Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Three dimensional heat flow in the direct chill casting of non-ferrous metals Venkateswaran, V. 1980

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

Item Metadata

Download

Media
831-UBC_1980_A1 V46.pdf [ 13.19MB ]
Metadata
JSON: 831-1.0079161.json
JSON-LD: 831-1.0079161-ld.json
RDF/XML (Pretty): 831-1.0079161-rdf.xml
RDF/JSON: 831-1.0079161-rdf.json
Turtle: 831-1.0079161-turtle.txt
N-Triples: 831-1.0079161-rdf-ntriples.txt
Original Record: 831-1.0079161-source.json
Full Text
831-1.0079161-fulltext.txt
Citation
831-1.0079161.ris

Full Text

THREE DIMENSIONAL HEAT FLOW IN THE DIRECT CHILL CASTING OF NON-FERROUS METALS by V. VENKATESWARAN •Sc.  ( H o n s ) , Bangalore U n i v e r s i t y ,  B.E.,  Indian I n s t i t u t e  India,  of Science,  A. S c . , The U n i v e r s i t y of B r i t i s h  1969  1972  C o l u m b i a , 1976  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n the  Department of  Metallurgical  We accept t h i s  Engineering  t h e s i s as c o n f o r m i n g  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA July ©  1980  V. Venkateswaran, 1980  ABSTRACT  A t h r e e dimensional  mathematical  model has been  developed t o study heat f l o w and s o l i d i f i c a t i o n  in  Direct  rectan-  Chill  c a s t i n g of n o n - f e r r o u s metals w i t h  g u l a r as w e l l  as i r r e g u l a r  cross-sections.  The model  which i s based on an a l t e r n a t i n g d i r e c t i o n , finite-difference  implicit  numerical method i s capable of  heat f l o w both i n the steady s t a t e and t r a n s i e n t the c a s t i n g o p e r a t i o n .  The v a l i d i t y  simulating part  of  o f the model has been  v e r i f i e d by comparing p r e d i c t e d pool p r o f i l e s depths w i t h i n d u s t r i a l  the  and pool  measurements.  The model has been used to study the importance heat f l o w s  i n t h e v a r i o u s d i r e c t i o n s , and the  of using t w o - d i m e n s i o n a l  limitations  heat f l o w models are b r o u g h t  This study has shown t h a t a t w o - d i m e n s i o n a l  of  out.  model which  n e g l e c t s heat f l o w normal t o the narrow f a c e can be used to s i m u l a t e the s o l i d i f i c a t i o n greater  than 2 . 5 ,  D.C. c o o l i n g .  of slabs w i t h aspect  cast under c o n d i t i o n s  of  lations  conventional  F u r t h e r i t was demonstrated t h a t w i t h  duced secondary c o o l i n g , a t w o - d i m e n s i o n a l neglects axial  ratios  heat c o n d u c t i o n  show t h a t  model  is p r e f e r a b l e .  in cooling large sections  s t a t e can occupy around 25% o f the t o t a l  i i  that  Model  the  casting  re-  calcu-  unsteady cycle.  The f o r m a t i o n of cracks  i n jumbo i n g o t s o f  Prime  Western Grade z i n c has been i n v e s t i g a t e d w i t h the a i d the mathematical  model.  It  has been shown t h a t  i s caused by r e h e a t i n g of the s u r f a c e below the c o o l i n g zone, i f high w a t e r f l u x . strains  this  attempts  cracking  spray  zone is s h o r t and c h a r a c t e r i z e d by a  The s u r f a c e r e h e a t i n g g e n e r a t e s  at the s o l i d i f i c a t i o n  by the presence of lead regions.  the  of  f r o n t where c r a c k i n g  rich liquid  i n the  tensile is  aided  inter-dendritic  A new spray assembly has been d e s i g n e d which to cool  the c a s t i n g more u n i f o r m l y from, the  o f the mould to the bottom of the l i q u i d p o o l .  This new  spray system has been t e s t e d i n - p l a n t f o r c a s t i n g Western Grade z i n c and r e s u l t s ness i n p r e v e n t i n g crack  have proven i t s  formation.  top  Prime  effective-  TABLE OF CONTENTS Page Abstract Tab! e of Contents L i s t o f Tables L i s t o f Figures Acknowledgements  ii iv vi viii xiv  Chapter 1  2  3  Introduction  1  1.1  2  Review of the L i t e r a t u r e  3  2.1 2.2 2.3  3 4  Introduction D.C. Casting Review of Mathematical i n D.C. Casting  Models  8  Development o f the Heat Flow Model  16  3.1 3.2 3.3  16 17  3.4 3.5 3.6 4  O b j e c t i v e s of the Present Work  Introduction Assumptions Made i n the Model Heat Flow Equations and Boundary Conditions Method o f Sol u t i o n Mathematical Check f o r I n t e r n a l Cons i stency of the Computer Program Flow Chart of the Computer P r o g r a m . .  V a l i d a t i o n of the Results Mathematical Model 4.1 4.2  From t h e  4.3 4.4 4.5  34 44 47  Introduction Conventional D.C. C a s t i n g of Aluminium - Alcan 4 . 2 . 1 Aluminium I n g o t s 381 x 4 . 2 . 2 Aluminium I n g o t s 457 x mm 4 . 2 . 3 Aluminium Ingots 305 x mm 4 . 2 . 4 Aluminium I n g o t s 229 x 4 . 2 . 5 Summary of C o n v e n t i o n a l Casting S i m u l a t i o n s  19 23  47 48 991mm. 1143 1010 813mm. D.C.  Reduced Secondary C o o l i n g - B r i t i s h Al umi n i um Zinc-Jumbo Casting - Cominco Summary o f V a l i d a t i o n Runs iv  48 56 59 59 63 66 76 92  Chapter 5  Page Effect 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9  6  Introduction E f f e c t of Aspect Ratio E f f e c t of A x i a l Conduction Importance o f Unsteady S t a t e E f f e c t o f S e c t i o n Size E f f e c t of Superheat E f f e c t of Cooling C o n d i t i o n s E f f e c t of L a t e n t Heat Release on Model C a l c u l a t i o n s Summary  Use o f the Heat Flow Model to Solve a C r a c k i n g Problem i n the D.C. C a s t i n g of Prime Western Grade Jumbo I n g o t s 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8  7  of Casting V a r i a b l e s on Heat Flow.  Introduction I n t e r n a l Cracks i n the D.C. C a s t i n g of Prime Western Grade Zinc Heat Flow A n a l y s i s Metal 1ographic A n a l y s i s Mechanism of Crack Formation Design of the New Cooling System . . . . T e s t i n g o f the New C o o l i n g System . . Summary  Summary and Conclusions  97 98 104 110 115 119 119 121 122  124 124 125 134 140 144 150 154 155 157  Symbols Bibliography Appendi ces Al  97  160 162  Development of F i n i t e D i f f e r e n c e Equation A l . l Alternating Direction Finite f e r e n c e Equations f o r Three Dimensional Problems  170 Dif171  Al .2 S t a b i l i t y C r i t e r i o n f o r E x p l i c i t F i n i t e D i f f e r e n c e Using Convective Type Boundary C o n d i t i o n s  181  A2  Source L i s t i n g o f the Computer Program . .  183  A3  Three Dimensional Temperature b u t i o n i n the C a s t i n g f o r the Runs  227  v  DistriDifferent  LIST OF TABLES  Tables I  II  III  IV  V VI  VII VIII IX  X  Page Comparison between numerical and a n a l y t i c a l values o f temperatures (°C) at the c e n t r e of a cube, as a f u n c t i o n of time f o r d i f f e r e n t s i z e s o f nodes  37  Comparison between numerical and a n a l y t i c a l values o f c e n t r e temperatures (°C) o f a cube, as a f u n c t i o n of time f o r d i f f e r e n t time steps  39  Thermophysical p r o p e r t i e s of a l u m i n i u m used i n c o n v e n t i o n a l f l o o d c o o l i n g simulations  49  Comparison between c a l c u l a t e d and measured pool depths f o r aluminium s e c t i o n s c a s t at di f f e r e n t speeds  64  Thermophysical p r o p e r t i e s o f a l u m i n i u m used i n reduced secondary c o o l i n g s i m u l a t i o n s . . .  67  H e a t - t r a n s f e r c o e f f i c i e n t s used as a f u n c t i o n o f p o s i t i o n below the l i q u i d s u r f a c e f o r reduced secondary c o o l i n g  68  Thermophysical p r o p e r t i e s jumbo i n g o t s i m u l a t i o n Heat-transfer coefficients jumbo c a s t i n g  o f z i n c used  in 79  used i n  zinc  Comparison between t h r e e dimensional and two dimensional pool depths f o r reduced secondary c o o l i n g of 254 x 690 mm aluminium i n g o t Steady s t a t e pool depths f o r aluminium and z i n c i n g o t s c a l c u l a t e d w i t h and w i t h o u t t h e a x i a l conduction  vi  81  0  3  ^^  Tab 1es XI XII  XIII  XIV  XV  XVI  XVII  XVIII  A3.1  A3.II  A3.Ill  Page C a s t i n g c o n d i t i o n s f o r the d i f f e r e n t d u r i n g the e x p e r i m e n t a l campaign  runs 126  L o c a t i o n of cracks i n jumbo c r o s s - s e c t i o n s taken at v a r i o u s p o i n t s along the l e n g t h o f s t r a n d s A and B (Run 1) . ..  128  L o c a t i o n o f cracks in jumbo c r o s s - s e c t i o n s taken at v a r i o u s p o i n t s along the l e n g t h o f s t r a n d s A and B (Run 2)  129  L o c a t i o n o f cracks i n jumbo c r o s s - s e c t i o n s taken a t v a r i o u s p o i n t s along the l e n g t h o f s t r a n d s A and B (Run 3)  130  L o c a t i o n o f cracks i n jumbo c r o s s - s e c t i o n s taken at v a r i o u s p o i n t s along the l e n g t h o f s t r a n d s A and B (Run 4)  131  L o c a t i o n of cracks in jumbo c r o s s - s e c t i o n s taken at v a r i o u s p o i n t s along the l e n g t h of Strands A.and B (Run 5)  132  L o c a t i o n o f cracks i n jumbo c r o s s - s e c t i o n s taken at v a r i o u s p o i n t s along the l e n g t h o f s t r a n d s A and B (Run 6)  133  C a l c u l a t e d values of reheat at d i f f e r e n t p o i n t s on the s u r f a c e o f the jumbo s e c t i o n . Top and bottom correspond to m i d - f a c e on the non-notched s u r f a c e s  148  T h r e e - d i m e n s i o n a l temperature d i s t r i b u t i o n i n 381 x 991 mm aluminium i n g o t c a s t a t 1.775 mm/s ( c o n v e n t i o n a l c o o l i n g ) . .••  228  T h r e e - d i m e n s i o n a l temperature d i s t r i b u t i o n i n 254 x 690 mm aluminium i n g o t c a s t at 0.833 mm/s (reduced secondary c o o l i n g )  234  T h r e e - d i m e n s i o n a l temperature d i s t r i b u t i o n i n z i n c jumbo i n g o t cast at 1.27 mm/s  238  vi i  LIST OF FIGURES  Figure  2.1 3.1  Page  A schematic diagram of the openhead D i r e c t C h i l l c a s t i n g process  5  The e f f e c t of the number o f nodes on p e r cent e r r o r at the end of d i f f e r e n t time intervals  38  Comparison between a n a l y t i c a l and n u m e r i c a l c a l c u l a t e d s u r f a c e temperatures f o r l a r g e values o f time i n t e r v a l  42  S t a b l e o s c i l l a t o r y nature of the numerically c a l c u l a t e d surface temperatures f o r l a r g e values of time i n t e r v a l  43  3.4(a)  Flow c h a r t of the computer program  45  3.4(b)  Flow c h a r t of the computer program ( c o n t i n u e d from F i g . 3 . 4 ( a ) )  46  Comparison between the p r e d i c t e d and measured pool p r o f i l e s f o r 381 x 991 mm aluminium i n g o t cast at 1.778 mm/s ( o b t a i n e d at the m i d - p l a n e p a r a l l e l t o the narrow f a c e )  51  Steady s t a t e pool p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l mid-planes f o r 381 x 991 mm aluminium i n g o t cast at 1.778 mm/s  53  Comparison of the pool p r o f i l e s from the t w o - d i m e n s i o n a l and t h r e e - d i m e n s i o n a l c a l c u l a t i o n s f o r c a s t i n g 381 x 991 mm aluminium i n g o t a t 1.778 mm/s  55  3.2  3.3  4.1  4.2  4.3  4.4  Comparison between the measured and c a l c u l a t e d pool depths f o r 381 x 991 mm aluminium i n g o t cast a t d i f f e r e n t speeds  viii  ...  57  Fi gure  Page  4.5  C a l c u l a t e d pool p r o f i l e s f o r 381 x 991 mm aluminium i n g o t c a s t at d i f f e r e n t speeds . . .  58  4.6  Comparison between the c a l c u l a t e d and measured pool depths f o r 457 x 1143 mm aluminium i n g o t cast at d i f f e r e n t speeds  ...  60  Comparison between the c a l c u l a t e d and measured pool depth f o r 305 x 1010 mm aluminium i n g o t c a s t at d i f f e r e n t speeds  ...  61  Comparison between the c a l c u l a t e d and measured pool depths f o r 229 x 813 mm aluminium i n g o t cast at d i f f e r e n t speeds  ...  62  Time r e q u i r e d f o r the pool p r o f i l e s t o reach steady s t a t e f o r aluminium i n g o t s o f v a r i o u s s i z e s as a f u n c t i o n of c a s t i n g speed  65  L i q u i d u s and s o l i d u s p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l mid-planes of 254 x 690 mm aluminium i n g o t cast under reduced c o o l i n g c o n d i t i o n s at 0.833 mm/s, a f t e r 312 s from s t a r t  69  L i q u i d u s and s o l i d u s p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l mid-planes of 254 x 690 mm aluminium i n g o t cast under reduced c o o l i n g c o n d i t i o n s at 0.833 mm/s, a f t e r 624 s from s t a r t  70  4.7  4.8  4.9  4.10  4.11  4.12  4.13  4.14  L i q u i d u s and s o l i d u s p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l mid-planes o f 254 x 690 mm aluminium i n g o t cast under reduced secondary c o o l i n g c o n d i t i o n s at 0.833 mm/s, a f t e r 1 248 s from s t a r t  71  Comparison of steady s t a t e pool p r o f i l e s f o r the t w o - d i m e n s i o n a l and the t h r e e dimensional c a l c u l a t i o n s of 254 x 690 mm aluminium i n g o t cast under reduced secondary c o o l i n g c o n d i t i o n s at .833 mm/s  73  Three-dimensional v i s u a l i z a t i o n of l i q u i d pool s u r f a c e o f 254 x 690 mm aluminium i n g o t c a s t at .833 mm/s under reduced secondary c o o l i n g , as seen from t h e d i f f e r e n t angles ix  74  Fi gure 4.15  Page A cross s e c t i o n o f the jumbo i n g o t w i t h the v a r i o u s dimensions given i n mm  77  4.16  D i s c r e t i z a t i o n of the jumbo i n g o t f o r finite-difference calculations  78  4.17  Comparison o f the measured and c a l c u l a t e d t e m p e r a t u r e p r o f i l e s f o r the z i n c jumbo i n g o t c a s t at 1.69 mm/s  82  A t h r e e - d i m e n s i o n a l view o f the jumbo i n g o t showing the l o n g i t u d i n a l s e c t i o n s where the pool p r o f i l e s have been o b t a i n e d . .  83  Pool p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l s e c t i o n s shown i n F i g . 4.18 f o r c a s t i n g z i n c jumbo i n g o t at 1.27 mm/s, 278 s a f t e r the s t a r t  84  Pool p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l s e c t i o n s shown i n F i g . 4.18 f o r c a s t i n g z i n c jumbo i n g o t at 1.27 mm/s, 557 s a f t e r the s t a r t  85  4.18  4.19  4.20  4.21  4.22  4.23  4.24  4.25  4.26  the  Steady s t a t e pool p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l s e c t i o n s shown in F i g . 4.18 f o r c a s t i n g z i n c jumbo i n g o t at 1.27 mm/s . . Comparison between the c a l c u l a t e d and measured pool p r o f i l e s f o r the c a s t i n g z i n c jumbo i n g o t at 1.27 mm/s  of  8 6  88  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 of the l i q u i d pool s u r f a c e o f z i n c jumbo i n g o t c a s t at 1.27 mm/s  89  Comparison between the c a l c u l a t e d and measured pool depths o b t a i n e d at d i f f e r e n t times from the s t a r t o f c a s t i n g o f z i n c jumbo i n g o t cast at 76 mm/min  90  Comparison between the c a l c u l a t e d and measured pool depths o b t a i n e d at d i f f e r e n t times from the s t a r t o f c a s t i n g of z i n c jumbo i n g o t cast at 102 mm/min  91  Freezing p r o f i l e s cross-section  93  as seen on a jumbo x  Fi gure 4.27  4.28  5.1  5.2  5.3  5.4  5.5  5.6  5.7 5.8  5.9  Page Macrostructure o f the zinc jumbo c r o s s s e c t i o n showing the f r e e z i n g l i n e s seen i n F i g . 4.26  94  Comparison o f the c a l c u l a t e d pool p r o f i l e s o b t a i n e d w i t h and w i t h o u t the n o t c h f o r c a s t i n g z i n c a t 1.69 mm/s  95  E f f e c t o f aspect r a t i o s on the pool depths i n c a s t i n g 381 mm and 457.2 mm t h i c k aluminium slabs at 1.778 mm/s  99  E f f e c t o f aspect r a t i o s on the pool depths i n c a s t i n g 381 mm t h i c k z i n c slabs a t 1 .78 and 2.2 mm/s  101  C a l c u l a t e d s u r f a c e temperature p r o f i l e s o b t a i n e d at the m i d - f a c e o f 457 x 457 mm aluminium i n g o t cast at .974 mm/s, w i t h and w i t h o u t the a x i a l c o n d u c t i o n  105  C a l c u l a t e d c e n t r e temperature p r o f i l e s o b t a i n e d f o r 457 x 457 mm aluminium i n g o t c a s t at .974 mm/s, w i t h and w i t h o u t the axial conduction  107  C a l c u l a t e d s u r f a c e temperature p r o f i l e s the i n i t i a l and steady s t a t e s l i c e s f o r c a s t i n g aluminium jumbo i n g o t ( a t the bottom m i d - f a c e of a jumbo s e c t i o n )  112  Time r e q u i r e d f o r reach steady s t a t e aluminium and z i n c aspect r a t i o s cast  for  the pool p r o f i l e s to f o r 381 mm t h i c k slabs of d i f f e r e n t at 1.778 mm/s  ,. . .  114  Steady s t a t e pool depths f o r square s e c t i o n s of aluminium and zinc cast at 1.778 mm/s . . .  116  C a l c u l a t e d steady s t a t e pool p r o f i l e s f o r 381 mm square s e c t i o n s of aluminium and z i n c c a s t at 1.778 mm/s  117  Time r e q u i r e d f o r the pool p r o f i l e s to reach steady s t a t e f o r square s e c t i o n s o f aluminium and z i n c cast at 1.778 mm/s  118  xi  Fi gure 5.10  6.1 6.2  Page E f f e c t o f the h e a t - t r a n s f e r c o e f f i c i e n t on the steady s t a t e pool depths f o r c a s t i n g 61 0 x 546 mm z i n c i n g o t at 1 mm/s  120  A c r o s s - s e c t i o n of Prime Western Grade jumbo i n g o t showing the i n t e r n a l c r a c k s  135  ....  C a l c u l a t e d temperature p r o f i l e s f o r the d i f f e r e n t nodes f o r z i n c jumbo i n g o t c a s t at 1 .69 mm/s  1 37  Growth o f the s h e l l as a f u n c t i o n o f d i s t a n c e i n the a x i a l d i r e c t i o n f o r z i n c jumbo i n g o t c a s t at 1.69 mm/s. F i g u r e on the r i g h t shows the s u r f a c e t e r m p e r a t u r e p r o f i l e at the bottom m i d - f a c e o f a jumbo section  139  6.4  A macro-photograph of the cracked M a g n i f i c a t i o n 1.3 X  141  6.5  Scanning e l e c t r o n micrograph of a c r a c k e d surface. M a g n i f i c a t i o n 200 X  6.6(a)  Scanning e l e c t r o n micrograph of a c r a c k e d s u r f a c e r e v e a l i n g the smooth n a t u r e o f the surface. M a g n i f i c a t i o n 1 000 X  6.3  surface.  1 43  6.6(b)  Pb x - r a y p i c t u r e  6.7(a)  Scanning e l e c t r o n micrograph of a c r a c k e d s u r f a c e r e v e a l i n g the smooth n a t u r e o f the surface. M a g n i f i c a t i o n 1000 X  1 45  6.7(b)  Pb  145  6.8  Phase diagram o f Pb- Zn  6.9  E f f e c t o f c a s t i n g speed on the s u r f a c e r e h e a t i n g at the bottom m i d - f a c e o f a jumbo s e c t i o n  149  E f f e c t o f the h e a t - t r a n s f e r c o e f f i c i e n t i n the sub-mould r e g i o n on the s u r f a c e r e h e a t i n g at the bottom m i d - f a c e o f a jumbo section  I '  6.10  of F i g . 6 . 6 ( a )  1 42  x - r a y p i c t u r e of F i g . 6 . 7 ( a )  xi i  system (89)  143  146  5  Fi gure 6.11  6.12  A1.1 A1.2  Page Arrangement of spray nozzles i n t h e new c o o l i n g assembly f o r the bottom s u r f a c e of a jumbo s e c t i o n A c r o s s - s e c t i o n of Prime Western Grade jumbo i n g o t cast w i t h the new c o o l i n g system  153  ;...  156  Dotted r e g i o n i s the volume over which c a l c u l a t i o n s are performed  172  D i s c r e t i z a t i o n of the r e c t a n g u l a r piped showing the s u r f a c e nodes  173  xiii  paralleol-  ACKNOWLEDGEMENTS  The a u t h o r wishes to express h i s s i n c e r e  gratitude  t o Dr. J . K. Brimacombe f o r h i s guidance t h r o u g h o u t course of t h i s  research.  acknowledge the e f f o r t s  The a u t h o r a l s o would l i k e  for  results.  initiating  the p r o j e c t  Mr. P. J . P l a y e r ,  on the c r a c k i n g problem and t o  Dr. G. W. Toop, Mr. M. L.  stages of t h i s  The d i s c u s s i o n s  help and  and a s s i s t a n c e o f f e l l o w  The a u t h o r i s g r a t e f u l  Fellowship.  graduate  have been i n -  to the U n i v e r s i t y  Columbia f o r p r o v i d i n g f i n a n c i a l  form o f a UBC Graduate  co-operation  project.  s t u d e n t s , f a c u l t y members and t e c h n i c i a n s  British  Connolly,  Mr. J . Newton, Mr. C. A. Johnson and  o f Cominco f o r t h e i r  d u r i n g the v a r i o u s  valuable.  experi-  Thanks are expressed t o Mr. Roger Watson  Mr. C. A. S u t h e r l a n d ,  Mr. R. Joseph  to  of Mr. J . S u t h e r l a n d o f Alcan and  Mr. D. D. B e a t t i e o f B r i t i s h Aluminium i n p r o v i d i n g mental  the  support  in  of the  Chapter 1  INTRODUCTION  Over the l a s t h a l f c e n t u r y a number o f new processes have been developed i n both the f e r r o u s industries.  and n o n - f e r r o u s  The m a j o r i t y of these new processes have been  brought about t o i n c r e a s e p r o d u c t i o n and t o e f f e c t ment i n o v e r a l l  efficiency.  c a s t i n g of s t e e l ,  Direct Chill  m e t a l s , BOF and Q-BOP s t e e l processes have a l l efficiency continually  Examples are t h e  making e t c .  been c o m m e r c i a l i s e d ,  continuous  non-ferrous Although  these  improvement  in  sought.  This r e q u i r e s an u n d e r s t a n d i n g o f  Mathematical  models are v a l u a b l e t o o l s  the process and o f s t u d y i n g the e f f e c t on the o v e r a l l  is the  between fundamental and process in  r e g a r d as they p r o v i d e an i n e x p e n s i v e way o f l e a r n i n g of process  this about  variables  operation.  A number o f mathematical f o r the i n v e s t i g a t i o n ferrous  the  o f p r o d u c t i o n and the q u a l i t y o f the p r o d u c t  process and the r e l a t i o n s h i p s variables.  c a s t i n g of  improve-  continuous  models have been developed  of heat f l o w i n f e r r o u s  casting.  and non-  However o n l y a h a n d f u l  these have been used beyond the developmental predict  solidification  ditions  f o r the e l i m i n a t i o n o f i n t e r n a l  stage  s t r u c t u r e and to d e t e r m i n e  1  cracks.  of to  conAlmost  all  2  the models developed i n t h i s area to date are e i t h e r or two dimensional  in nature.  Although t h e y are adequate  f o r the s i m u l a t i o n of the c o n t i n u o u s c a s t i n g o f their  use i s r a t h e r l i m i t e d f o r the a n a l y s i s  i n D.C. c a s t i n g of r e c t a n g u l a r  sections of  metals w i t h low aspect r a t i o s .  made i t  1.1  steel,  of heat  of high-speed d i g i t a l  flow  non-ferrous  Three-dimensional  models are r e q u i r e d under these c o n d i t i o n s . ments i n the f i e l d  one  heat  flow  Rapid d e v e l o p -  computers  have  p o s s i b l e t o develop these models.  Objectives  of the Present Work  The p r i m a r y o b j e c t i v e o f t h i s s t u d y was t o a fully  three-dimensional  Direct Chill  nature of t h i s process, i t  to i n c l u d e the t r a n s i e n t the c a l c u l a t i o n s . validate  heat f l o w model t o s i m u l a t e  c a s t i n g of n o n - f e r r o u s m e t a l s .  semi-continuous  develop  initial  the  Because o f was a l s o  desired  p o r t i o n o f the c a s t i n g  The second o b j e c t i v e o f t h i s  the model developed w i t h i n d u s t r i a l  measurements  components of heat f l o w s under d i f f e r e n t  conditions.  Finally  the most i m p o r t a n t o b j e c t i v e  study was to use the model to a s s i s t  in  s t u d y was t o  and then use the model t o demonstrate t h e importance o f different  the  the  casting of  this  in s o l v i n g a cracking  problem i n the D.C. c a s t i n g of Prime Western Grade  zinc.  Chapter 2  REVIEW OF THE LITERATURE  2.1  Introduction Direct Chill  c a s t i n g , more commonly known as D.C.  c a s t i n g , was developed i n the l a t e 1 930' s a n d , s i n c e t i m e , has become the "work horse" of a modern casting plant.  that  non-ferrous  In a d d i t i o n t o being r e l i a b l e ,  it  has  proved t o be a very economical p r o d u c t i o n t e c h n i q u e c a s t i n g n o n - f e r r o u s metals such as a l u m i n i u m , magnesium and z i n c . either  circular  for  copper,  Most s e c t i o n s t h a t are D.C. c a s t  forextrusion applications  or  rectangular  f o r r o l l i n g a p p l i c a t i o n s ; b u t o t h e r shapes are a l s o duced.  Examples are the T - i n g o t s  i n the a l u m i n i u m  d u s t r y and the jumbo i n g o t of s p e c i a l industry.  shape i n the  As an example, the s l a b  inzinc  casting  at Alcan i n Oswego, New York can p r o d u c e . 5 , 7 or  9 slabs a t a time of 460 mm t h i c k n e s s , w i d t h and 5100 mm l e n g t h , e q u i v a l e n t drop r a n g i n g from 35 to 45 tonnes  It  pro-  Modern day D.C. c a s t i n g machines have enormous  production c a p a c i t i e s . facility  are  is  interesting  2200 mm maximum  to a c a s t w e i g h t  per  (3).  to compare D.C. c a s t i n g o f non-  f e r r o u s metals w i t h the c o n t i n u o u s c a s t i n g of 3  steel.  Two i m p o r t a n t d i f f e r e n c e s  that  can be seen are the  of the mould and the c a s t i n g speed. are c h a r a c t e r i z e d by t h e i r  D.C. c a s t i n g  short lengths.-  It  is  common t o f i n d moulds o n l y 20 to 50 mm long f o r aluminium and z i n c  The c a s t i n g  encountered i n D.C. c a s t i n g s are an o r d e r of lower than those i n s t e e l .  of i t s  very casting  D.C. c a s t i n g  speeds  magnitude  One o t h e r f a c t o r which  i s the s e m i - c o n t i n u o u s  dis-  nature  o p e r a t i o n , a l t h o u g h i n r e c e n t years a c o n t i n u o u s  horizontal  c a s t e r has been developed  horizontal  D.C. c a s t i n g however is r e s t r i c t e d  smaller sections developed in t h i s  D.C.  (4-7).  The use of to  and a l l o y s o f low s t r e n g t h .  casting  The model  study has been used f o r t h e a n a l y s i s  heat f l o w i n v e r t i c a l  2.2  moulds  i n comparison w i t h 600-900 mm long  moulds used i n t h e case of s t e e l .  tinguishes  length  D.C. c a s t i n g  of  operations.  Casting  A schematic diagram of the D i r e c t process i s shown i n F i g . 2 . 1 .  casting  As i s obvious from the name  the s u r f a c e o f the c a s t i n g in t h i s  operation  by the mould c o o l i n g water which e x i t s impinges d i r e c t l y  Chill  on the s o l i d i f y i n g  is  the mould and  metal.  Basically,  c o o l i n g o f the c a s t i n g i n a c o n v e n t i o n a l  D.C.  process  primary  is c a r r i e d out in t h r e e s t a g e s :  chilled  casting cooling  i n the w a t e r j a c k e t e d mould, secondary c o o l i n g by the  flood  5  Floating , Distributor 11  —Liquid Metal =  Solid Stool  Schematic of Open-head D.C. Casting  Fig.  2.1  A schematic diagram of the openhead C h i l l C a s t i n g Process.  Direct  6  water i m p i n g i n g on the c a s t i n g and f i n a l l y by the s t a g n a n t water pool some s p e c i a l practice  applications  tertiary  in a w a t e r - c o l l e c t i o n  cooling  pit.  In  minor v a r i a t i o n o f the above  has been noted ( 2 8 - 3 0 ,  33).  The sequence of o p e r a t i o n s c a r r i e d out d u r i n g c a s t i n g are as f o l l o w s .  The s t o o l  or bottom b l o c k  D.C.  is  r a i s e d i n t o the mould to cover the opening and l i q u i d i s pumped i n t o a certain rate.  the mould.  level  Once the l i q u i d metal  metal  reaches  the bottom block i s lowered at a c o n t r o l l e d  The l e v e l  o f metal  i n the mould i s very  critical  from the s t a n d p o i n t o f s u r f a c e q u a l i t y o f the c a s t i n g and i s m a i n t a i n e d c o n s t a n t t h r o u g h the use o f a f l o a t  valve  The exact value o f the metal head i n the mould w i l l on a number o f f a c t o r s cast,  assembly. depend  i n c l u d i n g the c a s t i n g speed, the  the s e c t i o n s i z e and the p o u r i n g t e m p e r a t u r e .  When  the l e n g t h o f the c a s t i n g reaches the bottom of the p i t casting is terminated.  The e n t i r e o p e r a t i o n  once the c a s t i n g i s taken out of the p i t . continuous  D.C. c a s t i n g o p e r a t i o n  p l i e d on an i n t e r m i t t e n t of a c a s t i n g .  is  the  repeated  In a semi-  lubricant  is normally  basis to the mould at the  However c o n t i n u o u s a p p l i c a t i o n  is p r a c t i c e d i n some h o r i z o n t a l  alloy  of  ap-  beginning  lubricant  machines.  Even though the d e s c r i p t i o n  given  in the  preceding  paragraph i s a t y p i c a l  D.C. c a s t i n g sequence, t h e  is often modified f o r special  cases.  For example,  c a s t i n g aluminium slabs f o r deep drawing casting  procedure while  application  the  i s s u b j e c t e d to a reduced secondary spray c o o l i n g  place of c o n v e n t i o n a l  f l o o d c o o l i n g to o b t a i n a l a r g e r  structure  (28-30, 33).  Similarly  operation  the w a t e r j a c k e t e d mould i s r e p l a c e d by a  mould which i s cooled by water  i n a z i n c D.C.  solid  sprays.  aspects of the D.C. c a s t i n g o p e r a t i o n  with  (1-48).  Emley (2) has r e c e n t l y p u b l i s h e d a complete survey o f continuous similar  c a s t i n g of aluminium i n c l u d i n g  account f o r  e t al  alloys  c a r r i e d out p r i o r effort quality  A  has  (17).  Since the s u r f a c e and s u b - s u r f a c e q u a l i t y D.C. c a s t i n g o t s  the  D.C. c a s t i n g .  the case of copper and i t s  been p u b l i s h e d by K r e i l  cell  casting  A number of papers have been p u b l i s h e d d e a l i n g the v a r i o u s  in  of  the  determines the amount of s c a l p i n g t o be to f a b r i c a t i o n ,  considerable  research  has been d i r e c t e d towards i m p r o v i n g the ( 1 3 , 18, 2 0 - 2 2 , 24, 4 4 ) .  moulds, which r e s u l t s  The use o f  in a higher q u a l i t y  surface  shorter  surface  i s made  p o s s i b l e by i n s u l a t i n g the top p o r t i o n o f the normal with insulating marinite  lining.  mould  Recently an e l e c t r o -  magnetic mould has been developed i n Russia (18)  i n which  the metal  does not c o n t a c t the mould s u r f a c e .  been shown to y i e l d high q u a l i t y fabricated without  technique  scalping  operation.  in a c a s t i n g p l a n t  S w i t z e r l a n d has been d e s c r i b e d by Meier et a l  Like o t h e r a l s o been f e l t  in  (44).  the i n f l u e n c e o f computers has  i n D.C. c a s t i n g o p e r a t i o n s , where f o r  micro-processors  2.3  industries  has  i n g o t s which c o u l d be  the i n t e r m e d i a t e  Adoption of t h i s novel  This  are being used f o r o n - l i n e  control  Review o f Mathematical  Models i n D.C.  The f i r s t  model of heat f l o w i n  mathematical  c a s t i n g was p u b l i s h e d by Roth (26) i n 1943. solve the mathematical  (43).  casting D.C.  In o r d e r  equations a n a l y t i c a l l y  this  author  simplifying  g i b l e heat t r a n s f e r  in the mould r e g i o n of the c a s t i n g and  surface  the s i m p l i s t i c  like  to  had to make s e v e r a l  constant  assumptions,  exampl  temperature below the mould.  n a t u r e of t h i s  However i n s p i t e of the r a t h e r  operation within  Because of  model very poor agreement was  o b t a i n e d between p r e d i c t e d and measured s h e l l  c r e d i t must be g i v e n f o r  thicknesses.  crude n a t u r e of t h i s  a t t e m p t i n g to p l a c e the  a mathematical  model,  casting  framework.  A more r e f i n e d model employing a n u m e r i c a l was proposed by Adenis et al  negli-  (27)  in the e a r l y  solution  1960's.  The  9  model was developed to s i m u l a t e steady s t a t e heat f l o w casting c y l i n d r i c a l rather  magnesium a l l o y  l o n g mould used i n t h e i r  ingots.  study  metal  and mould, the i n t e r m e d i a t e  of o i l finally  Because of  the  (240' mm), the heat  f l o w i n the mould r e g i o n was d i v i d e d i n t o the top zone having p e r f e c t c o n t a c t  in  three  zones,  between the  molten  zone having a f i l m  between the c a s t i n g and i n s i d e of the mould and the bottom zone having an a i r gap.  These  authors  r e p o r t e d good agreement between measured pool depths and values p r e d i c t e d using t h e i r model.  However they have  used the l i q u i d u s  temperature  than the s o l i d u s  this  The main u n c e r t a i n t y  comparison.  associated with c h a r a c t e r i z i n g  rather  in  i n t h e i r model was  boundary c o n d i t i o n s  in  the  1ong mould r e g i on.  Kroeger and Ostrach s t a t e temperature natural  as w e l l  (31) have s i m u l a t e d the as f l u i d  flow r e s u l t i n g  from  c o n v e c t i o n d u r i n g the c a s t i n g o f a c y l i n d r i c a l  of a pure m e t a l .  Assuming c o n s t a n t t e m p e r a t u r e  boundary c o n d i t i o n on the s u r f a c e of the i n g o t f l o w model, these authors velocities  have shown t h a t  can develop from n a t u r a l  was also shown t h a t effect  steady-  as the for  field  on the l o c a t i o n of the s o l i d - l i q u i d  the  substantial  convection.  the s t r o n g v e l o c i t y  ingot  had  heat fluid  However  it  negligible  interface.  10  The mathematical  model d e s c r i b e d by Peel  ( 2 8 , 29) was also developed f o r heat f l o w i n i n g o t s as w e l l  as s t e a d y - s t a t e c o n d i t i o n s .  long moulds used and small  and  Pengelly  cylindrical Because o f  diameters c a s t . t h e  bottom of  the pool was very close to the bottom of the m o u l d . a substantial  p o r t i o n of the t o t a l  the  Since  heat was removed in  the  mould r e g i o n , these authors have f o r m u l a t e d a v a r i a b l e r e s i s t a n c e model f o r d e s c r i b i n g heat f l o w t h r o u g h t h e gap.  Experimental  the model.  r e s u l t s were l a t e r  An a t t e m p t to p r e d i c t  used f o r  the c e l l  "fine  structure  the average c o o l i n g r a t e i n the 1 i q u i d u s - s o l i d u s range was not s u c c e s s f u l  although b e t t e r  air tuning" using  temperature  agreement was  o b t a i n e d w h i l e u s i n g the g r a d i e n t of the c o o l i n g curve the s o l i d u s t e m p e r a t u r e .  Even w i t h t h i s  new t e c h n i q u e  a match between the measured and c a l c u l a t e d  values of  s i z e could o n l y be c o n s i d e r e d q u a l i t a t i v e .  The  o f the p r o p e r c h a r a c t e r i z a t i o n been w e l l  emphasized i n t h e i r  stresses  and s t r e s s d i s t r i b u t i o n  has  work.  o f both heat  during continuous c a s t i n g .  w a s . f o r m u l a t e d to c a l c u l a t e  cell  importance  of boundary c o n d i t i o n s  Mathew (32) has made an a n a l y s i s and thermal  at  the s t e a d y - s t a t e  in a c y l i n d r i c a l  p r e v i o u s models a f i n i t e - e l e m e n t  T h i s model  temperature  ingot.  numerical  flow  Unlike  the  p r o c e d u r e was  adopted f o r s o l v i n g the heat f l o w and s t r e s s e q u a t i o n s ;  and  11 the elegance of the f i n i t e  element method i n h a n d l i n g  types o f boundary c o n d i t i o n s  various  has been s t r e s s e d i n t h i s  work.  Although the model was set up f o r c a l c u l a t i n g heat f l o w i n a cylinder,  it  has a l s o been used f o r square s e c t i o n s .  stress calculations doubtful  c a r r i e d out i n t h i s  value i n a q u a n t i t a t i v e  certainty  in  the  metals near t h e i r  high  s t u d y are  of  sense because of the un-  temperature mechanical  melting  The  properties  point.  C o n t i n u i n g on the same l i n e as Peel and P e n g a l l y 29), Beattie  ( 3 0 , 33) has developed a model  heat f l o w i n r e c t a n g u l a r p l e s h i s model takes  slabs.  for  Based on s t e a d y - s t a t e  direction.  Careful  work was undertaken t o c h a r a c t e r i z e boundary The, t e m p e r a t u r e s measured by i m p l a n t i n g locations  the heat f l u x  boundary c o n d i t i o n s .  f o r p r e d i c t i n g the c e l l  structure  r e p o r t e d by Peel and P e n g e l l y . gradient  at the s o l i d u s  the g r a d i e n t  at a temperature  numerical  sizes.  employed  from  I n s t e a d of u s i n g  that  the  a u t h o r has used  solidus-1iquidus calculated  The mesh s i z e employed i n  model had to be much f i n e r  at  calculating  r a n g e , i n o r d e r t o o b t a i n a good match between and p r e d i c t e d c e l l  the  conditions.  is d i f f e r e n t  in the  princi-  experimental  The parameter  temperature, t h i s  to  thermocouples  i n the mould were used i n  (28,  calculating  i n t o account heat f l o w normal  broad face and i n the a x i a l  various  of  than t h a t  his  used by  12 Peel and Pengelly . ( 2 . 5 mm i n s t e a d of 10 mm) to reasonable c e l l  size predictions.  obtain  In s p i t e o f t h e s e  finements major d i s c r e p a n c i e s were observed i n cell  structures  for  a rectangular  predicting  s l a b , 690 x 250 mm, sub-  j e c t e d to reduced secondary c o o l i n g . cause heat f l o w p e r p e n d i c u l a r  re-  This may a r i s e be-  to the narrow face has been  neglected.  Fossheim and Madsen (48) have developed heat models f o r c y l i n d r i c a l  as w e l l  as r e c t a n g u l a r  r e c t a n g u l a r model again corresponded to a two heat f l o w i n the a x i a l discretization  flow  ingots.  Their  dimensional  and one t r a n s v e r s e d i r e c t i o n .  i n t h e i r model was based on a box  Space  integration  method and the t i m e i n t e g r a t i o n on an e x p o n e n t i a l  trans-  f o r m a t i o n of the heat c o n d u c t i o n e q u a t i o n w i t h an a l t e r n a ting direction  implicit  technique.  As w i l l  Chapter 5, the use of a t w o - d i m e n s i o n a l heat f l o w i n a r e c t a n g u l a r 1.5)  contour p r o f i l e s  a u t h o r s have made an i n c o r r e c t dimension f o r heat f l o w . o f the two s e c t i o n s , calculations  model f o r  simulating  slab 381 x 250 mm ( a s p e c t  can lead t o c o n s i d e r a b l e e r r o r .  l o o k i n g at t h e i r  be shown i n  it  ratio  In a d d i t i o n t o appears  that  this,  the  choice r e g a r d i n g t h e  I n s t e a d of c o n s i d e r i n g the  second smaller  namely 250 mm, they have performed  f o r a t h i c k n e s s o f 381 mm.  the  13 J o v i c et al  (35) have r e c e n t l y developed a t w o -  dimensional model f o r c a l c u l a t i n g  the t e m p e r a t u r e f i e l d  c o n t i n u o u s l y c a s t r e c t a n g u l a r slabs o f a l u m i n i u m as w e l l  as p r e d i c t i n g the c e l l  structure.  conditions  the  these authors have made an i n c o r r e c t  one o f t h e i r  assumptions.  alloys,  Although  s i d e r a b l e care was taken i n c h a r a c t e r i z i n g  In c a l c u l a t i o n s  con-  boundary choice  two t r a n s v e r s e  neglect  d i r e c t i o n and c o n s i d e r e d o n l y  directions.  in  involving a  r e c t a n g u l a r s l a b 360 x 1600 mm, they have chosen t o heat f l o w i n the a x i a l  in  the  Since the c a s t i n g speed employed  i n the s i m u l a t i o n was very low (1 mm/s), a x i a l  heat con-  d u c t i o n cannot be n e g l e c t e d e s p e c i a l l y when c a s t i n g a m a t e r i al l i k e hi - "\% Mn which has a high thermal better  conductivity.  choice would have been to n e g l e c t heat f l o w  in a  d i r e c t i o n normal  t o the narrow face s i n c e t h e a s p e c t  was f a i r l y  (4.4).  high  Weckman e t al  uses the f i n i t e - e l e m e n t andeutectic  alloys.  The model  The pool p r o f i l e p r e d i c t e d by the model  d u r i n g the c o n t i n u o u s  pool p r o f i l e s  c a s t i n g of a zinc  In order  the c r o s s - s e c t i o n a l  ingots.  state  method and can o n l y treat pure metals  has been compared to e x p e r i m e n t a l  cross-section.  ratio  (45) have also developed a steady  model based on axisymmetry f o r c y l i n d r i c a l  A  obtained  i n g o t w i t h a square  to make the comparison  areas o f the c y l i n d r i c a l  meaningful,  and square  14 i n g o t s were set equal  t o each o t h e r .  A simple  o f the s u r f a c e area to volume r a t i o s  indicates  a difference  o f 11% between the two cases.-  calculation that there  Thus t h i s  can only have l i m i t e d use i n s t u d y i n g heat f l o w gular  in  model  rectan-  sections.  Szargut et al  (36) have p u b l i s h e d a paper on s t e a d y -  s t a t e heat f l o w in the continuous  casting of a c y l i n d r i c a l  copper i n g o t .  Other than the f a c t t h a t t h e i r model i s  on an e x p l i c i t  finite-difference  stability their  is  scheme w i t h  problems, no new m a t e r i a l  its  based  associated  has been p r e s e n t e d  in  paper.  Recently Jensen (37) has developed a model t i n g heat f l o w i n c y l i n d r i c a l  ingots  s t a t e p a r t o f the c a s t i n g o p e r a t i o n . tion for  information  unsteady-  The r e s u l t s  of a simula-  i n g o t are  has been made to check the v a l i d i t y  Further  o p e r a t i o n and c y l i n d r i c a l  geometry.  calculations.  models have been  to s i m u l a t e heat f l o w i n D.C. c a s t i n g .  of these models are based on assumptions o f  numerical  no comparison  of the model  In summary, a number of mathematical written  reported  is given r e g a r d i n g the  procedure adopted i n the c a l c u l a t i o n .  simula-  i n c l u d i n g the  the c a s t i n g of 381 mm diameter  but very l i t t l e  for  The m a j o r i t y  steady-state  The r e m a i n i n g models  1 5  which apply t o r e c t a n g u l a r sional  s e c t i o n s are a l s o o n l y two dimen-  w i t h heat f l o w n e g l e c t e d e i t h e r  verse d i r e c t i o n s  or the a x i a l  i n one of t h e  trans-  direction.  The p r e s e n t work has been undertaken t o d e v e l o p a truly  t h r e e dimensional  model to s i m u l a t e b o t h unsteady and-  steady s t a t e heat f l o w i n c a s t i n g square and sections.  F u r t h e r as w i l l  f o r t e s t i n g the v a l i d i t y under s p e c i a l  limiting  rectangular  be shown t h i s model has been used  o f u s i n g a two d i m e n s i o n a l  cases.  version  Chapter 3  DEVELOPMENT OF THE HEAT FLOW MODEL  3.1  Introduction The problem o f d e v e l o p i n g a model f o r  of heat f l o w i n D.C. c a s t i n g e s s e n t i a l l y s o l u t i o n of a p a r t i a l  differential  unsteady heat c o n d u c t i o n .  the  analysis  involves  equation  the  describing  Because of the slow  casting  speeds employed i n t h i s o p e r a t i o n and the high thermal ductivity  o f the m a t e r i a l  negligible  heat c o n d u c t i o n  i n the models f o r valid.  cast the normal assumption i n the a x i a l  of  d i r e c t i o n , made  the continuous c a s t i n g o f s t e e l  The problem of c o n s i d e r i n g heat f l o w  is  in a l l  directions  i s made s i m p l e r  ingot.  axisymmetry can be assumed the heat f l o w  If  con-  not three  i n the case o f a c y l i n d r i c a l  i s reduced to two d i r e c t i o n s  namely r a d i a l  problem  and a x i a l  ( 27-29 , 31 , 32 , 36 , 45 ) .  When d e v e l o p i n g heat f l o w models f o r sections  such a s i m p l i f i c a t i o n  rectangular  i s not p o s s i b l e .  However  order to reduce the problem also to two dimensions  it  is  usual  to  the  to n e g l e c t  narrow face  heat f l o w i n the d i r e c t i o n  ( 3 0 , 37, 4 8 ) .  assumption f o r  Although t h i s  normal  i s not a bad  slabs which are t h i n and w i d e , i t 16  leads  to  in  considerable error  f o r s e c t i o n s w i t h a small  aspect  ( r a t i o of the t r a n s v e r s e d i m e n s i o n s ) , as w i l l Chapter 5.  In such cases i t  flow in a l l  three  n a t u r e o f the  i s i n t e r e s t i n g also t o s t u d y  p o r t i o n o f the c a s t i n g as i t  sizeable f r a c t i o n  of the t o t a l  model undertaken i n t h i s three-dimensional  in heat  directions.  casting operation, i t  3.2  be seen  i s necessary to c o n s i d e r  Because o f the s e m i - c o n t i n u o u s  unsteady-state  ratio  heat  D.C. the  c o u l d occupy a  casting time.  Thus  the  study i s based on unsteady  state,  flow.  Assumptions Made i n the Model The f o l l o w i n g assumptions have been made i n  the  development o f the model. 1.  In t h e case o f the s i m u l a t i o n o f r e c t a n g u l a r  or  square s e c t i o n s t w o - f o l d symmetry has been assumed for  the mid face p l a n e s , and t h e c a l c u l a t i o n s  performed only f o r one q u a r t e r o f the' c a s t i n g . the case o f jumbo z i n c  i n g o t s , because o f  presence o f o n l y one symmetry p l a n e , are made f o r o n e - h a l f of the 2.  are In  the  calculations  casting.  M i x i n g i n the l i q u i d pool has been n e g l e c t e d , a s t a g n a n t pool has been assumed.  i.e.  This has been  18 shown e x p e r i m e n t a l l y  (29) and t h e o r e t i c a l l y  be a reasonable assumption f o r n o n - f e r r o u s with shallow pools.  involving  castings  e1ectro-magnetic  In such cases the thermal  conductivity  the l i q u i d c o u l d be i n c r e a s e d t o r e f l e c t  the  3.  of  stirring  as has been done i n the case o f the c o n t i n u o u s of s t e e l  to  The assumption may not be v a l i d  however i n s i t u a t i o n s stirring.  (31)  casting  (71 ) .  In the s i m u l a t i o n o f z i n c - j u m b o c a s t i n g the  thermal  conductivity  o f z i n c has been assumed c o n s t a n t and  the same f o r  both l i q u i d and s o l i d .  physical  properties  thermal  conductivity  From the  given by T o u l o u k i a n  thermo-  (51),  the  of s o l i d z i n c has a value  twice  t h a t of the l i q u i d .  Thus a s s i g n i n g a c o n s t a n t  solid  thermal  value f o r  conductivity  as the s o l i d corresponds the l i q u i d p o o l . c o n s t a n t thermal small  since  it  the l i q u i d as w e l l  to a very m i l d s t i r r i n g  The e r r o r  i n t r o d u c e d by assuming  conductivity  for solid  zinc  is  changes o n l y by 20% between room  t e m p e r a t u r e and i t s m e l t i n g p o i n t .  Moreover  and P e n g e l l y  the e f f e c t  thermal  (29) have claimed t h a t  conductivity  aluminium c a s t i n g . stant  is n e g l i g i b l e In t h i s  Peel of  i n the case of  study a p p r o p r i a t e  values have been used f o r  regions.  in  s o l i d and  con-  liquid  The value used i n the mushy r e g i o n on  1 9  the r e l a t i v e  amount of s o l i d and l i q u i d  assuming e q u i l i b r i u m s o l i d i f i c a t i o n .  fractions  It  is  s i b l e to take i n t o account v a r y i n g thermal ductivity puter 4.  t h r o u g h minor m o d i f i c a t i o n  con-  o f the com-  program.  The s p e c i f i c  heat i s a l l o w e d to vary as a f u n c t i o n  of temperature.  However no i t e r a t i v e  are made w i t h i n a time i n t e r v a l . specific  calculations  The value  of  heat i s e v a l u a t e d at the b e g i n n i n g o f a  time i n t e r v a l  based on the t e m p e r a t u r e s  from the p r e v i o u s  interval  ing t h a t p a r t i c u l a r  3.3  pos-  obtained  and i s kept c o n s t a n t  interval.  The t e m p e r a t u r e  pendence o f s p e c i f i c  heat i s a d e q u a t e l y  through use of small  time  described  Conditions  The u n s t e a d y - s t a t e , h e a t - c o n d u c t i o n e q u a t i o n conductivity  c o - o r d i n a t e system can be w r i t t e n 2  2  k (i-J-) + k 0  where  x,y,z, k,  1  )  = pc *z  temperature  the t h r e e  the thermal  three  a Cartesian  as,  + k  3y^  T denotes the  for  in  2  ( ^ - o  *  de-  intervals.  Heat Flow Equation and Boundary  dimensions w i t h c o n s t a n t thermal  dur-  directions  conductivity  . . . 6  3.1  p,  the  c,  the s p e c i f i c  t,  the  density  time.  A detailed l i s t Appendix 1 . conditions Initial  heat  o f the symbols used is a l s o g i v e n  In o r d e r to solve Eq. ( 3 . 1 )  initial  before and boundary  are r e q u i r e d .  condition: Physically,the  ditions  initial  condition describes  the con-  e x i s t i n g a t the b e g i n n i n g of c a s t i n g o p e r a t i o n when  the bottom b l o c k i s r a i s e d to cover the mould o p e n i n g , and the mould i s f i l l e d w i t h molten m e t a l .  Since the  o f the mould i s c a r r i e d out i n a very s h o r t time (usually  around 1 m i n u t e ) ,  it  i s assumed t h e r e  heat l o s s d u r i n g t h i s p r o c e d u r e .  Thus the i n i t i a l  t u r e i s taken to have a c o n s t a n t u n i f o r m value the c a s t i n g and equal  is  to the p o u r i n g  filling interval negligible tempera-  throughout  temperature.  T = Tp at t = o, o^x-sX, o^y^Y, o^z^Z where  Tp i s the p o u r i n g  Top Boundary  speed.  3.2  temperature.  Condition:  When the c a s t i n g o p e r a t i o n l i q u i d metal  ...  into  i s s t a r t e d , the f l o w  the mould is a d j u s t e d t o match t h e  Since t h i s metal  of  casting  n o r m a l l y comes from a h o l d i n g  f u r n a c e , the temperature o f the metal  remains s t e a d y  a c a s t i n g o p e r a t i o n . Thus to s i m u l a t e t h i s  top  boundary  c o n d i t i o n , the top p o r t i o n o f the c a s t i n g i s always a t the p o u r i n g  if  t>o,  is possible T = T  kept  temperature.  T = Tp  It  during  z = 0,  o«x<X,  o^y^Y  ...  3.3  ...  3.4  t o have (x,y),  t>o, z = o  the t e m p e r a t u r e d i s t r i b u t i o n  at the top i s known more  accurately. Bottom Boundary  Condition:  This i n v o l v e s  the t r a n s f e r o f heat from t h e  o f the i n g o t t o the p l a t t e n  w i t h which i t  is in  contact.  Since the p l a t t e n i s not p r o v i d e d w i t h any c o o l i n g , a small  amount o f heat f l o w s through i t .  bottom  I n the  only  present  work t h i s boundary c o n d i t i o n has been handled t h r o u g h use of a h e a t - t r a n s f e r 9T  -k —  coefficient.  = h (T-T, ) , t > o , z =  <j Z  D  A c o n s t a n t value o f  o«y.$Y  Z, O « X J ? X ,  2  .209 kW/m K ( .005 c a l / c m  2  . ° c s)  been employed t h r o u g h most o f the c a l c u l a t i o n s in t h i s work.  Although t h i s  have some e f f e c t  the  heat-transfer  d u r i n g the s t a r t  ...  3.!  has  presented  coefficient  up o f t h e c a s t i n g , i t  might has  no e f f e c t on the s t e a d y - s t a t e sion o f the s u b j e c t w i l l  operation.  be reserved f o r  A further a later  s e c t i o n on  the importance o f unsteady heat f l o w i n Chapter Side Boundary  5.  Conditions:  This i s ditions  discus-  the most i m p o r t a n t of a l l  s i n c e the i n g o t  again a h e a t - t r a n s f e r  the boundary con-  i s cooled only from the s i d e s .  coefficient  type o f boundary  Here  condition  has been employed. -k U  = h(z)  -k f j - = h ( z )  h(z)  t > o , x = X, o«ysY, o<z«Z  ...  3.6  (T-T )  t > o , y = Y, o«x«X, o<z«Z  ...  3.7  W  ay  where  (T-T) W  oX  i s the o v e r a l l  heat-transfer  coefficient  which  i s a f u n c t i o n o f t h e p o s i t i o n along the z - a x i s .  Thus the  heat-transfer  have a d i f -  ferent  coefficient  used i n the mould w i l l  value than the one used i n the spray r e g i o n .  acterization  of t h i s h e a t - t r a n s f e r  coefficient  p o r t a n t to the accuracy o f the s i m u l a t i o n s . used w i l l  is  Char-  very  The e x a c t  be g i v e n i n the s e c t i o n d e a l i n g w i t h the  imvalue  valida-  t i o n o f the model. Centre Boundary  Conditions:  Wherever p o s s i b l e ,  symmetry  i n s i m p l i f y i n g the p r o b l e m .  c o n d i t i o n s were  Thus i n the case o f a  applied  23 rectangular  slab w i t h two symmetry p l a n e s , zero heat  flux  has been assumed a t the m i d - f a c e planes and t h e  calcula-  t i o n s are performed only f o r one q u a r t e r o f the  casting,  as mentioned -k — o  earlier. = o  t>o,  x = o,  o«y«cY,  o<z«Z  ...  3.8  = o  t>o,  y = o,  o^x-sX,  o<z$Z  ...  3.9  X  -k —  The d i f f e r e n t i a l initial  and boundary c o n d i t i o n s  a complete mathematical  3.4  e q u a t i o n Eq. ( 3 . 1 )  Method o f  Eqs. ( 3 . 2 )  together with to  (3.9)  Solution methods to s o l v e Eq. ( 3 . 1 )  p r e c l u d e d by the complex n a t u r e o f the boundary as the growth o f the i n g o t  Similarly  comprise  s t a t e m e n t o f the p r o b l e m .  The use o f a n a l y t i c a l  as w e l l  semi-analytical  direction.  integral-  ( 8 0 ) , although  useful  m o d e l l i n g heat f l o w i n the c o n t i n u o u s c a s t i n g of s t e e l , l i m i t e d use i n the p r e s e n t problem due t o t h e c o n d u c t i o n and the unsteady s t a t e .  equation.  of time and e f f o r t  i n v o l v e d in the development o f choice o f the  have  A numerical partial  Because of the c o n s i d e r a b l e  computer program, the i n i t i a l  in  importance  method has t h e r e f o r e been adopted f o r s o l v i n g the differential  is  conditions  i n the c a s t i n g  methods, l i k e the  p r o f i l e method developed by H i l l s  of axial  the  amount  the  numerical  24 method i s very i m p o r t a n t .  The method chosen s h o u l d be a b l e  to y i e l d a reasonable l e v e l puter  o f accuracy w i t h moderate com-  requirements.  Two o f the most  popular  methods f o r s o l v i n g  r e l a t e d to heat f l o w and s o l i d i f i c a t i o n and f i n i t e - d i f f e r e n c e  techniques.  problems  are f i n i t e  The use o f  element  finite-  element methods f o r s o l v i n g s o l i d i f i c a t i o n  problems  are  d i s c u s s e d by s e v e r a l  Although  this  authors  method i s e l e g a n t l y  (32, 72-74).  s u i t e d f o r handling steady s t a t e  and has an edge over f i n i t e - d i f f e r e n c e complex geometry, dimensional  it  method  in  has a r a t h e r l i m i t e d use i n  handling three  t r a n s i e n t problems from the s t a n d p o i n t  cost of computation. finite-element  Emery and Carson (74)  and f i n i t e - d i f f e r e n c e  t i o n o f a two dimensional that  the f i n i t e - e l e m e n t  methods i n e f f i c i e n c y  a finite-difference  the  have compared  heat f l o w p r o b l e m .  It  the f i n i t e - d i f f e r e n c e  problems i n t h r e e d i m e n s i o n s .  of  methods f o r the  from t h i s a n a l y s i s  problems,  for  is  soluclear  method exceeds transient  Thus i t was d e c i d e d t o use  method f o r s o l v i n g Eq.  A good t r e a t m e n t o f the f i n i t e  (3.1).  difference  methods  can be found i n any s t a n d a r d t e x t book on n u m e r i c a l  methods  ( 6 4 , 6 5 , 78, 7 9 ) .  general  be c l a s s i f i e d  Finite-difference  methods can i n  i n t o t h r e e broad c a t e g o r i e s  namely  explicit,  25 fully  implicit  and. e x p l i c i t - i m p l i c i t  The e x p l i c i t  finite  techniques.  d i f f e r e n c e methods are  s i m p l e s t o f the t h r e e types and do not r e q u i r e solution of equations.  the  simultaneous  In t h i s method the f u t u r e  tempera-  t u r e of a node i s c a l c u l a t e d based on the p r e s e n t  tempera-  t u r e of t h a t p a r t i c u l a r three-dimensional  node and s u r r o u n d i n g nodes.  p r o b l e m , an i n t e r i o r  rounded by s i x nodes.  imposed on the t i m e steps  explicit that  be used f o r a given node s i z e , f o r p r o p e r s t a b i l i t y  the numerical interval  method.  The r e s t r i c t i o n  i s kept the same.  the case o f z i n c and f o r For a D i r i c h l e t  stability  criterion  kAt , 1 ~T pc Ax (  where  k is  ,  1  a small  This i s  of  time  three problem,  illustrated  node s i z e used i n  type boundary c o n d i t i o n ,  for  this  the  is ,  1 x  ~2  ~ 2  Ay  AZ  the thermal  p i s the  <  K  1  2  conductivity  density  c i s the s p e c i f i c AX,  the  case as compared to the one d i m e n s i o n a l  the node s i z e  work.  imposed on the  i n c r e a s e s by a f a c t o r of t h r e e f o r  dimensional if  be s u r -  The main d i s a d v a n t a g e of the  method i s the r e s t r i c t i o n can  node w i l l  In a  heat  A y , A z are node s i z e s  At i s the time  interval.  i n x, y and z dimensions  26 Substituting  the a p p r o p r i a t e  values f o r the  different  v a r i ables , k = 113 W/m K, p = 7140 k g / m ,  c = .3830 J/gK  Ax = 15.24 mm,  Az = 20 mm  3  Ay = 15.24 mm,  one o b t a i n s At <? 1.08  s.  When the above-mentioned c a l c u l a t i o n s  are r e p e a t e d w i t h a  c o n v e c t i o n type boundary c o n d i t i o n a lower v a l u e , 0.55 (see end of Appendix 1 f o r t h i s for At.  Other f i n i t e  are not s u b j e c t At, eight  calculation)  t o any s t a b i l i t y  criterion  the e x p l i c i t  t o play i n t h r e e dimensional  method, the f u t u r e  implicit  and v a l u e s  method, u n l i k e  the  explicit  direction,  the i m p l i c i t method  Although most of the elements  the  tempera-  essentially equations.  problem w i t h ten nodes i n each  i t would mean s o l v i n g one thousand  simultaneously.  to  temperature  the s o l u t i o n of a system of s i m u l t a n e o u s  For a t h r e e - d i m e n s i o n a l  role  problems.  Since these s u r r o u n d i n g  t u r e s are not known a p r i o r i  it  methods have a l i m i t e d  temperature of a node i s r e l a t e d  o f the s u r r o u n d i n g nodes.  of methods  In c o n c l u s i o n  p r e s e n t t e m p e r a t u r e of t h a t node and the f u t u r e  involves  below  imposed by e x p l i c i t  have been commonly used i n t h i s work.  In the f u l l y  obtained  d i f f e r e n c e methods d e s c r i b e d  f o l d g r e a t e r than t h a t  can be s a i d t h a t  is  s,  equations in  the  coefficient matrix w i l l  be zeros  (there w i l l  be o n l y  non zero terms per row) and need not be s t o r e d , the s o l u t i o n procedure l i k e  the Gauss-Seidel  In t h i s  regard the f u l l y  methods c o u l d be compared to the f i n i t e However t h e r e are no s t a b i l i t y fully  implicit  implicit  stability  simultaneous  procedures  (77).  It  +  has an u n c o n d i t i o n a l e r r o r o f the  originally stability  order  (At) ]. 2  In t h i s method w i t h i n each time i n t e r v a l calculations  are performed i n t h r e e s t a g e s .  stage the c a l c u l a t i o n s and e x p l i c i t by s i m i l a r  called  methods have been developed  and converges w i t h d i s c r e t i z a t i o n 2  of  The procedure adopted i n t h i s work was  proposed by B r i a n  [ ( A X )  with  methods and problems  i n i m p l i c i t methods, s p e c i a l  alternating direction  0  implicit  methods.  a s s o c i a t e d w i t h s o l v i n g a l a r g e number o f  (75-77).  would  problems a s s o c i a t e d  encountered i n e x p l i c i t  equations  still  element methods.  In o r d e r to overcome the d i f f i c u l t i e s conditions  best  method would  take up c o n s i d e r a b l e computer time as the s o l u t i o n have to be i t e r a t e d .  seven  are made i m p l i c i t  i n the y and z d i r e c t i o n s .  procedures  A t , the  In the  i n the This i s  i n the y and z d i r e c t i o n .  first  x-direction followed Finally  the new t e m p e r a t u r e s at the end o f the t i m e i n t e r v a l  is  28 calculated  u s i n g an e x p l i c i t  * -  T  T  n  -  ? x  6  formula  * T  +  6  ? y n  ?  T  +  z  5  T  n  3  '  1  0  At/2  T - T  _  n  ^  6  -  x  T  +  6  ^  T  +  y  6  z  T  ...3.11  n  At/2  T  -Tp  «  =  x  T  + 6*  *  9 +  T  y  + 5% T  ...  3.12  At/2  T  ,i-T  n+1  9 6^  _  n  x  T  6  T  y  **  +  6  9  ,  z  T  ***  At where  6  2 x  operators  ,  6  2 2 and 6 y z  defined  are  the c e n t r a l  difference  by  2 x  6  T  i,j,k  T  i-i,j,k "  2  T  i , j ,-k * A X  T  i+1 ,j,k  2  2 y  6  T  i . j  .k  =  T  i  .3-1  ,k  "  2  T  <  4  ..i . k  y  *  i  +  1 ,k  2  2 6  z i . j .k T  =  T  i,,i,k-1 '  2 T  i,,i,k AZ  2  *  T  i ,.i , k + l  ...  3.13  29 i,  j , k being the l e t t e r s  used f o r numbering  the  nodes i n the x , y and z d i r e c t i o n s . T*  5  T**,  j***  temperatures  are the i n t e r m e d i a t e  c a l c u l a t e d which do n o t have any  special  meaning.  T  n +  n  fictitious  and T -j are the temperatures  and the end o f a time i n t e r v a l are the d i s t a n c e  at t h e  At.  beginning  A X , Ay and A Z  between nodes i n x , y and z  directions .  S u b s t i t u t i o n of f i n i t e  difference  to 3.13 i n place o f the p a r t i a l sults  differential  i n a set o f simultaneous e q u a t i o n s  diagonal  coefficients  procedure e x i s t s . calculations this  equations  method.  equation  involving  f o r which a very e f f i c i e n t  Thus i n s p i t e of the t h r e e  tri-  solution  interval,  procedure than the f u l l y  implicit  D e t a i l s o f the procedure by which the c a s t i n g  divided into d i f f e r e n t nodal e q u a t i o n s  nodes and the s e t t i n g up o f  are presented i n Appendix  case o f a r e c t a n g u l a r p a r a l 1 el piped in a l l  the t h r e e dimensions  ferent  types o f nodal  is  the  For a simple  with regular  there w i l l  equations.  1.  re-  sets'of  to be c a r r i e d out w i t h i n each t i m e  i s a more e f f i c i e n t  Eqs . 3.10  node  size  be t w e n t y - s e v e n  dif-  30  One o f the problems w i t h the a l t e r n a t i n g technique  i s the t r e a t m e n t of the r a d i a t i o n  dition. Introduction directly  boundary  of the r a d i a t i o n boundary  T  h  equations  av  = oe (  T  n^TT  ••• '  )  3  1 4  ct  assuming an average value f o r the s u r f a c e  h  f l v  is  calculated  temperature.  i s the t e m p e r a t u r e o f the ambient medium, a i s  S t e f a n - B o l t z m a n n c o n s t a n t and E , the e m i s s i v i t y . the low temperatures  boundary  relationship  where the average heat t r a n s f e r c o e f f i c i e n t  g  the  In o r d e r to overcome the p r o b l e m , the  c o n d i t i o n can be l i n e a r i z e d using the 4 4  con-  condition  i n t o the heat balance e q u a t i o n r e n d e r s  non-linear.  T  implicit  the Because  encountered i n aluminium and z i n c  i n g , t h i s boundary c o n d i t i o n w i l l  Here  have a n e g l i g i b l e  The growth o f the i n g o t i s s i m u l a t e d t h r o u g h  of  cast-  effect.  periodic  a d d i t i o n o f a s e t o f nodes at the p o u r i n g t e m p e r a t u r e t o  the  top p o r t i o n of the c a s t i n g , as has been c a r r i e d o u t by Ballantyne  (66).  The time i n t e r v a l  over which a .row o f  nodes i s added i s c a l c u l a t e d from the c a s t i n g speed and the node s i z e i n the c a s t i n g d i r e c t i o n .  Thus f o r a c a s t i n g  speed o f v and node s i z e A Z , an a d d i t i o n  i s made every  At  a  given by  31 The time i n t e r v a l s e l e c t e d such t h a t At  At f o r model c a l c u l a t i o n s  will  be an i n t e g r a l  is  multiple  of At  a At  a  = N (At)  where N i s an i n t e g e r . control  o f At and Az.  ... Thus t h e r e i s no independent  Initially  performed to check the e f f e c t  numerical  heat d u r i n g  To overcome t h i s  employed by o t h e r workers study.  c  =  m  c  L  solidification  finite  difference  problem, a technique  commonly  heat is r e l e a s e d l i n e a r l y  over  temperature range as f o l l o w s : c +  m  c  work  ( 2 9 , 33, 66) has been adopted  The l a t e n t  the 1 i q u i d u s - s o l i d u s  where  ...  L T  V  T T  s  denotes the s p e c i f i c  heat i n the mushy  region  the average s p e c i f i c  heat e v a l u a t e d a t  the  solidus  temperatures.  and l i q u i d u s  the l a t e n t  T , T  $  of  8.  poses some problems i n the i m p l i c i t  in t h i s  results.  node s i z e and  Values of N used i n t h i s  The r e l e a s e of l a t e n t  technique.  were  and thus the value  based on c a s t i n g speed,  the cost of c o m p u t a t i o n . range from 2 to  calculations  of N, on the p r e d i c t e d  The e f f e c t was found to be very small N was a r r i v e d a t ,  3.16  heat of  the l i q u i d u s  solidification and s o l i d u s  temperatures.  3.17  This method r e q u i r e s undergoing s o l i d i f i c a t i o n solidus If  interval  t h a t the t e m p e r a t u r e o f  fall  within  the  liquidus-  a t some p o i n t d u r i n g the  calculation.  however l a r g e heat flows take place i t  is  some nodes to jump from above the l i q u i d u s s o l i d u s w i t h i n one time i n t e r v a l .  because o f the high heat t r a n s f e r  will  cast.  When t h i s  t o below  o f D.C.  for  the often  casting  coefficients  from f l o o d c o o l i n g and the high thermal the m a t e r i a l  possible  This problem i s  encountered i n heat f l o w c a l c u l a t i o n s  nodes  resulting  conductivity  of  happens the l a t e n t , heat  not be r e l e a s e d from t h a t node.  This p o i n t can be  a p p r e c i a t e d from the simple c a l c u l a t i o n amounts to over 60% of the t o t a l  that  latent  heat  heat removed from the  top  o f the c a s t i n g to the bottom of the p o o l .  In o r d e r to avoid t h i s  problem a p o s t - i t e r a t i v e  c o r r e c t i o n procedure has been employed. the t e m p e r a t u r e  In t h i s  technique  of the nodes which jumped from one phase  to another i s m o d i f i e d using a heat b a l a n c e a p p r o a c h . example f o r a node going from above the l i q u i d u s the s o l i d u s Let  to  An  below  i s given below.  T.j  be the temperature o f node b e f o r e a t i m e  T^  the l i q u i d u s  T  s  the s o l i d u s  T  9  the t e m p e r a t u r e a f t e r  temperature temperature the time  interval  interval  33 c.j , the s p e c i f i c  heat e v a l u a t e d  p  the d e n s i t y of the  Cr,,  c  the s p e c i f i c  3  at  material  heat of the m a t e r i a l  in  the  mushy zone and e v a l u a t e d at the s o l i d u s v  temperature  the volume of the node  The net change i n the heat c o n t e n t o f t h a t = vpc-j  node  (T^-Tg)  The f o l l o w i n g comparison decides whether t h e node w i l l end up i n the mushy r e g i o n or i n the s o l i d r e g i o n . pv  C l  then the node w i l l  (T T ) r  2  >  P V C  1  ( T  1  - T J  +  v c (T -T ) P  end up i n the s o l i d zone.  corrected temperature w i l l  T  3  "  2  £  s  Otherwise  be i n the mushy r e g i o n .  c o r r e c t e d t e m p e r a t u r e when the above-mentioned is s a t i s f i e d  If  the  The  inequality  i s g i v e n by  T  s  " " W S J " V V (  J  •••  3  -  1 8  S i m i l a r e q u a t i o n s are o b t a i n e d f o r nodes j u m p i n g from the liquid  to mushy and mushy to s o l i d phases.  the c a s t i n g undergoes r e h e a t i n g t h i s in  In cases where  procedure  is  repeated  reverse.  This c o r r e c t i v e latively  procedure i s very handy i n  long time i n t e r v a l s  a l l o w e d by the  using  unconditional  34 stability  o f the a l t e r n a t i n g d i r e c t i o n  difference  scheme.  implicit  Howevera 1arge i n c r e a s e  finite  i n time  interval  to decrease the c o s t o f c o m p u t a t i o n i s not recommended on two c o u n t s . perties  Firstly  the values of t h e r m o - p h y s i c a l  t h a t change w i t h temperature are e v a l u a t e d a t  b e g i n n i n g o f a time i n t e r v a l interval.  that  Thus having a very l a r g e time step may c o u n t e r a c t  Secondly when a l a r g e time i n t e r v a l iterative  the  and kept c o n s t a n t d u r i n g  the accuracy sought w i t h v a r y i n g thermophysica1  field  pro-  properties.  i s u s e d , the  c o r r e c t i o n procedure a l t e r s  the  post-  temperature  so much t h a t the program goes i n t o an u n s t a b l e mode.  Thus a l t h o u g h t h e r e are no c o n s t r a i n t s s i z e of time i n t e r v a l s  regarding  a r i s i n g from the s o l u t i o n  the  procedure,  c a u t i o n should be e x e r c i s e d i n s e l e c t i n g the time  interval  both from the v i e w p o i n t o f the accuracy o f s o l u t i o n  and t h e  cost of computation.  ranging  In t h i s  s t u d y , time i n t e r v a l s  from 2 t o 1 0 seconds have been used. s e l e c t e d depended on the metal  cast, size of  c a s t i n g speed and s e v e r i t y o f c o o l i n g  3.5  Mathematical  Check f o r  The e x a c t  Internal  value  section,  conditions.'  Consistency  of  the Computer Program Although a m a j o r i t y of mathematical numerical  models r e q u i r e a  method t o o b t a i n a s o l u t i o n , a n a l y t i c a l  methods  35 are very u s e f u l  t o . c h e c k out the numerical  complex problem using f i n i t e  differences,  which o f t e n have a n a l y t i c a l son to the numerical  techniques.  solution  results.  limiting  t o o l s f o r debugging the computer program i n the stages o f model this  development.  check, s o l i d i f i c a t i o n  It  compari-  valuable initial  should be noted t h a t  ( l a t e n t heat)  phenomena of the model are not  cases  can be used f o r  They are a l s o very  In a  in  and growth  included.  The s o l u t i o n of the t h r e e dimensional  heat  flow  e q u a t i o n d e f i n e d i n the r e g i o n - a « x « a , - b « y « b , - c ^ z ^ c  with  zero i n i t i a l  is  t e m p e r a t u r e and u n i t s u r f a c e t e m p e r a t u r e  given by Carslaw  =  1  and Jaeger (83) as  _ 64 n  E°° E°° Ii SL = O m=o n = o 00  3  cos'(2£+l)nx  cos (2m+l)ny cos  2a  where  (3 =  an r( 2  4  ( - 1 ) (2£+l)  2b  2 + 1)  a  2  2  n  (2m+l)  (2n+1)nze  (2n+!) •Bl ,m ,n t n  2c  +(2 +l) m  W  b  2  2  +(  2n + l)  V  2  j  3  ] g  and a is the thermal diffusivity, t is the time, 2 a , 2b, 2c the dimensions of rectangular paral1 el piped , (x,y,z) is the location of where the temperature is calculated.  36 In t h i s  specific  example heat f l o w i n a cube o f  609.6 mm was c o n s i d e r e d .  The thermal  diffusivity  of  side  the  2 material  used i s 12.95 mm / s .  uniformly all  Initially  the m a t e r i a l  is  at a t e m p e r a t u r e of 260°C and s u b s e q u e n t l y  for  time t>o the s u r f a c e s o f the cube are m a i n t a i n e d  537.7°C.  The c a l c u l a t e d temperatures  at the c e n t r e  the cube are p r e s e n t e d i n Table  I  analytical  shown i n t h i s  methods.  been c a l c u l a t e d different  The r e s u l t s  for  As can be seen the  c a l c u l a t e d values approach the a n a l y t i c a l number o f nodes used i n the c a l c u l a t i o n the r e s u l t s  from t h i s  table  for this  and a n a l y t i c a l  time i n t e r v a l  the numerical  results  analytical  increases.  Some o f  in a d i f -  The p e r c e n t e r r o r on t h e y  results.  It  constant.  can be seen, t h a t side, the  results .  between numerical II.  between  numbers very c l o s e t o  The i n f l u e n c e of the time i n t e r v a l  Table  but  as the  of 36 s , using 21 nodes per  solution yields  have  numerically  a x i s has been c a l c u l a t e d as the p e r c e n t d i f f e r e n c e the numerical  and  o f 36 s ,  t a b l e have been p l o t t e d  ferent fashion in Fig. 3 . 1 .  of  both numerical  f o r a c o n s t a n t time i n t e r v a l  node s i z e s .  at  and a n a l y t i c a l  on the  comparison  method i s p r e s e n t e d  in  For these runs the node s i z e has been kept It  can be seen from t h i s  s i z e c o n s i d e r e d , the time i n t e r v a l  table that  for  had a v e r y small  the node effect  37  Num aer of Nodes Per Edqe A n a l y t i cal  Ti me (s)  N = 9  360  N = 13  N = 17  N = 21  268. 87  265.51  264.30  263.73  262 .66  720  311 . 01  305 .41  303.31  302.31  300 .57  1080  365 94  362.1 7  360.83  360.21  357 .48  1440  414 .07  412.06  411.37  411 .05  410.51  1800  450 .67  449 .69  449.35  449 .20  448.95  2160  476 .01  476.57  476 .42  476.35  476.24  2520  495 .55  495.38  495 .33  495.30  495 .27  2880  508 .48  508.45  508.44  508.44  508.44  3240  517 .47  517.50  517.52  517.52  517.54  3600  523 .70  523 . 76  523.79  523.80  523.81  Table  I  Comparison between n u m e r i c a l analytical  values of  and  temperatures  (°C) at the c e n t r e o f a c u b e , as a f u n c t i o n of time f o r d i f f e r e n t of nodes .  sizes  F i g . 3.1  The e f f e c t  o f the number of nodes on p e r c e n t  e r r o r at the end of d i f f e r e n t  time  intervals.  39  Ti me (s)  | A t = 180s  360  264.71  263.73  263 .70  262 .66  720  300.83  302.31  302 .35  300.57  1080  359 .52  360.21  360 .23  357.48  1440  410.80  411 .05  411 .06  410.51  1800  449 .1 2  449 .20  449.20  448.95  2160  476.32  476 .35  476.35  476 .24  2520  495.30  495 .30  495.30  497 .27  2880  508.44  508.44  508.44  508.44  3240  517.52  517.52  517.52  517 .54  3600  523.80  523.80  523.79  523.81  Table I I  = 36 s  At  At  = 18 s A n a l y t i c a l  Comparison between numerical analytical  values of  temperatures  centre  (°C) o f a cube, as  of a f u n c t i o n of t i m e , f o r time steps .  and  different  40 on the r e s u l t s  o f the c a l c u l a t i o n .  However t h i s  comment  holds good o n l y w i t h r e s p e c t to c e n t r e t e m p e r a t u r e s . on the o t h e r hand comparisons were made f o r away from the c e n t r e then the c a l c u l a t i o n s time i n t e r v a l s w i l l  show a p p r e c i a b l e  performed w i t h c o a r s e r time Convective Boundary  temperatures done w i t h  difference  developed f o r a l l  procedure was f o l l o w e d .  helped i n  interior  t o zero i n two d i r e c t i o n s  the t h r e e dimensional  The a n a l y t i c a l  In  order  different  coefficient  and a f i n i t e  The r e s u l t  value  obtained  model should y i e l d a zero  i n two o f the  checking  nodes.  The heat t r a n s f e r  i n s e r t e d i n the t h i r d d i r e c t i o n .  gradient  that  Conditions:  to check the e q u a t i o n s o f the s u r f a c e node a  was set equal  to  finer  steps.  The above-mentioned comparisons the e q u a t i o n s  If  from  temperature  directions.  solution  i n a slab o f t h i c k n e s s  f o r one-dimensional  2L, i n i t i a l l y  s u b j e c t e d to c o n v e c t i v e  at a u n i f o r m  type boundary c o n d i t i o n  is  heat  flow  temperature given  by (83) 8( x , t ) e  6  _ 2  E-j  e  -6*  n  6  n  2  Sin 6 cos U x/L) 6 + Sin 6 cos 6^ n n n n  0  tan  (at/L )  3.20  3.21 k  41 e (x,t)  = T(x,t)  - T  e  = T  - T  o  Where  o  e  0  T  and T  are the i n i t i a l  and ambient  temperatures  0  oo  and h, the heat t r a n s f e r  coefficient,  the value o f x i s  measured w i t h r e s p e c t t o c e n t r e of the t h i c k n e s s . check again the thermal 12.95 m m / s . 2  diffusivity  o f the m a t e r i a l  Other values used are T  5 3 7 . 7 ° C , L = 305.8 mm, h = 565.4 Comparison between the numerical are presented i n F i g . 3 . 2 .  It  ference between the numerical  In  used i s  = 260°C, T  W/m K, k = 46.7 W/m K. 2  and a n a l y t i c a l  results  can be seen t h a t the and a n a l y t i c a l  dif-  methods  crease i n going from the c e n t r e o f the s l a b to the However w i t h s h o r t e r time i n t e r v a l s can be o b t a i n e d between the two. intervals  some t i m e .  time  finite  This dif-  are s t a b l e and w i l l  The time i n t e r v a l s  study were s e l e c t e d to avoid t h i s  agreements  at the s u r f a c e .  U n l i k e the e x p l i c i t  ference methods these o s c i l l a t i o n s dampen a f t e r  very c l o s e  in-  surface.  The use o f l o n g e r  may generate o s c i l l a t i o n s  i s shown i n F i g . 3 . 3 .  this  used i n  this  problem i n most of  the  cases . The procedure mentioned above was r e p e a t e d f o r o t h e r two d i r e c t i o n s y i e l d i n g  identical  results  v e r i f y i n g the e q u a t i o n s developed f o r a l l  the  and  nodes,  the thereby  42  Fig.  3.2  Comparison between a n a l y t i c a l calculations  of temperatures  and n u m e r i c a l i n the  slab.  Fig.  3.3  Stable o s c i l l a t o r y  nature o f t h e  c a l c u l a t e d surface temperatures values of time  interval.  numerically for  large  44 i n c l u d i n g the s u r f a c e  3.6  nodes.  Flow Chart o f the Computer Program A f l o w c h a r t o f the computer program i s  Figs.  3.4a  and 3 . 4 b .  F o r t r a n IV l a n g u a g e .  given  in  The program has been w r i t t e n The b a s i c v e r s i o n developed  in  originally  was f o r t h e a n a l y s i s o f heat f l o w i n r e c t a n g u l a r  or  shaped c a s t i n g s .  sub-  stantially  for  However t h i s  has been m o d i f i e d  use i n i r r e g u l a r l y  square  shaped z i n c jumbo  ingots.  A copy of the source program f o r jumbo shaped c a s t i n g p r e s e n t e d i n Appendix  t o stop the computer run at any i n t e r m e d i a t e and r e s t a r t  cedure was very u s e f u l  it  in detecting abortive  470/V-6-II  is  possible  point,  from the same p o i n t .  s t a n d p o i n t o f s a v i n g computer t i m e . was an Amdahl  is  2.  The program has been w r i t t e n such t h a t  the r e s u l t s  This  study pro-  runs from t h e  The computer  under the MTS s y s t e m .  used The program  r e q u i r e s r o u g h l y 0.8 Mega Bytes of memory and t a k e s  about  80 seconds o f CPU time i n p e r f o r m i n g 160 i t e r a t i o n s . this particular  the  case the a r r a y dimensions v a r i e d  ( 1 0 , 16, 3) to ( 1 0 , 16, 43) at the end o f the  from  run.  In  45  /START/  READ  INPUT  PARAMETERS  DISCRETIZE CASTING  READ T E M P . , N O D E  ARE THE CALCULATIONS FROM  A  CONTINUED^  MIDDLE  OF  PREVIOUS RUN  A '.UN  No INITIALIZE  ARRAY  TO POURING  f  TEMP.  INITIALIZE  ALL  NODES AS  LIQUID  C L A S S I F Y T H E NODES D E P E N D I N G ON T H E I R POSITION IN THE  CASTING  , CALCULATE  VARIOUS  AREA AND VOLUME TERMS  EVALUATE PHYSICAL  THERMO PROPERTIES  T  Fig.  3.4(a)  0  Flow c h a r t of the computer  ID'S  FROM T H E END OF A  progra  46  0  IMPLICIT IN  THE  CALCULATIONS X  DIRECTION  BOUNDARY CONDITIONS  **  IMPLICIT IN T H E Y  CALCULATIONS DIRECTIONS  BOUNDARY CONDITIONS  \ IMPLICIT IN T H E Z  CALCULATIONS DIRECTION  COMPUTE NEW  CHECK  Fig.  3.4(b)  TEMPS.  WHETHER ANY  NODE  Flow c h a r t o f the computer program ( c o n t i n u e d from F i g . 3 . 4 ( a ) )  BOUNDARY CONDITIONS  Chapter 4  VALIDATION OF THE RESULTS FROM THE MATHEMATICAL MODEL  4.1  Introduction Before the mathematical model c o u l d be used w i t h  c o n f i d e n c e i n a p r e d i c t i v e mode i t tion.  required careful  This was accomplished by comparing  pool p r o f i l e s identical  to i n d u s t r i a l  model-predicted  measurements o b t a i n e d  casting conditions.  A similar  In D.C. c a s t i n g the pool p r o f i l e  e x p e r i m e n t a l l y by adding an a l l o y towards  (28-30,  the end o f Later  sectioning, polishing  the c a s t i n g o t .  In some cases the pool p r o f i l e s  seen immediately  after  sectioning without  t i o n because of the d i f f e r e n c e tracer  36,  may be o b t a i n e d  c a s t i n g when steady s t a t e has been r e a c h e d . contour i s r e v e a l e d a f t e r  under  validation  t e c h n i q u e has been r e p o r t e d i n o t h e r s t u d i e s 48, 6 6 ) .  valida-  the  and  etching  can be  surface  prepara-  i n the m a c h i n a b i 1 i t y  of  the  a l l o y and the p a r e n t m e t a l .  In a d d i t i o n t o t r a c e r a d d i t i o n , d i p - s t i c k  measure-  ments have been made i n which a rod was lowered i n t o molten p o o l , to o b t a i n the maximum pool d e p t h . simple but very u s e f u l  This i s a  t e c h n i q u e and the v a l u e o f 47  the  dip-stick  48 measurements  increases  c o n s i d e r a b l y when a p p l i e d to  i n g o t s which r e q u i r e c o n s i d e r a b l e c u t t i n g  Three separate v a l i d a t i o n s study.  and m a c h i n i n g .  have been made i n  Two i n v o l v e the D.C. c a s t i n g of a l u m i n i u m  using c o n v e n t i o n a l  f l o o d c o o l i n g and reduced  ingot  The  third  v a l i d a t i o n was made f o r the c a s t i n g o f z i n c jumbos  4.2  this  secondary  c o o l i n g i n the sub-mould r e g i o n r e s p e c t i v e l y .  special  large  of  shape.  Conventional 4.2.1  D.C.  Casting o f Aluminium -  Aluminium I n g o t s  : 381 x 991 mm  In t h i s c o n v e n t i o n a l  form o f D.C. c a s t i n g  s u r f a c e o f the c a s t i n g i s f l o o d e d w i t h  the  water from the mould, thereby p r o d u c i n g c o o l i n g i n the sub-mould r e g i o n . tions  p r e s e n t e d , the m a t e r i a l  the t h e r m o p h y s i c a l Table I I I .  Alcan  properties  In a l l  the  cooling  intense the  c a s t i s pure  simula-  aluminium,  of which are g i v e n  in  The working l e n g t h of the mould used i s  63.5 mm so t h a t w i t h a node t h i c k n e s s i n the c a s t i n g d i r e c t i o n ,  of 15.87 mm  f o u r node s l i c e s  t a i n e d i n the mould. The h e a t - t r a n s f e r  are con-  coefficient  used f o r the top two s l i c e s i n the mould are 1256 2 2 W/m K and 1047 W/m K r e s p e c t i v e l y . The use o f these heat-transfer  coefficients  result  i n heat f l u x  values  Speci f i c Heat of Sol i d  -  0.934 J / g K  Speci f i c Heat of  -  0.934 J / g K  L a t e n t Heat of  Liquid  387 J/g  Fusion  Li qui dus Temperature  631 °C  S o l i dus Temperature  630 °C  Dens i t y o f S o l i d  2700 kg/m  Densi t y of  2700 k g / m  Liquid  Thermal Conducti v i t y  -  209.3 W/m K  Ambient Temperature  -  15°C  Table  III  Thermophysical  Properties  Aluminium used i n Flood Cooling  3  of  Conventional  Simulations.  50 o b t a i n e d from temperature measurements i n t h e mould (33, 84).  In o r d e r to account f o r  formation transfer  the  air-gap  i n the lower p a r t o f the m o u l d , t h e  heat-  c o e f f i c i e n t was decreased t o 209 W/m K f o r  the s u r f a c e nodes i n the r e m a i n i n g two s l i c e s the mould.  For a l l  s u r f a c e nodes below the m o u l d , a  very high h e a t - t r a n s f e r to r e f l e c t zone.  coefficient  has been used  the i n t e n s e c o o l i n g w i t h i n  The value of t h i s  which i s  inside  the  heat-transfer  14.65 kW/m K was a r r i v e d a t ,  flood-water  coefficient by t r i a l  and  e r r o r methods, by comparing measured and p r e d i c t e d pool mm/s.  depths f o r a s e c t i o n 381 x 991 mm c a s t at From these runs i t was very c l e a r t h a t  heat-transfer  coefficient  had a very small  effect  because o f  on the pool  its  the  high  depths.  1.778  value, Thus  i t was p o s s i b l e to check the model by comparing pool depths o b t a i n e d f o r d i f f e r e n t  casting  the  speeds  f o r the same set of boundary c o n d i t i o n s .  Finally  i n these c a l c u l a t i o n s  released  the l a t e n t  heat was  over a 1°C i n t e r v a l .  For the 381 x 991 mm s e c t i o n t h e cas t i n g speeds ranged from 1.185 mm/s to 2.116 mm/s. F i g . 4.1 shows  the  comparison between c a l c u l a t e d and measured pool profiles  for  a c a s t i n g speed of 1.778 mm/s.  measured p r o f i l e s  The  have been o b t a i n e d by a d d i n g  zinc  51  F i g . 4.1  Comparison between the p r e d i c t e d and measured pool p r o f i l e s  for  381 x 991 mm aluminium  ingot  c a s t at 1.778 mm/s ( o b t a i n e d at the m i d - p l a n e parallel  to the narrow  face).  52 and lead t r a c e r  (84) t o the molten p o o l .  As can be  seen t h e r e i s an e x c e l l e n t agreement between the two p r o f i l e s .  It  should be noted t h a t the  p a t t e r n of the c a l c u l a t e d pool  staircase  p r o f i l e which  from the coarseness of the f i n i t e - d i f f e r e n c e and the narrow range over which the l a t e n t  transfer coefficient felt  of the high  mesh  heat i s  leased has been smoothened out i n F i g . 4 . 1 . also be noted t h a t the e f f e c t  results  I t may heat-  i n the sub-mould r e g i o n  is  h i g h e r up i n the c a s t i n g because o f a x i a l  conduction. start  Of the t o t a l  temperature  distribution  i n the c a s t i n g c a l c u l a t e d from the model i s i n Table A 3 . I o f Appendix 3.  after  steady s t a t e  n o t a t i o n used f o r  z-axis.  conditions  mid-face  planes  have been r e a c h e d .  the v a r i o u s d i r e c t i o n s  The  are as  The c a s t i n g d i r e c t i o n was always taken as In the t r a n s v e r s e plane x - a x i s was taken  pendicular  to the broad face and y - a x i s  t o the narrow f a c e . files  presented  F i g . 4 . 2 shows pool  o b t a i n e d at the l o n g i t u d i n a l  follows.  less  removed i n the mould r e g i o n .  The t h r e e - d i m e n s i o n a l  profiles  heat  heat removed from the  of c o o l i n g t o the bottom of the p o o l ,  than 5% i s  re-  In a l l  perpendicular  the c a l c u l a t e d pool  pro-  shown in t h i s w o r k , the top two node s l i c e s  at the p o u r i n g t e m p e r a t u r e ;  and t h i s  per-  are  should be taken  Fig.  4.2  Steady s t a t e pool longitudinal  profiles  mid-planes f o r  obtained at  the  381 x 991 mm  aluminium i n g o t cast at 1.778 mm/s.  54 note of when c a l c u l a t i n g contours.  The p r o f i l e  mid-plane p a r a l l e l  the pool depth from  o b t a i n e d i n the  these  longitudinal  t o broad face i s i n  agreement  w i t h t h e bucket shaped pool observed across the  width  of the s e c t i o n .  In o r d e r to study the e f f e c t o f heat i n the second t r a n s v e r s e d i r e c t i o n , on the heat t r a n s f e r , two-dimensional  overall  the computer program was run i n a mode.  Here heat f l o w  to the narrow face ( y - d i r e c t i o n  perpendicular  ) was n e g l e c t e d .  c a l c u l a t i o n s were performed by s e t t i n g transfer  conduction  coefficient  the  equal t o zero i n the  The  heaty-direction  and at the same time r e d u c i n g the number o f nodes i n the y - d i r e c t i o n  to a minimum of t h r e e .  F i g . 4.3  shows the comparison between the two and t h r e e dimensional  c a l c u l a t i o n s , o b t a i n e d i n the  mid-plane p a r a l l e l  t o the narrow f a c e .  that there is n e g l i g i b l e difference pool p r o f i l e s .  As w i l l  It  longitudinal is  seen  between the two  be shown i n Chapter 5, i f  the  aspect r a t i o exceeds 2.5 f o r c o n v e n t i o n a l  D.C.  t h e r e i s no need t o i n c l u d e the heat f l o w  in a d i r e c -  tion perpendicular  to the narrow face  i n the h e a t - f l o w c a l c u l a t i o n s .  casting,  (y-direction)  T h i s i s an i m p o r t a n t  r e s u l t because computing costs f o r  the  three  55  F i g . 4.3  Comparison of the pool two-dimensional tions  profiles  from the  and t h r e e - d i m e n s i o n a l  calcula-  f o r c a s t i n g 381 x 991 mm a l u m i n i u m  at 1.778 mm/s.  ingot  56 dimensional  model are about 5 to 8 times more than  f o r the two dimensional  case.  Therefore  considerable  computer c o s t s can be saved by safe.ly making assumption o f two dimensional all  the v a l i d a t i o n  conventional ratios  heat c o n d u c t i o n .  runs presented i n t h i s  D.C. c a s t i n g , wherever t h e  are g r e a t e r  In  chapter  for  aspect  than 2 . 5 , the model has been run  o n l y i n a two dimensional  mode.  F i g . 4.4 shows a comparison between measured and c a l c u l a t e d pool o f c a s t i n g speed f o r In a l l  the  the  depths as a f u n c t i o n  381 x 991 mm a l u m i n i u m  the cases the pool  section.  depth has been measured  by u s i n g a s t e e l w i r e to probe the bottom o f  the  p o o l ; the accuracy of these measurements  is  ± 10 mm ( 8 4 ) .  calculated  The agreement between the  and measured pool profiles  o b t a i n e d at the d i f f e r e n t  are shown i n F i g .  4.2.2  depths are e x c e l l e n t .  The pool  casting  speeds  4.5.  Aluminium I n g o t s  : 457 x 1143 mm  The boundary c o n d i t i o n s  used here are  to those given i n s e c t i o n 4 . 2 . 1 . of the f o u r  within  sections  This i s the  s i m u l a t e d under t h i s  identical largest  category.  The c a s t i n g speeds used range from 0.974 mm/s  to  600  Casting Speed F i g . 4.4  (mm/s)  Comparison between the measured and c a l c u l a t e d pool 381 x 991 mm aluminium i n g o t c a s t at d i f f e r e n t  depths  speeds.  for tn  F i g . 4.5  C a l c u l a t e d pool p r o f i l e s f o r 381 x 991 mm aluminium i n g o t cast at d i f f e r e n t speeds.  2.117 mm/s.  F i g . 4.6 shows the comparison between  the c a l c u l a t e d and measured pool depths f o r different  the  five  c a s t i n g speeds. Although the match i s  p e r f e c t the d i f f e r e n c e between the c a l c u l a t e d measured pool  depths was only 30 mm or 4% o f  measured sump d e p t h .  For t h i s s e c t i o n  not  and the  s i z e w i t h an  aspect r a t i o o f 2 . 5 , a small d i f f e r e n c e was observed i n the pool  depths o b t a i n e d between a two  and t h r e e dimensional 2.117 mm/s. at lower  4.2.3  calculations  dimensional  at a speed of  However no d i f f e r e n c e s were observed  speeds.  Aluminium I n g o t s For t h i s  : 305 x 1010 mm.  s e c t i o n s i z e the comparison  between  the c a l c u l a t e d and measured pool  depth i s shown i n  F i g . 4 . 7 ; and l i k e the p r e v i o u s  c a s e s , c l o s e match  i s o b t a i n e d between the two.  4.2.4  Aluminium I n g o t  : 229 x 813 mm  This i s the s m a l l e s t of the f o u r  sections  s i m u l a t e d and h i g h e r c a s t i n g speeds have been employed, the maximum of which i s 2.794 mm/s.  Fig.  4 . 8 shows the comparison between the measured and c a l c u l a t e d pool  depths.  60  850  1  1  aluminum ingot - 457 mm X 1143 mm conventional d.c. cooling • calculated o measured 750  E E 650 JC  a. Q  "5 o Q_  E 550 E X  o  450  35  F i g . 4.6  1.2 Casting Speed  2.2  1.7 (mm/s)  Comparison between the c a l c u l a t e d and measured pool  depths f o r 457 x 1143 mm a l u m i n i u m  cast at d i f f e r e n t  speeds.  ingot  61  l 5  °|.0  1.5  Casting  Fig.  4.7  2.0  Speed  2.5  (mm/s)  Comparison between the c a l c u l a t e d and measured pool depth f o r  305 x 1010 mm a l u m i n i u m  c a s t at d i f f e r e n t  speeds.  ingot  62  450  1  1  aluminum ingot - 229 mm X 813 mm conventional d.c. cooling • calculated o measured  E E 350 sz  a o o  Q_  E 250 E X D  I50  1  i  Casting Speed  Fig.  4.8  2.4 (mm/s)  2.9  Comparison between the c a l c u l a t e d and measured pool depths f o r 229 x 813 mm aluminium i n g o t cast at d i f f e r e n t speeds.  63 4.2.5  Summary o f Conventional Table IV  pool  D.C. C a s t i n g  Simulations  summarises the c a l c u l a t e d and measured  depths o b t a i n e d  f o r the v a r i o u s  Thus i n the case of c o n v e n t i o n a l  casting conditions.  D.C. c o o l i n g , com-  p a r i s o n has been made f o r f o u r d i f f e r e n t  sections  r a n g i n g i n s i z e from 229 x 813 mm to 457 x 1153 mm. In a d d i t i o n  f o r each s e c t i o n  at l e a s t f o u r  different  c a s t i n g speeds have been used to check the of the model.  In a l l  validity  cases good agreement has been  o b t a i n e d between measured and p r e d i c t e d v a l u e s of pool  the  depths.  Figure 4.9 shows a p l o t o f the time  required  t o reach steady s t a t e a g a i n s t the c a s t i n g speed the v a r i o u s s e c t i o n s i z e s .  The time t o reach  for  steady  s t a t e i s d e f i n e d as the time taken f o r the m o l t e n pool  t o cease growing from the s t a r t  229 mm and 381 mm t h i c k s e c t i o n s  of c a s t i n g .  the c a s t i n g  has no e f f e c t on the time t o reach s t e a d y However f o r t h i c k e r s e c t i o n s causes a small  state.  time.  be noted t h a t  from 125  s to 450 s .  It  steady  should  i n any normal c a s t i n g o p e r a t i o n  maximum c a s t i n g speed i s reached not but over a f i n i t e  speed  Going from 229  to 457 mm t h i c k s e c t i o n s , the time t o reach state increases  speed  increasing casting  i n c r e a s e in t h i s  For  the  instantaneously  amount o f t i m e , u s u a l l y  around 1  64  Pool Depth Meas ured (mm)  Pool Depth Calculated (mm)  S e c t i on Si ze  Casting Speed (mm/s)  229mm x 813mm  1 .524  1 84  1 75  1 .905  216  222  2.328  260  270  2.794  305  318  1 .100  216  1 98  1 .481  267  270  1 .862  343  341  2.159  381  396  1 .185  330  31 8  1 .439  394  397  1 .778  470  476  2.116  559  572  0 .974  368  365  1 .312  489  492  1 .566  584  587  1 .947  699  730  2.117  762  778  305mm x 1010mm  381mm x 991mm  457mm x 1143mm  Table  IV  Comparison between c a l c u l a t e d and measured pool depths f o r Aluminium s e c t i o n s speeds .  c a s t at  different  ouu  1  1  aluminum ingot conventional d.c. cooling  -  (/>  I  1  — 4 5 7 mm X 1143 mm  400  -  CD O  00 >v  TJ O  300  CU  mm X 991 mm  6H\  00  o  E I— 2 0 0  3 0 5 mm X 1010 mm  2 2 9 mm X 813 mm ion 0.9  •  i  1.4  i  1.9  2.4  Casting Speed Fig.  4.9  Time r e q u i r e d  for  for  ingots  aluminium  casting  speed.  the  pool of  •  (mm/s)  profiles  various  2.9  to  sizes  reach  steady  as a f u n c t i o n  state of  CTi  cn  66 to 2 minutes..  Thus i n the case of t h i n n e r  by the time the steady c a s t i n g speed i s  sections  reached,  the c a s t i n g would have s e t t l e d down' t o steady  state  conditions.  in-  The importance of unsteady s t a t e  creases w i t h i n c r e a s i n g s e c t i o n t h i c k n e s s e s . further  4.3  d i s c u s s i o n of t h i s  i s made i n Chapter  Reduced Secondary Cooling - B r i t i s h In t h i s  cooling practice  A  Aluminium  the mould w a t e r  a p p l i e d i m m e d i a t e l y below the mould.  In i t s  is  not  place a m i l d  c o o l i n g is e f f e c t e d by the use o f f i n e a i r - a t o m i s e d This r e s u l t s  i n a very coarse c e l l  deep drawing a p p l i c a t i o n s . sured  the h e a t - t r a n s f e r  coefficients  various operating conditions physical  properties  i n Table  V,  i n Table  VI.  structure  B e a t t i e e t al for  these sprays  are  used are  f o r the s o l i d and l i q u i d  values of thermal  given  presented to  conductivity  regions.  The s e c t i o n s i z e used in t h i s  simulation  690 mm, w h i l e the c a s t i n g speed is 0.833 mm/s. ment of the molten pool  under  The thermo-  The computer program has been a l t e r e d  accommodate the d i f f e r e n t  for  (30) have mea-  simulation  w h i l e boundary c o n d i t i o n s  sprays.  suitable  i n the l a b o r a t o r y .  employed i n t h i s  5.  computed as a f u n c t i o n  presented i n F i g s . 4.10 to 4 . 1 2 , which show the  is  254 x  The d e v e l o p of time pool  is  Specific  Heat of  S p e c i f i c Heat of L a t e n t Heat of Liquidus Solidus  Solid  -  1.13 J/g K  Liquid  -  1.13 J / g K  -  387 J / g  -  658°C  -  635°C  -  2700 k g / m  -  222 W/mK  -  105 W/mK  Fusion  Temperature Temperature  Dens i ty Thermal C o n d u c t i v i t y Solid  of  Thermal C o n d u c t i v i t y Liquid  of  Table V  Thermophysica1  Properties  3  of  Aluminium used i n Reduced Secondary Cooling  Simulations.  68  Zone No.  P o s i t i o n Below Liquid Surface (mm)  ' Heat T r a n s f e r C o e f f i c i ent (W/m  K)  2  0 -  52  921  2  52 -  78  1 842  3  78 - 104  9 38  4  104 - 156  663  5  156 - 208  622  6  208 - 286  580  7  286 - 364  538  8  364 - 442  953  9  442 - 520  290  10  520 - downwards  166  1 (mould)  Tab!e VI  Heat T r a n s f e r  Coefficients  f u n c t i o n of p o s i t i o n surface,  for  used as a  below the  reduced secondary  liquid cooling.  69  Distance 0 oi  F i g . 4.10  80 1  along 160 1  Y-axis (mm) 240 1  320 n  L i q u i d u s and s o l i d u s p r o f i l e s longitudinal  X-axis(mm) 0  4 0 8 0 120 I — • — ' — H  o b t a i n e d at  mid-planes of 254 x 690 mm  aluminium i n g o t c a s t under reduced conditions start.  at 0.833 mm/s, a f t e r  cooling  312 s from  the  70  F i g . 4.11  L i q u i d u s and s o l i d u s p r o f i l e s longitudinal  mid-planes  o b t a i n e d at  of 254 x 690 mm  aluminium i n g o t cast under reduced conditions start.  at 0.833 mm/s, a f t e r  cooling  624 s from  the  71  F i g . 4.12  L i q u i d u s and s o l i d u s p r o f i l e s longitudinal  o b t a i n e d at  mid-planes of 254 x 690 mm aluminium  i n g o t cast under reduced secondary ditions  the  at 0.833 mm/s, a f t e r  c o o l i n g con-  1248 s from  start.  72 profiles  at the l o n g i t u d i n a l  s t a t e pool  mid-face planes.  depth obtained was 470 mm i n comparison w i t h  measured value o f 510 mm. ture d i s t r i b u t i o n s Table A 3 - I I  The t h r e e - d i m e n s i o n a l  computed from the  i n Appendix 3.  (y-direction)  o b t a i n e d i n the l o n g i t u d i n a l  c o o l i n g an e f f e c t  of n e g l e c t i n g  the  narrow  mid-plane  face pool  parallel  owing to heat c o n d u c t i o n i n both  D.C.  the  is seen even though the aspect  The c a l c u l a t i o n s  revealed t h a t f o r  in  heat  U n l i k e the case o f c o n v e n t i o n a l  transverse d i r e c t i o n s is 2.79.  tempera-  i s shown i n F i g . 4 . 1 3 , i n the form o f  to the narrow f a c e .  small  to  the  model are p r e s e n t e d  The e f f e c t  c o n d u c t i o n i n a d i r e c t i o n normal  profiles  The steady  this  ratio  rather  s e c t i o n , steady s t a t e is reached o n l y 15 minutes  from the s t a r t  of c o o l i n g .  the pool s u r f a c e  A three dimensional  i s shown i n F i g .  view  of  4.14.  An a t t e m p t t o s i m u l a t e an i n c r e a s e d c a s t i n g speed of 1.266 mm/s w i t h the same boundary c o n d i t i o n was successful.  Here the pool  t i o n was 1014 mm.  depth o b t a i n e d from the  pool  (85).  254 x 690 mm s e c t i o n  at at  1.266  were encountered i n o b t a i n i n g a s t a b l e  Even i n three o f the more s u c c e s s f u l  ments the pool  simula-  When experiments were c a r r i e d o u t  B r i t i s h Aluminium to cast t h i s mm/s d i f f i c u l t i e s  less  experi-  depths ranged from 760 mm t o 860 mm.  the c a l c u l a t e d value from the model f a l l s h i g h e s t pool depth measured.  s h o r t of  Thus the  73  F i g . 4 . 13  Comparison of steady s t a t e pool p r o f i l e s two-dimensional  for  and the t h r e e - d i m e n s i o n a l  calcula-  t i o n s o f 254 x 690 mm aluminium i n g o t c a s t reduced secondary c o o l i n g c o n d i t i o n s  at  the  under  .833 mm/s.  74  F i g . 4.14  Three-dimensional  visual ization  of l i q u i d  pool  s u r f a c e of 254 x 690 mm aluminium i n g o t c a s t .833 mm/s under seen from the  reduced secondary c o o l i n g , as  d i f f e r e n t angles.  of  75 The reason f o r the d i f f e r e n c e and c a l c u l a t e d pool certainty  between the measured  depths can be a t t r i b u t e d  t o the un-  i n the p r e c i s e value o f the h e a t - t r a n s f e r  efficients.  Numerical  calculations  value o f the h e a t - t r a n s f e r depth i s a very s e n s i t i v e  where the heat t r a n s f e r on the pool d e p t h .  in t h i s  variable.  in conventional  The u n c e r t a i n t y  the  showed t h a t the pool  function of t h i s  coefficient  stems from the f a c t t h a t ,  performed by v a r y i n g  coefficients  is i n c o n t r a s t to the c o n d i t i o n s  co-  This  D.C.  cooling,  had a v e r y small  effect  i n the boundary  simulation  condition  the s u r f a c e  the c a s t i n g stays a t 360°C even at the bottom o f the  of  strand.  Thus f o r c o n s i d e r a b l e l e n g t h o f the c a s t i n g from the top  of  the mould, the heat t r a n s f e r mechanism would be i n the  film  b o i l i n g regime.  the  Depending on the t e m p e r a t u r e at which  mechanism changes to n u c l e a t e b o i l i n g , the h e a t - t r a n s f e r efficient  will  change i n magnitude.  these c o n d i t i o n s transfer  it  It  i s very d i f f i c u l t  coefficients  a priori  is f e l t  to p r e d i c t  co-  that  under  the  heat-  from the l a b o r a t o r y  experi-  ments, and measurements would have t o be c a r r i e d o u t i n to c h a r a c t e r i z e  the boundary c o n d i t i o n s  The model c a l c u l a t i o n s cooling conditions,  it  both t r a n s v e r s e d i r e c t i o n s one o f the d i r e c t i o n s  properly^  show t h a t under these  is e s s e n t i a l  slow  to c o n s i d e r heat f l o w  and should not be n e g l e c t e d  as i n the model of B e a t t i e  even though the aspect r a t i o s  plant  are g r e a t e r  than  (30,  2.5.  in 32),  in  4.4  Zinc - Jumbo Casting - Cominco F i g . 4.15 shows the c r o s s - s e c t i o n o f a jumbo  and the v a r i o u s  dimensions of the c a s t i n g .  absence o f symmetry elements i n one o f the direction, a casting.  calculations In t h i s  Because o f  a full  three  model i s r e q u i r e d because, the aspect r a t i o  one-half  of  dimensional is  material  cast i s high grade z i n c f o r which t h e  physical  properties  1.07.  The  thermo-  are given i n Table V I I .  The use o f the f i n i t e - d i f f e r e n c e regular  the  transverse  must be performed ^ o r  simulation  ingot  method f o r  ir-  geometries such as t h a t o f the jumbo does not pose  any s p e c i a l  problems i n p r i n c i p l e .  However a new set  of  e q u a t i o n s had to be i n c l u d e d i n the computer program t o take care o f the i r r e g u l a r l y  shaped nodes.  which the c a s t i n g has been d i s c r e t i z e d Fig. 4.16. tions  is presented  Since i t was r e q u i r e d to p e r f o r m the  f o r one-half of a c a s t i n g ,  computer c o s t s s l i g h t l y , in this  The manner  simulation.  in  in  calcula-  i n o r d e r t o c u t down the  a v a r i a b l e node s i z e has been used  The s u b - d i v i s i o n of t h e jumbo has been  made such t h a t a f i n e r  node e x i s t s  at the s u r f a c e and  coarsernodesatthecentre.  The boundary c o n d i t i o n s  for this  simulation  were  o b t a i n e d by f r e e z i n g i n thermocouples near the s u r f a c e  of  77  C  F i g . 4.15  A  B  C  D  E  F  546  508  216  114  114  38  A cross s e c t i o n of the jumbo i n g o t w i t h various  dimensions given in mm..  the  78  Fig.  4.16  Discretization  o f the jumbo i n g o t f o r  finite-difference  calculations.  the  Density of  Liquid  -  6620 k g / m  3  Dens i t y of  Solid  -  6981 k g / m  3  Specific  Heat of  Liquid  Specific  Heat of  Solid  -  -480  J/g K  Zinc - 0.343 + .154 ( I O ) - 3  Thermal  Conducivity  L a t e n t Heat o f Liquidus Solidus  Fusion  Temperature Temperature  Table V I I  T  J/g K  -  113 W/m K  -  113 J/g  -  420°C  -  410°C  2  Thermophysical  Properties  in Jumbo I n g o t  Simulation.  of  Zinc  the jumbo (~20 mm) d u r i n g a c a s t i n g and l e t t i n g w i t h the descending i n g o t .  The exact p o s i t i o n of  thermocouple t i p was l o c a t e d s u b s e q u e n t l y by the i n g o t .  From the t e m p e r a t u r e - t i m e  surface h e a t - t r a n s f e r a trial  coefficient  the  sectioning  p l o t measured, the  c o e f f i c i e n t was b a c k - c a l c u l a t e d by  and e r r o r p r o c e d u r e .  transfer  them drop  Table V I I I gives the  used i n t h i s s i m u l a t i o n .  compares the measured t e m p e r a t u r e w i t h the  heat  F i g . 4~.17  calculated  values.  The pool p r o f i l e was o b t a i n e d f o r a c a s t i n g speed o f 1.27 mm/s by adding an a l l o y sump a f t e r  of 10% c o p p e r - i n - z i n c  steady s t a t e c o n d i t i o n s  to  had been r e a c h e d .  the a d d i t i o n , - the pool was s t i r r e d using a p a d d l e , by a power d r i l l . tudinally  The i n g o t was l a t e r  at the m i d - p l a n e p e r p e n d i c u l a r  notched f a c e s .  After  driven  longi-  to the two non-  The s u r f a c e then was g e n t l y p o l i s h e d  remove the machine marks.  The pool boundary was  d e l i n e a t e d by e t c h i n g w i t h a 4% N i t a l tudinal  sectioned  the  solution.  s e c t i o n s where the c o n t o u r p r o f i l e s  to  finally The l o n g i -  have been  o b t a i n e d from the model are shown i n F i g . 4 . 1 8 , as shaded. F i g s . 4.19 to 4 . 2 1 , shows the c o n t o u r p r o f i l e s at these two s e c t i o n s , at d i f f e r e n t o f the c a s t i n g .  times from the  F i g . 4.21 correspond to steady  c o n d i t i o n s whereby the pool p r o f i l e  obtained start  state  remains the same w i t h  J 81  Distance  Heat  from  Transfer  Coefficients (W/m K)  the top o f Mould (mm)  2  0 - 212  20934  212-216  4187 125  1450 - 1454  Between 212 and 1454 mm the Heat Transfer C o e f f i c i e n t  has been  dropped i n an .exponent! cal  fashion.  Heat T r a n s f e r  used i n  -  Table V I I I  Coefficients  Zinc Jumbo C a s t i n g .  Time (s) 4.17  Comparison o f the. measured and c a l c u l a t e d temperature p r o f i l e s cas t at 1.69 mm/s .  for  the z i n c jumbo  ingot  y  F i g . 4.18  A three-dimensional  view of the jumbo  showing the l o n g i t u d i n a l pool  profiles  ingot  s e c t i o n s where the  have been o b t a i n e d .  Distance along Y-axis 0  T  Fig.  80 1  4.19  160 1  240 ;  1  320 1  (mm) 400  X-axis(mm) 480  1——i—I  0  r — — i  80  Pool p r o f i l e s o b t a i n e d at the l o n g i t u d i n a l  160  1  sections  shown i n F i g . 4.18 f o r c a s t i n g zinc jumbo i n g o t 1.27 mm/s, 278 s a f t e r the  start.  240 i —  at  00  85  Fig.  4.20  Pool p r o f i l e s sections  o b t a i n e d at the  longitudinal  shown i n F i g . 4.18 f o r c a s t i n g  jumbo i n g o t at 1.27 mm/s, 557 s a f t e r  zinc  the  start.  Distance  0  Op  1  7 2 0  80  160  1  along Y-axis (mm)  240  i——T  320 4 0 0 1  X - axis ( m m )  4 80  — n  I  0  80  160  240  •""— — ~ r  T  f  800880960-  4.21  s t a t e pool p r o f i l e s o b t a i n e d a t the 1ongi t u d i nal s e c t i o n s shown i n F i g . 4.18 fo c a s t i n g z i n c jumbo i n g o t at 1.27 mm/s.  Steady  87 further  increase in time.  The t h r e e d i m e n s i o n a l  s t a t e temperature d i s t r i b u t i o n  steady  i s g i v e n i n Table A 3 - 1 1 1 -  Comparison between measured and c a l c u l a t e d pool files  f o r t h e z i n c jumbo i s seen i n F i g . 4 . 2 2 .  pro-  As can be  seen t h e r e i s a good agreement between the t w o .  The  s h i f t i n g o f the bottom o f the sump to an a s y m m e t r i c a l position  can be c l e a r l y  mensional  seen i n t h i s  figure.  A three  view o f the pool s u r f a c e i s p r e s e n t e d i n  The development o f the pool f o r t h i s has also been m o n i t o r e d by d i p p i n g a s t e e l The r e s u l t s  di-  Fig.4.23.  casting  speed  rod i n t o  o b t a i n e d are p r e s e n t e d i n F i g . 4 . 2 4 .  the  In  pool.  this  graph the pool depth i s p l o t t e d a g a i n s t t h e l e n g t h o f  the  c a s t i n g at any i n s t a n t .  really  Note t h a t the s h e l l  commence growth at the bottom u n t i l  does not  a f t e r .500 mm o f  casting.  This i s due t o the f a c t t h a t the heat has t o d i f f u s e  from  the c e n t r e o f the c a s t i n g o n l y through the s i d e s as no c o o l i n g i s p r o v i d e d on the b o t t o m .  F i g . 4.25 shows a s i m i l a r p l o t o b t a i n e d f o r c a s t i n g speed o f 1.693 mm/s.  As i n the p r e v i o u s  case,  good agreement i s o b t a i n e d between the c a l c u l a t e d measured v a l u e s . ditions  a higher  and  The time r e q u i r e d f o r s t e a d y s t a t e  can be c a l c u l a t e d by d i v i d i n g the c a s t  length  con-  Comparison between the c a l c u l a t e d and measured pool  profiles  f o r the c a s t i n g of z i n c  i ngot at 1.27 mm/s .  jumbo  Fig.  4.23  Three-dimensional pool mm/s .  visualization  of the  liquid  s u r f a c e of z i n c jumbo i n g o t cast a t  1.27  Length  I500r  20  of  casting  40  60  Casting  80  speed - 7 6 m m / m i n High  ~  (In)  grade  (3in/min)  zinc  lOOOf  40  -  c  E Q.  a>  Q  xr-cr •Q-Q.  TJ  r> n  o  a.  o  cu  o o  0.  o o  500  — o  500  1000 Length  F i g . 4.24  Calculated  a.  Measured  1500 of  20  2000  casting (mm)  Comparison between the c a l c u l a t e d and measured pool depths o b t a i n e d at d i f f e r e n t times from the s t a r t of c a s t i n g of zinc jumbo i n g o t cast at 76 mm/mi n.  o  Length 1500  20  of  casting (in )  40  60  ——r  I  ——  Casting speed I02mm/min (4in/min) Prime western grade zinc  1000  o o  500  0 0 F i g . 4.25  o  500  Calculated Measured  1000 1500 Length of casting ( m m )  Comparison between the c a l c u l a t e d and measured pool o b t a i n e d at d i f f e r e n t  1  2000 depths  times from the s t a r t of c a s t i n g  z i n c jumbo i n g o t cast at 102 mm/min.  of  where the curve becomes f l a t  by the c a s t i n g speed.  seen from t h i s and the p r e v i o u s  Fig. 4.24, that  it  j u s t over 10 minutes f o r steady s t a t e c o n d i t i o n s Since the t o t a l  d u r a t i o n of the c a s t i n g  m i n u t e s , t h i s means t h a t  is  takes  to  prevail.  i s o n l y around 40  f o r 25% of the t o t a l  the c a s t i n g i s i n an unsteady  It  casting  time  state.  F i g . 4.26 shows the f r e e z i n g o f the i n g o t as seen from a t r a n s v e r s e s e c t i o n . different  position  Here the s o l i d u s i s o t h e r m  along the c a s t i n g d i r e c t i o n  been compressed on to the plane of the p a p e r . o f the f r e e z i n g f r o n t seen as f a i n t  rings  o b t a i n e d i n an a c t u a l  in Fig. 4.27.  are caused by the i n t e r m i t t e n t the c a s t i n g o f a d i l u t e  Finally the c a l c u l a t i o n profiles  have  a l l o y of lead i n  all  The shape  c a s t can be in Fig.  s t i r r i n g of the pool  4.27 during  zinc.  the importance o f i n c l u d i n g the notch  in  i s shown i n F i g . 4 . 2 8 , which shows pool  c a l c u l a t e d w i t h and w i t h o u t the n o t c h .  be seen t h e r e i s a s u b s t a n t i a l pool  The r i n g s  for  effect  As can  of the notch on the  depths .  4.5  Summary of V a l i d a t i o n  o  Runs  The model has been v a l i d a t e d using measurements  from  93  F i g . 4.26  Freezing p r o f i l e s cross-section .  as seen on a jumbo  F i g . 4.27  Macrostructure  o f the z i n c jumbo c r o s s - s e c t i o n  freezing lines  seen i n F i g .  4.26.  showing  the  95  Distance 80  160  along  240  Y-axis  320  400'  480  160  320  4 80  640  800  960  I I 20h  Zinc ingot Casting speed 1.69 m m / s Jumbo Jumbo  1  Fig.  4.28  1  ingot ingot  1  with notch without notch  i  i  Comparison of the c a l c u l a t e d pool o b t a i n e d w i t h and w i t h o u t z i n c at 1.69 mm/s.  profiles  the notch f o r  casting  96 t h r e e D.C. c a s t i n g o p e r a t i o n s : and one f o r the case o f z i n c .  two f o r  the case o f  E x c e l l e n t agreement i s  t a i n e d between c a l c u l a t e d and measured pool aluminium i n g o t s  aluminium  cooled by c o n v e n t i o n a l  profiles  obin  the  f1ood c o o l i n g .  the case of aluminium b l o c k s s u b j e c t e d to reduced  In  secondary  c o o l i n g the s i m u l a t i o n from the model was l e s s s u c c e s s f u l h i g h e r c a s t i n g speeds. certainty  for  This has been a t t r i b u t e d t o un-  i n c h a r a c t e r i z i n g the boundary c o n d i t i o n s  the r e s u l t s o b t a i n e d i n the l a b o r a t o r y e x p e r i m e n t s .  from In t h e  case o f z i n c jumbo good agreement has been observed between measured and p r e d i c t e d  pool  profiles.  Chapter 5  EFFECT OF CASTING VARIABLES ON HEAT FLOW  5.1  Introduction In t h i s  chapter a d e s c r i p t i o n  model, developed i n the p r e v i o u s  s e c t i o n , was used i n a  p r e d i c t i v e mode to study the e f f e c t v a r i a b l e s on the o v e r a l l \  calculations  heat f l o w .  meaningful, a l l  i s g i v e n of how the  of d i f f e r e n t  In o r d e r to make these  the s i m u l a t i o n s  presented  t h i s s e c t i o n have been c a r r i e d out under c a s t i n g obtainable  in i n d u s t r i a l  practice.  casting  conditions  In t h i s way the  tance of c o n s i d e r i n g heat f l o w i n the t r a n s v e r s e is brought o u t , with s p e c i f i c of v a r i o u s of a x i a l casting speeds.  sections  conduction  The  o f thermal  conductivity  Finally  of  casting  importance the  casting  on the  heat f l o w i s s t u d i e d by comparing s i m u l a t i o n s and z i n c c a s t i n g .  dimensions  i n comparison w i t h b u l k motion of  i s demonstrated by c o n s i d e r i n g d i f f e r e n t The e f f e c t  impor-  examples i n v o l v i n g the  of aluminium and z i n c .  in  overall  aluminium  some comments are made r e g a r d i n g  v a r i a b l e s which have n e g l i g i b l e  effect  on heat f l o w but a  very profound i n f l u e n c e on the m e t a l l u r g i c a l  97  structure.  5.2  E f f e c t of Aspect  Ratio*  In the m o d e l l i n g of heat f l o w i n r e c t a n g u l a r it  has been common p r a c t i c e  transverse d i r e c t i o n , that ( 30 , 33, 4 8 ) . computational  slabs  to n e g l e c t heat f l o w i n is, parallel  to the broad face  In t h i s way the programming e f f o r t costs are reduced c o n s i d e r a b l y  to the case o f a t h r e e - d i m e n s i o n a l  the  and.the  as compared  model.  The importance o f heat f l o w i n the t r a n s v e r s e r e c t i o n has been i n v e s t i g a t e d w i t h s p e c i f i c  examples o f  c a s t i n g o f aluminium and z i n c i n g o t s w i t h d i f f e r e n t ratios.  In these c a l c u l a t i o n s  are s i m i l a r conventional  di-  aspect  the c o o l i n g c o n d i t i o n s  used  to those discussed i n Chapter 4 p e r t a i n i n g D.C. c o o l i n g .  1.778 mm/s f o r a l l  the  the  to  The c a s t i n g speed employed was  calculations.  F i g . 5.1 shows the s t e a d y - s t a t e  pool  depth  i n c a s t i n g aluminium i n g o t s w i t h v a r i o u s s e c t i o n The bottom curve corresponds  obtained sizes.  to s e c t i o n s o f 381 x 381 mm,  381 x 571 mm, 381 x 762 mm and f i n a l l y aluminium siab of i n f i n i t e w i d t h .  a 381 mm t h i c k  The 1 a t t e r  two-dimensional  c a l c u l a t i o n was c a r r i e d out by r e d u c i n g the number of nodes  *The aspect r a t i o i s d e f i n e d as the r a t i o between the two t r a n s v e r s e dimensions w i t h the l a r g e r value taken as the numerator.  f 300'  1 2  I  Fig.  5.1  Aspect  ratio  1 3  4  E f f e c t o f aspect r a t i o s on the pool depths i n c a s t i n g 381 mm and 457.2 mm t h i c k aluminium slabs at 1.778 mm/s.  CO CO  100 i n the w i d t h d i r e c t i o n coefficient  in t h i s  no g r a d i e n t s cal  to t h r e e w h i l e the  heat-transfer  d i r e c t i o n was set equal  to z e r o .  are imposed i n the w i d t h d i r e c t i o n  and i d e n t i -  temperatures are c a l c u l a t e d by the program f o r  three  rows o f  nodes.  seen t h a t the e f f e c t diminishes  From F i g . 5.1 i t  Thus  the  can be c l e a r l y  of the second t r a n s v e r s e  as the aspect r a t i o exceeds 2 . 5 .  dimension Thus f o r  flow c a l c u l a t i o n s  i n r e c t a n g u l a r slabs w i t h aspect  g r e a t e r than 2 . 5 ,  it  heat  ratios  is adequate to c o n s i d e r two dimensions  only.  The top curve i n F i g . 5.1 corresponds  to s e c t i o n s  of  457 x 457 mm, 457 x 686 mm, 457 x 914 mm, 457 x 1143 mm and a 457 mm t h i c k  slab o f i n f i n i t e  the lower curve the t r a n s i t i o n to t w o - d i m e n s i o n a l  is n e g l i g i b l e  as f o r s e c t i o n s  three-dimensional  But even i n t h i s  higher  case the  beyond an aspect r a t i o o f 2 . 5 .  conclusion is v a l i d for a l l one used i n t h i s  Compared t o  heat f l o w occurs at s i i g h t l y  values o f the aspect r a t i o . effect  from  width.  c a s t i n g speeds lower than  calculation,  namely 1.778 mm/s,'as  s m a l l e r than 457 mm i n t h i c k n e s s .  addition this w i l l  This  well In  also apply to c o o l i n g c o n d i t i o n s  are more i n t e n s e than those used i n the p r e s e n t  the  which  calculations.  F i g . 5.2 shows the c a l c u l a t e d s t e a d y - s t a t e  pool  2.2 mm/s I200h  e E f.  CD TJ  1.78 mm/s  lOOOf  o o CL  Zinc ingot 381mm thick section Conventional D.C. cooling O - 2-dimensional  800  600  1  Aspect Fig.  5.2  O  Effect  ratio  o f aspect r a t i o s on the pool depths in  381 mm t h i c k  z i n c slabs at 1.78 and 2.2 mm/s.  casting  depth p l o t t e d a g a i n s t the aspect r a t i o f o r two speeds. 381 mm.  The t h i c k n e s s of the s e c t i o n  The top curve corresponds  zinc cast  at  considered  is  to a c a s t i n g speed o f  mm/s w h i l e the bottom curve corresponds %o 1.778 mm/s. comparison w i t h aluminium the o n - s e t of heat f l o w i s  ratios  exceeding 2 . 0 .  Thus i t  heat f l o w f o r  The same r e s u l t  In  twoTdimensional  seen a t even lower aspect r a t i o s .  adequate to c o n s i d e r t w o - d i m e n s i o n a l  2.2  is  aspect  a l s o holds f o r  higher  c a s t i n g speeds.  The above d i s c u s s i o n on the r e l a t i v e two-and t h r e e - d i m e n s i o n a l respect  to c o n v e n t i o n a l  importance  heat f l o w has been made w i t h  D.C. c o o l i n g .  However the  t i o n of a reduced secondary c o o l i n g p r a c t i c e  applica-  such as  f o l l o w e d a t B r i t i s h Aluminium ( 3 0 , 85) changes t h i s  Two-and t h r e e - d i m e n s i o n a l  calculations  performed on a 250 x 690 mm s e c t i o n w i t h these c o o l i n g c o n d i t i o n s .  c a s t i n g speeds:  i s seen t h a t  there  two c a l c u l a t i o n s  tions  IX  presents  2.76)  the  pool  for  two  0.833 mm/s and 1.26 mm/s.  It  difference  between the  even though the aspect r a t i o exceeds  Thus e r r o r  increases  picture.  (aspect r a t i o of  Table  is a s i g n i f i c a n t  and the d i s c r e p a n c y creased.  is  have been  depth o b t a i n e d from the two sets of c a l c u l a t i o n s different  of  i n c r e a s e s as the c a s t i n g speed i s introduced into  the p o o l - d e p t h  2.5 in-  calcula-  from 10% a t 0.833 mm/s t o 20.5% at 1.26 mm/s.  103  A A Ingot Reduced Speed mm/s  Table  IX  254 x 690 mm  Secondary  Pool Depth 3-Dimensional  Cooling Pool Depth 2-Dimensional  0.833  468 mm  520 mm  1 .266  1014 mm  1222 mm  Comparison between t h r e e dimensional two dimensional  pool depths f o r  and  reduced  secondary c o o l i n g of 254 x 690 mm aluminium ingot.  104  As w i l l for  be seen i n the f o l l o w i n g s e c t i o n  this  appears  c o o l i n g c o n d i t i o n to n e g l e c t a x i a l  c o n s i d e r heat f l o w i n both t r a n s v e r s e  5.3  it  E f f e c t of Axial It  practice  conduction.  i n almost a l l c a s t i n g , to  In-order  to check  computer program was m o d i f i e d by bypassing the i n the z - d i r e c t i o n .  t h e r e i s o n l y one s l i c e cation  In these  i n the z - d i r e c t i o n  i s f o l l o w e d as t h i s  slice  downwards at the c a s t i n g speed.  travels  include  the  implicit  calculations and the  solidifi-  from the meniscus  This i s the  f o l l o w e d i n models of the c o n t i n u o u s  the  the  importance o f the z-component o f heat c o n d u c t i o n ,  calculations  but  Conduction  has been t r a d i t i o n a l  of a x i a l  conduction  directions.  models developed t o d a t e f o r n o n - f e r r o u s the e f f e c t  better  procedure  c a s t i n g process  for  steel.  F i g . 5.3 shows the s u r f a c e t e m p e r a t u r e at t h e m i d face o f a 457 mm square aluminium i n g o t vs d i s t a n c e the meniscus c a l c u l a t e d w i t h and w i t h o u t a x i a l  conduction.  As e x p e c t e d , the curve o b t a i n e d by i n c l u d i n g the c o n d u c t i o n i s the smoother of the t w o . surface temperature f o r  below  axial  The rebound o f  the case of no a x i a l  conduction  caused by the f o r m a t i o n o f an a i r gap i n the mould.  the is  Distance from the meniscus (m m ) F i g . 5.3  C a l c u l a t e d s u r f a c e temperature p r o f i l e s 457 x 457 mm aluminium i n g o t cast at axial  conduction.  o b t a i n e d at the m i d - f a c e  .974 mm/s, w i t h and w i t h o u t  of the  o  106 Furthermore the curve w i t h o u t  the a x i a l  conduction  also  shows a much s t e e p e r t e m p e r a t u r e g r a d i e n t c o r r e s p o n d i n g direct  chilling  i n the secondary  to  zone.  F i g . 5.4 shows the c e n t r e t e m p e r a t u r e of a 457 mm square aluminium i n g o t vs d i s t a n c e below t h e meniscus. As b e f o r e the two c o n d i t i o n s axial  conduction.  axial  conduction  mm, t h a t  It  c o n s i d e r e d are w i t h ^ a n d  can be seen t h a t by not i n c l u d i n g  the pool  depth i n c r e a s e s from 254 t o  the  302  i s by about 20%.  Table  X. shows  the  importance o f i n c l u d i n g  conduction f o r various casting c o n d i t i o n s . points  without  can be drawn from t h i s  importance o f a x i a l c a s t i n g speed. differences  table.  It  Several  axial important  i s seen t h a t  c o n d u c t i o n decreases w i t h  the  increase  Thus f o r 457 mm square aluminium i n g o t  between the two c a l c u l a t i o n s  in the  decrease from 20%  at a c a s t i n g speed of 0.974 mm/s, t o 3% at 2.116 mm/s. In the case of slabs the e f f e c t l e s s pronounced. difference  o f c a s t i n g speed i s much  For a 457 mm t h i c k  between the two c a l c u l a t i o n s  and 4% at 2.116 mm/s.  Since b i l l e t s  speeds than s l a b s , t h i s means i t axial  aliminium slab  c o n d u c t i o n i n the b i l l e t  been a b e t t e r  is  9% at 0.974 mm/s  are c a s t at  is possible  case.  the  to  faster neglect  Thus i t would have  assumption f o r J o v i c et al  (35)  in  modelling  Aluminum  Distance  F i g . 5.4  from  the  ingot  457  X 4 5 7 mm  m e n i s c u s (mm)  C a l c u l a t e d c e n t r e temperature p r o f i l e s 457 x 457 mm aluminium i n g o t cast at and w i t h o u t  the a x i a l  conduction.  obtained  for  .974 mm/s, w i t h  108  Material  Section Size (mm)  Cooling  Casting Speed (mm/s)  Pool Depth with Axial Conduction  Pool Depth without Axial Conduction  (mm)  ( rnrr, )  A£  457 x 457  Conventional  0.974  254  302  Ail  457 x 457  Conventional  2.116  572  587  A£  457 x  Conventional  0.974  365  397  A*  457 x °°  Conventional  2.116  794  826  Zn  457 x 457  Conventional  0.974  746  810  A*  254 x 690  Reduced Secondary  0.833  468  494  Table X  00  Steady s t a t e pool depths f o r a l u m i n i u m and z i n c ingots  c a l c u l a t e d w i t h and w i t h o u t t h e  conduction.  axial  109 heat f l o w i n a 360 x 1600 mm s e c t i o n to i n c l u d e d u c t i o n , but n e g l e c t c o n d u c t i o n i n a d i r e c t i o n to the narrow f a c e .  The e f f e c t  o f thermal  to decrease the importance of a x i a l crease i n thermal  conductivity.  axial  con-  perpendicular  conductivity  is  c o n d u c t i o n w i t h a de-  Thus f o r 457 mm square  i n g o t s of aluminium and z i n c c a s t at 0.974 mm/s, the ferences between the case of a x i a l  dif-  c o n d u c t i o n and no a x i a l  are 20% and 9% r e s p e c t i v e l y .  Finally  the importance o f a x i a l  heat  conduction  f o r the case of reduced secondary c o o l i n g has been  studied  f o r the c a s t i n g of a 254 x 690 mm aluminium s e c t i o n 0.833 mm/s. Table  .VI.,  difference  The boundary c o n d i t i o n s in  Chapter 4 .  at  employed are g i v e n  Compared to p r e v i o u s cases  in  the  between the two pool d e p t h s , namely 468 mm and  49'4 mm, i s o n l y 5%. even at t h i s low c a s t i n g speed. h i g h e r speeds the d i f f e r e n c e s would be even less  At  signifi-  cant.  The r e s u l t s  obtained in t h i s  o p p o s i t e t o the e f f e c t s  p a r t o f the work  r e p o r t e d i n s e c t i o n 5.2 where  n e g l e c t i n g c o n d u c t i o n i n one o f the t r a n s v e r s e was d i s c u s s e d .  In t h a t case i t was seen t h a t  the c a s t i n g speed increased the e r r o r dimensional  are  calculations.  directions an i n c r e a s e  i n the two-  T h e r e f o r e when d e v e l o p i n g  in  110 two-dimensional neglect axial transverse  5.4  models i n these systems  it  is safer  c o n d u c t i o n but c o n s i d e r heat f l o w i n  both  directions.  Importance o f Unsteady  State  In most o f the models developed t o d a t e , those f o r c y l i n d r i c a l state conditions  shapes w i t h a x i a l  have been assumed.  t o note t h a t u n l i k e  the c o n t i n u o u s  the c a s t i n g o p e r a t i o n  is t r u l y  including  symmetry,  However i t  steady-  is  c a s t i n g of s t e e l  important where  c o n t i n u o u s , the v e r t i c a l  casting is only semi-continuous initial  to  in nature.  Therefore  D.C. the  t r a n s i e n t p a r t o f the c a s t i n g may be a s i g n i f i c a n t  fraction  of the c a s t i n g c y c l e .  Since the model  i n t h i s work can be used to study t r a n s i e n t importance of unsteady s t a t e has been  developed  effects  investigated.  The unsteady s t a t e has been s t u d i e d  experimentally  by S e r g e r i e and Bryson (14) who have discussed the of ingot  "bowing" observed i n the t r a n s i e n t  l a r g e sheet i n g o t c a s t i n g .  This r e s u l t s  s h o r t moulds i n which the f i r s t cap f e e l s  the s t r o n g e f f e c t s  i t would i f  the  section  problem of  from the use of  metal f r e e z i n g on the  of a d i r e c t  l a r g e r moulds were used.  quench sooner  During  'bowing'  occurs as the b u t t emerges from the mould, the ends o f butt shrink  upwards o f f  the s t o o l  stool than which the  cap and inwards away from  111  the ends o f the mould.  The a u t h o r s have p a t e n t e d a " p u l s e d  c o o l i n g " method to reduce the heat t r a n s f e r r a t e transient  p o r t i o n by using a pulsed water  in  the  spray.  Recently Yu (43) has t a c k l e d the same problem by drastically  changing the heat t r a n s f e r mechanism at  i n g o t s u r f a c e through the use o f water c o n t a i n i n g carbon d i o x i d e . As the c o o l i n g water e x i t s  the  dissolved  f r o m the mould  the d i s s o l v e d gas evolves as m i c r o n - s i z e bubbles f o r m i n g a temporary, e f f e c t i v e casting,  insulation  l a y e r on t h e s u r f a c e o f  t h e r e b y r e d u c i n g the heat t r a n s f e r .  The importance  o f c o o l i n g i n t h e t r a n s i e n t p o r t i o n can be a p p r e c i a t e d the f a c t  the  from  t h a t both o f the above-mentioned ideas have been  patented.  The t e m p e r a t u r e f i e l d the c a s t i s d i f f e r e n t  i n the i n g o t a t the s t a r t  from the t e m p e r a t u r e  s t a t e p a r t of the c a s t i n g which f o l l o w s .  "in the Thus  measures may need be taken to p r e v e n t c r a c k i n g transient  p a r t o f the c a s t ,  of  steady-  special i n the  since the c r a c k , once  early  initiated,  can c o n t i n u e to propogate i n the steady s t a t e even though the s t e a d y - s t a t e per se. profiles  cooling conditions  may not g e n e r a t e  F i g . 5 . 5 . shows the c a l c u l a t e d s u r f a c e  cracks  temperature  at the m i d - p l a n e on the bottom f a c e of a jumbo  cross-section.  The metal  cast i s aluminium at a speed o f  112  Time F i g . 5.5  below  the  meniscus  (s )  Calculated surface temperature p r o f i l e s initial  and steady s t a t e s l i c e s  aluminium jumbo i n g o t a j umbo s e c t i o n ) .  (at  for  for  the  casting  the bottom m i d - f a c e  of  113 1.35 mm/s. histories  The two p r o f i l e s o f two s l i c e s  correspond to the  l e a v i n g the mould at  t i m e s , one c o r r e s p o n d i n g to the f i r s t mould (dashed l i n e )  conditions  be noted t h a t s t e e p e r a x i a l  and t h i s  slice exiting  slice  (solid l i n e ) .  surface  profile  the metal  cast,  transient  time was presented as a f u n c t i o n of  speed f o r c a s t i n g aluminium slabs o f v a r i o u s I t was seen t h a t  sections  it  had a small  F i g . 5.6 shows the t r a n s i e n t sections  o f aluminium and z i n c .  of thermal  conductivity  The e f f e c t  slice;  reach including This  casting thicknesses  for  on t h i s  smaller  time  while  effect.  time f o r c a s t i n g  various  Because of the lower  in the case of z i n c  l o n g e r time f o r steady s t a t e c o n d i t i o n s  the t r a n s i e n t  to  s e c t i o n s i z e and c a s t i n g speed.  s e c t i o n s , c a s t i n g speed had no e f f e c t  set-  cracks.  depend on a number of f a c t o r s  i n F i g . 4.6 of Chapter 4 .  should  are  compared to the s t e a d y - s t a t e  The time r e q u i r e d f o r the pool  for thicker  the  It  temperature gradients  p o s s i b l y could i n i t i a t e  steady s t a t e w i l l  different  and the o t h e r c o r r e s p o n d i n g t o a s l i c e  cast under s t e a d y - s t a t e  up i n the i n i t i a l  temperature  value  it.takes, a  to be a c h i e v e d .  of bottom h e a t - t r a n s f e r  coefficient  p o r t i o n was s t u d i e d w i t h the model.  The  on  114  800  <D  E 4 00 i-  Al  38.1 cm  200  Aspect F i g . 5.6  ratio  Time r e q u i r e d f o r the pool state for  381 mm t h i c k  different  aspect r a t i o s  profiles  t o reach  steady  aluminium and z i n c slabs cast at 1.778 mm/s.  of  115 value o f h e a t - t r a n s f e r had n e g l i g i b l e e f f e c t  coefficients  2 as high as 418 w/m K  in conventional  D.C. c o o l i n g ,  while  the e f f e c t was g r e a t e r w i t h reduced secondary  cooling.  The value o f bottom h e a t - t r a n s f e r  did  coefficient  affect  the steady s t a t e t e m p e r a t u r e  5.5  E f f e c t of S e c t i o n The e f f e c t  not  field.  Size  o f s e c t i o n s i z e has been i n v e s t i g a t e d  c a s t i n g square s e c t i o n s o f aluminium and z i n c .  in  The s e c t i o n  s i z e s c o n s i d e r e d were 305 x 305 mm, 381 x 381 mm and 457 x 457 mm.  In a l l  at 1.778 mm/s.  the cases the c a s t i n g has been m a i n t a i n e d F i g . 5.7 shows the s t e a d y - s t a t e  pool  o b t a i n e d i n c a s t i n g s e c t i o n s mentioned above under cooling conditions.  sible for  conductivity  the much s t e e p e r i n c r e a s e  increasing section size. conductivity  on the pool  i n the pool  for  time r e q u i r e d f o r steady s t a t e  conductivity,  respon-  depth  with  thermal  381 x 381 mm s e c t i o n s  to be a c h i e v e d .  the t r a n s i e n t in a l i n e a r  of  F i g . 5.9 shows the It  i n the case of a l u m i n i u m , because o f i t s  section size r e s u l t i n g two.  of  is  is  shape can be a p p r e c i a t e d f r o m F i g .  aluminium and z i n c are a l s o p r e s e n t e d .  thermal  of zinc  The d r a m a t i c e f f e c t  5 . 8 , where the pool p r o f i l e s  seen t h a t  D.C.  Comparing aluminium and z i n c i t  found t h a t the low thermal  depths  can be high  time keeps pace w i t h relationship  between  the  116  35  45  Square section size (cm) F i g . 5.7  Steady s t a t e pool depths f o r  square s e c t i o n s  aluminium and z i n c cast at 1.778 mm/s.  i  of  117  F i g . 5.8  C a l c u l a t e d steady s t a t e pool p r o f i l e s f o r 381 mm square s e c t i o n s of aluminium and z i n c cast at 1.778 mm/s.  118  Fig,  5.9  Time r e q u i r e d f o r steady s t a t e  the pool p r o f i l e s  f o r square s e c t i o n s of  and z i n c c a s t at 1.778 mm/s.  to  reach  aluminium  119 5.6  E f f e c t o f Super Heat In o r d e r t o compensate f o r the drop i n t e m p e r a t u r e  the molten metal of superheat analysis effect  it  during t r a n s f e r  i n t o the m o u l d , a q u a n t i t y  is p r o v i d e d to the l i q u i d m e t a l . has been found  of  In  this  t h a t super heat has a very  on the l o c a t i o n of the s o l i d u s  isotherm.  minor  I t was a l s o  seen t h a t most o f the superheat was removed i n the very  top  p o r t i o n of the c a s t i n g .  5.7  Effect  of Cooling  Conditions  The e f f e c t o f sub-mould c o o l i n g c o n d i t i o n s has a l r e a d y been c o n s i d e r e d i n Chapter 4 i n c o n n e c t i o n w i t h of the model where the d i f f e r e n c e  validation  between c o n v e n t i o n a l  c o o l i n g and reduced secondary c o o l i n g of aluminium was seen. transfer  F i g . 5.10 shows the r e l a t i o n s h i p coefficient  below the mould and the  efficient  In these c a l c u l a t i o n s ,  l e n g t h o f the s t r a n d .  the h e a t - t r a n s f e r increase  coefficient  i n the pool  depth.  sections  the h e a t - t r a n s f e r  It  heat-  steady-state  is kept c o n s t a n t , at the a p p r o p r i a t e  the e n t i r e  slabs  between the  pool depth f o r the c a s t i n g o f 610 x 546 mm z i n c at 1 mm/s.  D.C.  value,  i s observed t h a t  cofor as  i s d e c r e a s e d , -there i s a steep  E E  -  SZ  zinc ingot- 610 mm x 546 mm speed - I mm /s 10001  QL 00  Q o o CL  E E  900  X D  800  Fig.  1  0  5.10  4.186  Effect  8.373 12.560 16.747 Heat Transfer Coefficient (kw/m °C)  o f the h e a t - t r a n s f e r  coefficient  depths f o r c a s t i n g 610 x 546 mm zinc  20.934  on the steady s t a t e  i n g o t ' a t 1 mm/s.  pool ro o  121 5.8  Effect  of L a t e n t Heat Release on Model  Calculations  In the s e c t i o n d e a l i n g w i t h the development o f mathematical  the  model a d e t a i l e d account was g i v e n o f the man-  ner i n which l a t e n t heat i s r e l e a s e d . runs were undertaken i n the i n i t i a l ment to t e s t the s e n s i t i v i t y  A number of  stages o f model  of the c a l c u l a t i o n s  parameter.  In a l l  the c a l c u l a t i o n s  temperature  range over which the l a t e n t  had a small  effect  on the l o c a t i o n  computer develop-  to  this  i t was seen t h a t  heat was r e l e a s e d  of the s o l i d u s  However, as might be e x p e c t e d , the l o c a t i o n i s o t h e r m was changed c o n s i d e r a b l y .  the  isotherm.  o f the  liquidus  For example, i n two  c a l c u l a t i o n s . of the c a s t i n g of z i n c where the l a t e n t  heat  was r e l e a s e d over a range of 10°C between 420°C and 410°C and over a range of 1 °C between t h e r e was a d i f f e r e n c e bottom o f the s o l i d u s 110 mm d i f f e r e n c e  and 420°C " r e s p e c t i v e l y -  of o n l y 10 mm i n the p o s i t i o n o f i s o t h e r m , whereas t h e r e was c l o s e  i n the case o f the l i q u i d u s  e f f e c t o f the manner of l a t e n t i n c r e a s i n g thermal  419  isotherm.  heat r e l e a s e decreases  conductivity  conductivity,  from above the l i q u i d u s iterative  it  The  with  simulating  i n the D.C. c a s t i n g o f n o n - f e r r o u s metals  high thermal  to  of the m e t a l .  I t was discussed i n C h a p t e r ' 3 t h a t w h i l e heat f l o w  the  with  i s p o s s i b l e f o r nodes to jump  to below the s o l i d u s .  A post  c o r r e c t i o n procedure has been used to overcome  122 this  problem.  However use o f t h i s  correction routine  the program i n c o n j u n c t i o n w i t h having a l a r g e time val  can c r e a t e s t a b i l i t y  the l i q u i d u s  problems.  and the s o l i d u s  the s t a b i 1 i t y ' problems.  in inter-  A w i d e r spread between  temperature help's i n  improving  In most of the c a l c u l a t i o n s  the  l a t e n t heat has been r e l e a s e d over a range o f e i t h e r  1°C  or 10°C.  In the case o f s i m u l a t i o n s  d e a l i n g w i t h the  casting  of reduced secondary cooled aluminum slabs a w i d e r range of 23°C has been used.  5 .9  Summary In t h i s  chapter,  the r e s u l t s  the importance o f the d i f f e r e n t been r e p o r t e d f o r  of the model study on  heat f l o w v a r i a b l e s  D.C. c a s t i n g under a range of i n D.C. c a s t i n g  casting  conditions.  I t was shown t h a t  conventional  secondary c o o l i n g , where the aspect  exceeds 2 . 5 ,  it  involving ratio  i s adequate t o c o n s i d e r heat f l o w  i n two  d i m e n s i o n s , namely, the s h o r t e r of the t r a n s v e r s e and the a x i a l  direction.  c o o l i n g the a x i a l  In the case of reduced  dimensions secondary  c o n d u c t i o n was seen to p l a y a minor  compared to the t r a n s v e r s e heat f l o w s . heat f l o w s  Thus i n  part  modelling  i n these systems, the c o m p l e x i t y o f the  may be reduced c o n s i d e r a b l y by n e g l e c t i n g the a x i a l conduction.  have  problem heat  The importance of the unsteady s t a t e and the  effect  o f s e c t i o n s i z e and c o o l i n g c o n d i t i o n s  demonstrated using s p e c i f i c and z i n c .  have been  examples of c a s t i n g  aluminium  Chapter 6  USE OF THE HEAT FLOW MODEL TO SOLVE A CRACKING PROBLEM IN THE D.C. CASTING OF PRIME .WESTERN GRADE JUMBO INGOTS  6 .1  Introduction Formation o f cracks has long been r e c o g n i z e d as a  problem i n the D.C. c a s t i n g of metals such as high aluminium a l l o y s  ( 1 2 , 42, 4 7 ) .  strength  U n l i k e the c o n t i n u o u s  casting  of s t e e l where many cracks are generated m e c h a n i c a l l y , by mould o s c i l l a t i o n cracks thermal  and bending and s t r a i g h t e n i n g  in non-ferrous  e t al  i n c a s t i n g copper and i t s  alloys.  are also o t h e r papers d e a l i n g w i t h  alloys  (55, 56).  literature  Similarly  formed  Dieffenbach  measures f o r s o l v i n g the  problems faced i n c a s t i n g d i f f e r e n t  aluminium a l l o y s  casting.  (17) d i s c u s s the v a r i o u s cracks  (12) proposed remedial  operations,  D.C. c a s t i n g are most o f t e n caused by  s t r e s s e s generated d u r i n g  Kreil  e.g.  cracking  aluminium a l l o y s . internal!cracks  There  in  ( 3 2 , 40, 42, 47, 6 1 - 6 3 , 67) and copper No r e f e r e n c e s  have been found i n  however concerned w i t h the f o r m a t i o n o f  i n the D.C. c a s t i n g of z i n c and i t s  124  alloys.  the cracks  125 6.2  Internal  Cracks i n the D.C. C a s t i n g of  Prime  Western Grade Zinc Prime Western Grade z i n c  i s an a l l o y o f z i n c  ing a p p r o x i m a t e l y  1 wt%  applications.  has been found t h a t  It  lead which i s used in  contain-  galvanizing  D.C. c a s t i n g o f  particular  a l l o y can give r i s e  problems.  In o r d e r to solve t h i s problem i t was necessary  to study the cracks  to severe i n t e r n a l  this  cracking  i n d e t a i l , e . g . crack l o c a t i o n  and  f r e q u e n c y , morphology of crack s u r f a c e s  and thermal  ditions  formation.  p r e v a i l i n g at the time o f crack  Thus an e x p e r i m e n t a l a t Cominco L t d . , T r a i l  campaign was conducted  where a two s t r a n d D.C.  con-  in-plant  casting  machine has been i n o p e r a t i o n f o r many years f o r the d u c t i o n o f Special cross-section  pro-  High Grade z i n c jumbos w i t h a notched  (Fig. 4.15).  In t h i s  study a t o t a l  of  c a s t i n g s o f Prime Western Grade z i n c were made w i t h  seven the  e x i s t i n g c o o l i n g assembly which i s c h a r a c t e r i s e d by a s h o r t i n t e n s e c o o l i n g zone below the mould; and the of the v a r i o u s  runs are presented i n Table  XI.  details  Four of  the runs were cast at 1.69 mm/s and the o t h e r t h r e e  at  1.27 mm/s.  below  normal  These correspond to normal and s l i g h t l y  c a s t i n g speeds r e s p e c t i v e l y  c a s t i n g of High Grade z i n c  t h a t apply f o r  (99.99%).  In the  the  existing  126 I Spray Pressure  I I Spray Pressure  427°C  289.5 kPa  262  kPa  1.27 mm/s  426°C  289.5 kPa  262  kPa  3  1.69 mm/s  430°C  303.4 kPa  310.3 kPa  0 310.3 kPa**  4  1.27 mm/s  424°C  310.3 kPa  310.3 kPa  317.2 kPa**  5  1.69 mm/s  425°C  275.8 kPa  275.8 kPa  344.7 kPa  6  1.69 mm/s  425°C  248.2 kPa  262.0. kPa  262.0 kPa  7  1.27 mm/s  422°C  206.8 kPa  206.8 kPa  275.8 kPa  Casting Speed  Pouring Temp  1  1.69 mm/s  2  Run No.  * **  I I I Spray Pressure  *  0  *  E x i s t i n g spray arrangement i n both s t r a n d s AandB Only 8 o f the 16 nozzles i n the t h i r d i n the t h i r d  r i n g used  ring f o r strand B  3 - 7  No sprays  3,4,5  T h i r d s e t o f sprays placed 457 mm below the  centre  of second r i n g , i n s t r a n d A 6,7  T h i r d set o f sprays placed 125 mm below t h e  centre  of the second r i n g i n s t r a n d A  Table XI  C a s t i n g speed c o n d i t i o n s  f o r the d i f f e r e n t  d u r i n g the e x p e r i m e n t a l  campaign.  runs  12 7 c o o l i n g ' assembly each of  the two s t r a n d s  i s cooled by a  set o f two spray r i n g s one i m p i n g i n g d i r e c t l y (flat  spray)  on the mould  and the second l o c a t e d j u s t below the mould.  In some o f the runs e x t r a c o o l i n g was a p p l i e d , to one of s t r a n d s by a t h i r d spray r i n g . mal c o n d i t i o n s  that exist  In o r d e r t o o b t a i n the  i n the c a s t i n g machine,  couples were i n s e r t e d i n the l i q u i d  pool  the ther-  thermo-  from the top and  f r o z e n i n near the s u r f a c e of the c a s t i n g ' d u r i n g  steady-  s t a t e o p e r a t i o n , and then were allowed to descend w i t h the jumbo.  Thus i t was p o s s i b l e to c h a r a c t e r i z e  extraction  rates  the  i n the mould and sub-mould r e g i o n as a  f u n c t i o n o f d i s t a n c e below the mould.  In most o f the  the l e n g t h o f the jumbo c a s t was about 3600 mm. was f i n a l l y  cut i n t o s e c t i o n s  610 mm.long and the  Whenever a major crack was seen i t s Tables  XII  location  runs  The c a s t i n g  verse s e c t i o n s were i n s p e c t e d f o r the presence o f  were measured.  heat  transcracks.  and l e n g t h  t o XVII give the r e s u l t s  these measurements f o r each run r e s p e c t i v e l y .  All  of  the  cracks mentioned i n these t a b l e s had a crack w i d t h  ranging  from 0.5 mm to 2.0 mm.  hair-  In a d d i t i o n  to t h e s e , f i n e  l i n e cracks were a l s o observed but are not important, to X V I I .  and t h e r e f o r e  considered  are not i n c l u d e d i n Tables  S e c t i o n s c o n t a i n i n g cracks w e r e ' a l s o  and taken to the l a b o r a t o r y  f o r metal 1 o g r a p h i c  collected examination.  Some o f the s e c t i o n s were macro-etched to study the structure.  XII  grain  1 28  Section No.  Bottom  Notch(Left)  Notch(Right)  1A2  Surface - 160 mm  Surface -• 160 mm  -  1A5  25 - 140 mm  30 -• 130 mm  -  30 •- 125 mm  -  30 •- 120 mm  -  1A6 1A7  35 - 125 mm  1B2  Surface - 155 mm  -  1B3  30 - 150 mm  -  1B4  30 - 140 mm  -  1B5  28 - 140 mm  -  1B6  28 - 147 mm  -  Table X I I  L o c a t i o n of cracks taken at v a r i o u s  i n jumbo  cross-sections  p o i n t s along the l e n g t h  strands A and B (Run 1 )  of  1 29  Section No.  Bottom  2A2  -  2A3  -  2A4  -  2A5  48 - 112 mm  2A7  2B2  Notch(Left)  50 - 90 mm  -  20 - 130 mm  35 - 150 mm  35 - 125 mm  3 0 - 1 3 2 mm  -  50 - 115 mm  2B3  -  2B4  -  2B6  -  -  L o c a t i o n of cracks  55 - 105 mm _  40 - 135 mm  2B7  Table X I I I  Notch(Right)  in jumbo  4 5 - 1 4 5 mm  -  cross-sections  taken at v a r i o u s p o i n t s along the l e n g t h s t r a n d s A and B (Run 2 ) .  of  1 30  Section No.  Bottom  Notch(Left)  Notch (Right)  3A2  40 •- 150 mm  3A3  40 -- 150 mm  40 -• 135 mm  3A4  40 •- 155 mm  35 -- 140 mm  Surface •- 120 mm 1  3A5  40 -• 130 mm  25 -- 185 mm  3A6  Surface -• 75 mm  30 -- 120 mm  3A7  -  3B2  Surface •- 150 mm  Surface -• 125 mm  3B3  40 -• 130 mm  50 -- 120 mm  3B4  35 -- 160 mm  3B5  25 -• 150 mm  3B6  30 -- 140 mm  3B7  Table XIV  L o c a t i o n of cracks taken a t v a r i o u s  in jumbo  points  s t r a n d s A and B (Run 3) .  cross-sections  along the l e n g t h  of  1 31  Section No.  Bottom  Notch(Left)  Notch(Right)  4A2  -  Surface - 140 mm  Surface - 145 mm  4A3  -  Surface - 130 mm  Surface - 140 mm  4A5  -  Surface - 120 mm  30 - 140 mm  4A6  -  Surface - 100 mm  40 - 115 mm  4A7  -  -  4B2  -  Surface -  90 mm  -  4B4  -  Surface -  90 mm  •-  4B5  -  Surface -  85 mm  -  4B6  -  Surface -  85 mm  -  4B7  -  -  Table XV  -  •-  L o c a t i o n o f cracks in jumbo c r o s s - s e c t i ons taken at v a r i o u s p o i n t s along the l e n g t h s t r a n d s A and B (Run 4) .  of  1 32  Section No.  •  . 5A1  Bottom  Surface -  90 mm  Notch(Left)  Surface - . 75 mm  Notch(Right)  Surface -  -  75 mm  5A2  Surface - 150 mm  5A3  Surface - 140 mm  5A4  Surface - 140 mm  -  Surface - 140 mm  5A5  Surface - 140 mm  -  Surface - 120 mm  5A6  -  5B1  Surface -  5B2  -  -  90 mm  90 mm  Surface -  -  Surface - 105 mm  -  5B3  -  40 - 140 mm  -  5B4  -  45 - 150 mm  -  5B5  -  Surface - 110 mm  —  —  5B6  Table XVI  L o c a t i o n of cracks  Surface -  in jumbo  taken at v a r i o u s p o i n t s s t r a n d s A and B (Run 5 ) .  80 mm  —  cross-sections  along the l e n g t h  of  1 33  Section No.  Bottom  Notch(left)  Notch(Right)  6A1  25 - 160 mm  5 0 - 1 2 0 mm  25 -  6A2  25 - 165 mm  30 - 155 mm  30 - 150 mm  6A3  30 - 148 mm  30 - 135 mm  30 - 150 mm  50 mm  6A4  -  6A5  -  6A6  -  -  -  6B1  -  Surface - 135 mm  Surface - 150 mm  6B2  -  25 - 130 mm  Surface - 165 mm  6B3  -  25 - 110 mm  Surface - 135 mm  6B4  -  2 0 - 1 3 5 mm  Surface - 130 mm  6B5  -  -  -  6B6  -  -  -  Table XVII  35 - 130 mm  -  L o c a t i o n of cracks taken at v a r i o u s  i n jumbo  points  s t r a n d s A and B (Run 6) .  cross-sections  along the l e n g t h  of  134 A few general  comments can be made r e g a r d i n g  the  occurrence of the l a r g e c r a c k s , an example o f which can be seen i n the t r a n s v e r s e s e c t i o n of a Prime Western Grade jumbo i n g o t shown i n F i g . 6 . 1 .  It  i s seen t h a t  these  tend to form normal to the s u r f a c e p r e d o m i n a n t l y  in  cracks  three  areas near the two notches and next to the bottom s u r f a c e . These cracks are s i m i l a r  i n many r e s p e c t s  cracks o f t e n seen i n t r a n s v e r s e cast steel  billets  (86, 87).  sections  t o the mid-way of  continuously  A f t e r m a c r o - e t c h i n g the  trans-  verse s u r f a c e s of the jumbo s e c t i o n s , these c r a c k s were seen to occupy i n t e r zinc..  granular  r e g i o n s , between columnar g r a i n s  Although the cracks observed i n F i g . 6.1  to the s u r f a c e ,  penetrate  t h i s was not always the c a s e , but  when the c r a c k i n g was very  only  severe.  S i x t y - f i v e : s e c t i o n s were v i s u a l l y  inspected f o r  cracks  and f i f t y - o n e ,  or 78%> were found to c o n t a i n one or more  severe c r a c k s .  In some runs a crack c o u l d be seen r u n n i n g  through the e n t i r e  6.3  l e n g t h o f the  casting.  Heat Flow A n a l y s i s A heat f l o w a n a l y s i s o f the jumbo c a s t i n g was p e r -  formed by using the heat f l o w model the measured temperature p r o f i l e in thermocouples.  Since i t  of  in conjunction  o b t a i n e d w i t h the  with frozen-  was not p o s s i b l e to f r e e z e  in  Fig.  6.1  A c r o s s - s e c t i o n of Prime Western Grade j umbo i n g o t showing the i n t e r n a l c r a c k s .  1 36 the thermocouple at the s u r f a c e o f the c a s t i n g , the heat-transfer  conditions existing  i n the machine were back  c a l c u l a t e d using the t h r e e - d i m e n s i o n a l  heat f l o w model by  trial  and e r r o r .  The boundary c o n d i t i o n s were  until  a match was o b t a i n e d between the measured and  c a l c u l a t e d temperature  surface  adjusted  profiles.  F i g . 6.2 shows the t e m p e r a t u r e p r o f i l e s  calculated  from the model f o r the v a r i o u s nodes as shown i n the  insert.  These have been o b t a i n e d f o r a c a s t i n g speed o f 1.69 mm/s with heat-transfer  conditions  i n the e x i s t i n g c o o l i n g  w i t h a s h o r t i n t e n s e spray i n the sub-mould r e g i o n . be seen t h a t beyond 120 s.  the s u r f a c e o f the jumbo undergoes • T r a n s l a t i n g the time a x i s  system It  can  reheating  into a distance  a x i s using the c a s t i n g speed, t h i s corresponds t o  the  bottom of the second spray r i n g below the mould.  Surface r e h e a t i n g below the sprays i s because i t  results  i n crack f o r m a t i o n  important  in the f o l l o w i n g  R e h e a t i n g , which is a maximum at the s u r f a c e causes s u r f a c e to expand more than the i n t e r i o r solidified  s h e l l ; and thus the s u r f a c e  put i n t o c o m p r e s s i o n , w h i l e a t e n s i 1 e at the s o l i d i f i c a t i o n  front.  r e g i o n of  way.  the the  i s c o n s t r a i n e d and strain  These t e n s i l e  r e s p o n s i b l e f o r the f o r m a t i o n o f the c r a c k s .  is  generated  strains  are  A similar  1 37  500.  I  400  o CD  i_  300  TL  Surface node (I) 23mm from surface (2) 46 mm from surface (3)  \  Z3  o Q.  E 200  100  0 0  240  480  720  Time (s) Fig.  6.2  C a l c u l a t e d t e m p e r a t u r e p r o f i l e s f o r the d i f f e r e n t nodes f o r z i n c jumbo i n g o t c a s t at. 1.69 mm/s. ,  138 mechanism has been proposed f o r cracks  the f o r m a t i o n of mid-way  i n c o n t i n u o u s l y cast s t e e l  billets  The p o s i t i o n below the l i q u i d l e v e l internal  (86).  a t which the  cracks are generated can be determined from the  heat f l o w model i f  it  can be assumed t h a t the cracks  close to the s o l i d i f i c a t i o n  front,  because then the depth  o f the crack beneath the s u r f a c e gives the s h e l l at the time of crack f o r m a t i o n .  form  The s h e l l  thickness  thickness  calcu-  l a t e d f o r a 1.69 mm/s c a s t i n g speed i s shown i n F i g . and the accompanying diagram gives the s u r f a c e profile.  termperature  The band on the l e f t - h a n d s i d e has been drawn  from the measured l o c a t i o n of the i n n e r t i p o f the for  6.3,  the v a r i o u s  sections  I t can be seen from t h i s initiated  after  i n s p e c t e d i n the t e s t  crack  campaign.  f i g u r e t h a t cracks are always  the reheat event has taken  place.  The magnitude of r e h e a t i n g r e q u i r e d t o cause cracks i s b e l i e v e d to be 45-50°C which i s lower than the value 100-150°C quoted f o r s t e e l on the c o e f f i c i e n t line  zinc  (86).  This comparison i s  o f l i n e a r expansion which f o r  i s 39.7 ( 1 0 " ) / ° C 6  steel.  The c r i t i c a l  tearing  in steel  similar  criterion  as compared to  value o f t e n s i l e  17  strains  based  polyerystal-  (10" )/°C 6  for  t h a t cause h o t -  has been e s t i m a t e d at around 0.2%. is applied f o r zinc,  of  If a  then the r e h e a t  of  1 39  Distance 0  F i g . 6.3  160  along  Y-axis 320  (mm) 480  Growth of the s h e l l i n the a x i a l at 1.69 mm/s.  Temperature (°C)  as a f u n c t i o n o f  direction  distance  f o r z i n c jumbo i n g o t  Figure on the r i g h t shows the  face t e m p e r a t u r e p r o f i l e of a jumbo s e c t i o n .  cast sur-  at the bottom m i d - f a c e  140 45°C observed i n the p r e s e n t work i s adequate f o r hot to o c c u r . critical  This i s o n l y a rough comparison because the s t r a i n t h a t i s r e q u i r e d would v e r y much depend on  the cohesion between g r a i n s which i n t u r n i s a f f e c t e d the presence o f l i q u i d  6.4  tears  Metallographic  f i l m s between columnar  dendrites.  Analysis  In o r d e r to i n v e s t i g a t e tion further  by  the mechanism of crack  a metal 1ographic e x a m i n a t i o n was c a r r i e d  formaout  on the cracked s u r f a c e .  F i g . 6.4 shows a macro-photograph  o f the cracked s u r f a c e .  In t h i s p a r t i c u l a r  was seen to go a l l section  sample the  the way through the t h i c k n e s s o f  crack  the  (100mm). The bottom face of the jumbo s e c t i o n i s on the  left-hand parallel obviously  side o f the f i g u r e .  The growth o f  dendrites  to the d i r e c t i o n o f heat f l o w is q u i t e the a x i a l  evident;  component o f heat c o n d u c t i o n i s  very  i m p o r t a n t owing t o the i n t e n s e spray c o o l i n g below the m o u l d .  F i g . 6.5 shows a scanning e l e c t r o n m i c r o g r a p h o f cracked s u r f a c e taken at a h i g h e r m a g n i f i c a t i o n . dendritic  The  nature of these cracks i s c l e a r from t h i s  The i d e n t i t y  of the w h i t e p a r t i c l e s  s u r f a c e was i n v e s t i g a t e d  F i g . 6.6  observed on the  the  inter-  picture. fractured  i n some d e t a i l .  (a) shows the scanning e l e c t r o n  micrograph  Fig.  6.4  A macro-photograph of the cracked Magni f i c a t i o n 1 . 3 X.  surface.  142  Fig.  6.5  Scanning e l e c t r o n micrograph o f a cracked surface. Magnification  200 X.  (a)  (b)  Fig.  6.6(a)  Scanning e l e c t r o n micrograph o f a cracked face r e v e a l i n g the smooth n a t u r e of t h e Magnification  Fig. 6.6(b)  1000 X.  Pb x - r a y p i c t u r e  of F i g .  6.6(a).  sur-  surface.  144 of the cracked s u r f a c e at a much h i g h e r m a g n i f i c a t i o n . smooth n a t u r e o f the s u r f a c e s t r o n g l y of l i q u i d f i l m s 6.6 (b)  points  t o the  a t the i n t e r f a c e and hot t e a r i n g .  presence Fig.  shows the Pb x - r a y scan o b t a i n e d from, the same a r e a ;  and thus the w h i t e p a r t i c l e s  seen i n F i g . 6.6  are a l e a d - r i c h second phase.  (a) and 6.5  S i m i l a r o b s e r v a t i o n s were  made w i t h r e s p e c t to o t h e r areas o f the s u r f a c e b).  The  The presence o f the l e a d - r i c h  also  (Fig.6.7a and  phase can be e x p l a i n e d  by examining the phase diagram f o r the Zn-Pb system shown i n Fig. 6.8.  It  can be seen t h a t z i n c has an e x t r e m e l y low  solubility  f o r l e a d , e . g . 0.5 - 0.9 wt % a t the  temperature of 417.8°C. it  monotectic  Thus i n Prime Western Grade z i n c  is p o s s i b l e to have l e a d - r i c h  dendritic  solid  l i q u i d p r e s e n t i n the  r e g i o n , thereby d r a s t i c a l l y  decreasing  inter-  cohesion  betweengrains.  6.5  Mechanism o f Crack  Formation  Based on the preceding r e s u l t s can be proposed f o r crack f o r m a t i o n Prime Western Grade z i n c . of cracks i s  incorrect  i s seen t h a t the s h o r t practice  mechanism  i n the D.C. c a s t i n g  The p r i m a r y cause f o r the  of.  formation  c o o l i n g p r a c t i c e below the mould. i n t e n s e c o o l i n g adopted i n the  in the expansion o f  It  current  leads to r e h e a t i n g o f the jumbo s u r f a c e below  second spray r i n g which r e s u l t s surface.  the f o l l o w i n g  the  the  Because the s u r f a c e heats and expands more than  (b) F i g . 6 . 7 ( a ) Scanning e l e c t r o n micrograph of a cracked face r e v e a l i n g the smooth n a t u r e of the M a g n i f i c a t i o n 1 0 0 0 X. Fig.  6.7(b)  Pb  x-ray  picture  of  Fig.  6.7(a).  sur-  surface.  146  WEIGHT 10  i  1000  20  l  30  l  40 I  50  SO  CENT  LEAD  80  85  J  I  I  BOILING  906°  PER  70  I  1  ,,I 95  (REF.5)  900  «  HASS, JELLINEK,  •  WARING  «  KLEPPA,  A . S E I T H ,  2 8 ( 5 5 1 - 7 9 8 °  800  90 ,  REF.9  ET A L . , REF. REF.  -  1  2  JOHNEN,  REF.3  '700  TWO  MELTS  « 6 O 0  " 5 0 0  417.6°  419.5'  - 94 . (98)  400^0.3(0.91  327°  318.2°  300  ( 9 9 . 5 F  200 0 Zn  F i g . 6.8  20  30  4 0 ATOMIC  Phase diagram of  50 PER  60 CENT  70  60  9 0  Pb-Zn  100  Ph  LEAD  system  (89).  147 the i n t e r i o r  o f the s o l i d i f i e d  s h e l l , a compressive  is generated at the s u r f a c e and a t e n s i l e solidification  front.  The s t r a i n  s t r a i n at  is s u f f i c i e n t  strain the  (-.2  to  to cause d e n d r i t e s  separated by l i q u i d  the s o l i d i f i c a t i o n  f r o n t to open up and form a c r a c k .  morphology o f the crack s u r f a c e s  is  .3%)  f i l m s o f lead near  indicative  The  of such a  hot t e a r i n g mechanism.  Tab! e X V I I I shows the maximum s u r f a c e reheat u s i n g the model at the v a r i o u s  locations  calculated  of the c a s t i n g .  nodes r e f e r r e d to as top and bottom correspond to the nodes a t the m i d - p l a n e of non-notched s u r f a c e s  The  surface  and the  corresponds t o the node at the bottom o f the n o t c h .  notch  Compared  to o t h e r s u r f a c e nodes these t h r e e show the maximum r e h e a t . Further  it  can be noted from t h i s  t a b l e t h a t the  reheat  o f the top node was much l e s s compared to the bottom and notch nodes.  These p r e d i c t i o n s match the o b s e r v a t i o n s  crack l o c a t i o n s ;  cracks were c o n f i n e d t o the  of  mid-plane  r e g i o n a d j a c e n t to the bottom and notched f a c e s .  F i g . 6.9 shows the e f f e c t  of c a s t i n g speed on the  surface  reheat phenomenon as p r e d i c t e d by the heat  flow  model.  The t h r e e speeds c o n s i d e r e d are 1.69 mm/s,  1.27  mm/s and 0.85 mm/s.  I t can be seen t h a t  the r e h e a t i n g  reduced c o n s i d e r a b l y when c a s t i n g at lower speeds.  is  This  suggests t h a t c r a c k i n g should be less severe at lower  speeds  148  Node L o c a t i o n  Table X V I I I  Reheat  Bottom  (1)  64°C  Top  (2)  35.5°C  Notch  (3)  53°C  C a l c u l a t e d values of r e h e a t different  at  p o i n t s on the s u r f a c e  the jumbo s e c t i o n .  of  Top and bottom  correspond to m i d - f a c e on the nonnotched  surfaces.  149  500  960  Fig.  6.9  E f f e c t of c a s t i n g speed on the s u r f a c e r e h e a t i n g at the bottom m i d - f a c e o f a jumbo section.  150 which i s  in l i n e with operator experience.  Although i t  is  p o s s i b l e to reduce the s e v e r i t y of the c r a c k i n g problem by d e c r e a s i n g the c a s t i n g speed, i t  i s n o t a' p r a c t i c a b l e  s o l u t i o n owing t o the lower p r o d u c t i o n r a t e s : quality  o f the s u r f a c e d e t e r i o r a t e s  Cold shuts  and s u r f a c e  further  at l o w e r / c a s t i n g  laps are o f t e n seen on the  the  speeds. surface  o f the i n g o t s c a s t at low speeds.  6.6  Design of New Cooling System Having thus a s c e r t a i n e d t h a t s u r f a c e  the sprays was the cause of cracks  reheating  below  i n Prime Western Grade  z i n c jumbos the design of a new spray system which would minimize t h i s  phenomenon was u n d e r t a k e n .  A number o f  runs were made using the h e a t - f l o w model and the of h e a t - t r a n s f e r  coefficient  at v a r i o u s  s t r a n d on the r e h e a t values was s t u d i e d . the s u r f a c e temperature p r o f i l e s heat-transfer  coefficients  p o i n t s along  the  F i g . 6.10 shows  obtained with  different  i n the sub-mould r e g i o n .  values 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 s 2 kW/m K c o r r e s p o n d i n g to the e x i s t i n g 2 and 41.86 kW/m K.  effect  The  used were 20.93 set-up,  The value o f heat t r a n s f e r  2 10.46 kW/m K coefficients  used below the second spray have been kept the same f o r all  the t h r e e r u n s .  heat-transfer  It  can be seen t h a t d e c r e a s i n g  coefficient  in the sub-mould r e g i o n  the  results  151  500  400h  o o  300  CU  o CD  Q. E  CD H  2004-  \00W  F i g . 6.10  Effect  of the h e a t - t r a n s f e r  coefficient  the sub-mould r e g i o n on the s u r f a c e at the bottom m i d - f a c e of a jumbo  in  reheating  section.  . i n a decrease i n the r e h e a t .  '  .  1  5  2  I t was very c l e a r from these  runs t h a t the reheat could be minimized by d e c r e a s i n g intensity  o f the sprays i n the sub-mould r e g i o n and by  i n c r e a s i n g the c o o l i n g below the second spray r i n g additional  the  sprays.  with  ^.  Thus the new c o o l i n g arrangement was designed to maintain uniform cooling a l l  the way down t o the bottom o f  the l i q u i d pool to ensure t h a t was minimized p r i o r  r e h e a t i n g o f the  surface  to complete s o l i d i f i c a t i o n .  accomplished by r e d i s t r i b u t i n g  the t o t a l  used over a wide area of the s u r f a c e .  T h i s was  amount of  The arrangement  the sprays i n the new c o o l i n g assembly i s shown i n 6.11.  This corresponds  notched jumbo s e c t i o n . with similar flat  water of  Fig.  to the bottom face o f the nonOther faces were a l s o  arrangement.  provided  As i n the e x i s t i n g  spray nozzle was used i n the top r i n g  the mould to ensure adequate s o l i d i f i c a t i o n and a minimum o f b r e a k - o u t s .  practice.a  i m p i n g i n g on i n the mould  However i n o r d e r to  decrease  the q u a n t i t y of. water which ul t i matel y f a l l s from the mould through the sub-mould sprays the nozzle s e l e c t e d f o r top r i n g had o n e - h a l f spray n o z z l e .  the c a p a c i t y of the e x i s t i n g  the flat  For c o o l i n g i n the sub-mould r e g i o n a t o t a l  of f o u r spray r i n g s were designed w i t h 8 nozzles per  spray  ring  four  (two per f a c e ) .  The nozzles s e l e c t e d f o r these  153  F i g . 6.11  Arrangement of spray nozzles  i n the new c o o l i n g  assembly f o r the bottom s u r f a c e of a jumbo section.  154 r i n g s were d i f f e r e n t  from the nozzles used i n the o l d  design.  The new nozzles are of the wide angle type which p r o v i d e a u n i f o r m water f l u x d i s t r i b u t i o n  over a given face  (88).  Thus  the use of new spray nozzles r e s u l t s i n a w a t e r f l u x o f 2 2 0.9 £/m s i n comparison w i t h 4.8 £/m s. Although t h e r e i s a large difference  between the two v a l u e s , the t o t a l  water i n the new design i s only m a r g i n a l l y value.  amount o f  l e s s than the  old  The new spray assembly was c o n s t r u c t e d u s i n g 38 mm  diameter g a l v a n i z e d pipes w i t h threaded c o n n e c t i o n s ' .  A few  minor m o d i f i c a t i o n s were r e q u i r e d t o the c a s t i n g assembly  to  accommodate the new spray system.  6.7  T e s t i n g o f the New C o o l i n g System The new spray assembly was t e s t e d i n - p l a n t  a total  of four runs.  In the f i r s t  was high grade z i n c at 1.27 mm/s. out e s s e n t i a l l y  run the m a t e r i a l This run was  cast  carried  to check out the new assembly as w e l l  gain c o n f i d e n c e o f the o p e r a t o r s ficult  by making  as t o  f o r c a s t i n g the more  Prime Western Grade z i n c a l l o y .  Since t h i s  dif-  initial  run d i d not pose any p r o b l e m , Prime Western Grade z i n c was cast at speeds of 1.27 mm/s and 1.48 mm/s i n t h r e e  runs.  As b e f o r e t r a n s v e r s e s e c t i o n s were i n s p e c t e d f o r the sence of c r a c k s .  Of the t w e n t y - t w o  sections  pre-  inspected  between two s t r a n d s i n two runs no s e c t i o n showed any major cracks as seen i n the p r e v i o u s  campaigns.  As b e f o r e  in  155 some s e c t i o n s , e x t r e m e l y f i n e h a r i 1 i n e observed. It  cracks were a l s o  F i g . 6.11 shows a s e c t i o n from the new campaign.  should be noted t h a t the f i n e h a i r l i n e - crack  present  was l e s s v i s i b l e b e f o r e e t c h i n g w i t h h y d r o c h l o r i c  acid.  It  was the o p i n i o n o f the o p e r a t i n g people t h a t t h e q u a l i t y  of  the Prime Western Grade z i n c c a s t using the new design was at l e a s t as good as the q u a l i t y of high grade z i n c cast w i t h the o l d  6.8  jumbos  set-up.  Summary The c r a c k i n g problem encountered i n c a s t i n g  Western Grade z i n c has been i n v e s t i g a t e d , the r e s u l t  of the s h o r t i n t e n s e  and shown to be.  spray c o o l i n g  practice  employed which generates s u r f a c e r e h e a t i n g and strains  at the s o l i d i f i c a t i o n  front.  between a d j a c e n t columnar d e n d r i t e s of these s t r a i n s liquid  Prime  tensile  Opening of under the  cracks  influence  i s enhanced by the presence o f  lead-rich  fi1ms.  A new spray system has been designed w i t h the a i d o f the t h r e e - d i m e n s i o n a l  heat f l o w model to overcome the  of s u r f a c e r e h e a t i n g by m a i n t a i n i n g bottom of the l i q u i d p o o l . in-plant  u n i f o r m c o o l i n g to  The new assembly has been  problem the tested  f o r the c a s t i n g of Prime Western Grade z i n c and  shown to t o t a l l y  eliminate  the severe  cracks.  1 56  F i g . 6.12  A c r o s s - s e c t i o n o f Prime Western Grade jumbo i n g o t w i t h the new c o o l i n g  system.  J  Chapter 7  SUMMARY AND CONCLUSIONS  A fully  t h r e e dimensional  model has been developed  to s i m u l a t e heat f l o w and s o l i d i f i c a t i o n  i n the D i r e c t  c a s t i n g of n o n - f e r r o u s metals w i t h r e c t a n g u l a r irregular  notched c r o s s - s e c t i o n s .  alternating-direction  implicit  Chill  as w e l l  as  The model employs an  finite-difference  method  to s o l v e the g o v e r n i n g heat c o n d u c t i o n e q u a t i o n and can take i n t o account both s t e a d y - s t a t e initial  transient  extensively for  conditions.  internal  o p e r a t i o n as w e l l  The model has  and pool depths w i t h i n d u s t r i a l o f aluminium and z i n c . gate the r e l a t i v e  validity  o f pool  has  profiles  data f o r the D.C.  The model has been used t o  importance o f the i n d i v i d u a l  of heat c o n d u c t i o n and has r e v e a l e d the 1.  been tjested  c o n s i s t e n c y and i t s  been checked by comparing p r e d i c t i o n s  as  casting investi-  components  following:  For t h i c k aluminium s l a b s e . g . 381 mm and, 457 mm thickness,  s u b j e c t e d to c o n v e n t i o n a l  two-dimensional  model i n w h i c h , h e a t  D.C. c o o l i n g , a flow p a r a l l e l  the b r o a d - f a c e i s n e g l e c t e d can be used when the aspect r a t i o exceeds 2.  For t h i c k  2.5.  z i n c slabs of 381 mm t h i c k n e s s  157  the same  to  158 two-dimensional exceeding 3.  model i s adequate f o r aspect  ratios  2.0.  For the case of reduced secondary c o o l i n g i t p r e f e r a b l e t o n e g l e c t heat f l o w i n t h e d i r e c t i o n , but c o n s i d e r both the two  is  axial  transverse  d i r e c t i o n s when f o r m u l a t i n g a t w o - d i m e n s i o n a l model even though the aspect r a t i o exceeds 4.  The model has shown the f i r s t  25% o f the  c a s t i n g c y c l e i n the p r o d u c t i o n o f z i n c sections  i s i n the unsteady s t a t e .  2.5.  total jumbo  The t r a n -  s i e n t p a r t o f the c a s t i n g i s also very  important  when reduced secondary c o o l i n g i s employed. conventional  For  D.C. c o o l i n g o f aluminium the un-  steady s t a t e i s i m p o r t a n t o n l y i n c a s t i n g  large  s e c t i o n s , e . g . 457 x 1143 mm.  A c r a c k i n g problem encountered i n the D.C.  casting  of Prime Western Grade z i n c has been i n v e s t i g a t e d w i t h a i d o f the mathematical  model.  It  has shown t h a t the c r a c k i n g  i s caused by the use of s h o r t - i n t e n s e sub-mould r e g i o n . lations  the  spray c o o l i n g i n  Temperature measurements and model  have r e v e a l e d t h a t t h i s c o o l i n g p r a c t i c e  the calcu-  results  in  the r e h e a t i n g o f the s u r f a c e below the spray r e g i o n , which i n t u r n generates t e n s i l e  strains  at the  solidification  159  f r o n t where f i l m s o f l e a d - r i c h and f a c i l i t a t e  l i q u i d separate  the opening up o f  Based on t h i s a n a l y s i s  dendrites  cracks.  a new spray c o o l i n g  assembly  has been designed f o r c a s t i n g Prime Western Grade jumbo i n g o t s .  This design a t t e m p t s to cool the s u r f a c e  the jumbo more u n i f o r m l y over the e n t i r e liquid  length of  pool and also p r e v e n t s u r f a c e r e h e a t i n g .  ments have been undertaken i n - p l a n t and r e s u l t s effective  zinc  the  Experi-  to t e s t the new s y s t e m ,  have shown t h a t the new c o o l i n g assembly  in preventing  cracks.  of  is  160 SYMBOLS Symbol  Description  c  specific heat  Units ,  Jg  K"  1  1  -2 -1 h  heat transfer coefficient  Wm  k  thermal conductivity  Wm~^  L  latent heat of s o l i d i f i c a t i o n  J g~^  thickness of slab  K  mm  * ** *** T,T^,T ,T,T 2  ,T  temperature  C  t  time  s  v  casting speed  mm s~^ 3  volume of a node  mm  x  x-direction  dimensionless  y  y-direction  dimensionless  z  z-direction  dimensionless  X  length in X-direction  mm  Y  length in Y-direction  mm  Z  length in Z-direction  mm  2 -1 a  thermal d i f f u s i v i t y  mm s  Ax  distance step in x-direction  mm  Ay  distance step in y-direction  mm  Az  distance step in z-direction  mm  At p  time step density  skgm  e e  temperature emissivity  a  Stefan-Boltzmann constant  dimensionless -2 kWm  _3  Subscripts £  -  liquidus  s  -  solidus  m  -  mushy  n  -  present time interval  n+1  -  future time interval  i  -  node identification in x-direction  j  -  node identification in y-direction  k  -  node identification in z-direction  -  Base  -  average  b av  '  BIBLIOGRAPHY  1.  Lewis, D.M. and Savage, J . , "The P r i n c i p l e s o f Continuous Casting of M e t a l s " , M e t a l l u r g i c a l Reviews, v o l . 1 , Part I , 1956, pp. 65-115.  2.  Emley, E . F . , "Continuous Casting o f A l u m i n i u m " , I n t e r n a t i o n a l Metals Reviews, June 1976, Review No. 206, pp. 75-115.  3.  Nussbaum, A . I . , "Recent Developments i n SemiContinuous Casting of Aluminium B i l l e t s and S l a b s " , Paper presented at the Annual AIME Meeting 1 9 7 1 , TMS Paper No. A71-42.  4.  Spear, R.E. and B r o r d y k e , K . J . , "Continuous Casting of A l u m i n i u m " , Paper presented at the AIME Annual Meeting 1970, TMS Paper No. A70-30.  5.  M o r i t z , G., " H o r i z o n t a l S t r i p Casting i n D.C. Moulds, C a s t i n g Speed o f 8 F . P . M Paper p r e s e n t e d at the Annual AIME M e e t i n g , 1970, TMS Paper No. A70-29.  6.  Powers, R . B . , "The Alcoa H o r i z o n t a l Continuous C a s t i n g P r o c e s s " , L i g h t Metal Age, v o l . 33, no. 12, December 1975, pp. 5 - 7 .  7.  S p a u l d i n g , H . S . , "The Kaiser Aluminium Process f o r H o r i z o n t a l Continuous C a s t i n g Aluminium I n g o t " , v o l . 33, no. 12, Dec. 1975, pp. 8 - 1 1 .  8.  A i t c h i s o n , L. and K o n d i c , V . , The C a s t i n g o f Non Ferrous I n g o t s , Macdonald and Evans L t d . , 1953.  9.  T a y l o r , A . T . , Thompson, D.H. and Wegner, " D i r e c t C h i l l Casting o f Large Aluminium Metal P r o g r e s s , Nov. 1957, pp. 7 0 - 7 4 .  J.J., Ingots",  10.  Wickle K.G., "Semi-Continuous Casting of Copper". Metal P r o g r e s s , A p r i l 1958, pp.  Beryllium 85-89.  11.  Nussbaum, A . I . , " F u l l y and Semi-Continuous C a s t i n g of Copper and Copper Base A l l o y B i l l e t s and S l a b s " , Proceedings of the Continuous Casting Symposium of the 102 AIME M e e t i n g , Chicago, 1973.  12.  D i e f f e n b a c h , R. P.. , " P r a c t i cal Problems i n C a s t i n g Aluminium D.C. I n g o t " , Paper presented at the Annual AIME M e e t i n g , 1 9 7 1 , TMS Paper No. A71-40. 162  163 13.  B e n n e t t , C.G., "The Mode o f Formation o f S t r i a t e d Surface Defects on S e m i - C o n t i n u o u s l y Cast Aluminium A l l o y B i l l e t " , Proceedings o f the Continuous C a s t i n g Symposium o f the 102 AIME M e e t i n g , C h i c a g o , 1973, pp. 237-245.  14.  S e r g e r i e , F.A. and B r y s o n , N . B . , " R e d u c t i o n o f I n g o t Bottom Fowing and Bumping i n Large Sheet I n g o t C a s t i n g " , Paper p r e s e n t e d at the AIME Annual Meeting 1974, TMS Paper No. A74-56.  15.  B i n c z e w s k i , G . J . and Kramer, W.K., " S i m u l t a n e o u s C a s t i n g o f A l l o y C o m p o s i t e s " , Paper p r e s e n t e d a t the AIME Annual Meeting 1972, TMS Paper No. A72-82.  16.  Thomson, R. and E l l w o o d , E.C., "Closed-Head Continuous C a s t i n g . Part I - Processes and A p p l i c a t i o n s " , The B r i t i s h Foundryman, v o l . 65, Part 4 , . A p r i l 1972, p p . 1 38-45 .  17.  K r e i l , A . , V o s s k u h l e r , H. and W a l t e r , K . , ' " T h e Continuous Casting o f Copper and i t s A l l o y s , M e t a l l u r g i c a l Reviews, v o l . 5, no. 2 0 , 1960, p p . 41 3-446 .  18.  G e t s e l e v , Z . N . , " C a s t i n g i n an E l e c t r o m a g n e t i c F i e l d " , Journal o f M e t a l s , O c t o b e r , 1 9 7 1 , p p . 3 8 - 3 9 .  19.  T r a p i e d , G, " M o d i f i e d Semi-Continuous C a s t i n g P r o c e s s e s " , Journal o f the I n s t i t u t e o f M e t a l s , 92, 1963-64, pp. 305-312.  20.  C o l l i n s , D.L.W., "A New E x p l a n a t i o n o f the S u r f a c e S t r u c t u r e s o f D.C. I n g o t s " , Metallurgia, vol.76, O c t . 1967, pp. 137-144.  21.  Bergmann, W . J . , " S o l i d i f i c a t i o n i n Continuous C a s t i n g o f A l u m i n i u m " , Paper p r e s e n t e d at the Annual AIME Meeting 1970, TMS Paper No. A70-27. - •  22.  Bergmann, W . J . , "High S u r f a c e . Q u a l i t y Aluminium I n g o t " , Proceedings of the Continuous C a s t i n g Symposium o f the 102 AIME M e e t i n g , Chicago, 1973, pp. 247-256.  23.  Bhamra, H . S . , Garber, S . , Kondic, V. and H i l l , J . F . ' , "The M e t a l l o g r a p h y o f Semi-Continuous and S t a t i c Cast 70:30 Brass S l a b s " , P r a c t i c a l M e t a l l o g r a p h y , v o l . 12, 1975, pp. 128-147.  vol.  164 24.  Bergmann, W . J . , " S u r f a c e S t r u c t u r e s o f C o n t i n u o u s l y Cast A l u m i n i u m " , Paper presented at the Annual AIME Meeting 1 9 7 1 , TMS Paper No. A71-58.  25.  L e w i s , D.M., "Techniques f o r the I n v e s t i g a t i o n o f Thermal C o n d i t i o n s i n Continuous C a s t i n g " , J o u r n a l of the I n s t i t u t e of M e t a l s , v o l . 82, V953-54, pp. 395-413.  26.  Roth, W., " C o o l i n g the Casting i n ' C h i l l A l u m i n i u m , v o . 25, 1943, pp. 2 8 3 - 2 9 1 .  27.  A d e n i s , D . J . - P . , Coats, K.H. and Ragone, D.V., "An A n a l y s i s of the D i r e c t - C h i l l C a s t i n g Process by Numerical Methods", Journal o f the I n s t i t u t e o f M e t a l s , v o l . 9 1 , 1962-63, pp. 395-403.  28.  P e e l , D.A., P e n g e l l y , A . E . and P i l k i n g t o n , S . , " P i l o t P l a n t Studies of Heat T r a n s f e r S o l i d i f i c a t i o n and R e s u l t a n t S t r u c t u r e of C o n t i n u o u s l y Cast A l u m i n i u m " , Paper presented at the Annual AIME M e e t i n g , TMS Paper No. A70-49.  29.  P e e l , D.A. and P e n g e l l y , A . E . , " H e a t - t r a n s f e r , S o l i d i f i c a t i o n and M e t a l l u r g i c a l S t r u c t u r e i n the Continuous C a s t i n g of A l u m i n i u m " , Mathematical Models i n M e t a l l u r g i c a l Process Development, 1970, London, The I r o n and Steel I n s t i t u t e p p . ' 1 8 6 - 1 9 6 .  30.  B e a t t i e , D.D., Davies G . J . and T i z a r d , A . H . , " C o n t r o l of P e r i p h e r a l Zone S t r u c t u r e i n D i r e c t C h i l l Cast A l u m i n i u m " , Paper presented at U.K. Automation Councils 5th C o n t r o l C o n v e n t i o n , U n i v e r s i t y of B a t h , 17-20th o f Sept. 1973.  31.  Kroeger, P.G. and O s t r a c h , S . , "The S o l u t i o n of a Two' Dimensional Freezing Problem I n c l u d i n g Convect i o n E f f e c t s in the L i q u i d R e g i o n " , I n t e r n a t i o n a l J o u r n a l o f Heat and Mass T r a n s f e r , v o l . 17, 1974, pp. 1191-1207.  32.  Mathew, J . , " S i m u l a t i o n of Heat Flow and Thermal Stresses i n Continuous C a s t i n g : A p p l i c a t i o n of the Model to C r a c k i n g " , Ph.D. T h e s i s , Dept. o f M e t a l l u r g i c a l E n g i n e e r i n g , U n i v . of P i t t s b u r g h , 1977.  33.  B e a t t i e , D.D., "Mathematical Model S t u d i e s M e t a l l u r g i c a l S t r u c t u r e s in Aluminium D.C. Metals Technology, v o l . 4, March 1977, pp.  Casting'",  of Castings", 147-152.  165 34.  J i n , I . and S u t h e r l a n d , J . G . , "Thermal A n a l y s i s o f S o l i d i f i c a t i o n o f Aluminium A l l o y s During Continuous C a s t i n g " , Paper p r e s e n t e d at the I n t e r n a t i o n a l S o l i d i f i c a t i o n Conference, S h e f f i e l d , J u l y , 1977.  35.  J o v i c , L . , S p a s o j e v i c , D., S t e f a n o v i c , V. and Novakovic, M., "Study of Heat T r a n s f e r and M e t a l l u r g i c a l S t r u c t u r e i n the Continuous C a s t i n g o f Aluminium and Aluminium A l l o y s " , Paper p r e s e n t e d at the I n t e r n a t i o n a l Conference on Heat and Mass T r a n s f e r i n M e t a l l u r g i c a l Systems, D u b r o v n i k , Yugoslavia,Sept.1979.  36.  S z a r g u t , J . and Skorek, J . , " A n a l y s i s o f I n g o t temperature f i e l d i n Continuous C a s t i n g of Copper", Metals Technology, v o l . 7, Part I , Jan. 1980, pp. 3 6 - 4 0 .  37.  Jensen, E.K., "Mathematical Model C a l c u l a t i o n s i n Level Pour D.C. C a s t i n g of Aluminium E x t r u s i o n I n g o t s " , L i g h t M e t a l s , 1 9 8 0 , P u b l i s h e d by the M e t a l l u r g i c a l S o c i e t y of AIME, pp. 631-642.  38.  Yu, Ho, "A Process to Reduce I n g o t B u t t Curl S w e l l " , L i g h t M e t a l s , 1980, pp. 613-628.  39.  S c h a r f , G. and Lossack, E., "The I n f l u e n c e o f Aluminium A l l o y C a s t i n g S t r u c t u r e s on Homogenising and E x t r u d a b i 1 i t y " , Proceedings of Second I n t e r n a t i o n a l Aluminium E x t r u s i o n Technology Seminar, vol . 1 , 1 977 , pp. 311-20.  40.  F o r t i n a , G. and A n s e l m i , A . , "Parameters f o r A n a l y s i n g D.C. C a s t i n g Behaviour of A l . A l l o y s " , L i g h t M e t a l s , 1 977 , vol . 2, pp. 207-221 .  41.  Buxmann, K., " S o l i d i f i c a t i o n C o n d i t i o n s and M i c r o s t r u c t u r e i n C o n t i n u o u s l y Cast A l u m i n i u m " , ' Presented at the I n t e r n a t i o n a l Mass and Heat T r a n s f e r Conf e r e n c e , D u b r o v n i k , Y u g o s l a v i a , Aug. 1979.  42.  Nawata, S . , Kubota, M. and Y o k o t a , K., "High Speed C a s t i n g of AA-6063 E x t r u s i o n I n t o t " , American I n s t i t u t e of M i n i n g , M e t a l l u r g i c a l and Petroleum E n g i n e e r s , 1975, pp. 161-174.  43.  T h e l e r , J . J . , M e i e r , H.A., and L e c o n t e , G.B. , "OnLine C o n t r o l of a D.C. Casting Machine Using a M i c r o p r o c e s s o r " , L i g h t M e t a l s , 1977. v o l . 2 , p.p. 271 - 2 7 7 .  and  1 66 M e i e r , H.A., L e c o n t e , G.B. and Odok, A . M . , A l u s u i s s e Experience w i t h E l e c t r o Magnetic M o u l d s " , L i g h t M e t a l s , v o l . 2 , 1977, p p . 2 2 3 - 2 3 3 . Weckman, D . C , P i c k , R.J. and N i e s s e n , P., "A Numerical Heat T r a n s f e r Model o f the D.C. Continuous C a s t i n g P r o c e s s " , Z e i t s c h r i f t f u r Metal 1kunde, v o l . 7 0 , no. 1 1 , Nov. 1979, pp. 750-757. E r i c k s o n , W.C., "Computer S i m u l a t i o n o f S o l i d i f i c a t i o n " , AFS I n t e r n a t i o n a l Cast Metals J o u r n a l , Mar. 1980, pp. 30-40. Bryson, N . B . , " I n c r e a s i n g the P r o d u c t i v i t y o f Aluminium D.C. C a s t i n g " , Paper p r e s e n t e d at the Annual AIME Meeting 1972, TMS Paper No. A72-42. Fossheim, H. and Madsen, E . E . , " A p p l i c a t i o n of a Mathematical Model i n L e v e l - P o u r D.C. C a s t i n g of Sheet I n g o t s " , Paper presented at the Annual Meeting of AIME, 1979, TMS Paper No. 79-40. Also L i g h t Metals 1979, pp. 695-720. M a t y j a , H . , Giessen, B.C. and G r a n t , N . J . , "The E f f e c t of Cooling Rate on the D e n d r i t e Spacing in S p l a t - C o o l e d Aluminium A l l o y s " , J o u r n a l of the I n s t i t u t e of M e t a l s , v o l . 96, 1968, p p . 3 0 - 3 2 . S l a t e , P.M. and W h i t a k e r , M., "The Nature o f P a r t i c l e Size of I n t e r m e t a l 1 i c s in Two Types o f Aluminium A l l o y " , Journal of I n s t i t u t e of M e t a l s , v o l . 92, 1963-64, pp. 7 0 - 7 7 . T o u l o u k i a n , Y.S. and Buyko, E . H . , "Thermophys i cal Properties of M a t t e r " , I F I / P l e n u m , New York Washington, 1970. B a l a n d i n , G.F. and Y a k o v l e v , Yu. P . , "The Use of V i b r a t i o n i n the Continuous C a s t i n g o f Non-Ferrous Metals and A l l o y s " , Tsevtnye M e t a l l y , v o l . 2, Issue no. 1 , 1 9 6 1 , pp. 7 8 - 8 1 . B a k h t i a r o v , R.A., Pokrovskaya, G.N. and Kraeva, T . M . , " I n t e n s i f y i n g the Semi c o n t i n u o u s C a s t i n g of Aluminium Bronze I n g o t s " , S o v i e t Journal of Non-Ferrous M e t a l s , v o l . . 1 4 , no. 9, pp. 5 4 - 5 7 . E l l i s o v , V.S. and Stepanov, Yu. A . , " E f f e c t of Operat i n g C o n d i t i o n s of the Semi-Continuous C a s t i n g Machine Subassemblies on the Q u a l i t y of A l u m i n i u m - A l l o y I n g o t s " S o v i e t Journal of Non-Ferrous M e t a l s , v o l . 12, no. 5, pp. 77-80.  167 S h a d r i n , G.G. and Cherepok, G.V., "Some Aspects o f Crack Formation When Producing Round I n g o t s w i t h Large Cross S e c t i o n s " , S o v i e t J o u r n a l of Non-Ferrous M e t a l s , Dec. 1965, pp. 7 5 - 8 1 . Dmitrieva, Goloveshko, Continuous Journal of pp. 8 1 - 8 3 .  G . S . , S h l e p t s o v , V . F . , G a l d o b i n , V.G. and V . F . , "Crack Formation During the SemiCasting o f F l a t Copper I n g o t s " , S o v i e t Non-Ferrous M e t a l s , v o l . 1 2 , no. 5, '  E l i s o v . V . S . , S k u c h i l o v , A . I . , S h i p i l o v , V . S . and S a v u s h k i n , V. I . , " E f f e c t of C a s t i n g Parameters on Force I n t e r p l a y o f Aluminium I n g o t s w i t h M o u l d s " , S o v i e t J o u r n a l of Non-Ferrous M e t a l s , v o l . 10, 1968, pp. 114-116. F o r t i n a , G. and G a t t o , F., "Thermal D i s t r i b u t i o n During I n g o t S o l i d i f i c a t i o n i n D.C. C a s t i n g " , Paper presented at the Annual Meeting of AIME 1978, TMS Paper No. 7 8 - 4 9 , Putman, J . L . , "The Use o f A u t o r a d i o g r a p h y f o r F i n d i n g the S o l i d i f i c a t i o n Boundary i n C o n t i n u o u s l y Cast A l u m i n i u m " , Journal o f the I n s t i t u t e o f M e t a l s , 1953-54, v o l . 82, pp. 414-416. Lucas, G., "Auto C o n t r o l of D.C. C a s t i n g O p e r a t i o n s " , L i g h t M e t a l s , 1974, Published by AIME, pp. 6 3 5 - 6 3 8 . Dodd, R.A., "Residual Stresses i n C h i l l - C a s t and C o n t i n u a l l y Cast Aluminium A l l o y B i l l e t s " , J o u r n a l o f I n s t i t u t e of M e t a l s , v o l . 8 0 , 1951-52, pp. 4 9 3 - 5 0 0 . Levy, S . A . , Zinkham, R.E. and Carson, J . W . , " R e s i d u a l S t r e s s Measurements f o r S t u d y i n g I n g o t C r a c k i n g " , L i g h t Metals 1 974 , P u b l i s h e d by the M e t a l l u r g i c a l S o c i e t y of AIME, pp. 571-585. M o r i c e a u , J . , "Thermal Stresses i n Continuous D.C. C a s t i n g of Aluminium A l l o y s , D i s c u s s i o n . o f Hot T e a r i n g Mechanisms", L i g h t Meta1s 1975, P u b l i s h e d by M e t a l l u r g i c a l S o c i e t y o f AIME, v o l . 2, pp. 11.9-133. S m i t h , G.C., Numerical S o l u t i o n o f P a r t i a l E q u a t i o n s , Clarendon P r e s s , O x f o r d , 1978.  Differential  Carnahan, B . , L u t h e r , H . A . and W i l k e s , J . O . , A p p l i e d Numerical Methods, John.Wiley & Sons, I n c . , 1969.  168 66.  B a l l a n t y n e , A . S . , "Heat Flow i n Consumable E l e c t r o d e Remelted I n g o t s " , Ph.D T h e s i s , U n i v . o f B r i t i s h Columbia, 1978.  67.  P e l l i , T. and Z o l 1 e r , H . , "The Development of I n t e r n a l Stresses and S t r u c t u r a l Deformations During the Continuous Casting of Aluminium A l l o y s " , M e t a l l , v o l . 26, no. 3, 1972, pp. 214-218.  68.  B r a c a l e , G . , Fommei, F. D.C. Type o f Casting of ings of the Continuous AIME M e e t i n g , Chicago,  69.  Weinberg, F., "Continuous C a s t i n g " , Metals vol .6 , .February 1 979 , pp. 2 8 - 5 5 .  70.  S i n g l e t o n , J r . , O.R., "An A n a l y s i s of New Quenchants f o r A l u m i n i u m " , Journal of M e t a l s , v o l . 2 0 , Nov. 1968, pp. 6 0 - 6 7 .  71.  M i z i k a r , E.A., "Mathematical Heat T r a n s f e r Model f o r S o l i d i f i c a t i o n of C o n t i n u o u s l y Cast Steel S l a b s " , T r a n s a c t i o n s of AIME, v o l . 299, 1967, pp. 1747-1753.  72.  C o m i n i , G . , Del G u i d i c e , S . , Lewis, R.W. and Z i e n k i e w i c z , " F i n i t e Element S o l u t i o n of. NonL i n e a r Heat Conduction Problems w i t h Special Ref e r e n c e to Phase Change", I n t e r n a t i o n a l Journal f o r Numerical Methods i n E n g i n e e r i n g , v o l . 8, 1974, pp. 6 1 3 - 2 4 .  73.  Ohnaka, I . and Fukusako, T . , " C a l c u l a t i o n of S o l i d i f i c a t i o n of Molten Metal by F i n i t e Element M e t h o d " , T e c h n i c a l Report, Osaka U n i v e r s i t y , v o l . 24, 1974, pp. 461-475.  74.  Emery, A . F . and Carson, W.W., "An E v a l u a t i o n of the Use of the F i n i t e Element Method i n the Computation of T e m p e r a t u r e " , T r a n s a c t i o n s of ASME, Journal of Heat T r a n s f e r , v o l . 9 3 , May 1 9 7 1 , pp. 136-145.  75.  Peaceman, D.W. and R a c h f o r d , H . H . , "The Numerical S o l u t i o n of P a r a b o l i c and E l l i p t i c D i f f e r e n t i a l E q u a t i o n s " , J . Soc. I n d u s t . A p p l . M a t h . , v o l . 3, no. 1 , March 1 955 , pp . 28-41 .  76.  Douglas, J r . , J . and Peaceman, D.W., "Numerical S o l u t i o n o f Two Dimensional Heat Flow P r o b l e m s " , A . I . Ch.E. J o u r n a l , v o l . 1 , no. 4 , pp. 505-512.  and A l t i , A . , " R e s e a r c h on Aluminium A l l o y s " , ProceedCasting Symposium of the 102 1973, pp. 353-372. Technology,  1 69 77.  B r i a n , P . L . T . , "A F i n i t e D i f f e r e n c e Method o f High Order Accuracy f o r the S o l u t i o n of Three Dimensional T r a n s i e n t Heat Conduction P r o b l e m s " , A . I . C h . E . J o u r n a l , v o l . 7, no. 3, pp. 367-370.  78.  Myers, G. E . , A n a l y t i c a l Methods i n Conduction Heat T r a n s f e r , McGraw-Hill, 1971.  79.  D u s i n b e r r e , G.M., "Heat T r a n s f e r C a l c u l a t i o n s by Finite Differences", I n t e r n a t i o n a l Text Book Company, P e n n s y l v a n i a , 1961.  80.  H i l l s , A . W . D . , "A G e n e r a l i s e d I n t e g r a l P r o f i l e Method f o r the A n a l y s i s of U n i d i r e c t i o n a l Heat Flow During S o l i d i f i c a t i o n " , T r a n s a c t i o n s o f TMS AIME, v o l . 246, J u l y 1969, pp. 1471-79.  81.  K a r l e k a r , B.V. and Desmond, R.M., E n g i n e e r i n g Heat T r a n s f e r " , West P u b l i s h i n g C o . , 1 977 .  82.  O l s o n , F.C.W. and S c h u l t z , O.T.-, "Temperatures i n S o l i d s During Heating and C o o l i n g " , I n d u s t r i a l Engin e e r i n g C h e m i s t r y , v o l . 34, 1942, pp. 8 7 4 - 7 7 .  83.  Carslaw, H.S. and Jaeger, J . C , "Conduction o f Heat i n S o l i d s " , Second E d i t i o n , Oxford U n i v . Press 1959.  84.  S u t h e r l a n d , J . G . , Personal  85.  Beattie,  86.  Van Drunen, G., Brimacombe, J . K . and Weinberg, F . , " I n t e r n a l Cracks i n S t r a n d - c a s t B i l l e t s , Ironmaking and Steelmaking ( Q u a r t e r l y ) , v o l . 2 , pp. 1 2 5 - 3 3 , 1 975.  87.  A g a r w a l , P.K., "Case Study o f Spray Design f o r a Continuous B i l l e t C a s t e r " , M.A. Sc. T h e s i s , U n i v . of B r i t i s h Columbia, Dec. 1979.  88.  Prabhakar,  D.D.,  B.,  Personal  Personal  Communications.  Communications.  Communications..  Hansen, M., " C o n s t i t u t i o n of McGraw H i l l , 1958.  Binary  AlToys",  .  APPENDIX 1  DEVELOPMENT OF FINITE DIFFERENCE EQUATIONS  1 70  171 APPENDIX  Al .1  Alternating  Direction  Finite  f o r Three Dimensional  The f i n i t e unsteady p a r t i a l  1  Difference  Equations  Problems.  d i f f e r e n c e e q u a t i o n s which r e p l a c e differential  e q u a t i o n have been o b t a i n e d  using a heat balance approach.  In t h i s method the  material  being analysed i s d i v i d e d i n t o a number o f d i s c r e t e of f i n i t e  dimensions.  In a t h r e e dimensional  problem o f a r e c t a n g u l a r  the  heat  elements flow  para 11 el p i p e d , c a l c u l a t i o n s  performed o n l y f o r one q u a r t e r o f the c a s t i n g as  are  indicated  i n F i g . A l - 1 , because of the symmetry p r e s e n t at t h e m i d planes.  In the z - d i r e c t i o n  which i s a l s o the  casting  d i r e c t i o n , the whole c a s t i n g had t o be analysed because o f the d i f f e r e n t bottom.  boundary c o n d i t i o n s  i n v o l v e d at the t o p and  The s u b - d i v i s i o n of the c a s t i n g i n t o a number of  elements i s shown i n F i g . A l - 2 .  H a l f nodes are p r e s e n t  the s u r f a c e and c e n t r e w i t h r e s p e c t to x and y  directions  and top and bottom w i t h r e s p e c t t o z - d i r e c t i o n . A l - 2 o n l y the s u r f a c e nodes are v i s i b l e . c u l a r problem t h e r e are a l t o g e t h e r of nodes depending on t h e i r  location  In  In t h i s  27 d i f f e r e n t i n the  at  Fig.  parti-  types  casting.  1 72  oo'x'x, oyy'o' - zero heat flux boundary condition y y ' d ' d , dd'x'x, o ' y ' d ' x ' - heat-transfer coefficient boundary condition oydx - constant temperature boundary condition  Fig.  Al .1  Dotted r e g i o n i s the volume over which t i o n s are p e r f o r m e d .  calcula-  Fig. Al.. 2  Discretization of the rectangular parallelpiped showing the surface nodes.  1  The g e n e r a t i o n o f the simultaneous e q u a t i o n s i l l u s t r a t e d below f o r the case o f an i n t e r i o r surface  74  is  node and a  node.  Interior  Node:  Let i , j , k  be the i n d i c e s o f the node under  t i o n and Ax, Ay, Az be i t s  d i m e n s i o n s , as w e l l  investiga-  as d i s t a n c e  between nodes . Stage I :  Implicit  in  x-Direction  Rate o f Heat i n by c o n d u c t i o n in x - d i r e c t i o n  Rate of Heat out by c o n d u c t i o n  j * * = -kAyAz ( i , j , k ~ i - l , j , k ) Ax T  -kAyAz ( i + 1 , j , l ~ i , ; i , k ) AX T  in  x-direction  Rate o f Heat i n  in  the y - d i r e c t i o n  Rate o f Heat out the  y-direction  Rate of heat i n the  in  z-direction  in  T  = +kAxAz  ( i,j-l,k" i,j,k) Ay  = -kAxAz  ( i,j+l,k~ i,j,k) Ay  +kAxAy  ( i,j,k-l~ i,j,k) AZ  T  T  T  T  T  T  175 Rate o f Heat out i n =  the z - d i r e c t i o n  - kAxAy (T.  .  -T  k + 1  J  . .)  I!  ? J  AZ  Rate o f heat consumption  = 0  Rate o f heat g e n e r a t i o n  = 0  Rate o f heat a c c u m u l a t i o n  = pc AxAyAz (T. . . - T . .' . ) At/2  ' '  '  '  * i s t h e t e m p e r a t u r e a t t h e end o f t i m e Step At/2  T  From Energy Balance, (Rate o f heat i n ) - (Rate o f heat o u t ) + (Rate o f heat generation)  k  k  ^  (  - ^  T  (  Ay  k  ^  (  T  - (Rate o f heat  i - i , j , k -  T  i , J - i v k -  i , j , k - l  2  i , j , k  T  2  -  T  Z T  i , j , k  1.].k  +  T  consumption) Rate-of Accumulation  i i,j,k  )  +  +  +  i,:  T  T  + 1  ,k>  +  1J.k l> +  - 2 a x a y t z pc ( T * .  t  -T, ,  k  )  * Since o n l y T s are unknown, they are kept on the left  hand s i d e and e v e r y t h i n g e l s e moved t o the r i g h t  Si mpli f y i ng we g e t ,  176  ^  g  ^ f  (  p  *  T  i . J , k  i , j , k - i  T  -  ^f  +  2  T  (  i . j . k  T  +  T  i j - i  ;  i . j . k  k -  +  i  2  T  i , j , k  +  T  i ,  j  +  i , k >  +  •••  >  The form i n which Eq. A l - 1 i s presented i s very u s e f u l i n cases where i t i s d e s i r e d t o have a change i n t h e node s i z e s in the d i f f e r e n t  directions.  Implicit  in y - d i r e c t i o n :  F o l l o w i n g on t h e same, l i n e s  -  /AXAZx  T.  + T.  i . j - l . k (-^y-)  2 A X  k Af y  P C  ( T  where T  T  i>J»k  .  .  ,2AxAZ  as b e f o r e r e s u l t s i n ,  +  2AxAyAZPCx  i.J.M-^-  ^f  +  i,j\k-l -  2 T  are the unknown  ( T  kit  i-l,j,k "  i,j,k  +  T  2 T  i,j,k l +  temperatures  i,j,k )  )  +  T  1 l,j,k> +  +  ... Al-2  Implicit  i n the z - d i r e c t i o n :  *** T. .  *** AZ  kAt  "kick  ( " g f " ' l.J.k C  2  vi.j.k  ) +  ^  T  +  ^'  **  i";j-i.k-  ( T  ( T  l-i.j.k  2 T  **  i"j.k  +  -  T  2 T  i.j,k  *!j i,k^  Al-3  +  where T*** are t h e unknown temperatures Explicit  formula f o r c a l c u l a t i n g  end o f a time i n t e r v a l n+1 i »jik  T  »  f T  A t i s given by  i-i,j,k-  ** ^  ^  ( T  <  T  i , j , k - l  /AXAyAZ PCX At '  [  .n + 1 where T. .  u  1 »J »K  Z T  2 T  -  +T  i,j,k  2 T  i,j,k  i, ,k Vi,j,k> t  j  **  i,M,k-  the new t e m p e r a t u r e a t the  ** +  T  i,j l,k'  +  +  +  T  i,j,k l')/  i.j.k  i s the new t e m p e r a t u r e  +  Al -4  If  the nodes were indexed i = 1,2  ...  L i n the x D i r e c t i o n  j = 1,2  ...  M i n the y  k = 1,2  ...  N i n the z Di r e c t i on  Direction  then the above e q u a t i o n s are v a l i d f o r nodes having indices  i,j,k  Equations  where  i = 2,3  ...  L-1  j = 2,3  ...  M-1  k = 2,3  ...  N-l  ^8  the  f o r a node l y i n g on the boundary w i t h r e s p e c t  to  x d i r e c t i o n , but i n t e r i o r w i t h r e s p e c t t o y and z d i r e c t i o n s .  Dimensions o f the node  A_x, Ay, A Z 2  Nodal i ndi ces ( L , j , k ) Implicit  -  T  L -  h AyAZ\ k '  ( T  ^  1  i n the  ,  j  , k O  +  T  L ,  , k ( ^  j  = AxAyAzpe T . . + h_ T kAt ' ' k L  L,j-l,k " L,j,k 2 T  (T  x-direction:  L,3,k-l -  2 T  +  T  J  K  L,j,k  +  T  }  ^ p  +  AyAz + AxAz 2Ay  M  L,j+l,k  +  +  L,j,k l) +  ' "  Implicit  in y - d i r e c t i o n :  • l** j l b /AXAZ^ + T.** . . /AXAZ  **  T.  . , , . ,AXAZ  N  L,j+l,k  (^y-)  AXAyAZ kAt  h  T  (T,  A Z A X  . + AZAV_ ( T , , , . - T. . . ) Ax ' ' ' '  ' ' J  L  L,j,k  +  '  T  Implicit  J  ,  kAx  2 T  . '  J  -  JA  K  L,j,k+l  in  (T.  A X A X 2AZ  . L  '  J  J  .  J  K  ,  K  1  )  z-direction  _i f  » + ,  +  A Z A Z  Ak  ** L,j,k  + AxAyAz PC  (§f4  L.j.k 1  AxAyAZ kAt  . . ,AXAy  T  *** T, .  2 T  L  "kick k  h  +  M  ~kieje  •T.  AxAyAZ PC\  +  T  T  l  i k ' '  (Ti  . L , J  +  +  (T  AX  J  T  '  k  K  -  T.) H  ** L,j+l,k  }  L-1 i k ~ L i ' ' ' T  J >  +  AXAZ  2Ay  (T.  J  '  , J  ,  '  .  )  Explicit  formula  r  n+l L,j,k  hPfL ( T  ^  (  L-l,j,k  T  L > j > k  - T ) A  ** T  Ax£z ( T  ***  L,J l,k) +  +  §^  I^AxAyAz pc^  where h i s  +  " W  ( T  L i j  - 2 T** u j  > k  - L j,k 2 T  i k  +  *** +  T  s  L,j,k l) +  Al-8  L,j ,k  the heat t r a n s f e r  the x d i r e c t i o n , and  ** . ,  ***  L,j,k-l  +T  +  is  coefficient  the ambient  at the boundary  temperature  in  Al .2  Stability  Criterion  for  Explicit  Using Convective Type Boundary  The most s t r i n g e n t  criterion  Finite  Difference  Conditions.  will  be a p p l i e d f o r a  s u r f a c e node, which i s on the bottom c o r n e r of the This p a r t i c u l a r  node has t h r e e of i t s  to heat t r a n s f e r w i t h the  Let LMN stand f o r  k  subjected  outside  the node  identification  Doing a heat balance f o r t h i s difference  s i x faces  ingot.  node i n e x p l i c i t  finite  form  (T  L - 1 , M , N ' - L,M,N fgp. T  AXAZ + k (T  4 Ay (T  h  L > M j N  3  (T  }  L,M,N-1  A  T  }  k  L,M,N  " T ) A^AZ - h  L,M,N " A  +  0  (T  A Z  (T  L,M-1,N "  \  ^  4 Az  L > M ) N  - T ) AXAZ A  If*.  n:+l L,M,N " L,M,N r  P C  (T  T  }  . Al-9  • 1 82 where h-j , h,,, h x,  y and z  Here  3  are  the heat  coefficients  direction.  the s t a b i l i t y  ( 1 - 2  condition  At k , 2 AX pc  2  _ 2h At  _ 2h At  pCAX  pC Ay  pC AZ  1  2  substituting  get  for  explicit  method  A t k - 2 At k 2 2 Ay pc AZ PC  2h At  we  transfer  3  0  k  =  -113  W/m K .  P  =  7140  kg/m  c  =  .3830  J/g K  3  h  1  =  9210.9 W/m K  h  2  =  921 0.9 W/m K  h  3  =  209.34 W/m K •  2  2  2  Ax =  Ay = 1 5 . 2 4 mm  Az =  20 mm  At = < 0 . 5 5  seconds  is  in  the  APPENDIX 2  SOURCE LISTING OF THE COMPUTER PROGRAM  1 83  184  C C C C C  A EBCGFAM W F I T T EN I N FOBTEAN TO S I M U L A T E THREE DIRENSIONAL HEAT TLOV AND S O I I D I F I C A T I O S I H C A S T I N G JUMBO S E C T I O N S OF Z I N C . B E C A U S E OF T H I SYBMETBY THE C A L C U L A T I O N S HAVE BEEN P E f i F C E H E D ONLY F O E ONE HALF O F THE C A S T I N G  C C  c  C C C C C C C C C C C C C C C C C C C C C C C C C C C C  NST AFT  - 0  CORRESPONDS TO S T A R T I N G FBOB THE B E G I N N I N G 1 FROM I N T E B M E 1 A T E T I M E N I T E - NOME EE OF T I R E S C A L C U L A T I O N S ABE PEEFOBMED HE1CT -NI7K EEB OF T 1 H E S F L O T S ABE R E Q U I R E D RNOE - NUM EEB OF S T E P S A F T EE HHICH A NEW SLICE I S ADDED TO T H E Z - D I R E C T I O N Z - T H I C K N E S S OF THE S L I C E I N T H E 2 D I B E C T I O N RKAY - THERMAL C O N C D C T 1 V I T Y OF T H E M A I E E I A L I N CGS DT - T I R E I N T E B V A L OVER WHICH C A I C U L A T I O N S ABE DONES TAD - THE TOT AI C A S T I N G T I M E AT THE END OF EACH C A L C t J L T I O N L,R,N - T H E NUMBER C F KCDES I N T H E X Y Z D I R E C T I O N - 1 F O F JUMEC C A L C D 1 T I O N S L=9 AND M=15 . THE D I SCfi E H Z A T I O N I S CONE AS P E F EAT A S U B R O U T I N E AB EV CL TECE1 - FOUEING TEMPERATDBE DEG C TEME2 - SOME S B A L L T E M P E E A T O B E AS DUMMY V A L U E TO F I L L T H E GBID I N THE NOTCH AREA FOB THE P L O T T I N G S U B R O U T I N E DEKI - D E N S I T Y OF T H E L I Q U I D G/CM3 DENS - D E N S I T Y OF T H E S O L I D G/CM3 TIIC " L I C U I E U S T E H P E B A T U B E DEG C TSCL S O L I E U S T E M P E E A T U P E DEG C CEI - S F E C I F I C HEAT O F T H E L I Q U I E C A L / G C ELHT - L A T E N T HEAT OF S O U CI F I C A T I C N C A L /G H3 - EOTIOM BEAT TRANS FEB C O E F F I C I E N T CGS UNITS S P E C I F I C HEAT CI. S O L I D INCORPORATED AS A A F D N T I C N OF T E H P E B A T U B E IN T H E F U N C T I O N S U E B O U T I N E CP  C  c c c DIBENSION T (10, 1 6 , 9 1 ) , T 1 ( 10,16,91) , T 2 ( 1 0 , 1 6 , 9 1 ) , 1 T3(10,16,91) ,18(10,16,91) D I f E N S I C N A (101) ,E (101) ,C (101) ,D (10 1) , T P E I M E (101) DI RENSICN AEX ( 16, 1 1,2) , AEY ( 1 1 , 1 6 , 2 ) ,AiiZ (1 1 , lb) DIMENSION TS(38,30,50) CORKON/C1/DX,EY,CZ,DT,BKAY COBMON/C2/S4(10,16,91) C 0 H B 0 N / C 3 / N T Y P E ( 1 0 , 16,9 1) , L P S ( 1 0 , 1 6 , 9 1 ) C0HB0N/C4/L,H,N COHBOK/CS/NCH,TIHP1 COBMON/C6/H3 C0BM0N/C7/T,T 1,T2,T3,TN COBBON/C8/PHY(10,16,91) COEROK/C9/PHYM,FHYS,PHYl COBMON/C10/TLI0,TSOI,DENS,DENL COBBCN/C11/CPL,CSP,NUME0N,ELHT C C B R O N / C 1 2 / B I N P I ( 2 0 ) , C P f (20) C0BB0N/C13/TA0 C0BB0N/C16/NNUH,NUB2 CCBBCN/C17/ABX,ABY,ABZ  COBBON/C18/X1,T1,X2,Y2,Y3,Y4,TH C 0 I ! E 0 N / C 1 9 / A B E A 1 , AB IA2 , AEEA3,ABEAU,ABEA5,ABEA6 CCBECN/C21/TS NAKILIST/LISTA/Z,BKAY,DT HAEELIST/LISTB/TSOL,TLIQ,EENL,DENS,CPL,BLfiI,d3 C C C  BEADIKG INPUT  95  90 C C C 100 200  C C C C  DATA TO THE PBOGRAE  BEAT ( 5 , 2 0 0 ) NUBKUN BEAD ( 5 , 9 0 ) (BINPT ( I ) , 1 = 1 , 2 0 ) EEAD ( 5 , 9 5 ) NSTABT,NITE,NPLOT,NNUM ro£nAi(i4) * BEIE ( 5 , 1 0 0 ) Z , B K A Y , E T . T A U BEAE(5,2CC)L,«,N B E I E ( 5 , 100)TEMP 1,TEBP2 H E A £ ( 5 , 10C) D E N L , D E N S , T L I C , T S 0 1 , C P l , B L H T , H 3 BIAE(5,90) (CPF(I) , 1 = 1 , 2 0 ) FOEBAT(20A4) CALL GSET ( T S , 1 0 4 0 , 5 0 , 1 0 4 0 , 0 . ) ECHO INPUT WBI1E ( 6 . L I S T A ) HB1TE(6,LISTB) FCEPAT (SF 1 0 . 4 ) FOBMAT (3 ( 1 4 , 1 X ) ) PLC ATL= L FLC A T I = H F L C AT N= N . DZ=2/FLOAIN CSE= EZ/(ET*NNUM) L = L+1 B = B+1 H = N+1 CALCULATION OF THEBflO PHYSICAL FOB BUSHY ZCNE DEKE= ( D E N S + D E N L ) / 2 . CEB= (CPL+CP(TSOL) ) / 2 FHYE=DFNM»CEH PHYL=CPL*DENL EHYS=DENS * C P ( T S O I )  C C C  101 105  INITIALISATION  +  FBOPIBHES  (BLHT/(ILIQ-ISOL)  PEOCEEUBE  I F ( N S T A B T . N E . 0 ) G O TO 105 H0H2=0 C A I I I K I T 12 ( T , T E a P 1 , T E H P 2 ) CALL I N I T I 2 ( T 1 , T E H P 1 , T E P P 2 ) C A I L I N I T 12 (T 2,TEHP 1 ,TEHP2) CALL I N I T I 2 ( T 3 , I E K P 1 , T E B F 2 ) CAII INITI2(TN,TEMPI,TEHP2) DO 101 1 = 1 , L DO 101 J= 1,H DC 101 K = 1 , N LPS(I,J,K) =1 CONTINUE GO TO 114 CONTINUE  )  1 86  108  113 114 C C C C  DC 108 K = 1 , « DO 108 J = 1 , f i BE AC (8) (T ( I , J , K) , 1 = 1 , 1 . ) , (LPS ( I , J . K ) , 1 = 1 , L ) CONTINUE DC 113 1 = 1 , L DO 113 J = 1 , H T1 ( I . J , 1)=T ( I . J . 1) T2 ( I , J , 1 ) = T ( I , J , 1 ) T3 ( I , 0 , 1 ) = T ( I , J , 1 ) TN ( I . J , 1) =T ( I . J . I ) CONTINUE NUB2 = 0 BCE=1 CONTINUE CA1L CUTPDT (T) CALL SUBFT NCDE SOETING C A I L NDSOET LL=L-1 BB=B-1 NN=N-1  C C C  DISCEITIZATION 111  C C C  OP THE CASTING  CALL ARIVOL CChTINUE CALL ODTPT2  STAET  Cf  CALCULATIONS  DO 1001 K J I = 1,N PLOT DO 1002 K J J = 1 , N I T E TAC=TAU+DT C A I I EHYPEP DO 600 J = 1 , 4 DO 600 K = 2 , N CAIL C C E F I I ( 1 , L , J , K , A , B , C , D ) CAII TBIDAG(1,L,A,E,C,E,TPRISE) DO 550 1 = 1 , L 11 » I , J , K ) = TPBIHE(I) 550 CONTINOE 600 CCMINOE C CALCOLATICN AT THE NCTCH C u=e JI =5 DO €15 K I J = 1 , « DC 610 K = 2 , N CALL C C I E f I ( 1 , 1 J , J I , K , A , B , C , D ) CAII TEIDAG(1,1J,A,B.C,E.TPBIME) DO 630 1 = 1 , I J Tl (I,JI,K)=TPE1«E(I) 630 CONTINUE 610 CCKTINUF JI=JI+1 IJ=IJ+1 6 1 5 CONTINOE  C C  CALCOIATICN  DO 640 J = 9 , f l CO £45 K = 2 , N C A I I CCEFEI ( 1 , L , J , K, A, £ , C. D) CALL TBICAG ( 1 , L , A , B , C , D , T F B I H E ) DO 660 1 = 1 , L T1 ( I , J , K ) = T F B I H E ( I ) CC KTIND E CONTINUE CCKTINUE  660 645 640 C C C c  EELOS THE NOTCH  IHPLICIT  WITH BESPICT TO J  DIBECTION  DC ECC 1 = 1 , 6 DC 600 K=2,N CAIL C C E F E J ( 1 , B , I , K , A , B , C , D ) CALL T F I D 8 G ( 1 , R , A , fi.C, D . I P E I M E ) CO 750 J = 1 , B T2 ( I , J , K) = TPBIHE (J) 750 CONTINUE 800 CCKTINUE C CALCOIATICN AT I BE NOTCH C DO 76C K = 2 , N DO 770 1 = 7 , L CAIL C C I E I J ( 1 , 4 , I , K , A , B , C , D ) CALL T B I D A G ( 1 , 4 , A , B , C , D , T P B I H E ) DO 780 J = 1 , 4 T2 ( 1 , J , K ) = T F K I B E { J ) 7 8 0 CONTINUE 7 7 0 COKTINOE 7 6 0 CONTINUE IJJ=7 JII=6 DC 799 J K I = 1 , 4 DO 790 K = 2 , N C A I I COEFFJ ( J I I . H . I J J . K , A , B , C , D ) CALL T E I C A G ( J I 1 , B , A , B , C , C , T F B I f i E ) DC 765 J = J I I , f l T2 ( I J J , J , K ) = I F B I H E ( J ) 765 COKTINOE 790 CONTINUE 795 CCKTINUE JII=JII+1 IJJ=IJJ*1 799 CONTINUE C C C C  I8ELICII  850 900  WITH EESPECT TO Z  DIBECTION  DO 900 1 = 1 , L DO SOO J=1,4 CAII CCEFFK(2,N,I,J,A,B,C,D) CALL I B I C A G ( 2 , N , A , B , C , D , T P E I H E ) DC 650 K = 2 , N T3 ( I , J . K ) = T P E I B E ( K ) CCKTINUE CONTINUE  C C  CALCUITICN  920 915  AT THE NOTCH  IJK =6 JIK =5 DC 910. K K K = 1 , 4 DO 915 1 = 1 , I J K C A I I CCEFFK ( 2 , N , I , J I K , A , B , C , D ) CAIL T B I E A G ( 2 , N , A , B , C , D , T P E I H E ) DC 920 K = 2 , N 1 3 < I , J I K , K ) = T P B I H E (K) CONTINOE CONTINOE IJF=IJK+1 JIK=JIK+1 CONTINUE  910 C C C A I C D I A T I C N EELOW THE NOTCH DO 910 1 = 1 , L DO 950 J = 9 , H C A I L CCEFFK ( 2 , N , I , J , A , B , C , D ) C A I I T EID AG (2 , N , A, E , C, E, TFR IK E) DO 960 K = 2 , N T3 ( I , J , K ) = T E B I M E (K) 960 CONTINOE 950 CCKIIRUE 940 CONTINUE C C C C f l f U I I THE TEBPEBATUBES AT THE END CF A T I H E C CALL COHPUT ( 1 , L , 1 , « , N ) C A I L CCBPUT ( 1 , 6 , 5 , 5 , N ) CALL C O S P D T ( 1 , 7 , 6 - , 6 , N ) CAII CCBEUT(1,8,7,7,N) CALL COMPUT ( 1 , 9 , 8 , 8 , N ) C A I I CCBPUT ( 1 , L , 9 , f l , N ) C C COBBECT THE TEHPEBATUBES CALCULATED FOB THE C BELEASE OF LATENT HEAT •c CALL L A 1 H E T ( 1 , L , 1 , 4 , N ) CAII IATHET(1,6,5,5,N) CALL LATHET ( 1 , 7 , 6 , 6 , N ) C A I I LATHET ( 1 , 8 , 7 , 7 , N ) CALL LATHET ( 1 , 9 , 8 , 8 , N ) CAII LATHET(1,L,9,H,N) C C R E I N I T I A L I S E T HATBIX C DC 1110 1 = 1 , L DO 1110 J = 1 , H DC 1110 K = 2 , N T(I,J,K)=IN(I,J,K) 1110 COMINUE CALL CHECK (MCH) I F ( B C H . E Q . 0 ) G O TO 109 CALL AEENOD CALL COTPOT(T) C A I L SUBFT 109 CCKTINUE 1002 CONTINOE v  INTEBVAL  189  1001 901  C C C  C A I I GEAEfc CAII FILEIN(4) CC*IIN0E CALL 7SARAY C A I I ELCTND STCP EN I I M P L I C I T CALCULTIONS I N THE X DIBECTION CALCLLATINS OF THE I R I D I A G O N A L COEFFICIENTS'  SDEFCUTINE C 0 E E F 1 ( I B , I E , J , K , A,B,C,D) EIMENSION A ( 1 ) ,E (1) ,C ( 1 ) , D (1) DIf.ENSICN A B X ( 1 6 , 1 1 , 2 ) , ABY ( 1 1 , 1 6 , 2 ) ,ABZ (1 1 , 16) E1EINSICN I ( 1 0 , 1 6 , 9 1) , T 1 ( 1 0 , 1 6 , 9 1 ) 1 ,72(10,16,91) ,33(10,16,91),TN(10,16,91) CCEECN/C1/EX,EY,E2,ET,fiKAY CCrECN/C17/ABX,ARY.ARZ CCEECN/C7/T.T1,T2,T3,TN COBBON/C2/S4 ( 1 0 , 1 6 , 9 1 ) CCEEON/C3/NTYPE ( 1 0 , 1 6 , 9 1 ) , L P S ( 1 0 , 1 6 , 9 1 ) CCEECN/C5/NCH,TIHP1 CCEECN/C6/H3 COKBON/C8/PHY ( 1 0 , 1 6 , 9 1 ) CCMKON/C18/X1,Y 1 , X 2 , Y 2 , Y 3 , Y 4 , TH CCf r C N / C 1 9 / A B E A 1 , A R E A 2 , A B E A 3 , AREA4 , ABEA5, AH EA6 EC 1000 I = I E , I I IJK=NTYPE(I,J,K) GO TO ( 1 0 , 1 1 , 1 2 , 1 3 , 14, 15, 1 6 , 1 7 , 1 6 , 1 9 , 2 0 , 2 1 , 2 2 , 123,24,25,26,27,28,29, 2 3 0 , 3 1 , 3 2 , 3 3 , 3 4 , 3 5 , 3 6 , 3 7 , 3 6 , 3 9) , I J K 10 B (I)=ABX ( J , I , 2 ) + PHY(I,J,K) * S4(I,J,K) C ( I ) = -AFX ( J , I , 2) D(I)= A B Z ( I , J ) * ( T ( I , J , K - 1 ) - 2 . * I ( I , J , K ) + I ( I , J , K + 1) ) 1 • ( AFY(I,J,2)) * ( T (I,J+1,K)-T (I,J,K) ) 2 • PHY(I,J,K) * S4(I,J,K) * 1(1,J,K) 8 •(ABE(K)-T(I,J,K) ) * (H4 (K) * ARY ( I , J , 2) *Y 2/BK A Y) GC TC 1000 11 A ( 1 ) = - A E X ( J , I , 1 ) B(I) = ABX(J,I,1)+ARX ( J , l , 2 ) + S4(I,J,K) * PHY(I,J,K) C ( I ) = -ABX ( J , I , 2 ) D ( l ) = A E Y ( I , J , 2 ) * ( T ( I , J + 1,K) - T ( I , J , K ) ) • 1 ( *EZ(I,J)) * ( T(I,J,K-1)-2.*T(I,J,K)*T(I,J,K*1) ) • 2 EHY(I,J,K) * S4(I,J,K) * T ( I , J , K ) 8 • (»BE(K) -T ( I , J , K ) ) » (K4(K)*AEY(I,J,2)*Y2/EKAi) GC TO 1000 12 A(I) = -ABX(J,I,1) E ( I ) = ( A B X ( J , I , 1 ) ) • ( S4 ( I , J , K ) * P H Y ( I , J , K ) ) * 1 (H1(K)*ARX ( J , I , 1 ) *X1/BKAY) D ( I ) = ( A E Z ( I , J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J ,K) *T ( I , J , K + 1 ) 1 (AEY(I,J,2)) * ( I ( I , J + 1 , K ) - T ( I , J , K ) ) t 2 (S4(I,J,K)) * PHY(I,J,K) * T ( I , J , K ) * 3 (H1 (K) *AEX ( J , I , 1) * X 1 / B K A Y * A B D (K) ) 8 • (ABE ( K ) - T ( 1 , J , K ) ) * (H4 (K) *ARY ( I , J , 2) * Y 2 / B K A Y ) GC TO 1000 13 E(I) = ABX(J,I,2) • S4(I,J,K) * PHY(I,J,K) C(I)= -ABX(J,I,2) D(I)= A BY(I,J,1)»(T(I,J-1,K)-T(I,J,K))*ARY (I,J,2) * 1 (I (I,J»1,K)-T(I,J,K) ) •  1 90  14  15  16  17  18  19  20  3 ( A E Z ( I . J ) ) * ( T ( l , J , K - 1 ) - 2 . * T ( I , J , K ) • T ( I , J , K + 1) ) • 2 SI ( I , J . K ) * FflY(I.J.K) * T ( I . J . K ) GC TO 1000 A(I) = -»2X(J,I,1) E ( I ) = (ABX ( J . I . 1 ) ) +ABX ( J . I , 2 ) • S 4 ( I , J , K ) * P H Y ( I , J , K) C(I) = -ARX(J,I,2) D(I)= ABI(I,J, 1)*(T(I,J-1,K)-T(I,J,K))+ABY(I,J,2) » 1 (T ( I , J + 1 ,K) - I ( I , J . K ) ) • 2 AE2(I,J) * ( T ( I , J,K-1)-2.*T (I,J.K)+T(1,J,K*1) ) • 2 S«(I,J,K) * PEY(l.J.K) »T(I,J,K) GC TO 1000 A(I) = - ABX(J.I.I) B(l) = ( ABX(J.I.I)) • (S4(I,J,K) * PHY(I,J,K) ) • 1 (fc 1 (K) *AEX ( J . I , 1 ) +X1/BKAY) D ( I ) = (H1 (K) *AEX ( J . I , 1) * X 1 / R K A Y * A » B (K) ) • 1S4 ( J . J . K ) * P H Y ( I , J , K ) * T ( I , J , K ) • 2 (ABY(I,J,1) ) * (T(1,J-1,K)-T(I,J,K> )• 2 (AIY(I,J,2)) * (T(I,J*1,K)-T(I,J,K) ) • 3 ( A B Z ( I . J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + T ( I , J , K + 1) ) GC TO 1000 E(I) = (AEX(J,I,2)) • (SU(I.J.K)) * PHY(I.J.K) C ( I ) = -ARX ( J . I , 2 ) D(I) = (ARY(I,J,1)) * ( T(1,J-1,K) -T(I.J.K) ) • 1 ( H2 ( K ) * A B Y ( I , J , 1 ) *Y1/BKAY ) * (AflE(K) - T ( l . J . K ) )• 2 (AFZ(I.J)) * ( T(I,J,K-1)-2.*T(I,J,K)+T(I,J,K*1) ) 3 +| S4(I,J,K) ) * PEY(I.J.K) • T ( I , J , K ) GC TO 1000 A(I) = -AEX(J,I,1) B(I) = ARX ( J . I . I ) +A2X ( J . I , 2 ) • S 4 ( I , J , K ) » P U Y ( I , J , K ) C(I) = -ABX(J,I,2) D ( I ) = SU ( I , J , K ) *FHY ( I , J . K ) *T ( I , J . K ) • 1 ( A P Y ( I . J . I ) ) * (T ( I , J - 1 , K ) - I ( I , J , K ) )• 2 (AIZ(I,J)) * ( T(1,J,K-1)-2.*T(1,J,K) •T(I,J,K*1) ) 3 (K2 (K) *AEY ( I . J , 1) * Y 1 / B K A Y ) * ( 1 ( 1 , J . K ) - A (IB (K) ) GC TC 10CO A ( I ) = -ABX ( J . I , 1) B ( I ) = ( A B X ( J , I , 1 ) ) • H I ( K ) * A B X ( J . I . 1 ) * X 1 / B K AY * 1 (SO ( I , J , K ) ) * f H Y ( I , J . K ) D ( I ) = H i ( K ) * A R X ( J , I , 1) * X 1 / E K A Y * A R E ( K ) • 1 (AEY(I.J.I)) * ( 1(1,J-1,K) - T(I,J,K) ) • 2 (AFZ(I.J)) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + T ( I , J , K » 1 ) ) 3 (F.2 (K) *AEY ( I . J . 1 ) * Y 1 / E K A Y ) * (T ( I , J , K) - ABB (K) ) • 4 (S4(I,J,K)) • PHY(I.J.K) * T ( I . J . K ) GC TC 1000 E ( I ) = ( A B X ( J , I , 2 ) / 2 . ) • S4 ( I , J , K) *PHY ( I , J , K) C ( I ) = - ABX ( J , 1 , 2) / 2. E(I) = (ABY(I,J.2)/2.) * ( T(I,J+1,K)-T(I,J,K) ) • 1 (ABZ(I,JJ) * ( T ( I , J , K - 1 ) - T ( I , J , K ) ) 2 ( H 3 * A E Z ( I , J ) * D Z / B K A Y ) * ( T ( I . J . K ) - A B B (K) ) • 3 ( SU ( I , J . K ) * P H Y ( I , J . K ) »T ( I , J . K ) ) 8 • ( A E Y ( I , J , 2 ) *H4 (K) * Y 2 / E K A Y / 2 . ) * (A B3 ( K ) - T ( 1 , J , K) ) GC TO 1000 A(I) = - A R X ( J , I , 1 ) / 2 . B ( I ) = (ARX ( J . I . I ) * AKX ( J . I , 2 ) ) / 2 . • (S4 ( I , J . K ) * PHY ( I , J . K ) ) C(1) = - A R X ( J . I ,2)/2. D(I) = S4(I,J,K) * PHY(I,J,K) * T ( I , J , K ) • 1 (AFY ( I , J , 2) / 2 . ) * ( T ( I , J * 1 , K ) - T ( I . J . K ) ) + 2 (AEZ(I.J) ) * ( T(I,J,K-1) - T(I.J.K) ) 3 ( H 3 * A E 2 ( I , J ) *DZ/EKAY ) * t T ( I . J . K ) - ABB (K) ) 8 • IABY ( I , J . 2 ) *H4 (K) * Y 2 / E K A Y / 2 . ) * (ABE (K) - T ( I . J . K ) )  > '  GC 10 1000 A ( I ) = -ABX ( J , I , 1 ) / 2 . E ( I ) = (ARX ( J , I , 1 ) / 2 . ) • (H 1 (K) *ABX ( J . I , 1) * X 1 / E K A Y / 2 . ) «• 1 S 4 ( I , J , K) *FB Y ( 1 , J , K) D ( J ) = ( H I (K) *ABX ( J , l , 1) *X 1/BKAY*AM£ (K) /2. ) • 1 (JH(I J,2)/2.)» ( T(I,J*1,K)-T(I,J,K) ) • 2 (AFZ(I.J)) * ( I (1,J,K-1)-T (1,J,K) ) 3 (H3*ABZ ( I , J ) * D Z / B K A Y ) * (T ( I , J , K) - ABB (K) ) + 4 S 4 ( I , J , K) *PHY ( I , J , K ) * T ( 1 , J , K) 8 + (AEY ( I . J , 2 ) *hU (K) * Y 2 / 6 K A Y / 2 . ) * (ABB (K) - T , ( I . J - , K ) ) GO TO 1000 22 B ( I ) = (ABX ( J . 1 , 2 ) / 2 . ) • ( S 4 ( I , J , K ) * P h Y ( I , J . K ) ) C ( I ) = -ABX ( J , 1 , 2 ) / 2 . D ( I ) = (S4 ( I , J . K ) ) * P H Y ( I . J . K ) * T ( I , J , K ) • 1 (AEY ( I . J , 1 ) / 2 . ) * (T ( I . J - 1 , K ) - T ( I . J , K ) ) • 1 ( J F Y ( I , J , 2 ) / 2 . ) * (T ( I , J * 1 , K ) - T ( I , J , K ) ) • 2 (AFZ(I,J)) * ( T ( I , J , K - 1 ) - T ( I , J , K ) ) 3 ( H 3 * A R Z ( I , J ) * D 2 / B K A Y ) * ( T ( I , J , K ) - ABB (K) ) GC TC 1000 23 A ( I ) = -AFX ( J , I , 1 ) / 2 . B(I) = (ABX ( J , I , 1) *AEX ( J , I , 2 ) ) / 2 . + S 4 ( I , J , K ) * P u Y ( I , J . K ) C ( I ) = - ABX ( J , 1 , 2) / 2. D ( I ) = S 4 ( I , J , K ) * PHY ( I , J . K ) * T ( I . J . K ) + 1 (ABY(I,J,1)/2.)*(KIfJ-1»K)-I(I,J,K) ) + 1 (AFY ( I , J , 2 ) / 2 . ) * ( T ( I , J * 1 , K ) - T ( 1 , J , K ) ) • 2 |AFZ(I,J)) * (T(I,J,K-1) - T(I,J,K) ) 3 ( B 3 * A F Z ( I , J ) * E Z / R K A Y ) * ( T ( I , J , K ) - ABE (K) ) GC TC 1000 24 A ( I ) = -ABX ( J , I , 1) / 2 . B ( I ) = (ABX ( J , I , 1 ) / 2 . ) • . (S4 ( I , J , K ) j * P H Y ( 1 , J , K ) •• 1 ( E l (K)*AEX ( J . I . I ) * X 1 / E K A Y / 2 . ) E(I) = S4(I,J,K) * PhY(l.J.K) * T (I.J.K) • 1 (AFY(I,J,1)/2.)*(T (I,J-1,K)-T(I,J,K) ) • . 1 ( A E Y ( I , J , 2 ) / 2 . ) * (T ( I , J +1 , K ) - T ( I , J , K ) ) • 2 (AFZ(I.J)) * (T(I,J.K-1) - T(I,J,K) ) 3 (H3*ABZ ( I , J) * E Z / B K A Y ) • ( T ( I , J , K ) - AHB (K) )'• 4 (H1(K)*AEX(J,I,1)*X1/EKAY*ABB(K)/2. ) GC TO 1000 25 E ( I ) = (AEX ( J . I , 2 ) / 2 . ) • S 4 ( I , J , K ) • P H Y ( I , J , K ) C ( I ) = - AEX ( J , 1 , 2 ) / 2 . E ( I ) = ( S 4 ( I , J , K ) * PHY ( I , J . K ) * T ( I . J . K ) ) • 1 ( I F Y ( I , J , 1 ) / 2 . ) * ( T ( I , J — 1 , K ) - T ( I , J , K) ) 2 (E2(K) * A F Y ( I , J , 1 ) * Y 1 / B K A Y / 2 . ) * ( I ( I , J , K ) - A H B ( K ) ) • 3 (AEZ(I,J)) » ( T ( I , J , K - 1 ) - T ( I , J , K ) ) 4 ( H3*AEZ ( I , J ) *CZ/EKAY ) * ( T ( I , J , K ) - AB3 (K) ) GC TC 1000 26 A(I) =-ARX(J,I,1)/2. E ( I ) = (AEX ( J . I , 1) + ABX ( J . I , 2) ) /2.* (S4 ( I . J , K ) *PtiY ( I , J . K ) ) C ( I ) = -AEX ( J , I , 2 ) / 2 . D(I) = S4(I,J,K) * PHY(I,J,K) * T ( I , J , K ) 1 (F.2 (K) *ABY ( I , J , 1) *Y 1 / B K A Y / 2 . ) * (T ( I , J . K ) - A H B (K) ) 2 ( E 3 » A B Z ( I , J ) * E Z / B K A Y ) * (T ( I , J , K ) - A B B (K) ) • 3 (JFY(I,J,1)/2. ) » (T(I.J-1.K)-T(I.J.K) ) • 4 ( A E Z ( I . J ) ) * (T ( I . J . K - 1) - T ' ( I . J . K ) ) GC TO 1000 27 A ( I ) = -AEX ( J . I , 1 ) / 2 . E ( I ) = ( A B X ( J , 1 , 1 ) / 2 . ) + S4 ( I . J . K ) * P H Y ( I , J . K ) • 1 H 1 (K) *AEX ( J . I , 1) *X 1 / E K A Y / 2 . D ( l ) = ( S 4 ( I , J , K ) *FHY ( I , J , K ) * T ( I , J . K ) ) + 1 (H 1 ( K ) » ABX ( J , I , 1 ) * X 1 / E K A Y * A B £ ( K ) / 2 . ) • 21  r  2 3 4 5  ( I B Y ( I , J . 1 ) / 2 . ) » (T ( I , J - 1 . K ) - T ( I , J . K ) ) • (ABZ(I.J)) * ( T ( I . J . K - I ) - T ( I . J . K ) ) (E2 (K) *ABY ( I , J . 1 ) * Y 1 / E K A Y / 2 . ) * (T ( I , J , K ) - A B u (K) ) ( B 3 * A E Z ( I . J ) * E Z / B K A Y ) * (T ( I , J , K ) -AMB (K) ) GO 10 1000 28 M I ) = -AEX(J,I,1) B ( I ) = (ABX ( J , 1 , 1 ) ) + AEX ( J . I , 2 ) +S4 ( I , J . K ) * P H Y ( I , J . K ) 1 • B7 (K) * AE E A 1 C(I) = -ABX(J,I,2) D { I ) = AR Y ( I , J , 1) * (T ( I , J - 1 . K ) - T ( I . J . K ) ) +ASY ( I . J , 2) * 1 ( 1 ( I . J + 1 ,K) - T ( I , J . K ) ) • 2 A F Z ( I . J ) * ( T ( I . J . K - 1 ) - 2 . *T ( I , J . K ) + T ( I . J . K + 1) ) • • 2 S4(I,J,K) » PHY ( I . J . K ) * T ( 1 , J , K ) 3 • (H7 ( K ) * A B E A 1*AMB (K) ) 4 * ( H 5 ( K ) * A B E A 4 ) * (ABB ( K ) - T ( I , J . K ) ) GC TC 1000 29 »(I) = -ABX(J,I,1)/2. B ( I ) = (AFX ( J . I , 1) *ABX ( J , I , 2) ) / 2 . +S4 ( I . J . K ) • P a Y ( I . J . K ) 1 • (H7(K)*ABEA1/2.) C ( l ) = -ABX ( J , I , 2 ) / 2 . D ( I ) = S 4 ( I , J , K ) * PHY ( I . J . K ) * T ( I , J , K ) + 1 (AFY(I,J,1)/2.)*(T (I,J-1,K)-T(I,J,K) ) + 1 ( I F Y ( I , J , 2 ) / 2 . ) * (T ( I , J + 1 , K ) - T ( I , J , K ) ) + 2 (ABZ(I,J)) * (T(I,J,K-1) - T(I.J.K) ) 3 ( H 3 * A E Z ( I , J ) » E Z / B K A Y ) * ( T ( I . J . K ) - AHB(K) ) 4 + (E7(K) • A E E A 1 * A H E ( K ) / 2 . ) 4 • (H5 (K) * A E E A 4 / 2 . ) * ( A B B ( K ) - T ( I , J . K ) ) GC TO 1000 30 A ( l ) = - AEX(J,1,1) B(I) = ( A B X ( J . I . I ) ) • ( S 4 ( I , J , K ) * PHY ( I . J . K ) ) + 1 ( B 7 ( K ) * A E E A5) 2 • (H6 (K) * A E E A 2 * S I N (TB) ) D ( I ) = (H7 ( K ) * A B E A 5 * A B E ( K ) ) • 1S4 ( I . J . K ) * P H Y ( I . J . K ) * T ( I , J . K ) + 2 (AEY(I,J,1) ) * (T(I,J-1,K)-T(I,J,K) )• 2 (AEY(I,J,2)) * (T(I,J»1,K)-T(I,J,K) ) • 3 (AEZ(I,J)) * ( T ( I , J , K - 1 ) - 2 . *T(I,J,K) +T(I,J K*1) ) 4 • (H6 (K) »AEEA2»SIN ( T E ) * A H B ( K ) ) 5 «• (H6 (K) *AEEA2*C0S (TE) ) * (ABB (K) - T ( I . J . K ) ) GC TO 1000 31 A ( I ) = -ABX ( J . I , 1 ) / 2 . E(I) = ( A B X ( J , I , 1 ) / 2 . ) • ( S 4 ( I , J , K ) ) »PHY(1,J,K) • 1 {E7 ( K ) * A F E A 5 / 2 . ) 2 • (H6 ( K ) * A E E A 2 * S I N ( T H ) / 2 . ) D ( I ) = £4(1,J.K) » P H Y ( I . J . K ) * 1(1,J.K) • 1 (IEY(I,J,1)/2.)*(KI.J-1,K)-T(I,J,K) ) • 1 ( A E Y ( I , J , 2 ) / 2 . ) * ( T ( I , J + 1 , K ) - T ( I , J , K ) ) «• 2 (AEZ(I,J)) * (1(I,J,K-1) - T ( l . J . K ) ) 3 (H3*ABZ ( I . J ) * E Z / E K A Y ) * ( T ( I , J , K ) - ABB (K) ) «• 4 (H7(K)*ABEA5*ABB(K)/2.) 5 + (H6 (K) * A E I A 2 * A H E ( K ) * S l N ( T H ) / 2 . ) 6 • (H6 (K) *ABEA2*COS ( T K ) / 2 - ) * (ABB ( K ) - T ( I , J , K) ) GC TO 1000 32 A ( I ) = -ABX ( J . I . I ) E ( l ) = A B X ( J . I , 1) +ABX ( J . I , 2 ) + S 4 ( I , J . K ) * P H Y ( I , J . K ) C ( I ) = - A EX ( J . I . 2 ) D ( I ) = S4 ( I . J . K ) *PHY ( I , J . K ) *T ( I , J . K ) • 1 ( * E Y ( I , J , 1 ) ) * (1 ( I , J - 1 , K ) - T ( I , J , K ) )• 2 (AFZ(I,J)) * ( 1 ( I , J , K - 1 ) - 2 . * T ( I , J , K ) +T(1,J,K*1) ) 3 (H5 (K) *AEY ( I , J , 1) * Y 3 / E K A Y ) • ( T ( I . J . K ) - AMB (K) ) r  33  34  35  36  37  38  GC TO 1000 M I ) = -AEX ( J , I , 1 ) / 2 . B(I) = (ABX(J,I,1)+ ARX(J.I,2))/2.•<S4(I.J,K)*PHY(I,J.K)) C ( I ) = -ABX ( J . l , 2 ) / 2 . E(I) = S4(I,J,K) * PHY(I,J,K) * T ( I , J , K ) 1 ( E 5 ( K ) *AEY ( I , J , 1 ) * Y 3 / B K A Y / 2 . ) * (T ( I , J , K) - ABB (K) ) 2 ( B 3 * A E Z ( I , J ) *EZ/BKAY ) * (T ( I , J , K ) - A M B (K) ) • 3 ( 1 I Y ( I . J . I ) / 2 . ) * (T ( I , J - 1 , K ) - T ( I . J . K ) ) + 4 ( A F Z ( I . J ) ) * (T ( I . J . K - I ) - T ( I . J . K ) ) GC TC 1000 A(I) = -ABX(J,I,1) B(I) = (ABX(J,I,1)) • ( S4 ( I , J . K ) *PHY ( I , J . K ) ) • 1 (H6 (K) *ARX ( J . 1 , 1 ) * X 2 / B K A Y ) D ( I ) = (ABZ ( I , J ) ) * (T ( I , J , K - 1 ) - 2 . * I ( I , J , K ) +T ( 1 , J , K + 1 ) ) • 1 ( A F Y ( I , J , 2 ) ) • ( T ( I , J • 1,K) - T ( I , J , K) ) • 2 (£4(1,J,K)) * PHY(I,J,K) * T(I,J,K) + 3 ( B 6 ( K ) * A F X ( J , I , 1) * X 2 / B K A Y * A B B ( K ) ) 8 • (ABE ( K ) - T ( I . J . K ) ) * (H6 (K) * AEY ( I , J , 2) * Y4/BKA Y) GC TO 1000 A ( I ) = -ABX ( J , I , 1 ) / 2 . £ ( ! ) = (ABX ( J , I , 1 ) / 2 . ) + (H6 (K) *AEX ( J , I , 1) * X 2 / i i f U Y/2 . ) * 1 54 ( I . J . K ) * E H Y ( 1 , J , K ) D ( I ) = (H6 ( K ) * A B X ( J . I , 1 ) * X 2 / B K A Y * A H E ( K ) / 2 . ) • 1 (AIY(I.J.2)/2.)* ( T(I,J*1,K)-T(1,J,K) ) • 2 (lEZ(I.J)) * ( T(I,J,K-1)-T(I,J,K) ) 3 (H3*ABZ ( 1 , J ) * C Z / B K A Y ) * (T ( I , J , K) - A BB (K) ) • 4 S4 ( I , J . K ) *PHY ( I , J . K ) * T ( I . J . K ) 8 • (AFY ( I , J , 2) *E6 (K) * Y 4 / E K A Y / 2 . ) * (AMB (K) - T ( I , J , K ) ) GC TO 1000 AU) = -AEX(J.I.I) E ( I ) = ( A R X ( J , I , 1 ) ) + ( S4 ( I , J , K) * PHY ( I , J , K) ) • 1 ( B 1 ( K ) *AE£A6) 2 • (H6 ( K ) * A E E A 3 * S I N ( T H ) ) D ( I ) = (ABZ ( 1 , J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) * T ( I , J , K * 1 ) ) + 1 (AIY(I,J,2)) * ( T(I,J*1,K)-T(I,J,K) ) • 2 (54(1,J,K)) * PHY(I,J,K) * T(I,J,K) + 3 (H1 (K) *AEEA6»ABE ( K ) ) 6 • (H6 (K) *AEEA3*COS (TH) ) * (ARB ( K ) - T ( I , J , K ) ) 5 • ( B 6 ( K ) * A B E A 3 * £ I N (TH)*ABB ( K ) ) GC TO 1000 A(I)= -ABX(J,I, 1)/2. B ( I ) = ( A E X ( J , I , 1) /2.) • ( H 1 ( K ) * A E E A 6 / 2 . ) • 1 S4 ( I , J , K ) *FHY ( I , J , K ) 2 • (H6 (K) * A B E A 3 * S I N ( T H ) / 2 . ) E(I) = (Hl(K)*ABEA6*AMB(K)/2. ) • 1 ( » E Y ( I , J , 2 ) / 2 . ) * < T ( I , J +1 ,K)-T (I.J.K) ) • 2 (AEZ(I.J)) * ( T ( I . J . K - I ) - T ( I , J , K ) ) 3 (E3*ABZ ( I , J ) * E Z / B K A Y ) * (T ( I , J , K) - AMB (K) ) • 4 S4 ( I . J . K ) * P H Y ( I , J , K ) * T ( I . J . K ) 2 • (H6 (K) * A E E A 3 * S I N (TB) *AMB (K) / 2 . ) 3 • (H6 (K) * ABE A3 *COS ( T H ) / 2 . ) * (A BB ( K ) - T ( I , J , K) ) GC TC 1000 A(I) = -ARX(J,I,1) E ( I ) = ( A B X ( J , I , 1 ) ) • H 1 ( K ) * A B X ( J , I , 1)»X1/BKAY • 1 (£4(I,J,K))*FHY(1,J,K) D ( I ) = HI (K)*AEX ( J . I , 1 ) * X 1 / B K A Y * A B B ( K ) • 1 (*EY(I,J,1)) * ( T(I,J-1,K) - 1(1,J.K) ) • 2 (AFZ(I.J)) * ( T (I,J,K-1)-2. *T(I,J,K) • T(I,J,K*1) ) 3 (H5 (K) *AFY ( I , J , 1 ) * Y 3 / B K A Y ) * (T ( I . J . K ) - A M B (K) ) • 4 (£4(1,J.K)) * PHY(I.J.K) • T(I,J,K)  1 94  GC 30 1000 = -AEX ( J , I , 1 ) / 2 . B ( I ) = (ABX ( J , I , 1 ) / 2 . ) *S4 ( I , J , K ) * F H Y ( 1 , J , K) • 1 H I (K) * A F X ( J , I , 1) *X 1 / B K A Y / 2 . D ( I ) = (S4 ( I . J . K ) * F H Y ( I , J . K ) • T ( I , J , K ) ) + 1 (E1(K)*A6X(J,I,1)*X1/EKAY*ABE (K)/2.) + 2 (AEY ( I , J , 1 ) / 2 . ) * ( T ( I . J - I . K ) - T ( I , J , K ) ) • 3 (AFZ(I,J)) * ( T ( I , J , K - 1 ) - T ( I . J . K ) ) 4 (fiE (K) *AFY ( I , J , 1) * Y 3 / B K A Y / 2 . ) * < I ( I . J . K ) - A B B (K) ) 5 ( K 3 * A R Z ( I , J ) * E Z / E K A Y ) * (T ( I , J , K ) -AHB (K) ) GO 10 1000 1000 CCKTINUE EE3CBN ENI C C I B F L I C I T CALCULATIONS I N T H E Y DIRECTION C C C SUEFCUTINE C O E P F J ( I E , I E , I , K , A , B , C , D ) EIRENSION A ( 1 ) , E ( 1 ) , C ( 1 ) . D ( 1 ) DIKENSICN ABX ( 1 6 , 1 1,2) ,ABY ( 1 1 , 1 6 , 2 ) ,ARZ ( 1 1 , 1 6 ) EISINSION T ( 1 0 , 16,91) ,T 1 ( 1 0 , 16,91) 1 ,32(10, 16,91),13(10,16,91),TN(10,16,91) CCFRCN/C 1 / C X , I Y , E Z , E T , E K A Y CCrCCN/C17/ABX,AEY,AEZ COKBON/C7/T,I 1,32,13,TN CORBON/C2/S4 ( 1 0 , 1 6 , 5 1 ) CCBKCN/C3/KTYPI ( 1 0 , 1 6 , 9 1 ) , L F S ( 1 0 , 1 6 , 9 1 ) COBKCN/C5/KCh,IEHP1 CCKECN/C6/H3 COBBON/C18/X1,Y1,X2,Y2,Y3,Y4,TH CCBBCN/C8/PHY( 1 0 , 1 6 , 9 1 ) CCePCN/C19/ABEA1,AEEA2,AEEA3,ABEA4,ABEA5,ABEA6 EC 2000 J = I E , I E IJK= NTY?E(I,J,K) GOTO (110,111,112,113,114,115,116,117,118, 1 115,120,121,122,123,124,125,126,127,128,129,130,. 2 13 1 , 13 2 , 1 3 3 , 13 4 , 1 3 5 , 13 6 , 1 3 7 , 13 6 , 13S) , I J K 110 B ( J ) = ( A R Y ( I , J , 2 ) ) *S4 ( I , J , K ) * F H Y ( I , J , K ) 1 • AFY ( I , J , 2 ) * B 4 ( K ) * Y 2 / B K A Y C(J) = - A B Y ( I , J , 2 ) E ( J ) = S4 ( I , J . K ) * P H Y ( I , 0 , K ) * T ( I , J . K ) • 1 (AFZ(I.J)) * ( T ( I , J , K-1)-2. *T(I,J,K) •1(I,J,K+1) ) • 2 (AIX(J,I,2)) • ( T1(I«1,J,K) - 11(1,J.K) ) 8 • <AFY(I,J,2) *B4(K)*Y2*ABB(K)/BKAY) GO 30 2000 111 B(J) = A E Y ( I , J , 2 ) • PHY ( I , J , K) * S 4 ( I , J , K ) 1 • iFY ( I , J . 2 ) *H4 ( K ) * Y 2 / E K A Y C ( J ) = -AEY ( I , J , 2 ) D ( J ) = S4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I , J , K ) • 1 (!FZ(I,J)) * ( 3 (I,J,K-1)-2. *T(I,J,K) •T(I,J,K*1) ) • 2 ( » F X ( J , I , 1 ) ) * (T 1 ( I - I . J . K ) - T I ( I . J . K ) ) • 2 (AEX(J,I,2)) * (T1(If 1,J,K)-T1(I,J,K) ) 8 • (AEY ( I , J , 2 ) » H 4 ( K ) * Y 2 * A E B ( K J / B K A Y ) GC TC 2 0 0 0 112 E(J) = ( A B Y ( I , J , 2 ) ) * S 4 ( I , J , K ) * F H Y ( I , J , K ) 1 • A E Y ( I , J , 2 ) * H 4 ( K ) *Y2/BKAY C (J) = -AKY ( I , J , 2 ) E ( J ) = 34 ( I , J , K ) » P H Y ( I , J , K ) * T ( I , J , K ) • 39  Ml)  113  114  115  116  117  116  119  1(AFZ(I,J)) * ( T(I,J,K-1)-2.*T(I,J,K) + 1(I,J,K+1) ) • 2 (*BX(J,I,1)) * ( 11(1-1,J,K)- T1(I,J,K) ) 3 (E1 ( K ) * A E X ( J , I , 1 ) * X 1 / B K A Y ) * ( T 1 ( I , J , K ) - AHB (K) ) 8 * (AEY ( I , J , 2 ) * E 4 ( K ) » Y 2 » A H B ( K ) / B K A Y ) GO 30 2 0 0 0 A(J) = - A E Y ( I , J , 1 ) B(J) = A B Y ( I , J , 1 ) • A B Y ( I , J , 2 ) • S 4 ( I , J , K ) » PflY(I,J,K) C(J) = - A E Y ( I , J , 2 ) B ( J ) = S U ( I , J , K ) * PHY ( I , J , K ) * T ( I , J , K ) + 1 ( A B Z ( I , J ) ) * ( T ( I . J . K - 1 ) - 2 . * T ( I , J , K ) + T ( I , J , K + 1) ) • 2 AFX(J,I,2) » ( T1(i+1,J,K) - T1(1,J,K) ) GO 10 2000 A(J) = -AEY(I,J,1) b ( J ) = A B Y ( I , J , 1 ) + A B Y ( I , J , 2 ) + S4 ( I , J , K) * PH Y ( I , J , K) C(J) = -IBY ( 1 , J.2) C(J) = S 4 ( I , J , K ) * P H Y ( I . J . K ) * T ( I . J . K ) + 1 ABZ(I.J) * ( 1 ( I , J , K - 1 ) - 2 . * T ( I , J . K ) + T ( I , J , K + 1) ) + 2 AFX ( J . I , 1 ) * (T 1 ( I - 1 . J . K ) - T 1 ( I . J , K ) ) • 3 A F X ( J , I , 2 ) * (11 (1 + 1 , J , K ) - H ( I . J . K ) ) GC I C 200C A ( J ) = -AEY ( I . J , 1) E ( J ) = AEY ( I , J , 1 ) +AEY ( I , J , 2 ) • S 4 ( I , J , K ) * PHY ( I . J . K ) C ( 0 ) = -AFY ( I . J , 2 ) D(J) = S 4 ( I , J , K ) * PHY(I,J,K) * T ( I , J , K ) + 1 ( A I Z ( I . J ) ) * ( 1 ( I , J , K - 1 ) - 2 . * T ( I , J . K ) + T ( I , J . K + 1) ) + 2 (AIX(J.I.I)) * (T1(I-1,J,K)-T1(I,J,K) ) 3 (E1 ( K ) * A R X ( J , 1 , 1 ) * X 1 / B K A Y ) • ( T 1 ( I , J , K ) - A3E (K) ) GC I C 2000 A ( J ) = -AEY ( I , J , 1) B ( J ) = ( A E Y ( I , J , 1 ) ) • (EHY ( I , J . K ) *S4 ( I , J . K ) ) + • 1 (E2 ( K ) * A F Y ( I , J , 1 ) * Y 1/BKAY) C(J) = £ 4 ( 1 , J , K ) * FHY(1,J,K) * T ( I , J , K ) • 1 < 1 B X ( J , I , 2 ) ) * ( T1 (1 + 1 , J . K ) - T 1 ( I , J , K ) ) • 2 (AEZ(I.J)) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) +I(I,J,K+1) ) • 3 (E2(K)*AEY(1,J,1)»Y1/BKAY*AHE(K) ) GC I C 2000 A ( J ) = -AEY ( I . J , 1) B ( J ) = ( A E Y ( I , J , 1 ) ) • PEY ( I . J . K ) » S 4 ( I . J . K ) • 1 (E2(K)•AFY(I,J,1)*Y1/EKAY ) D(J) = S4(X,J,K) * PHY(I.J.K) * T ( I , J , K ) • 1 ( A E X ( J , I , 1 ) ) • (11 ( I - 1 , J , K ) - T 1 ( I . J . K ) )• 1 (AFX ( J , 1 , 2 ) ) * ( T l ( 1 + 1 , J . K ) - T 1 ( 1 , J . K ) ) • 2 ( A E Z ( I . J ) ) * (1 ( I . J . K - 1 ) - 2 . * T ( I , J , K ) + T ( I , J , K + 1 ) ) • 3 ( B 2 ( K ) * A E Y ( I , J , 1) *Y1/EKAY*AMB (K) ) GO 10 2 0 0 0 A(J) = - A B Y ( I , J , 1 ) B ( J ) = ( A E Y ( I , J , 1 ) ) • (S4 ( I , J , K ) *PHY ( I , J , K ) ) • 1 (B2(K) » A E Y ( 1 , J , 1 ) *Y1/BKAY) D(J) = S 4 ( I , J , K ) » P H Y ( I , J , K ) * T ( I , J , K ) • 1 (AEX(J,I,1)) * ( T1(I-1,J,K) -T1(I,J,K) ) 2 ( B 1 ( K ) * A E X ( J , I , 1 ) * X 1 / E K A Y ) * ( T 1 ( I , J , K ) - ASE(K) ) + 3 (*FZ(I,J) ) * ( T (I.J.K- 1)-2.*T (I.J.K)+T (I,J.K+1) ) • 4 ( B2 ( K ) * AR Y ( I , J , 1 ) * Y 1/BKAY*ARB(K) ) GO 10 2 0 0 0 B ( J ) = (AEY ( I . J , 2 ) / 2 . ) • S4(I,J,K) • PHY(I,J,K) 1 + AEY(I.J.2)*H4(K)*Y2/FKAY/2. C ( J ) = -AEY ( I , J . 2 ) / 2 . D(J) = S 4 ( I , J , K ) * PHY(I.J.K) » T ( I , J , K ) • 1 ( A E X ( J , I , 2 ) / 2 . ) * (T1 ( 1 + 1 , J , K ) - T 1 ( I . J . K ) ) + 2 ( ABZ(I.J)) • ( T ( I , J , K - 1 ) - T(I.J.K) ) -  3 ( E 3 * A R Z ( I , J ) * E Z / B K A Y ) * (T (1, J . K ) — A B B (K) ) 8 • (AFY ( I , J , 2) * E-4 ( K ) * Y 2 * A B E ( K ) / B KA Y/2. ) GC TO 2 0 0 0 120 E ( J ) = (ABY ( I , J , 2) / 2. ) • ( S 4 ( I , J , K ) * P H Y ( I . J . K ) ) 1 • ABY ( I . J , 2 ) * K 4 ( K ) * Y 2 / B K A Y / 2 . C (J) = -ARY ( I , J . 2 ) / 2 . D(J) = £4(1,J.K) * PHY(1,J,K) * T ( I , J , K ) • 1 (APX(J,I,1)/2.)*(11(I-nJ#K}-T1(I,J,K) ) • 1 ( I B X ( J , I , 2 ) / 2 . ) * (T1 (1 + 1 , J , K ) - T 1 ( I , J , K ) ) + 2 (AFZ(I,J) ) * ( T(I,J,K-1) -T(I.J.K) ) 3 (6 3*ABZ ( I . J ) * C2/B KAY ) * ( T ( I . J . K ) - A H 3 ( K ) ) 8 • (AEY ( I , J . 2 ) *H4 (K)*Y2«AI'.B ( K ) / R K A Y / 2 . ) GO TC 2 0 0 0 121 E ( J ) = (AEY ( I . J , 2 ) / 2 . ) + S 4 ( I , J , K ) * E H Y ( 1 , J , K ) 1 • ABY ( I , J , 2 ) *B4 ( K ) * Y 2 / B K A Y / 2 . C ( 0 ) = -ARY ( I , J , 2) / 2 . E ( J ) = £4(1,J.K) * E H Y ( I , J , K ) * T ( I , J , K ) • 1 (AEX ( J . I , 1 ) / 2 . ) * ( T 1 ( I - 1 , J , K ) - T 1 ( I . J . K ) ) 2 <H1 ( K ) * A B X ( J . 1 , 1 ) * X 1 / B K A Y / 2 . ) * (T 1 ( I . J . K ) - A B E (K) ) + 3 ( A E Z ( I . J ) ) • ( T(l.J.K-l) - T ( I . J . K ) ) 4 ( E 3 * A E Z ( I . J ) * E Z / B K A Y ) * ( I ( I , J , K ) -ABB (K) ) 8 • (AEY ( I . J . 2 ) *U4 (K) *Y2*Af<E ( K ) / E K * Y / 2 . ) GC TO 2 0 0 0 122 A(J)= -ABY(I,J,1)/2. E ( J ) = ( A E Y ( l . J . I ) • ABY ( I . J , 2 ) ) / 2 . • S 4 ( i , J , K ) * E H Y ( X , J , K ) C ( J ) =-ARY ( I . J . 2 ) / 2 . D(0) = S 4 ( I , J , K ) * P H Y ( I . J . K ) * T ( I , J , K ) • 1 (AEX ( J . I , 2 ) / 2 . ) * (T 1 ( I * I . J . K ) - T 1 ( I . J . K ) ) + 2 (AEZ(I.J)) * (T(I,J,K-1) -T(I.J.K) ) 3 (H3*AEZ ( I , J ) »CZ/RKAY ) * ( T ( I , J , K ) - ABE (K) ) GO TC 2 0 0 0 123 A ( J ) = -ABY ( I . J , 1) / 2 . E ( J ) = (ARY ( I , J , 1 ) + A B Y ( I , J , 2 ) ) / 2 . *S4 ( I . J . K ) * P H Y ( I . J . K ) C ( J ) = - I E Y ( I , J , 2) / 2. E ( J ) = £4(1,J,K) * E H Y ( I , J , K ) *T(1,J,K) • 1 ( ABX ( J . I . 1) / 2 . ) * (T1 ( 1 - 1 , J . K J - T 1 ( I . J . K ) ) • 1 (AEX ( J , 1 , 2 ) / 2 - ) * (T 1 ( I t 1 , J , K ) - T 1 ( I . J . K ) ) • 2 (AEZ(I.J)) » (T(I,J,K-1) -T(I.J.K) J 3 - (H3*AEZ ( I . J ) * E Z / E K A Y ) * ( T ( l . J . K ) - ABB (K) ) GO TO 2 0 0 0 124 A ( J ) = -ABY ( I , J , 1 ) / 2 . B ( J ) = (ABY ( I . J , 1 ) + A R Y ( 1 , J , 2 ) ) / 2 . + S4(I.J.K)*PHY(I,J.K) C ( J ) = -AFY ( I , J . 2 ) / 2 . D(J) = £4(1,J.K) * I H Y ( I , J , K ) * T ( I , J , K ) • 1 ( A I X ( J , I , 1 ) / 2 . ) * (T1 ( l - I . J . K ) - T I ( I . J . K ) ) 2 (H 1 ( K ) * A B X ( J , 1 , 1 ) * X 1 / B K A Y / 2 . ) * ( T1 ( I . J . K ) - A f l d (K) ) • 3 (ABZ(I.J) ) * ( T(I,J,K-1) -T(I,J,K) ) 4 ( B 3 * A B Z ( I , J ) *EZ/BKAY ) * (T ( I , J , K) - A8 B (K) ) GC TC 200 0 125 A (J) = - A B Y ( I , J . l ) / 2 . E ( J ) = ( A B Y ( I , J , 1 ) / 2 . ) • (S4 ( I , J , K ) * P H Y ( I . J . K ) ) • 1 (E2(K) * A F Y ( I . J , 1 ) * Y 1 / E K A Y / 2 . ) E(J) = S4(I,J,K) * PBY(I.J.K) * T(I.J.K) • 1 (AFX ( J . I . 2 J / 2 . ) * ( T 1 ( I * 1 , J , K ) - T 1 ( I , J , K ) ) • 2 (AFZ(I.J) ) * ( T(I,J,K-1) - T(I,J,K) ) 3 (H3*ABZ ( I , J ) *CZ/EKAY ) * ( T ( I , J , K ) - ABB (K) ) • 4 (K2(K)*AEY(I,J,1) •Y1/BKAY*ABB(K)/2. ) GC TO 2 0 0 0 126 A ( J ) = -AFY ( I . J , 1 ) / 2 . B ( J ) = ( A R Y ( I , J , 1 ) / 2 . ) • (S4 ( I . J . K ) * P H Y ( I , J . K ) ) •  197  ( E 2 (K) * A F Y  1  E(0)  =  ( I , J , 1 ) * Y  S 4 ( I , 0 , K )  1  < A B X ( 0 , I ,  1  (1FX  1/BKAY/2.)  *1(I.0,K)  ( 1 - 1 , 0 , K J - T 1  ( J , I , 2 ) / 2 . )  *  ( T l  (It1,0,K)-T1(I,J,K)  (JFZ(I.J)  (E3*AEZ(I,J)•EZ/EKAY  U  (K2 (K)*»EY TO  )  *  ( T ( 1 , 0 , K - 1 ) - I  -AEY  (1,0,1)/2.  B(J)  =  (AEY  ( I , 0 , 1 ) / 2 .  ( H 2 (K)  *AEY(1,0,1)  =  ( A E X ( 0 , I , 1 ) / 2 .  2  ( B 1 (K) * A E X  3  (*EZ(I,J)}  U  ( H 3 * A B Z (1,0)  5  ( H 2 (K) * A F Y GO  10 =  B(0)  =  • C  » )  *  (0)  1) * Y  (1,0,  -ARY  =  S4(1,J,K)  A I X ( 0 , 1 , 2 )  5  •  TC  )*  *  T ( I , J , K )  , 0 , K ) - T 1  )  •  )  *  •  ) -  (I,J,K)  ( 1 1 ( 1 , 0 , K ) -  (1,0,K)  )  (T(I,0,K)  (1,0,2)  »  PHY  ( I , J , K -  (1  *  •  AMB ( K )  )  •  -  -AMB  SI  (K)  )  •  ( I , 0 , K ) * PHY  (1,0,K)  *  T ( I , 0 , K )  1 ) - 2 . *T ( I , 0 , K )  ,0 ,K)-T1  (1,0,K)  (1 + 1 , 0 , K ) - T 1  (11  =  B(J)  =  - A E Y  ( I , 0 , K )  )  •  •  (1,0,K)  K ) - A M B (K)  +  * T ( I , 0 , K + 1)  )  )  )  • (E5(K) C(0)  =  D(0)  =  ( 1 , 0 , 1 ) / 2 .  ( A E Y ( 1 , 0 , 1 ) + A K Y ( 1 , 0 , 2 ) ) / 2 .  +SU(I,J,K)  *  P H Y ( I , J , I  *AEEAU/2.) - A E Y ( 1 , 0 , 2 ) / 2 .  S 4 ( I , 0 , K )  1  ( i F X ( 0 , 1 , 1 ) / 2 . )  1  (AFX ( 0 , I , 2 ) / 2 .  2  ( A E Z ( I . O ) )  •  *IEY(1,J,K) * )  (T1 *  -  <H3*ARZ(I,J)  4  -  (B7 (K) * A E E A 1 / 2 . )  * T ( I , 0 , K )  (1-1,0,K)-11  *EZ/RKAY  »  • )  •  )  ( T ( I , J , K ) -  * (T 1 ( I , 0 , K ) - A H B  )  ( I , 0 , K )  - T ( I , J , K )  )  •  (1,0,K)  (T 1 ( 1 + 1 , J , K ) - 1 1  (1(I,J,K-1)  3  (K)  AME(K)  )  )  (B5(K)*AREA4*AMB(K)/2.) TO  2000  A(0)  =  B(0)  =  •  H6  - A B Y ( 1 , 0 , 1 ) ARY(1,0,1)  + ARY  (1,0,2)  C (0)  =  -ARY  D(0)  =  S 4 ( I , 0 , K )  2  ( A F X ( 0 , I , 1 ) )  3  (H7 (K) * A E I A 5 )  *  (  * T  *  PHY(1,0,K)  (1,0,  K - 1 ) - 2 .  * T ( I , 0 , K ) *T  ( T 1 ( I - 1 , 0 , K ) - T 1 *  (  T 1 ( I , 0 , K )  ( H 6 ( K ) * A 8 E A 2 * S I N ( T H ) ) R6(K)*AFEA2*C0S  *  PHY(I,J,K)  *  (TH)*ARE  -  ( I , J,K)  +  •T(I,J,K*1)  (I,J,K) AHB(K)  )  )  •  -  )  ( T 1 ( I , 0 , K ) - A M B ( K ) ) (K)  2000  10  A(0)  =  E(0)  =  •  S « ( I , 0 , K )  (1,0,2)  ( A E Z ( I , 0 ) )  U-  •  (K)*AEEA2*C0S(TF)  1  1  (I,0,K)  2000  A (0)  •  ( I , 0 , K ) * P H Y  ( K 5 ( K ) * A R E A U *A ME ( K ) )  GC  131  (1-1  <H7 ( K ) * A E E A 1) * ( T 1 ( 1 , 0 ,  GC  •  (1,0,2)  »  -  5  )  1/EKAY*AMB(K)/2.)  1) + A E Y  =  3  •  (K)  /2.)  - A E Y ( I , J , 1 ) AEY  *»  GO  (SU  P H Y ( I , 0 , K )  * E Z / E K AY  (1,0,  A F X ( 0 , I , 1 ) * (T 1 ( I - 1  1  •  (1,0,K-1J-T  (1  2  130  )  ( J , l , 1) * X 1 / R K A Y / 2 .  A F Z ( I , 0 )  1  -  (T ( 1 , J , K ) - A H B  l/BKAY/2.)  » (T1  1  1  )  +  (B5(K)•AEEAU)  0(0)  129  ( I , J,K)  ) •) •  2000  A(0)  1  *Y  SU(1,0,K)  1  128  •  (1,0,K)  2000 =  D(J)  )  (1,0,1}*Y1/BKAY*AMB(K)  A(J)  1  •  (Tl  3  GO  PHY(I,0,K)  *  2  127  *  1 ) / 2 . )  -ARY  ( 1 , 0 , 1 ) / 2 .  ( A E Y ( 1 , J , 1) + A R Y (1, J , 2) ) / 2 .  E 6 (K)  •  SU  (1,0  , K) * P H Y ( i , 0 ,  * A E £ A 2 * C 0 S ( T H ) / 2 .  C(0)  =  -AEY  D(0)  =  S 4 ( I , J , K )  ( 1 , 0 , 2 ) / 2 . *EHY(1,0,K)  1  ( A F X ( J , I ,  1 ) / 2 . )  2  (H7(K)*AFEA5/2.)  3  ( A E Z ( I , 0 )  )  *  *  (  (T1 •  (  *T(I,J,K)  ( 1 - 1 , 0 , K ) - T 1  •  (1,0,K)  1 1 ( I , J , K) - A MB <K)  T ( I , 0 , K - 1 )  -1(1,0,K)  )  ) )  • -  -  K)  198  4  ( H 3 * A B Z ( I . J )  5  -  6  •  TO =  - A B Y ( I . J . I )  E ( J )  =  ( A R Y ( I , J , 1 ) )  (B5  (K)  E ( J ) 1  *AEY  =  )  3  (HE  (K) TO  ( I , J , K) - A H E ( K )  =  B ( J )  =  ( I , J , K ) * S 4  P H Y ( I , J , K )  (11  *  - A B Y ( 1 , J , 1 ) (ABY  ( I , J ,  )  •  =  S 4 ( I , J , K )  * T ( 1 , J , K )  »  1  ( A E X ( J , I , 2 ) / 2 . )  *  (T1  (1 +1 , 0 ,  4  ( H 5 ( K ) * A F Y ( I . J , 1 ) * Y 3 / B K A Y * A K B GC  TC  E ( J ) 1  )  *  ( I . J ) * E Z / 3 K A Y  ( A E Y ( I , J , 2 ) )  AFY  (J)  ( I , J , 2 ) =  D ( J )  =  -ABY  *H6  (HE •  (K)  *  *ABX  (AEY  GO  TO  B ( J )  (  +  *  ( J . I ,  (AEY  AEY =  -AFY  =  S 4 ( I , J , K )  3  ( A B Z ( I . J ) )  •  (K)  *AFX  (AEY TO  H6  C (J)  -  -  / 2. )  *EEY *  ( J , I , *  (  (  +  E H Y ( I . J . K )  ( I , J , K ) ( I . J ,  + K) •  T ( I , J . K  T 1 ( I , J , K )  *  (  )  T 1 ( I , J , K ) -  + 1)  )  (K)  )  •  AHB  (KJ/BKAY)  S 4 ( I , J , K) * F H Y ( I , J ,  *  T ( l . J . K )  ( I - 1 , J , K ) - T 1  T ( I , J , K - 1 ) )  *  -  K)  )  *  •  ( I . J . K )  )  -  ( 1 1 ( I . J . K ) - A H S ( K )  T ( I , J , K )  (T  (K)*Y4*AHB  (ABY  = =  ( I . J , 2 )  )  )  )  •  -  ( I , J , K ) - A H B  (K)  )  ( K ) / B K A Y / 2 . )  •  S 4 ( I , J , K )  (TH)  ( I , J , K ) * P H Y *  (  P U Y ( I . J . K )  ( I , J , K ) * T ( I , J , K )  •  1 ( I , J , K - 1 ) - 2 . * T ( I , J , K ) *  (K)*ABEA6)  10  *  )  - A B Y ( I , J . 2 ) S4  ( *  1 1 ( 1 - 1 , J . K ) (  H6 ( K ) * A E E A 3 * C O S  B ( J )  *  *T  ( I . J . K ) T1  T I ( I . J . K ) (TH)»AHB  (H6 ( K ) * A E E A 3 * S I N ( T H )  GO  1) - 2 .  1 ) * X 2 / B K A Y / 2 .  (K) * A E E A 3 * C O S  ( * I X ( J , I , 1 ) ) (HI  •  ( K ) * Y 4 / B K A Y / 2 .  ( I , J . 2 ) * E 6  1 ( A F Z ( I , J ) )  • (  )  2000 =  D ( J )  S 4 ( I , J , K )  (K)*Y4*AHB  ( E 3 * A B Z ( I , J ) » E Z / R K A Y  •(  (K)  -  ( I , J , 2 ) / 2 .  (b6  GO  • ) •  ( K ) / 2 . )  T 1 ( I - 1 , J , K ) -  ( I , J , 2 )  D ( J )  E ( J )  )  ( I , J , K ) - A H 5  (T  1) * X 2 / E K A Y )  ( I , J . 2 ) * K 6  C (J)  •  *  )  K)  2000 =  2  1  (  ( I , J , 2 ) * h 6  ( J I X ( J , I , 1 ) / 2 . )  137  •  (K)*Y4/EKAY  1 ( I , J , K  1  4  )  ( I , J , 2 )  ( A F X ( J , 1 , 1 ) )  3  )  E 4 ( I , J , K ) » P F . Y ( 1 , J . K ) * T  1 ( A E Z ( I , J ) )  3  •  2000 =  • C  3  )  •  ( I , J ,  ( T ( 1 , J , K - 1 ) - T ( I , J , K )  ( I I Z ( I . J ) (H3**EZ  2  ( I , J , K)  1 ( 1 , J , K )  K)-11  3  2  K) * P H Y  1 , J , K ) - T  2  136  •  P d Y ( I , J , K )  (J-  4  )  * T ( I , J . K + 1 )  )  ( I , J ,  (S4  (T1  8  ) +  / 2 .  1 ) / 2 .  »  1  (K)  1)»Y3/BKAY*AMB  ( J , I , 1 ) / 2 . )  135  '  •  ( I , J , K )  * I ( I , J , K )  (1 ( I , J , K - 1 ) - 2 .  ( 1 , 0 ,  •  T ( I , J , K )  ( 1 - 1 , J , K ) - T 1  (AFX  8  ( I , J , K)  )  ( T 1 ( I * 1 , J , K ) - T 1 ( I , J , K )  *  *AFY  FHY  ( B £ ( K ) * A R Y ( 1 , J , 1 ) • Y 3 / B K A Y / 2 . ) D(J)  2  )  ( K ) / 2 .  1  134  )  - AHB(K)  2000  A(J)  1  * AHB  •  * *  ( A F X ( J , I , 2 ) ) * ( I F Z ( I , J ) )  GO  (T  1) * Y 3 / B K A Y  S 4 ( I , J , K )  2  133  ( I , J ,  ( « F X ( J , I , 1 )  1  •  2000  A{0)  1  )  H6 ( K ) * A E E A 2 * C 0 S ( T H )  GO 132  *EZ/RKAY  ( K ) * A E E A 2 * S I N ( I f i ) / 2 . ) * ( T l ( I , J , K )  (H6  )*  (T  T1 AHE  +  T ( I , J . K + 1 )  ( I . J . K ) (K)  )  -  )  (K)) 1 ( I , J , K ) - A H B  (K))  2000 =  (AEY  ( I , J , 2 ) / 2 . )  • (B6 ( K ) * A B E A 3 * C O S C(O)  =  -ABY  D ( J )  =  S 4 ( I , J , K )  S 4 ( I , J , K) * P h Y ( I , J ,  K)  )  ( I . J , 2 ) / 2 . • E B Y ( I , J , K )  1  ( H X ( J , 1 , 1 ) / 2 . )  *  2  (HI  )  (K)*A  •  (IH)/2«  F E A 6 / 2 . *  (  (  * ( T 1  ( A B Z ( I . J ) )  4  ( H 3 * A E Z ( I , J ) * E Z / B K A Y  T ( I . J . K )  ( l . J . K ) - A H B  T ( I . J . K - I )  3  *  T 1 ( I - 1 , J , K ) - T 1  )  *  (T  •  ( I . J . K ) (K)  T ( I , J , K )  )  •  )  -  ( I , J , K ) - A H B ( K )  )  )  -  ')  •  1 99  • (f.6 ( K )  1 2  GO  138  1C  A (J)  =  -ABY ( I , J , 1 )  =  < A E Y ( I , J , 1 ) )  (H5  (K)*AEY  E ( J )  £"{!,J,K)  =  2  (61  3  (AEZ (  U  (K) * A E X  (I.J)  1C  (  )  *  (  T  (K)  Y ( I , J , K ) )  *  )  T ( I . J . K )  •  - T 1 ( I , J , K )  )  *  ( T I ( l . J . K )  (1,J,K)  ( 1 , 0 , 1 ) * Y 3 / E K A Y * A H B ( K )  -ABY (1,J,1)/2. ( A B Y ( 1 , 0 , 1 ) / 2 .  (K)*AEY =  (1EX  )  •  -  A R B (K)  +T'(I, J . K + 1 )  1 ) / 2 .  (t  3  ( A E Z ( I , 0 ) )  * )  (H3*ABZ  5  ( B 5 (K) TO  )  •  (SU  ( I , J , K)  )  *  *  1,0,K)  * X 1 / E K A Y / 2 .  ( T ( 1 , J , K - 1 ) - T  ( I , J ) * E Z / R K A Y  *AEY  ( 1 -  ) *  (T  J , K )  •  1 ( I , 0 , K )  )  1(1,  - T  )  ( I . O . K )  )  ( I , J , K )  -AHE  AHB  (K)  )  •  ( I , J , 1 ) * S 3 / B K A Y * A H E ( K ) / 2 . )  2000  BETUEK EN E C I f l l L I C I T  CALCULATIONS  IN  THE  Z  DIRECTION  C C C C S U E E C U T I K E  COEIEK  DIRINS1CN  A (1)  D I E E N S I C N  ABX  CIBENSICN  1 ( 1 0 ,  , 1 2 ( 1 0 ,  ,E  ( I E , I E , 1 , J , A , B , C , D )  (1)  ( 16,  ,C  (1)  1 1, 2)  16,91)  , D ( 1 )  , ARY  , 1  ( 1 1,  1 ( 1 0 ,  1 6 , 2)  , AHZ  ( 1 1,  16)  16,91)  1 6 , 9 1 ) , 1 3 ( 1 0 , 1 6 , 9 1 ) , 1 N ( 1 0 , 1 6 , 9 1 )  C C f B C N / C 1 / E X , I Y , E Z , C T , B K A Y COKECN/C17/ARX,AB  Y , A E Z  C 0 B H 0 N / C 7 / T , T 1 , 1 2 , 1 3 , T N C 0 B H 0 N / C 2 / S 4 ( 1 0 , 1 6 , 9 1 ) C 0 E B C N / C 3 / N 1 Y P I ( 1 0 , 1 6 , 9 1 )  , L P S ( 1 0 , 1 6 , 9 1 )  C O B E 0 N / C 5 / N C H . T I H P 1 C0BBCN/C6/H3 C C H K 0 N / C 1 8 / X 1 , Y 1 , X 2 , Y 2 , Y 3 , Y 4 , T H C C H E C N / C 8 / P H Y ( 1 0 , 1 6 , 9 1 ) C C B E C N / C 1 9 / A R I A 1 , A B E A 2 , A R E A 3 , A R E A 4 , A B E A 5 , A B E A 6 DC  3000  K = I E , I E  I J K =NTYPE GC  TO  ( I , J . K )  ( 2 1 0 , 2 1 1 , 2 1 2 , 2 1 3 , 2 1 4 , 2 1 5 , 2 1 6 , 2 1 7 , 2 1 8 ,  1  2 1 5 , 2 2 0 , 2 2 1 , 2 2 2 , 2 2 3 , 2 2 4 , 2 2 5 , 2 2 6 , 2 2 7 , 2 2 8 , 2 2 9 , 2 3 0 ,  2  2 2 1 , 2 3 2 , 2 3 3 , 2 3 4 , 2 3 5 , 2 3 6 , 2 3 7 , 2 3 8 , 2 3 9 ) , I O K  210  A(K)  =  - A B Z  B(K)  =  (AEZ  (1,0)  C ( K )  =  A (K)  D ( K)  =  S U ( 1 , 0 , K ) * P H Y ( 1 , J , K ) * T  ( I , J ) * 2 . )  •  S« ( I . J . K ) * P H Y  1  ( A E X ( 0 , I , 2 ) )  *  (  T l  2  ( A I Y ( I , J , 2 ) )  *  (  T 2 ( 1 , J + 1 , K )  8  • I F GO  (1 + 1 , 0 , K )  ( A F Y ( I , J . 2 ) * K 4 ( K ) * Y 2 / R K A Y ) ( K . E Q . 2 ) D 10  +  3000  •  -  (11(1,J,K)-  *  CONTINUE  1  •  )  )  ( I , J , K ) * P H Y  P H Y ( I , 0 , K )  * ( T l  1 ( K ) * A B X ( J . I , 1 )  4  C  )  ( 1 , 0 , 1 ) * Y 3 / E F A Y / 2 . )  S 4 ( I , 0 , K )  ( J , I ,  2  GO  - A B S  2000  =  D(0)  ( I . J . K )  ( 1 , J , K - 1 ) - 2 . * 1  =  1  ) ( I . J . K )  ( I , J , K ) * P H  1) * X 1 / E K A Y  A(J)  (HE  (SU  1 1 ( 1 - 1 , J , K )  B ( J ) 1  2000  +  *'PHY  *  ( J . I ,  H5 ( K ) * A B Y  GO  (K) ( T l  ) *  ( I , J , 1 ) * Y 3 / B K A Y )  ( 1 R X ( J , I , 1 ) )  1  (TH) / 2 . * A H B (TH) / 2.  200C  B(J) 1  139  * A E E A3 * C O S  (H6 (K) * A B E A 3 * £ I H  (K)=C  (K)•  (AEZ  ( I . J . K ) -T1 -  ( I . J , K )  •  ( I , J . K )  )  1 2 ( 1 , J . K )  * (ABB  ( I . J ) ) * T  • )  ( K ) - 1 2 ( I . J . K ) ) ( I . J , 1 )  (K)  ) +  200  211  A ( K )  =  B(K)  - A B Z ( I , J )  =  A E Z ( I , J ) * 2 .  C ( K )  =  A(K)  D(K)  =  S 4 ( I , J , K )  ( A E X ( J , I , 2 )  1  )  A F Y ( I , J , 2 )  2 8  •  (K.F.0.2)  GO  TC  *  *  ( A F Y ( I , J , 2 )  I F  T  (T 1 ( 1 + 1 , J , K ) - T 1  *fc4(K)  ( I , J . K )  ( I . J . K )  ( A E Z ( I , J ) * 2 .  C ( K )  =  A (K)  C(K)  =  S U ( I . J . K ) * P H Y ( I . J . K ) * T  ( H i (K) * A B X  3  ( * F Y ( I , J , 2 ) )  *  TC  »  SU ( I . J . K )  *  P H Y ( I . J . K )  ( I . J . K )  T 1( I - I . J . K ) - T 1 *  ( -  +  ( I . J . K )  A B Z ( I , J ) * 2 .  C(K)  =  A (K)  C(K)  =  S U ( I . J . K )  ( A B B ( K ) - 1 2  3  ( A F Y ( I , J , 2 )  ) *  10  PHY ( I . J . K )  *  ( I . J . K )  1 ( I . J . K )  E ( K ) = E ( K ) • ( A E Z ( l . J ) ) * T ( I . J , 1)  A (K)  =  - A E Z ( I , J )  =  A E Z ( I , J ) * 2 .  C(K)  =  A (K)  D(K)  =  S U ( I , J , K )  •  *FHY  SU ( I . J . K ) * P K Y  ( I . J . K )  ( I , J . K )  * T ( I , J , K )  )  *  (T 1 ( 1 - 1 , J . K ) - T 1  ( l . J . K )  1  ( i E X ( J , I , 2 )  )  *  ( T l ( 1 + 1 , J . K ) - T 1  ( I . J . K )  2  ( A E Y ( I , 0 , 1 ) ) ( I B Y ( I , J , 2 )  * )  (T2 ( I , J - 1 , K ) - T 2 *  ( T 2 ( I , J + 1 , K )  I F ( K . E C . 2 ) D ( K ) = E ( K ) + A B Z 10 =  - A E Z ( I , J )  B(K)  =  A E Z ( I , J ) * 2 .  ) • )  ( I . J . K )  +  ) •  - T 2 ( I , J , K )  )  ( I , J ) * T ( I , J , 1 )  •  S 4  ( 1 , J . K ) * P H Y ( I , J . K )  =A(K)  D(K)  =  S 4 ( I , J , K )  1  ( A B X ( J , I , 1 ) )  2  (E1(K)  2  ( A I Y ( I , J , 1 ) )  *AEX  ( K . E C . 2 ) 10  * E E Y ( I , J , K )  *  (  • )  *  T ( I , J , K )  1 1 ( 1 - 1 , J , K ) -  ( J . I , 1 )  ( A E Y ( I , J , 2 )  *X1/BKAY)  *  •  ) -  ( T 1 ( I , J , K )  ( T 2 ( I , 0 + 1 , K ) - T 2 ( I , J , K )  D (K) = t (K) • ( A B Z ( I , J )  •  T 1 ( I , J , K )  ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K )  )  *  -  A B B (K)  )  ) • )  T ( I , J , 1 )  3000  A (K)  =  - A B Z ( I , J )  B(K)  =  (ABZ ( I , J) * 2 . )  C(K)  =  A (K)  C(K)  =  S 4 ( I , J , K )  1  ( A E X ( J , I , 2 )  2  ( A F Y ( I . J . I ) )  3  <B2 ( K ) * A F Y  )  * *  *  •  S 4 ( I , J , K )  F H Y ( I , J , K )  10  *  *  P H Y ( I , J , K )  T ( I , J , K )  ( T l (1 + 1 , J , K ) - T 1 ( I , J . K ) (  T 2 ( I , J - 1 , K )  ( I , J , 1) * Y 1 / E K A Y )  I F ( K . E Q . 2 ) D ( K ) = D ( K ) • ( A B Z GO  •  3000  A (K)  GO  +  )  3000  ( * F X ( J , I , 1 )  216  • )  ( T 2 ( I , 0 + 1 , K ) - T 2 ( I , J , K )  £(K)  IF  • )  ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K )  1  2  •  ( I , J , K ) )  * T ( I , J , K )  (T 1 ( 1 + 1 , J . K ) - T )  (K.EC..2)  + S4 ( I . J . K ) * P H Y  *  *  ( A E Y ( I , J , 1 )  C(K)  )  - A E Z ( I . J )  A F X ( J , I , 2 )  GO  A H B (K)  )  300C  2  215  -  T 2 ( I , J , K )  * E U ( K ) » Y 2 / E K A Y ) »  =  2  )  T I ( I . J . K )  D ( K ) =D ( K ) • ( A E Z ( I . J ) ) * T ( I . J , 1)  B(K)  IF  . •  ( T 2 ( I , J + 1 , K )  =  GO  )  ( J . I , 1) * X 1 / B K A I )  (AEY ( I , J , 2 ) ( K . F Q . 2 )  (  A (F)  1  •  - A F Z ( I . J )  ( A F X ( J » I , 1 ) )  GO  )  )  300C  =  •  ( I , J , K )  ( I , J . K )  * Y 2 / B K A Y ) * ( A A B ( K ) - T 2 ( I , J . K ) )  B(K)  I F  •.  •  ( A B Z ( I , J ) ) * T ( I . J , 1)  +  =  8  217  *  ( I 2 ( I , J + 1 , K ) - I 2  D (K)=D( K )  2  214  P H Y ( I , J , K )  A(K)  1  213  *  S4 ( I , J , K ) * F H Y  ( l F X ( J , l , 1 ) ) » ( I 1 ( I - 1 , J , K ) - - r l ( I . J , K ) )  1  212  •  *  • )  T 2 ( 1 , J , K )  ( T 2 ( I , J , K )  ( 1 , J ) ) * T  ( I , J , 1 )  3000  A(K)  =  B(K)  =  - A B Z ( I , J ) A B Z ( I , J ) * 2 .  + S 4 ( I , J , K )  * P H Y ( 1 , J , K )  + )  -  - A H B (K)  )  •  21E  219  220  221  222  223  C ( K ) = A (K) D (K) = S a ( I , J , K ) * F H Y ( I , J , K ) * T ( I , J , K ) • 1 ( A E X ( J . I . I ) ) * ( I 1 ( I - 1 , J , K) - T 1 ( 1 , 0 , K) ) • 1 ( A E X ( J , I , 2 ) ) * (T1 ( I O , J . K ) - T 1 ( I . J . K ) ) 2 (ABY(I,J,1) ) • ( T2(I,0-1,K)-T2(l.J.K) ) 3 ( h2 (K) * AE Y ( I , J , 1) * Y 1/EKAY) * ( T 2 ( I , J , K ) - AHB(K) ) If ( K . E C . 2 ) D ( K ) = D ( K ) • (AEZ ( I . J ) ) * T ( I . J , 1) GO 10 3000 A(K) = - A R Z ( I , J ) B ( K ) = (AEZ ( I , J ) * 2 . ) • SU(I,J.K)*£HY(I,J.K) C ( K ) = A(K) D(K) = S 4 ( I , J , K ) * E H Y ( I , 0 , K ) * T ( I , 0 , K ) • 1 (AFX(J,I,1)) * (11(1-1,J.K) -T1(1,J,K) )2 ( H I (K) * A B X ( J . I , 1 ) * X 1 / B K A Y ) * ( T I ( I . J . K ) -AHE(K) )• 3 (IFY(I,J,1) ) • ( T2(I,J-1,K ) -12(1,J.K) ) 1 ( H 2 ( K ) *AEY ( 1 , 0 , 1 ) * Y 1 / E K A Y ) * ( T 2 ( I , J , K ) -AHB (K) ) IF(K.EC.2)D(K)=C(K) + (ABZ<I,J))*T(I,J,1) GC 10 3 0 0 0 A ( K ) = -AEZ ( I . J ) B(K) = ( A E Z ( I . J ) ) • ( S U ( l . J . K ) * P H Y ( I . J . K ) ) • 1 (B3*ABZ ( I . J ) * D Z / B K A Y ) D(K) = S 4 ( I , J , K ) * P H Y ( I , J . K ) * T ( I . J . K ) • 1 (JEX ( J , 1 , 2 ) / 2 . ) * ( T I ( I O . J . K ) - T1(I,J,K) ) • 2 ( ABY(I,J,2)/2. ) * { 12(I,JO,K)-T2(I,J,K) ) • 3 ( B 3 * A B Z ( I , J ) *rZ/BKAY*AHB(K) ) 8 • (ARY ( I , J . 2 ) * H U ( K ) * I 2 / B K A Y / 2 . ) * IAHB(K) - T 2 ( I . J . K ) ) GO 1C 3000 A (K) = -AEZ ( I . J ) E(K) = ( A E Z ( I . J ) ) • (SU ( I , J , K ) * P H Y ( I . J . K ) ) • 1 (E3*ABZ ( I . J ) * E Z / B K A Y ) 0 (K) = S U ( I , J , K ) * P H Y ( I . J . K ) * T ( I , J , K ) + 1 ( l £ X ( J , I . 1 ) / 2 . ) * (11 ( 1 - 1 , J , K ) - 1 1 ( I . J . K ) ) • 1 (AFX ( J . I , 2 ) / 2 . ) * ( T l ( I * I . J . K ) - T 1 ( I . J , K) ) + 2 (AEY ( I . J , 2 ) / 2 . ) * (T2 ( I , J O , K ) - T 2 ( I , J , K ) )• 3 ( H 3 * A E Z ( I . J ) * E Z / f i K A Y * A H E (K) ) 8 • (AEY ( I , J . 2 ) * H4 (K) * Y 2 / E K A Y / 2 . ) * (AHB (K) - T 2 ( l . J . K ) ) GO 10 3C00 A (K) = - A E Z ( I . J ) £ (K) = ( A E Z ( I . J ) ) • (S4(I,J,K) * EHY(I.J.K) ) • 1 (E3*AEZ ( I , J ) * E Z / B K A Y ) D ( K ) = S < » ( I , J . K ) * PHY ( I . J . K ) * T ( I . J . K ) • 1 (*FX(J,I,1)/2. ) * ( T1(I-1,J,K)-11(I,J,K) ) 2 (B1 (K) * A E X ( J , I , 1 ) * X 1 / E K A Y / 2 . ) * ( T l ( l . J . K ) - AHB (K) ) • 3 (AEY ( I , J . 2 ) / 2 . ) * ( 1 2 ( 1 , J O , K ) - T 2 ( I , J , K ) ) • 3 ( B 3 * A E Z ( I , J ) *EZ/EKAY*AHE ( K ) ) 8 • ( A F Y ( I , J . 2 ) * E U ( K ) * Y 2 / F K A Y / 2 . ) • (AHB ( K ) - T 2 ( I , J . K ) ) GO 10 3 0 0 0 A (K) = - ABZ ( I , J ) E ( K ) = ( A E Z ( l . J ) ) • SU ( l . J . K ) * P H Y ( I . J , K ) • 1 (E3*AEZ ( I , J ) * E Z / B K A Y ) D(K) = S U ( I . J . K ) * £ H Y ( I , 0 , K ) * T ( I . O . K ) • 1 (HEX ( J . I , 2 ) / 2 . ) * { T 1 ( l O . J . K ) - H ( I . J . K ) ) • 2 (AFY(I,J,1)/2.)*(12(I,J-1,K)-T2(I,J,K))* 2 (AEY ( I . J . 2 ) / 2 . ) * (12 ( I . J O , K ) - T 2 ( I . J . K ) )• 3 ( B 3 * A F Z ( I . J ) * r Z / B K A Y * A H E (K) ) GO 10 3 0 0 0 A (K) = -ABZ ( I . J ) B(K) = A E Z ( I . J ) • S U ( I . J . K ) * P H Y ( I . J . K ) • 1 (E3*ABZ ( I , J ) * E Z / B K A Y ) D(K) = S U ( I . J . K ) * P H Y ( I . J . K ) * T ( I . J . K ) •  (AFX  2  ( AEY ( 1 , 0 , 1 ) / 2 . ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I ,  2  * ( H  d *  -T  H3*AEZ GO  22<4  IC  =  ( A E Z ( I , J ) )  +  r(K)  =  * ( S 4 ( 1 , J , K )  S U ( I , J , K )  *  P H Y ( I , J , K ) )  ( H 1 (K)  3  (AEY  ( I , J ,  3  (AEY  ( I , J , 2 ) / 2 . ) * ( T 2 ( I , J * 1 , K )  «  (E 3 * A BZ ( I , J )  * AEX ( J , I , 1 ) / 2 .  *  *  T ( I , J , K )  11(1-1,J,K)  2  10  )  P H Y ( I , J , K )  ( I E X ( J , I , 1 ) / 2 .  A(K)  *  (  1) * X 1 / B K A Y / 2 . )  )*  ( I , J -  (12  -  * (11  1 , K ) - T 2  ( I , J , K )  ( I , J , K )  - T 2 ( I , J . K )  * E Z / B K A Y * A BB ( K )  +  =  E(K)  )  •  )  ( A E Z ( I , J )  =  )  •  (SU ( I , J ,  K) * P H Y  ( I ,  J . K )  )  S U ( I . O . K )  * E H Y ( I , J , K )  *  11(1*1,  T ( I . J . K )  J . K )  •  )  •  (  2  ( A I Y ( I , J , 1 ) / 2 .  )  »  (T2  3  ( E 2 ( K ) * A E Y ( I , 0 , 1 ) * Y 1 / E K A Y / 2 . ) * ( T 2 ( I , J , K )  4  (E2*ABZ  (K)  ( I . J . K ) )  =  - A E Z ( I , J )  B(K)  =  ( A E Z ( I . J ) )  (E3»AEZ E(K)  - A B B (K) )  =  +  ( S U ( 1 , 0 , K )  *£HY  ( I . J . K )  £4(1,0,K)  *  P H Y ( I . J . K )  (11  (1-1  *  X ( I . J . K )  ( A f X ( J . I . 2 ) / 2 . ) * ( I  1(1*  (  AEY  *  3  (  H2 ( K ) * » B Y ( I . J , 1 ) * Y 1 / B K A Y / 2 . ) * ( T 2 ( I . J . K )  TO  (1,J,  1) / 2 .  A (K)  = =  =  )  £ 4 ( 1 , J , K )  2  (H 1 (K) * A B X  3  ( A F Y ( I , J , 1 ) / 2 .  4  (12  5  (E3*AFZ  •  * )  ( S 4 ( 1 , J , K )  *  (T1  )  *  •  T ( I , J , K )  (T2  TO  ( I , J , K )  C(K)  =  A  E(K)  =  S U ( I , J , K )  •  SU  ( I , J , K )  *PHY  ( I ,  * F B I ( 1 , J , K )  ( A F X ( J , I , 1 )  )  *  (T  1  ( A E X ( J , I , 2 )  )  *  (T 1 ( I *  2  ( A F Y ( I , 0 , 1 ) ) ( * F Y ( I , J , 2 )  * )  (T2 *  * T ( I , J , K )  1 ( 1 - 1 , J , K ) - T 1  1,0,K)  - T l  ( I , J - 1 , K ) - T 2  ( 1 2 ( 1 , 0 * 1 , K )  1  ( H 5 ( K ) * AE E A U ) (K. E C . 2 ) D ( K ) = E  *  ( A H B ( K ) - T 2  (K)  +ABZ(1,0)  *  ( 1 , J , K )  (1,0,K)  ) • )  ( I . J . K )  - T 2 ( I , 0 , K )  - ( H 7 ( K ) * A B E A 1 ) * ( T 1 ( I , 0 , K ) - A K B ( K )  229  J , K )  (K)  (1,0  *  ) • )  )  , K) )  *T (I ,0  ,  1)  3000  (1,0)  A(K)  =  -AEZ  £(K)  =  A E Z ( I , 0 )  (H3*AEZ  •  S U ( I , 0 , K )  ( I , J ) * E Z / B K A Y )  *  PHY  ( I , J . K )  •  •  -  - AHB(K)  )  - A E Z ( I , J )  1  10  *  -  )  3000  A E Z ( I , J ) * 2 .  GO  )  1 ( 1 , J , K ) - A H B ( K ) )  ( I , J - 1 , K ) - 1 2  ( I , J ) * I Z / B K A Y * A B E ( K )  =  •  -  +  ( I , J , K ) )  ( K ) * A F Y ( 1 , 0 , 1 ) * Y l / B K A Y / 2 . ) * ( T 2 ( 1 , 0 , K )  B(K)  I E  )  - A H B ( K ) )  P H Y ( I , J , K )  ( I - 1 , 0 , K ) - T 1  1 ) * X 1 / E K A Y / 2 . ) * ( I  (0,1,  =  5  •  T 2 ( I , J , K )  )  F B Y ( 1 , J , K ) *  A (K)  2  -  ( I , J ) * I Z / B K A Y )  ( A I X ( J , I , 1 ) / 2 .  U  ( l . J . K ) )  (1,0)  ( A B Z ( I , J )  1  GO  1 , J . K ) - T l  ( T 2 ( I , J - 1 , K )  ( I . J ) * E Z / B K A Y * A B E ( K )  - A E Z  (K3*AEZ  228  )  3000  B(K)  D(K)  •  •  1  (H3*ABZ  )  (I,J.K))*  , J , K ) - I 1  2  1  •  )  ( 1 , J ) * E Z / B K A Y )  ( A E X ( J . I , 1 ) / 2 . ) »  227  -  3000  10  GO  +  - T I ( I . J . K ) )  ( I , J - 1 , K ) - I 2  (1,0)*DZ/BKAY*ABB  A (K)  4  •  ( I . J ) * E 2 / B K A Y )  ( A B X ( 0 , I , 2 ) / 2 .  1  ) •.  ) •  1  1  -Afl£(K)  3 00 0  (E3*AEZ  GO  ) -  T 1 ( I , J , K )  = - A B Z ( I , J )  B(K)  226  •  ( I , J ) * E Z / E K A Y  1  1  •  ( I , J ) * E Z / B K A Y * A B E < K )  H3*AEZ  GC  •  )  J , K )  = - A B Z ( I , J )  B(K)  225  )  3000  A (K)  1  1 ( 1 , 0 , K )  -12(1,J,K))  ( I F Y ( I , J , 2 ) / 2 . ) * ( 1 2 ( I , J * 1 , K )  3  1  1 , 0 , F ) - T  •  (I,J,K))  ( A E X ( J , I , 1 ) / 2 . ) * ( T  1  ( 0 , 1 , 2 ) / 2 . )  1 ( I -  1,0,K)  1  ) •  C (K)  =  S 4 ( I , J , K )  *  PHY ( I . J , K )  *  T ( I . J . K )  •  < A E X ( J , I , 1 ) / 2 . ) * ( I 1 ( I - 1 , J . K ) - T 1 ( I , J , K ) ) *  1 1  ( A I X ( J , I , 2 ) / 2 . ) * ( 1 1 ( I * 1 , J , K ) - T 1  ( l . J . K ) )  •  2  (ARY ( I . J , l ) / 2 . ) * ( T 2 (I , 0 - 1 . K ) - T 2  ( I , J . K ) )  •  ( / I Y ( 1 , J , 2 ) / 2 . ) * { T 2 ( I , J + 1 , K ) - T 2  ( I , J , K ) )  •  2 3  H3*ABZ  IJ  -  5  (B7 (K) * A R E A l / 2 .  •<E5 GC  230  ( I , J ) » E Z / B K A Y * A H E ( K ) ) * (11 ( I . J . K ) - A K E  ( K ) *ABEA4/2.)  1C  *  =  - A B Z ( I . J )  E  =  A E Z ( I , J ) * 2 .  (K)  C(K)  =A(K)  E(K)  =  ( A B X ( J , I , 1 ) )  2  ( H 7 ( K ) * A F E * 5 )  2  ( A E Y ( I . J . I ) )  2  ( A B Y ( I , J , 2 )  •  (  4  • . (H6 (K) * A R E A 2 * C O S  -  AEB(K)  )  (TB) ) * (11 ( I . J . K )  D (K)  )  ( T H ) ) * (AMB (K) - 1 2 ( I , J . K )  )  (K)=C ( K ) • ( A B Z  ( I , J ) )  *  T  ( I . J . I )  ( A E Z ( I , J ) )  • < S 4 ( 3 , J , K )  *  P H Y ( I , J , K ) )  »  T ( l . J . K )  =  S 4 ( I , J , K )  *  (E7 ( K ) * A E E A 5 / 2 . )  3  ( I E Y ( I , J , 1 ) / 2 . ) * (T2 ( I , J - 1.K) - T 2  (1,  3  (AFY ( 1 , J , 2 ) / 2 . ) * ( I 2 ( I , J - H , K ) - T 2  ( I , J . K )  4  (E3*APZ  (1,J)  )  PHY ( 1 , J . K )  2  •  (  1 1 ( 1 - 1 , J . K )  *  (  11 ( I . J . K )  * f Z / F K A Y * A £ E  (K)  3  -  (H6 (K) * A R E A 2 * S I K  ( T H ) / 2.  1  •  (H6(K)  ( T b ) / 2 . )  1C  *AEEA2*C0S  -  - A ME ( K )  =  - A B Z ( I . J )  B(K)  =  A F Z ( I , J ) * 2 .  C(K)  =  A (K)  C(K)  =  S U ( I . J . K )  )  •  1 ( I , J . K ) - AMB (K)  ) * (1  * (ABB (K) -12  •S4( 1,J,K)  * E H Y ( 3 , J , K )  * T ( I , J , K )  ( T l ( 1 - 1 , J , K ) - T 1  ( l . J . K )  1  ( I F X ( J , I , 2 )  )  *  (11 ( I M . J . K ) - T I  ( l . J . K )  2  ( A F Y ( 1 , J , 1 )  )  *  (  3  (  H 5 ( K ) * A B Y ( I , J , 1 ) (K. E C 10  )  • ) • )  +  1 2 ( 1 , J - 1 . K ) - T 2 ( I , J , K ) *Y3/EKAY)  2) E (K) = E (K) •  *  )  -  (T2 ( I . J . K ) - A B B  (AEZ ( I . J ) ) * 1  (K) )  ( I . J . 1)  3000  A(K)  =  B(K)  =  - A E Z ( I . J ) (AEZ ( I . J ) )  (h3*AEZ =  •  ( S U ( I . J . K )  * E h Y ( i , J , K )  S 4 ( I , J , K )  (ABX ( J . I . I )  *  P H Y ( 3 , J . K )  *  / 2 . ) * ( T 1 ( I - l . J . K )  l ( I . J . K )  - T l ( I , J . K )  +  1  ( I I X ( J , I , 2 ) / 2 . ) • ( 1 1 ( 1 (  AEY ( I . J , 1 ) / 2 .  3  (  H 5 ( K ) * AB Y ( I . J , 1 ) * Y 3 / E K A Y / 2 . ) * ( 1 2 ( I . J . K )  4  (B3 * ABZ ( I . J ) * E Z / B K A Y * A H E  )  *  +1 . J . K ) - 1 1 ( I . J . K )  • ) •  2  1C  )  ( I . J ) * E Z / P K A Y )  1  GC  )  ( l . J . K )  * P H Y ( 1 , J , K )  ,  •  234  -  )  )  0(K)  )  •  J . K ) ) •  ( A E X ( J , I , 1 )  1  )  3000  A ( K )  GO  •  T I ( I . J . K )  1  233  •  ( I . J ) * D Z / B K A Y  (AEX ( J . I , 1 ) / 2 .  I F  )  - AMB(K)  1  232  ) •  ( I . J . K )  = - A B Z ( I , J ) =  H3*ABZ  GC  ) -  +  3000  A (K) B(K) 1  •  T1 ( I . J . K )  ( 1 2 ( I , J * 1 , K ) - 1 2  (H6 (K) * A B E A 2 * S I N  TC  1 ( 1 , J . K )  ( I . J - 1.K) - 1 2 ( 1 , J . K )  (12 *  -  ( K . E Q - 2 ) D  *  , J . K ) -  ( 1 1 ( 1 , J . K )  *  -  ( l . J . K ) * P H Y ( l . J . K )  11 (1-1  •  )  S4  * P K Y ( I , J , K )  *  3  231  )  ^  S U ( I . J . K )  1  I E  (K)  ( I . J . K ) )  3000  A (K)  GC  (AMB ( K ) - 1 2  ( 1 2 ( 1 , J - 1 , K)  (K)  -  )  •  T 2 ( I , J , K )  )  - ABB(K)  )  )  3000  A(K)  =  B(K)  =  —ABZ  ( A E Z ( I , J ) * 2 .  ( I . J )  C(K)  =  A (K)  D(K)  =  £ 4 ( I , J , K ) * P H Y ( I , J , K ) * T  1  ( I E X ( J , I , 1 ) )  2  (H£ (K)»ABX  3  ( A E Y ( I , J , 2 ) )  •  (  )  •  S 4 ( I , J , K )  *  ( I . J . K )  T 1 ( 1 - l . J . K ) - T l  ( J . I , 1) * X 2 / B K A Y ) * ( T 2 ( I , J * 1 , K )  *  (  P H Y ( I . J . K )  •  ( I . J . K )  )  T 1 ( I , J , K )  -  - T 2 ( I , J , K )  AMB (K) )  )  8  •  (ABY  I F GC 235  I C  A(K)  =  - A B Z ( I . O )  =  ( A E Z ( I , J ) )  ( E 3* D(K)  =  2  (B6  3  (IFY  (K) »ABX  •  ( J . I ,  I C  *  P B Y ( I . J . K )  ( I . J . K )  (  T  1(1-  *  1) * X 2 / B K A Y / 2 . ) )  *  (  T2  I | 1 , J , K )  1 , J . K )  * (11  - A B Z  =  A (K)  D (F )  =  S 4 ( I , J , K ) * P E Y ( I , J , K ) * T ( I , J , K )  (H1(K)*AFEA6)  3  ( A F Y ( I . J , 2 ) )  4 +(B6(K)  *  (  )  11  *  (  *  •  1 1 ( 1 , J . K )  ( T 2 ( I , J * 1 , K ) ) * (AHB  ( H 6 ( K ) * A E E A 3 * S 1 N ( T R )  10  E(K)  =  E (K) =E (K)  -ABZ  *  ) *  + (AEZ  =  +  £ 4 ( 1 , J , K )  ( J E X ( J , I , 1 ) / 2 .  2  ( E l  (K)  *  (AEY  3  ( H 3 * A E Z ( I . J )  ABB  -  T 2 ( I , J , K )  ( S 4 ( I , J . K )  *  ( I . J , 2 )  * )  )  /2.  (  T  * *  1 (1-1,  *  ( 1 1 ( 1 , J . K ) {  ( T H ) / 2 .  ) *  (ABE  )*  (11  A(K)  = =  C ( K )  =  A  D(K)  =  £ 4 ( 1 , J . K )  (AEZ  ( I . J ) * 2 . )  •  S4  * E B Y ( 1 , J , K )  ( E 1 ( K ) » A B X ( J , I , 1 ) * X 1 / B K A Y )  3  ( A F I d . J . I ) ( H 5 (K)  *  )  •  (  )  (K)  )  ( I , J ,  1) * Y 3 / F K A Y )  (K)=E  (K)  TO  • (AEZ  •  -TUl.J.K) *  ) -  ( T 1 ( I , J , K )  T 2 ( 1 , J - 1 , K  *ABY  )  *  -ABB(K)  - T 2 ( I , J , K )  ( T 2 ( I , J , K )  ( I , J ) ) * T  ( I , J ,  )  )+  -  -AHB(K)  )  1)  3000  A (K)  =  B(K)  =  —ABZ  =  ( I , J )  ( A F Z ( I , J )  (B3*ABZ  )  +  ( S 4 ( 1 , J , K )  S 4 ( I , J , K )  ( J , I ,  1 ) / 2 .  ( K ) * ABX ( I , J ,  PHY •  ( 1 , 0 , K )  (T 1 ( I -  •  (T2  FETUEK  *  1,0,K)  T ( I , J , K )  •  - I  )  ( I , J - 1 , K ) - 1 2  (hf  3000  ( I , 0 ,  )  ( E 3 *A R Z ( I , J ) * D Z / B K A Y * A f B TC  P K Y ( I , J , K )  )  1 (1,0,K)  •  -  ( J , I , 1 ) * X 1 / E K A Y / 2 . ) * ( T 1 ( I , J , K ) - A B £ ( K )  1 ) / 2 .  (K)*AEY  * )  4  COKTINOE  *  ( I , J ) * D Z / B K A Y )  5  EN E  ( K ) - T 2 ( I , J , K )  (I , J , K ) - A B B  * T ( I , J , K )  ( 1 1 ( 1 - 1 , J . K )  ( K . E C . 2 ) D  GO  •  ( I , J . K ) * F H Y ( I , J . K )  2  (AEY  )  (K)  ( A E X ( J , I , 1 ) )  3  -  - A K Z ( I . J )  1  (E1  ) •  3000  B(K)  2  •  (K))  ( T H ) / 2 .  •ABEA3*SIS  )  T 2 ( I , J + 1 , K ) - T 2 ( I , J , K )  *EZ/BKAY*ABE  (H6(K)*ABEA3*COS  TO  )  +  ( I , J . K )  A B B (K)  (H6 (K)  (AEX  (K))  1)  T ( I . J . K )  J . K J - 1 1  •  D(K)  )  F H Y ( I . J . K )  -  1  -  •  ( I , J , K ) )  ) *T ( I , J ,  4  1  )  ( I , J , K ) - A H B  ( I . J )  5  GC  (K)  )  -  P B Y ( I , J , K )  )  * AE E A 6 / 2 .  3  239  •  ( I , J . K )  ( K ) - T 2  (T1  P H Y ( I . J . K )  ( I , J ) * E Z / F K A Y )  1  I F  ( I , J . K ) )  ( I . J )  ( A E Z ( I . J ) )  (B3»AEZ  GC  ( K ) - 1 2  3000 =  E(K)  S 4 ( 1 , J , K )  ( 1 - 1 , J . K ) - 1 1  • A E E A 3 * C C S ( I B )  (K. E C - 2 )  A(K)  1  •  •  ( I . J )  ( A E Z ( I , J ) * 2 .  2  4  (ABB  C(K)  GO  )  )  * B 6 ( K ) * Y 4 / E K A Y / 2 . ) *  •=  -  )  - A B E (K))  ( I . J * 1.K)-12 ( I . J . K )  =  IF  •  •  ( I , J . K )  ( I . J . K )  B(K)  ( A E X ( J , I , 1 ) )  236  -11  A (K)  5  )  300 0  1  237  PHY *  ( I , J ) * D 2 / B K A Y * A J E ( K )  ( A E Y ( I , J . 2 )  GC  ( 1 , J ) ) * T ( I , J , 1 )  ( S 4 ( 1 , J , K )  * )  ( I , J , 2 ) / 2 .  (H3*ARZ  6  •  S 4 ( I , J , K )  ( A B X ( J , I , 1 ) / 2 .  3  *(AEZ  A E Z ( I . J ) * E 2 / E K A Y )  1  236  E(K)  3000  B(K) 1  3000  ( 1 , 0 , 2 ) * B 6 ( K ) * Y 4 / E K A Y ) * ( A H B ( K ) - T 2 ( 1 , 0 , K ) )  ( K . I C . 2 ) C ( K ) =  1) * Y 3 / E K A Y / 2 . ) * (K)  )  ( I , J , K )  )  )  •  )  •  -  ( 1 2 ( I , 0 , K ) - A H E ( K )  205  S U E E O 0 I I N E  c C  S O E B C U T I H E  C  EKD  CE  A  C 0 S P U T ( 1 1 , L 2 , B 1 , E 2 , K 1 )  TO  C A L C U L A T E  T I H E  THE  TEBPEBAT  UEES  AT  THE  STEP  C  c E I E E H S I O N T 3  1  ( 1 0 ,  T ( 1 0 ,  16,91)  , T 1 ( 1 0 , 1 6 , 9 1 )  , 1 2 ( 1 0 , 1 6 , 9 1 )  ,  1 6 , 9 1 ) , I H ( 1 0 , 1 6 , 9 1 )  D I E I N S I O N  A ( 1 0 1 ) , B ( 1 0  E I E E N S I G N  AEX  1 ) , C ( 1 0 1 ) , D ( 1 0  ( 1 6 , 1 1 , 2 )  1) , T P R I M E (1 0 1)  , A B Y < 1 1 , 1 6 , 2 )  , A 82  ( 1 1 , 1 6 )  C O E E C N / C 1 / E X , E Y , C Z , E T , B K A Y C O H B O N / C 2 / S 4 ( 1 0 , 1 6 , 9 1 ) CO E E O N / C 3 / N T Y P E ( 1 0 ,  16,9  1 ) , L P S ( 1 0 , 1 6 , 9 1 )  C G E E O N / C 4 / L , H , N C O E E C K / C 5 / N C H , T I H P 1 COBBON/C6/H3 C O E E O N / C 7 / T . T 1 . T 2 . T 3 . T N C O H H O N / C 8 / P E Y ( 1 0 , 1 6 , 9 1 ) CC E E C N / C 9 / E H Y H , E H Y S , P  HYL  C O E H O N / C 1 0 / T L I C , T S 0 1 , D E K S , D E N 1 C O B E C N / C 1 1 / C E L , C S P , N U M R U N , E L h T C O E K O N / C 1 2 / B 1 N P I ( 2 0 ) COE ECK/C13/T  , C P I  (20)  AU  C O E E O N / C 1 6 / N N B E . N U E 2 C C E E C N / C 1 7 / A B X , A B Y ,  AEZ  C O E B O N / C 1 8 / X 1 , Y 1 , X 2 , Y 2 , Y 3 , Y U , T H C O E E C N / C 1 9 / A B E A 1 , A B I A 2 , A E E A DO  1000  I = L 1 , L 2  DO  1000  J = B 1 , H 2  DO  10C0  K=2,K1  3 , A B E A 4 , A B E A 5 ,  AEEA6  I J K = K T Y E E ( I , J , K ) GO 1 2 310  10  ( 3 1 0 , 3 1 1 , 3 1 2 , 3 1 3 , 3 1 4 , 3 1 5 , 3 1 6 , 3 1 7 ,  3 18,3  19,3 2 0 , 3 2  3 3 0 , 3 3 1 , 3 3 2 , I N ( I , J , K )  =  (  ( A B Y ( I , J , 2 )  1 4  (ABY  1 , 3 2 2 ,  3 2 3 , 3 2 4 ,  ( A E X ( J , I , 2 ) ) * < T )  *  (12  ( I . J . 2 ) * H 4 ( K ) » Y 2 / £ K A Y ) *  ( A E Z ( I . J ) ) * ( 1 3 ( I , J . K - 1 ) - 2 .  3  ( < £ 4 ( I , J . K ) / 2 . ) * P H Y ( I , J , K ) GO  TO  =  (ARX  (  1  ( A E X ( J , 1 , 2 J )  1  ( A B Y ( I , J , 2 ) ( A E Y ( I , J , 2 )  *  ( J , l , 1 )  (T  TO  ( I . J . K ) )  +  J , K * 1)  ) ) /  ) *  (  T 2 ( I , J *  *H4(K)  * ( T 3  1,K)  *Y2/EKAY)  - T 1 )  - T 2 ( I , J , K )  *  (ABB  +  )  ( K ) - 1 2  ( I , J , K - 1 ) - 2 . * T 3 ( I , J , K )  ( I , J  ,K)  )  •  • +  ( I , J , K ) )  * T 3 ( I , J , K  + + 1) ) )  /  T ( I , J , K )  = ((AEX  (J , I ,  1) ) * ( T 1 ( I - 1 , J , K )  - T  1 ( I , J , K))  -  ( E 1 ( K ) * A E X ( J , I , 1 ) * X 1 / E K A Y ) * ( T 1 ( I , J , K ) - A B E ( K ) ) • ( A E Y ( I , J , 2 )  8  (AEY  3  (AEZ  4  ( GO  ( I , J)  (S4 TO  ) *  (T2  ( I , J *  1 . K ) - T 2  ( 1 , 0 , 2 ) * H 4 ( K ) * Y 2 / B K A Y ) * ) * (13  ( I , J , K )  ) +  ( A B B ( K ) - 1 2 ( 1 , 0 , K )  ( I , J , K - 1 ) - 2 . * T 3  ( I , J , K ) / 2 . ) * E B Y ( I , J , K ) )  ( I , 0 , K )  t T 3  ( I , J . K  ) + + 1) ) )  /  • T ( l . J . K )  1000  T N ( I , J , K ) 1  •  1000  T B ( I , J , K )  313  )  •  T ( I . J . K )  ) * (T 1 ( 1 - 1 , J , K )  ( < S 4 ( I , O , K ) / 2 . ) * P H Y ( I , 0 , K ) )  2  •  9,  ( I , J , K )  )  ( K ) - 1 2  6,32  , I J K  J , K ) + T 3 (I,  1 (I +1 , J , K ) - T 1 ( I . J . K )  3  1  (4MB  )  ( A E Z ( I . J ) )  GO  ( I , J , K )  3 ( I ,  «1  2  312  7 , 3 2  1000  T N ( I , 0 , K )  8  6,32  1 ( 1 + 1 , J , K ) - T l  ( I , J + 1 , K ) - T 2  2  311  3 2 5 , 3 2  3 3 3 , 3 3 4 , 3 3 5 , 3 3 6 , 3 3 7 , 3 3 8 , 3 3 9 )  = ( (ABX  ( A E Y ( 1 , J , 1 ) )  * *  ( J , l , 2 )  )*  (T  ( I . J - 1  , K ) - T 2 ( I , J , K )  (T2  ( 1 , 0 * 1 , K ) - 1 2 ( 1 , 0 , K )  1  ( A E Y ( I , 0 , 2 ) )  2  (AEZ  3  ( ( « 4 ( I , J , K ) / 2 . ) * P H Y ( I , 0 , K ) )  ( I , J)  ) * (13  1 ( I * 1, J , K )  (T2  ( I , J , K -  1 ) - 2 .  »13 •  ( I , J . K )  -T  1 ( I , J . K ) )  ) )  •  * T 3 ( I ,  T . ( I , J , K )  •  •  0,K*1)  ) )  /  GC 314  1C  1000  T N d . J . K )  =  (  (ABX  1  ( A E X ( J , I , 2 )  )  2  ( I F Y ( I . J . l )  )  2  <*EY(I,J,2))  2  •  ( J . I ,  *  1 ) ) * (T 1 ( 1 - 1 , J . K )  (I +1 , J , K ) - T 1  (11  - I 2 ( I , J , K )  (<£4(I,J,K)/2.)*Pb5f(I#JrK) GO  315  10  (  2  = ( ( A B X ( J , I ,  HI ( K ) * A E X ( J . I , 1 )  ( A F Y ( I , J , 2 ) )  4  ( ( S 4 ( I , J , K ) / 2 . ) * P H Y ( 1 , J , K ) ) GO  10  1  *  =  ( (ABX ( J . I . 2 )  ( A F Y ( I . J . I )  )  *  )*  ( H 2 (K) * A E Y (AEZ(I,J))  4  ( < £ « ( I , J , K ) / 2 . ) * P H Y ( I , J , K ) GC  10  Til 1  *(T3  )  =  (  ( A B X ( J . I . I ) *  (T1  (AEZ  5  ( ( £ 4 ( I , J , K ) / 2 . ) * P K Y ( I , J , K ) )  (I.J)  *  A H E (K) )  ( J . I ,  (J , 1 , 1 )  -T1  ( i . J . K ) )  ( I , J.K)  •  ) + )  -  +T3 ( I , J.K  )  •  + 1) )  ) /  • T . ( I . J . K )  J E Y ( I , J , 1 )  (1 1 ( I - 1 , J . K ) - T 1  (B2 (K) * A £ Y  4  (AEZ  5  ( ( S 4 ( I , J , K ) / 2 . ) * P H Y ( I , J , K ) )  ( I . J )  )  1) ) *  *  (  T2  ( I , J - 1 , K ) - T 2  ( I . J , 1) * Y 1 / R K A Y )  ) * (13  (I.J.K)  * X 1 / E K A Y ) * (1 1 ( I , J . K ) - A H E  (  *  ( I . J . K - 1 ) - 2 -  (I.J.K)  (T2 ( l . J . K )  *T 3 ( I . J . K )  ) -  (K)) + )  -  - AHB (K)  )  +T 3 ( I , J . K + 1 )  • )  ) /  +T(I,J,K)  1000  1N(1,J,K)  =  ( (AEX ( J . I , 2 ) / 2 .  ( H Y ( I , J , 2 ) / 2 . )  *  (  T 2  )*  (I.J+  (Tl  (1+1,  1.K)  -12  J . K ) - T 1  (I.J.K)  (I.J.K)  )  *  (I.J)  (  T 3 ( I , J , K - 1 )  *CZ/SKAY)  »  - T 3 ( I , J . K )  3  (B3*ARZ  4  ( ( E 4 ( I , J , K ) / 2 . ) * P E Y ( 1 , J , K ) )  (T  3(1,J.K)  )  ) +  +  ( A E Y ( I , J . 2 ) * B 4 ( K ) * Y 2 / B K A Y / 2 . ) * ( A R B ( K ) - T 2 ( I . J . K ) ( A i Z ( I . J ) )  ) •  -  - AflB(K))  ) /  + T ( I , J , K )  1000  T N ( I . J . K ) 1  /  1000  2  1C  +  ( I , J . K + 1 ) ) )  ( T 2 ( I , J , K) - A H B ( K )  ( I , J , K - 1 ) - 2 . * T 3  3  GC  •  -  ( 1 - 1 , J . K )  ( l . J . K )  ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K )  I I . J , K)= ( (AEX  (£1 (K)*AEX  2  ) ) /  T ( I . J . K )  (Tl  ( I , J , 1) » Y 1 / E K A Y )  ) * (13  1  8  +  ) *  4  1  + 1)  ( I , J.K))  )  ( T 2 ( I , J , K ) -  (1+1 , J , K ) - T 1  <E2 ( K ) * A B Y  10  •  1(1,J,K)  *  )  ( I F Y ( I . O . l ) ) *  GO  )  1000  (I.J.K)  TO  ) (K)  •  (I.J.K - 1)-2.*13(I,J,K)+13  3  GO  AHB  •  (1+1, J . K ) - T 1  (11  ( I , J , 1) * Y 1 / E K A Y  ( A B X ( J , I , 2 ) )  TN  ) ) /  ( I , J,K)  ( I . J . K ) + T 3 ( I , J . K +  2  =  (  (ABX  ( J . I , 1 ) / 2 . ) * ( T 1  ( 1 - 1 , J . K J - T 1 ( I . J . K )  ) •  ( t I X ( J , I , 2 ) / 2 . ) * ( T 1 ( I + 1 , J , K ) - T 1 ( l . J . K ) ) •  2  ( A £ Y ( I , J , 2 ) / 2 . )  8  *  (ABY ( 1 , J . 2 ) * H 4  3  ( I f Z ( I . J )  )  *  (  12  (I,J+  ( T 3 J I . J . K - 1 )  (B3*AEZ  5  ( ( S 4 ( I , J , K ) / 2 . ) * P M ( I , J , K ) GC  1C  ( I , J) * E Z / B K A Y  )  *  (T3  T 3 ( I , J , K ) ( I . J . K ) - A H B  )  •  =  (  ( A B X ( J , I . 1 ) / 2 .  (AEY(I.J,2)  2  (AFY  *H 4 ( K )  (I.J,2)/2.)  T  ) •  -  (K)  ) ) /  (I.J.K)  (  )*  - T 1  ( 1 1 ( 1 , J . K ) -  (I.J.K) A H B ( K) )  * Y 2 / F K A Y / 2 . ) * (AHB ( K ) - T 2 ( I , J , K ) (T2 ( I . J + 1 . K J - T 2  3  (AFZ(I.J))  (H3*AEZ (I,J)*EZ/RKAY  5  ( ( £ 4 ( 1 , J . K ) / 2 . ) * P H Y ( I , J , K ) ) TC  *  *  4  GC  • (I.J.K)  )  ) * (T 1 ( 1 - 1 , J . K )  ( E l ( K ) * A F X ( J . 1 , 1 ) * X 1 / B K A Y / 2 .  8  ( I , J . K ) )  (AHB ( K ) - T 2  1000  T N ( l . J . K ) 1  1 , K ) - T 2  ( K ) * Y 2 / E K A Y / 2 . ) *  4  T 3 ( I , J , K - 1 ) ) *  -  ( I . J . K ) )  T 3 ( I , J , K )  ( T 3 ( I , J , R ) •  -  )+  •  )  -  A H B (K)  )  ) /  T ( I . J . K )  1000  T N ( l . J . K ) 2  - T l  ( T 2 ( I , J - 1 . K ) - T 2 ( I , J , K )  2  321  J,K)  ( T I ( I . J . K ) -  ( T 2 ( I , J + 1 , K ) - T 2 ( 1 , J , K ) )  3  317  + 1)  1000  1JJ(I,J,K)  322  *  ( A E Z ( I . J ) ) * ( 1 3 ( I . J . K - 1 ) - 2 . * 1 3  320  + ( I , J . K  T ( I , J , K )  1) ) * ( T 1 ( 1 - 1 ,  *X1/BKAY)  2  319  +  )  3  318  )  ( A E Y ( I , J , 1 ) ) » ( 1 2 ( I , J - 1 , K ) - 1 2 ( I , J , K ) )  316  )•  1000  1 N U , J , K ) 1  ( l . J . K )  ) •  ( T 2 ( I , J - 1 . K ) - T 2 ( I , J , K ) )  * ( 1 2 ( 1 , J + 1.K)  A E Z ( I , J ) * ( T 3 ( I . J . K - 1 ) - 2 . * 1 3 ( I , J . K ) + T 3  3  - T l  (I.J.K)  = ( (ARX ( J , I , 2 ) / 2 . ) *  (11  (1 +  1 , J . K ) - T 1 ( I . J . K ) ) •  ( * F Y ( I , J , 1 ) / 2 . ) » ( 1 2 ( I » J - 1 # K ) - 1 2 ( 1 , J , K ) )  •  )  •  2  (*f Y ( I , J , 2 ) / 2 . ) * ( 1 2  2  ( A B Z ( I . J )  3  <B3*AEZ  tl  ( (£<4 ( I , J , K ) / 2 . ) GO  323  10  .  *  ( I , J)  ( I , j + 1 , R )  - T 2 ( I , J , K )  ( T 3 ( I , J , K - 1 ) - T 3  *EZ/BKAY *PHY  )  »  ( I , J , K )  ) * )  ( I . J . K )  )  = ( (ABX  ( J .  1,  1 ) / 2 .  ) * (T  1 ( 1 -  1, J . K )  - T l  ( A I X ( J , I , 2 ) / 2 . ) * ( 1 1 ( I * 1 . J , K ) - 1 U I , J , K ) )  2  3  ( A F Z ( I . J )  4  (E3*ABZ  5  ( | S U ( I , J . M / 2 . ) * P H ! ( I , J , K ) ) » GO  10  1  ( E l  2  )  ) *  ( T 2 ( I , J , K )  = ( ( A B X ( J , I , 1 ) / 2 . ) *  (R) * A B X  ( 0 , 1 , 1 )  s  -AHE(K)  )  ) /  I ( I , J , K |  (T 1 ( I - 1, J , K ) - T  * X 1 / H K A Y / 2 . )  1 ( 1 , J , K) )  * (1 1 ( I , J , K ) - A H B  (K)  ) •  ( A I Y ( I , 0 , 2 ) / 2 . ) * ( 1 2 ( I , 0 * 1 , K ) - T 2 ( I , J , K ) ) • ( A B Z ( I , 0 ) ) *  4  (H3*ABZ  5  ( (£4 GO  32 5  TN  10  (1JY  (13  (1,0)  ( I , 0 , K - 1 ) - T 3  *EZ/BKAY  )»  K) = ( ( A B X  ( 1 , 0 ,  (0,  1) / 2 . )  1,2)  / 2.  4  (E3*ABZ  5  ( ( S 4 ( 1 , J , K ) / 2 . ) * P E Y ( I , J . K ) ) 10  *  ( 1 , J )  * Y1/BK A Y / 2 .  (13  1,0,K)  )*  (12  - T l  ) * (T3  ( 1 , 0 , •  )  = ( ( A B X ( 0 , I , 1 ) / 2 . ) » )* (11  ( A F Y ( I , J , 1 ) / 2 . ) * ( T 2 ( I , 0 - 1 , R ) - T 2 (B2  4  ( A F Z ( I , J ) )  (K)  *AEY  5  (K3»AFZ  6  ( (£4  K) - A H B  (K)  ( I , J , *  (  1) * Y 1 / B K A Y / 2 . T3  ( 1 , 0 , K -  ( I , J ) * D Z / B F A Y  )  *  (12  ( (AEX  ( 0 , 1 , 1 ) / 2 . ) *  ( * F Y ( I , J , 1 ) / 2 . ) (H2  4  ( A E Z ( I , J ) )  5  (B3*ABZ  6  ( < S 4 ( I , J , K ) / 2 . ) * P H Y ( I , 0 , K ) )  (K)*AFY  10  *  ( I , J ,  ( 1 , 0 , R )  ,K)  ) •  )  (K)  ) •  ) AHB  (K)  )  ) /  • T ( I . J . K )  (T  1 ( 1 - 1 , 0 , K ) - T l  *  (T2  ( I , 0 - 1 , K ) - T 2  1 ) * Y 1 / £ h A Y / 2 .  )»  (1,J,K)  - ABB  ( I , J , K )  )  ( 1 , 0 , K )  ( T 3 ( X , J , K ) •  )  (K)  ) )  -  ) * ( 1 2 ( 1 , 0 , K ) - A f l B  ( 1 3 ( 1 , 0 , R - 1 ) - I 3  ( I , J ) * E Z / R K A Y  ( K ) ) •  -  -AME(K)  ) ) /  T ( I . J . K )  1000  ( I . J . K )  =  (  (ABX  1  ( A F X ( J , I , 2 )  )  2  ( A F Y ( 1 , J , 1 )  )  2  ( A F Y ( I , 0 , 2 ) )  * * *  6  -  (H7  7  •  (H5 ( K ) * A B E A U )  2  ABZ  (K)  ( J . I ,  1 ) ) * ( T 1 ( I - 1 , J , K ) - T 1  (11  ( 1 * 1 , 0 , R ) - T 1  ( 1 , 0 , K )  (12  ( 1 , 0 - 1  ( I , J , K ) )  , K ) - T 2  * A R E A 1) * ( T l *  ( I , J , K ) - A F B  ( AME ( K ) - T 2  ) •  •  (K)  )  ( I , 0 , K ) ) •  ( I , J ) * ( 1 3 ( I , J , K - 1 ) - 2 . * T 3 ( I , J , K )  1C  ( I . . J , K )  ) •  ( T 2 ( 1 , J * 1 , K ) - T 2 ( I , J , K ) )  ( ( £ 4 ( I , J , K ) / 2 . ) * P E Y ( I , J , K )  )  +  +T 3 ( I , J , K •  1)))  /  1 ( 1 , J . K )  1000  ( 1 , J , K ) =  ( JEX  (I , J  1000  ( 1 , 0 , K ) =  3  1  ) ) /  ( I , J . K ) - A M B  ( 1 3 ( 1 , 0 , R ) -  ( I , J , K ) / 2 . ) * P B Y ( I , J , K ) )  10  ( I , 0 , K )  )*  1) - T 3  2  GC  •  (1,0,K) ) •  ( E 1 ( K ) * A B X ( 0 , 1 , 1 ) * X 1 / B K A Y / 2 . ) * ( 1 1 ( I , 0 , K )  TN  (K))  -  |T 1 ( I - 1 , J , K ) - 1 1  ( 1 + l . J . K ) - T l  3  GO  ) •  -  T f l . J . K )  2  IN  ( 1 , J , K )  )  ( I , J , K) - A K 3  ( 1 , 0 , K - 1 ) - T 3 ( 1 , 0 , K )  *EZ/FKAY  ( A FX ( 0 , 1 , 2 ) / 2 .  1  ) /  1000  T N ( I . 0 , K )  GO  )  + T ( 1 , J , K )  ) * (T 1 ( 1 +  ( A E Z ( I . J ) )  TN  -  * ( T 2 ( 1 , 0 - 1 , K ) - T 2 ( I , 0 , K )  ( H 2 ( K ) * A E Y ( I , J , 1 )  1  )  - AHB (K)  K)  1000  2  GO  (I, J ,  )  3  326  ( 1 , 0 , K )  (T3  ( I , 0 , K ) / 2 . ) * P H Y ( I , J . K )  ( I , J ,  1  ( <ABX  ( J . I . 1 )  ( J . I , 2 ) / 2 . ) *  (1  / 2 .  ) • (11  (AFY  ( I . J ,  1 ) / 2 . )  * ( 1 2  ( I . J - 1 . K )  2  (AEY  ( I . J , 2 ) / 2 . )  * (12  (I  7  -  (H7 (K) * A B I A  e  f  (H5  (K)  ( A E Z ( I . J )  1/2.  * ABE A 4 / 2 . )  * (13  ) * )  ( 1 - 1 , J . K )  1 ( 1 * 1 , J . K ) - T l  2  3  •  -  ( * F Y ( I , J , 1 ) / 2 . ) * ( 1 2 ( I , J - 1 , K ) - T 2 ( 1 , J , K ) ) *  2  329  )  1000  3  3  ( I . J . K )  +  )  • ( 1 3 ( I , J , K - 1 ) - T 3 ( I , J , K )  ( I , J ) » C Z / E K A Y  T N ( I , J , K )  328  ) /  ( A F Y ( I , J , 1 ) / 2 . ) * ( 1 2 ( I , 0 - 1 , K ) - T 2 ( I , 0 , K ) ) * ( I F Y ( I , 0 , 2 ) / 2 . ) * ( T 2 ( I , 0 + 1 , K ) - T 2 ( I , J , K ) )  327  )  • T ( I . J . K )  2  324  -  (T 3 ( I , J , K ) - A H B ( K )  1000  m | I , J , K ) 1  )  ,  - T 1  ( 1 , J . K ) ) •  - T 2  ( I , J . K )  ) •  J + 1 , K ) - T 2 ( I , J . K ) )  (T 1 ( I . J . K ) - A H E * (ABB  (KJ-12  ( I , J , K - 1 ) - T 3  (K)  )  ( I . J . K )  ( I , J . K )  ) )  • -  ( I . J . K )  ) •  4  (H3»ABZ  5  < < S M I , J , K ) / 2 . ) GC  330  30  ( I , J ) * E Z / B K A Y  ) *  ( T 3 ( I , J , K )  *Pfn  > •  -AHB  = ( (AEX  ( 0 , 1 ,  E7(K)*ABEA5)  *  1) ) *  <T 1 ( 1 - 1 . 0 , K )  1  (  2  ( A F Y ( I , J , 1 ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) )  2  ( A B Y ( I , J , 2 ) )  ( T I ( I . J . K ) -  •  5  -  (H6 ( K ) * A B E A 2 * S I N  ( T E ) ) *  +  (H6  (IB)  (K) * A E E A 2 * C O S (33  (T1  (ABZ  ( 1 , 0 ) ) *  4  ( (£4  ( I , J . K ) / 2 . ) * P B Y ( I , J . K ) )  TO  - T l  (K)  )  ( I . J . K )  •  ( I . J . K ) - A H B (K) - 1 2  (K))  ( I . J . K )  ) •  •  = ( (ABX  ( J , I , 1 ) / 2 . ) *  2  (11Y  ( I . J ,  (T1  ( 1 - 1 , J , K )  • ( T 1 ( I , J , K ) - A H E ( K )  1 ) / 2 . ) *  ( T 2 ( I , J - 1 , K ) - T 2  - T  )  1 ( Z . J . K )  ( I , J . K ) ) •  ( A E Y ( I , J , 2 ) / 2 . ) * ( T 2 ( I , J * 1 , K ) - T 2 ( l . J . K ) ) -  (H6  7  •  ( H 6 ( K ) » A E E A 2 * C O S ( T H ) / 2 . ) * ( A P . B ( K ) - I 2 ( I , 0 , K i )  3  (K)*ABEA2*SIR  ( A F Z ( I . J ) ) *  ( T H J / 2 . ) *  (E3*ABZ  5  ( ( S 4 ( I , 0 , K ) / 2 . ) * P S Y ( I , J , K ) ) GC  332  TN 1  TC  ( I , J ) * E Z / B K A Y  ) *  (T3  (1,0,  )  =  (  ( A B X ( J , I , 1 ) *  K) - A H B  (K)  ) *  ( A E Y ( I , J , 1 ) )  3  (EE  4  (AIZ  5  ( ( S 4 ( I , J , K ) / 2 . ) * P E Y ( I , J , K ) )  (K)  *AEY  ( I , J)  IO  (Tl  ( 1 - 1 , J . K )  *  (T2  ( I , J ,  ) * (T3  ( I , J -  1,K)  - T 2 ( I , J , K )  1)*Y3/EKAY)  *  ( I , J . K - 1 ) - 2 .  * T 3 ( I ,  )  J , K )  ( A F Y ( I , J , 1 ) / 2 . ) * ( I 2 ( I , J - 1 , K ) - T 2 ( I , J , K )  4  ( A I Z ( I . J ) )  ( I , J , 1 ) »(  13  * Y 3 / E K A X / 2 . ( I , J , K - 1 ) - T 3  5  ( H 3 * A E Z ( I , J )  6  ( ( £ « ( ! , 0 , K ) / 2 . ) * P H Y ( I , 0 , K ) ) 30  (B6(K)  GO TN 1  3  (AEX  ( J . I ,  *  ) *  ( 1 , 0 ) ) * ( I 3  1) ) »  TC  -  (I, J , K )  - A B £  (K) '  ) •  ) AHB(K)  )  ) /  + T ( I , J , K )  *  K) - 3 1  ( I . J . K )  ( T 1 ( I , 0 , K ) -  )  -  (K)  AHD  ) +  ) + + T 3 ( I , J . K  + 1 ) ) ) /  • T ( I . J . K )  = ( (AEX  ( J , I , 1 ) / 2 . )  * (T  1 ( 1 - 1 , J , K )  - T 1  ( I . J . K )  )  ( K ) * A E X ( J . I , 1 ) * X 2 / E K A Y / 2 . ) * ( T 1 ( I . J . K ) - A B B ( K ) ) ( I , J , 2 ) / 2 . )  (AEY  ( 1 , 0 , 2 )  IC  (1,0)  *  *  *H6 (  ( T 2 ( I , 0 + 1 , K ) - I 2 ( I , 0 , K )  (K) * Y 4 / B K A Y / 2 . )  T 3 ( I , 0 , K - 1 )  * r z / B K A Y  ) *  ( H 1 (K)  =  ( ( A B X ( J , I ,  » AB E A 6 )  ( A F Y ( I , J , 2 )  ) *  *  *ABEA3*COS  ( T H ) ) *  6  -  (H6  *AEEA3*SIN  (TE)  ( I . J . K )  -  (K)  ( I . J )  ) * (T3  1000  1 ( I - I . J . K ) AHB  ( I , J , K -  (AEE  ) * (T1 1 ) - 2 .  (K)  )  ( K ) - T 2  )  { 1 , 0 , K )  ) •  -  AHB (K)  )  ) /  - T l  ) -  )  ( I , J , K ) - A H B  *T  ( I . J . K )  •  ( I . J . K ) )  «T3 ( I , J , K )  ( £ 4 ( I , J , K ) / 2 . ) * F H Y ( I , 0 , K ) ) TO  T 3 ( I , 0 , K )  ( T 2 ( I , J * 1 , K ) - T 2 ( I , J , K )  (H6(K)  (  )  • (AHB ( K ) - 1 2  • T ( I . J . K )  1) ) * ( T  •  4  (T3  ( 1 1 ( 1 , J . K ) -  5  (AFZ  -  -  •  1000  T N ( I . J . K )  3  /  ( 1 , 0 , K) ) •  )  ( I , J , K )  ( I , J , K - 1 ) - 2 . * T 3 ( I , J , K )  (B3*AEZ  GO  • ) )  +  ( K ) * Y 4 / E K A Y ) * ( A H E ( K ) - T 2 ( I , J . K ) )  ( ( £ 4 ( I , J , K ) / 2 . ) • P B Y ( I . O . K ) )  2  - T l  ( T 2 ( I , J 4 1 , K ) - T 2 ( I , 0 , K )  5  1  )  1000  ( A E Z ( I , 0 ) )  GC  (12  (T 1 ( 1 - 1 , 0 ,  4  336  ) *  ( 1 3 ( I , J , K ) -  1)*X2/BKAY)  ( 1 , 0 , 2 ) * H 6  ( 1 , J , K )  (AFY *  )  ( £ 4 ( I , J , K ) / 2 . ) * P H Y ( I , J , K ) )  (B6  2 8  =(  * A F X ( J , I ,  (ABY  (AFZ  33 5  *EZ/RKAY  ( i I Y ( I , J , 2 )  (  +  1000  T N ( I , J , K )  4  (K)  ( I , J , K * 1 )  ( I , J , K ) )  (EE(K)*AEY  3  K))  1C0C  2  +  ( I , J ,  -  +13  3  2  ) /  • T ( I . J . K )  ( A I X ( J , I , 2 ) / 2 . ) * ( I 1 ( I 4 1 , J , K ) - T 1  1  - T l  ( T 2 ( I , J , K ) - A H E  T N ( I , J , K ) = ( ( A f i X ( J , I , 1 ) / 2 . ) * ( " l ( I - l » 0 , K )  GO  )  • T ( I , 0 , K )  ( T 1 ( I t 1 , J , K ) - T 1 ( I , J , K ) ) +  2  1  •  -  1000  ( I . J . K )  ( A I X ( J , I , 2 ) )  GC  ( T l ( I . J . K ) - A H B ( K ) )  ( T 3 ( I , J , K - 1 ) - T 3 ( 1 , J , K )  4  8  ) -  •  6  334  ) ) /  T ( 1 , J , K )  1000  ( I , J , K )  (H7 ( K ) * A E £ A 5 / 2 . )  333  ) -  •  ( I , J , K - 1 ) - 2 . * T 3 ( I , J . K ) * T 3 ( I , J . K * 1 )  1  2  AHB  ) * (AHB  3  GO  ) /  ( T 2 ( I , J + 1 , K ) - I 2 ( I , J , K ) )  6  IN  )  1000  I N ( I , 0 , K )  331  (K)  T ( I , 0 , K )  (K)  *T3  ( I . J . K )  )'*  ( I , J . K * 1 )  ) )  /  •  337  (I,J,K)  TU 1 2  = ( (AEX ( J . I . 1 )  (B1 ( K ) * A E E A 6 / 2 .  ) *  ( A F Y ( I , J , 2 ) / 2 . )  *  / 2 . ) * ( T 1 ( 1 - 1 , J.K)  (11 ( I . J . K ) -  •  (H6 (K) » A B E A 3 * C 0 S  ( T H ) / 2 .  ) * <*BB ( K ) - T 2  7  -  (H6 (K) * A E E A 3 * S I N  (T E ) / 2 .  ) *  | A B 2 ( I , J ) )  *  (  T3(I,J,K-1)  4  (H3*AB2  ( ( S 4 ( I , J , K ) / 2 . > * P B Y ( I , J , K ) ) GO  TO  ( I , J) » E 2 / B K A Y  = ( (ABX ( J . I ,  ( B 1 (K) *AEX (  (B5 (K)*AEY  4  ( l I 2 ( l , J ) ) * ( T 3 ( I , J , K - 1 ) - 2 .  •  A E Y ( I , J , 1 )  TO  )  *  (  T 2 (I.J-1,K)-T2  ( I , J , 1)*Y3/BKAY)  = ( ( A E X (J, 1 , 1 ) / 2 . )  (BI ( K ) * A E X ( J . I ,  )  )/  - T 1( I , J . K )  *  -  )  AHE(K)  (I.J.K)  )  -  ) •  -  (T2 ( I , J . K ) - A H B  (K)  • T 3 ( I , J , K ) + T 3 ( I , J , K  )  •  + 1 ) ) ) /  +T ( I . J . K )  * (T 1 ( 1 - 1 , J . K ) - T 1 ( 1 , J . K )  1) * X 1 / B K A Y / 2 .  ) *  (T 1  (I,J.K)  ( A B Y ( I , J , 1 ) / 2 . ) * C r 2 ( I , J - 1 , K ) - T 2 ( I , J , K )  3  ( B 5 (K) * A E Y  4  (AEZ(I.J))  5  (B3*ABZ  6  ( ( S 4 ( I , J , K ) / 2 . ) * P E Y ( I . J . K ) ) GO  ) +  -  A H B (K)  2  1000  )  1000  ( I , J . K )  TH 1  -  )  AHE(K)  -  T ( I . J . K )  * ( T l (I.J.K)  ( ( S 4 ( I , J , K ) / 2 . ) * F B Y ( I , J . K ) )  339  ( I , J . K )  J , K)  1 ) ) *(T 1 ( 1 - 1 . J . K )  ( J . J . 1) * X 1 / B K A Y )  3  GO  (I ,  ( T 3 ( I , J , K )  2  5  ) -  •  1000  T H ( I , J , K ) 1  ) »  (11  - T 3 ( I , J , K )  5  338  )  ( T 2 ( I , J * 1 , K ) - T 2 ( I , J , K ) )  6  3  - T l (I.J.K)  A E B (K)  10  ( I , J , 1) * Y 3 / E K A Y / 2 . *  )* (12 (I.J.K)  ( T 3 ( I , J , K - 1 ) - T 3 ( I , J , K )  ( 1 , J) * E 2 / E K A Y  ) *  )  ( T 3 ( I , J , K ) •  )  -  ) -  )  AHB(K)  -  A H B (K)  )  •  -  - A H B (K)  ) ) /  T ( l . J . K )  1000  COKTINOE BEIURN ENE  C C C  S D E E C C T I N E  C  SIBUI1ANEOUS  C  C C I E I J C I E N T  C  FOB  SDEEO01INE  C  C  CCHPOIE  EEIA  A  CF  IINEAB  T B I E l AGON A L  TBIDAG  ( I F . L . A . B . C . D . V )  I N I E F H E D I  ITE  AERBAYS  OF  BETA  , BETA  AND  ( 1 0 1 ) , G AHH A ( 1 0 1 )  GAHHA  ( I F )=E ( I E )  I F H  = 1  I F  ( I E ) / B E T A  ( I F )  +1  I = I F P 1 , L  BET A (I) 1  SYSTEH  HAVING  A ( 1 ) , E ( 1 ) , C ( 1 ) , D ( 1 ) , V ( 1 )  GABE A ( I f ) = D  DO  A  HATI! I X  DIEENSICN  C  SOLVING  EQUATIONS  *  G A E E A ( l )  E ( I ) =  - A (3) * C ( 3 - 1 ) / B E T A ( I - 1 )  (E ( I ) - A (I)*GAEEA  ( 1 - 1 ) ) / E E T A ( I )  C COBECTE  C  C  SCLDTICN  V ( I )  =G AHB A ( L )  LAST  =  DO  K  I 2  F I N A L  2 =  L  CF  VECTCE  V  - I F  = 1 , L A S T  L-K  V (I)=GAMHA  (I)  -  C ( I )  *  V (1+1)/BETA  (I)  BETUIN  C  ENE  C  SUEBCDTINE  TO  C  AKE  TEBBS  VCIUEE  C  D I S C B E T I Z A T I O N  C  IHEOUGH  C  T B I  CALCULATE  OF  EATA  IOB THE  THE  T E E  DIFFEBEKT  ABE  ELEMENTS  CASTING  STATEMENTS  I S  ALEEADY  BUILT  I N  •  210  SUEFODTINE  AEEVCL  DIBENSION  ARX  E I E I N S I C N  XY  (10 ,2)  ( 16,1  1,2)  DIBENSION  XX  ( 1 6 ) , Y Y  , A E Y ( 1 1 , 1 6 , 2 )  ,YX  (  A B Z ( 1 1 , 1 6 )  (16,2)  ( 1 0 ) , Z Z  ( 1 6 , 1 0 )  C C E E C N / C l / C X , t Y , E Z , C T , B K A Y C O E E O N / C 2 0 / X X , Y Y , Z Z C O E E C N / C 1 7 / A R X , A R Y , A R Z COBBON/C2/S4  ( 1 0 , 1 6 , 9 1 )  CCEECN/C3/NTYPE  ( 1 0 , 1 6 , 9 1 ) , L F S ( 1 0 , 1 6 , 9 1 )  COEECN/C4/L,H,N C C E E C N / C 1 8 / X 1 , Y 1 , X 2 , Y 2 , Y 3 , Y 4 , T H C O E E C N / C 1 9 / A R I A 1 , A B DATA EAT A 1  E A 2 , A B E A 3 , A F E A 4 , A R E A 5 , A B E A 6  X Y / 0 . , 1 . 9 7 5 , 8 * 1 . 4 2 5 , 2 * 1 . 9 7 5 , 7 * 1 . 4 2 5 , 0 . / Y X / 0 . ,  1 . 7 5 , 2 . , 2 . 0 5 , 9 * 1 . 7 5 , 3 * 1 .  1 9 , 2 * 1 .  7 5 , 2 . , 2 . 0 5 ,  9 * 1 . 7 5 , 2 * 1 . 1 9 , 0 . / T H - . 8 8  74 1  L L = L - 1 flfl=E-1 NN=N-1 C  C A I C O I A T I N G  C  G F I C  THE  COCEEINAT  ES  OF  THE  X  AND  Y  EOINTS  X X ( 1 ) = 0 . DO  1  J = 2 , B  XX(J) = 1  YY  (1)=0.  CO YY 2  1400 1300  1600 1500  Y X ( J - 1 , 2 ) + Y X ( J , 1 ) + X X ( J - 1 )  CONTINUE  2  I = 2,L  <I) = X Y ( 1 - 1 , 2 )  DC  1300  1 = 1 , L L  DO  14C0  J = 1 , «  AEX  1700  ( J , I , 2 ) =  190C 4  1500  1=2,L  CO  1600  0 = 1 , B  ABX  +YY(I-1)  ( J ,  1)  +YX  ( 0 , 2 ) )  * C Z / ( X Y ( I , 2 )  +XY (1+  1,1))  ( 0 , I , 1 ) = A B X  (0,1-1,2)  C O M I N U E CONTINUE DC  1700  CC  1800  J = 1 , B B 1=1,L ( X Y ( I , 1 )  + X Y ( 1 , 2 ) ) * D Z / ( Y X  CONTINUE C O M I N U E CC  19C0  0=2,M  DO  2000  1 = 1 , L  ( I , J , 1 ) = A E Y ( I , J - 1 , 2 )  CON1INUE CONTINUE  ABX  2100  1=6,LL  ( 4 , 1 , 2 ) = A B X ( 4 , 1 , 2 ) / 2 .  C C M I N U E CC AFX  2200  1)  C O M I N U E  DC  2100  (YX  DO  ABY 2000  ( I ,  CONTINUE  A B Y ( 1 , 0 , 2 ) = 180C  +XY  CONTINUE  2200 (4,1,  1=7,L 1)=ABX  ( 4 , 1 - 1 , 2 )  CONTINUE ABY  ( 6 , 4 , 2 )  = ABY  ( 6 , 4 ,  2)  / 2 .  A B Y ( 6 , 5 , 1 ) = A B Y ( 6 , 5 , 1 ) / 2 . A R ! M = Y X AB E A 4 = X Y  (4,2)  »EZ/RKAY  ( 6 , 2 ) * D Z / R K A Y  ( J , 2 ) • Y X  (J+  1,1))  211  AEEA5=YX  (5,  1)*CZ/RKAY  A B 1 A 2 = Y X ( 5 , 2 ) • D Z / E K A Y / S I K A B 1 1 3 = Y X ( 9 , 1 ) * D Z / B K A Y / S I ABIA6=YX  ( 9 , 2)  (7,2)  + XY  (1,1)  (8,1)  Y 1 = I X ( B - 1 , 2 ) + Y X Y2=YX  (TH)  *EZ/BKAY  X1 =X Y ( L - 1 , 2 ) + X Y X 2 = XY  (TH) *  (1,2)+YX  (fi,1)  (2,  1)  Y4 = Y X ( 7 , 2 ) + Y X ( 8 , 1 ) Y3=YX  2300  CO  2400  ABZ 2400  ( 3 , 2 ) • Y X  DO  (4,1)  y  .  1=1,L 0=1,fl  ( I , J)  = (XI  ( 1 ,  1) + X Y ( 1 , 2 )  ) * (YX  ( J ,  1) + Y X ( 0 , 2 )  )  CONTINUE  2300  CONTINUE DO  2500  ABZ 2500  1 = 7 , L  (1,1)/2.  ( I , 4 ) = A B Z  CCS1INUE  1  AEZ  ( 6 , 1 ) = A B Z ( 6 , 4 ) - ( Y X ( 4 , 2 ) * X Y  AEZ  (6,5)  = ( X Y (6 ,  XY  (6,1)  • XY ( 6 , 2 )  ABZ  (7,6)  = (YX  (6,  1) + Y X  ( 6 , 2 )  )*  (XY  (7 ,  1) +XY  ABZ  (8,7)  = (YX  (7,  1)  (7,2)  ) * (XY  (8,  1)  AEZ  (9,8)  = (YX  ( 8 ,  1) + Y X  ) * (XY  ( 9 ,  AEZ  (1C.9)  2700  (5,  ( 6 , 2 ) )  1) ) • ( Y X ( 6 , 2 ) / 2 . ) *  2600  1=1,1  CC  270C  J = 1 , H  (XY  (6,1)  +  )  +YX  (6  ,2)  (7,2)  +XY ( 6 , 2 )  1) + X Y ( 9 , 2 )  = ( Y X ( 9 , 1 ) + Y X ( 9 , 2 ) ) * X Y ( 1 0 , 1 ) - ( 0 .  DO  ABZ  1) * Y X  ) / 2 . ) )  /2. / 2 .  5 * X Y ( 1 0 , 1 )  * Y X ( 9 ,  1) )  (1,0)=AEZ(1,0)/IZ  CONTINUE  2600  CCFTINOE CO  11C0  1=1,1  DC  1100  J = 1 , H  DO  11C0  K=2,NN  S4 1100  < I , 0 , K ) = ABZ  ( I , J )  * E Z * C Z » 2 . / ( D T * B K A Y )  CONTINUE DC  1200  1=1,L  DO  1200  0=1,B  S4 1200  (1,0,N)=S4  (1,0,NN)/2.  CONTINOE F.ETDEK END  C C C SOEBCCTINE  C  10  PEIKT  THE  BATEIX  Of  TEBPEBATOEES  C C  c SOEEOCTINE D I E I N S I C N  OU1PUT(1) T  (10,  1 6 , S 1)  C C E E O N / C 4 / I , H , N C C F E C N / C 1 / C X , C Y , C Z , E T , E K A Y COEH0N/C13/TAU C C E E C N / C 1 0 / T L I C , T S C I , D E N S , D E N 1 H B I 1 E ( 6 , 1 0 0 ) I A D 100  FOFEAT 1  F 8 .  ('  1 ' ,  ZT= EZ*F1C HBIIE 145  1 0 X , ' T E B P E B A T U B E S  1 , 1 X , ' S E C O N D S ' AT  ( 6 , 1 4  FOFEA1  AT  TEE  END  OF  T I M i ' ,  )  ( N - 1)  5)ZT  ( 1 0 X , • S I Z E  OF  THE  INGCT  I N  Z  C I E E C T I O N • , F 1 0 . 2 , • C B S  •)  150  FO E E AT(5 X , *  K = • , 2X  , 1 3 )  B B I 1 E ( 6 , 1 8 5 ) 185  FOEEAT  ( 2 X , ' C E N T R E  S B I T I ( 6 , 1 8 0 )  (T  I I N E  ( 1 , 9 , K )  1 E E EE £ ATU F E S  •)  , K = 1 , N )  W E I 1 E ( 6 , 1 9 0 ) 190  P O I E A T ( 1 X , / )  180  FOEEAT K1  ( 1 X , 1 1 ( F 1 0 . 1 , 1 X )  WBI1E EC  ( 6 , 1 5 0 ) K  210  1  J = 1 , H  K B I 1 E ( 6 , 1 8 0 ) 210  )  = ( N / 2 ) * 1  (1  ( I , J . M )  , 1 = 1 , 1 )  C O M I N D E W B I 1 E ( 6 , 1 9 0 ) R2=F CO  1C9  K=1,N  KK=K-K+1 I F < l ( 1 , 9 , N K ) . L E . I S O L ) K 2 = N K 109  C C M 1 N 0 E I F  ( K 2 . E C . N ) K 2 = K2-1  K3=K2+1 CC  3C0  K=K2,K3  W R I T E ( 6 , 1 5 0 ) K DO  31C  WR H E 310  J = 1 , H  ( 6 ,  180)  (T  ( 3 , J . K )  , 1 = 1 ,  L)  CON1INEE  300  C O M I N U E BEICRN Et<T  c C C  HEAT  C  VABICOS  1 E A N S E E 5  C  D E S C E I E I N G  C  THE  C  BEES  C O E E F I C I E N 1  S U E E C U T I N E S ETC  S U F F A C E S I I  CF  FOE  A l THE  HI  I K E  S U B E C U 1 I N E S . T H E  THBCOGH VARIOUS  C A S T I N G .  S Y B K E 1 E I C A I  b7  ABE  FCR1IOK  THEY  ALL  FOB CF HAVE  CCC1ING  C C  c FUNC1ION  H1(K)  C C K E O N / C 1 / E X , C Y . E Z . D T . B K A Y I F  ( K . G 1 . 1 3 )  GO  TC  10  H 1 = . 5 BE10RN 10  H 1 = 0 . 3 4 1 7 8 * E X P  (-(  ( ( K - 1 ) * D Z -  ( D Z / 2 . ) ) * . 0 2 4 3 5 ) )  EE1CEN E N I F D S C 1 1 0 H  H2(K)  C C E E C N / C 1 / E X , C Y . E Z . D T . R K A Y I F ( K . G 1 . 1 3 ) G O  TO  10  H2 = . 5 BEICEN 10  H2 = 0 . 3 4  178*EXP  (-(  ( (K-  1 ) * D Z -  ( D Z / 2 . ) ) * . 0 2 4 3 5 ) )  EE1CEN INC FUNCTION  H4  (K)  C O E E O N / C 1 / D X , t Y , E Z , D T , B K A Y I f  ( K . G T .  13)  GO  TC  10  H 4 = . 5 BET 10  CBN  H4 = 0 . 3 4 1 7 6 * E X P ( - ( ( ( K - 1 ) * D Z -  ( D Z / 2 . ) ) * . 0 2 4 3 5 )  )  BETUBN EN C F U K C T I C N  H5  (K)  C O E B O N / C 1 / E I , E I , E Z , D T , B K A Y IF  ( K . G T . 1 3 ) G O  TO  10  H5=.5 EE1UFN 10  H 5 = C . 3 4 1 7 6 * E X P ( - ( ( ( K - 1 ) * D Z -  ( D Z / 2 . ) ) * . 0 2 4 3 5 )  )  EETOEN EN E FU KCTICN  H6  (K)  C C B E C N / C 1 / E X , E X , C Z , E I , E K A Y I F  ( K . G T .  13)  GO  TO  10  '  H6=.5 EETOEN 10  H 6 = C . 3 4 1 7 8 * E X P ( - ( ( ( K - 1 ) * D Z EETOEN  ( D Z / 2 . ) ) * . 0 2 4 3 b ) ) '  ENE F U f C T I C N  H7  (K)  C O B E C N / C 1 / D X , E * , E Z , D T , B K A * I F  ( K . G T .  13)  GO  TO  10  B7 = . 5 EETOEN 10  H 7 = C . 3 4  178*EXP  (-(  { (K-1)  * E Z -  EEIDEN ENE C C  c FUNCTION  AHB(K)  ABE=5. BETDFN  ENE C C  I M T I A I I S A T I C N  E O U T I N I  C C SUEFOOTINE E I B E NS I O N  I N I T I A T  ( 1 0 ,  C O B B C K / C 4 / L , H , N I t = I - 1 H B = fi- 1 KN=N-1 DO  100  1 = 1 , L L  CO  100  J=1,HM  DO  1C0  K=2,NN  T ( I , J , K ) = T 1 100  CONTINUE DC  150  DO  150  T  ( I , J ,  1 = 1 , L L J = 1 , H B 1)  T ( I , J , N ) 150  T1  =  T1  CCKTINUE EC  160  K=2,NN  EC  160  J = 1 , B B  1 ( 1 . J , K ) 160  =  =  T1  C O M I N U E EO  170  1 = 1 , L L  EC  170  K=2,KN  T ( I , B , K )  =  T2  16,  (T , T 1 9 1)  , T2)  ( D Z / 2 . ) ) * . 0 2 4 3 5 ) )  170  C O I I I N U E DO  190  J = 1 , H H  T ( I , J ,  1)=T1  I ( I , J , N ) = T 1 190  C O I T I F O E DO  210  1 = 1 , L L  T ( I , E , 1 )  =  T ( I , H , N ) 210  T2  =  T2  CONTINUE DO  220  K=2,NN  T ( 1 , f l , K ) T ( L ,  =  T ( I , E , K ) 220  T 2  1,K)=T1 =  T2  COBTINOE X ( I . 1 , 1 ) = I 1 T ( L , H ,  1)  =T2  T ( 1 , B , 1 )  =T2  T ( L , 1 , N ) = T 1 I ( I , f l . N ) = T 2 T{  1 , H , N ) = T 2  EETUFK ENE C  c c c SOEBCOTINE  LATHET  (L  1 , L 2 , B 1 , B 2 , K 1 )  C C  SUBRCUTINE  C  AT  T E E  C  THIS  C  EE1NC  TO  E E L E AS E  E I F F E E E N T  ALSO  THE  CHARACTERIZES  L I Q U I D ,  LATENT  HEAT  OF  S O L I D I F 1 C A H O N  NCEES  BUSHY  TEE  CE  PHYSICAL  S C U D  STATE  OF  EACH  NODE  EEG10N  C DIBENSION  T  D I E E N S I C N  T 3 ( 1 0 , 1 6 , 9 1 )  ( 1 0 ,  C O E B O N / C 7 / T , T  16,9  , T 1  ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 ,  ,TN  (1C,  16,91)  1 6 , 9 1 )  1 , T 2 , T 2 , T N  COBEON/C2/NTYPE CCEHCN/C8/PHY  1)  ( 1 0 , 1 6 , 9 1 ) , L E S ( 1 0 , 1 6 , 9 1 )  ( 10,  16,91)  C O B E C N / C 9 / P E Y B , P H Y S , P H Y I C C E E C N / C 1 0 / T L I C , 1 S O L , D E N S , C E N I C O B B O N / C U / L , H , N DO  1000  I = L 1 , L 2  CO  10C0  J = B 1 , H 2  DO  1000  K=2,K1  LNE GC  = TO  L P S  ( I , 0 , K )  ( 1 0 , 2 0 ,  1 0 0 0 ) , L N D  C HCDE  C  I N I T I A L L Y  IN  L I C U I E  C 10  I F  (TN ( I , 3 , K ) . G E . T I I C )  GO  TO  1000  IN  BUSfcY  C C  CHANGE  CE  STATE  C C  DCES  F I N A l  I E B P  ENE  DP  ZONE?  C I F  ( ( I L I C - T N  ( I , J , K ) ) * P H Y I -  ( T L I C - I S C L ) * P H Y M . G l . 0 . ) G O  C C  BODE  ENDS  DP  IN  HOSEY  BEGJCN  C T N ( I , J , K ) =  T L I C -  ( I L I Q - T N  ( I , J , K )  )*EHYL/PHYfl  TO  215  LPS  ( I , J , K )  GO  c  10  RODE  C  =2  1000  HOVES  INTO  SOLID  C 10  TN ( I , J , K ) = T S C L - ( 1  * EEYB LPS  ) /  <I,0,K)  GC  TO  ( T L I Q - T N < I , 0 , K ) ) * E H Y L - ( T L I C - T S O L )  EHYS =  3  1000  C  / HCDI  C  I K I T I A L L Y  IK  THE  BUSHY  EEGION  C  20  I P (  TK ( I , 0 , K ) . G I . T S C L )  GC T C 1 0 0 0  C NODE  C  BOVES  INTO  SCLID  C TK  ( I , J , K ) = T S O L -  IES 1000  (TSOL-TK  ( 1 , 0 , K ) )  *FHYH/PHYS  ( I , 0 , K ) = 3  CONTINUE FETUFN EN I  C C  c SUEECOTIRI  OUTINT  ( I I , N N )  C C  SUEBCOTIKE  C  AEEAY  TO  PEINT  CONTAINING  CUT  THE  I NT E G E E  T KB EE  DIBENSIONAL  NUHEEES  C DIBFNSICN  I I  ( 1 0 , 1 6 , S 1)  C O B K O N / C 4 / L , B , N EC  1000  S B I T E 900  K=1,N,NN  ( 6 , 9 0 0 ) K  F O E B A T ( 5 X , • K = » , 2 X , 1 4 / / ) DO  6C0  0 = 1 , B  H B I T I ( 6 , 8 5 0 ) £50  FOEBAT  ( I I ( I , J , K )  ( 5 X , 11 ( J K , 2 X )  800  C C M I N U E  1000  CONTINUE  , 1 = 1 , L )  )  EE1UBN  ENE  C C FUNCTION  CP(T)  C C  FOKTICN  BCUTINE  C  C i  AT  C  ONITS  ZINC CF  ANY  CALCULATES E A E T I C U L A E  THI  S P E C I F I C  TEHEEEATUEE  CAL/G.C  C T K = I + 2 7 3 . CP  =  0 . 0 8 1 8 4  + 0 . 0 3 6 7  * 1 . E - 0 3 * T K  EE1UFN  EN C  C C SOEFCOTINE  PHYPEP  C C  C A L C O I A T I S  C  S P E C I F I C  C  STCBES  I T  THE  HEAT IN  PBCEUCT FOE  THE  A l l A B B AY  OF THE PKY  DENSITY NODES  AND  AND  IN  HEAT  •  •  DIEENSION  T ( 10,  DIBENSION  1 6 , S 1) , 1 1 ( 1 0 , 1 6 , 9  1)  , 7 2 ( 1 0 , 1 6 , 9 1 )  1 3 ( 1 0 , 1 6 , 9 1 ) , 1 K ( 1 0 , 1 6 , 9 1 )  C O B B C N / C 7 / l , T 1 , T 2 , T 3 , T N C O B B C N / C 4 / L , f l , N C O B B O N / C 3 / N T Y P E ( 1 0 , 1 6 , 9 1 ) , C C E E C N / C 8 / P H Y ( 1 0 , 1 6 , 9  L F S ( 1 0 , 1 6 , 9 1 )  1)  C O B B O N / C 9 / F E Y B , P H Y S , P H Y I C C F . E C N / C 1 0 / T L I C , T S O L , D I N S , D E N I DO  1CC0  DO  1000  EC  10C0  INC  =  GO  1 = 1 , 1 J = 1 , H K=2,N  1 E S ( I , J , K )  10  ( 1 0 , 2 0 , 3 0 ) , I . ND  C BCEES  C  AEOVE  LIQUIDUS  C 10  PHY  (I , J ,  GO  10  K)=FHYL  1000  C SCEES  C  IN  THE  BUSHY  EEGION  C 20  EHY GO  ( I , J , K ) = E H Y H 10  1000  C NODES  C  IN  THE  SOLID  REGICK  C 30 1000  P H Y ( I , J , K ) = C E  (I  ( I , J , K )  )*EENS  COK1INUI RETURN END  CC  c C  SDEECUTINE C  S0EBCU1INE  C  BESUI1ING  C  THE  Z  AICNCE  FOE IBOH  ADDING THE  D I R E C T I O N .  ONEXTEA  GEOWTH  ALSC  OF  SOBE  SET I E E  CF  KODES  INGOI  IN  I N I T I A L I S A T I O N  C  c DIBENSION  T ( 1 0 , 1 6 , 9 1 )  DIHENSION  1 3 ( 1 0 , 1 6 , 9 1 )  , 1 1 ( 1 0 ,  16,91)  , 1N ( 1 0 ,  , 1 2 ( 1 0 , 1 6 , 9 1 )  1 6 , 9 1)  C O E B O N / C 7 / 1 , ! 1 , 1 2 , 1 3 , T N C O E r C K / C V L . B j N COBHON/C3/N1YPE  ( 1 0 , 1 6 , 9 1 ) , L F S  C O E E C R / C 5 / N C B , T E B P  ( 1 0 ,  16,91)  1  C O B B O N / C 2 / S 4 ( 1 0 , 1 6 , 9 1 )  C C  HAKE  C  AS  C  I B I S  THE  T E E  TOP  S L I C E  SECOND  1 I H I  S L I C E  IN  THE  EEOH  EEEVICOS I E E  STEP  C CO  10  CO  10  K=1,N J = 1 , B  CO  10  1= 1  ,L  SC=»*2-K  10  T ( I , J , N C ) = 1  ( I , J . N C - 1 )  L F S ( I , J , N C )  =  CONTINUE  KN=*-1  LES  ( I , J ,  NC-1)  TOP  I N T E S V A L  FOE  21 7  DO  15  DO  15  CC  15  K=1,2 1=1,  L  J = 1 , B  KC=K+2-K SU  ( 1 , 0 , N C ) = S 4 ( I , J , K C - 1  H T X t E ( I , 0 , K C ) = N T Y P E 15  )  ( 1 , J , N C -  1)  CCKTINUE  C C  I N I T I A L I S A T I O N  OF  THE  NEWLY  ADDED  S L I C E  C  < • CO  20  1 = 1 , L  DC  20  J = 1 , B  1(1,J,1) LES 20  =1(1,0,2)  ( 1 , 0 , 1 )  = L P S ( I . J , 2 )  C0N1INUE N= K • 1 B E 1 CBN ENE  C C  C O F Y I K G  CE  T E E  C  C A S 1 I K G  IN  EINABY  C  ST A F T I K G  THE  1 EM F E B A 1 U B E FOE  F I E L D  IN  SUESECUENT  1HE  USE  IK  PBOGEAH  C SUEFCUT1NE  F I L  DIMENSION  T  D I B F N S I C N  EIN(NI)  ( 1 0 , 1 6 , 9 1 ) , 1 1 ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 , 1 6 , 9 1 )  T 3 ( 1 0 , 1 6 , S 1 ) , 1 N ( 1 0 , 1 6 , 9 1 )  C O B B G N / C 7 / T , T l , T 2 , I 3 , T N COI!EON/'C«/L,M,^• CO!!EON/C3/NTYPE CO  1C  K=1,N  DC  10  J = 1 , B  KBIIE(NI) 10  ( 1 0 , 1 6 , 9 1 )  , L P S ( 1 0 ,  16,91)  (T(I,0,K) ,1=1, 1 ) , ( 1 F S ( I , 0 , K )  ,I=1,L)  C 0 K 1 I N U E BE1CFN ENE  C  c c S U E F C 0 T 1 N E  GBAPH  C C  T H I S  C  1BE  S D E E C U T I N E T I B P E E A T U R E  F10TS  T E E  CCNIOUE  P B O E I L E S  OF  E 1 E L E  C DIBENSION  T  C I E E K S I O N  XX (  ( 1 0 ,  1 6 , y i )  , T 1  ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 , 1 6 , 9 1 )  DIHENSION  1 3 ( 1 0 , 1 6 , 9 1 ) , 1 N ( 1 0 ,  C I B E N S I O N  ZP1  1 6 ) , Y Y ( 1 0 ) , Z 2 ( 1 6 , 1 0 )  ( 1 C 1 ,  10)  , Z E 2  (101  16,91) ,16)  , X E  (101)  , Y E 1  (11)  , Y P 2 ( 1 6 )  C O R E C N / C 1 / E X , I Y , E Z , E T , E K A Y C C B B O N / C 7 / T , I 1 , T 2 , T 3 , T N C O B B O N / C 2 0 / X X , Y Y , Z Z C O B B C N / C 1 ( » / S X , S Y 1 , S Y 2 , D X X , D Y Y 1 , D Y Y 2 , D X X F , D Y Y 1 P , D Y Y 2 P C C E E C N / C 4 / L , B , N C C B R C N / C 1 0 / T L I Q , T S O L , D E N S , D E N L C O B E C N / C 1 1 / C P L , C S P , N U B E U N , E I H T DO  10  1=1,L  DC  10  0 = 1 , N  Z E 1 (0,1) 10  =  COKTINOE DO  20  1=1,H  DC  20  0 = 1 , N  1(1,9,0)  20 C C C  ZE2 < J , I ) = CONTINUE  T(1,I,J)  SCALIKG THE X AXIS X AN I Y AXES )  (COBEESFONDS TO ORIGINAL  C  15 16  DYY1P=4. DYY2F=«. SY 1 = YY( 1 0 ) / E Y Y 1 E SY2 = XX ( 1 6 ) / C Y Y 2 E CO 15 I = 1 , L YF1 ( I ) = YY ( I ) / E Y Y IP CONTINOE DC 16 J = 1 , f l YP2 (J) =XX ( J ) / D Y Y 2 P CCMINOE SCALING THE X AXIS  40  (ORIGINALLY  2 AXIS)  DXX=CZ/4. SX=rXX*FLOAT ( N - 1 ) DXXI=U. XP(1) = 0 . NX = N - 1 DO 4C 1 = 1 , N X XP ( I + l ) = XP ( I ) • CXX CONTINUE CCNTCl'E OETAINEE EEOK SECTION FER PE KCICUL AE TC Y AXIS XZ ELANE C A I I F L C T E I ( ' X S I Z E • , 128) CAII ELC1EL('YS1ZE ,66) C A I I FFAME1 C A I I CNTCUB ( X F , N , Y P 1 , L , Z E 1, 10 1, T L I C , 3 . , T L I C ) C A I I CNTCOF. ( X P , N , YP 1 , 1 , Z P1 , 1 01 , T S C L , 3 . , T S O L ) XB=SX+4. CALL PLOT ( X f l , 0 . , - 3 ) ,  CCNTCOB CETAINEE FROE SECTION PEB PE KCICULAB TC X AXIS YZ PLANE C A I I FFABE2 CN^TLIQ C A I I C N T O U B ( X P , N , Y P 2 , B , Z P 2 , 10 1 , T L I C . 3 . . 1 L I Q ) CALL CNTCUB (XP,N , Y P 2 , B , Z E 2 , 10 1 , T S C I , 3 . , T S C L ) XB=SX+4. CAII ELCT(XB,0.,-3) C A I I XYFLAN C A I I CtJTLNE XH 1 = SY2 + 4 . CAII ELCT(XM1,0.,-3) BETCRN EN C  SUEFCDTINI OUTPT2 CCrKON/C1C/TLIC,TSCl,D£KS,DEKI COEECN/C1/EX,CY,EZ,ET,BKAY  219  C 0 H E C N / C 4 / L , B , N C 0 E E C N / C 1 1 / C P L , C S P , N D H B U N , B L B T COEEON/C6/B3 C 0 E E C N / C 1 2 / B I N P T ( 2 0 ) , C P I ( 2 0 ) W B I T E ( 6 , 1 0 ) 10  F C E E AT ( *  (RINFT  S E I T E ( 6 , 2 0 ) 20  FOIEAT  ( 5 X , ' T H E B B O  B B I T E FOBHAT  •  •  /  , 2 X , I 4 / / / )  PHYSICAL  ( 1 0 X , ' L I Q O I D O S  ( 6 , 5 0 )  F 0 F E A 1 ( BB H E  6 0  •  P E C P E B T I E S ' / )  ( 6 , 4 0 ) T L I C  B B I T E 50  :  NO  ( 6 , 3 0 )  FOEEAT  40  , 1 = 1 , 2 0 )  NOHBUN  ( 5 5 X , • E U N  B B I T E 30  (I)  1 * , 2 6 X , 2 0 4 4 / / )  T E E I E E A T U B E  =•,  I X , F 7 . 1 , 2 X  , ' D E G  C )  7SCL  1 C X , ' S O L I D U S  ( 6 , 6 0 )  TEHFEB  ATOEE  = • , 1 X , F 7 . 1 , • D E G  C )  DENI  F O E B 4 T ( 1 0 X , • C E N  SITY  OE  T E E  L I Q U I D  ( 1 0 X , ' D E N S I T Y  OF  THE  SOLID  = • , 1 X , F 5 . 1 , 1 X , • G / C H 3 • )  B B I T I ( 6 , 7 0 ) C E N S 70  FOFBAT  ='  ,1X  , F 5 .  1,•G/CM3')  B B I T E ( 6 , 8 0 ) C E L 80  F O F B A T ( 1 0 X , ' S P E C I F I C B B I T E ( 6 , S 0 )  90  FOEBST  (CPF  FOFBAT  HEAT  BEAT  ( 1 0 X , ' T H E B B A L  «CAL/ B B I T E  130  THE  L I Q U I D  OF  THE  SOLID  = ' , I X , F 5 . 2 , • C A L / G B • )  ' , 1 X , 2 0 ( A 4 ) /  )  CF  S O L I D I F I C A T I O N  =  • , F 6 . 1 , ' C A L / G f l ' )  ( 6 , 1 2 0 ) B K A Y  FCEBAT 1  OF  1,20)  BLET  ( I O X , ' L A T E N T  NBITE 120  HEAT  ,1=  ( I O X , ' S P E C I F I C  B £ I T E ( 6 , 1 0 0 ) 100  (I)  ( 6 ,  F C F E A T  OF  THE  L I Q U I D  OF  THE  S O L I D  = ' , F 5 . 2 ,  130)BKAY  ( 1 0 X , ' I H E B H A i  I ' C I I /  C O N I U C I 1 V I T I  C B . D E G . C . S E C ' )  CONDUCTIVITY  = ' , F 5 . 2 ,  C B . C E G . C . S E C . • )  B E I 1 E ( 6 , 1 4 0 ) 140  F O B B A T ( 1 0 X , / / / / , 5 X , ' C A S T I N G  CONDITIONS  * , F 5 . 2 , ' C H S / S E C )  B E I T I ( 6 , 1 5 0 ) C S P 150  F O F E A I  ( I O X , ' C A S T I N G  BBITE 160  S P E E D  = ' , F 5 . 2 , ' C B S / S E C • )  ( 6 , 1 6 0 )  F O E E A T ( I O X , ' H E A T  TBANSFEE  C O E F F I C I E N T S  U S E D ' )  H E I T E ( 6 , 1 7 0 ) H 3 170  EOEBAT  ( 1 5 X , ' B C 1 T C B  HEAT  1 E A K S F E B  C O E F F I C I E N T  ' , 2 X , F 1 0 . 4 )  EETUEN ENE C C C SOEEOUTINE  FBABE1  C O E B C N / C 1 4 / S X , S Y 1 , S Y 2 , C X X , D Y Y 1 , D Y Y 2 , D X X P , D Y I I P  ,DYY2P  C O E E C N / C 1 1 / C P I , C S P , N D H F 0 l i , E L H T C C E B C N / C 1 3 / T C A L L  AU  F L C T E L ( ' B E T E I C ' , 1 )  C A I I  A X C I F L  C A I I  A X C I B L  C A L L  AX E L C T  (•DIST  ALONG  C A I I  AXCTFL  (• S I D E '  ,  C A L L  AXPLOT  ( ' D I S T  ALONG  C A I I  ( ' S I D E '  , - 1 )  ( ' D I G I T S ' , 1 ) Z - A X I S  (CHS)  X - A X I S  (CBS)  ; », 0 .  , S i , 0.  ,  DXXP)  1) ; ' , 9 0 . , S Y 1 , 0 . , D Y Y 1 P )  E L C T ( 0 . , 0 . , 3 )  C A I I  A X C T F L ( ' I O E I G I N ' , S Y 1 )  C A L L  A X C T R L ( ' S I D E ' , 1 )  C A I I  AX E L C T ( ' D I S T  C A I I  FLCT  ( 0 . , 0 .  ALONG  Z - A X I S  ,3)  C A L L  A X C T E L ( • Y O B I G I N ' , 0 . )  C A I I  A X C T E L ( ' X O E I G I N ' , S X )  (CBS)  ; '  , 0 . , S X , 0 . , D X X P )  C A I I C A I I  A X C T E L ( ' S I E E ' , - 1 ) A X F L C I  (• D I S T  ALONG  X - A X I S  C A I I  A X C T E I i ' X C E l G I N V O . )  C A I I  A X C I B I . (  ,  (CES)  ; • ,90 .  ,SY  1 , 0 .  , DY X 1 P )  I O B I G I N ' , 0 . )  FLCAT=NOflEON C A I I  SYBECL  ( 1 . , 1 . , 0 . 4 , ' E U N  C A I I  NUBEEE  ( 1 . , 2.  C A I I  SYBEOL  ( 1 . 5 ,  C A I I  NUBEEE  ( 1 . 5 , 3 . 0  5 , 0 .  ' , 9 0 . , 4 )  4 , E L O A T , 9 0 . , -  1 . 0 , 0 . 4 , , 0 . 4  ,  1)  T I E E = * , S 0 . , 5 )  , T A U , 9 0 . , 0 )  EETOEN EN I  S 0 E E O 0 T I N E  IBABE2  C C E E C N / C 1 4 / S X , S Y 1 » S Y 2 , D X X , D Y Y 1 , D Y Y 2 , D X X P , D Y Y J P , D Y Y 2 P C O E B C N / C 1 1 / C F L , C S P , K U B E U K , E I H T CCEBON/C13/T  AO  CAIL  E L C T E L (  B E T B I C  •,  C A I I  A X C T E L (  S I t E '  - 1)  CALL  A X C T E L (  D I G I T S  C A I I  A X F L C T (  DIST  ALONG  S H E '  ,  ALONG  C A I I  AXCTEL  (  CALL  A X P L C T (  DIST  C A I I  F L O T ( 0 .  0 . , 3 )  C A I I CALL  A X C T E L ( AXCTEL  (  C A I I  AX E L C T  (  C A I I  ELCT  (0.  ,  1)  • , 1)  DIST 0 .  ALONG  Z - A X I S  Y O E I G I N • , 0 . )  A X C T E L (  X C E I G I N ' , S X )  CALL  A X C T E L (  C A I I  AXCTEL  (  C A I I FLC  A X C T E L (  SYHECL  C A I I  (  NUEEEE(  CAL1  SYBEOL  (  C A L L  NUflEEE  (  EETUEN  S I C E ' DIST  ; •  . , 1 . , 0 . 4 , « E U K  GEN  BCB=1 NUB2=C BETOEK  • , 9 0 .  ,4)  . , 2 . 5 , 0 . 4 , E L O A T , 9 0 . , - I ) •  5 , 1 . 0 , 0 . 4 , ' T I E E ^  . 5 , 3 . 0 , 0 . 4 ,  CHECK  ( B C E)  ( F U B 2 . E C . K N U B ) G O  RCE=0  END  (CBS)  Y C E I G I N ' , 0 . )  BUE2=NDB2 + 1  BET  Y - A X I S  X O E I G I N ^ O . )  C O E E C N / C 1 6 / K N U E , N U E 2  I F  ; ' , 0 . , S X , 0 .  ,DXXP)  , - 1 ) ALONG  t  SUEFCDTINE  0 . , S Y 2 , 0 . , D Y Y 2 F )  (CBS)  AT= NUB EON  CALL  EN  ( C B S ) ; • , 9  ,3)  A X C T E L (  (  Y - A X I S  Y C E 1 G I N « , S Y 2 )  C A I I  AX E L C T  ( C B S ) ; ' , 0 . , S X , 0 . , D X X P )  S I D E * , 1 )  CALL  C A I I  Z-AX1S  1)  TO  10  , 9 0 .  T A U , 9 0 . , 0 )  ,5)  , 9 0 . , S Y 2 , 0 . , D Y Y 2 P )  221  S 0 B B C 0 1 I N E  C  C  TO  P E I H T  SUEECUTINE  OUTPT  EIBENSION  T E E  HATBIX  1 (T)  T ( 1 0 , 1 6 , 9 1 )  COeEC»/C«/L,H,H C C 8 B O N / C 1 / E X , E X , D Z , D T , B K A Y COBBCN/C13/IAC C O E B C N / C 1 0 / 1 L I Q B B I 1 I  to  100 1  ( 6 ,  ft.  11  ( ' 0 '  F 6 .  1,  I X , ' S E C C N E S '  , 10X ,  Z T = E Z * F I C A T HBITE 145  (N-  501  HBITE EC  ,  K =  502  EHD  OF  T I H E ' ,  CCKTINUE  501  CONTINOE F C I E A T  ,  Of  T B E  JHGOT  IN  Z  D I B E C T I O N  ,  , F 1 0 . 2 , * C B S ' )  , 2 X , I 3 )  150)K  J=1,fl  502  I F  TBE  K=1,H  ( 6 ,  •  HBI1E(6,180)  160  AT  )  f)  ( I O X , ' S I Z E  F O B B A T ( 5 X , DO  'TEHFERATORES  ( 6 . 1 4 5 ) 2 1  FOEBAT  150  , T S C I , D E S S , D E N I  100)TAU  (T(1,J,K) ,1 = 1,L)  ( 1 X , 1 1 ( E 1 0 . 1 , 1 X )  ( K . E C . 1 0 0 ) G C  10  )  1  BEIOFN 1  K=11 B B I I E DO  (6,  503  150)K  J = 1 , 1 1  W B I I £ ( 6 , 1 8 0 ) 503  (T(I,J,K)  , 1 = 1 , L )  CONTINOE DC  504  HBI1E DO  K=20,21  ( 6 ,  505  150)K  0 = 1 , 1 1  H B I 1 E ( 6 , 1 8 0 ) 505  ( 1 ( 1 , 0 , K )  ,1 = 1,1)  CCKTINUE  SOU  C0N3INDE BEIOFN EBC  C  C C  c  S O E I C O T I N E  XXFIAN  C  S O E B O C I I N E  C  BCIE  C  T E E  GE1E  I S  1BBEGDLABLY  C  OVEBCCEE  BI  GIVING  C  G£ID  TEAT  IC  PICT  THE  COHTOOBS  CONTODfi  IN  PBOGEAB  THE  SHAPED  DOBBI  XX  tOES .  T H I S  VALUES  TO  PLANE  NOT  H O B EC CAN  NON  I F  BE  E X I S T I N G  F C I N T S  DIBENSION  T ( 1 0 , 1 6 , 9  EIBENSION  1 3 ( 1 0 , 1 6 , 9 1 )  1 ) , T 1 ( 1 0 , 1 6 , 9 1 ) , I 2 ( 1 0 , 1 6 , 9 1 )  D I E E K S I C N  XX ( 1 6 ) , Y Y  DIBENSICN  X X I ( 1 6 ) , Y Y  (10)  , 1 8 ( 1 0 , 1 6 , 9 1 ) , Z Z  1 ( 10)  ( 1 6 , 1 0 )  , Z Z 1 ( 1 6 ,  10)  C O B B C N / C 2 0 / X 1 , 1 J , Z 2 C O B B C N / C 1 4 / S X , S Y  1 , S Y 2 , D X X , D Y Y 1 , D Y Y 2 , D X X P , B Y Y 1 ? , D Y Y 2 P  C O E 8 0 N / C 7 / 1 , 1 1 , 1 2 , 1 3 , I B C 0 B K C N / C 4 / L , B , N C O B E C N / C 1 0 / 1 L I C , T S O L , D E K S , D E N 1 COHFCN/C13/T  C C  AU  C O B B O N / C 1 1 / C P L . C S P , N U B B 0 B , B L H T  G B I C  F C I B T S  ON  X  AXIS  (ORIGINALLY  I  AXIS)  222  C  S C A I I N G DO XX  100  1C0  J=1,fl  1 ( J ) = XX  DC  200  1 = 1 , L  Y Y 1 ( I ) = Y Y 200 C  ( J ) / E Y Y 2 P  CONTINUE  ( I ) / D Y Y 1 P  C C M I N O I  D E A N I KG  A  CEOSS  SECTION  OF  THE  JOEBC  INGOT  C  10  C A I L  P L C T E L  C A I I  AXCTFL  ( * H E T B I C ' , 1 )  C A I I  A X E L C 3 C C 1 S T  ALONG  Y  AXIS  C A I I  A X E L C T ( ' D I S T  ALONG  X  AXIS  ( ' S H E ' ,  (XX 1 ( 1 )  0)  C A I I  ELCT  CALL  F L O T ( X X 1 ( 4 ) , Y Y 1  , Y Y 1 ( 1 0 )  C A I I  F L C T ( X X I ( 4 ) , Y Y  CALL  PLOT  (XX1  (5)  C A I I  ELCT  (XX1  (9) (16)  , 0.  , XXI  ,3)  1 ( 6 ) , 2 ) (6)  ,2)  4  , Y Y 1 ( 1 0 ) , 2 )  CALL  FLOT  (XXI  FLCT  (XX 1 ( 16)  CALL  PLOT  (XX1  ,YY1 , YY  ,YY1  (10)  1 ( 1) (1)  ,2) ,2)  ,3)  F L C AT= N U B B U N C A I I  S Y B E C L ( 1 . , 1 . , 0 . 4 , « E U N  C A I I  K U B E I E ( 1 . , 2 . 5 , 0 . 4 , E L O A T , 9 0 .  C A I I  ' , 9 0 . , 4 ) , - 1 )  S Y B B O L ( 1 . 5 , 1 . 0 , 0 . 4 , ' I I E E = ' , S O . , 5 )  C A I I  N O B E E P ( 1 . 5 , 3 . 0 , 0 . 4  , T A U , 9 C . , 0 )  EETOEN C  F I L L I N G  THE  GRID  KITH  T E B P E EATUEE  VALUES  C ENTIY K2=  CUTLNE  (N/2)  • 1  DO  3C0  1 = 1 , L  DO  350  J = 1 , 4  ZZ1  ( J , I ) = T ( I , J , K 2 )  350  C C M I N U E  3C0  CONTINUE CO  400  ZZ1 400  ZZ  450  ( I , 6 , K 2 )  CONTINUE DO  500  1=1,8  ( 7 , I ) = 1  ( 1 , 7 , K 2 )  CONTINUE CO ZZ  600  ( I , J , K 2 )  1 = 1 , 7  1 ( 6 , I ) = I  Z Z 1 500  1,6  CCKTINUE DO  450  1=  ( 5 , I ) = T  £C0  1=1,9  1 ( 8 , I ) = T  ( 1 , 8 , K 2 )  CONTINOE DC  700  1 = 1 , L  CO  800  J = 1 , 1 6  Z Z 1 ( J , I ) = T ( I , J , K 800 700  CONTINUE C N T O U E ( X X 1 , 1 6 , Y Y 1 , 1 0 , Z Z 1 , 1 6 , T L I Q , 3 . , T L 1 Q )  C A I I CNTOUE T- B n EETUBN EN C C C C  2)  CONTINUE  C A I L  ( X X I , 1 6 . Y Y 1 ,  (16)  , 0 .  ,DYY2P)  ( C B S ) ; ' , 9 0 . , Y Y 1 ( 1 0 ) , 0 . , D Y Y 1 P )  ( 1 0 ) , 2 )  ,YY1  C A I I  (1)  ( C B S ) ; '  1 C , 2 Z 1 , 1 6 , I S O L , 3 . , I S O L )  223  SOEECOTINI C  SUEBOC1ISE  C  IN  THI  CASE  C  E I I E E I T S  C  TC  I N I T I A L I Z E  OF  A  JUBEC  OUTSIDE  rORHY  A  I N I T I 2 ( 1 , T E B E , D U K Y )  TC  TE I  VALOE  TO  TEE  INGOT  JDHBC ENABLE  TEMEEBATUSE  F I E L E  .  THE  NOTE  EAVI THE  THAT  BEEN USE  I N I T I A L I S E D  OF  CONTOUB  PBOGBAE  C  DIEENSION  T  (10,  1 6 , S 1)  C O E E O N / C 4 / L , H , N C A I I  S999  DO  150  DC T 200 150  K=1,N 1= 1,  2C0  J = 1 , 4  CONTINUE CCF.1INUE  T  25C  1=1,6  ( 1 , 5 , K ) =  TEBF  CONTINUE DO  300  1 = 1 , 7  T ( I , 6 , K ) = 300  DO  4C0  1 = 1 , 8  ( I , 7 , K ) = T E B P  CONTINUE DC I  500  500  1 = 1 , 9  ( I , 8 , K ) =  1EHP  COK1INUE CO  6C0  1 = 1 , L  DC  650  J = 9 , B  I  ( I , J , K ) = 1 E B P  650  COK1INUE  600  COH1INUI  9999  TEMF  CONTINUE  T 100  /  L  ( 1 , J , K ) = T E H E  CO  250  G S E T ( T , 1 6 0 , 9 1 , 1 6 0 , D U B Y )  DO  COS1INUE BE1UBN EN I  C  c  c  SDBICUT1NE  C  TC  SCBT  CUT  T E E  E1FFEBENT  T Y P E S  C c SUEECUTINI  NESOBT  C C B E C N / C 3 / N T Y P I  ( 1 0 , 1 6 , 9 1 ) , L P S ( 1 0 , 1 6 , 9 1 )  C G E E C N / C 4 / L , B , N I L = I - 1 BH=B-1 NN=5-1 CO  S999  1 = 1 , L  DC  9999  J = 1 , B  DO  S999  I F  ( 1 . E C .  I F  ( I . E C . L )  K=1,N 1)  IF  ( J . E C .  I F  ( J . E C . K )  IF  (K.  1)  GO  TO  100  GO  TO  200  GO  TO  300  GO  TC  400  EC-N)  N T Y I I ( I , J , K ) GC 500  IF  10  TO  500  =5  9999  N I Y P E ( I , J , K ) GO  100  IC  GO  =  14  9999  (J.EC..  1) G O  TO  600  OF  NODES  I F ( J . E C . f l ) IF  ( K . E C .  GO N)  N T Y E E ( I , J , K ) GC 800  TC  ( J . E C . 1 )  GO  TO  900  GO  TO  1000  ( K . E C - N )  GO  TO  TO  =  GC  GO  GO  TC  TC  6 0 0  I F  GO 1400  700  I F  TO  GO  TO  900  I F  ( K . F C . N ) (I  TC  GC 1600  =  17  GO  TO  , J ,K)  =  TO  =  10  GC  TO  1000  TO  ( K . E C . N )  I f  TC  GO  999S  GO  TC =  3  =  12  GO  TO  =  9  9999 =  18  1600  K=1,NN  Y F E ( 6 , 4 , K ) = 1 9  CONTINOE EC  1350  HTYEE  K=1,NN  { 6 , 5 , K ) = 2  1  CONTINCE NTYEE  ( 6 , 4 , N ) = 2 0  N T Y F E ( 6 , 5 , N ) = 2 2 BC  1900  K=1,NN  DO  2CC0  1 = 7 , I I  N T Y I E ( I , 4 , K ) = 2 3 2000 1900  CONTINOE C O M I N U E DO  21C0  NTYEE 2100  1 = 7 , L L  ( I , 4 , N )  = 24  CONTINUE DC  1600  9999  CONTINOE DC  1350  16  9999  N I Y F F ( I , J , K )  HT 1800  =  ( I , J , K )  ( K . E C . N )  TO  1500  7  999S  N T Y F E ( I , J , K )  1700  =  9999  N T Y F E ( I , J ,K) GO  1400  1  9999  ( K . E C - N )  TC  1300  8  9999  TC  NTYFE  TO  9999  N I Y F E ( I , J , K ) GO  1200  9999  N T Y E E ( 1 , J , K )  1500  11  =  N T Y F I ( I , J , K ) GO  =  GO  N T Y E E ( 1 , J , K )  NT Y F E  2  9999  N T Y I E ( I , J , K )  GC  TO =  99S9 ( I , J ,K)  I F ( K . I C . N )  GO  15  9999  ( K . E C - N )  NTYFE  1100  6  9999  N T Y E E ( I , J . K )  I F  TC  =  N T Y E f ( I , J , K )  1300  13  ( J . F C . M )  GO  tOO  =  800  IF  GO  1200  4  9999  N T Y I E ( I , J , K )  300  =  700  I F  IF  1100  TO  9999  N T Y P E ( I , J , K ) GC  200  TC  TO  GO  2200  K=1,NN  1700  NTYPE (7,6,K) =25 BTYEE (8,7,K)=25 BTYFE(9,8,K)=25 2200 CONTINUE HTYIE(7,6,N)=26 NTYIE (8,7,N) =26 BTYEE (9,6,N)=26 DO 2300 K=1,NN HT Y IE ( I , 9,K) = 27 2300 CONTINUE NTYEE (1,9,N)=28 DO 2400 K=1,NN BT I IE (1,4,K)=29 2400 COKTINOE BTYEE ( I , 4, N)=30 BET  /  CBN  END  C  C  C  SOEECUTINE SUBIT C SDEECOT1KI TC S10EI THE SU B E AC E TEKFEEATUBES CP DIFTEfiENT C ELEEEK1S IN AN ABE AY. TE1S IS FCB STUDYING THE UKSXEADY C STATE ECEIION OF THE CASTING. 0  C C  10 20  21  25  30  DIEENSION 1(10,16,91) ,11(10,16,9 1),12(10,16,-91) DIEFNSICN T3(10,16,91) ,1K(10, 16,91) ,TS (33,30,50) COEECR/C4/L,M,N CCEECN/C7/T,T1,I2,T2,TN COHEON/C21/1S IF (K.GT.52)GO TC 100 K1=N-2 IF(K.G1.30)GO TC 10 K2 = N GO TC 20 K2 = 30 DC 100 KK=1,K2 1=1 K3=K-KK+1 CO 21 J=1,1C TS (I,KK,K 1) =T ( J , 1, K3) 1=1*1 CCKTINUE DO 25 J=2,4 IS (I,KK,K 1) = T (10> J,K3) 1=1*1 CONTINUE CO 30 J=1,4 I1=10-J TS (I,KK ,K1)=1 (11 ,4, K3) 1= I • 1 CONTINUE TS ( I , KK , K1) =T (6, 5, K 3) 1=1*1 TS (I,KK,K1)=T (7,6,K3) 1 = 1*1 TS (1,KK,K1) =T (8,7,K3) 1=1*1 TS (1, KK , K 1) =T (9 , 8, K 3)  /  226 I=I + I 35  DC  J = 9 , 1 6  ( I , K K , K 1 ) = T ( 1 0 , J , K 3 )  IS  1=1+1 35  CONTINUE DO  40  J = 1 , 9  12 = 1 C - J (1,KK,K1)=T  TS  1=1+ 40 100  (12,  16.K3)  1  C O M I N U E CONTINUE BETUBN  EN I C C  SUE ECUTINE TSAEAY C C  SUERCDIINE  C  ONTO  *  TC  F I L E  COEY  IN  THE  SUEFACE  BINARY  C DIHENSION  T S ( 3 8 , 3 0 , 5 0 )  COF.rCK/C4/L,H,N  C0BH0N/C21/T S DC  10  K=1,N  DO  1C  J = 1 , 3 0  WEITF(7) 10  CONTINUE BETUBN ENL  ( T S ( I , J , K )  , 1 = 1 , 3 6 )  TEHPEBATURE  AR&A Y  APPENDIX 3  THREE DIMENSIONAL TEMPERATURE DISTRIBUTION IN THE CASTING FOR THE DIFFERENT RUNS  227  228  DIST. FBOH THE HOOLD 700. 7CC. 700. 7CC. 700. 70C. 700. 70C. 70C. 7 0C. 700.  700. 700. 700. 700. 700. 70C. 700. 700. 700. 70C. 700.  DIST.  FECH THE BOOLD  691. 6S1. 6S1. 6S1. 6S1. 6 SC. 6£S. 6£4. 673. 64C. 476.  691. 691. 691. 691. 690. 690. 638. 684. 673. 64C. 478.  700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700.  689. 689. 689. 689. 689. 6es. 687. 633. 672. 639. 478.  700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700.  687. 687. 687. 687. 687. 636. 685. 681. 67 1 . 639. 478.  DIST. FBCH THE HOOLD 676. 67£. 676. 676. 675. 674. 670. 661. 637. 546. 147.  Table A3.1  675. 675. 675. 675. 674. 673. 670. 661. 636. 546. 147.  672. 67 2 . 672. 672. 671. 670. 667. 659. 636. 543. 146.  667. 667. 667. 667. 666. 665. 663. 655. 634. 521. 144.  0.0  CKS  70C. 700. 700. 700. 700. 700. 70 0 . 700. 700. 700. 700. 6.3 683. 683. 683. 683. 683. 682. 681. 678. 668. 639. 477. 12.7 659. 659. 659. 659. 659. 658. 656. 650. 631. 502. 139.  70C. 700. 70C. 700. 70C. 700. 70C. 700. .700. 700. 700.  700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700.  700. 700. 700. 700. 700. 700. 700. 700. 700. 70J. 700.  700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700.  700. 700. 700. 700. 700. 700. 700. 7C0. 700. 700. 700.  70 0 . 70C. '700. 70 0 . 70 0 . 700. 700. 700. 70 0 . 700. 700.  669. 669. 669. 669. 669. 669. 668. 666. 659. 636. 472.  658. 658. 658. 658. 658. 657. 657. 656. 651. 634. 466.  644. 630. 644. 630. 644 . 630. 644. 630. 644. . 630. 644. 630. 64<4 . 630. 643. 630. 641. ' 630. 631. 430. 453. 4 13.  521. 521. 52 1 . 521. 521. 521. 52 1 . 52 1 . 52 0 . 51 1 . 373.  631. 631. 631. 631. 631. 631. 631. 631. 630. 451. 126.  571. 571. 571. 571. 571. 570. 569. 563. 524. 396. 113.  461. 461. 461 . 461. 460.. 460. 457. 450. 420. 321. 95.  149. 149. 149. 149. 149. 149. 148. 14 5 . 136. 108. 4 0.  CBS 677. 677. 677. 677. 677. 677. 676. 673. 665. 638. 475. CHS 64E. 643. 64£. 648. 648. 647. 646. 642. 631. 481. 134.  Three-dimensional  320. 320. 320. 320. 319. 319. 317. •3 1 1 . 291. 226. 71.  temperature d i s t r i b u t i o n  381 x 991 mm. aluminium i n g o t c a s t a t 1.775 mm/s ( c o n v e n t i o n a l  cooling).  in  229  D I S T .  FECH  6 5 6 .  6 5 7 .  6 5 4 .  6 4 8 .  64  6 3 1 .  5 5 4 .  4 6 3 .  6 5 8 .  6 5 7 .  6 5 4 .  6 4 8 .  6 4 0 .  63  5 5 4 .  '463.  THE  HOU1D  1 9 . 0  CBS  0.  1.  3 5 8 . .  3 5 8 .  2 4 C .  1 1 2 .  2 4 0 .  112.  6 5 6 .  6 5 7 .  6 5 4 .  6 4 3 .  6 4 0 .  6 3 1 .  5 5 4 .  4 6 3 .  3 5 8 .  2 4 0 .  6 5 6 .  112.  6 5 7 .  6 5 3 .  6 4 7 .  6 4 0 .  63  5 5 4 .  4 6 3 .  3 5 6 .  2 4 0 .  6 5 7 .  112.  6 5 6 .  6 5 3 .  6 4 7 .  6 3 9 .  6 3 1 .  5 5 4 .  4 6 2 .  3 5 7 .  2 3 9 .  6 5 5 .  6 5 4 .  112.  6 5 1 .  6 4 6 .  6 3 9 .  6 3 1 .  5 5 2 .  4 6 0 .  3 5 5 .  2 3 8 .  11  6 3 5 .  6 3 0 .  5 4 6 .  4 5 4 .  3 5 0 .  2 3 4 .  110. 10  1.  1.  6 5 1 .  6 5 C .  6 4 8 .  6 4 3 .  6<42.  6 4 2 .  6 4 0 .  6 3 5 .  6 3 1 .  5 8 9 .  5 1 8 .  4 3 2 .  3 3 3 .  2 2 3 .  6 3 1 .  6 3 1 .  6 3 1 .  6 3 0 .  5 7 4 .  5 1 3 .  4 4 8 .  3 7 4 .  2 8 9 .  194.  9 3 .  4 0 2 .  4 0 0 .  3 9 4 .  382.  3 6 2 .  3 3 4 .  2 9 8 .  2 5 2 .  197.  1 3 5 .  6 7 .  1C4.  1 0 4 .  1 0 2 .  9 9 .  9 5 .  8 9 .  8 0 .  7 0 .  5 7 .  4 3 .  2  7.  2 0 0 .  9  5.  2 0 0 .  9 5 .  D I S T .  FBCH  6 4 3 .  6 4 2 .  6 3 9 .  6 3 4 .  6 3 0 .  5 5 4 .  4 7 8 .  3 9 4 .  3 0 1 .  6 4 3 .  6 4 2 .  6 3 9 .  6 3 4 .  6 3 0 .  5 5 5 .  4 7 9 .  3 9 4 .  301  THE  HOU1D  2 5 . 4  5.  CMS  .  6 4 3 .  6 4 2 .  6 3 9 .  6 3 4 .  6 3 0 .  5 5 5 .  4 7 8 .  3 9 4 .  3 0 0 .  2 0 0 .  9 5 .  6 4 3 .  6 4 2 .  6 3 9 .  6 3 4 .  630.  5 5 4 .  4 7 7 .  3 9 3 .  3 0 0 .  2 0 0 .  6 4 2 .  6 4 1 .  9 5 .  6 3 6 .  6 3 4 .  6 3 0 .  5 5 3 .  4 7 6 .  3 9 1 .  2 9 9 .  199.  64  9  6 4 0 .  6 3 7 .  6 3 3 .  5 4 9 .  4 7 1 .  3 8 7 .  2 9 5 .  1 9 6 .  9 3 .  1.  .  6 3 0 .  4 .  6 3 6 .  6 3 7 .  6 3 4 .  6 3 1 .  5 9 1 .  5 3 1 .  4 5 8 .  3 7 5 .  2 8 6 .  1 9 0 .  9 0 .  63  63  63  6 3 0 .  5 5 7 .  4 9 1 .  4 2 1 .  3 4 5 .  2 6 3 .  1 7 5 .  8 4 .  1.  1.  1.  5 1 7 .  5 1 2 .  4 9 9 .  4 7 4 .  4 3 9 .  3 9 4 .  3 4 1 .  2 8 2 .  2 1 6 .  145.  3 2 0 .  317.  3 0 9 .  2 9 5 .  2 7 5 .  2 4 9 .  2 1 8 .  1 6 2 .  1 4 1 .  9 7 .  5 0 .  6 4 .  8 4 .  8 2 .  7 9 .  7 4 .  6 8 .  5 3 .  4 4 .  34.  2  D I S T .  FBCH  THE  BOU1D  3 1 . 7  6  1.  7  1.  3.  CHS  6 3 4 .  6 3 3 .  6 3 2 .  6 3 0 .  5 6 2 .  4 9 7 .  4 2 5 .  3 4 7 .  2 6 4 .  1 7 5 .  6 3 4 .  6 3 3 .  6 3 2 .  6 3 0 .  5 6 2 .  4 9 7 .  4 2 5 .  3 4 7 .  2 6 4 .  1 7 5 .  8 4 .  6 3 3 .  6 3 2 .  6 3 0 .  5 6 2 .  4 9 6 .  4 2 5 .  3 4 7 .  2 6 3 .  1 7 5 .  8 4 .  6 3 3 .  6 3 2 .  6 3 0 .  56  4 9 5 .  4 2 4 .  3 4 6 .  2 6 2 .  175.  8 4 .  6 3 4 .  6 3 3 .  6 3 2 .  6 3 0 .  5 5 9 .  4 9 2 .  4 2 0 .  3 4 3 .  2 6 0 .  173.  8 3 .  6 3 3 .  6 3 2 .  63  6 0 0 .  5 4 6 .  4 8 3 .  4  3 3 6 .  2 5 4 .  1 6 9 .  8 1 .  6 3 4 . 6 3 4 .  '  1.  1.  12.  8 4 .  6 3 1 .  6 3 1 .  6 3 1 .  5 7 7 .  5 2 1 .  4 5 9 .  3 9 1 .  3 1 9 .  2 4 2 .  1 6 1 .  7 8 .  5 7 6 .  5 8 7 .  5 5 1 .  5 0 8 .  4 6 0 .  4 0 6 .  3 4 7 .  2 8 3 .  2 1 5 .  1 4 4 .  7 1 .  4 2 9 .  4 2 3 .  4 0 8 .  3 8 4 .  3 5 3 .  3 1 5 .  2 7 2 .  2 2 4 .  1 7 1 .  2 5 8 .  2 5 5 .  2 4 7 .  2 3 5 .  2 1 7 .  196.  170.  1 4 2 .  7C.  6 9 .  6 8 .  6 5 .  6 1 .  5 6 .  5 0 .  4 4 .  1  16.  5 6 .  1 1 0 .  7 7 .  4  3 7 .  2 9 .  2 1 .  1.  230  DIST.  FECH TBI HOULD  63 1. 63 1. 631. 631. 6 31. 631. 630.  631. 631. 63 1. 631. 631. 631. 589. 470. 346. 208. 58.  lilt.  350. 2 1C. 5S.  38. 1  630. 57 1. 63C. 571. 630. 570. 630. 568. 6 19. 564. 5S6. 545. 549. 503. 450. ' 420. 334. 314. 201. 191. 57. 55.  DIST. FECH THE HOOID 631. 631. 631. 631. 631. 574. 484. 3SC. 287. 173. 51.  630. 63C. 630. £3C. 60 1. 555. 476. 385. 284. 171. 50.  578. 577. 576. 571. 556. 520. 454. 371. 274. 166. 49.  566. 554. 542. 52S. 4SS. 454. 393. 32C. 237. 143. 44.  544. 541. 530. 516. 491. 447. 388. 317. 234. 142. 44.  512. 510. 504. 492. 468. 429i 374. 306. 227. 138. 43.  472. 471. 466. 455. 435. 401. 352. 289. 214. 131. 4 1.  514. 514. 513. 510. 505. 489. 452. 382. 289. 176. 51.  44. 4  527. 527. 525. 520. 507. 477. 422. 34 8. 259. 157. 47.  DIST. FBCfi T BE HOULD  CHS  426. 425. 421. 412. 394. 365. 322. 266. 198. 121. 39.  385. 385. 384. 382. 376. 364. 337. 290. 222. 138. 43.  313. 313. 312. 310. 306. 295. 2 74. 237. 183. 115. 38.  237. 237. 236. 235. 231. 223. 206. 180. 141 . 90.  158. 156.' 157. 156. 154. 149. 139. 121. 96. 63. 26.  76. 76. 76. 76. 75. 73. 6 8. 61. 50. 3 6. 20.  352. 351. 350. 346. 337. 319. 269. 243. 184. 114. 37.  286. 285. 284. 230. 273. 259. 236. 200. 152. 96. 33.  216. 21o. 215. 212. 207. 197. 179. 153. 118. 75. 29.  1 44. 143. 142. 138. 132. 1^0. 104. 61. 54. 24.  70. 70. 70. 69. 66. 65. 6 0. 53. 4 3. 32. 19.  319. 318. 315. 309. 297. 276. 246. 205. 154. 96. 33.  259. 258. 25b. 251. 242. 226. 2 02. 169. 128. 81. 30.  196. 196. J94. 190. 183. 172. 154. • 130. 100. 64. 26.  131. 131. 130. 126. 123. 116. 104. 89. 69. 47. 22.  65. 65. 64. 63. 61. 56. 53. 4 7. 3 8. 29. 18.  32.  CHS  473. 472. 470. 465. 454. 429. 383. 319. 236. 145. 44.  50.8  452. 452. 45 1. 449. 442. 428. 396. 339. 256. 159. 47.  4 14. 414. 412. 407. 397. 376. 338. 283. 212. 131. 41.  1HH.  CHS 374. 373. 370. 362. 346. 323. 286. 237. 178. 110. 36.  231  D I S T .  FRCH  THF  H0U1D  5 7 . 1 .  CHS  4 5 5 .  4 4 9 .  4 3 2 .  1 0 5 .  3 7 0 .  4 5 3 .  4 4 7 .  4 3 0 .  4 0 4 .  3 6 9 .  3 2 9 .  4 4 6 .  4 4 0 .  4 2 4 .  3 9 8 .  3 6 4 .  3 2 3 .  4 3 2 .  4 2 6 .  4 1 1 .  3 8 6 .  3 5 2 .  3 1 4 .  321.  4 C 6 .  4 0 3 .  3 8 S .  3 6 6 .  3 3 5 .  2 9 8 .  3 7 2 .  3 6 8 .  3 5 5 .  3 3 5 .  3 0 8 .  2 7 4 .  3 2 4 .  3 2 1 .  3 1 0 .  2 9 3 .  2 7 0 .  2 6 5 .  2 6 3 .  2 5 4 .  2 4 1 .  2 2 2 .  19S.  1 S 7 .  1 9 5 .  1 8 9 .  179.  166.  1 4 9 .  120.  119.  115.  110.  102.  9 3 .  3 S .  3 8 .  D I S T .  FECH  3 8 .  THE  3 6 .  HOUID  24  3 5 .  6 3 . 5  1.  3 2 .  2 8 2 .  2 3 0 .  175.  1 16.  5 9 .  2 8 1 .  22  1 7 5 .  1  5 9 .  2 7 7 .  2 2 6 .  172.  1 16.  5 9 .  2 6 9 .  2 2 0 .  1 6 8 .  1 1 3 .  5 7 .  2 5 6 .  2 1 0 .  1 6 0 .  1 0 9 .  5 5 .  2 3 6 .  1  1 4 9 .  10  5 2 .  2 0 8 .  172.  132.  9 0 .  4 7 .  173.  1 4 3 .  1 1 1 .  7 7 .  4 2 .  130.  1 0 3 .  6 5 .  6 0 .  34.  8 2 .  6 9 .  5 6 .  4 1 .  2 6 .  3 0 .  2 7 .  24.  2 1 .  16.  2 0 2 .  3 8 4 .  3 8 C .  3 6 7 .  3 4 6 .  3 1 9 .  2 8 5 .  2 4 5 .  3 7 7 .  3 6 5 .  3 4 4 .  3 1 7 .  2 6 3 .  2 4 4 .  3 7 5 .  3 7 C .  3 5 6 .  3 3 8 .  3 1 1 .  2 7 6 .  2 4 0 .  36  356.  3 4 6 .  3 2 7 .  30  2 6 9 .  2 3 2 .  34 3  94.  1.  1.  155.  1 0 5 .  5 4 .  1 5 4 .  1 0 4 .  5 3 .  1 9 7 .  151.  I  1 0 3 .  5  1 4 7 .  1 0 0 .  201. SI  -  1.  3 3 7 .  3 2 6 .  3 0 6 .  2 8 4 .  2 5 4 .  2 1 9 .  1 8 1 .  10.  3 0 7 .  2 9 7 .  28  2 5 9 .  2 3 2 .  2 0 1 .  1 6 6 .  1.  16.  CHS  3 6 2 .  1.  9 .  139.  12a.  3.  1.  9 5 .  5 4 9 .  8 6 .  4 6 .  2 7 1.  2 6 6 .  2 5 9 .  2 4 6 .  2 2 7 .  2 0 4 .  177.  146.  2 2 2 .  2 1 9 .  2 1 3 .  2 0 2 .  187.  166.  146.  121  1 6 5 .  163.  158.  150.  1 3 9 .  1 2 6 .  110.  10  9 2 .  101.  7 3 .  98.  52.  3  9 3 .  8 7 .  7 9 .  7 0 .  6 0 .  4 9 .  3 4 .  3 7 .  2 4 .  3 3 .  3 1 .  2 9 .  2 7 .  2 5 .  2 3 .  2 0 .  17.  1.  3 4 .  DI  ST.  3 4 .  FEOH  THE  HOUID  6 9 . 8  .  1 1 3 .  7 8 .  4  9 5 .  6 6 .  3 7 .  2.  1.  CHS  3 2 6 .  3 2 4 .  3 1 4 .  2 9 7 .  2 7 4 .  2 4 6 .  2 1 3 .  1 7 6 .  3 2 5 .  1 3 5 .  3 2 2 .  3 1 2 .  9 3 .  2 9 5 .  4 8 .  2 7 2 .  2 4 4 .  2 1 1 .  175.  3  1 3 5 .  3 1 5 .  3 0 5 .  9 2 .  4 6 .  2 8 9 .  2 6 7 .  2 3 9 .  2 0 7 .  1 7 1 .  3 0 7 .  132.  3 0 3 .  2 9 4 .  SO.  4 7 .  278.  2 5 7 .  23  2 0 0 .  2 6 6 .  1 6 5 .  1 2 7 .  2 8 5 .  2 7 6 .  8 6 .  2 6 1 .  4 6 .  24  2 1 7 .  1 8 8 .  2 6 2 .  1 5 6 .  120.  2 5 9 .  25  8 3 .  4 4 .  2 3 8 .  2 2 0 .  198.  172.  1 4 2 .  2 2 6 .  1 1 0 .  2 2 6 .  2 1 9 .  7 7 .  2 0 8 .  4  1 9 2 .  1 7 3 .  151.  1 2 5 .  187.  185.  9 6 .  180.  6  171.  3 8 .  156.  1 4 3 .  125.  1 0 4 .  1 2 S .  8 2 .  1 3 6 .  1 3 4 .  5 8 .  128.  119.  1 0 7 .  9 4 .  7 9 .  8 7 .  6 3 .  8 6 .  8 4 .  4 6 .  2 8 .  8 0 .  7 5 .  6 8 .  6 1 .  5 2 .  3 1 .  3 1 .  4 3 .  3 3 .  2 3 .  3 0 .  2 8 .  2 7 .  2 5 .  2 3 .  2 1 .  19.  17.  IS.  1.  3 0 .  2.  1.  8.  2.  3 4 .  232  D I S T .  FECH  2 8 3 .  2 8 0 .  27  2 5 7 .  2 3 6 .  2 1 4 .  185.  1 5 4 .  1 1 9 .  8 2 .  2 E 1 .  2 7 6 .  2 6 9 .  2 5 5 .  2 3 6 .  2 1 2 .  184.  152.  118.  8 2 .  2 7 4 .  2 7 2 .  2 6 3 .  2 4 9 .  23  2 0 7 .  180.  1 4 9 .  1 16.  6 0 .  4 3 .  T H I  HOOLD  1.  3 . 2  CHS  1.  4 4 . .  4 4 .  2 6 3 .  2 6 1 .  2 5 3 .  2 3 9 .  2 2 2 .  1 9 9 .  173.  1 4 4 .  1 1 1 .  7 7 .  4  2 4 7 .  2 4 4 .  2 3 7 .  2 2 4 .  2 0 6 .  187.  1 6 3 .  1 3 5 .  1 0 5 .  7 3 .  2 2 4 .  4 0 .  2 2 2 .  2 1 5 .  2 0 4 .  1 8 9 .  1 7 0 .  148.  1 2 3 .  9 6 .  6 7 .  3 8 .  1 9 5 .  193.  187.  178.  165.  14S.  130.  1 0 8 .  8 5 .  6 0 .  3 4 .  16C.  156.  154.  146.  136.  1 2 3 .  1 0 7 .  9 0 .  7 1 .  5 1 .  3 1 .  120.  119.  1 15.  110.  102.  9 3 .  82.  6 9 .  5 6 .  4 1 .  2 6 .  2.  7 5 .  7 5 .  7 3 .  7 0 .  6 5 .  6 0 .  5 3 .  4 6 .  3 8 .  3 0 .  2 2 .  2 9 .  2 8 .  2 8 .  2 7 .  2 6 .  2 5 .  2 4 .  2 2 .  2 0 .  1 6 .  16.  4 0 .  iC H S  D I S T .  FECH  2 4 8 .  2 4 5 .  2 3 8 .  225.  2 0 9 .  186.  163.  1 3 6 .  1 0 6 .  7 3 .  2 4 £ .  2 4 3 .  2 3 6 .  2 2 4 .  2 0 7 .  1 8 6 .  16  1 3 5 .  1 0 5 .  7 3 .  4 0 .  2 4 0 .  2 3 7 .  2 3 0 .  2 1 8 .  2 0 2 .  182.  156.  1 3 2 .  1 0 3 .  7 2 .  3 9 .  2 3 C .  2 2 7 .  22  2 0 9 .  194.  1  152.  1 2 6 .  9 9 .  6 9 .  3 8 .  2 1 5 .  2  2 0 6 .  196.  161.  164.  143.  1 1 9 .  9 3 .  6 5 .  3 7 .  1  SS.  13.  T H E  HOOLD  1.  !.  5  7 5 .  2.  1 9 3 .  187.  178.  165.  1 4 S .  1 3 0 .  1 0 9 .  8 5 .  6 0 .  3 4 .  170.  168.  163.  155.  144.  130.  114.  9 5 .  7 5 .  5 4 .  3 2 .  1 3 S .  138.  1 3 4 .  127.  119.  1 0 7 .  9 4 .  8 0 .  6 3 .  4 6 .  2 8 .  105.  104.  101.  9 6 .  9 0 .  8 2 .  7 2 .  6 2 .  5 0 .  3 8 .  2 5 .  6 7 .  6 6 .  6 4 .  6 2 .  5 8 .  5 3 .  4 8 .  4 2 .  2 7 .  2 6 .  2 6 .  2 5 .  2 5 .  2 4 .  2 2 .  21  .  3 5 .  2 8 .  2 1 .  2 0 .  16.  16.  3 7 .  D I S T .  FECB  2 2 1 .  2 1 9 .  2 1 3 .  2 0 2 .  187.  168.  147.  1 2 2 .  9 5 .  6 7 .  2 2 C .  2 1 7 .  2 1 1 .  2 0 0 .  1 8 5 .  1 6 7 .  146.  1 2 1 .  9 5 .  6 7 .  3 7 .  2  2 1 2 .  2 0 6 .  195.  18  163.  142.  1 1 9 .  9 3 .  6 5 .  3  2 0 3 .  197.  187.  173.  1 5 6 .  136.  114.  8 9 .  6 3 .  3 6 . 3 4 .  14.  2 C 5 .  T H I  MOOLE  I  . 9  CHS  1.  7.  1S1.  190.  184.  175.  162.  146.  128.  1 0 7 .  8 4 .  5 9 .  1 7 4 .  1 7 2 .  167.  159.  147.  1 3 3 .  116.  9 7 .  7 7 .  5 5 .  1 5 1 .  150.  145.  138.  126.  1 1 6 .  102.  6 6 .  6 3 .  1 2 4 .  1 2 3 .  120.  114.  106.  9 6 .  8 5 .  7 2 .  5 8 .  4 3 .  2 7 .  9 4 .  9 3 .  9 0 .  8 6 .  8  7 4 .  6 5 .  5 6 .  4 6 .  3 5 .  2 4 .  £ C .  6 0 .  5 8 .  5 6 .  5 3 .  4  4 4 .  3 9 .  3 3 .  2 6 .  2 0 .  2 5 .  2 5 .  2 5 .  2 4 .  23.  2 3 .  2 2 .  2 0 .  1 9 .  18.  1 6 .  J  1.  9 .  .  4 9 .  3 2 . 3 0 .  233  D I S T .  FECH  THE  HOULD  9 5 . 2  CHS  2 0 0 .  2 0 2 .  1 9 6 .  186.  172.  1 5 5 .  1 3 5 .  1 1 3 .  2 C 2 .  2 0 0 .  194.  184.  17  154.  134.  1 1 2 .  1 S 7 .  1 9 5 .  1 8 S .  180.  167.  1 5 0 .  1 3 1 .  1 1 0 .  8 6 .  6 1 .  3 5 .  166.  187.  18  172.  160.  144.  126.  1 0 5 .  8 3 .  5 9 .  3 4 .  1.  1.  8 9 . .  8 8 .  6 3 .  3 5 .  6 2 .  3 5 .  1 7 6 .  1 7 4 .  1 6 9 .  161.  149.  1  118.  9 9 .  7 8 .  5 6 .  3 2 .  1 5 9 .  158.  153.  146.  135.  123.  1 0 7 .  9 0 .  7 1 .  5 1 .  3 1 .  1 3 9 .  136.  134.  127.  118.  1 0 7 .  9 4 .  7 9 .  6 3 .  4 6 .  2 8 .  1  113.  1 10.  105.  9 6 .  8 9 .  7 9 .  6 7 .  5 4 .  4 0 .  2 6 .  14.  35.  6 7 .  8 6 .  8 4 .  8 0 .  7 5 .  6 8 .  6 1 .  5 2 .  4 3 .  3 3 .  2  5 6 .  5 6 .  54.  5 2 .  4 9 .  4 6 .  4 1 .  3 6 .  3 1 .  2 5 .  19.  2 4 .  2 4 .  2 4 .  2 3 .  2  2 2 .  2 1 .  2 0 .  19.  17.  16.  3.  3.  234  Table  DIST.  FBOrt I H E  7C0. 700. 700. 700. 7C0. 700. 700. 700. 700. 700. 700.  700. 700. 700. 700. 700. 700. 7C0. 700. 700. 700. 700.  CIST.  IBOa  678. 678. 678. 678. 678. 677. 675. 671. 662. 647. 557.  677. 677. 677. 677. 676. 676. 674. 670. 662. 646. 555.  DIST.  FECK  662. 662. 662. 66 1 . 661. 660. 659. 657. 641. 569. 516.  661. 661. 661. 661. 660. 660. 659. 657. 635. 565. 512.  A3.II  HCULD  700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700. THE  TH*  700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700.  RCULD  673. 673. 673. 673. 673. 672. 670. 667. 659. 64 1 . 547.  0.0  700. 700. 700. 700. 700. 700. 700. 7C0. 700. 700. 700.  10.4  666. 666. 666. 666. 666. 665. 664. 66 1. 656. 6 11. 531.  654. 654. 654. 654. 654. 652. 648. 622. 584. 536. 489.  700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700.  700. 700. 700. 700. 70U. 70U. 700. 700. 70U. 700. 70 0 .  620. 620. 620. 620. 620. 619. 617. 610. 589. 543. 477.  57 1 . 571. 571. 57 1 . 570. 56 S . 56 7. 56 0 . 539. 499. 439.  603. 603. 603. 602. 5*8. 592. 580. 560. 529. 491 . 450.  57 1 . 571. 57 0 . 56S. 566. 560. 549. 530. 501. 465. 426.  CHS  65o. 656. 656. 656. 656. 656. 655. 653. 642. 56 1 . 507.  KCULD 2 0 . 8  659. 659. 658. 658. 658. 658. 657. 65 1. 610. 553. 503.  CfiS  CMS  636. 636. 636. 635. 628. 623. 613. 5S1. 557. 515. 471.  Three-dimensional  temperature  distribution  i n 254 x 690 mm aluminium i n g o t c a s t a t 0.833 mm/s (Reduced Secondary  Cooling).  235  DIST. FBOM THE MOULD 31.2 CMS 658. 658. 658. 658. 658. 657. 649. 586. 535. 491. 451.  657. 657. 657. 6 57. 657. 65564 1. 580. 532. 488. 449.  653. 653. 653. 652. 650. 642. 6 1 1. 566. 522. 480. 442.  637. 6C7. 577. 54 9. 637. 6 06. 577. 549. 636. 6C4. 575. . 54 b., 627. 599. 570. 54 2. 62 1. 59 1. 562. 53 a. 607. 577. 548. 52 1. 554. 528. 502. 582. 546. 524. 500. 476. 507. 4 89. 458. 446. 468. 453. 434. 414. 417. 401. 382. 431.  CIST. FEOM THE MOULD 41.6 CMS 655. 655. 654. 649. 605. "566. 529. 4 92. 455. 415. 370.  650. 649. 647. 638. 5S8. 561. 525. 489. 452. 413. 368.  622. 621. 6 16. 603. 577. 547. 515. 481. 445. 407. 363.  592. 590. 584. 572. 553. 528. 499. 467. 433. 397. 354.  560. 558. 552. 542. 525. 504. 478. 449. 4 17. 382. 341.  525. 486. 524. 485. 518. 480. 509. 47 1. 494. 458. 475. 44 0. 451 . 41 y. 425. 394. 395. 367. 363. 337. 324. 301.  EI ST. IEOK THE MOULD 52.0 CM S 558. 554. 544. 529. 510. 489. 466. 442. 4 19. 398. 380.  554. 546. 551. 543. 534. 541. 527. 521. 508. 503. 487. 483. 464. 460. 44 1. 437. . 4 18. 4 14. 397. 39a. 379. 376.  1  535. 532. 524. 511. 495. 475. 454. 431. 409. 386. 371.  522. 520. 512. 500. 485. 466. 445. 423. 402. 382. 365.  509. 496. 507. 494. 500. 487. 489. 476. 4 74. 4b2. 456. 445. 436. 425. 415. 405. 3S4. 384. 374. 365. 358. 3*9.  risi.  FROM  439. 487. 482. 473. 461. 447. 432. 4 16. 402. 388. 376.  488. 486. 481. 472. 460. 446. 431. 4 16. 401. 388. 377.  DIST.  FBCM THE  443. 442. 438. 432. 424. 4 14. 393. 382. 371. 362.  443. 44 1. 438. 431. 423. 4 14. 403. 392. 381. 371. 362.  DIST.  FECM THE  405. 404. 4C1. 396. 391. 383. 376. 367. 358. 350. 341.  404. 403. 400. 3S6. 390. 383. 375. 367. 358. 349. 34 1 .  4C4.  THE  ECU LB 6 2 . 4  486. 484. 478. 470. 458. 444. 429. 4 14. 399. 386. 375.  482. 460. 474. 466. 454. 44 1 . 426. 4 11. 396. 383. 373.  477. 475. 469. 46 1 . 450. 436. 422. 407. 392. 379. 369.  KCULD 7 2 . 8  44 1. 440. 436. 430. 42 2 . 4 12. 402. 39 1 . 380. 370. 360.  438. 437. 433. 427. 419. 4 10. 399. 388. 377. 367. 358.  403. 402. 399. 394. 389. 382. 374. 365. 357. 348. 339.  400. 399. 396. 392. 386. 379. 371. 363. 354. 346. 337.  470. 469. 463. 455. 444. 431. 416. 402. 388. 375. 364.  464. 462. 457. 44>3. 438. 425. 411. 3ii6. 38 2 . 37 0 . 35 9 .  429. 428. 424. 418. 411. 401 . 391. 380. 370. 360. 351.  423. 422. 418. 413. 405. 396. 386. 375. 365. 355. 346.  392. 391. 389. 364. 379. 372. 364. 356. 347. 339. 331.  33 7. 386. 383. 379. 373. 367. 359. 351. 343. 334. 326.  CMS  434. 433. 42y. 423. 415. 406. 396. 365. 374. 364. 355.  MOULD 8 3 . 2  236  CMS  CMS  397. 396. 393. 389. 383., 376. 368. 360. 351. 343. 334.  237 DIST.  F B C H . THE  370. 369. 367. 363. 359. 353. 347. 340. 333. 325. 317.  369. 369. 366. 363. 358. 353. 347. 340. 332. 325. 3 17.  DIST.  FROM T i i E  337. 337. 335. 332. 328. 324. 3 18. 312. 306. 299. 292.  337. 336. 335. 332. 328. 323. 3 18. 312. 305. 298. 29 1 .  MODLC 9 3 . 6  368. 367. 365. 362. 357. 352. 345. 338. 33 1 . 324. 3 16.  366. 365. 363. 359. 355. 349. 343. 336. 329. 322. 3 14.  363. 362. 360. 356. 352. 346. 340. 333. 326. 319. 311.  MOOLD104.0  336. 335. 333. 331. 327. 322. 3 17. 311. 304. 2S7. 290.  334. 333. 32 1 . 329. 325. 320. 3 15. 309. 302. 296. 289.  CHS 358., 358. 356. 352. 348. 343. 336., 330. 323. 315. 308.  354. 353. 351. 348. 343. 33 a . 332. 325. 316. 311. 304.  CMS  331. 330. 32S. 326. 322. 3 17. 3 12. 306. 300. 293. 266.  327. 327. 325. 322. 318. 314. 308. 303. 296. 290. 263.  323. 322. 320. 318. 314. 309. 304. 299. 292. 286. 279.  CIST FSOM THE MOULD 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  risi  JfrCK THE MOULD  32. 3 1. 2 2 C. 2 1 5 . 378. 376. 42C. 4 2 0 . 425. 423. 427. 425. 429. 428. 430. 429. 431. 430. 430. 430. 427. 427. 422. 422. 4 1 2 . 4 12. 26C. 2 7 9 . 162. 162. 32. 32.  Table  435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  30. 201. 344. 416. 4 18. 42C. 424. 426. 428. 428. 426. 422. 4 12. 279. 16 1 . 32.  A3.Ill  435. 435. 435. 4 35. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  27. 182. 301. 334. 349. 396. 4 18. 423. 425. 426. 425. 421. 4 12. 277. 160. 32.  0. 0 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  14.2 24. 154. 2 32. 213. 224. 292. 357. 417. 42 1. 424. 423. 420. 411. 272. 156. 31.  CMS 435. 435. 435. 435.  435. 435. 435. 435.  435. 435. 435„ 435.  435. 435. 435. 4 35.  435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435. 435.  435. 435. 435. 4 35. 435. 435. 435. 435.  CMS  20. 123. 160. 4b. 39. 161. 261. 347. 4 15. 420. 420. 419. 366. 255. 148. 30.  17. 96. 114. 20.  14. 71. 80. 14.  10. 43. 47. 10.  26. 147. 251. 334. 384. 413. 410. 324. 224. 130. 27.  24. 139. 228. 281. 296. 288. 242. 172. 101. 22.  115. 157. 170. 166. 141. 102. 61. 15.  22.  6. 11. 11. 6.  18. 28. 30. 29. 25. 20. 13. 6.  Three-dimensional temperature d i s t r i b u t i o n i n z i n c jumbo i n g o t c a s t a t 1.27 mm/s.  239 CI £ 3 F E C K  THI  46. 15 1 . 26 1 . 365. 419. 420. 421. 423. 423. 422. 42C. 4 12. 307. 213. 133. 52.  44. 143. 246. 344. 415. 419. 420. 422. 422. 422. 420. 4 12. 3C6. 2 12. 133. 52.  39. 125. 21C. 285. 337. 373. 414. 420. 42 1. 4 2 1. 420. 4 10. 30C. 207. 129. 5 1.  CIST  FECK  THE  44. 115. 19C. 269. 342. 415. 42C. 420. 42C. 420. 414. 337. 255. 181. 12G. 57.  42. 109. 179. 252. 317. 374. 415. 4 19. 420. 420. 413. 333. 251. 178. 118. 56.  37. 94. 153. 213. 268. 3 16. 355. 385. 4 16. 4 18. 385. 316. 239. 170. 112. 54.  HCULD 33. 104. 170. 222. 264. 308. 355. 410. 418. 420. 419. 379. 287. 199. 124. 49.  MOU1E  31. 78. 124. 169. 213. 258. 300. 336. 363. 372. 347. 291. 222. 158. 105. 50.  28.3 27. 80. 123. 146. 175. 227. 283. 336. 375. 4 14. 414. 350. 266. 184. 115. 46.  20. 57. 78. 56. 61. 135. 202. 262, 308. 339. 339. 300. 233. 163. 102. 41.  42.5  25. 60. 91. 115. 147. 192. 238. 278. 306. 316. 299. 256. 198. 142. 95. 46.  CMS  15. 36. 47. 22.  11. 25. 29. 13.  8. 16. 17. 9.  43. 119. 183. 233. 263. 267. 241. 190. 134. 85. 35.  39. 106. 156. 165. 191. 175. 140. 100. 64. 27.  34. 85. 109. 115. 1C7. 86. 62. 41. 18.  27. 38. 41. 38. 32. 24. 17. 9.  14. 29. 36. 22.  10. 19. 22. 13.  8. 13. 14. 9.  6. 8. 8. 6.  47. 104. 151. 183. 196. 194. 173. 137. 100. 67. 33.  41. 90. 124. 141 . 141. 127. 102. 75. 51. 26.  35. 71. 67. 89. 81 . 66. 49. 34. 18.  26. 38. 39. 36. 30. 23. 17. 10.  6.  fa.  9. 6.  CMS  19. 43. 59. 52. 63. 120. 171. 2 14. 245. 256. 248. 2 16. 170. 123. 82. 40.  EIST  FBOfl  TBI  MOULD  56.6  44.  42.  26.  88.  37. 77.  32.  S3.  65.  147. 2C7.  139.  121.  196.  169.  100. 137.  51. 76.  250.  215. 26C.  265. 322. 380. 419. 420. 416. 349. 285. 217.  302. 354. 410. 4 16. 411. 339. 276. 211. 154.  303. 338. 3 5 1. 343. 304. 254.  107.  196. 144. 100.  61.  60.  56.  EIST  FBCM  158. 109.  44.  42.  8C.  76. 1 14. 157.  120. 165. 2 10.  200.  251.  239.  290. 323. 338.  275. 303. 314.  316.  299.  278. 232. 181. 137.  266.  Ttil  176. 216. 254. 285. 300. 296. 270. 229. 179. 132. 92. 52.  MOULD  38. 67.  33. 58.  100.  85.  130. 175. 210.  1 14. 146.  243. 267. 277. 267.  97. 126. 164. 202. 232. 249. 250. 232. 20U.  23.  14.  148. 179. 199. 203. 193. 168. 134.  30. 38. 39. 36. 30. 24.  56. 43. 31„  18.  48. 73.  40.  126. 130.  SO. 97.  58. 66.  40.  125.  95.  137.  111.  40. 37.  111. 86.  91. 71.  85. 71.  66. 61.  55.  51. 40.  31. 25. 20. 14.  169.  127.  231. 241.  191.  150. 164.  100.  63. 74 o 75. 68.  66. 110.  178. 208.  133.  45. 27.  37.  15.  49. 65. 96.  1t>3. 131.  82. 62.  9. 7.  9.  7.  85. 110. 140.  148. 1 13.  81. 56. 34.  112. 101.  8.  13.  9. 13. 14. 10.  21.  186.  158. 152. 135. 106.  45. 80. 1 04. 114.  7.  CM c  35. 46.  163.  51. 94. 128. 150.  8. 12.  12.  27.  176.  15.  19.  46. 66.  224.  59.  50. 65. 1 08.  70.8  187.  63.  32.  71. 40.  2 16.  98.  51.  11 . 18. 20.  82. 47.  242. 2C6.  64.  26.  100.  236.  100.  20. 37.  158. 117.  203. 201.  124. 9 1.  CMS  165. 156.  16. 25. 30. 25.  12. 18. 20.  9. 10. 6.  54.  84.  74.  64.  53.  42.  31.  54.  48.  42.  35.  28.  21.  33.  CIST  FfiOM THE  45. 70. 100. 133. 165. 192. 214. 229. 234. 227. 209. 181. 148. 116. 91. 65.  43. 68. 96. 128. 158. 184. 206. 221. 226. 220. 202. 176. 144. 1 14. 86. 63.  EIST  FBOM THE  45. 63. 84. 108. 130. 148. 163. 173. 176. 172. 161. 143. 12 0 . 99. 81. 64.  43. 61. 81. 104. 125. 143. 158. 168. 171. 167. 156. 139. 118. 97. 80. 62.  39. 61. 85. 1 13. 14 1 . 165. 187. 202. 207. 202. 187. 164. 134. 106. 83. 60.  39. 55. 73. 94. 113. 131. 145. 155. 159. 156. 146. 13 1. 111. 91. 75. 59.  MOULD  85.0  34. 53. 73. 96. 120. 144. 164. 179. 186. 183. 170. 150. 123. 98. 77. 55.  29. 43. 59. 74. 95. 117. 137. 153. 160. 159. 150. 133. 110. 87. 69. 50.  MOULD  99  35. 49. 64. 81. 99. 116. 130. 140. 145. 143. 135. 121. 102. 85. 70. 55.  30. 4 1. 53. 65. 82. S7. 112. 122. 128. 127. 120. 109. 92. 77. 63. 50.  23. 34. 43. 48. 64. 67. 108. 124. 133. 134. 127. 114. 95. 76. 60. 43.  1  241  CMS 18. 25. 30. 26.  14. 16. 21. 17.  11. 14. 15. 12.  9. 10. 11. 9.  49. 67. ' 42. 79. 55. 84. 61 . 82. 61. 75. 56. 63. 46. 51. 39. 41. 31. 30. 24.  35. 41. 41. 36. 33. 27. 22. 17.  19. 25. 29. 27.  15. 19. 21. 18.  12. 15. 16. 14.  10. 12. 13. 11.  «; e 7^ 0w .. 62. 89. 91. 88. 80. 69. 58. 48. 3£.  49. 62. 70. 73. 72. 66. 57. 48. 40. 32.  43. 52. 56„ 56. 52. 46. 39. 32. 26.  37. 42. 42. 39. 34. 29. 25. 20.  55. 78. S5. 105. 109. 104. S4. 79. 64. 50. 37.  CMS  24. 33. 41. 46. 60. 77. 91. 102. 109. 109. 105. 95. 81. 68. 56. 44.  242 CIST  FFCM  THE  MOULD  113.3  CMS  44.  42.  39.  35.  3 1.  26.  \ 21.  17.  14.  57.  55.  50.  45.  39.  32.  26.  12.  20.  16.  72. 89. 104.  69. 86. 101.  63. 78. 93.  56. 69. 63.  48. 58. 70.  38. 43. 56.  29. 28.  22. 20.  17.  14. 14.  15.  13.  117.  113.  1C5.  95.  82.  68.  53.  127.  124.  1 15.  105.  92.  78.  63.  134. 136. 134. 126. 115. 99. 85.  130. 133. 131. 123. 112. 97. 63.  122. 125. 123. 116. 106. 92. 79.  112. 1 15. 114. 106. 99. 86. 74.  99. 103. 103. 96. 90. 79. 66.  86. 90. SO. 87. 80. 70. 61.  71. 76. 77. 75. 70. 62. 53.  43. 50. 53.  39. 42.  73. 61.  72. 60.  66. 57.  64. 53.  59. 49.  53. 44.  46. 39.  LIST  FFCM  42.  41.  5C.  49. 59. 71.  61. 73. 84.  THE  MOULD  49. 57. 62. 65. 63. 59. 52. 46. 40. 34.  33. 2d.  38. 46.  34. 41.  30. 36.  26. 31.  22. 26.  18. 21.  14.  55.  16. 13.  50.  36.  "66. 76.  59. 69.  40.  26. 26.  22.  82.  43. 51. 6 1.  18. 17.  16. 14.  50.  21.  93.  91.  85.  78.  69.  100.  98.  85.  76.  67.  57.  105. 107.  1C3. 105. 1 G3.  92. 97.  4 7.  90. 92. 91.  81. 84.  72. 75. 75.  62. 65. 66.  52. 56. 57.  73. 68.  64. 60.  61.  55.  55. 49.  49. 44.  43.  39.  99. 98.  84.  100.  se.  94.  88.  81.  93.  91.  81.  83. 73.  81.  87. 77.  75. 67. 60.  65. 57.  42. 40. 36. 31. 27. 24.  CM c  127.4  52. 60.  105.  52. 49. 44. 38.  72. 64.  69.  73. 65.  61.  58.  56.  54.  51.  54. 47.  43. 47.  15.  49.  39. 4 1.  56,  46.  41.  53. 48. 43.  46. 42. 37.  3S. 36.  39. 34.  34. 30.  29. 26.  32.  EIS1 FEOM THE HCU1D J8. 43. 51. 59. 67. 7 3. 79. 82. 84. 83. 79. 74. 67. 61. 56. 51.  37. 42. 49. 58. 65. 72. 77. £1. 82. 61. 78. 73. 66. 60. 55. 50.  34. 39. 46. 54. 61. 68. 73. 76. 78. 77. 74. 70. 63. 56. 53. 48.  141.6  28. 32. 36. 32. 37. 42. 49. 43. 57. 51. 57. 63. 68. 62. 66. 72. 68. 73. 73. 67. 70. 65. 66. . 61. 60. 56. 55. 51. 47. 50. 46. 43.  CBS  25. 28. 32. 36. 45. 51. 56. 59. 61. 61. 60. 56. 52. 47. 43. 39.  21. 24. 26. 26.  18. 20. 21. 20.  16. 17. 18. 17.  14. 15. 16. 15.  /  45. 49. 52. 54. 55. 54. 51. 47. 43. 39. 36.  43./ 46. 48. 49. 48. 45. 42. 38. 35. 32.  39. 42. 43. 42. 40. 37. 34. 32. 29.  37. 38. 38. 36. 33. 31. 26. 2b.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0079161/manifest

Comment

Related Items