d i s t o r t i o n c a r r i e d ou t on the b i l l e t c a s t e r a t S t e l c o ' s Edmonton Works s u p p o r t t h i s p roposed mechan i sm. The p e r i o d i c n a t u r e of the s i g n a l o f t r a n s d u c e r 2 l o c a t e d ~ 65 mm below the m e n i s c u s , shown i n F i g . ' s 45 -50 , s u gge s t t h a t b o i 1 i n g / n o - b o i 1 i n g c y c l e s are r e p e a t e d l y o c c u r -r i n g c a u s i n g a p e r i o d i c movement o f the mould w a l l . T r a n s -duce r 3 on the o t h e r hand , wh i ch was l o c a t e d 100 mm f rom the bo t tom of the mou l d , and t h e r e f o r e w e l l o u t s i d e the b o i l i n g r e g i o n , d i s p l a y e d a s t e ad y s i g n a l , e x c ep t f o r the e f f e c t s o f mould o s c i l l a t i o n . The h e a t - f l o w a n a l y s i s p e r f o rmed f o r t h i s p a r t i c u l a r b i l l e t mou l d , c o n f i r m s t h a t b o i l i n g c o u l d i n i t i a t e i n the mould ( F i g . 5 2 ) . F o c u s s i n g a t t e n t i o n on the s t r a n d , i t ' i s e v i d e n t t h a t d u r i n g the i n t e r m i t t e n t b o i l i n g c y c l e , each t ime the mould w a l l moves away f rom the s t r a n d , the r e d u c t i o n i n h e a t -e x t r a c t i o n r a t e s wou ld cause the s t r a n d s u r f a c e t e m p e r a t u r e s to r e bound . The s t eep t empe r a t u r e g r a d i e n t s e s t a b l i s h e d in the s o l i d s h e l l d u r i n g the p r e v i o u s h a l f o f the t he rma l c y c l e m a i n t a i n s the h i g h h e a t - f l o w r a t e s f rom the i n t e r i o r to the s u r f a c e . Due to the r e d u c t i o n i n h e a t - e x t r a c t i o n a t the s u r -f a c e , a b u i l d - u p of hea t o c c u r s i n the s u r f a c e l a y e r s l e a d i n g to a r e h e a t i n g o f the s u r f a c e o f the s h e l l . T h i s c o n t i n u e s u n t i l the t empe r a t u r e g r a d i e n t s i n the s t r a n d r e a d j u s t to the new r a t e s o f h e a t - e x t r a c i o n . I t i s r e a s o n a b l e to e x p e c t t h a t 261 the t empe r a t u r e e x c u r s i o n i n the s u r f a c e l a y e r s o f the s t r a n d wou ld be h i g h e r than i n the mould due to the low t he rma l d i f f u s i v i t y o f the s t e e l s h e l l . T h e r e f o r e the s u r f a c e l a y e r s o f the s o l i d i f i e d s h e l l a l t e r n a t e l y coo l and r e h e a t i n u n i s on w i t h the mould w a l l t empe r a t u r e e x c u r s i o n i n t he men i s cus r e g i o n . The p e r i o d i c n a t u r e o f the b o i l i n g / n o - b o i l i n g c y c l e l e a d s t o t he p o s s i b i l i t y o f o u t - o f - p h a s e b o i l i n g on the f o u r f a c e s o f the mou l d . Th i s wou ld c l e a r l y r e s u l t i n uneven c o o l i n g of the b i l l e t as i t moves t h r ough the upper r e g i o n s o f the mou l d , s i n c e i t has been shown t h a t the h e a t - e x t r a c t i o n r a t e s c o u l d v a r y w i d e l y d u r i n g the b o i l i n g c y c l e . F o l l o w i n g the i d e a t h a t i n t e r m i t t e n t b o i l i n g can l e a d to uneven c o o l i n g , the nex t s t e p , i s to i l l u s t r a t e how t h i s e v en t c o u l d l e ad to r h o m b o i d i t y i n the mould and s o l i d i f i e d s t e e l s h e l l . Tab l e XXXI11 p r e s e n t s the t empe r a t u r e d i s t r i -b u t i o n a c r o s s the mould w a l l a t the upper and l owe r t empe r a -t u r e l i m i t s o f the b o i l i n g / n o - b o i l i n g c y c l e , a t a depth 27 .7 mm below the men i s cus wh i ch c o r r e s pond s to the l o c a t i o n o f peak t e m p e r a t u r e s . The p l a n e - s t r e s s f i n i t e - e l e m e n t model d e ve l oped e a r l i e r to c a l c u l a t e the t he rma l e x p a n s i o n o f a t r a n s v e r s e s l i c e o f the mould i n c h a p t e r 4.0 i s u t i l i z e d f o r t h i s a n a l y s i s . The e n t i r e c r o s s - s e c t i o n o f the mould was d i s c r e t i z e d i n t o a s e r i e s o f t r i a n g u l a r e l emen t s as shown 262 Tab l e XXX I I I Tempera ture D i s t r i b u t i o n A c r o s s the T h i c k n e s s  o f the Wa l l a t the Upper and Lower L i m i t o f  B o i l i n g D i s t a n c e f rom the Co l d Face (mm) Tempera tu re D i s t r i b u t i o n A c r o s s the Wal l 27 .7 mm below the meni scus (°C) Upper L i m i t Lower L i m i t 0 .0 156 .0 137 .0 4 .7625 193 .0 167 .0 9 .525 231 .0 198 .0 263 i n F i g . 82 the boundary c o n d i t i o n s were s e l e c t e d so as to ensu re t h a t the sys tem c o u l d no t undergo r i g i d - b o d y t r a n s -l a t i o n o r r o t a t i o n . Four p a r t i c u l a r cases were i n v e s t i -g a t e d , and a re l i s t e d be l ow . ( i ) two o p p o s i t e f a c e s a t the upper l i m i t o f the b o i l i n g / n o - b o i l i n g c y c l e w i t h h o t - and c o l d - f a c e t empe r a t u r e s assumed to be equa l to the h i g h e r v a l u e s i n Tab l e X X X I I I , and the o t h e r two f a c e s a t the n o - b o i l i n g l i m i t o f the c y c l e w i t h the l ower t empe r a t u r e s g i v e n i n the same t a b l e . ( i i ) two a d j a c e n t f a c e s b o i l i n g as above , and the o t h e r two f a c e s q u i e s c e n t . ( i i i ) one f a c e b o i 1 i n g , t h r e e f a c e s n o - b o i l i n g . ( i v ) t h r e e f a c e s b o i l i n g , one f a ce n o - b o i l i n g . F i g . 83 s c h e m a t i c a l l y shows the mould c o n f i g u r a t i o n s f o r a 2 130 x 130 mm mould w i t h the r e l a t i v e movement o f each c o r n e r i n p a r e n t h e s i s . The d i f f e r e n c e i n the l e n g t h o f the d i a g o -n a l s e x p r e s s e d as a p e r c en t age o f the o r i g i n a l l e n g t h o f the d i a g o n a l s f o r each case i s 0 . 0 , 0 . 0 5 4 , 0 . 0 , and 0 .0 r e -s p e c t i v e l y . I t i s e v i d e n t t h a t o n l y i f b o i l i n g t a ke s p l a c e on two a d j a c e n t f a c e s wou ld the two d i a g o n a l s d i f f e r i n 264 Y • X F i g . 82 D i s c r e t i z a t i o n o f a T r a n s v e r s e S e c t i o n o f the Mould Cont inuum to Compute the D i s t o r t i o n Due to Non Un i f o rm Tempera tu res i n t he Mould by a P l a n e - S t r e s s F i n i t e - E l e m e n t A n a l y s i s . 265 (0 .027 ,0 .384) H (0.027,0.012) (0.010,0.315) J-Q ( 0 , 0 ) ( 0 3 2 7 , 0 3 8 4 ) ( i i ) ( 0 3 2 7 , 0 0 1 2 ) (0.375.Q376) (Q3I5,0.010) 83 A S chema t i c Diagram o f a 130x130 mm B i l l e t Mould C o n f i g u r a t i o n s f o r ( i ) Two O p p o s i t e Faces o f the Mould a t the Upper Tempera tu re L i m i t o f the I n t e r m i t t e n t B o i l i n g C y c l e and the Othe r Two a t t h e Lower Tempera tu re L i m i t . ( i i ) B o i l i n g on Two A d j a c e n t F a c e s , No B o i l i n g on the Other Two. ( c o n t . ) ( -a i85 ,CX327) H -11- (OJ 85 ,0 .327) C ~ ( - 0 1 5 5 , 0 . 0 2 3 ) -a— C ( i i i ) (0.155 , a 0 2 3 ) (-0.181,0.385), H (0.181,0385) H: H (-0.151,0.016) —11-C ( i v ) (0.151,0.016) 83 A S chema t i c D iagram o f a 130x130 mm' C o n f i g u r a t i o n s f o r ( i i i ) B o i l i n g Only on One Face ( i v ) B o i l i n g on Three F a c e s . B i l l e t Mould 267 l e n g t h , wh i ch wou ld r e s u l t i n a change i n c o r n e r gap s i z e . The impo r t an ce o f the s i z e o f c o r n e r gaps f rom the s t a n d -p o i n t o f c o r n e r c r a c k i n g w i l l be d i s c u s s e d i n the subsequen t s e c t i o n . A l s o i t i s o n l y case ( i i ) t h a t l e a d s to two ob tu se o p p o s i t e c o r n e r s o f the s e c t i o n . S i n c e c o r n e r c r a c k s most 29 77 12 3 o f t e n appear on the ob tuse o p p o s i t e ang l e c o r n e r s , ' ' i t may be assumed t h a t case ( i ) , ( i i i ) and ( i v ) have ve r y l i t t l e e f f e c t on r h o m b o i d i t y l e a d i n g to c o r n e r c r a c k s . S t r a n d s u r f a c e t empe r a t u r e s f o l l o w a t empe r a t u r e e x c u r s i o n s i m i l a r to the mou l d , as d i s c u s s e d e a r l i e r , e x c e p t t h a t due to the low the rma l d i f f u s i v i t y o f s t e e l , the magn i -tude o f the r e l a t i v e changes i n t empe r a t u r e wou ld be h i g h e r i n the s t r a n d . T h e r e f o r e c o n s i d e r i n g the case where b o i l i n g i s o c c u r r i n g on two a d j a c e n t f a c e s o f the mou l d , the shape o f the t r a n s v e r s e s e c t i o n o f the s o l i d i f i e d s h e l l c o u l d be e x p e c t e d to r e semb le t h a t o f the mould e x c e p t t h a t the r e l a t i v e changes i n the d i a g o n a l s wou ld be g r e a t e r . F i g . 84 s c h e m a t i c a l l y shows the e x p e c t e d s t r a n d and mould shapes due to b o i l i n g on two a d j a c e n t f a c e s o f the mou l d . The d i s -t o r t i o n has been e x a g g e r a t e d to i l l u s t r a t e t h a t the r e s u l t i n g r h o m b o i d i t y o f the s t r a n d and mould w a l l wou ld i n c r e a s e the c o r n e r gaps a t ob tu se ang l e c o r n e r s and dec r ea se the gaps a t a cu t e ang l e c o r n e r s . The impo r t an ce o f t h i s f rom the s t a n d p o i n t o f c o r n e r c r a c k i n g w i l l be d i s c u s s e d l a t e r . 268 F i g . 8 4 A S chema t i c Diagram o f B i l l e t and Mou ld C o n f i g u r a t i o n s Wi th and W i t hou t Uneven C o o l i n g , ( i ) No Rhombo i d i t y ( i i ) Rhombo i d i t y o f S t r a n d and Mould Showing an I n c r e a s e i n Co rne r Gaps a t Obtuse Ang l e C o r n e r s . 269 6 .3 Mechanism f o r the Fo rma t i on o f L o n g i t u d i n a l  Co rne r C rack s For c r a c k s to o r i g i n a t e a t the c o r n e r s o f the s o l i d i -f i e d s h e l l , t e n s i l e s t r e s s e s t h a t exceed the s t r e n g t h of the s h e l l must be g e n e r a t e d . F i g . 85 s c h e m a t i c a l l y i l l u s t r a t e s t h r e e e ven t s i n the mould t h a t c o u l d l e a d to t e n s i l e s t r e s -ses i n t he s h e l l . F i g . 8 5 ( i ) d e p i c t s the t e n s i l e s t r e s s e s i n t he s h e l l nea r the s o l i d i f i c a t i o n f r o n t , owing to the combined e f f e c t o f f e r r o s t a t i c p r e s s u r e o ve r the un suppo r t ed c o r n e r r e g i o n and s h r i n k a g e due t o c o o l i n g . F i g . 8 5 ( i i ) shows the n a t u r e o f the s t r e s s e s t h a t a r i s e a t the ob tuse a n g l e d c o r n e r s o f the rhombo id s h e l l t h a t r e s u l t s f rom un -even c o o l i n g . Here aga i n as the ang l e a t the c o r n e r i n c r e a s e s t e n s i l e s t r e s s e s a re g ene r a t e d i n the l a y e r s a d j a c e n t t o the s o l i d i f i c a t i o n f r o n t w h i l e c omp r e s s i v e s t r e s s e s are g e n e r a t e d i n t he s u r f a c e l a y e r s . I t was p o i n t e d out i n the p r e c e d i n g s e c t i o n t h a t s i n c e the r h o m b o i d i t y o f the s o l i d i f i e d s t e e l s e c t i o n wou ld be g r e a t e r than t h a t o f the mou l d , gaps a t the o b t u s e ang l e c o r n e r s wou ld i n c r e a s e w i t h i n c r e a s i n g r h o m b o i d i t y w h i l e gaps a t the a cu t e a n g l e d c o r n e r s wou ld de-c r e a s e . Th i s wou ld l e a d t o r e h e a t i n g o f the s u r f a c e l a y e r s a t ob tuse ang l e d c o r n e r s wh i ch wou ld r e s u l t i n a d d i t i o n a l t e n s i l e s t r e s s e s near the s o l i d i f i c a t i o n f r o n t . Th i s i s s c h e m a t i c a l l y shown i n F i g .85 ( i i i ) o f the t h r e e f a c t o r s / / / ^ U t T I T r T I H (1) ( i i ) UU ' ( i i i ) 85 A S chema t i c D iagram I l l u s t r a t i n g F a c t o r s t h a t Cause T e n s i l e S t r e s s e s Near the S o l i d i f i c a t i o n F r o n t ( i ) F e r r o s t a t i c P r e s s u r e Over the Unsuppo r t ed Co rne r and S h r i n k a g e o f the S h e l l ( i i ) the I n c r e a s e i n Ang l e a t Obtuse Ang l e Co r ne r s ( i i i ) R ehea t i n g o f the S h e l l a t Obtuse Ang le Co rne r s Due to the I n c r e a s e i n Co rne r Gap S i z e . 271 l e a d i n g to t e n s i l e s t r e s s e s i n the s o l i d i f i e d s h e l l , the l a s t two c e r t a i n l y appear to have a s i g n i f i c a n t e f f e c t as the s e v e r i t y o f l o n g i t u d i n a l c o r n e r c r a c k i n g i n c r e a s e s w i t h 2 9 7 7 8 0 8 1 r h o m b o i d i t y , ' and the c r a c k s tend to appea r a t N 29 7 7 12 3 ob tuse ang l ed c o r n e r s o f the c a s t s e c t i o n . 5 ' 29 Perminov e t a l . have i n v e s t i g a t e d the p rob lem o f rhom-b o i d i t y i n a 280 x 280 mm b loom and have r e p o r t e d t h a t the ave rage t h i c k n e s s o f the b i l l e t s h e l l a t the ob tuse ang l e s i s 10 ~ 15% l e s s than t h a t i n the a cu te a n g l e s , and a l s o o b s e r v ed t h a t the s u r f a c e t empe r a t u r e o f the ob tu se c o r n e r s o f the b loom a t the mould e x i t was much h i g h e r (1100 -1300°C) than a t the a cu te a n g l e s (850 - 1000°C ) . These o b s e r v a t i o n s s u p p o r t the t h e o r y t h a t t he gaps a t ob tu se a n g l e d c o r n e r s are g r e a t e r than a t the a cu te a ng l e d c o r n e r s l e a d i n g to r e h e a t i n g o f the s u r f a c e l a y e r s and a t h i n n e r s h e l l due t o the reduced hea t e x t r a c t i o n r a t e s . Based on t h i s q u a l i t a t i v e a n a l y s i s o f the d i s t o r t i o n o f t e n s i l e s t r e s s e s i n the s o l i d i f i e d l a y e r , and the causes o f t h e i r o r i g i n , i t i s p r oposed t h a t l o n g i t u d i n a l c o r n e r c r a c k s a re a r e s u l t o f ho t t e a r i n g i n i n t e r d e n d r i t i c r e g i o n s v e r y c l o s e to t he s o l i d i f i c a t i o n f r o n t due to h i gh t e n s i l e s t r e s s e s . S u p p o r t i v e e v i d e n c e f o r t h i s ho t t e a r i n g mechanism i s p r o v i d e d by the o b s e r v a t i o n s o f many wo rke r s i n the f i e l d . 1 25 Ake t a and U s h i j i m a c onduc t ed meta l 1 og r a ph i c e x a m i n a t i o n o f l o n g i t u d i n a l c o r n e r c r a c k s and have r e p o r t e d t h a t c r a c k s 272 appea r between p r ima r y d e n d r i t e s and the s u r f a c e s o f c r a c k s a re smooth wh i ch i s i n d i c a t i v e o f h o t - t e a r i n g a t h i g h -t e m p e r a t u r e s . In some cases they ob se r ved t h a t the c r a c k s 123 were f i l l e d w i t h l i q u i d m e t a l . Baumann and Lopmann, i n c o n n e c t i o n w i t h c r a c k s a p p e a r i n g a t ob tuse ang l e c o r n e r s due to r h o m b o i d i t y , have found t h a t the c r a c k s were f i l l e d w i t h a meta l phase h a v i n g a h i g h c o n c e n t r a t i o n o f e l emen t s 29 t h a t t end to s e g r e g a t e . Perminov e t a l . a l s o ob se r ved a h i g h c o n c e n t r a t i o n o f i n c l u s i o n s and s e g r e g a t e s i n the i n -t e r n a l c o r n e r c r a c k s accompany ing r h o m b o i d i t y i n b l ooms . T h i s e v i d e n c e c l e a r l y i n d i c a t e s t h a t c r a c k s o c cu r i n i n t e r -d e n d r i t i c r e g i o n s where l i q u i d f i l m s c o n t a i n i n g a h i g h c o n -c e n t r a t i o n o f e l emen t s t h a t s e g r e g a t e are p r e s e n t . These l i q u i d f i l m s , wh i ch do not f r e e z e u n t i l t empe r a t u r e s w e l l be low the s o l i d u s a re r ea ched a re t hough t to l owe r the s t r e n g t h and d u c t i l i t y o f the s t e e l r e n d e r i n g i t h i g h l y s u s c e p t i b l e to c r a c k f o r m a t i o n . C rack s wh i ch form near the s o l i d u s i n i n t e r d e n d r i t i c r e g i o n s wou ld t h e r e f o r e be e x pe c t e d to have smooth s u r f a c e s due to the p re sence o f l i q u i d f i l m s as has been ob s e r v ed w i t h l o n g i t u d i n a l c o r n e r c r a c k s . 1 2 ^ Othe r e v i d e n c e wh i ch i n d i c a t e s t h a t l o n g i t u d i n a l c o r n e r c r a c k s o r i g i n a t e a t the i n t e r i o r o f the s h e l l i s the ob s e r v ed r e l a t i o n s h i p between the w i d t h o f the c r a c k a t the s u r f a c e to i t s d e p t h . I t has been r e p o r t e d t h a t c r a c k s t h a t appear wide a t the s u r f a c e are s h a l l o w , i n c o n t r a s t to the ve r y f i n e c r a c k s wh ich are deep and o f t e n d i f f i c u l t 122 to d e t e c t and remove. Th i s s ugge s t s t h a t the c r a c k s a re w i d e r a t the i n t e r i o r o f the s h e l l than a t the s u r f a c e . T h i s i s c l e a r l y i l l u s t r a t e d i n F i g . 86 wh i ch i s a t r a n s v e r s e s e c t i o n o f a b i l l e t showing a l o n g i t u d i n a l c o r n e r c r a c k . These o b s e r v a t i o n s l e nd s u p p o r t to the p r oposed t h e o r y t h a t c r a c k s o r i g i n a t e a t the i n t e r i o r o f the s h e l l ; i f the magn i -tude o f the t e n s i l e s t r e s s e s a re s u f f i c i e n t l y h i g h they p r opaga t e a c r o s s the t h i c k n e s s to the s u r f a c e o f the b i l l e t . The nex t s t e p i s to de t e rm i ne the p o s i t i o n i n the mould a t wh i ch t he se c r a c k s o r i g i n a t e . Assuming t h a t the i n s i d e t i p o f the c r a c k i s l o c a t e d a t the s o l i d i f i c a t i o n f r o n t , the l e n g t h o f the c o r n e r c r a c k i s a measure o f the s h e l l t h i c k -ness a t the t ime o f i n i t i a t i o n . L o n g i t u d i n a l c o r n e r c r a c k s 122 a r e commonly o f the o r d e r o f 1.6 mm (1_") deep- F i g . 87 8 shows a p l o t o f c a l c u l a t e d s h e l l t h i c k n e s s o f a b i l l e t a t 8? the m i d - p l a n e as a f u n c t i o n o f t ime based on Savage and P r i t c h a r d s h e a t - f l u x c u r v e . A s h e l l t h i c k n e s s o f the o r d e r o f 1.6 mm c o r r e s pond s to a t ime o f 2 s e e s . S i n c e S i n gh and B l a z e k ' s h e a t - f l u x p r o f i l e used i n t h i s a n a l y s i s i s h i g h e r than Savage and P r i t c h a r d ' s , the s h e l l t h i c k n e s s i n the men i s cus r e g i o n i s u n d e r p r e d i c t e d i n F i g . 87 . T h e r e f o r e the t ime f o r c r a c k f o r m a t i o n i s l e s s t han 2.0 s e e s . For example f o r mould J , m u l t i p l y i n g by the c a s t i n g speed o f 2.032 m/min, c r a c k i n i t i a t i o n i s p i n - p o i n t e d t o w i t h i n a F i g . 86 Photograph of a Transverse Sec t ion of a B i l l e t Showing a 133 L o n g i t u d i n a l Corner Crack i n the S h e l l . F i g . 8 7 P r e d i c t e d B i l l e t S u r f a c e Tempera tu re and S h e l l T h i c k n e s s as a F u n c t i o n o f T i m e - i n - t h e - M o u l d (Dwe l l T i m e ) . 8 2 depth o f a p p r o x i m a t e l y 65 mm below the m e n i s c u s . From F i g . 79 wh i ch i s a p l o t o f d i s t o r t i o n as a f u n c t i o n o f d i s t a n c e be low the mould f o r J , i t appea r s t h a t t h i s l o c a t e s c r a c k i n i t i a t i o n i n the r e g i o n where r h o m b o i d i t y due to i n t e r m i t -t e n t b o i l i n g and uneven c o o l i n g c o u l d be e x p e c t e d to o c c u r by the mechanism p roposed e a r l i e r . I t i s a l s o e v i d e n t t h a t t h i s r e g i o n i s the l ower end o f the " n e g a t i v e zone " o f the d i s t o r t e d mou l d . The a i r gap between t he s t r a n d and the mould wou ld p r o g r e s s i v e l y i n c r e a s e o v e r t h i s r e g i o n due t o bo th the " n e g a t i v e - t a p e r " e f f e c t as w e l l as the s h r i n k a g e o f the s t r a n d ow ing to c o o l i n g . These e v e n t s c o u l d l e a d t o a r e d u c t i o n i n hea t e x t r a c t i o n r a t e s and poor s u p p o r t o f the s t r a n d by the mou l d , wh i ch may f a c i l i t a t e c o r n e r c r a c k i n g . Exam in ing b i l l e t s u r f a c e t e m p e r a t u r e s i n t h i s r e g i o n ( Tab l e XXXII) i t i s e v i d e n t t h a t t hey f a l l w i t h i n the h i g h -t empe r a t u r e zone o f low d u c t i l i t y . (142.0°C to s o l i d u s ) . In t h i s z o n e , sma l l t e n s i l e s t r a i n s o f 0 .2 ~ 0.3% a re s u f -f i c i e n t t o cause h o t - t e a r i n g , t h e r e b y i n c r e a s i n g the s u s c e p t i b i l i t y to c r a c k i n g . 6 6 .4 E f f e c t o f C o o l i n g - w a t e r V e l o c i t y on Rhombo i d i t y  and L o n g i t u d i n a l Co rne r C r a c k i n g In many b i l l e t c a s t i n g o p e r a t i o n s a common p r a c t i c e i s to reduce wa t e r f l ow when c a s t i n g h i g h c a r bon g rades o f 277 84 8ft s t e e l (>0.4%). For t hese g rades o f s t e e l the use o f normal f l o w r a t e s r e s u l t s i n a h i g h e r i n c i d e n c e o f l o n g i -t u d i n a l c o r n e r c r a c k s and r h o m b o i d i t y , w h i l e r educed f l o w -r a t e s have been found to g i v e b e t t e r q u a l i t y b i l l e t s . T h i s p r a c t i c e has been found to be s u c c e s s f u l o n l y i f w a t e r q u a l i t y was good because o t h e r w i s e heavy d e p o s i t i o n o f s c a l e and accompany ing mould d i s t o r t i o n l e d t o a wo r s en i ng o f the p r o b -84 l e m . Th i s s ugge s t s t h a t a t low f l o w - r a t e s b o i l i n g must be o c c u r r i n g i n the wa t e r c h a n n e l ; c l e a r l y t h e r e f o r e c o o l i n g -wa t e r f l o w - r a t e s have a s i g n i f i c a n t i n f l u e n c e on q u a l i t y v i a i t s e f f e c t on b o i l i n g i n the c h a n n e l s . For mould J , a s t a n d a r d wa te r v e l o c i t y o f 8 .2 m/sec was found to y i e l d u n s a t i s f a c t o r y s u r f a c e q u a l i t y and a wa t e r v e l o c i t y o f 4 .58 m/sec was employed when c a s t i n g g rades o f s t e e l g r e a t e r than 0.4% c a r b o n , w i t h a r e d u c t i o n i n the i n c i d e n c e o f l o n g i t u d i n a l c o r n e r c r a c k s and rhom-b o i d i t y . A h e a t - f l o w and d i s t o r t i o n a n a l y s i s p r e v i o u s l y c a r r i e d ou t f o r mould J w i t h the s t a n d a r d wa te r v e l o c i t y o f 8 .2 m/sec f o r a h i gh ca rbon s t e e l has shown t h a t i n t e r -m i t t e n t b o i l i n g c o u l d t a ke p l a c e i n the channe l l e a d i n g to uneven c o o l i n g , r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k i n g . In an a t t emp t to unde r s t and why the use o f low f l o w -r a t e s has p roven s u c c e s s f u l f rom the s t a n d p o i n t o f q u a l i t y , a h e a t - f l o w a n a l y s i s f o r mould J w i t h the r educed f l o w - r a t e was p e r f o r m e d . The s o l i d l i n e s i n F i g . 88 a re h o t - a nd n oo co Cu "O r t -S fD r— C L 0 -•• s. o r+ S fD CU C L r+ fD Z3Z ~i O 1 r+ m i o s. zo Cu r+ fD to Cu 3 C L o o _ J C L I -n cu o fD fD • 3 fD d cu Q r+ - i E Q 3 s 5* ~S O fD to - h O -s C L CD o O Distance from the top of the mould ( m m ) CD CD - N ! cn O l OJ ro — O o o o o o o o o o o o o o o o o o o 8ZZ 279 c o l d - f a c e t empe r a t u r e p r o f i l e s f o r the upper and l owe r t he rma l f i e l d a t the i n i t i a t i o n o f b o i l i n g . The c o r r e s p o n d -i ng d i s t o r t i o n p r o f i l e s a re shown i n F i g . 89 . The r e d u c -t i o n i n hea t e x t r a c t i o n i n the men i scus r e g i o n a s s o c i a t e d w i t h l o c a l i z e d mould w a l l movement due to b o i l i n g as de-s c r i b e d e a r l i e r was c a l c u l a t e d and the r e s u l t s a re p r e s e n t e d i n Tab le XXXIV. . The reduced h e a t - f l u x p r o f i l e so o b t a i n e d i s compared w i t h the t i m e - a v e r a g e d h e a t - f l u x p r o f i l e f o r a h i g h ca rbon hea t i n F i g . 90 . A second h e a t - f l o w a n a l y s i s u s i n g the r educed h e a t - f l u x p r o f i l e r e v e a l s t h a t b o i l i n g w i l l c o n t i n u e to t a ke p l a c e and the r e s u l t i n g h o t - and c o l d -f a c e t empe r a t u r e p r o f i l e s f o r t h i s case a re shown i n F i g . 88 as d o t t e d l i n e s f o r c ompa r i s on w i t h the p r e v i o u s r e s u l t s . Under t he se c i r c u m s t a n c e s b o i l i n g wou ld be c o n t i n u o u s , and the t empe r a t u r e o s c i l l a t i o n s a s s o c i a t e d w i t h the b o i l i n g e v e n t , wou ld be p r o g r e s s i v e l y damped out e l i m i n a t i n g the p o s s i b i l i t y o f an i n t e r m i t t e n t b o i l i n g sequence i n the mou l d . The l a t t e r was shown to be u n d e s i r a b l e as i t c o u l d l e a d t o o u t - o f - p h a s e b o i l i n g o f a d j a c e n t f a c e s o f the mould r e s u l t i n g i n uneven c o o l i n g , r h o m b o i d i t y , and l o n g i t u d i n a l c o r n e r c r a c k i n g . Thus low w a t e r - f l o w r a t e s c o u l d be e xpe c t ed to improve the q u a l i t y o f the b i l l e t s . I f wa te r q u a l i t y i s poor then d e p o s i t s would form a t the c o o l i n g - w a t e r i n t e r f a c e d u r i n g the c o n t i n u o u s b o i l i n g c y c l e and i t has been shown i n the p r e v i o u s c h a p t e r t h a t the l o c a l r e d u c t i o n i n h e a t -t r a n s f e r c o e f f i c i e n t accompany ing s c a l e f o r m a t i o n would l e ad Table XXXIV Reduced H e a t - f l ux Owing to a Change in Gap Size f o r High Carbon S tee ls a t Low Water F low- ra tes d men t Tm T s * Ag * q q red Aq/q (mm) (sees) (°c) (°C) (mm) (mm) (kW/m ) (kW/m2) % 4.49 0.13 200.2 1506.0 .029 .0700 3690 2161.3 41.0 11.61 0.343 200.0 1504.0 .019 .0708 3645 2666.8 27.0 23.22 0.685 199.0 1500.0 .024 • 0716 3594 2389.3 34.0 34.83 1.020 198.9 1496.9 .027 .0724 3545 2222.9 37.0 46.44 1.371 198.8 1492.8 .029 .0729 3510 2113.7 40.0 58.05 1.714 198.8 1488.8 .028 .0734 3476 2150.0 38.0 69.66 2.056 198.8 1467.2 .028 .0729 3440 2107.7 39.0 81.27 2.399 198.8 1440.1 .028 .0723 3397 2081.4 39.0 92.88 2.742 198.8 1413.1 .026 .0716 3354 2136.0 36.0 104.49 3.048 198.5 1385.3 .022 .0710 3306 2281.6 31.0 116.10 3.428 197.9 1358.7 .018 .0707 3246 2419.58 25.0 127.71 3.770 197.2 1331.3 .013 .0704 3186 2597.68 18.0 139.91 4.131 198.5 1304.2 .013 .0704 3108 2534.0 18.0 150.93 4.456 203.8 1276.4 .017 .0706 3005 2281.4 24.0 162.54 4.799 201.4 1249.4 .015 .0715 2902 2293.2 21.0 174.15 5.142 201.2 1246.6 .008 .0742 2786 2485.6 11.0 185.76 5.485 198.2 1243.7 .004 .0778 2658 2521.3 5.0 Ag 9 o - change in gap width - o r ig ina l gap width ro CO o 0 0.04 0.08 0.12 0.16 0.20 Distortion (mm) 89 Thermal D i s t o r t i o n P r o f i l e s o f Mould J A l ong the Cen t r e o f the Wa l l a t the Ho t - F a ce at the I n c e p t i o n o f B o i l i n g f o r Low Wa te r - F l ow R a t e s . 282 4000 3000 E X 3 zz\ 2000 o X 1000 1 1 i i i i Heat flux profiles — \ High carbon Reduced heat — ft \ flux owing to ' t \ boiling cycle ' » / • \ v \ / \ — \ — — — i i i i i i 8 12 16 Dwell time ( s ) 2 0 24 F i g . 90 A Compar i son o f the T ime -Ave raged H e a t - F l u x P r o f i l e as a F u n c t i o n o f T i m e - i n - t h e - M o u l d (Dwe l l Time) and the Reduced H e a t - F l u x P r o -f i l e A s s o c i a t e d w i t h the Tempera tu re Ove r -shoo t a t the I n c e p t i o n o f B o i l i n g . 283 t o a marked i n c r e a s e i n mould w a l l t e m p e r a t u r e s . The e x c e s -s i v e d i s t o r t i o n and i n c r e a s e d s e v e r i t y o f c r a c k i n g ob s e r v ed i n p r a c t i c e under t hese c i r c u m s t a n c e s stems f rom the h i gh w a l l t e m p e r a t u r e s . An a l t e r n a t i v e t e c h n i q u e to p r e v e n t uneven c o o l i n g and the a t t e n d a n t p r o b l e m s , i s to r a i s e c o o l i n g w a t e r v e l o c i t i e s w e l l above the c r i t i c a l v a l u e a t wh i ch the c o l d -f a c e p e a k - t e m p e r a t u r e c o u l d i n i t i a t e b o i l i n g . A p r e l i m i n a r y s u r v e y o f s e v e r a l b i l l e t c a s t i n g o p e r a t i o n s s u g g e s t s t h a t such a p r a c t i c e i s b e n e f i c i a l . For i n s t a n c e , a t c a s t e r ' K ' ( see p r e v i o u s c h a p t e r ) , o p e r a t i n g p e r s onne l have i n d i c a t e d t h a t t h e i r b i l l e t s are r e l a t i v e l y f r e e f rom l o n g i t u d i n a l c o r n e r c r a c k s and r h o m b o i d i t y . They a l s o r e p o r t l o ng mould l i v e s and t h e i r moulds show ve r y l i t t l e e v i d e n c e o f permanent d i s t o r t i o n . A h e a t - f l o w a n a l y s i s f o r t h i s p a r t i c u l a r mould o p e r a t i o n was pe r f o rmed i n c h a p t e r 5 and the r e s u l t s r e v e a l e d t h a t b o i l i n g wou ld not o c c u r i n the c o o l i n g - w a t e r channe l owing to the h i gh f l ow v e l o c i t i e s employed (10 .71 m / s e c ) . Fu r t he rmo re f rom F i g . 76 , wh i ch i s a c ompos i t e p l o t o f d i s -t o r t i o n p r o f i l e s o f s e v e r a l o p e r a t i o n s , i t i s e v i d e n t t h a t mould K e x h i b i t s the l e a s t amount o f d i s t o r t i o n . Company G wh i ch a l s o employs h i g h - w a t e r v e l o c i t i e s , a l s o r e p o r t s a r e l a t i v e l y t r o u b l e f r e e o p e r a t i o n . As a r e s u l t o f the p r o f ound i n f l u e n c e wa te r v e l o c i t y appea r s to have on c o r n e r c r a c k i n g , r h o m b o i d i t y and mould d i s t o r t i o n i t i s s ugge s t ed 284 t h a t a s u f f i c i e n t l y h i g h - w a t e r v e l o c i t y be employed to a v e r t b o i l i n g i n the c h a n n e l . To s i m p l i f y the task o f s e l e c t i n g the a p p r o p r i a t e v e l o c i t y , the c r i t i c a l wa te r v e l o c i t y f o r d i f f e r e n t mould wa t e r p r e s s u r e s and wa te r i n l e t t empe r a t u r e s have been computed and the r e s u l t s a re g r a p h i c a l l y shown i n F i g . 9 1 . These r e s u l t s a re a p p l i c a b l e to a mould w i t h a w a l l t h i c k n e s s o f 9 .525 mm and a channe l gap o f 5.0 mm wh i ch r e p r e s e n t s s t a n d a r d i n d u s t r i a l v a l u e s . I t i s e v i d e n t t h a t wa t e r i n l e t t empe r a t u r e s have o n l y a sma l l e f f e c t on the c r i t i c a l wa t e r v e l o c i t y a t wh i ch the b o i l i n g / n o - b o i l i n g t r a n s i t i o n o c c u r s , i n c o n t r a s t to mould wa te r p r e s s u r e s . In c u r r e n t p r a c t i c e , a back p r e s s u r e o f 240 kPa i s m a i n t a i n e d on the mould to ensu re t h a t the wa te r channe l i s f i l l e d d u r i n g o p e r a t i o n . At t hese wa te r p r e s s u r e s the t r a n s i t i o n o c c u r s a t a wa t e r v e l o c i t y o f 8 .2 ~ 8.6 m/sec . Water v e l o c i t i e s commonly employed i n b i l l e t moulds are i n the 7 ~ 8 m/sec range wh ich p l a c e s the sys tem ve r y c l o s e to the b o i l i n g / n o - b o i l i n g t r a n s i t i o n on the c h a r t . The c l o s e r the sys tem i s to the t r a n s i t i o n the more s u s c e p t i b l e i t i s to an i n t e r m i t t e n t b o i l i n g c y c l e d e s c r i b e d e a r l i e r , wh i ch c o u l d l e a d to r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k i n g . Both i n c r e a s i n g and d e c r e a s i n g wa te r v e l o c i t i e s above or be low the c r i t i c a l v e l o c i t y en su r e s a s t a b l e o p e r a t i o n e x c e p t t h a t i n the l a t t e r case mould pe r fo rmance c o u l d d e t e r i o r a t e i f w a t e r q u a l i t y i s poor and s c a l e forms on the mou l d . 285 in E o o <u > co o 5 c o o o 8 Water inlet temperature • I5°C o 30°C A 45°C Boiling \ D No boiling ^ \ 1 1 150 350 1 550 750 F i g . 91 Mould water exit pressure (kPa) A C o o l i n g - W a t e r V e l o c i t y / W a t e r E x i t P r e s s u r e Diagram Demarca t i ng t h e B o i l i n g / N o - B o i l i n g Regions f o r B i l l e t Mould O p e r a t i o n when C a s t i n g High Carbon Grades o f S t e e l . 6.5 The I n f l u e n c e o f S t e e l C ompo s i t i o n on L o n g i t u d i n a l Co rne r C r a c k i n g Carbon c o n t e n t o f the s t e e l has been found t o have an e f f e c t on the i n c i d e n c e o f l o n g i t u d i n a l c o r n e r c r a c k s and r h o m b o i d i t y . I t has been r e p o r t e d t h a t the p rob l em 80 i n c r e a s e s i n s e v e r i t y w i t h i n c r e a s i n g c a r b o n . The i n -f l u e n c e o f c a rbon on r h o m b o i d i t y and c o r n e r c r a c k i n g i s e x p l a i n e d on the b a s i s o f the ob se r ved i n f l u e n c e o f c a r bon on the h e a t - e x t r a c t i o n r a t e s . A h e a t - f l o w a n a l y s i s f o r mou ld J f o r a 0.1% ca rbon s t e e l , based on t h e • h e a t - f 1 u x p r o f i l e i n F i g . 19 , has shown t h a t c o l d - f a c e t e m p e r a t u r e s a r e w e l l be low the t empe r a t u r e r e q u i r e d to i n i t i a t e b o i l i n g . The h o t - and c o l d - f a c e t empe r a t u r e p r o f i l e s can be seen i n F i g . 9 2 . The d i s t o r t i o n p r o f i l e f o r t h i s t he rma l f i e l d was a l s o computed and i s compared w i t h t h a t o b t a i n e d f o r the "same c o n d i t i o n s f o r a h i gh ca rbon hea t i n F i g . 9 3 . I t was f ound t h a t f o r the 0.1% ca rbon s t e e l the mould d i d no t de fo rm p l a s t i c a l l y . A mould wa t e r v e l o c i t y p r e s s u r e d i ag ram d e -m a r c a t i n g the b o i l i n g / n o - b o i l i n g t r a n s i t i o n i s shown i n F i g . 94 . Thus i t i s seen t h a t a t s t a n d a r d wa t e r v e l o c i t i e s o f 7 ~ 8 m/sec the sys tem i s f a r f rom the b o i l i n g r e g i o n . T h e r e f o r e the i n i t i a t i o n o f an i n t e r m i t t e n t b o i l i n g c y c l e i s h i g h l y u n l i k e l y . Th i s a n a l y s i s e x p l a i n s the low i n -c i d e n c e o f l o n g i t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y when c a s t i n g low ca rbon g rades o f s t e e l ( 0 . 1%) . 0 50 100 150 200 250 .92 Temperature (°C) Ho t - and Co l d - F a c e Temperature P r o f i l e s o f a B i l l e t Mould f o r a Low Carbon (0 .1%) S t e e l . 288 0-04 F i g . 9 3 0.20 0.08 0.12 0.16 Distortion (mm) The In f l uence of Steel Composition on the Thermal D i s t o r t i o n of Mould J at the Centre of the Wall as a Funct ion o f Distance Down the Mould. 289 in o o CD > CD a 5 CP c o o o i — r Water inlet temperature • — — I5°C 0 — 30 °C A — — 45°C Boiling \ \ X No boiling 1 1 1 150 350 550 750 Mould water pressure (kPa) F i g . 94 A C o o l i n g - W a t e r V e l o c i t y / W a t e r E x i t P r e s s u r e Diagram Demarca t i ng the B o i l i n g / N o - B o i l i n g Reg ions f o r B i l l e t Mould O p e r a t i o n When C a s t i n g Low-Carbon. Grades o f S t e e l . 290 6.6 The Use . o f S o f t - c o o l i n g P r a c t i c e s to C o n t r o l Rhombo id i t y and L o n g i t u d i n a l Co rne r C r a c k i n g A s o f t - c o o l i n g p r a c t i c e has been deve l oped by 83 Funabash i S t e e l Works to c o n t r o l l o n g i t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y . Th i s i n v o l v e s the mach i n i ng o f h o r i z o n t a l g rooves i n the mould w a l l a t the c o o l i n g -wa t e r i n t e r f a c e nea r the men i s cus . The s u c ce s s o f t h i s p r a c t i c e can be e x p l a i n e d on the b a s i s o f the p roposed mechanism f o r l o n g i t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y . The g rooves on the mould w a l l wou ld de c r ea se the h y s t e r e s i s i n the b o i l i n g cu r ve by p r o v i d i n g vapour t r a p -p i ng c a v i t i e s o r by a c t i n g as n u c l e a t i n g s i t e s . Th i s f o l l o w s f rom o b s e r v a t i o n s made by s e v e r a l i n v e s t i g a t o r s t h a t an i n c r e a s e i n roughness o f b o i l i n g s u r f a c e s d e c r e a s e s the i n i t i a l s u pe r h ea t r e q u i r e d to i n i t i a t e b o i l i n g on aged s u r -97 98 faces. ' Under such c i r c u m s t a n c e s the t empe r a t u r e o v e r -shoo t a t the i n c e p t i o n o f b o i l i n g i s s u b s t a n t i a l l y r e d u c e d . A h e a t - f l o w a n a l y s i s f o r mould J w i t h o u t the i n c u b a -t i o n b a r r i e r o f 1 5 . 0 °C , employed to s i m u l a t e h y s t e r e s i s , r e v e a l e d t h a t t h e r e are no tempera tue o s c i l l a t i o n s a t the i n i t i a t i o n o f b o i l i n g , and the the rma l f i e l d conve rges to a " s t a b l e b o i l i n g " c o n d i t i o n . The h o t - and c o l d - f a c e t empe ra -t u r e p r o f i l e s f o r t h i s case are shown i n f i g . 95 . The F i g . 95 H o t - and Co ld-Face Tempera tu re P r o f i l e s P r e d i c t e d f o r a S o f t - C o o l e d B i l l e t Mou ld . 292 absence o f l a r g e o v e r s h o o t s i n t empe ra t u r e a t the i n i t i a -t i o n o f b o i l i n g p r e c l u d e s the p o s s i b i l i t y o f an i n t e r m i t t e n t b o i l i n g c y c l e t h a t c o u l d l e a d to uneven c o o l i n g and r e l a t e d q u a l i t y p r o b l e m s . The r e d u c t i o n i n the i n c i d e n c e o f l o n g i -t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y i n t h i s ca se c o u l d be a t t r i b u t e d to the r e l a t i v e l y s t a b l e mould shape owing t o c o n t i n u o u s b o i l i n g . 6 .7 The I n f l u e n c e o f S e c t i o n S i z e on Rhombo i d i t y  and L o n g i t u d i n a l Co rne r C r a c k i n g I t has been remarked t h a t r h o m b o i d i t y and c o r n e r c r a c k i n g a re more s e ve r e f o r s m a l l e r s e c t i o n s i z e s . A g e n e r a l o b s e r v a t i o n t h a t can be made w i t h r e g a r d to mou lds used f o r c a s t i n g s m a l l e r b i l l e t s , i s t h a t the w a l l t h i c k n e s s i s o f t e n l e s s t han f o r l a r g e r s e c t i o n s . I t i s t h o u g h t t h a t the r ea son f o r the i n c r e a s e i n r h o m b o i d i t y and l o n g i -t u d i n a l c o r n e r c r a c k i n g f o r s m a l l s e c t i o n b i l l e t moulds i s due t o the t h i n w a l l r a t h e r than the e f f e c t o f s e c t i o n s i z e i t s e l f . Th i s t o p i c i s examined be l ow . F i r s t the e x t e n t o f r h o m b o i d i t y i s c a l c u l a t e d f o r a mould w i t h a l a r g e s e c t i o n s i z e m a i n t a i n i n g the w a l l t h i c h n e s s the same as b e f o r e . F i g . 96 s c h e m a t i c a l l y shows the mould c o n f i g u r a t i o n o f a rhombo id b i l l e t o f s e c t i o n s i z e 2 181 x 181 mm due to b o i l i n g on two a d j a c e n t f a c e s . The (0.0I0,0435)_ 0 (0,0) (0.521,0.521) (0.434,0.005) F i g . 96 A Schemat i c Diagram o f a 181x181 mm1" B i l l e t Mould C o n f i g u r a t i o n Showing the R e l a t i v e Movement o f the Co rne r s Due to B o i l i n g on Two A d j a c e n t Faces o f the Mould N o - B o i l i n g on the O the r Two. ro CO CO 294 t empe r a t u r e d i s t r i b u t i o n o f the two h o t - f a c e s i s assumed to c o r r e s p o n d to the upper t empe r a t u r e d i s t r i b u t i o n i n Tab l e X X X I I I . T h i s c o r r e s pond s to the upper t empe r a t u r e l i m i t o f the mould d u r i n g the b o i l i n g c y c l e . The t empe r a t u r e d i s t r i -b u t i o n o f t h e two c o l d - f a c e s were a l s o o b t a i n e d f rom Tab l e XXX I I I wh i ch i s the l ower t empe r a t u r e l i m i t o f the b o i l i n g c y c l e . The d i f f e r e n c e i n the l e n g t h o f the d i a g o n a l s ex -p r e s s e d as a p e r c en t age o f the o r i g i n a l d i a g o n a l l e n g t h i s 0.053% wh i ch i s o n l y s l i g h t l y d i f f e r e n t f rom the v a l u e o f 0.05% c a l c u l a t e d p r e v i o u s l y f o r a 130 x 130 mm mou l d . T h e r e f o r e i t appea r s t h a t s e c t i o n s i z e by i t s e l f does not have a marked i n f l u e n c e on r h o m b o i d i t y . R e t u r n i n g to the s u b j e c t o f t h i n mou l d s , a h e a t -f l o w and d i s t o r t i o n a n a l y s i s was pe r f o rmed f o r a 111 x 111 2 mm mould h a v i n g a 5.875 mm w a l l . A l l o t h e r c a s t i n g c o n -d i t i o n s were assumed t o be i d e n t i c a l t o t h o s e o f mould J . The h o t - and c o l d - f a c e t empe r a t u r e p r o f i l e s can be seen i n F i g . 97 . B o i l i n g i s i n i t i a t e d i n the c o o l i n g - w a t e r c h anne l s and as b e f o r e a t empe r a t u r e o s c i l l a t i o n r e s u l t s i n t he men i s cus r e g i o n . Owing to the t h i n n e r w a l l , t h e d i f f e r e n c e i n the t empe r a t u r e s between the upper and l owe r t empe r a t u r e d i s t r i b u t i o n d u r i n g b o i l i n g was found to be g r e a t e r than f o r the t h i c k e r w a l l . The the rma l d i s t o r t i o n o f the t h i n mould has been c a l c u l a t e d f o r t he two t e m p e r a t u r e f i e l d s and i s shown i n F i g . 9 8 . The l o c a l i z e d ou tward movement o f the w a l l 0 50 100 150 200 250 Temperature (°C) Fig.97 Hot- and Cold-Face Temperature P ro f i l e s Predicted fo r a B i l l e t Mould with a Wall Thickness of 5.875 mm with a l l Other Conditions Corresponding to Mould J. 296 100 2 0 0 3 0 0 £ E ZJ I 4 0 0 c o 5 0 0 | 6 0 0 b 7 0 0 8 0 0 9 0 0 Meniscus Steel side of mould D i s t o r t i o n p r o f i l e s - U p p e r temperature limit at incept ion of boi l ing •Lower t e m p e r a t u r e limit at incept ion of boil ing 1 1 0.04 0.08 0.12 Distortion (mm) 0.16 F i g . 98 Thermal D i s t o r t i o n P r o f i l e s o f t he Th in Mould A l ong the Cen t re o f the Wa l l a t t he Ho t - F a c e a t the I n c e p t i o n o f B o i l i n g . Q20 d u r i n g the b o i l i n g c y c l e r e s u l t s i n an i n c r e a s e d gap . T ab l e XXXV p r e s e n t s the change i n h e a t - t r a n s f e r r a t e a t d i f f e r e n t p o s i t i o n s a l o ng the mould wh i ch shows t h a t r e d u c t i o n s o f 30% a re p o s s i b l e wh i ch i s s i g n i f i c a n t l y g r e a t e r than the maximum r e d u c t i o n o f 18% i n h e a t - t r a n s f e r r a t e s , in . the t h i c k e r mou l d . Th i s i s f u r t h e r i l l u s t r a t e d i n F i g . 99 where the r educed h e a t - f l u x p r o f i l e i s compared w i t h t h a t o b t a i n e d f o r the 9 .525 mm t h i c k mou l d . The t h i r d cu r ve i n the F i g . 99 c o r -r e sponds to the t ime ave raged h e a t - f l u x p r o f i l e . T h e r e f o r e r e h e a t i n g i n the s t e e l s h e l l a s s o c i a t e d w i t h the sudden r e d u c t i o n i n h e a t - e x t r a c t i o n r a t e s owing t o b o i l i n g wou ld be h i g h e r f o r t h i n n e r mou l d s . Under these c i r c u m s t a n c e s i f b o i l i n g o c c u r s on two a d j a c e n t f a c e s o f the mould the r h o m b o i d i t y o f the b i l l e t s e c t i o n wou ld be g r e a t e r than f o r a t h i c k e r mould owing t o the l a r g e r d i f f e r e n c e i n t e m p e r a t u r e between the hot and c o l d - f a c e s o f the s t r a n d . The h o t - and c o l d - f a c e t empe r a t u r e p r o f i l e s a re p l o t t e d i n F i g . 100 , f o r the r educed h e a t - f l u x p r o f i l e and i t i s seen t h a t b o i l i n g i s s u p p r e s s e d . The mould r h o m b o i d i t y was computed as b e f o r e based on the h i g h - and l o w - t e m p e r a t u r e l i m i t s o f the b o i l i n g c y c l e . F i g . 101 s c h e m a t i c a l l y shows the mould c o n f i g u r a t i o n w i t h the r e l a t i v e movement o f the c o r n e r s i n p a r e n t h e s i s . The r h o m b o i d i t y o f the mould e x p r e s -sed as the d i f f e r e n c e between the l e n g t h o f the d i a g o n a l s Table XXXV Reduced H e a t - f l u x Owing to a Change in Gap Size f o r a Thin Wall Mould . ^men (mm) t (sees) T m (°C) T (4) * (mm) * 9 o (mm) q (kW/mZ) q red (kW/m2) Aq/q % 4.49 0.13 186.06 1506.0 .014 .0708 3690 2960.3 20 11.61 0.343 187.72 1504.0 .015 .0714 3645 2879.2 21 23.22 0.685 185.81 1500.0 .020 .0724 3594 2601.2 28 34.83 1.020 187.97 1496.9 .022 .0731 3545 2478.1 30 46.44 1.371 189.96 1492.8 .022 .0735 3510 2459.4 30 58.05 1.714 190.66 1488.8 .019 .0739 3476 2582.3 26 69.66 2.056 190.21 1467.2 .015 .0735 3440 2737.9 20 81.27 2.399 189.64 1440.1 .013 .0728 3397 2790.3 18 92.88 2.742 189.01 1413.1 .010 .0722 3354 2889.4 14 104.49 3.048 188.16 1385.3 .008 .0717 3306 2937.1 11 116.10 3.428 187.27 1358.7 .006 .0714 3246 2973.2 8 127.71 3 770 185.67 1331.3 .004 .0712 3186 3007.0 6 139.91 4.131 181.21 1304.2 .002 .0715 3108 3021.0 3 150.93 4.456 176.13 1276.4 .002 .0725 3005 2922.10 3 162.54 4.799 170.74 1249.4 .001 .0735 2902 2862.5 1 174.15 5.142 164.65 1246.6 0 .0768 2786 2786.0 0 185.76 5.485 158.35 1243.7 0 .0808 2658 2658.0 0 9 0 _ o r ig ina l gap width Ag - change in gap width ro CO 299 4000 3000 CVJ E 3 2000 o cu 1000 0 T ~ ~ I — r ~ f J I L I I I I I 1 1 H e a t flux p r o f i l e s High c a r b o n Reduced heat f l u x for 9 . 5 2 5 m m t h i c k mould wal l — • — R e d u c e d heat f lux for 5 . 8 7 5 m m th ick mould wall I I i I 1 I I L 8 12 16 Dwell time (s ) 20 24 F i g . 99 A Compar i son o f the Reduced H e a t - F l u x P r o f i l e s P l o t t e d as a F u n c t i o n o f T i m e - i n - t h e - M o u l d (Dwe l l Time) f o r B i l l e t Moulds o f Two D i f f e r e n t Wa l l T h i c k n e s s e s . 9 0 0 l I I 1 1 I I 1 I I I 0 5 0 1 0 0 150 2 0 0 2 5 0 Temperature (°C) CO F i g . 1 0 0 Ho t - and C o l d - F a c e Tempera ture P r o f i l e s o f a Th in Wa l l Mould A s s o c i a t e d § w i t h the Reduced H e a t - F l u x P r o f i l e Owing to I n t e r m i t t e n t B o i l i n g . ( - 0 . 0 1 0 , 0 2 2 0 ) ( 0 , 0 ) ( 0 . 2 8 0 , 0 . 2 7 9 ) ( 0 . 2 2 0 , - 0 . 0 1 0 ) F i g . 101 A Schemat i c Diagram o f a 111x111 mm B i l l e t Mould Con-f i g u r a t i o n Showing the R e l a t i v e Movement o f the Co rne r s Due to B o i l i n g on Two Ad j a c en t Faces o f the Mou l d , No-B o i l i n g on the Other Two. co o 302 d i v i d e d by t h e . o r i g i n a l l e n g t h o f a d i a g o n a l ( e x p r e s s e d as a p e r c e n t a g e ) was found to be 0 .0450 wh i c h i s s l i g h t l y s m a l l e r than the v a l u e s o f 0 . 053 and 0:050 c a l c u l a t e d f o r 2 2 181 x 181 mm and 130 x 130 mm square s e c t i o n s . In g e n e r a l i f the r h o m b o i d i t y o f the mould r e l a t i v e t o the r h o m b o i d i t y o f the s t r a n d i s s m a l l , then c o r n e r gaps a t ob tu se ang l e s wou ld be h i g h e r than i f the mould r h o m b o i d i t y were com-p a r a t i v e l y l a r g e . T h e r e f o r e a h i g h e r i n c i d e n c e o f c o r n e r c r a c k s c o u l d be e x p e c t e d i n the f o rmer c a s e . S umma r i z i n g , the r e s u l t s o f t h i s a n a l y s i s i n d i c a t e t h a t t h i n n e r moulds have l a r g e r w a l l t empe r a t u r e o s c i l l a t i o n s d u r i n g b o i l i n g than t h i c k e r ones wh i ch l e a d to l a r g e r changes i n h e a t - e x t r a c t i o n r a t e s . Th i s may be p r i m a r i l y r e s p o n s i b l e f o r the i n c r e a s e i n r h o m b o i d i t y and h i g h e r i n c i d e n c e o f c o r n e r c r a c k s ob se r ved i n p r a c t i c e . 6 .8 The I n f l u e n c e o f O the r F a c t o r s on L o n g i t u d i n a l Co rne r C r a c k i n g I t has been s t a t e d t h a t mould l i f e i n c r e a s e s rhom-29 77 b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k i n g . ' Tn c h a p t e r 4 . 0 , i t was demons t r a t ed t h a t a f t e r a s i n g l e t he rma l c y c l e , the permanent d i s t o r t i o n was g r e a t e r at t he c o r n e r s than a t the m i d - f a c e . Th i s o f c ou r se stems f rom the h i g h e r r e s t r a i n t to t he rma l e x p a n s i o n a t the c o r n e r s wh ich r e s u l t s i n y i e l d i n g 303 and g r e a t e r p l a s t i c s t r a i n s . Th i s a n a l y s i s s ugge s t s t h a t the shape o f the c o r n e r s may be p r o g r e s s i v e l y a l t e r e d w i t h i n -c r e a s i n g s e r v i c e l i f e chang i ng the gap w i d t h ove r t h a t r e g i o n . Th i s may be the reason f o r a h i g h e r i n c i d e n c e o f c o r n e r c r a c k s w i t h mould l i f e . O the r f a c t o r s t h a t have been r e c o g n i z e d as h a v i n g a d e l e t e r i o u s e f f e c t on l o n g i t u d i n a l c o r n e r c r a c k i n g a re im -2 79 12 5 p r ope r c o r n e r r a d i i ' * and h i g h e r c a s t i n g t empe ra -t u r e s . ' The h i g h e r i n c i d e n c e o f c o r n e r c r a c k i n g a s s o c i a -t ed w i t h these two mould c o n d i t i o n s does no t seem to be l i n k e d to the rhombo id c o n d i t i o n o f the b i l l e t . T h e r e f o r e bo th h i gh p o u r i n g t empe r a t u r e s and h i gh c o r n e r r a d i i must somehow a l t e r c o r n e r gaps enhanc i ng the r e h e a t i n g e f f e c t wh i ch c o u l d l e a d to h i gh t e n s i l e s t r e s s e s near the s o l i d i -. f i c a t i o n f r o n t . Measurements o f s h e l l t h i c k n e s s i n the t r a n s v e r s e s e c t i o n s o f b r e a k - o u t s h e l l s o f b i l l e t s nea r the mould e x i t f o r t h r e e p o u r i n g t empe r a t u r e s (1577 , 1593 , 1622°C) have 21 • • been p e r f o r m e d . I t has been ob se r ved t h a t an i n c r e a s e i n p o u r i n g t empe r a t u r e r e s u l t e d i n a t h i n n e r s h e l l a t the 21 c o r n e r s than a t the m i d - f a c e , i n d i c a t i v e o f r educed h e a t -e x t r a c t i o n r a t e s . However i n the absence o f more d e t a i l e d knowledge on the v a r i a t i o n s o f h e a t - f l o w i n the c o r n e r r e g i o n s i t i s no t p o s s i b l e to c a l c u l a t e the gap w i d t h 304 v a r i a t i o n s or advance more d e f i n i t i v e e x p l a n a t i o n s on the e f f e c t o f h i gh p o u r i n g t e m p e r a t u r e . A d d r e s s i n g the p rob l em o f imp rope r c o r n e r r a d i i , i t i s p o s s i b l e t h a t w i t h d e c r e a s ed c o r n e r r a d i i , the e f f e c t s o f t w o - d i m e n s i o n a l h e a t - f l o w may be enhanced as compared w i t h the more rounded c o r n e r s . In the f o rme r c a s e , t empe ra -t u r e s a t the c o r n e r may d e c l i n e f a s t e r than i n the l a t t e r case r e d u c i n g the t ime the s h e l l s u r f a c e t e m p e r a t u r e s a re i n the h i g h t empe r a t u r e zone o f low d u c t i l i t y (50 ~ 100 be low s o l i d u s ) . 6 Once ou t o f t h i s r e g i o n , i n wh i ch the s t e e l i s h i g h l y s u s c e p t i b l e to c r a c k i n g , the c o r n e r i s a b l e to w i t h s t a n d h i g h e r s t r a i n s t h a t may be caused by r e h e a t i n g r e s u l t i n g f rom l owe r h e a t - e x t r a c t i o n r a t e s . 6 . 9 Summary I t has been shown t h a t f o r c e r t a i n mould w a t e r - f l o w c o n d i t i o n s an i n t e r m i t t e n t b o i l i n g c y c l e can be i n i t i a t e d i n the mou l d . The p e r i o d i c n a t u r e o f the e ven t may r e s u l t i n o u t - o f - p h a s e b o i l i n g on the f o u r f a c e s o f the mou l d . Due t o the wide v a r i a t i o n i n h e a t - e x t r a c t i o n r a t e s i n the men i s cus r e g i o n d u r i n g the b o i l i n g c y c l e , the f o u r f a c e s o f the b i l l e t wou ld be c o o l e d u n e v e n l y . I t has been shown t h a t when two a d j a c e n t f a c e s o f the mould a re c o o l e d more than the o t h e r two the mould c a v i t y becomes rhombo id i n s h ape . R a t i o n a l i z i n g 305 the e ven t s i n the mould d u r i n g an i n t e r m i t t e n t b o i l i n g c y c l e , i t has been p o s t u l a t e d t h a t the s t r a n d s u r f a c e t e m p e r a t u r e s i n the men i s cus r e g i o n f o l l o w a t e m p e r a t u r e e x c u r s i o n p a t t e r n s i m i l a r to t he mould due to the v a r y i n g h e a t - e x t r a c t i o n r a t e s and r e s u l t a n t r e h e a t i n g o f the s u r f a c e . T h i s wou ld cause the s t r a n d to a l s o a t t a i n a rhombo id shape wh ich wou ld be more p ronounced than the mou l d , s i n c e due t o the l owe r t he rma l d i f f u s i v i t y o f the s t e e l , h i g h e r t e m p e r a t u r e d i f f e r e n t i a l s wou ld e x i s t between a d j a c e n t f a c e s owing t o the d i f f e r e n t c o o l i n g p a t t e r n s . These e ven t s can l e a d t o an i n c r e a s e i n c o r n e r gaps a t d i a g o n a l l y o p p o s i t e ob tuse a n g l e c o r n e r s o f the rhombo id s e c t i o n and a de c r ea se i n c o r n e r gaps a t a cu t e ang l e c o r n e r s . T e n s i l e s t r e s s e s are g ene r a t e d n e a r the s o l i d i f i c a t i o n f r o n t due t o r e h e a t i n g o f the ob tuse c o r n e r s owing to i n -c r e a s e i n c o r n e r gaps and a l s o due to the i n c r e a s e i n a n g l e . F e r r o s t a t i c p r e s s u r e a c t i n g ove r the u n s u p p o r t e d c o r n e r , as w e l l as s h r i n k a g e of the s h e l l , i n c r e a s e s t h e t e n s i l e s t r e s s l e v e l nea r the s o l i d i f i c a t i o n f r o n t . Based on t h i s a n a l y s i s and the o b s e r v a t i o n s o f o t h e r wo rke r s i n the f i e l d , i t i s p roposed t h a t l o n g i t u d i n a l c o r n e r c r a c k s o c cu r by h o t - t e a r i n g i n i n t e r d e n d r i t i c r e g i o n s ve r y c l o s e to the s o l i d i f i c a t i o n f r o n t , when the b i l l e t i s 306 i n the h i g h - t e m p e r a t u r e zone o f low d u c t i l i t y . The c r a c k s wh i ch have been ob se r ved to be o f the o r d e r o f 1.6 mm deep , most p r o b a b l y o r i g i n a t e w i t h i n 2 sees o f the m e n i s c u s , w h i c h c o r r e s pond s to the r e g i o n where uneven c o o l i n g and r h o m b o i d i t y wou ld be e xpe c t ed to o c cu r by the p r opo sed mechani sm. C o o l i n g - w a t e r v e l o c i t y has been shown to have a p r o -found i n f l u e n c e on r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k i n g . A mould wa t e r v e l o c i t y / p r e s s u r e d i ag ram demarca -t i n g the b o i l i n g / n o - b o i l i n g r e g i o n s has been d e v e l o p e d . I t has been shown t h a t i f the mould i s o p e r a t i n g a t a wa t e r v e l o c i t y nea r the t r a n s i t i o n zone the sys tem c o u l d go i n t o an i n t e r m i t t e n t b o i l i n g sequence t h a t c o u l d r e s u l t i n uneven c o o l i n g l e a d i n g to r h o m b o i d i t y and c o r n e r c r a c k i n g . The ob s e r v ed i n f l u e n c e o f ca rbon and mould w a l l t h i c k n e s s on r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k i n g have been r a t i o n a l i z e d on the b a s i s o f the p roposed mechan i sm. The l owe r h e a t - t r a n s f e r r a t e s a s s o c i a t e d w i t h the 0.1% ca rbon s t e e l , r e s u l t i n low mould w a l l t empe r a t u r e s f o r the c o o l i n g -wa te r v e l o c i t i e s o f 7 ~ 8 m/sec c u r r e n t l y emp l o y ed . The sys tem has t h e r e f o r e been shown to be w e l l ou t o f t he b o i l i n g r e g i o n , e l i m i n a t i n g the p o s s i b i l i t y o f an i n t e r m i t t e n t b o i l i n g c y c l e and the a t t e n d a n t p r o b l ems . Th in w a l l moulds used f o r c a s t i n g sma l l s e c t i o n s , t end to a m p l i f y the 307 t empe r a t u r e o s c i l l a t i o n s d u r i n g ' b o i 1 i n g . Th i s r e s u l t s i n w i d e r v a r i a t i o n s i n h e a t - e x t r a c t i o n r a t e s d u r i n g the b o i l i n g c y c l e . The s t r a n d tends to become more rhombo id under t he se c i r c u m s t a n c e s , w i t h a r e s u l t a n t i n c r e a s e i n c o r n e r gaps a t ob tu se ang l e c o r n e r s . Th i s c o u l d r e s u l t i n a h i g h e r i n -c i d e n c e o f c o r n e r c r a c k i n g and a g r e a t e r degree o f rhom- . b o i d i t y . 7.0 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 7 .1 Summary Two -d imens i ona l h e a t - f l o w models based on the f i n i t e -d i f f e r e n c e and f i n i t e - e l e m e n t methods have been d e ve l o ped t o c a l c u l a t e the s t e a d y - s t a t e t empe r a t u r e d i s t r i b u t i o n i n the l o n g i t u d i n a l m i d - p l a n e o f the w a l l o f a b i l l e t or s l a b mou l d . A t r a n s i e n t t w o - d i m e n s i o n a l h e a t - f l o w model wh ich employs the i m p l i c i t a l t e r n a t i n g d i r e c t i o n f i n i t e - d i f f e r e n c e method has been f o r m u l a t e d e x p r e s s l y to s t udy the i n f l u e n c e o f l o c a l s u r f a c e b o i l i n g a t the c o o l i n g - w a t e r i n t e r f a c e on w a l l t empe r a -t u r e s i n the c e n t e r - p l a n e o f a b i l l e t m o u l d . The models were v a l i d a t e d by compar ing p r e d i c t e d w a l l t e m p e r a t u r e s w i t h e x p e r i -m e n t a l l y measured t empe r a t u r e s i n a b i l l e t mould r e p o r t e d i n the 1 i t e r a t u r e . Based on the p r e d i c t e d the rma l f i e l d s two p r e l i m i n a r y a n a l y s e s were conduc t ed to o b t a i n a b e t t e r u n d e r s t a n d i n g o f the c h a r a c t e r i s t i c s o f the the rma l d i s t o r t i o n o f b i l l e t mou l d s . To a s s e s s the impo r t an ce o f e l a s t i c b end i ng on the o v e r a l l d i s t o r t i o n p r o f i l e , a p l a t e a n a l y s i s f o r t he mould w a l l was f i r s t c a r r i e d o u t . The r e s u l t s were compared w i t h the ave rage t he rma l e x p a n s i o n o f a t r a n s v e r s e s l i c e o f the mould c a l c u -l a t e d u s i n g a p l ane s t r e s s f i n i t e - e l e m e n t m o d e l . Based on the 308 309 r e s u l t s o f t hese two a n a l y s e s a t h r e e - d i m e n s i o n a l e l a s t o -p l a s t i c model was s e l e c t e d f o r the f i n a l d i s t o r t i o n c a l c u l a -t i o n s . B i l l e t mould w a l l movements were m o n i t o r e d u s i n g L i n e a r V a r i a b l e D i s p l a c emen t T r an sdu ce r s f o r s e v e r a l h e a t s i n s u c c e s s i o n i n an i n d u s t r i a l i n s t a l l a t i o n . Mould d i s t o r t i o n s c a l c u l a t e d by the t h r e e - d i m e n s i o n a l e l a s t o - p l a s t i c model were compared to the e x p e r i m e n t a l measurements and f ound to be i n f a i r l y good ag reement . The e x p e r i m e n t a l measurements l e d to the h y p o t h e s i s t h a t n u c l e a t e b o i l i n g may be o c c u r r i n g Q a t the m o u l d / c o o l i n g - w a t e r i n t e r f a c e . The i n f l u e n c e of a v a r i e t y o f o p e r a t i n g v a r i a b l e s on the t he rma l f i e l d i n s l a b and b i l l e t moulds has been i n -v e s t i g a t e d . . The r e s u l t s o f t h i s s t udy s e r v e d as a b a s i s f o r i d e n t i f y i n g mould c o n d i t i o n s t h a t c o u l d l e a d to a l o s s o f s t r e n g t h and e x c e s s i v e d i s t o r t i o n t h rough s o f t e n i n g . G u i d e -l i n e s f o r m a t e r i a l s e l e c t i o n f o r b i l l e t and s l a b moulds have been o u t l i n e d on the b a s i s o f t hese r e s u l t s . A c o m p a r a t i v e s t udy o f the d i s t o r t i o n o f s i x i n d u s t r i a l b i l l e t moulds was c a r r i e d out to e x e m p l i f y the combined e f f e c t o f mould p a r a -me te r s and o p e r a t i n g c o n d i t i o n s on the t he rma l d i s t o r t i o n o f b i 1 1 e t moulds . The r e s u l t s o f the h e a t - f l o w and d i s t o r t i o n a n a l y s i s f o r b i l l e t moulds has i n d i c a t e d t h a t l o n g i t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y may be l i n k e d to the b o i l i n g pheno-menon. Th i s has been f o r m a l l y i n v e s t i g a t e d t h r ough a s e r i e s of. c a l c u l a t i o n s f o r a b i l l e t mou l d . In the l i g h t o f the r e s u l t s o f t h i s s t udy a mechanism f o r the f o r m a t i o n o f t h e se d e f e c t s has been p o s t u l a t e d . The o b s e r v a t i o n s o f o t h e r wo r ke r s p e r t a i n i n g to t h i s q u a l i t y p rob lem have been r a t i o n a l -i z e d on the b a s i s o f the p roposed mechan ism. 7 . 2 C o n c l u s i o n s (1) H e a t - f l o w eve rywhere i n the mould i s e s s e n t i a l l y u n i -d i m e n s i o n a l e x c ep t nea r the c o r n e r s and in .and above the men i s cus r e g i o n where t w o - d i m e n s i o n a l h e a t - f l o w o c c u r s . Owing to the l o n g i t u d i n a l h e a t - f l o w i n the l a t t e r c a s e , the hea t f l u x d i s t r i b u t i o n i s p r o g r e s -s i v e l y d i s t o r t e d as i t t r a n s v e r s e s f rom the h o t - t o the c o l d - f a c e w i t h the r e s u l t t h a t the h e a t - f l u x p r o f i l e measured a t the c o l d - f a c e has a peak s e v e r a l c e n t i m e t e r s below the m e n i s c u s . The h e a t - f l u x d i s t r i -b u t i o n a t the h o t - f a c e g e n e r a l l y has a peak a t the meni s c u s . (2) The p r e sence o f a doub le peak i n the h e a t - f l u x p r o -f i l e measured a t the c o l d - f a c e o f some moulds i n d i -c a t e s t h a t n u c l e a t e b o i l i n g i s t a k i n g p l a c e a t the 311 mould c o o l i n g - w a t e r i n t e r f a c e . (3) Under s t a n d a r d o p e r a t i n g c o n d i t i o n s where wa t e r v e l o c i t i e s are i n the 7 ~ 8 m/sec r a n g e , n u c l e a t e b o i l i n g can o c cu r i n b i l l e t mou l d s . T h i s e v e n t i s h i g h l y u n l i k e l y i n s l a b mou l d s , ow i ng to the l o w e r c o l d - f a c e t empe r a t u r e s r e s u l t i n g f rom t h i c k e r w a l l s . T h e r e f o r e s e a l e f o r m a t i o n i s more o f a p rob l em i n b i l l e t moulds than i n s l a b mou l d s . On the o t h e r hand owing to the h i g h e r h o t - f a c e t e m p e r a t u r e s i n s l a b mou l d s , s o f t e n i n g and d i s t o r t i o n due to the l o s s o f s t r e n g t h i s more l i k e l y i n s l a b moulds p a r t i c u l a r l y i f a l o w - m e l t i n g powder i s u s e d . (4) Owing to the g r e a t e r r i s k o f s o f t e n i n g i n s l a b moulds i t i s recommended t h a t coppe r a l l o y s c o n t a i n i n g e l emen t s such as Ag wh i ch are known to r e t a r d the r e c r y s t a l 1 i z a t i o n k i n e t i c s be u s ed . In b i l l e t moulds the peak t empe r a t u r e s c a l c u l a t e d f o r a v a r i e t y o f o p e r a t i n g c o n d i t i o n s were l e s s than 250°C . Under t he se c i r c u m s t a n c e s e x c e s s i v e d i s t o r t i o n due to a l o s s o f s t r e n g t h i s i m p r o b a b l e . Based on t h i s i t i s , s ugge s t ed t h a t the r e s i s t a n c e to t he rma l d i s t o r t i o n and the l i f e o f moulds in b i l l e t mach ines may be imp roved .by work h a r d en i n g the moulds o r u s i n g h i gh s t r e n g t h a l l o y s i n t h e i r m a n u f a c t u r e , r a t h e r than 312 a l l o y i n g w i t h e l emen t s such as Ag t h a t may o n l y a f f e c t the r e c r y s t a l 1 i z a t i o n k i n e t i c s . (5) The r e s u l t a n t t he rma l d i s t o r t i o n i n b i l l e t moulds i s i n the form o f an ou tward bu l ge i n the men i s cus r e g i o n w i t h a peak i n the .1 ~ .3 mm r a n g e , a p p r o x i -ma t e l y 6 cms be low the m e n i s c u s . The d i s t o r t e d shape has a n e g a t i v e t a p e r o f a p p r o x i m a t e l y 1 ~ 2% /m above the peak and a p o s i t i v e t a p e r o f ~0.3%/m be l ow . The dominant component o f the t he rma l d i s t o r t i o n a t any t r a n s v e r s e s e c t i o n i s the the rma l e x p a n s i o n due to the ave rage w a l l t e m p e r a t u r e . The c o n t r i b u t i o n due to bend ing o f the w a l l r e s u l t i n g f rom the t h r ough t h i c k -ness g r a d i e n t s i s r e l a t i v e l y s m a l l . The main f a c t o r s l e a d i n g to permanent d i s t o r t i o n o f t h e mou ld w a l l a re the type o f mould c o n s t r a i n t , i t s l o c a t i o n r e l a t i v e to t he s t e e l l e v e l , and the g r a d i e n t s a c r o s s the t h i c k n e s s o f the w a l l . The permanent d i s t o r t i o n can be r educed by l o w e r i n g the s t e e l l e v e l w i t h r e s p e c t t o the p o s i t i o n o f the mould c o n s t r a i n t s o r m o d i f y i n g the method o f c o n s t r a i n t so as to r educe the r e -s t r a i n t s on the mould to the rma l e x p a n s i o n . P l a s t i c f l o w a lways i n i t i a t e s a t the c o r n e r s because o f the g eome t r i c r e s t r a i n t o f the square mould c a v i t y . T h e r e -f o r e permanent d i s t o r t i o n i s h i g h e r a t t he c o r n e r s than at the m i d - f a c e . The r e s u l t s o f the a n a l y s i s i n d i c a t e t h a t the i n c i -dence o f r h o m b o i d i t y and l o n g i t u d i n a l c o r n e r c r a c k i n g i n b i l l e t s i s r e l a t e d to an i n t e r m i t t e n t b o i l i n g c y c l e a t the m o u l d / c o o l i n g - w a t e r i n t e r f a c e . I f two a d j a c e n t f a c e s o f the mould are b o i l i n g i n t e r m i t t e n t l y and o u t - o f - p h a s e w i t h the o t h e r two , uneven c o o l i n g o f the b i l l e t wou ld r e s u l t . Th i s c auses r h o m b o i d i t y o f the s t r a n d and the mou l d , l e a d i n g to an i n c r e a s e i n c o r n e r gaps a t d i a g o n a l l y o p p o s i t e ob t u s e ang l e c o r n e r s . I t i s t hough t t h a t the t e n s i l e s t r e s s e s g e n e r a t e d nea r the s o l i d i f i c a t i o n f r o n t a t the ob tuse c o r n e r s by the r e h e a t i n g even t c o u p l e d w i t h the i n -c r e a s e i n ang l e are s u f f i c i e n t t o i n i t i a t e h o t -t e a r i n g i n i n t e r d e n d r i t i c r e g i o n s . The c r a c k s so formed may p r opaga t e t h rough the t h i c k n e s s and appea r on the s u r f a c e . C o o l i n g - w a t e r v e l o c i t y i s the most i m p o r t a n t c o n t r o l pa r ame te r f rom the s t a n d p o i n t o f p r e v e n t i n g the o c c u r r e n c e o f these d e f e c t s . In t he moulds i n v e s t i -g a t ed r a i s i n g the wa te r v e l o c i t y i n the c h anne l s to ~ 11 m/sec wou ld p r e v e n t i n t e r m i t t e n t b o i l i n g and the a t t e n d a n t q u a l i t y p r o b l ems . The low i n c i d e n c e o f c r a c k i n g i n t he low ca rbon (.0.1%) s t e e l s i s t hough t to be r e l a t e d to the low 314 h e a t - e x t r a c t i o n r a t e s and the r e s u l t i n g absence o f b o i l i n g . The h e a t - e x t r a c t i o n r a t e i n c r e a s e s w i t h ca rbon and f o r the h i g h e r ca rbon g rades (>0.4%) an i n t e r m i t t e n t b o i l i n g c y c l e may be i n i t i a t e d . 7 .3 Recommendat ions f o r Fu tu r e Work An i m p o r t a n t d i r e c t i o n f o r f u r t h e r r e s e a r c h i s the t e s t i n g o f the p roposed mechanism th rough a s e r i e s o f c a r e -f u l l y p i anned e x p e r i m e n t s . P roposed i n s t r u m e n t a t i o n f o r t he b i l l e t mould wou ld i n c l u d e t r a n s d u c e r s to m o n i t o r movement on a l l f o u r f a c e s o f the mou l d , and a s e r i e s o f t he rmo-c o u p l e s f o r t empe ra t u r e measurement . A s e r i e s o f t e s t s s h o u l d be pe r f o rmed f o r d i f f e r e n t wa te r v e l o c i t i e s i n the 4 to 11 m/sec range and f o r s t e e l s o f d i f f e r e n t c a rbon c o n -t e n t s . The mould w a l l movement and t empe r a t u r e s h o u l d be c o n t i n u o u s l y m o n i t o r e d t h r o ughou t the t r i a l s . The q u a l i t y o f the b i l l e t a s s o c i a t e d w i t h each f l o w c o n d i t i o n and s t e e l c o m p o s i t i o n s h o u l d be e v a l u a t e d f rom the s t a n d p o i n t o f l o n g i -t u d i n a l c o r n e r c r a c k i n g and r h o m b o i d i t y . The r . e su l t s o f such an i n v e s t i g a t i o n may he l p c o n c l u s i v e l y e s t a b l i s h the v a l i d i t y o f the p roposed mechan i sm. SYMBOLS A Area A.j . . .Ag Area x y i j i j i j X- and Y- coordinate directions dx,dy,dz Incremental distances in the X-, Y- and Z- cartesian coordinate directions e D d e . . , d e . . , d e . . Increments of t o t a l , elastic and plastic strain respectively Dh Hydraulic diameter of water channel e 0 e c Emissivities p e Plastic component of strain tensor i j — P e Cumulative plastic strain e Direct strain in X-coordinate direction x Fj_i Shape factor between i and j surface m2 2 m a Coefficient of thermal expansion m/m C , C Plate curvatures with respect to the ' m Cp Specific heat kJ/kg°C C ,C C . Specific heats of water, mould and kJ/kg°C Pw Pm l iqu id respectively d Thickness of mould wall mm d Width of water channel gap mm w dt Increment of time sees mm mm E Youngs Modulus MPa i - j Generalized Lateral Force at i node in p late analys is Generalized Moment at i node about X-coordinate d i rec t ion in plate analys is Generalized Moment at i node about Y-coordinate d i rec t i on in p late analys is Accelerat ion due to gravi ty and grav i ta t iona l constant E f fec t i ve radiant heat-transfer coe f f i c i en t Local and Average mould heat-t ransfer coe f f i c i en ts Forced convection heat-transfer coe f f i c i en t Heat-transfer coe f f i c i en t at mould/ water inter face Heat-transfer coe f f i c i en t at i ^ node on mould/water inter face Latent heat fo r steam Slope of cumulative p l a s t i c stress s t r a i n curve Temperature and displacement funct ionals for heat flow and plate bending var ia t iona l problems th I n t r i n s i c thermal conduct iv i ty of i component in a mixture Thermal conduct iv i ty of mixture Thermal conduct iv i ty of mould Length and increment of length Length of a segment between nodes i and j i natural coordinate 317 M . ,M . xi y i M. i vw N i P V p w p * ' p v % q f c ' q i n ' q t r : > q b q i , l q o J q s qw q j ' q i q r i ' q s i cr-| ,0^ a e f ' ° Direct moments in the X and Y kN/m coordinate d i rect ions and tw is t ing moment in the X-Y plane Moments in the X- and Y- coordinate kN/m d i rect ions at the i t h node j_r_ Molecular weight of i component kg in the mixture V i scos i ty of water kg/m sec Interpolat ion polynomial for node i Pressure kPa 3 Densit ies of the mould and water kg/m 3 Densit ies of water in the l i q u i d and kg/m vapour phase at saturat ion temperature Average heat- f lux kW/m 2 Heat f luxes associated with forced kW/m convection, i nc ip i en t b o i l i n g , t r ans i t i on region, and pool bo i l i ng regions of the bo i l i ng curve. 2 Heat f luxes in and out of body No. 1 kW/m 2 Heat extract ion from steel kW/m 2 Heat transferred to mould coo l ing water kW/m 2 Input heat- f lux associated with kW/m nodes j , and i 2 Input heat- f lux due to r ad i a t i on , and kW/m so l i d i f y i n g strand at node ( i , j ) P r i n c i p l e stresses MPa Ef fec t ive stress MPa Y i e l d stress in simple tension MPa 3 1 8 a..,o'. Stress tensor and deviatoric MPa 1 J 1 J stress tensor a ,o Direct stress in the X-coordinate MPa x x x direction a ,a Direct stress in the Y-coordinate MPa ^ y y direction o Direct stress in the Z-coordinate MPa direction T v \ / ' T 7 55 7 Shear stress components MPa xy x z y z a Surface tension kN/m x Temperature dependent component MPa of y ie ld stress T $ Yield stress in simple shear MPa T Temperature independent y ie ld MPa component of y ie ld stress t Time sees t Time-on-the mould (dwell time) sees m T. , ,T" ,,T?+!,T* Temperatures °C T »T Ambient temperatures °C Tw Cooling-water temperatures °C T . Cooling-water temperature at i node °C T , + Saturation temperature of water °C S a l u,v,w Components of displacement in the three X-, Y- and Z- coordinate directions m v f >v g Specific volumes of water in the f l u i d m3/kg vc Casting speed m / m i n Vw Cooling-water velocity m/sec v Poisson Ratio w l Lateral displacement of plate w. Lateral displacement of i t h node X. Molar f rac t i on of i component in the mixture X Mould wal l thickness m X X-coordinate d i rec t ion Y Y-coordinate d i rec t ion Z Z-coordinate d i rec t ion Z m > Z f Mould length and depth of free board' Special Symbols [ ] Matr ix { } Vector V .V - J JV , , D i f fe ren t i a l operators A Difference operator a Pa r t i a l d i f f e r en t i a l operator d Total d i f f e r en t i a l operator Other Symobols [A] Coef f i c ien t matrix of f i n i t e -di f ference equations [D] E l a s t i c i t y matrix [D p ] E l a s to -p l a s t i c material matrix [K] Master s t i f f ness matrix [K^] Element convection matrix [Kj.] Element thermal s t i f f ness matrix [N] Matrix of shape functions in te rpo la t ion polynominals {a} Thermal coe f f i c i en t of expansion vector {B} Right hand side vector { e } S t ra in vector {F} Body force vector e th {F.} i component of load vector in element e {K t w> Influence vector fo r cool ing-water temperature {M} Interpolat ion polynomial vector fo r temperature {n^} Normal vector to surface {P} Right-hand side load vector in assembled f in i te-e lement equations {q} Input heat f l ux vector {s} Assembled nodal displacement vector e g th {s} , {s.} Nodal displacement vector for element and i component {T} Assembled nodal temperature vector {T} e Element nodal temperatures {u} Displacement vector Subscripts m mould w water l l i q u i d i , j nodes i and j Superscripts n n+1 n time step ( n + l ) t h time step intermediate time s 322 REFERENCES 1. 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