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

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THREE DIMENSIONAL HEAT FLOW IN THE DIRECT CHILL CASTING OF NON-FERROUS METALS by V. VENKATESWARAN •Sc. (Hons) , Bangalore U n i v e r s i t y , I n d i a , 1969 B.E. , Ind ian I n s t i t u t e of Sc ience, 1972 A. Sc . , The U n i v e r s i t y of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of M e t a l l u r g i c a l Engineer ing We accept t h i s t hes i s as conforming to the requ i red standard THE UNIVERSITY OF BRITISH COLUMBIA Ju ly 1980 © V. Venkateswaran, 1980 ABSTRACT A three dimensional mathematical model has been developed to study heat f low and s o l i d i f i c a t i o n in the D i r e c t C h i l l c a s t i n g of non- fe r rous metals w i t h r e c t a n -gu lar as we l l as i r r e g u l a r c r o s s - s e c t i o n s . The model which is based on an a l t e r n a t i n g d i r e c t i o n , i m p l i c i t f i n i t e - d i f f e r e n c e numerical method is capable of s i m u l a t i n g heat f low both in the steady s t a t e and t r a n s i e n t p a r t of 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 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 and pool depths w i t h i n d u s t r i a l measurements. The model has been used to study the importance o f heat f lows in the var ious d i r e c t i o n s , and the l i m i t a t i o n s of using two-dimensional heat f low models are brought o u t . This study has shown t h a t a two-d imensional model which neg lec ts heat f low normal to the narrow face can be used to s imu la te the s o l i d i f i c a t i o n of slabs w i t h aspect r a t i o s g rea te r than 2 . 5 , cast under c o n d i t i o n s of convent iona l D.C. c o o l i n g . Fur ther i t was demonstrated t h a t w i t h r e -duced secondary c o o l i n g , a two-dimensional model t h a t neg lects a x i a l heat conduct ion is p r e f e r a b l e . Model c a l c u -l a t i o n s show t h a t in coo l ing la rge sec t ions the unsteady s t a t e can occupy around 25% of the t o t a l c a s t i n g c y c l e . i i The fo rmat ion of cracks in jumbo i n g o t s of Prime Western Grade z inc has been i n v e s t i g a t e d w i t h the a id o f the mathematical model. I t has been shown t h a t the c rack ing is caused by rehea t ing of the sur face below the spray c o o l i n g zone, i f t h i s zone is shor t and c h a r a c t e r i z e d by a high water f l u x . The sur face rehea t ing generates t e n s i l e s t r a i n s at the s o l i d i f i c a t i o n f r o n t where c r a c k i n g is a ided by the presence of lead r i c h l i q u i d in the i n t e r - d e n d r i t i c r e g i o n s . A new spray assembly has been designed which at tempts to cool the cas t i ng more u n i f o r m l y from, the top of the mould to the bottom of the l i q u i d p o o l . This new spray system has been tes ted i n - p l a n t f o r c a s t i n g Prime Western Grade z inc and r e s u l t s have proven i t s e f f e c t i v e -ness in p reven t ing crack f o r m a t i o n . TABLE OF CONTENTS Page Abs t rac t i i Tab! e of Contents i v L i s t of Tables v i L i s t o f Figures v i i i Acknowledgements x i v Chapter 1 I n t r o d u c t i o n 1 1.1 Ob jec t ives of the Present Work 2 2 Review of the L i t e r a t u r e 3 2.1 I n t r o d u c t i o n 3 2.2 D.C. Cast ing 4 2.3 Review of Mathematical Models in D.C. Cast ing 8 3 Development o f the Heat Flow Model 16 3.1 I n t r o d u c t i o n 16 3.2 Assumptions Made in the Model 17 3.3 Heat Flow Equations and Boundary Condi t ions 19 3.4 Method of Sol u t i on 23 3.5 Mathematical Check f o r I n t e r n a l Con-s i stency of the Computer Program 34 3.6 Flow Chart of the Computer Program. . 44 4 V a l i d a t i o n of the Results From the Mathematical Model 47 4.1 I n t r o d u c t i o n 47 4.2 Conventional D.C. Cast ing of Aluminium - Alcan 48 4 . 2 . 1 Aluminium Ingots 381 x 991mm. 48 4 .2 .2 Aluminium Ingots 457 x 1143 mm 56 4 . 2 . 3 Aluminium Ingots 305 x 1010 mm 59 4 .2 .4 Aluminium Ingots 229 x 813mm. 59 4 .2 .5 Summary of Convent ional D.C. Cast ing S imula t ions 63 4.3 Reduced Secondary Cool ing - B r i t i s h Al umi n i um 66 4.4 Zinc-Jumbo Casting - Cominco 76 4.5 Summary o f V a l i d a t i o n Runs 92 i v Chapter Page 5 E f f e c t of Cast ing Var iab les on Heat Flow. 97 5.1 I n t r o d u c t i o n 97 5.2 E f f e c t of Aspect Rat io 98 5.3 E f f e c t of Ax ia l Conduction 104 5.4 Importance o f Unsteady State 110 5.5 E f f e c t of Sect ion Size 115 5.6 E f f e c t of Superheat 119 5.7 E f f e c t of Cool ing Condi t ions 119 5.8 E f f e c t of Latent Heat Release on Model Ca lcu la t i ons 121 5.9 Summary 122 6 Use of the Heat Flow Model to Solve a Cracking Problem in the D.C. Cast ing of Prime Western Grade Jumbo Ingots 124 6.1 I n t r o d u c t i o n 124 6.2 I n t e r n a l Cracks in the D.C. Cast ing of Prime Western Grade Zinc 125 6.3 Heat Flow Ana lys is 134 6.4 Metal 1ographic Analys is 140 6.5 Mechanism of Crack Formation 144 6.6 Design of the New Cool ing System . . . . 150 6.7 Tes t ing of the New Cool ing System . . 154 6.8 Summary 155 7 Summary and Conclusions 157 Symbols 160 B ib l i og raphy 162 Appendi ces Al Development of F i n i t e D i f fe rence Equation 170 A l . l A l t e r n a t i n g D i r e c t i o n F i n i t e D i f -ference Equations f o r Three Dimensional Problems 171 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 fe rence Using Convect ive Type Boundary Condi t ions 181 A2 Source L i s t i n g of the Computer Program . . 183 A3 Three Dimensional Temperature D i s t r i -b u t i o n i n the Cast ing f o r the D i f f e r e n t Runs 227 v LIST OF TABLES Tables Page I Comparison between numerical and a n a l y t i c a l values o f temperatures (°C) at the cen t re of a cube, as a f u n c t i o n of t ime f o r d i f -f e r e n t s izes of nodes 37 I I Comparison between numerical and a n a l y t i c a l values o f cent re temperatures (°C) o f a cube, as a f u n c t i o n of t ime f o r d i f f e r e n t t ime steps 39 I I I Thermophysical p r o p e r t i e s of a luminium used in convent iona l f l o o d coo l i ng s i m u l a t i o n s 49 IV Comparison between c a l c u l a t e d and measured pool depths f o r aluminium sec t ions cas t at di f f e rent speeds 64 V Thermophysical p r o p e r t i e s of a luminium used in reduced secondary coo l ing s i m u l a t i o n s . . . 67 VI 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 sur face f o r reduced secondary c o o l i n g 68 V I I Thermophysical p r o p e r t i e s of z inc used in jumbo i n g o t s i m u l a t i o n 79 V I I I H e a t - t r a n s f e r c o e f f i c i e n t s used in z inc jumbo c a s t i n g 81 IX Comparison between three dimensional and two dimensional pool depths f o r reduced secondary coo l ing of 254 x 690 mm aluminium ingo t 0 3 X Steady s t a t e pool depths f o r aluminium and z inc ingo ts c a l c u l a t e d w i t h and w i t h o u t the a x i a l conduct ion ^ ^ v i Tab 1es Page XI Cast ing c o n d i t i o n s f o r the d i f f e r e n t runs dur ing the exper imental campaign 126 X I I Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the l e n g t h of s t rands A and B (Run 1) . . . 128 X I I I Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the l eng th of s t rands A and B (Run 2) 129 XIV Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken a t var ious po in ts along the l e n g t h o f s t rands A and B (Run 3) 130 XV Locat ion o f cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the l eng th o f s t rands A and B (Run 4) 131 XVI Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the l e n g t h of Strands A.and B (Run 5) 132 XVII Locat ion o f cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the l e n g t h o f s t rands A and B (Run 6) 133 XVI I I Ca lcu la ted values of reheat at d i f f e r e n t po in t s on the sur face o f the jumbo s e c t i o n . Top and bottom correspond to mid- face on the non-notched surfaces 148 A3.1 Three-dimensional temperature d i s t r i b u t i o n i n 381 x 991 mm aluminium ingo t cas t a t 1.775 mm/s (convent iona l c o o l i n g ) . .•• 228 A 3 . I I Three-dimensional temperature d i s t r i b u t i o n in 254 x 690 mm aluminium ingo t cast at 0.833 mm/s (reduced secondary c o o l i n g ) 234 A 3 . I l l Three-dimensional temperature d i s t r i b u t i o n in z inc jumbo ingo t cast at 1.27 mm/s 238 vi i LIST OF FIGURES Figure Page 2.1 A schematic diagram of the openhead D i rec t C h i l l cas t i ng process 5 3.1 The e f f e c t of the number of nodes on per-cent e r r o r at the end of d i f f e r e n t t ime i n t e r v a l s 38 3.2 Comparison between a n a l y t i c a l and numerical c a l c u l a t e d sur face temperatures f o r l a rge values of time i n t e r v a l 42 3.3 Stable o s c i l l a t o r y nature of the numer i ca l l y c a l c u l a t e d surface temperatures f o r la rge values of t ime i n t e r v a l 43 3 .4(a) Flow char t of the computer program 45 3 .4(b) Flow char t of the computer program (cont inued from F i g . 3 . 4 ( a ) ) 46 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 f o r 381 x 991 mm aluminium ingo t cast at 1.778 mm/s (ob ta ined at the mid-plane p a r a l l e l to the narrow face) 51 4.2 Steady s t a t e pool p r o f i l e s ob ta ined at the l o n g i t u d i n a l mid-planes f o r 381 x 991 mm aluminium ingo t cast at 1.778 mm/s 53 4.3 Comparison of the pool p r o f i l e s from the two-dimensional and th ree-d imens iona l c a l c u l a t i o n s f o r cas t ing 381 x 991 mm aluminium ingo t a t 1.778 mm/s 55 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 ingo t cast a t d i f f e r e n t speeds . . . 57 v i i i Fi gure Page 4.5 Ca lcu la ted pool p r o f i l e s f o r 381 x 991 mm aluminium ingo t cast 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 ingo t cast at d i f f e r e n t speeds . . . 60 4.7 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 ingot cast at d i f f e r e n t speeds . . . 61 4.8 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 ingo t cast at d i f f e r e n t speeds . . . 62 4.9 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 aluminium ingo ts o f va r ious s izes as a f u n c t i o n of cas t ing speed 65 4.10 L iqu idus and so l idus p r o f i l e s ob ta ined at the l o n g i t u d i n a l mid-planes of 254 x 690 mm aluminium ingo 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 4.11 L iqu idus and so l i dus p r o f i l e s ob ta ined at the l o n g i t u d i n a l mid-planes of 254 x 690 mm aluminium ingo t cast under reduced coo l i ng 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 4.12 L iqu idus and so l idus p r o f i l e s ob ta ined at the l o n g i t u d i n a l mid-planes o f 254 x 690 mm aluminium ingot cast under reduced secondary coo l i ng cond i t i ons at 0.833 mm/s, a f t e r 1 248 s from s t a r t 71 4.13 Comparison of steady s ta te pool p r o f i l e s f o r the two-dimensional 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 ingo t cast under reduced secondary c o o l i n g cond i t i ons at .833 mm/s 73 4.14 Three-dimensional v i s u a l i z a t i o n o f l i q u i d pool sur face of 254 x 690 mm aluminium ingo t cast at .833 mm/s under reduced secondary c o o l i n g , as seen from the d i f f e r e n t angles 74 i x Fi gure Page 4.15 A cross sec t ion o f the jumbo ingo t w i t h the var ious dimensions given in 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 the f i n i t e - d i f f e r e n c e c a l c u l a t i o n s 78 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 f o r the z inc jumbo ingo t cast at 1.69 mm/s 82 4.18 A th ree-d imens iona l view of the jumbo ingo t showing the l o n g i t u d i n a l sec t ions where the pool p r o f i l e s have been o b t a i n e d . . 83 4.19 Pool p r o f i l e s obta ined at the l o n g i t u d i n a l sec t ions shown in F i g . 4.18 f o r c a s t i n g z inc jumbo ingo t at 1.27 mm/s, 278 s a f t e r the s t a r t 84 4.20 Pool p r o f i l e s obta ined at the l o n g i t u d i n a l sec t ions shown in F i g . 4.18 f o r c a s t i n g z inc jumbo ingo t at 1.27 mm/s, 557 s a f t e r the s t a r t 85 4.21 Steady s t a t e pool p r o f i l e s obta ined at the l o n g i t u d i n a l sec t ions shown in F i g . 4.18 f o r c a s t i n g z inc jumbo ingo t at 1.27 mm/s . . 8 6 4.22 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 of z inc jumbo ingo t at 1.27 mm/s 88 4.23 Three-dimensional v i s u a l i z a t i o n of the l i q u i d pool sur face of z inc jumbo i n g o t cast at 1.27 mm/s 89 4.24 Comparison between the c a l c u l a t e d and measured pool depths obta ined at d i f f e r e n t t imes from the s t a r t of c a s t i n g of z inc jumbo ingo t cast at 76 mm/min 90 4.25 Comparison between the c a l c u l a t e d and measured pool depths obta ined at d i f f e r e n t t imes from the s t a r t of cas t ing of z inc jumbo i n g o t cast at 102 mm/min 91 4.26 Freezing p r o f i l e s as seen on a jumbo c r o s s - s e c t i o n 93 x Fi gure Page 4.27 Macros t ruc ture o f the zinc jumbo c r o s s -sec t i on showing the f r e e z i n g l i n e s seen in F i g . 4.26 94 4.28 Comparison of the c a l c u l a t e d pool p r o f i l e s obta ined w i th and w i t h o u t the notch f o r c a s t i n g z inc at 1.69 mm/s 95 5.1 E f f e c t of 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 5.2 E f f e c t o f aspect r a t i o s on the pool depths in c a s t i n g 381 mm t h i c k z inc s labs a t 1 .78 and 2.2 mm/s 101 5.3 Ca lcu la ted sur face temperature p r o f i l e s ob ta ined at the mid- face o f 457 x 457 mm aluminium ingo t cast at .974 mm/s, w i t h and w i t h o u t the ax ia l conduct ion 105 5.4 Ca lcu la ted centre temperature p r o f i l e s obta ined f o r 457 x 457 mm aluminium i n g o t cast at .974 mm/s, w i th and w i t h o u t the a x i a l conduct ion 107 5.5 Ca lcu la ted sur face temperature p r o f i l e s f o r the i n i t i a l and steady s ta te s l i c e s f o r c a s t i n g aluminium jumbo ingo t (a t the bottom mid- face of a jumbo s e c t i o n ) 112 5.6 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 ta te f o r 381 mm t h i c k aluminium and z inc slabs of d i f f e r e n t aspect r a t i o s cast at 1.778 mm/s ,. . . 114 5.7 Steady s t a t e pool depths f o r square sec t i ons of aluminium and zinc cast at 1.778 mm/s . . . 116 5.8 Ca lcu la ted steady s ta te pool p r o f i l e s f o r 381 mm square sec t ions of aluminium and z inc cast at 1.778 mm/s 117 5.9 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 of aluminium and z inc cast at 1.778 mm/s 118 xi Fi gure Page 5.10 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 ta te pool depths f o r c a s t i n g 61 0 x 546 mm z inc ingo t at 1 mm/s 120 6.1 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 cracks . . . . 135 6.2 Ca lcu la ted 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 inc jumbo i n g o t cast at 1 .69 mm/s 1 37 6.3 Growth o f the s h e l l as a f u n c t i o n o f d is tance in the a x i a l d i r e c t i o n f o r z inc jumbo i n g o t cast at 1.69 mm/s. F igure on the r i g h t shows the sur face te rmpera tu re p r o f i l e at the bottom mid- face o f a jumbo sec t i on 139 6.4 A macro-photograph of the cracked s u r f a c e . 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 cracked s u r f a c e . M a g n i f i c a t i o n 200 X 1 42 6 .6 (a ) Scanning e l e c t r o n micrograph of a cracked sur face r e v e a l i n g the smooth nature o f the s u r f a c e . M a g n i f i c a t i o n 1 000 X 1 43 6 .6(b) Pb x - ray p i c t u r e of F i g . 6 .6 (a) 143 6 .7 (a) Scanning e l e c t r o n micrograph of a cracked sur face r e v e a l i n g the smooth nature o f the s u r f a c e . M a g n i f i c a t i o n 1000 X 1 45 6 .7(b) Pb x - ray p i c t u r e of F i g . 6 .7 (a) 145 6.8 Phase diagram of Pb- Zn system (89) 146 6.9 E f f e c t o f cas t i ng speed on the sur face rehea t ing at the bottom mid- face o f a jumbo s e c t i o n 149 6.10 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 in the sub-mould region on the sur face r e -hea t ing at the bottom mid- face of a jumbo sec t ion I 5 ' x i i Fi gure Page 6.11 Arrangement of spray nozzles in the new c o o l i n g assembly f o r the bottom sur face of a jumbo sec t ion 153 6.12 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 th the new c o o l i n g system ; . . . 156 A1.1 Dotted reg ion is the volume over which c a l c u l a t i o n s are performed 172 A1.2 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 p a r a l l e o l -piped showing the sur face nodes 173 x i i i ACKNOWLEDGEMENTS The author wishes to express h is s i nce re g r a t i t u d e to Dr. J . K. Brimacombe f o r h is guidance th roughou t the course of t h i s research . The author a lso would l i k e to acknowledge the e f f o r t s of Mr. J . Su ther land o f Alcan and Mr. D. D. B e a t t i e of B r i t i s h Aluminium in p r o v i d i n g e x p e r i -mental r e s u l t s . Thanks are expressed to Mr. Roger Watson f o r i n i t i a t i n g the p r o j e c t on the c rack ing problem and to Mr. C. A. Su the r l and , Dr. G. W. Toop, Mr. M. L. C o n n o l l y , Mr. P. J . P laye r , Mr. J . Newton, Mr. C. A. Johnson and Mr. R. Joseph of Cominco f o r t h e i r help and c o - o p e r a t i o n dur ing the var ious stages of t h i s p r o j e c t . The d iscuss ions and ass is tance of f e l l o w graduate s t u d e n t s , f a c u l t y members and techn ic ians have been i n -v a l u a b l e . The author is g r a t e f u l to the U n i v e r s i t y of B r i t i s h Columbia f o r p r o v i d i n g f i n a n c i a l suppor t i n the form of a UBC Graduate Fe l lowsh ip . Chapter 1 INTRODUCTION Over the l a s t h a l f century a number o f new processes have been developed in both the fe r rous and n o n - f e r r o u s i n d u s t r i e s . The m a j o r i t y of these new processes have been brought about to increase produc t ion and t o e f f e c t improve-ment in o v e r a l l e f f i c i e n c y . Examples are the cont inuous c a s t i n g of s t e e l , D i rec t C h i l l c a s t i n g of n o n - f e r r o u s meta ls , BOF and Q-BOP s tee l making e t c . A l though these processes have a l l been commerc ia l ised, improvement in the e f f i c i e n c y of p roduc t ion and the q u a l i t y o f the product i s c o n t i n u a l l y sought . This requ i res an unders tand ing o f the process and the r e l a t i o n s h i p s between fundamental and process v a r i a b l e s . Mathematical models are va luab le t o o l s in t h i s regard as they prov ide an inexpensive way o f l e a r n i n g about the process and of s tudy ing the e f f e c t of process v a r i a b l e s on the o v e r a l l o p e r a t i o n . A number o f mathematical models have been developed f o r the i n v e s t i g a t i o n of heat f low in f e r r o u s and non-fe r rous cont inuous c a s t i n g . However only a handful o f these have been used beyond the developmental stage to p r e d i c t s o l i d i f i c a t i o n s t r u c t u r e and to determine con-d i t i o n s 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 c r a c k s . Almost a l l 1 2 the models developed in t h i s area to date are e i t h e r one or two dimensional in n a t u r e . Although they are adequate f o r the s i m u l a t i o n of the cont inuous c a s t i n g of s t e e l , t h e i r 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 of heat f l ow in D.C. c a s t i n g of rec tangu la r sec t ions o f n o n - f e r r o u s metals w i t h low aspect r a t i o s . Three-d imensional heat f l ow models are r e q u i r e d under these c o n d i t i o n s . Rapid deve lop-ments in the f i e l d of high-speed d i g i t a l computers have made i t poss ib le to develop these models. 1.1 Ob jec t i ves of the Present Work The pr imary o b j e c t i v e o f t h i s s tudy was to develop a f u l l y th ree-d imens iona l heat f low model to s imu la te the D i r e c t C h i l l c a s t i n g of non- fe r rous m e t a l s . Because o f the semi-cont inuous nature o f t h i s process, i t was a lso d e s i r e d to i nc lude the t r a n s i e n t i n i t i a l p o r t i o n o f the c a s t i n g in the c a l c u l a t i o n s . The second o b j e c t i v e o f t h i s s tudy was to v a l i d a t e the model developed w i th i n d u s t r i a l measurements and then use the model to demonstrate the importance of the d i f f e r e n t components of heat f lows under d i f f e r e n t c a s t i n g c o n d i t i o n s . F i n a l l y the most impor tant o b j e c t i v e of t h i s study was to use the model to a s s i s t in s o l v i n g a c rack ing problem in the D.C. cas t ing of Prime Western Grade z i n c . Chapter 2 REVIEW OF THE LITERATURE 2.1 I n t r o d u c t i o n D i r e c t C h i l l c a s t i n g , more commonly known as D.C. c a s t i n g , was developed in the l a t e 1 930' s and, s ince t h a t t i m e , has become the "work horse" of a modern n o n - f e r r o u s c a s t i n g p l a n t . In a d d i t i o n to being r e l i a b l e , i t has proved to be a very economical p roduc t ion techn ique f o r c a s t i n g non - fe r rous metals such as a lumin ium, copper , magnesium and z i n c . Most sec t ions t h a t are D.C. cas t are e i t h e r c i r c u l a r f o r e x t r u s i o n a p p l i c a t i o n s or r e c t a n g u l a r f o r r o l l i n g a p p l i c a t i o n s ; but o ther shapes are a lso pro-duced. Examples are the T - i n g o t s in the aluminium i n -dus t ry and the jumbo i n g o t of spec ia l shape in the z inc i n d u s t r y . Modern day D.C. cas t i ng machines have enormous p roduc t ion c a p a c i t i e s . As an example, the s lab c a s t i n g f a c i l i t y at Alcan in Oswego, New York can produce.5, 7 or 9 slabs at a t ime of 460 mm t h i c k n e s s , 2200 mm maximum wid th and 5100 mm l e n g t h , e q u i v a l e n t to a cas t we igh t per drop ranging from 35 to 45 tonnes ( 3 ) . I t i s i n t e r e s t i n g 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 cont inuous c a s t i n g of s t e e l . 3 Two impor tan t d i f f e r e n c e s tha t can be seen are the leng th of the mould and the c a s t i n g speed. D.C. c a s t i n g moulds are c h a r a c t e r i z e d by t h e i r shor t l e n g t h s . - I t is very common to f i n d moulds only 20 to 50 mm long f o r c a s t i n g aluminium and z inc in comparison w i th 600-900 mm long moulds used in the case of s t e e l . The c a s t i n g speeds encountered in D.C. cas t ings are an order of magnitude lower than those in s t e e l . One other f a c t o r which d i s -t i n g u i s h e s D.C. c a s t i n g is the semi-cont inuous nature of i t s o p e r a t i o n , a l though in recent years a cont inuous h o r i z o n t a l cas te r has been developed ( 4 - 7 ) . The use of h o r i z o n t a l D.C. cas t ing however is r e s t r i c t e d to c a s t i n g smal le r sec t ions and a l l o y s of low s t r e n g t h . The model developed in t h i s study has been used f o r the a n a l y s i s of heat f low in v e r t i c a l D.C. cas t i ng o p e r a t i o n s . 2.2 D.C. Cast ing A schematic diagram of the D i r e c t C h i l l c a s t i n g process is shown in F i g . 2 . 1 . As is obvious from the name the sur face of the c a s t i n g in t h i s ope ra t i on is c h i l l e d by the mould c o o l i n g water which e x i t s the mould and impinges d i r e c t l y on the s o l i d i f y i n g m e t a l . B a s i c a l l y , coo l i ng of the c a s t i n g in a convent iona l D.C. c a s t i n g process is c a r r i e d out in three s tages: pr imary c o o l i n g in the water j a c k e t e d mould, secondary c o o l i n g by the f l o o d 5 Floating , —Liquid Metal Distributor 11 = Solid Stool Schematic of Open-head D.C. Casting F i g . 2.1 A schematic diagram of the openhead D i r e c t C h i l l Cast ing Process. 6 water imping ing on the cas t ing and f i n a l l y t e r t i a r y c o o l i n g by the stagnant water pool in a w a t e r - c o l l e c t i o n p i t . In some spec ia l a p p l i c a t i o n s minor v a r i a t i o n o f the above p r a c t i c e has been noted (28-30, 33) . The sequence of opera t ions c a r r i e d out du r ing D.C. c a s t i n g are as f o l l o w s . The s too l or bottom block is ra i sed i n t o the mould to cover the opening and l i q u i d metal i s pumped i n t o the mould. Once the l i q u i d metal reaches a c e r t a i n l eve l the bottom block is lowered at a c o n t r o l l e d r a t e . The leve l of metal i n the mould i s very c r i t i c a l from the s t a n d p o i n t of sur face q u a l i t y of the c a s t i n g and i s mainta ined constant through the use of a f l o a t va lve assembly. The exact value o f the metal head in the mould w i l l depend on a number of f a c t o r s i n c l u d i n g the c a s t i n g speed, the a l l o y c a s t , the sec t i on s ize and the pour ing tempera tu re . When the leng th o f the cas t i ng reaches the bottom of the p i t the c a s t i n g i s t e r m i n a t e d . The e n t i r e ope ra t i on is repeated once the c a s t i n g i s taken out of the p i t . In a semi-cont inuous D.C. c a s t i n g opera t ion l u b r i c a n t is normal ly ap-p l i e d on an i n t e r m i t t e n t basis to the mould at the beginning of a c a s t i n g . However cont inuous a p p l i c a t i o n of l u b r i c a n t is p r a c t i c e d in some h o r i z o n t a l machines. Even though the d e s c r i p t i o n given in the preceding paragraph is a t y p i c a l D.C. cas t ing sequence, the procedure is o f ten mod i f i ed f o r spec ia l cases. For example, w h i l e cas t i ng aluminium slabs f o r deep drawing a p p l i c a t i o n the cas t ing is sub jec ted to a reduced secondary spray c o o l i n g in place of convent iona l f l o o d coo l ing to o b t a i n a l a r g e r c e l l s t r u c t u r e (28 -30 , 33 ) . S i m i l a r l y in a z inc D.C. c a s t i n g opera t ion the water j acke ted mould is rep laced by a s o l i d mould which is cooled by water sprays . A number of papers have been pub l i shed d e a l i n g w i t h the var ious aspects of the D.C. cas t i ng o p e r a t i o n ( 1 - 4 8 ) . Emley (2) has r e c e n t l y pub l ished a complete survey o f the cont inuous c a s t i n g of aluminium i n c l u d i n g D.C. c a s t i n g . A s i m i l a r account f o r the case of copper and i t s a l l o y s has been pub l ished by K r e i l e t al ( 1 7 ) . Since the sur face and sub-sur face q u a l i t y o f the D.C. cast ingo ts determines the amount of s c a l p i n g to be c a r r i e d out p r i o r to f a b r i c a t i o n , cons iderab le research e f f o r t has been d i r e c t e d towards improving the sur face q u a l i t y (13, 18, 20-22, 24, 4 4 ) . The use of s h o r t e r moulds, which r e s u l t s in a h igher q u a l i t y sur face i s made poss ib le by i n s u l a t i n g the top p o r t i o n of the normal mould w i t h i n s u l a t i n g m a r i n i t e l i n i n g . Recent ly an e l e c t r o -magnetic mould has been developed in Russia (18) i n which the metal does not contac t the mould s u r f a c e . This has been shown to y i e l d high q u a l i t y ingots which could be f a b r i c a t e d w i t h o u t the in te rmed ia te s c a l p i n g o p e r a t i o n . Adoption of t h i s novel technique in a c a s t i n g p l a n t in Sw i t ze r land has been descr ibed by Meier et a l ( 4 4 ) . L ike o ther i n d u s t r i e s the i n f l u e n c e o f computers has a lso been f e l t in D.C. cas t i ng o p e r a t i o n s , where f o r exampl mic ro-processors are being used f o r o n - l i n e c o n t r o l ( 4 3 ) . 2.3 Review of Mathematical Models in D.C. c a s t i n g The f i r s t mathematical model of heat f l ow in D.C. c a s t i n g was pub l i shed by Roth (26) in 1943. In o rder to solve the mathematical equat ions a n a l y t i c a l l y t h i s author had to make severa l s i m p l i f y i n g assumpt ions, l i k e n e g l i -g i b l e heat t r a n s f e r in the mould reg ion of the c a s t i n g and constant sur face temperature below the mould. Because of the s i m p l i s t i c nature of t h i s model very poor agreement was obta ined between p r e d i c t e d and measured s h e l l t h i c k n e s s e s . However in s p i t e of the ra the r crude nature of t h i s model, c r e d i t must be given f o r a t tempt ing to p lace the c a s t i n g opera t ion w i t h i n a mathematical framework. A more r e f i n e d model employing a numer ical s o l u t i o n was proposed by Adenis et al (27) in the e a r l y 1960 's . The 9 model was developed to s imula te steady s t a t e heat f low in c a s t i n g c y l i n d r i c a l magnesium a l l o y i n g o t s . Because of the ra the r long mould used in t h e i r study (240' mm), the heat f low in the mould reg ion was d i v i ded i n t o th ree zones, the top zone having p e r f e c t contact between the molten metal and mould, the in te rmed ia te zone having a f i l m of o i l between the c a s t i n g and ins ide of the mould and f i n a l l y the bottom zone having an a i r gap. These authors repor ted 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 r a t h e r than the s o l i d u s in t h i s comparison. The main u n c e r t a i n t y in t h e i r model was assoc ia ted w i th c h a r a c t e r i z i n g boundary c o n d i t i o n s in the 1ong mould reg i on. Kroeger and Ostrach (31) have s imula ted the s teady-s t a t e temperature as we l l as f l u i d f low r e s u l t i n g from n a t u r a l convect ion dur ing the cas t ing of a c y l i n d r i c a l i n g o t of a pure m e t a l . Assuming constant temperature as the boundary c o n d i t i o n on the sur face of the i ngo t f o r the heat f low model, these authors have shown t h a t s u b s t a n t i a l f l u i d v e l o c i t i e s can develop from n a t u r a l c o n v e c t i o n . However i t was also shown t h a t the s t rong v e l o c i t y f i e l d had n e g l i g i b l e e f f e c t on the l o c a t i o n of the s o l i d - l i q u i d i n t e r f a c e . 10 The mathematical model descr ibed by Peel and Pengel ly (28, 29) was also developed f o r heat f low in c y l i n d r i c a l ingots as we l l as s t e a d y - s t a t e c o n d i t i o n s . Because of the long moulds used and small diameters c a s t . t h e bottom of the pool was very c lose to the bottom of the mould. Since a s u b s t a n t i a l p o r t i o n of the t o t a l heat was removed in the mould r e g i o n , these authors have fo rmula ted a v a r i a b l e res i s tance model f o r d e s c r i b i n g heat f low through the a i r gap. Exper imental r e s u l t s were l a t e r used f o r " f i n e t u n i n g " the model. An a t tempt to p r e d i c t the c e l l s t r u c t u r e us ing the average c o o l i n g ra te in the 1 i q u i d u s - s o l idus temperature range was not successfu l a l though b e t t e r agreement was obta ined wh i le using the g rad ien t of the c o o l i n g curve at the s o l i d u s tempera tu re . Even w i th t h i s new techn ique a match between the measured and c a l c u l a t e d values of c e l l s ize could only be considered q u a l i t a t i v e . The importance of the proper c h a r a c t e r i z a t i o n of boundary c o n d i t i o n s has been we l l emphasized in t h e i r work. Mathew (32) has made an ana lys is of both heat f l o w and thermal s t resses dur ing cont inuous c a s t i n g . This model was. fo rmula ted to c a l c u l a t e the s t e a d y - s t a t e temperature and s t ress d i s t r i b u t i o n in a c y l i n d r i c a l i n g o t . Un l i ke the prev ious models a f i n i t e - e l e m e n t numerical procedure was adopted f o r s o l v i n g the heat f low 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 in hand l ing v a r i o u s types o f boundary c o n d i t i o n s has been s t ressed in t h i s work . Al though the model was set up f o r c a l c u l a t i n g heat f low in a c y l i n d e r , i t has a lso been used f o r square s e c t i o n s . The s t ress c a l c u l a t i o n s c a r r i e d out in t h i s study are o f doub t fu l value i n a q u a n t i t a t i v e sense because of the un-c e r t a i n t y in the high temperature mechanical p r o p e r t i e s o f metals near t h e i r me l t i ng p o i n t . Cont inu ing on the same l i n e as Peel and Pengal ly ( 2 8 , 29) , B e a t t i e (30, 33) has developed a model f o r c a l c u l a t i n g heat f low in r e c t a n g u l a r s l a b s . Based on s t e a d y - s t a t e p r i n c i -ples h is model takes i n t o account heat f l ow normal to the broad face and in the a x i a l d i r e c t i o n . Carefu l exper imenta l work was undertaken to c h a r a c t e r i z e boundary c o n d i t i o n s . The, temperatures measured by imp lan t i ng thermocouples at var ious l o c a t i o n s i n the mould were used in c a l c u l a t i n g the heat f l u x boundary c o n d i t i o n s . The parameter employed f o r p r e d i c t i n g the c e l l s t r u c t u r e is d i f f e r e n t f rom t h a t repor ted by Peel and Penge l ly . Instead of us ing the g rad ien t at the s o l i d u s tempera ture , t h i s author has used the g rad ien t at a temperature in the s o l i d u s - 1 i q u i d u s range, in order to ob ta in a good match between c a l c u l a t e d and p r e d i c t e d c e l l s i z e s . The mesh s ize employed i n h i s numerical model had to be much f i n e r than t h a t used by 1 2 Peel and Pengel ly . (2 .5 mm ins tead of 10 mm) to o b t a i n reasonable c e l l s i ze p r e d i c t i o n s . In s p i t e o f these r e -f inements major d isc repanc ies were observed in p r e d i c t i n g c e l l s t r u c t u r e s f o r a rec tangu la r 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 . This may a r i s e be-cause heat f low perpend icu la r to the narrow face has been n e g l e c t e d . Fossheim and Madsen (48) have developed heat f low models f o r c y l i n d r i c a l as wel l as r e c t a n g u l a r i n g o t s . The i r r e c t a n g u l a r model again corresponded to a two dimensional heat f low in the a x i a l and one t ransverse d i r e c t i o n . Space d i s c r e t i z a t i o n in t h e i r model was based on a box i n t e g r a t i o n method and the t ime i n t e g r a t i o n on an exponen t ia l t r a n s -fo rmat ion of the heat conduct ion equat ion w i t h an a l t e r n a -t i n g d i r e c t i o n i m p l i c i t techn ique. As w i l l be shown in Chapter 5, the use of a two-dimensional model f o r s i m u l a t i n g heat f low in a r e c t a n g u l a r slab 381 x 250 mm (aspect r a t i o 1.5) can lead to cons iderab le e r r o r . In a d d i t i o n to t h i s , l ook ing at t h e i r contour p r o f i l e s i t appears t h a t the authors have made an i n c o r r e c t choice rega rd ing the second dimension f o r heat f l o w . Instead of c o n s i d e r i n g the smal le r o f the two s e c t i o n s , namely 250 mm, they have performed the c a l c u l a t i o n s f o r a th ickness of 381 mm. 1 3 Jov ic et al (35) have r e c e n t l y developed a two-dimensional model f o r c a l c u l a t i n g the temperature f i e l d in con t i nuous l y cast r e c t a n g u l a r slabs o f a luminium a l l o y s , as wel l as p r e d i c t i n g the c e l l s t r u c t u r e . A l though con-s i d e r a b l e care was taken in 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 these authors have made an i n c o r r e c t choice in one of t h e i r assumpt ions. In c a l c u l a t i o n s i n v o l v i n g a r e c t a n g u l a r s lab 360 x 1600 mm, they have chosen to neg lec t heat f low i n the a x i a l d i r e c t i o n and cons idered on ly the two t ransverse d i r e c t i o n s . Since the c a s t i n g speed employed in the s i m u l a t i o n was very low (1 mm/s), a x i a l heat con-duc t ion cannot be neglected 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 c o n d u c t i v i t y . A b e t t e r choice would have been to neg lec t heat f l ow in a d i r e c t i o n normal to the narrow face s ince the aspect r a t i o was f a i r l y high ( 4 . 4 ) . Weckman e t al (45) have also developed a steady s t a t e model based on axisymmetry f o r c y l i n d r i c a l i n g o t s . The model uses the f i n i t e - e l e m e n t method and can on ly treat pure metals a n d e u t e c t i c a l l o y s . The pool p r o f i l e p r e d i c t e d by the model has been compared to exper imental pool p r o f i l e s ob ta ined dur ing the cont inuous cas t ing of a z inc i n g o t w i t h a square c r o s s - s e c t i o n . In order to make the comparison m e a n i n g f u l , the c r o s s - s e c t i o n a l areas of the c y l i n d r i c a l and square 14 ingots were set equal to each o t h e r . A s imple c a l c u l a t i o n of the sur face area to volume r a t i o s i n d i c a t e s t h a t there is a d i f f e r e n c e o f 11% between the two cases.- Thus t h i s model can only have l i m i t e d use in s tudy ing heat f l ow in r e c t a n -gu la r s e c t i o n s . Szargut et al (36) have pub l ished a paper on s teady-s t a t e heat f low in the cont inuous c a s t i n g 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 based on an e x p l i c i t f i n i t e - d i f f e r e n c e scheme w i t h i t s assoc ia ted s t a b i l i t y problems, no new mate r ia l has been presented in t h e i r paper. Recently Jensen (37) has developed a model f o r s i m u l a -t i n g heat f low in c y l i n d r i c a l ingots i n c l u d i n g the unsteady-s t a t e pa r t of the cas t ing o p e r a t i o n . The r e s u l t s o f a s i m u l a -t i o n f o r the c a s t i n g of 381 mm diameter i n g o t are repo r ted but very l i t t l e i n f o r m a t i o n is given regard ing the numer ical procedure adopted in the c a l c u l a t i o n . Fu r the r no comparison has been made to check the v a l i d i t y of the model c a l c u l a t i o n s . In summary, a number of mathematical models have been w r i t t e n to s imu la te heat f low in D.C. c a s t i n g . The m a j o r i t y of these models are based on assumptions of s t e a d y - s t a t e o p e r a t i o n and c y l i n d r i c a l geometry. The remain ing models 1 5 which apply to r e c t a n g u l a r sec t ions are a lso only two dimen-s iona l w i t h heat f low neglected e i t h e r in one of the t r a n s -verse d i r e c t i o n s or the a x i a l d i r e c t i o n . The present work has been undertaken to develop a t r u l y th ree dimensional model to s imula te both unsteady and-steady s t a t e heat f low in c a s t i n g square and r e c t a n g u l a r s e c t i o n s . Fur ther as w i l l be shown t h i s model has been used f o r t e s t i n g the v a l i d i t y of using a two d imensional v e r s i o n under spec ia l l i m i t i n g cases. Chapter 3 DEVELOPMENT OF THE HEAT FLOW MODEL 3.1 I n t r o d u c t i o n The problem of developing a model f o r the a n a l y s i s of heat f low in D.C. cas t ing e s s e n t i a l l y i nvo l ves the s o l u t i o n of a p a r t i a l d i f f e r e n t i a l equat ion d e s c r i b i n g unsteady heat conduc t i on . Because of the slow c a s t i n g speeds employed in t h i s opera t ion and the high thermal con-d u c t i v i t y of the ma te r i a l cast the normal assumption o f n e g l i g i b l e heat conduct ion in the a x i a l d i r e c t i o n , made in the models f o r the cont inuous c a s t i n g of s t e e l i s not v a l i d . The problem of cons ider ing heat f low in a l l th ree d i r e c t i o n s is made s impler in the case of a c y l i n d r i c a l i n g o t . I f axisymmetry can be assumed the heat f low problem is reduced to two d i r e c t i o n s namely r a d i a l and a x i a l ( 27-29 , 31 , 32 , 36 , 45 ) . When develop ing heat f low models f o r r e c t a n g u l a r sec t ions such a s i m p l i f i c a t i o n is not p o s s i b l e . However in order to reduce the problem also to two dimensions i t is usual to neg lec t heat f low in the d i r e c t i o n normal to the narrow face (30 , 37, 4 8 ) . Although t h i s is not a bad assumption f o r s labs which are t h i n and w ide , i t leads to 16 cons iderab le e r r o r f o r sec t ions w i th a small aspect r a t i o ( r a t i o of the t ransverse d imens ions) , as w i l l be seen i n Chapter 5. In such cases i t i s necessary to cons ider heat f low in a l l th ree d i r e c t i o n s . Because o f the semi-cont inuous nature o f the D.C. c a s t i n g o p e r a t i o n , i t i s i n t e r e s t i n g also to study the uns teady-s ta te p o r t i o n o f the c a s t i n g as i t cou ld occupy a s i zeab le f r a c t i o n of the t o t a l cas t ing t i m e . Thus the model undertaken in t h i s study is based on unsteady s t a t e , th ree-d imens iona l heat f l o w . 3.2 Assumptions Made in the Model The f o l l o w i n g assumptions have been made in the development o f the model. 1 . In the case o f the s i m u l a t i o n of r e c t a n g u l a r or square sec t ions t w o - f o l d symmetry has been assumed f o r the mid face p lanes , and the c a l c u l a t i o n s are performed only f o r one quar te r o f the' c a s t i n g . In the case of jumbo z inc i n g o t s , because of the presence o f only one symmetry p l a n e , c a l c u l a t i o n s are made f o r one -ha l f of the c a s t i n g . 2. Mix ing in the l i q u i d pool has been n e g l e c t e d , i . e . a s tagnant pool has been assumed. 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 (31) to be a reasonable assumption f o r non - fe r rous cas t ings w i t h sha l low poo ls . The assumption may not be v a l i d however in s i t u a t i o n s i n v o l v i n g e1ec t ro -magnet ic s t i r r i n g . In such cases the thermal c o n d u c t i v i t y of the l i q u i d could be increased to r e f l e c t the s t i r r i n g as has been done in the case o f the cont inuous c a s t i n g of s tee l (71 ) . 3. In the s i m u l a t i o n of z inc- jumbo c a s t i n g the thermal c o n d u c t i v i t y of z inc has been assumed cons tan t and the same f o r both l i q u i d and s o l i d . From the thermo-phys ica l p r o p e r t i e s given by Touloukian ( 5 1 ) , the thermal c o n d u c t i v i t y of s o l i d z inc has a value tw ice t h a t of the l i q u i d . Thus ass ign ing a constant s o l i d thermal c o n d u c t i v i t y value f o r the l i q u i d as we l l as the s o l i d corresponds to a very m i l d s t i r r i n g in the l i q u i d p o o l . The e r r o r i n t roduced by assuming constant thermal c o n d u c t i v i t y f o r s o l i d z inc i s small s ince i t changes only by 20% between room temperature and i t s me l t ing p o i n t . Moreover Peel and Pengel ly (29) have claimed t h a t the e f f e c t of thermal c o n d u c t i v i t y i s n e g l i g i b l e in the case of aluminium c a s t i n g . In t h i s study a p p r o p r i a t e con-s tan t values have been used f o r s o l i d and l i q u i d r e g i o n s . The value used in the mushy reg ion 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 f r a c t i o n s 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 . I t i s pos-s i b l e to take i n t o account va ry ing thermal con-d u c t i v i t y through minor m o d i f i c a t i o n o f the com-puter program. 4. The s p e c i f i c heat is a l lowed to vary as a f u n c t i o n of tempera tu re . However no i t e r a t i v e c a l c u l a t i o n s are made w i t h i n a t ime i n t e r v a l . The value o f s p e c i f i c heat i s eva luated at the beg inn ing o f a t ime i n t e r v a l based on the temperatures ob ta ined from the prev ious i n t e r v a l and is kept cons tan t dur -ing t h a t p a r t i c u l a r i n t e r v a l . The temperature de-pendence of s p e c i f i c heat is adequate ly desc r ibed through use of small time i n t e r v a l s . 3 . 3 Heat Flow Equation and Boundary Cond i t ions The u n s t e a d y - s t a t e , hea t -conduc t ion equa t ion in th ree dimensions w i t h constant thermal c o n d u c t i v i t y f o r a Car tes ian c o - o r d i n a t e system can be w r i t t e n as, 2 2 2 k ( i - J - ) + k ( ^ - o 1 ) + k = pc . . . 3 . 1 0 * 3 y ^ *z 6 where T denotes the temperature x , y , z , the three d i r e c t i o n s k, the thermal c o n d u c t i v i t y p , the dens i t y c, the s p e c i f i c heat t , the t i m e . A d e t a i l e d l i s t o f the symbols used is a lso g iven before Appendix 1 . In order to solve Eq. (3 .1 ) i n i t i a l and boundary c o n d i t i o n s are r e q u i r e d . I n i t i a l c o n d i t i o n : P h y s i c a l l y , t h e i n i t i a l c o n d i t i o n descr ibes the con-d i t i o n s e x i s t i n g at the beginn ing of c a s t i n g o p e r a t i o n when the bottom block i s ra ised to cover the mould open ing , and the mould is f i l l e d w i th molten meta l . Since the f i l l i n g o f the mould is c a r r i e d out in a very sho r t t ime i n t e r v a l ( u s u a l l y around 1 m i n u t e ) , i t is assumed there is n e g l i g i b l e heat loss dur ing t h i s procedure. Thus the i n i t i a l tempera-tu re is taken to have a constant un i fo rm value th roughout the c a s t i n g and equal to the pour ing tempera tu re . T = Tp at t = o, o^x-sX, o^y^Y, o^z^Z . . . 3.2 where Tp is the pour ing temperature . Top Boundary C o n d i t i o n : When the c a s t i n g opera t ion is s t a r t e d , the f low of l i q u i d metal i n t o the mould is ad jus ted to match the c a s t i n g speed. Since t h i s metal normal ly 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 teady d u r i n g a c a s t i n g o p e r a t i o n . Thus to s imula te 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 kept a t the pour ing tempera ture . T = Tp t > o , z = 0, o«x<X, o^y^Y . . . 3.3 I t i s poss ib le to have T = T ( x , y ) , t > o , z = o . . . 3.4 i f the temperature d i s t r i b u t i o n at the top i s known more a c c u r a t e l y . Bottom Boundary C o n d i t i o n : This i nvo lves the t r a n s f e r of heat f rom the bot tom of the i n g o t to the p l a t t e n w i t h which i t i s i n c o n t a c t . Since the p l a t t e n i s not prov ided w i t h any c o o l i n g , on ly a small amount of heat f lows through i t . In the p resen t work t h i s boundary c o n d i t i o n has been handled through the use of a h e a t - t r a n s f e r c o e f f i c i e n t . 9T -k — = h (T-T, ) , t > o , z = Z, O « X J ? X , o«y.$Y . . . 3. ! <j Z D 2 2 A constant value o f .209 kW/m K ( .005 ca l /cm .°c s) has been employed through most of the c a l c u l a t i o n s presented in t h i s work. Al though t h i s h e a t - t r a n s f e r c o e f f i c i e n t might have some e f f e c t dur ing the s t a r t up of the c a s t i n g , i t has no e f f e c t on the s t e a d y - s t a t e o p e r a t i o n . A f u r t h e r d i s c u s -sion of the sub jec t w i l l be reserved f o r a l a t e r s e c t i o n on the importance of unsteady heat f low in Chapter 5. Side Boundary C o n d i t i o n s : This is the most impor tant of a l l the boundary con-d i t i o n s s ince the ingo t is cooled only from the s i d e s . Here again a h e a t - t r a n s f e r c o e f f i c i e n t type of boundary c o n d i t i o n has been employed. -k U = h(z) ( T - T ) t>o , x = X, o«ysY, o<z«Z . . . 3.6 oX W -k f j - = h (z ) (T-T ) t > o , y = Y, o«x«X, o<z«Z . . . 3.7 ay W where h(z) i s the o v e r a l l h e a t - t r a n s f e r c o e f f i c i e n t which is a f u n c t i o n o f the p o s i t i o n along the z - a x i s . Thus the h e a t - t r a n s f e r c o e f f i c i e n t used in the mould w i l l have a d i f -f e r e n t value than the one used in the spray r e g i o n . Char-a c t e r i z a t i o n of t h i s h e a t - t r a n s f e r c o e f f i c i e n t i s very im-p o r t a n t to the accuracy o f the s i m u l a t i o n s . The exact value used w i l l be g iven in the sec t i on dea l ing w i t h the v a l i d a -t i o n o f the model. Centre Boundary C o n d i t i o n s : Wherever p o s s i b l e , symmetry c o n d i t i o n s in s i m p l i f y i n g the problem. Thus in the case were app l i ed of a 23 rec tangu la r slab w i t h two symmetry p lanes , zero heat f l u x has been assumed at the mid- face planes and the c a l c u l a -t i o n s are performed only f o r one quar te r o f the c a s t i n g , as mentioned e a r l i e r . -k — = o t>o , x = o, o«y«cY, o<z«Z . . . 3.8 o X -k — = o t>o , y = o, o^x-sX, o<z$Z . . . 3.9 The d i f f e r e n t i a l equat ion Eq. ( 3 . 1 ) t o g e t h e r w i t h the i n i t i a l and boundary cond i t i ons Eqs. (3 .2 ) to ( 3 . 9 ) comprise a complete mathematical statement of the prob lem. 3.4 Method o f S o l u t i o n The use of a n a l y t i c a l methods to so lve Eq. ( 3 . 1 ) i s precluded by the complex nature of the boundary c o n d i t i o n s as we l l as the growth of the ingo t in the c a s t i n g d i r e c t i o n . S i m i l a r l y s e m i - a n a l y t i c a l methods, l i k e the i n t e g r a l -p r o f i l e method developed by H i l l s ( 8 0 ) , a l though use fu l i n mode l l ing heat f low in the cont inuous c a s t i n g of s t e e l , have l i m i t e d use in the present problem due to the importance o f a x i a l conduct ion and the unsteady s t a t e . A numer ica l method has t h e r e f o r e been adopted f o r s o l v i n g the p a r t i a l d i f f e r e n t i a l e q u a t i o n . Because of the cons ide rab le amount of t ime and e f f o r t i nvo lved in the development o f the computer program, the i n i t i a l choice o f the numerical 24 method is very i m p o r t a n t . The method chosen should be able to y i e l d a reasonable l e v e l of accuracy w i t h moderate com-puter requ i rements . Two o f the most popular methods f o r s o l v i n g problems r e l a t e d to heat f low and s o l i d i f i c a t i o n are f i n i t e element and f i n i t e - d i f f e r e n c e techn iques . The use o f f i n i t e -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 discussed by severa l authors (32 , 72 -74 ) . A l though t h i s method i s e l e g a n t l y s u i t e d f o r hand l ing steady s t a t e problems, and has an edge over f i n i t e - d i f f e r e n c e method in hand l ing complex geometry, i t has a r a t h e r l i m i t e d use in th ree dimensional t r a n s i e n t problems from the s t a n d p o i n t o f the cost of computa t ion . Emery and Carson (74) have compared f i n i t e - e l e m e n t and f i n i t e - d i f f e r e n c e methods f o r the s o l u -t i o n of a two dimensional heat f low prob lem. I t i s c l e a r from t h i s a n a l y s i s t h a t the f i n i t e - d i f f e r e n c e method exceeds the f i n i t e - e l e m e n t methods in e f f i c i e n c y f o r t r a n s i e n t problems in th ree dimensions. Thus i t was decided to use a f i n i t e - d i f f e r e n c e method f o r s o l v i n g Eq. ( 3 . 1 ) . A good t rea tment o f the f i n i t e d i f f e r e n c e methods can be found i n any standard t e x t book on numer ica l methods ( 6 4 , 65, 78, 7 9 ) . F i n i t e - d i f f e r e n c e methods can i n general be c l a s s i f i e d i n t o three broad ca tegor ies namely e x p l i c i t , f u l l y i m p l i c i t and. e x p l i c i t - i m p l i c i t t e c h n i q u e s . 25 The e x p l i c i t f i n i t e d i f f e r e n c e methods are the s imp les t o f the three types and do not r e q u i r e s imul taneous s o l u t i o n o f equa t i ons . In t h i s method the f u t u r e tempera-tu re of a node is c a l c u l a t e d based on the present tempera-tu re of t h a t p a r t i c u l a r node and sur round ing nodes. In a th ree-d imens iona l problem, an i n t e r i o r node w i l l be su r -rounded by s i x nodes. The main disadvantage of the e x p l i c i t method is the r e s t r i c t i o n imposed on the t ime steps t h a t can be used f o r a given node s i z e , f o r proper s t a b i l i t y of the numerical method. The r e s t r i c t i o n imposed on the t ime i n t e r v a l increases by a f a c t o r of three f o r the th ree dimensional case as compared to the one dimensional p rob lem, i f the node s ize is kept the same. This i s i l l u s t r a t e d f o r the case of z inc and f o r a small node s ize used in t h i s work. For a D i r i c h l e t type boundary c o n d i t i o n , the s t a b i l i t y c r i t e r i o n is kAt , 1 , 1 , 1 x < 1 ( ~ T ~ 2 ~ 2 K 2 pc Ax Ay AZ where k is the thermal c o n d u c t i v i t y p i s the dens i t y c i s the s p e c i f i c heat A X , A y , A z are node s izes in x, y and z dimensions At i s the time i n t e r v a l . 26 S u b s t i t u t i n g the app rop r ia te values f o r the d i f f e r e n t va r i ables , k = 113 W/m K, p = 7140 k g / m 3 , c = .3830 J/gK Ax = 15.24 mm, Ay = 15.24 mm, Az = 20 mm one obta ins At <? 1.08 s. When the above-mentioned c a l c u l a t i o n s are repeated w i t h a convect ion type boundary c o n d i t i o n a lower v a l u e , 0.55 s , (see end of Appendix 1 f o r t h i s c a l c u l a t i o n ) i s ob ta ined f o r A t . Other f i n i t e d i f f e r e n c e methods descr ibed below are not sub jec t to any s t a b i l i t y c r i t e r i o n and values of A t , e i g h t f o l d g rea te r than t h a t imposed by e x p l i c i t methods have been commonly used in t h i s work. In c o n c l u s i o n i t can be sa id t h a t the e x p l i c i t methods have a l i m i t e d r o l e to play in th ree dimensional problems. In the f u l l y i m p l i c i t method, u n l i k e the e x p l i c i t method, the f u t u r e temperature of a node i s r e l a t e d to the present temperature of t ha t node and the f u t u r e temperature of the sur round ing nodes. Since these su r round ing tempera-tu res are not known a p r i o r i the i m p l i c i t method e s s e n t i a l l y invo lves the s o l u t i o n of a system of s imul taneous e q u a t i o n s . For a th ree-d imens iona l problem w i t h ten nodes in each d i r e c t i o n , i t would mean s o l v i n g one thousand equat ions s i m u l t a n e o u s l y . Although most of the elements i n the c o e f f i c i e n t m a t r i x w i l l be zeros ( the re w i l l be on ly seven non zero terms per row) and need not be s t o r e d , the best s o l u t i o n procedure l i k e the Gauss-Seidel method would s t i l l take up cons iderab le computer t ime as the s o l u t i o n would have to be i t e r a t e d . In t h i s regard the f u l l y i m p l i c i t methods could be compared to the f i n i t e element methods. However there are no s t a b i l i t y problems assoc ia ted w i t h f u l l y i m p l i c i t methods. In order to overcome the d i f f i c u l t i e s of s t a b i l i t y c o n d i t i o n s encountered in e x p l i c i t methods and problems assoc ia ted w i t h s o l v i n g a la rge number o f s imul taneous equat ions in i m p l i c i t methods, spec ia l procedures c a l l e d a l t e r n a t i n g d i r e c t i o n i m p l i c i t methods have been developed ( 7 5 - 7 7 ) . The procedure adopted in t h i s work was o r i g i n a l l y proposed by Br ian ( 7 7 ) . I t has an u n c o n d i t i o n a l s t a b i l i t y and converges w i t h d i s c r e t i z a t i o n e r r o r of the order 0 [ ( A X ) 2 + ( A t ) 2 ] . In t h i s method w i t h i n each time i n t e r v a l A t , the c a l c u l a t i o n s are performed in three s tages . In the f i r s t stage the c a l c u l a t i o n s are made i m p l i c i t i n the x - d i r e c t i o n and e x p l i c i t in the y and z d i r e c t i o n s . This i s f o l l o w e d by s i m i l a r procedures in the y and z d i r e c t i o n . F i n a l l y the new temperatures at the end of the t ime i n t e r v a l is 28 c a l c u l a t e d u s i n g an e x p l i c i t f o r m u l a * ? * ? ? T - T n - 6 x T + 6 y T n + 5 z T n 3 ' 1 0 At /2 T - T _ 6 ^ T + 6 ^ T + 6 T . . . 3 . 1 1 n - x y z n At /2 T -Tp = « x T + 6 * y T + 5% T . . . 3.12 A t / 2 9 * 9 * * 9 * * * T , i - T _ 6^ T + 6 T + 6 , T . . . 3.13 n+1 n x y z A t 2 2 2 where 6 , 6 and 6 are t he c e n t r a l d i f f e r e n c e x y z o p e r a t o r s d e f i n e d by 2 6 x T i , j , k T i - i , j , k " 2 T i , j ,-k * T i + 1 , j , k A X 2 2 6 y T i . j . k = T i .3-1 , k " 2 T < . . i . k * i + 1 , k 4 y2 2 6 z T i . j .k = T i , , i , k - 1 ' 2 T i , , i , k * T i ,.i , k + l A Z 2 29 i , j , k being the l e t t e r s used f o r numbering the nodes in the x, y and z d i r e c t i o n s . T* 5 T* * , j * * * are the i n te rmed ia te f i c t i t i o u s temperatures c a l c u l a t e d which do not have any spec ia l meaning. Tn and Tn +-j are the temperatures at the beg inn ing and the end of a t ime i n t e r v a l A t . A X , Ay and AZ are the d is tance between nodes in x , y and z d i r e c t i o n s . S u b s t i t u t i o n of f i n i t e d i f f e r e n c e equat ions Eqs . 3.10 to 3.13 in place o f the p a r t i a l d i f f e r e n t i a l equa t ion r e -s u l t s i n a set o f s imultaneous equat ions i n v o l v i n g t r i -diagonal c o e f f i c i e n t s f o r which a very e f f i c i e n t s o l u t i o n procedure e x i s t s . Thus in s p i t e of the th ree s e t s ' o f c a l c u l a t i o n s to be c a r r i e d out w i t h i n each t ime i n t e r v a l , t h i s i s a more e f f i c i e n t procedure than the f u l l y i m p l i c i t method. D e t a i l s o f the procedure by which the c a s t i n g is d i v i d e d i n t o d i f f e r e n t nodes and the s e t t i n g up of the nodal equat ions are presented in Appendix 1. For a simple case o f a r e c t a n g u l a r para l1 el piped w i t h r e g u l a r node s ize in a l l the three dimensions there w i l l be twenty-seven d i f -f e r e n t types o f nodal equa t ions . 30 One of the problems w i t h the a l t e r n a t i n g i m p l i c i t technique is the t rea tment of the r a d i a t i o n boundary con-d i t i o n . I n t r o d u c t i o n of the r a d i a t i o n boundary c o n d i t i o n d i r e c t l y i n t o the heat balance equat ion renders the equat ions n o n - l i n e a r . In o rder to overcome the prob lem, the boundary c o n d i t i o n can be l i n e a r i z e d using the r e l a t i o n s h i p T 4 T 4 h a v = o e (n^TT ) ••• 3 ' 1 4 ct where the average heat t r a n s f e r c o e f f i c i e n t h f l v i s c a l c u l a t e d assuming an average value f o r the sur face t e m p e r a t u r e . Here T g i s the temperature of the ambient medium, a i s the Stefan-Bol tzmann constant and E , the e m i s s i v i t y . Because of the low temperatures encountered in aluminium and z inc c a s t -i n g , t h i s boundary c o n d i t i o n w i l l have a n e g l i g i b l e e f f e c t . The growth o f the i ngo t i s s imu la ted through p e r i o d i c a d d i t i o n o f a set o f nodes at the pour ing temperature to 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 out by Ba l lan tyne ( 6 6 ) . The t ime i n t e r v a l over which a .row of 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 ize in 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 ize A Z , an a d d i t i o n i s made every At given by a 31 The t ime i n t e r v a l At f o r model c a l c u l a t i o n s i s se lec ted such t h a t At w i l l be an i n t e g r a l m u l t i p l e of At a At = N (At ) . . . 3.16 a where N is an i n t e g e r . Thus there is no independent c o n t r o l o f At and Az. I n i t i a l l y numerical c a l c u l a t i o n s were performed to check the e f f e c t of N, on the p r e d i c t e d r e s u l t s . The e f f e c t was found to be very small and thus the value of N was a r r i v e d a t , based on c a s t i n g speed, node s i ze and the cost of computa t ion . Values of N used i n t h i s work range from 2 to 8. The re lease of l a t e n t heat du r ing s o l i d i f i c a t i o n poses some problems in the i m p l i c i t f i n i t e d i f f e r e n c e techn ique . To overcome t h i s problem, a technique commonly employed by o ther workers (29, 33, 66) has been adopted in t h i s s tudy . The l a t e n t heat is re leased l i n e a r l y over the 1 i q u i d u s - s o l i d u s temperature range as f o l l o w s : c = c + T LT . . . 3.17 m V T s where c m denotes the s p e c i f i c heat in the mushy reg ion c the average s p e c i f i c heat eva lua ted at the so l i dus and l i q u i d u s tempera tures . L the l a t e n t heat of s o l i d i f i c a t i o n T , T $ the l i q u i d u s and so l i dus tempera tu res . This method r e q u i r e s tha t the temperature o f nodes undergoing s o l i d i f i c a t i o n f a l l w i t h i n the l i q u i d u s -s o l i d u s i n t e r v a l a t some p o i n t dur ing the c a l c u l a t i o n . I f however l a rge heat f lows take place i t i s p o s s i b l e f o r some nodes to jump from above the l i q u i d u s to below the so l i dus w i t h i n one t ime i n t e r v a l . This problem is o f t e n encountered in heat f low c a l c u l a t i o n s of D.C. c a s t i n g because o f the high heat t r a n s f e r c o e f f i c i e n t s r e s u l t i n g from f l o o d c o o l i n g and the high thermal c o n d u c t i v i t y of the m a t e r i a l c a s t . When t h i s happens the l a t e n t , heat w i l l not be re leased from tha t node. This p o i n t can be apprec ia ted from the simple c a l c u l a t i o n t h a t l a t e n t heat amounts to over 60% of the t o t a l heat removed from the top of the c a s t i n g to the bottom of the p o o l . In order 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. In t h i s technique the temperature of the nodes which jumped from one phase to another is mod i f ied using a heat balance approach. An example f o r a node going from above the l i q u i d u s to below the so l i dus is given below. Let T.j be the temperature of node be fo re a t ime i n t e r v a l T^ the l i q u i d u s temperature T s the s o l i d u s temperature T 9 the temperature a f t e r the t ime i n t e r v a l 33 c.j , the s p e c i f i c heat evaluated at p the dens i t y of the ma te r i a l Cr,, c 3 the s p e c i f i c heat of the m a t e r i a l i n the mushy zone and evaluated at the s o l i d u s temperature v the volume of the node The net change in the heat content o f t h a t node = vpc-j (T^-Tg) The f o l l o w i n g comparison decides whether the node w i l l end up in the mushy reg ion or in the s o l i d r e g i o n . I f p v C l ( T r T 2 ) > P V C 1 ( T 1 - T J + v P c 2 ( T £ - T s ) then the node w i l l end up in the s o l i d zone. Otherwise the co r rec ted temperature w i l l be i n the mushy r e g i o n . The co r rec ted temperature when the above-mentioned i n e q u a l i t y i s s a t i s f i e d is g iven by T 3 " T s " " W S J " ( V V J ••• 3 - 1 8 S i m i l a r equat ions are obta ined f o r nodes jumping from the l i q u i d to mushy and mushy to s o l i d phases. In cases where the c a s t i n g undergoes rehea t ing t h i s procedure i s repeated in r e v e r s e . This c o r r e c t i v e procedure is very handy i n using l a t i v e l y long t ime i n t e r v a l s a l lowed by the u n c o n d i t i o n a l 34 s t a b i l i t y of the a l t e r n a t i n g d i r e c t i o n i m p l i c i t f i n i t e d i f f e r e n c e scheme. Howevera 1arge increase i n t ime i n t e r v a l to decrease the cost of computat ion i s not recommended on two counts . F i r s t l y the values of the rmo-phys ica l p r o -p e r t i e s tha t change w i t h temperature are eva lua ted a t the beg inn ing of a t ime i n t e r v a l and kept cons tan t d u r i n g t h a t i n t e r v a l . Thus having a very la rge time step may c o u n t e r a c t the accuracy sought w i t h vary ing thermophysica1 p r o p e r t i e s . Secondly when a l a rge t ime i n t e r v a l i s used, the p o s t -i t e r a t i v e c o r r e c t i o n procedure a l t e r s the temperature f i e l d so much t h a t the program goes i n t o an uns tab le mode. Thus a l though there are no c o n s t r a i n t s rega rd ing the s ize of t ime i n t e r v a l s a r i s i n g from the s o l u t i o n p rocedure , cau t ion should be exerc ised in s e l e c t i n g the t ime i n t e r v a l both from the v iewpo in t of the accuracy of s o l u t i o n and the cost of computa t ion . In t h i s s tudy , t ime i n t e r v a l s rang ing from 2 to 10 seconds have been used. The exact value se lec ted depended on the metal c a s t , s ize o f s e c t i o n , c a s t i n g speed and s e v e r i t y of c o o l i n g c o n d i t i o n s . ' 3.5 Mathematical Check f o r I n t e r n a l Consis tency of the Computer Program Although a m a j o r i t y of mathematical models r e q u i r e a numerical method to ob ta in a s o l u t i o n , a n a l y t i c a l methods 35 are very useful to .check out the numerical t e c h n i q u e s . In a complex problem using f i n i t e d i f f e r e n c e s , l i m i t i n g cases which o f t e n have a n a l y t i c a l s o l u t i o n can be used f o r compar i -son to the numerical r e s u l t s . They are a lso very va luab le t o o l s f o r debugging the computer program in the i n i t i a l stages of model development. I t should be noted t h a t in t h i s check, s o l i d i f i c a t i o n ( l a t e n t heat) and growth phenomena of the model are not i n c l u d e d . The s o l u t i o n of the three dimensional heat f l ow equat ion de f ined in the reg ion -a«x«a, -b«y«b, -c^z^c w i t h zero i n i t i a l temperature and u n i t sur face temperature is given by Carslaw and Jaeger (83) as = 1 _ 64 E°° E°° Ii 0 0 ( - 1 ) W n n 3 SL = O m=o n = o (2£+ l ) (2m+l) (2n+! ) c o s ' ( 2 £ + l ) n x cos (2m+l)ny cos (2n+1)nze 2a 2b 2c •Bn t l ,m ,n where (3 = a n 2 r ( 2 + 1) 2 +(2m+l) 2 + ( 2n + l) 2 j 3 ] g 4 a 2 b2 V and a is the thermal diffusivity, t is the time, 2a , 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 s p e c i f i c example heat f low i n a cube o f s ide 609.6 mm was cons ide red . The thermal d i f f u s i v i t y o f the 2 mate r ia l used i s 12.95 mm / s . I n i t i a l l y the m a t e r i a l i s un i fo rm ly at a temperature of 260°C and subsequent ly f o r a l l t ime t>o the sur faces o f the cube are main ta ined at 537.7°C. The c a l c u l a t e d temperatures at the cent re o f the cube are presented i n Table I f o r both numerical and a n a l y t i c a l methods. The r e s u l t s shown i n t h i s t a b l e have been c a l c u l a t e d f o r a constant t ime i n t e r v a l o f 36 s , but d i f f e r e n t node s i z e s . As can be seen the n u m e r i c a l l y c a l c u l a t e d values approach the a n a l y t i c a l r e s u l t s as the number of nodes used in the c a l c u l a t i o n i n c r e a s e s . Some of the r e s u l t s from t h i s t a b l e have been p l o t t e d i n a d i f -f e r e n t fash ion in F i g . 3 . 1 . The percent e r r o r on the y ax is has been c a l c u l a t e d as the percent d i f f e r e n c e between the numerical and a n a l y t i c a l r e s u l t s . I t can be seen, t h a t f o r t h i s t ime i n t e r v a l of 36 s, using 21 nodes per s i d e , the numerical s o l u t i o n y i e l d s numbers very c lose to the a n a l y t i c a l r e s u l t s . The i n f l u e n c e of the time i n t e r v a l on the comparison between numerical and a n a l y t i c a l method is presented in Table I I . For these runs the node s ize has been kept cons tan t . I t can be seen from t h i s t ab le t h a t f o r the node s ize cons ide red , the t ime i n t e r v a l had a very small e f f e c t 37 Ti me (s) Num aer of Nodes Per Edqe A n a l y t i cal N = 9 N = 13 N = 17 N = 21 360 268. 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 numerical and a n a l y t i c a l values of temperatures (°C) at the cent re o f a cube, as a f u n c t i o n of t ime f o r d i f f e r e n t s izes of nodes . F i g . 3.1 The e f f e c t o f the number of nodes on percent e r r o r at the end of d i f f e r e n t t ime i n t e r v a l s . 39 Ti me | (s) A t = 180s A t = 36 s A t = 18 s A n a l y t i c a l 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 Comparison between numerical and a n a l y t i c a l values of cen t re temperatures (°C) o f a cube, as of a f u n c t i o n of t i m e , f o r d i f f e r e n t t ime steps . 40 on the r e s u l t s of the c a l c u l a t i o n . However t h i s comment holds good only w i t h respect to centre t empera tu res . I f on the o ther hand comparisons were made f o r temperatures away from the cent re then the c a l c u l a t i o n s done w i t h f i n e r t ime i n t e r v a l s w i l l show apprec iab le d i f f e r e n c e to t h a t performed w i t h coarser t ime s teps . Convect ive Boundary Cond i t i o ns : The above-mentioned comparisons helped in check ing the equat ions developed f o r a l l i n t e r i o r nodes. In o rder to check the equat ions o f the sur face node a d i f f e r e n t procedure was f o l l o w e d . The heat t r a n s f e r c o e f f i c i e n t was set equal to zero in two d i r e c t i o n s and a f i n i t e value i n s e r t e d in the t h i r d d i r e c t i o n . The r e s u l t ob ta ined from the three dimensional model should y i e l d a zero temperature g rad ien t in two of the d i r e c t i o n s . The a n a l y t i c a l s o l u t i o n f o r one-d imensional heat f low in a slab of th ickness 2L, i n i t i a l l y at a u n i f o r m temperature sub jec ted to convec t ive type boundary c o n d i t i o n i s g iven by (83) 8( x , t ) _ 2 E-j e e 0 - 6 * ( a t / L 2 ) Sin 6 n cos U x/L) 3.20 6 + Sin 6 cos 6^ n n n 6 tan 6 k 3.21 n n 41 e ( x , t ) = T ( x , t ) - To e = T - Te o 0 Where T and T are the i n i t i a l and ambient temperatures oo 0 and h, the heat t r a n s f e r c o e f f i c i e n t , the value o f x i s measured w i t h respect to centre of the t h i c k n e s s . In t h i s check again the thermal d i f f u s i v i t y of the m a t e r i a l used i s 12.95 mm 2 /s . Other values used are T = 260°C, T 537.7°C, L = 305.8 mm, h = 565.4 W/m2K, k = 46.7 W/m K. Comparison between the numerical and a n a l y t i c a l r e s u l t s are presented in F i g . 3 . 2 . I t can be seen t h a t the d i f -ference between the numerical and a n a l y t i c a l methods i n -crease i n going from the centre of the s lab to the s u r f a c e . However w i t h s h o r t e r t ime i n t e r v a l s very c lose agreements can be obta ined between the two. The use o f longer t ime i n t e r v a l s may generate o s c i l l a t i o n s at the s u r f a c e . This i s shown i n F i g . 3 .3 . Unl ike the e x p l i c i t f i n i t e d i f -ference methods these o s c i l l a t i o n s are s t a b l e and w i l l dampen a f t e r some t i m e . The t ime i n t e r v a l s used in t h i s study were se lec ted to avoid t h i s problem i n most of the cases . The procedure mentioned above was repeated f o r the other two d i r e c t i o n s y i e l d i n g i d e n t i c a l r e s u l t s and thereby v e r i f y i n g the equat ions developed f o r a l l the nodes, 42 F i g . 3.2 Comparison between a n a l y t i c a l and numer ica l c a l c u l a t i o n s of temperatures i n the s l a b . F i g . 3.3 Stable o s c i l l a t o r y nature o f the n u m e r i c a l l y c a l c u l a t e d sur face temperatures f o r l a r g e values of t ime i n t e r v a l . 44 i n c l u d i n g the sur face nodes. 3.6 Flow Chart o f the Computer Program A f l ow cha r t of the computer program i s g iven i n F i g s . 3.4a and 3.4b. The program has been w r i t t e n i n the For t ran IV language. The basic vers ion developed o r i g i n a l l y was f o r the a n a l y s i s of heat f low in r e c t a n g u l a r or square shaped c a s t i n g s . However t h i s has been m o d i f i e d sub-s t a n t i a l l y f o r use in i r r e g u l a r l y shaped z inc jumbo i n g o t s . A copy of the source program f o r jumbo shaped c a s t i n g i s presented in Appendix 2. The program has been w r i t t e n such t h a t i t i s p o s s i b l e to stop the computer run at any i n t e r m e d i a t e p o i n t , s tudy the r e s u l t s and r e s t a r t from the same p o i n t . This p r o -cedure was very usefu l in d e t e c t i n g a b o r t i v e runs from the s tandpo in t o f sav ing computer t i m e . The computer used was an Amdahl 4 7 0 / V - 6 - I I under the MTS system. The program r e q u i r e s r o u g h l y 0.8 Mega Bytes of memory and takes about 80 seconds of CPU t ime in per forming 160 i t e r a t i o n s . In t h i s p a r t i c u l a r case the ar ray dimensions v a r i e d f rom ( 1 0 , 16, 3) to ( 1 0 , 16, 43) at the end o f the r u n . 45 / S T A R T / R E A D I N P U T P A R A M E T E R S D I S C R E T I Z E C A S T I N G A R E T H E C A L C U L A T I O N S C O N T I N U E D ^ FROM A M I D D L E O F A ' . U N No I N I T I A L I Z E A R R A Y T O P O U R I N G T E M P . I N I T I A L I Z E A L L N O D E S AS L I Q U I D R E A D T E M P . , N O D E I D ' S F R O M T H E E N D O F A P R E V I O U S R U N f C L A S S I F Y T H E N O D E S D E P E N D I N G ON T H E I R P O S I T I O N I N T H E C A S T I N G , C A L C U L A T E V A R I O U S A R E A A N D V O L U M E T E R M S E V A L U A T E T H E R M O P H Y S I C A L P R O P E R T I E S T 0 F i g . 3 .4(a) Flow char t of the computer progra 46 0 I M P L I C I T C A L C U L A T I O N S B O U N D A R Y IN T H E X D I R E C T I O N C O N D I T I O N S ** I M P L I C I T C A L C U L A T I O N S B O U N D A R Y IN T H E Y D I R E C T I O N S C O N D I T I O N S \ I M P L I C I T C A L C U L A T I O N S B O U N D A R Y IN T H E Z D I R E C T I O N C O N D I T I O N S C O M P U T E NEW T E M P S . C H E C K W H E T H E R A N Y N O D E F i g . 3 .4(b) Flow char t of the computer program (cont inued from F i g . 3 . 4 ( a ) ) Chapter 4 VALIDATION OF THE RESULTS FROM THE MATHEMATICAL MODEL 4.1 I n t r o d u c t i o n Before the mathematical model could be used w i t h conf idence in a p r e d i c t i v e mode i t r e q u i r e d c a r e f u l v a l i d a -t i o n . This was accomplished by comparing m o d e l - p r e d i c t e d pool p r o f i l e s to i n d u s t r i a l measurements ob t a in ed under i d e n t i c a l c a s t i n g c o n d i t i o n s . A s i m i l a r v a l i d a t i o n technique has been repor ted in o ther s t u d i e s (28 -30 , 36, 48, 66 ) . In D.C. cas t i ng the pool p r o f i l e may be obta ined e x p e r i m e n t a l l y by adding an a l l o y towards the end o f cas t ing when steady s ta te has been reached. Later the contour is revealed a f t e r s e c t i o n i n g , p o l i s h i n g and e tch ing the cast i n g o t . In some cases the pool p r o f i l e s can be seen immediately a f t e r s e c t i o n i n g w i t h o u t sur face p repara -t i o n because of the d i f f e r e n c e in the m a c h i n a b i 1 i t y of the t r a c e r a l l o y and the parent me ta l . In a d d i t i o n to 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 in which a rod was lowered i n t o the molten p o o l , to ob ta in the maximum pool d e p t h . This i s a simple but very usefu l technique and the va lue o f d i p - s t i c k 47 48 measurements increases cons iderab ly when a p p l i e d to l a rge ingots which r e q u i r e cons iderab le c u t t i n g and mach in ing . Three separate v a l i d a t i o n s have been made i n t h i s s tudy . Two i n v o l v e the D.C. cas t ing of a luminium i n g o t using convent iona l f l o o d c o o l i n g and reduced secondary coo l i ng in the sub-mould reg ion r e s p e c t i v e l y . The t h i r d v a l i d a t i o n was made f o r the cas t i ng of z inc jumbos o f spec ia l shape. 4.2 Conventional D.C. Cast ing of Aluminium - Alcan 4 .2 .1 Aluminium Ingots : 381 x 991 mm In t h i s convent iona l form o f D.C. c a s t i n g the sur face of the c a s t i n g is f looded w i t h the c o o l i n g water from the mould, thereby producing in tense coo l i ng in the sub-mould r e g i o n . In a l l the s i m u l a -t i ons p resen ted , the ma te r ia l cast i s pure a lumin ium, the thermophysical p r o p e r t i e s of which are g iven in Table I I I . The working length of the mould used is 63.5 mm so t h a t w i t h a node th i ckness of 15.87 mm in the c a s t i n g d i r e c t i o n , four node s l i c e s are con-ta ined in the mould. The hea t - t rans fe r c o e f f i c i e n t used f o r the top two s l i c e s in 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 of these h e a t - t r a n s f e r c o e f f i c i e n t s r e s u l t i n heat f l u x values Speci f i c Heat of Sol id - 0.934 J /g K Speci f i c Heat of L iqu id - 0.934 J /g K Latent Heat of Fusion 387 J/g Li qui dus Temperature 631 °C So l i dus Temperature 630 °C Dens i ty of So l i d 2700 kg/m Densi ty of L iqu id 2700 kg /m3 Thermal Conducti v i ty - 209.3 W/m K Ambient Temperature - 15°C Table I I I Thermophysical P rope r t i es of Aluminium used in Convent ional Flood Cool ing S i m u l a t i o n s . 50 obta ined from temperature measurements in the mould (33, 8 4 ) . In order to account f o r the a i r - g a p fo rmat ion in the lower pa r t of the mould, the heat -t r a n s f e r c o e f f i c i e n t was decreased t o 209 W/m K f o r the sur face nodes in the remaining two s l i c e s i n s i d e the mould. For a l l sur face nodes below the mould, a very high h e a t - t r a n s f e r c o e f f i c i e n t has been used to r e f l e c t the in tense coo l i ng w i t h i n the f l o o d - w a t e r zone. The value of t h i s h e a t - t r a n s f e r c o e f f i c i e n t which is 14.65 kW/m K was a r r i v e d a t , 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 depths f o r a s e c t i o n 381 x 991 mm cas t at 1.778 mm/s. From these runs i t was very c l e a r t h a t the h e a t - t r a n s f e r c o e f f i c i e n t because o f i t s h igh v a l u e , had a very small e f f e c t on the pool dep ths . Thus i t was poss ib le to check the model by comparing the pool depths obta ined f o r d i f f e r e n t c a s t i n g speeds f o r the same set of boundary c o n d i t i o n s . F i n a l l y in these c a l c u l a t i o n s the l a t e n t heat was re leased 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 the 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 p r o f i l e s f o r a cas t i ng speed of 1.778 mm/s. The measured p r o f i l e s have been ob ta ined by adding z inc 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 f o r 381 x 991 mm aluminium i n g o t cast at 1.778 mm/s (ob ta ined at the mid-p lane p a r a l l e l to the narrow f a c e ) . 52 and lead t r a c e r (84) to the molten p o o l . As can be seen there is an e x c e l l e n t agreement between the two p r o f i l e s . I t should be noted t h a t the s t a i r c a s e p a t t e r n of the c a l c u l a t e d pool p r o f i l e which r e s u l t s from the coarseness of the f i n i t e - d i f f e r e n c e mesh and the narrow range over which the l a t e n t heat i s r e -leased has been smoothened out in F i g . 4 . 1 . I t may also be noted tha t the e f f e c t of the high heat -t r a n s f e r c o e f f i c i e n t in the sub-mould reg ion i s f e l t h igher up in the c a s t i n g because of a x i a l heat conduc t ion . Of the t o t a l heat removed from the s t a r t of c o o l i n g to the bottom of the p o o l , less than 5% is removed in the mould r e g i o n . The th ree-d imens iona l temperature d i s t r i b u t i o n in the c a s t i n g c a l c u l a t e d from the model i s presented in Table A3 . I of Appendix 3. F i g . 4 .2 shows pool p r o f i l e s obta ined at the l o n g i t u d i n a l mid- face planes a f t e r steady s ta te cond i t i ons have been reached. The n o t a t i o n used fo r the var ious d i r e c t i o n s are as f o l l o w s . The cas t ing d i r e c t i o n was always taken as z - a x i s . In the t ransverse plane x - a x i s was taken per-pend icu la r to the broad face and y - a x i s pe rpend icu la r to the narrow face . In a l l the c a l c u l a t e d pool p ro -f i l e s shown in t h i s work, the top two node s l i c e s are at the pour ing tempera tu re ; and t h i s should be taken F i g . 4.2 Steady s ta te pool p r o f i l e s ob ta ined a t the l o n g i t u d i n a l mid-planes f o r 381 x 991 mm aluminium ingo t cast at 1.778 mm/s. 54 note of when c a l c u l a t i n g the pool depth from these con tou rs . The p r o f i l e obta ined in the l o n g i t u d i n a l mid-plane p a r a l l e l to broad face is i n agreement w i t h the bucket shaped pool observed across the w id th of the s e c t i o n . In order to study the e f f e c t o f heat conduct ion in the second t ransverse d i r e c t i o n , on the o v e r a l l heat t r a n s f e r , the computer program was run in a two-dimensional mode. Here heat f l o w p e r p e n d i c u l a r to the narrow face ( y - d i r e c t i o n ) was n e g l e c t e d . The c a l c u l a t i o n s were performed by s e t t i n g the heat -t r a n s f e r c o e f f i c i e n t equal to zero i n the y - d i r e c t i o n and at the same time reducing the number of nodes in 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 th ree dimensional c a l c u l a t i o n s , ob ta ined i n the l o n g i t u d i n a l mid-p lane p a r a l l e l to the narrow f a c e . I t i s seen t h a t there i s n e g l i g i b l e d i f f e r e n c e between the two pool p r o f i l e s . As w i l l be shown in Chapter 5, i f the aspect r a t i o exceeds 2.5 f o r conven t iona l D.C. c a s t i n g , there is no need to inc lude the heat f low in a d i r e c -t i o n pe rpend icu la r to the narrow face ( y - d i r e c t i o n ) in the h e a t - f l o w c a l c u l a t i o n s . This 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 th ree 55 F i g . 4.3 Comparison of the pool p r o f i l e s from the two-dimensional and th ree -d imens iona l c a l c u l a -t i o n s f o r cas t ing 381 x 991 mm aluminium ingo t at 1.778 mm/s. 56 dimensional model are about 5 to 8 t imes more than f o r the two dimensional case. There fo re c o n s i d e r a b l e computer costs can be saved by safe. ly making the assumption of two dimensional heat c o n d u c t i o n . In a l l the v a l i d a t i o n runs presented i n t h i s chapter f o r convent iona l D.C. c a s t i n g , wherever the aspect r a t i o s are g rea te r than 2 . 5 , the model has been run only in a two dimensional mode. F i g . 4.4 shows a comparison between the measured and c a l c u l a t e d pool depths as a f u n c t i o n o f c a s t i n g speed f o r 381 x 991 mm aluminium s e c t i o n . In a l l the cases the pool depth has been measured by using a s tee l w i re to probe the bottom of the p o o l ; the accuracy of these measurements i s w i t h i n ± 10 mm ( 8 4 ) . The agreement between the c a l c u l a t e d and measured pool depths are e x c e l l e n t . The pool p r o f i l e s obta ined at the d i f f e r e n t c a s t i n g speeds are shown in F i g . 4 . 5 . 4 .2 .2 Aluminium Ingots : 457 x 1143 mm The boundary cond i t i ons used here are i d e n t i c a l to those given in sec t ion 4 . 2 . 1 . This i s the l a r g e s t of the f o u r sec t ions s imulated under t h i s c a t e g o r y . The c a s t i n g speeds used range from 0.974 mm/s to 6 0 0 Casting Speed (mm/s) F i g . 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 at d i f f e r e n t speeds. t n F i g . 4.5 Ca lcu la ted pool p r o f i l e s f o r 381 x 991 mm aluminium ingo 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 the f i v e d i f f e r e n t cas t i ng speeds. Although the match i s not 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 and measured pool depths was only 30 mm or 4% o f the measured sump depth . For t h i s s e c t i o n s ize 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 in the pool depths obta ined between a two d imensional and three dimensional c a l c u l a t i o n s at a speed of 2.117 mm/s. However no d i f f e r e n c e s were observed at lower speeds. 4 . 2 . 3 Aluminium Ingots : 305 x 1010 mm. For t h i s sec t i on s ize the comparison between the c a l c u l a t e d and measured pool depth i s shown in F i g . 4 . 7 ; and l i k e the previous cases, c lose match i s obta ined between the two. 4 . 2 . 4 Aluminium Ingot : 229 x 813 mm This i s the smal les t of the f o u r sec t i ons s imula ted and h igher c a s t i n g speeds have been employed, the maximum of which is 2.794 mm/s. F i g . 4 .8 shows the comparison between the measured and c a l c u l a t e d pool depths. 60 850 750 E E J C a. Q "5 o Q_ E E X o 650 550 450 35 1 1 aluminum ingot - 457 mm X 1143 mm conventional d.c. cooling • calculated o measured 1.2 1.7 Casting Speed (mm/s) F i g . 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 ingo t cast at d i f f e r e n t speeds. 2.2 61 l 5 ° | . 0 1.5 2.0 2.5 Casting Speed (mm/s) F i g . 4.7 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 ingo t cast at d i f f e r e n t speeds. 62 450 E E sz a o o Q_ E E X D 1 1 aluminum ingot - 229 mm X 813 mm conventional d.c. cooling • calculated o measured 350 250 I50 1 i 2.4 2.9 Casting Speed (mm/s) F i g . 4.8 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 ingo t cast at d i f f e r e n t speeds. 63 4 . 2 . 5 Summary o f Conventional D.C. Cast ing S imu la t ions Table IV summarises the c a l c u l a t e d and measured pool depths obta ined f o r the var ious c a s t i n g c o n d i t i o n s . Thus in the case of convent ional D.C. c o o l i n g , com-par ison has been made f o r four d i f f e r e n t sec t i ons ranging in s ize from 229 x 813 mm to 457 x 1153 mm. In a d d i t i o n f o r each sec t ion at l e a s t f o u r d i f f e r e n t c a s t i n g speeds have been used to check the v a l i d i t y of the model. In a l l cases good agreement has been obta ined between measured and p r e d i c t e d values of the pool depths. Figure 4.9 shows a p l o t o f the t ime r e q u i r e d to reach steady s ta te aga ins t the c a s t i n g speed f o r the var ious s e c t i o n s i z e s . The t ime to reach steady s t a t e i s de f ined as the t ime taken f o r the mol ten pool to cease growing from the s t a r t of c a s t i n g . For 229 mm and 381 mm t h i c k sec t ions the c a s t i n g speed has no e f f e c t on the t ime to reach steady s t a t e . However f o r t h i c k e r sec t ions i n c r e a s i n g c a s t i n g speed causes a small increase in t h i s t i m e . Going from 229 to 457 mm t h i c k s e c t i o n s , the t ime to reach steady s t a t e increases from 125 s to 450 s . I t should be noted t h a t in any normal c a s t i n g o p e r a t i o n the maximum c a s t i n g speed is reached not i n s t a n t a n e o u s l y but over a f i n i t e amount o f t i m e , u s u a l l y around 1 64 Sect i on Si ze Cast ing Speed (mm/s) Pool Depth Meas ured (mm) Pool Depth Ca lcu la ted (mm) 229mm x 813mm 1 .524 1 84 1 75 1 .905 216 222 2.328 260 270 2.794 305 318 305mm x 1010mm 1 .100 216 1 98 1 .481 267 270 1 .862 343 341 2.159 381 396 381mm x 991mm 1 .185 330 31 8 1 .439 394 397 1 .778 470 476 2.116 559 572 457mm x 1143mm 0 .974 368 365 1 .312 489 492 1 .566 584 587 1 .947 699 730 2.117 762 778 Table IV Comparison between c a l c u l a t e d and measured pool depths f o r Aluminium sec t ions cast at d i f f e r e n t speeds . (/> CD O 00 >v T J O CU 00 o E I— o u u -1 1 1 aluminum ingot conventional d.c. I cooling 4 0 0 — 4 5 7 mm X 1143 mm -3 0 0 6H\ mm X 991 mm 2 0 0 305 mm X 1010 mm 229 mm X 813 mm i o n • i i • 0.9 1.4 1.9 2.4 2.9 Casting Speed (mm/s ) F i g . 4 . 9 Time r e q u i r e d f o r the poo l p r o f i l e s t o reach s t e a d y s t a t e f o r a l u m i n i u m 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 o f c a s t i n g s p e e d . CTi cn 66 to 2 minutes.. Thus in the case of t h i n n e r sec t ions by the t ime the steady cas t ing speed i s reached, the c a s t i n g would have s e t t l e d down' to steady s t a t e c o n d i t i o n s . The importance of unsteady s t a t e i n -creases w i t h i n c r e a s i n g sec t ion t h i c k n e s s e s . A f u r t h e r d i scuss ion of t h i s i s made in Chapter 5. 4.3 Reduced Secondary Cool ing - B r i t i s h Aluminium In t h i s c o o l i n g p r a c t i c e the mould water is not app l ied immediately below the mould. In i t s p lace a mi ld coo l i ng 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 sp rays . This r e s u l t s in a very coarse c e l l s t r u c t u r e s u i t a b l e f o r deep drawing a p p l i c a t i o n s . B e a t t i e e t al (30) have mea-sured the h e a t - t r a n s f e r c o e f f i c i e n t s f o r these sprays under var ious o p e r a t i n g c o n d i t i o n s in the l a b o r a t o r y . The thermo-phys ica l p r o p e r t i e s employed in t h i s s i m u l a t i o n are given in Table V, wh i l e boundary c o n d i t i o n s used are presented in Table V I . The computer program has been a l t e r e d to accommodate the d i f f e r e n t values of thermal c o n d u c t i v i t y f o r the s o l i d and l i q u i d r e g i o n s . The sec t i on s ize used in t h i s s i m u l a t i o n i s 254 x 690 mm, wh i le the c a s t i n g speed is 0.833 mm/s. The develop-ment of the molten pool computed as a f u n c t i o n of t ime is presented in F igs . 4.10 to 4 .12 , which show the pool S p e c i f i c Heat of So l i d - 1.13 J/g K S p e c i f i c Heat of L i q u i d - 1.13 J/g K La ten t Heat of Fusion - 387 J/g L iqu idus Temperature - 658°C Sol idus Temperature - 635°C Dens i ty - 2700 kg /m 3 Thermal C o n d u c t i v i t y of S o l i d - 222 W/mK Thermal C o n d u c t i v i t y of L i q u i d - 105 W/mK Table V Thermophysica1 P r o p e r t i e s of Aluminium used in Reduced Secondary Cool ing S i m u l a t i o n s . 68 Zone No. P o s i t i o n Below L iqu id Surface (mm) ' Heat T r a n s f e r Coe f f i c i ent ( W / m 2 K) 1 (mould) 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 - down-wards 166 Tab!e VI Heat Trans fer C o e f f i c i e n t s used as a f u n c t i o n of 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 . 69 D i s t a n c e a l o n g Y-axis (mm) X - a x i s ( m m ) 0 8 0 160 2 4 0 3 2 0 0 4 0 8 0 1 2 0 oi 1 1 1 n I — • — ' — H F i g . 4.10 L iqu idus and so l i dus p r o f i l e s ob ta ined at the l o n g i t u d i n a l mid-planes of 254 x 690 mm aluminium ingo 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 . 70 F i g . 4.11 L iqu idus and so l i dus p r o f i l e s ob ta ined at the l o n g i t u d i n a l mid-planes of 254 x 690 mm aluminium ingot 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 . 71 F i g . 4.12 L iqu idus and so l idus p r o f i l e s ob ta ined at the l o n g i t u d i n a l mid-planes of 254 x 690 mm aluminium ingo t cast under reduced secondary c o o l i n g con-d i t i o n s at 0.833 mm/s, a f t e r 1248 s from s t a r t . 72 p r o f i l e s at the l o n g i t u d i n a l mid- face p l a n e s . The steady s t a t e pool depth obtained was 470 mm in comparison w i t h the measured value of 510 mm. The th ree -d imens iona l tempera-tu re d i s t r i b u t i o n s computed from the model are presented in Table A 3 - I I in Appendix 3. The e f f e c t of n e g l e c t i n g heat conduct ion in a d i r e c t i o n normal to the narrow face ( y - d i r e c t i o n ) is shown in F i g . 4 .13 , in the form o f pool p r o f i l e s obta ined in the l o n g i t u d i n a l mid-p lane p a r a l l e l to the narrow f a c e . Unl ike the case of conven t iona l D.C. coo l i ng an e f f e c t owing to heat conduct ion i n both the t ransverse d i r e c t i o n s is seen even though the aspect r a t i o is 2 .79. The c a l c u l a t i o n s revealed t h a t f o r t h i s r a t h e r small s e c t i o n , steady s t a t e is reached only 15 minutes from the s t a r t o f c o o l i n g . A three dimensional view of the pool sur face is shown in F ig . 4 .14 . An at tempt to s imula te an increased 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 less s u c c e s s f u l . Here the pool depth obta ined from the s i m u l a -t i o n was 1014 mm. When experiments were c a r r i e d out at B r i t i s h Aluminium to cast t h i s 254 x 690 mm s e c t i o n at 1.266 mm/s d i f f i c u l t i e s were encountered in o b t a i n i n g a s t a b l e pool ( 8 5 ) . Even in three of the more success fu l e x p e r i -ments the pool depths ranged from 760 mm to 860 mm. Thus the c a l c u l a t e d value from the model f a l l s s h o r t of the h ighest pool depth measured. 73 F i g . 4 . 13 Comparison of steady s t a t e pool p r o f i l e s f o r the two-dimensional and the th ree-d imens iona l c a l c u l a -t i o n s o f 254 x 690 mm aluminium i n g o t cast under reduced secondary coo l i ng c o n d i t i o n s at .833 mm/s. 74 F i g . 4.14 Three-dimensional v isua l sur face of 254 x 690 mm .833 mm/s under reduced seen from the d i f f e r e n t i z a t i o n of l i q u i d pool aluminium ingo t cast of secondary c o o l i n g , as ang les . 75 The reason f o r the d i f f e r e n c e between the measured and c a l c u l a t e d pool depths can be a t t r i b u t e d to the un-c e r t a i n t y in the prec ise value of the h e a t - t r a n s f e r co-e f f i c i e n t s . Numerical c a l c u l a t i o n s performed by va ry ing the value 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 s showed t h a t the pool depth i s a very s e n s i t i v e f u n c t i o n o f t h i s v a r i a b l e . This is in c o n t r a s t to the cond i t i ons in convent iona l D.C. c o o l i n g , where the heat t r a n s f e r c o e f f i c i e n t had a very small e f f e c t on the pool depth . The u n c e r t a i n t y in the boundary c o n d i t i o n stems from the f a c t t h a t , in t h i s s i m u l a t i o n the sur face o f the c a s t i n g stays at 360°C even at the bottom of the s t r a n d . Thus f o r cons iderab le length of the c a s t i n g from the top o f the mould, the heat t r a n s f e r mechanism would be in the f i l m b o i l i n g regime. Depending on the temperature at which the mechanism changes to nucleate b o i l i n g , the h e a t - t r a n s f e r co-e f f i c i e n t w i l l change in magnitude. I t i s f e l t t h a t under these c o n d i t i o n s i t is very d i f f i c u l t to p r e d i c t the heat -t r a n s f e r c o e f f i c i e n t s a p r i o r i from the l a b o r a t o r y e x p e r i -ments, and measurements would have to be c a r r i e d out in p l a n t 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 p r o p e r l y ^ The model c a l c u l a t i o n s show t h a t under these slow c o o l i n g c o n d i t i o n s , i t is e s s e n t i a l to cons ider heat f low in both t ransverse d i r e c t i o n s and should not be neg lec ted in one o f the d i r e c t i o n s as in the model of B e a t t i e (30 , 3 2 ) , even though the aspect r a t i o s are g rea te r than 2 . 5 . 4.4 Zinc - Jumbo Cast ing - Cominco F i g . 4.15 shows the c r o s s - s e c t i o n o f a jumbo ingo t and the var ious dimensions of the c a s t i n g . Because of the absence o f symmetry elements i n one of the t r a n s v e r s e d i r e c t i o n , c a l c u l a t i o n s must be performed ^ o r o n e - h a l f of a c a s t i n g . In t h i s s i m u l a t i o n a f u l l th ree dimensional model i s requ i red because, the aspect r a t i o i s 1.07. The mate r ia l cast is high grade z inc f o r which the thermo-phys ica l p r o p e r t i e s are given in 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 method f o r i r -regu la r geometries such as t h a t of the jumbo does not pose any spec ia l problems in p r i n c i p l e . However a new set of equat ions had to be inc luded in the computer program to take care o f the i r r e g u l a r l y shaped nodes. The manner in which the c a s t i n g has been d i s c r e t i z e d is presented in F i g . 4 . 1 6 . Since i t was requ i red to per form the c a l c u l a -t i o n s f o r o n e - h a l f o f a c a s t i n g , in order to cut down the computer costs s l i g h t l y , a v a r i a b l e node s i ze has been used in t h i s s i m u l a t i o n . The s u b - d i v i s i o n of the jumbo has been made such tha t a f i n e r node e x i s t s at the sur face and c o a r s e r n o d e s a t t h e c e n t r e . The boundary cond i t i ons f o r t h i s s i m u l a t i o n were obta ined by f r e e z i n g in thermocouples near the sur face of 77 C A B C D E F 546 508 216 114 114 38 F i g . 4.15 A cross sec t ion of the jumbo ingo t w i th the var ious dimensions given in mm.. 78 F i g . 4.16 D i s c r e t i z a t i o n o f f i n i t e - d i f f e r e n c e the jumbo ingo t f o r the c a l c u l a t i o n s . Density of L iqu id - 6620 kg/m 3 Dens i ty of So l i d - 6981 kg/m 3 S p e c i f i c Heat of L iqu id - -480 J/g K S p e c i f i c Heat of So l i d Zinc - 0.343 + .154 ( I O - 3 ) T J/g K Thermal Conduc iv i ty - 113 W/m2 K Latent Heat of Fusion - 113 J/g L iqu idus Temperature - 420°C Sol idus Temperature - 410°C Table V I I Thermophysical P roper t i es of Zinc in Jumbo Ingot S i m u l a t i o n . the jumbo (~20 mm) dur ing a cas t i ng and l e t t i n g them drop w i t h the descending i n g o t . The exact p o s i t i o n of the thermocouple t i p was loca ted subsequent ly by s e c t i o n i n g the i n g o t . From the tempera tu re- t ime p l o t measured, the sur face h e a t - t r a n s f e r 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 a t r i a l and e r r o r procedure. Table VI I I gives the heat t r a n s f e r c o e f f i c i e n t used in t h i s s i m u l a t i o n . F i g . 4~.17 compares the measured temperature w i t h the c a l c u l a t e d va lues . The pool p r o f i l e was obta ined 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 of 10% c o p p e r - i n - z i n c to the sump a f t e r steady s ta te c o n d i t i o n s had been reached. A f t e r the a d d i t i o n , - the pool was s t i r r e d using a padd le , d r i ven by a power d r i l l . The i ngo t was l a t e r sec t ioned l o n g i -t u d i n a l l y at the mid-plane perpend icu la r to the two non-notched faces . The sur face then was gen t l y po l i shed to remove the machine marks. The pool boundary was f i n a l l y d e l i n e a t e d by e tch ing w i t h a 4% N i t a l s o l u t i o n . The l o n g i -t u d i n a l sec t ions where the contour p r o f i l e s have been obta ined from the model are shown in F i g . 4 . 1 8 , as shaded. F igs . 4.19 to 4 . 2 1 , shows the contour p r o f i l e s ob ta ined at these two s e c t i o n s , at d i f f e r e n t t imes from the s t a r t of the c a s t i n g . F i g . 4.21 correspond to steady s t a t e c o n d i t i o n s whereby the pool p r o f i l e remains the same w i t h J 81 Distance from Heat Trans fe r the top of Mould C o e f f i c i e n t s (mm) (W/m2 K) 0 - 212 20934 2 1 2 - 2 1 6 4187 1450 - 1454 125 Between 212 and 1454 mm the Heat Transfer C o e f f i c i e n t has been dropped in an .exponent! - cal f a s h i o n . Table VI I I Heat T rans fe r C o e f f i c i e n t s used in Zinc Jumbo Cas t i ng . Time (s) 4.17 Comparison of the. measured and c a l c u l a t e d temperature p r o f i l e s f o r the z inc jumbo ingo t cas t at 1.69 mm/s . y F i g . 4.18 A th ree-d imens iona l view of the jumbo ingo t showing the l o n g i t u d i n a l sec t ions where the pool p r o f i l e s have been o b t a i n e d . Distance along Y-axis ( m m ) X-axis(mm) 0 80 160 240 320 400 480 0 80 160 240 T 1 1 ; 1 1 1 — — i — I r — — i 1 i — F i g . 4.19 Pool p r o f i l e s obta ined at the l o n g i t u d i n a l sec t ions shown in F i g . 4.18 f o r cas t ing zinc jumbo ingot at 1.27 mm/s, 278 s a f t e r the s t a r t . 00 85 F i g . 4.20 Pool p r o f i l e s obta ined at the l o n g i t u d i n a l sec t ions shown in F i g . 4.18 f o r c a s t i n g z inc jumbo ingot at 1.27 mm/s, 557 s a f t e r the s t a r t . Distance along Y-axis (mm) X - axis (mm) 0 80 160 240 320 4 0 0 4 8 0 0 80 160 240 Op 1 i — — T 1 — n I • " " — r — T ~ 1 7 2 0 f 800-880-960-4.21 Steady s ta te 1ongi t ud i nal c a s t i n g z inc pool p r o f i l e s obta ined at the sec t ions shown i n F i g . 4.18 fo jumbo ingo t at 1.27 mm/s. 87 f u r t h e r increase in t i m e . The three d imensional s teady s t a t e temperature d i s t r i b u t i o n i s g iven i n Table A 3 -111 -Comparison between measured and c a l c u l a t e d pool p r o -f i l e s f o r the z inc jumbo is seen in F i g . 4 . 2 2 . As can be seen there i s a good agreement between the two. The s h i f t i n g o f the bottom o f the sump to an asymmetr ica l p o s i t i o n can be c l e a r l y seen in t h i s f i g u r e . A th ree d i -mensional view of the pool sur face i s presented in F i g . 4 . 2 3 . The development o f the pool f o r t h i s c a s t i n g speed has also been moni tored by d ipp ing a s t e e l rod i n t o the p o o l . The r e s u l t s obta ined are presented i n F i g . 4 . 2 4 . In t h i s graph the pool depth is p l o t t e d aga ins t the l eng th o f the c a s t i n g at any i n s t a n t . Note t h a t the s h e l l does not r e a l l y commence growth at the bottom u n t i l a f t e r .500 mm o f c a s t i n g . This is due to the f a c t t h a t the heat has to d i f f u s e from the cent re o f the cas t i ng only through the s ides as no c o o l i n g is prov ided on the bot tom. F i g . 4.25 shows a s i m i l a r p l o t ob ta ined f o r a h igher c a s t i n g speed o f 1.693 mm/s. As i n the p rev ious case, good agreement i s obta ined between the c a l c u l a t e d and measured va lues . The t ime requ i red f o r steady s t a t e con-d i t i o n s can be c a l c u l a t e d by d i v i d i n g the cast l e n g t h 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 cas t i ng of z inc jumbo i ngot at 1.27 mm/s . F i g . 4.23 Three-dimensional v i s u a l i z a t i o n of the l i q u i d pool sur face of z inc jumbo ingo t cast a t 1.27 mm/s . I 5 0 0 r 2 0 L e n g t h of cast ing ( I n ) 4 0 6 0 8 0 Cast ing speed - 7 6 m m / m i n ( 3 i n / m i n ) High g r a d e z i n c ~ lOOOf E Q . a> T J o o 0. 5 0 0 xr-cr •Q-Q. Q r> n o o — C a l c u l a t e d o M e a s u r e d 5 0 0 1 0 0 0 1 5 0 0 L e n g t h of cast ing ( m m ) 4 0 -c a. cu 2 0 o o a. 2 0 0 0 F i g . 4.24 Comparison between the c a l c u l a t e d and measured pool depths obta ined at d i f f e r e n t times from the s t a r t of cas t ing of z inc jumbo ingot cast at 76 mm/mi n. o 1500 20 Length of casting (in ) 40 60 I — — r —— Casting speed I02mm/min (4in/min) Prime western grade zinc 1000 o o Calculated 500 o Measured 0 1 0 500 2000 1000 1500 Length of casting (mm) F i g . 4.25 Comparison between the c a l c u l a t e d and measured pool depths obta ined at d i f f e r e n t times from the s t a r t of cas t ing of z inc jumbo ingo t cast at 102 mm/min. where the curve becomes f l a t by the c a s t i n g speed. I t is seen from t h i s and the prev ious F i g . 4 . 2 4 , t h a t i t takes 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 to p r e v a i l . Since the t o t a l d u r a t i o n of the cas t i ng is on ly around 40 minutes , t h i s means t h a t f o r 25% of the t o t a l c a s t i n g t ime the c a s t i n g is in an unsteady s t a t e . F i g . 4.26 shows the f r e e z i n g of the i n g o t as seen from a t ransverse s e c t i o n . Here the s o l i d u s iso therm f o r d i f f e r e n t p o s i t i o n along the cas t i ng d i r e c t i o n have a l l been compressed on to the plane of the paper. The shape of the f r e e z i n g f r o n t obta ined in an ac tua l cast can be seen as f a i n t r i ngs in F i g . 4 .27 . The r i ngs i n F i g . 4.27 are caused by the i n t e r m i t t e n t s t i r r i n g of the pool du r ing the c a s t i n g o f a d i l u t e a l l o y of lead in z i n c . F i n a l l y the importance of i n c l u d i n g the notch in the c a l c u l a t i o n is shown in F i g . 4 .28 , which shows pool p r o f i l e s c a l c u l a t e d w i th and w i t h o u t the n o t c h . As can be seen there i s a s u b s t a n t i a l e f f e c t of the notch on the pool depths . 4.5 Summary of V a l i d a t i o n Runs o 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 as seen on a jumbo c r o s s - s e c t i o n . F i g . 4.27 Macros t ruc tu re o f the z inc 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 in F i g . 4 . 2 6 . 95 Distance along Y-axis 80 160 2 4 0 320 4 0 0 ' 4 8 0 160 3 2 0 4 8 0 6 4 0 8 0 0 9 6 0 I I 20h Zinc ingot Casting speed 1.69 m m / s Jumbo ingot with notch Jumbo ingot without notch 1 1 1 i i F i g . 4.28 Comparison of the c a l c u l a t e d pool p r o f i l e s obta ined w i t h and w i t h o u t the notch f o r c a s t i n g z inc at 1.69 mm/s. 96 three D.C. c a s t i n g o p e r a t i o n s : two f o r the case o f aluminium and one f o r the case of z i n c . E x c e l l e n t agreement i s ob-ta ined between c a l c u l a t e d and measured pool p r o f i l e s in the aluminium ingots cooled by convent iona l f1ood c o o l i n g . In the case of aluminium blocks sub jec ted to reduced secondary coo l ing the s i m u l a t i o n from the model was less successfu l f o r h igher c a s t i n g speeds. This has been a t t r i b u t e d to un-c e r t a i n t y in 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 from the r e s u l t s obta ined in the l a b o r a t o r y exper iments . In the case o f z inc jumbo good agreement has been observed between measured and p r e d i c t e d pool p r o f i l e s . Chapter 5 EFFECT OF CASTING VARIABLES ON HEAT FLOW 5.1 I n t r o d u c t i o n In t h i s chapter a d e s c r i p t i o n is g iven of how the model, developed in the previous s e c t i o n , was used in a p r e d i c t i v e mode to study the e f f e c t o f d i f f e r e n t c a s t i n g v a r i a b l e s on the o v e r a l l heat f l o w . In order to make these \ c a l c u l a t i o n s mean ing fu l , a l l the s i m u l a t i o n s presented in t h i s sec t i on have been c a r r i e d out under c a s t i n g c o n d i t i o n s ob ta inab le in i n d u s t r i a l p r a c t i c e . In t h i s way the impor-tance of cons ide r ing heat f low in the t ransverse dimensions is brought o u t , w i th s p e c i f i c examples i n v o l v i n g the c a s t i n g of var ious sec t ions of aluminium and z i n c . The importance of a x i a l conduct ion in comparison w i t h bulk motion of the cas t ing is demonstrated by cons ide r ing d i f f e r e n t c a s t i n g speeds. The e f f e c t of thermal c o n d u c t i v i t y on the o v e r a l l heat f low is s tud ied by comparing s i m u l a t i o n s of aluminium and z inc c a s t i n g . F i n a l l y some comments are made regard ing v a r i a b l e s which have n e g l i g i b l e e f f e c t on heat f low 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 s t r u c t u r e . 97 5.2 E f f e c t of Aspect Ra t io * In the model l ing of heat f low in r e c t a n g u l a r s labs i t has been common p r a c t i c e to neg lec t heat f low i n the t ransverse d i r e c t i o n , t h a t i s , p a r a l l e l to the broad face ( 30 , 33, 4 8 ) . In t h i s way the programming e f f o r t and. the computat ional costs are reduced cons ide rab ly as compared to the case o f a th ree-d imens iona l model. The importance of heat f low in the t ransve rse d i -r e c t i o n has been i n v e s t i g a t e d w i th s p e c i f i c examples of the cas t i ng of aluminium and z inc ingo ts w i t h d i f f e r e n t aspect r a t i o s . In these c a l c u l a t i o n s the c o o l i n g c o n d i t i o n s used are s i m i l a r to those discussed in Chapter 4 p e r t a i n i n g to convent iona l D.C. c o o l i n g . The c a s t i n g speed employed was 1.778 mm/s f o r a l l the c a l c u l a t i o n s . F i g . 5.1 shows the s t e a d y - s t a t e pool depth obta ined in c a s t i n g aluminium ingots w i th var ious s e c t i o n s i z e s . The bottom curve corresponds to sec t ions o f 381 x 381 mm, 381 x 571 mm, 381 x 762 mm and f i n a l l y a 381 mm t h i c k aluminium siab of i n f i n i t e w i d t h . The 1 a t t e r two-d imensional c a l c u l a t i o n was c a r r i e d out by reducing the number of nodes *The aspect r a t i o is de f ined as the r a t i o between the two t ransverse dimensions w i th the l a r g e r value taken as the numerator. f 300' 1 1 I 2 3 4 Aspect ratio F i g . 5.1 E f f e c t of aspect r a t i o s on the pool depths in cas t ing 381 mm and 457.2 mm t h i c k aluminium slabs at 1.778 mm/s. CO CO 100 in the w id th d i r e c t i o n to th ree wh i le the h e a t - t r a n s f e r c o e f f i c i e n t in t h i s d i r e c t i o n was set equal to ze ro . Thus no g rad ien ts are imposed in the wid th d i r e c t i o n and i d e n t i -cal temperatures are c a l c u l a t e d by the program f o r the three rows of nodes. From F i g . 5.1 i t can be c l e a r l y seen tha t the e f f e c t of the second t ransverse dimension d imin ishes as the aspect r a t i o exceeds 2 .5 . Thus f o r heat f low c a l c u l a t i o n s in r e c t a n g u l a r s labs w i t h aspect r a t i o s g rea te r than 2 . 5 , i t is adequate to consider two dimensions o n l y . The top curve in F i g . 5.1 corresponds to sec t ions 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 of i n f i n i t e w i d t h . Compared to the lower curve the t r a n s i t i o n from th ree-d imens iona l to two-dimensional heat f low occurs at s i i g h t l y h igher values of the aspect r a t i o . But even in t h i s case the e f f e c t i s n e g l i g i b l e beyond an aspect r a t i o of 2 . 5 . This conc lus ion i s v a l i d f o r a l l cas t i ng speeds lower than the one used in t h i s c a l c u l a t i o n , namely 1.778 mm/s, 'as we l l as f o r sec t ions smal ler than 457 mm in t h i c k n e s s . In a d d i t i o n t h i s w i l l a lso apply to coo l i ng c o n d i t i o n s which are more in tense than those used in the present c a l c u l a t i o n s . 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 I 2 0 0 h e E f . l O O O f CD T J 1.78 mm/s O o o CL 800 Zinc ingot 381mm thick section Conventional D.C. cooling O - 2-dimensional 6 0 0 1 Aspect ratio F i g . 5.2 E f f e c t of aspect r a t i o s on the pool depths in cas t ing 381 mm t h i c k z inc slabs at 1.78 and 2.2 mm/s. depth p l o t t e d aga ins t the aspect r a t i o f o r z inc cast at two speeds. The th ickness of the s e c t i o n cons idered i s 381 mm. The top curve corresponds to a c a s t i n g speed o f 2.2 mm/s wh i le the bottom curve corresponds %o 1.778 mm/s. In comparison w i th aluminium the on-set of t w o T d i m e n s i o n a l heat f low is seen at even lower aspect r a t i o s . Thus i t i s adequate to consider t w o - d i m e n s i o n a l heat f l o w f o r aspect r a t i o s exceeding 2 . 0 . The same r e s u l t a lso holds f o r h igher c a s t i n g speeds. The above d iscuss ion on the r e l a t i v e importance of two-and th ree-d imens iona l heat f low has been made w i t h respect to convent iona l D.C. c o o l i n g . However the a p p l i c a -t i o n of a reduced secondary coo l i ng p r a c t i c e such as is f o l l owed a t B r i t i s h Aluminium (30, 85) changes t h i s p i c t u r e . Two-and th ree-d imens iona l c a l c u l a t i o n s have been performed on a 250 x 690 mm s e c t i o n (aspect r a t i o o f 2.76) w i t h these coo l i ng c o n d i t i o n s . Table IX presents the pool depth obta ined from the two sets of c a l c u l a t i o n s f o r two d i f f e r e n t c a s t i n g speeds: 0.833 mm/s and 1.26 mm/s. I t is seen t h a t there is a s i g n i f i c a n t d i f f e r e n c e between the two c a l c u l a t i o n s even though the aspect r a t i o exceeds 2.5 and the d iscrepancy increases as the c a s t i n g speed i s i n -creased. Thus e r r o r in t roduced i n t o the poo l -dep th c a l c u l a -t i ons increases from 10% at 0.833 mm/s to 20.5% at 1.26 mm/s. 103 A A Ingot 254 x 690 mm Reduced Secondary Cool ing Speed mm/s Pool Depth 3-Dimensional Pool Depth 2-Dimensional 0.833 468 mm 520 mm 1 .266 1014 mm 1222 mm Table IX Comparison between three dimensional and two dimensional pool depths f o r reduced secondary coo l i ng of 254 x 690 mm aluminium i n g o t . 104 As w i l l be seen in the f o l l o w i n g sec t ion i t appears b e t t e r f o r t h i s coo l i ng c o n d i t i o n to neg lec t a x i a l conduct ion but cons ider heat f low in both t ransverse d i r e c t i o n s . 5.3 E f f e c t of Ax ia l Conduction I t has been t r a d i t i o n a l p r a c t i c e in almost a l l the models developed todate f o r non- fe r rous c a s t i n g , to i nc lude the e f f e c t of a x i a l conduc t ion . I n - o r d e r to check the importance of the z-component o f heat conduc t i on , the computer program was mod i f ied by bypassing the i m p l i c i t c a l c u l a t i o n s in the z - d i r e c t i o n . In these c a l c u l a t i o n s there i s only one s l i c e in the z - d i r e c t i o n and the s o l i d i f i -c a t i o n is fo l l owed as t h i s s l i c e t r a v e l s from the meniscus downwards at the c a s t i n g speed. This is the procedure fo l l owed in models of the cont inuous cas t i ng process f o r s t e e l . F i g . 5.3 shows the sur face temperature at the mid-face o f a 457 mm square aluminium ingo t vs d is tance below the meniscus c a l c u l a t e d w i th and w i t h o u t a x i a l c o n d u c t i o n . As expected, the curve obta ined by i n c l u d i n g the a x i a l conduct ion is the smoother of the two. The rebound of the sur face temperature f o r the case of no a x i a l conduct ion is caused by the fo rmat ion of an a i r gap in the mould. Distance from the meniscus (m m ) F i g . 5.3 Ca lcu la ted sur face temperature p r o f i l e s obta ined at the mid- face of 457 x 457 mm aluminium ingot cast at .974 mm/s, w i th and w i t h o u t the o a x i a l conduc t ion . 106 Furthermore the curve w i t h o u t the a x i a l conduct ion a lso shows a much steeper temperature g rad ie n t cor respond ing to d i r e c t c h i l l i n g in the secondary zone. F i g . 5.4 shows the cent re temperature of a 457 mm square aluminium ingo t vs d is tance below the meniscus. As before the two cond i t i ons considered are wi th^and w i t h o u t a x i a l conduc t ion . I t can be seen tha t by not i n c l u d i n g the a x i a l conduct ion the pool depth increases from 254 to 302 mm, t h a t i s by about 20%. Table X. shows the importance of i n c l u d i n g a x i a l conduct ion f o r var ious c a s t i n g c o n d i t i o n s . Several impor tan t po in ts can be drawn from t h i s t a b l e . I t i s seen t h a t the importance o f a x i a l conduct ion decreases w i t h increase in cas t i ng speed. Thus f o r 457 mm square aluminium i n g o t the d i f f e r e n c e s between the two c a l c u l a t i o n s decrease from 20% at a c a s t i n g speed of 0.974 mm/s, to 3% at 2.116 mm/s. In the case of s labs the e f f e c t of c a s t i n g speed i s much less pronounced. For a 457 mm t h i c k a l im in ium s lab the d i f f e r e n c e between the two c a l c u l a t i o n s is 9% at 0.974 mm/s and 4% at 2.116 mm/s. Since b i l l e t s are cas t at f a s t e r speeds than s l a b s , t h i s means i t is poss ib le to neg lec t a x i a l conduct ion in the b i l l e t case. Thus i t would have been a b e t t e r assumption f o r Jov ic et al (35) in mode l l i ng A l u m i n u m ingot 4 5 7 X 4 5 7 mm D i s t a n c e from the m e n i s c u s (mm) F i g . 5.4 Ca lcu la ted cent re temperature p r o f i l e s obta ined f o r 457 x 457 mm aluminium ingot cast at .974 mm/s, w i th and w i t h o u t the ax ia l conduc t ion . 108 Material Section Size (mm) Cooling Casting Speed (mm/s) Pool Depth with Axial Conduction ( m m ) Pool Depth without Axial Conduction ( rnrr, ) A£ 457 x 457 Conventional 0.974 254 302 Ail 457 x 457 Conventional 2.116 572 587 A£ 457 x 0 0 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 Steady s t a t e pool depths f o r a luminium and z inc ingo ts c a l c u l a t e d w i th and w i t h o u t the a x i a l conduc t ion . 109 heat f low in a 360 x 1600 mm s e c t i o n to i nc lude ax ia l con-d u c t i o n , but neg lec t conduct ion in a d i r e c t i o n p e r p e n d i c u l a r to the narrow f a c e . The e f f e c t of thermal c o n d u c t i v i t y i s to decrease the importance of a x i a l conduct ion w i t h a de-crease in thermal c o n d u c t i v i t y . Thus f o r 457 mm square ingo ts of aluminium and z inc cast at 0.974 mm/s, the d i f -ferences between the case of a x i a l conduct ion and no a x i a l are 20% and 9% r e s p e c t i v e l y . F i n a l l y the importance o f a x i a l heat conduct ion f o r the case of reduced secondary coo l i ng has been s tud ied f o r the cas t i ng of a 254 x 690 mm aluminium s e c t i o n at 0.833 mm/s. The boundary c o n d i t i o n s employed are given in Table .VI., in Chapter 4. Compared to prev ious cases the d i f f e r e n c e between the two pool depths , namely 468 mm and 49'4 mm, i s only 5%. even at t h i s low c a s t i n g speed. At h igher speeds the d i f f e r e n c e s would be even less s i g n i f i -can t . The r e s u l t s obta ined in t h i s pa r t of the work are oppos i te to the e f f e c t s repor ted in s e c t i o n 5.2 where n e g l e c t i n g conduct ion in one of the t ransverse d i r e c t i o n s was d iscussed. In t h a t case i t was seen t h a t an increase in the c a s t i n g speed increased the e r r o r in the two-dimensional c a l c u l a t i o n s . Therefore when develop ing 110 two-dimensional models in these systems i t is sa fer to neg lec t a x i a l conduct ion but cons ider heat f low in both t ransverse d i r e c t i o n s . 5.4 Importance of Unsteady State In most of the models developed t o d a t e , i n c l u d i n g those f o r c y l i n d r i c a l shapes w i t h ax ia l symmetry, s teady-s t a t e c o n d i t i o n s have been assumed. However i t i s impor tan t to note t h a t u n l i k e the cont inuous c a s t i n g of s tee l where the c a s t i n g opera t ion is t r u l y con t inuous , the v e r t i c a l D.C. c a s t i n g is only semi-cont inuous in n a t u r e . Therefore the i n i t i a l t r a n s i e n t pa 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 f r a c t i o n of the cas t i ng c y c l e . Since the model developed in t h i s work can be used to study t r a n s i e n t e f f e c t s the importance of unsteady s t a t e has been i n v e s t i g a t e d . The unsteady s ta te has been s tud ied e x p e r i m e n t a l l y by Serger ie and Bryson (14) who have discussed the problem of i ngo t "bowing" observed in the t r a n s i e n t s e c t i o n o f la rge sheet i ngo t c a s t i n g . This r e s u l t s from the use of shor t moulds in which the f i r s t metal f r e e z i n g on the s too l cap fee l s the s t rong e f f e c t s of a d i r e c t quench sooner than i t would i f l a r g e r moulds were used. During 'bowing ' which occurs as the b u t t emerges from the mould, the ends of the b u t t sh r ink upwards o f f the s too l cap and inwards away from 111 the ends o f the mould. The authors have patented a "pu lsed c o o l i n g " method to reduce the heat t r a n s f e r r a t e in the t r a n s i e n t p o r t i o n by using a pulsed water sp ray . Recent ly Yu (43) has tack led the same problem by d r a s t i c a l l y changing the heat t r a n s f e r mechanism at the ingo t sur face through the use of water c o n t a i n i n g d i s s o l v e d carbon d i o x i d e . As the c o o l i n g water e x i t s f rom 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 fo rming a temporary, e f f e c t i v e i n s u l a t i o n l aye r on the sur face o f the c a s t i n g , thereby reducing the heat t r a n s f e r . The importance o f coo l i ng in the t r a n s i e n t p o r t i o n can be app rec ia ted from the f a c t t h a t both o f the above-mentioned ideas have been pa ten ted . The temperature f i e l d i n the i n g o t a t the s t a r t o f the cast i s d i f f e r e n t from the temperature "in the s teady-s t a t e p a r t of the cas t ing which f o l l o w s . Thus spec ia l measures may need be taken to prevent c r a c k i n g in the e a r l y t r a n s i e n t pa r t o f the c a s t , s ince the c r a c k , once i n i t i a t e d , can cont inue 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 coo l i ng c o n d i t i o n s may not generate cracks per se. F i g . 5 .5 . shows the c a l c u l a t e d sur face temperature p r o f i l e s at the mid-plane on the bottom face of a jumbo c r o s s - s e c t i o n . The metal cast is aluminium at a speed of 112 Time below the meniscus ( s ) F i g . 5.5 Ca lcu la ted sur face temperature p r o f i l e s f o r 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 ingo t (a t the bottom mid- face of a j umbo s e c t i o n ) . 113 1.35 mm/s. The two p r o f i l e s correspond to the temperature h i s t o r i e s of two s l i c e s l eav ing the mould at d i f f e r e n t t imes , one corresponding to the f i r s t s l i c e e x i t i n g the mould (dashed l i n e ) and the o ther cor responding to a s l i c e cast under s t e a d y - s t a t e c o n d i t i o n s ( s o l i d l i n e ) . I t should be noted tha t s teeper ax ia l temperature g rad ien ts are s e t -up in the i n i t i a l s l i c e compared to the s t e a d y - s t a t e s l i c e ; and t h i s poss ib l y could i n i t i a t e sur face c r a c k s . The t ime requ i red f o r the pool p r o f i l e to reach steady s t a t e w i l l depend on a number of f a c t o r s i n c l u d i n g the metal c a s t , sec t i on s ize and c a s t i n g speed. This t r a n s i e n t t ime was presented as a f u n c t i o n of c a s t i n g speed f o r c a s t i n g aluminium slabs of var ious th icknesses in F i g . 4.6 of Chapter 4. I t was seen t h a t f o r sma l le r s e c t i o n s , c a s t i n g speed had no e f f e c t on t h i s t ime w h i l e f o r t h i c k e r sec t ions i t had a small e f f e c t . F i g . 5.6 shows the t r a n s i e n t t ime f o r c a s t i n g var ious sec t ions of aluminium and z i n c . Because of the lower value of thermal c o n d u c t i v i t y in the case of z inc i t . t a k e s , a longer t ime f o r steady s t a t e c o n d i t i o n s to be ach ieved. The e f f e c t of bottom 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 t r a n s i e n t p o r t i o n was s tud ied w i t h the model. The 114 800 <D E 4 00 i -Al 38.1 cm 200 Aspect ratio F i g . 5.6 Time requ i red f o r the pool p r o f i l e s to reach steady s t a t e f o r 381 mm t h i c k aluminium and z inc slabs of d i f f e r e n t aspect r a t i o s cast at 1.778 mm/s. 115 2 value 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 as high as 418 w/m K had n e g l i g i b l e e f f e c t in convent iona l D.C. c o o l i n g , w h i l e the e f f e c t was g rea te r w i t h reduced secondary c o o l i n g . The value of bottom h e a t - t r a n s f e r c o e f f i c i e n t d id not a f f e c t the steady s t a t e temperature f i e l d . 5.5 E f f e c t of Sect ion Size The e f f e c t of sec t ion s ize has been i n v e s t i g a t e d in c a s t i n g square sec t ions of aluminium and z i n c . The s e c t i o n s izes considered were 305 x 305 mm, 381 x 381 mm and 457 x 457 mm. In a l l the cases the c a s t i n g has been main ta ined at 1.778 mm/s. F i g . 5.7 shows the s t e a d y - s t a t e pool depths obta ined in c a s t i n g sec t ions mentioned above under D.C. coo l i ng c o n d i t i o n s . Comparing aluminium and z inc i t i s found tha t the low thermal c o n d u c t i v i t y of z inc is respon-s i b l e f o r the much steeper increase in the pool depth w i t h i n c r e a s i n g s e c t i o n s i z e . The dramat ic e f f e c t of thermal c o n d u c t i v i t y on the pool shape can be apprec ia ted f rom F i g . 5 .8 , where the pool p r o f i l e s f o r 381 x 381 mm sec t i ons of aluminium and z inc are a lso p resented . F i g . 5.9 shows the t ime requ i red f o r steady s ta te to be ach ieved. I t can be seen tha t in the case of a lumin ium, because of i t s h igh thermal c o n d u c t i v i t y , the t r a n s i e n t t ime keeps pace w i t h sec t i on s ize r e s u l t i n g in a l i n e a r r e l a t i o n s h i p between the two. 116 35 45 Square section size (cm) F i g . 5.7 Steady s t a t e pool depths f o r square sec t ions of aluminium and z inc cast at 1.778 mm/s. i 117 F i g . 5.8 Ca lcu la ted steady s t a t e pool p r o f i l e s f o r 381 mm square sec t ions of aluminium and z inc cast at 1.778 mm/s. 118 F i g , 5.9 Time requ i red f o r the pool p r o f i l e s to reach steady s ta te f o r square sec t ions of aluminium and z inc cast at 1.778 mm/s. 119 5.6 E f f e c t of Super Heat In order to compensate f o r the drop i n temperature o f the molten metal dur ing t r a n s f e r i n t o the mould, a q u a n t i t y of superheat is prov ided to the l i q u i d m e t a l . In t h i s ana l ys i s i t has been found t h a t super heat has a very minor e f f e c t on the l o c a t i o n of the so l i dus i so the rm. I t was a lso seen t h a t most of the superheat was removed in the very top p o r t i o n of the c a s t i n g . 5.7 E f f e c t of Cool ing Condi t ions 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 ready been considered in Chapter 4 in connect ion w i t h v a l i d a t i o n of the model where the d i f f e r e n c e between conven t iona l D.C. c o o l i n g and reduced secondary coo l i ng of aluminium slabs was seen. F i g . 5.10 shows the r e l a t i o n s h i p between the heat -t r a n s f e r c o e f f i c i e n t below the mould and the s t e a d y - s t a t e pool depth f o r the cas t i ng of 610 x 546 mm z inc sec t ions at 1 mm/s. In these c a l c u l a t i o n s , the h e a t - t r a n s f e r co-e f f i c i e n t is kept c o n s t a n t , at the a p p r o p r i a t e v a l u e , f o r the e n t i r e length o f the s t r a n d . I t is observed t h a t as the h e a t - t r a n s f e r c o e f f i c i e n t is decreased, - there i s a steep increase in the pool depth . E E - 10001 SZ Q L 00 Q o o CL E E X D zinc ingot- 610 mm x 546 mm speed - I mm /s 900 8 0 0 1 0 4.186 8.373 12.560 16.747 20.934 Heat Transfer Coefficient (kw/m °C) F i g . 5.10 E f f e c t of 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 610 x 546 mm zinc i n g o t ' a t 1 mm/s. r o o 121 5.8 E f f e c t of Latent Heat Release on Model C a l c u l a t i o n s In the s e c t i o n dea l ing w i th the development of the mathematical model a d e t a i l e d account was given of the man-ner in which l a t e n t heat is r e l e a s e d . A number of computer runs were undertaken in the i n i t i a l stages of model deve lop-ment to t e s t the s e n s i t i v i t y of the c a l c u l a t i o n s to t h i s parameter. In a l l the c a l c u l a t i o n s i t was seen t h a t the temperature range over which the l a t e n t heat was re leased had a small e f f e c t on the l o c a t i o n of the s o l i d u s i s o t h e r m . However, as might be expected, the l o c a t i o n of the l i q u i d u s isotherm was changed c o n s i d e r a b l y . 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 inc where the l a t e n t heat was re leased over a range of 10°C between 420°C and 410°C and over a range of 1 °C between 419 and 420°C " respect ive ly -there was a d i f f e r e n c e of on ly 10 mm in the p o s i t i o n o f the bottom of the so l i dus i s o t h e r m , whereas there was c lose to 110 mm d i f f e r e n c e in the case of the l i q u i d u s i s o t h e r m . The e f f e c t o f the manner of l a t e n t heat re lease decreases w i t h i n c r e a s i n g thermal c o n d u c t i v i t y of the m e t a l . I t was discussed in Chapter '3 t ha t w h i l e s i m u l a t i n g heat f low in the D.C. cas t i ng of non - fe r rous metals w i t h high thermal c o n d u c t i v i t y , i t i s poss ib le f o r nodes to jump from above the l i q u i d u s to below the s o l i d u s . A post i t e r a t i v e c o r r e c t i o n procedure has been used to overcome 122 t h i s problem. However use of t h i s c o r r e c t i o n r o u t i n e in the program in c o n j u n c t i o n w i t h having a la rge t ime i n t e r -val can create s t a b i l i t y problems. A wider spread between the l i q u i d u s and the so l i dus temperature help's in improving the s t a b i 1 i t y ' problems. 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 re leased over a range of e i t h e r 1°C or 10°C. In the case of s imu la t i ons dea l ing w i t h the cas t i ng of reduced secondary cooled aluminum slabs a wider range of 23°C has been used. 5 .9 Summary In t h i s chap te r , the r e s u l t s of the model study on the importance o f the d i f f e r e n t heat f low v a r i a b l e s have been repor ted f o r D.C. c a s t i n g under a range of c a s t i n g c o n d i t i o n s . I t was shown tha t in D.C. c a s t i n g i n v o l v i n g convent iona l secondary c o o l i n g , where the aspect r a t i o exceeds 2 .5 , i t i s adequate to consider heat f low in two d imensions, namely, the s h o r t e r of the t ransverse dimensions and the a x i a l d i r e c t i o n . In the case of reduced secondary coo l i ng the a x i a l conduct ion was seen to p lay a minor pa r t compared to the t ransverse heat f l o w s . Thus in mode l l i ng heat f lows in these systems, the complex i ty of the problem may be reduced cons iderab ly by n e g l e c t i n g the a x i a l heat conduc t ion . The importance of the unsteady s t a t e and the e f f e c t of sec t i on s ize and c o o l i n g c o n d i t i o n s have been demonstrated using s p e c i f i c examples of c a s t i n g aluminium and z i n c . 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 I n t r o d u c t i o n Formation of cracks has long been recognized as a problem in the D.C. c a s t i n g of metals such as high s t r e n g t h aluminium a l l o y s (12, 42, 4 7 ) . Unl ike the cont inuous c a s t i n g of s tee l where many cracks are generated m e c h a n i c a l l y , e . g . by mould o s c i l l a t i o n and bending and s t r a i g h t e n i n g o p e r a t i o n s , cracks in non- fe r rous D.C. c a s t i n g are most o f t e n caused by thermal s t resses generated du r ing c a s t i n g . K r e i l e t al (17) d iscuss the var ious cracks formed in c a s t i n g copper and i t s a l l o y s . S i m i l a r l y Dief fenbach (12) proposed remedial measures f o r s o l v i n g the c rack ing problems faced in c a s t i n g d i f f e r e n t aluminium a l l o y s . There are also o ther papers dea l ing w i t h i n t e r n a l ! c r a c k s in aluminium a l l o y s (32 , 40, 42, 47, 61-63, 67) and copper a l l o y s (55, 56 ) . No re ferences have been found in the l i t e r a t u r e however concerned w i th the fo rma t ion of cracks in the D.C. cas t i ng of z inc and i t s a l l o y s . 124 125 6.2 I n t e r n a l Cracks in the D.C. Cast ing of Prime Western Grade Zinc Prime Western Grade z inc is an a l l o y of z inc c o n t a i n -ing approx imate ly 1 wt% lead which is used in g a l v a n i z i n g a p p l i c a t i o n s . I t has been found tha t D.C. c a s t i n g of t h i s p a r t i c u l a r a l l o y can give r i s e to severe i n t e r n a l c rack ing problems. In order to solve t h i s problem i t was necessary to study the cracks in d e t a i l , e . g . crack l o c a t i o n and f requency , morphology of crack sur faces and thermal con-d i t i o n s p r e v a i l i n g at the t ime o f crack f o r m a t i o n . Thus an exper imenta l campaign was conducted i n - p l a n t a t Cominco L t d . , T r a i l where a two s t rand D.C. c a s t i n g machine has been in opera t ion f o r many years f o r the p ro -duc t ion of Special High Grade z inc jumbos w i t h a notched c r o s s - s e c t i o n ( F i g . 4 . 1 5 ) . In t h i s study a t o t a l o f seven cas t ings of Prime Western Grade z inc were made w i t h the e x i s t i n g coo l i ng assembly which is c h a r a c t e r i s e d by a shor t in tense c o o l i n g zone below the mould; and the d e t a i l s of the var ious runs are presented in Table X I . Four of the runs were cast at 1.69 mm/s and the o ther th ree at 1.27 mm/s. These correspond to normal and s l i g h t l y below normal c a s t i n g speeds r e s p e c t i v e l y t h a t apply f o r the cas t i ng of High Grade z inc (99.99%). In the e x i s t i n g 126 Run No. Casting Speed Pouring Temp I Spray Pressure I I Spray Pressure I I I Spray Pressure 1 1.69 mm/s 427°C 289.5 kPa 262 kPa * 0 2 1.27 mm/s 426°C 289.5 kPa 262 kPa * 0 3 1.69 mm/s 430°C 303.4 kPa 310.3 kPa 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 * E x i s t i n g spray arrangement in both s t rands AandB * * Only 8 o f the 16 nozzles in the t h i r d r i n g used 3 - 7 No sprays in the t h i r d r i n g f o r s t rand B 3,4,5 Th i rd set of sprays placed 457 mm below the cen t re of second r i n g , in s t rand A 6,7 Th i rd set o f sprays placed 125 mm below the cen t re of the second r i n g in s t rand A Table XI Cast ing speed c o n d i t i o n s f o r the d i f f e r e n t runs dur ing the exper imenta l campaign. 12 7 coo l i ng ' assembly each of the two s t rands i s cooled by a set of two spray r i ngs one imping ing d i r e c t l y on the mould ( f l a t spray) and the second loca ted j u s t below the mould. In some of the runs e x t r a coo l i ng was appl ied, to one of the st rands by a t h i r d spray r i n g . In order to ob ta in the t h e r -mal c o n d i t i o n s t h a t e x i s t in the c a s t i n g machine, thermo-couples were i n s e r t e d in the l i q u i d pool from the top and f rozen in near the sur face of the c a s t i n g ' d u r i n g s teady-s t a t e o p e r a t i o n , and then were al lowed to descend w i t h the jumbo. Thus i t was poss ib le to c h a r a c t e r i z e the heat e x t r a c t i o n ra tes in the mould and sub-mould reg ion as a f u n c t i o n o f d is tance below the mould. In most o f the runs the length of the jumbo cast was about 3600 mm. The c a s t i n g was f i n a l l y cut i n t o sect ions 610 mm.long and the t r a n s -verse sec t ions were inspected f o r the presence of c racks . Whenever a major crack was seen i t s l o c a t i o n and leng th were measured. Tables X I I to XVII give the r e s u l t s of these measurements f o r each run r e s p e c t i v e l y . A l l the cracks mentioned in these tab les had a crack wid th ranging from 0.5 mm to 2.0 mm. In a d d i t i o n to these , f i n e h a i r -l i n e cracks were a lso observed but are not cons idered i m p o r t a n t , and t h e r e f o r e are not inc luded in Tables X I I to X V I I . Sect ions c o n t a i n i n g cracks were 'a l so c o l l e c t e d and taken to the l a b o r a t o r y f o r metal 1 ographic examina t ion . Some of the sec t ions were macro-etched to study the g ra in s t r u c t u r e . 1 28 Section No. Bottom Notch(Left) Notch(Right) 1A2 Surface - 160 mm Surface -• 160 mm -1A5 25 - 140 mm 30 -• 130 mm -1A6 30 • - 125 mm -1A7 35 - 125 mm 30 • - 120 mm -1B2 Surface - 155 mm -1B3 30 - 150 mm -1B4 30 - 140 mm -1B5 28 - 140 mm -1B6 28 - 147 mm -Table XI I Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the length of strands A and B (Run 1 ) 1 29 Section No. Bottom Notch(Left) Notch(Right) 2A2 - 50 - 90 mm 2A3 - -2A4 -2A5 48 - 112 mm 20 - 130 mm 35 - 150 mm 2A7 35 - 125 mm 3 0 - 1 3 2 mm 2B2 50 - 115 mm - 55 - 105 mm 2B3 - - _ 2B4 - 40 - 135 mm 2B6 - - 4 5 - 1 4 5 mm 2B7 - -Table X I I I Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the length of s t rands A and B (Run 2 ) . 1 30 Section No. Bottom Notch(Left) Notch (Right) 3A2 40 • - 150 mm Surface1 • - 120 mm 40 -• 130 mm 3A3 40 -- 150 mm 40 -• 135 mm 3A4 40 • - 155 mm 35 -- 140 mm 3A5 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 Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the length of s t rands A and B (Run 3) . 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 Locat ion of cracks in jumbo c ross -taken at var ious po in ts along the s t rands A and B (Run 4) . s e c t i ons leng th of 1 32 Section No. Bottom Notch(Left) Notch(Right) • . 5A1 Surface - 90 mm Surface - . 75 mm 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 - 90 mm Surface - 90 mm Surface - 80 mm 5B2 - Surface - 105 mm -5B3 - 40 - 140 mm -5B4 - 45 - 150 mm -5B5 - Surface - 110 mm 5B6 — — — Table XVI Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the leng th of s t rands A and B (Run 5 ) . 1 33 Section No. Bottom Notch(left) Notch(Right) 6A1 25 - 160 mm 5 0 - 1 2 0 mm 25 - 50 mm 6A2 25 - 165 mm 30 - 155 mm 30 - 150 mm 6A3 30 - 148 mm 30 - 135 mm 30 - 150 mm 6A4 - 35 - 130 mm -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 Locat ion of cracks in jumbo c r o s s - s e c t i o n s taken at var ious po in ts along the leng th of s t rands A and B (Run 6) . 134 A few general comments can be made rega rd ing the occurrence of the large c racks , an example o f which can be seen in the t ransverse sec t i on of a Prime Western Grade jumbo ingo t shown in F i g . 6 . 1 . I t is seen t h a t these cracks tend to form normal to the sur face predominant ly in th ree 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 in many respects to the mid-way cracks o f t e n seen in t ransverse sec t ions of c o n t i n u o u s l y cast s tee l b i l l e t s (86, 8 7 ) . A f t e r macro-e tch ing the t r a n s -verse sur faces of the jumbo s e c t i o n s , these cracks were seen to occupy i n t e r g ranu lar r e g i o n s , between columnar g ra ins of z i n c . . Although the cracks observed in F i g . 6.1 pene t ra te to the s u r f a c e , t h i s was not always the case, but on ly when the c rack ing was very severe. S i x t y - f i v e : sec t ions 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 con ta in one or more severe c r a c k s . In some runs a crack could be seen runn ing through the e n t i r e length of the c a s t i n g . 6.3 Heat Flow Analys is A heat f low ana lys is of the jumbo c a s t i n g was per-formed by using the heat f low model in c o n j u n c t i o n w i t h the measured temperature p r o f i l e obta ined w i t h the f r o z e n -in thermocouples. Since i t was not poss ib le to f reeze in F i g . 6.1 A c r o s s - s e c t i o n of Prime Western Grade i ngo t showing the i n t e r n a l c racks . j umbo 1 36 the thermocouple at the sur face of the c a s t i n g , the sur face h e a t - t r a n s f e r c o n d i t i o n s e x i s t i n g in the machine were back c a l c u l a t e d using the th ree-d imens iona l heat f l ow model by t r i a l and e r r o r . The boundary c o n d i t i o n s were ad jus ted u n t i l a match was obta ined between the measured and c a l c u l a t e d temperature p r o f i l e s . F i g . 6.2 shows the temperature p r o f i l e s c a l c u l a t e d from the model f o r the var ious nodes as shown in the i n s e r t . These have been obta ined f o r a c a s t i n g speed of 1.69 mm/s w i th h e a t - t r a n s f e r c o n d i t i o n s in the e x i s t i n g c o o l i n g system w i t h a sho r t in tense spray in the sub-mould r e g i o n . I t can be seen t h a t the sur face of the jumbo undergoes rehea t ing beyond 120 s. • T r a n s l a t i n g the t ime axis i n t o a d is tance axis using the c a s t i n g speed, t h i s corresponds to the bottom of the second spray r i n g below the mould. Surface rehea t ing below the sprays i s impor tan t because i t r e s u l t s in crack fo rmat ion in the f o l l o w i n g way. Reheat ing, which is a maximum at the sur face causes the sur face to expand more than the i n t e r i o r reg ion of the s o l i d i f i e d s h e l l ; and thus the sur face is cons t ra ined and put i n t o compression, wh i le a tens i1e s t r a i n is generated at the s o l i d i f i c a t i o n f r o n t . These t e n s i l e s t r a i n s are respons ib le f o r the fo rmat ion of the c racks . A s i m i l a r 500. 1 37 4 0 0 I T L o CD i_ Z3 o Q. E \ 300 Surface node (I) 23mm from surface (2) 46 mm from surface (3) 200 100 0 0 240 480 Time (s) 720 F i g . 6.2 Ca lcu la ted 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 inc jumbo i n g o t cast at . 1.69 mm/s. , 138 mechanism has been proposed f o r the fo rma t ion of mid-way cracks in con t inuous ly cast s tee l b i l l e t s ( 8 6 ) . The p o s i t i o n below the l i q u i d l e v e l a t which the i n t e r n a l cracks are generated can be determined from the heat f low model i f i t can be assumed t h a t the cracks form close to the s o l i d i f i c a t i o n f r o n t , because then the depth o f the crack beneath the sur face gives the s h e l l t h i ckness at the t ime of crack f o r m a t i o n . The she l l t h i ckness c a l c u -l a t e d f o r a 1.69 mm/s c a s t i n g speed is shown in F i g . 6 . 3 , and the accompanying diagram gives the sur face te rmpera ture p r o f i l e . The band on the l e f t - h a n d s ide has been drawn from the measured l o c a t i o n of the inner t i p of the crack f o r the var ious sec t ions inspected in the t e s t campaign. I t can be seen from t h i s f i g u r e t h a t cracks are always i n i t i a t e d a f t e r the reheat event has taken p l a c e . The magnitude of rehea t ing requ i red to cause cracks is be l ieved to be 45-50°C which is lower than the value of 100-150°C quoted f o r s tee l ( 8 6 ) . This comparison i s based on the c o e f f i c i e n t o f l i n e a r expansion which f o r p o l y e r y s t a l -l i n e z inc is 39.7 ( 1 0 " 6 ) / ° C as compared to 17 ( 1 0 " 6 ) / ° C f o r s t e e l . The c r i t i c a l value of t e n s i l e s t r a i n s t h a t cause ho t -t e a r i n g in s tee l has been est imated at around 0.2%. I f a s i m i l a r c r i t e r i o n is app l ied f o r z i n c , then the reheat o f 1 39 Distance along Y-axis ( m m ) 0 160 3 2 0 4 8 0 Temperature (°C) F i g . 6.3 Growth of the she l l as a f u n c t i o n of d i s tance in the a x i a l d i r e c t i o n f o r z inc jumbo i n g o t cast at 1.69 mm/s. Figure on the r i g h t shows the sur-face temperature p r o f i l e at the bottom mid - face of a jumbo s e c t i o n . 140 45°C observed in the present work is adequate f o r hot tea rs to occur . This i s only a rough comparison because the c r i t i c a l s t r a i n t h a t is r e q u i r e d would very much depend on the cohesion between gra ins which in t u rn i s a f f e c t e d by the presence of l i q u i d f i l m s between columnar d e n d r i t e s . 6.4 Me ta l l og raph ic Ana lys is In order to i n v e s t i g a t e the mechanism of crack forma-t i o n f u r t h e r a metal 1ographic examinat ion was c a r r i e d out 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 sample the crack was seen to go a l l the way through the th ickness of the sec t i on (100mm). The bottom face of the jumbo sec t i on i s on the l e f t - h a n d side o f the f i g u r e . The growth o f dend r i t es p a r a l l e l to the d i r e c t i o n of heat f low is q u i t e e v i d e n t ; obv ious l y the a x i a l component of heat conduct ion i s very impor tan t owing to the in tense spray c o o l i n g below the mould. F i g . 6.5 shows a scanning e l e c t r o n micrograph of the cracked sur face taken at a h igher m a g n i f i c a t i o n . The i n t e r -d e n d r i t i c nature of these cracks is c l ea r from t h i s p i c t u r e . The i d e n t i t y of the whi te p a r t i c l e s observed on the f r a c t u r e d sur face was i n v e s t i g a t e d in some d e t a i l . F i g . 6.6 (a) shows the scanning e l e c t r o n micrograph F i g . 6 . 4 A macro-photograph of the cracked s u r f a c e . Magni f i ca t i on 1 . 3 X. 142 F i g . 6.5 Scanning e l e c t r o n micrograph o f a cracked s u r f a c e . M a g n i f i c a t i o n 200 X. (a) ( b ) F i g . 6 .6 (a) Scanning e l e c t r o n micrograph o f a cracked su r -face revea l i ng the smooth na tu re of the s u r f a c e . M a g n i f i c a t i o n 1000 X. F i g . 6 .6 (b) Pb x - r a y p i c t u r e of F i g . 6 . 6 ( a ) . 144 of the cracked sur face at a much h igher m a g n i f i c a t i o n . The smooth nature o f the sur face s t r o n g l y po in ts to the presence of l i q u i d f i l m s at the i n t e r f a c e and hot t e a r i n g . F i g . 6.6 (b) shows the Pb x - ray scan obta ined from, the same a r e a ; and thus the whi te p a r t i c l e s seen in F i g . 6.6 (a) and 6.5 are a l e a d - r i c h second phase. S i m i l a r observa t ions were a lso made w i t h respect to other areas of the sur face (Fig.6.7a and b ) . The presence o f the l e a d - r i c h phase can be exp la ined by examining the phase diagram f o r the Zn-Pb system shown i n F i g . 6 .8 . I t can be seen t h a t z inc has an ext remely low s o l i d s o l u b i l i t y f o r l e a d , e . g . 0.5 - 0.9 wt % at the monotect ic temperature of 417.8°C. Thus in Prime Western Grade z inc i t is poss ib le to have l e a d - r i c h l i q u i d present in the i n t e r -d e n d r i t i c r e g i o n , thereby d r a s t i c a l l y decreasing cohesion b e t w e e n g r a i n s . 6.5 Mechanism of Crack Formation Based on the preceding r e s u l t s the f o l l o w i n g mechanism can be proposed f o r crack fo rmat ion in the D.C. c a s t i n g o f . Prime Western Grade z i n c . The pr imary cause f o r the fo rma t ion of cracks is i n c o r r e c t coo l ing p r a c t i c e below the mould. I t is seen t h a t the shor t in tense coo l i ng adopted in the c u r r e n t p r a c t i c e leads to rehea t ing of the jumbo sur face below the second spray r i n g which r e s u l t s in the expansion of the s u r f a c e . Because the sur face 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 sur-face revea l i ng the smooth nature of the s u r f a c e . M a g n i f i c a t i o n 1000 X. F i g . 6.7 (b) Pb x - r a y p i c t u r e of F i g . 6.7 ( a ) . 146 1000 900 800 ' 7 0 0 « 6 O 0 " 5 0 0 4 1 9 . 5 ' 9 0 6 ° 300 200 1 0 2 0 3 0 4 0 5 0 i l l I I 4 0 0 ^ 0 . 3 ( 0 . 9 1 0 Zn W E I G H T P E R C E N T L E A D S O 7 0 8 0 8 5 9 0 9 5 J I I , 1 , , I B O I L I N G ( R E F . 5 ) 2 8 ( 5 5 1 - 7 9 8 ° T W O M E L T S 4 1 7 . 6 ° 3 1 8 . 2 ° « H A S S , J E L L I N E K , R E F . 9 -• W A R I N G E T A L . , R E F . 1 « K L E P P A , R E F . 2 A . S E I T H , J O H N E N , R E F . 3 - 9 4 . ( 9 8 ) ( 9 9 . 5 F 3 2 7 ° 2 0 3 0 4 0 5 0 6 0 A T O M I C P E R C E N T L E A D 7 0 6 0 9 0 1 0 0 Ph F i g . 6.8 Phase diagram of Pb-Zn system ( 8 9 ) . 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 s t r a i n is generated at the sur face and a t e n s i l e s t r a i n at the s o l i d i f i c a t i o n f r o n t . The s t r a i n i s s u f f i c i e n t ( - .2 to .3%) to cause dendr i tes separated by l i q u i d f i l m s o f lead near 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 rack . The morphology o f the crack sur faces is i n d i c a t i v e of such a hot t e a r i n g mechanism. Tab! e XV I I I shows the maximum sur face reheat c a l c u l a t e d us ing the model at the var ious l o c a t i o n s of the c a s t i n g . The nodes r e f e r r e d to as top and bottom correspond to the sur face nodes at the mid-plane of non-notched sur faces and the notch corresponds to the node at the bottom of the n o t c h . Compared to o the r sur face nodes these three show the maximum r e h e a t . Fur ther i t can be noted from t h i s tab le t h a t the reheat o f the top node was much less compared to the bottom and notch nodes. These p r e d i c t i o n s match the observa t ions of crack l o c a t i o n s ; cracks were conf ined to the mid-p lane reg ion ad jacent to the bottom and notched faces . F i g . 6.9 shows the e f f e c t of cas t i ng speed on the sur face reheat phenomenon as p r e d i c t e d by the heat f l ow model. The three speeds considered are 1.69 mm/s, 1.27 mm/s and 0.85 mm/s. I t can be seen t h a t the rehea t ing i s reduced cons iderab ly when c a s t i n g at lower speeds. This suggests t h a t c rack ing should be less severe at lower speeds 148 Node Locat ion Reheat Bottom (1) 64°C Top (2) 35.5°C Notch (3) 53°C Table XVI I I Ca lcu la ted values of reheat at d i f f e r e n t po in ts on the sur face of the jumbo s e c t i o n . Top and bottom correspond to mid- face on the non-notched su r faces . 149 500 960 F i g . 6.9 E f f e c t of cas t ing speed on the sur face r e -hea t ing at the bottom mid- face of a jumbo s e c t i o n . 150 which is in l i n e w i th opera to r exper ience. Al though i t i s poss ib le to reduce the s e v e r i t y of the c rack ing problem by decreasing the cas t i ng speed, i t i s not a' p r a c t i c a b l e s o l u t i o n owing to the lower p roduc t ion r a t e s : f u r t h e r the q u a l i t y of the sur face d e t e r i o r a t e s at l o w e r / c a s t i n g speeds. Cold shuts and sur face laps are o f ten seen on the sur face o f the ingots cast at low speeds. 6.6 Design of New Cool ing System Having thus ascer ta ined tha t sur face rehea t ing below the sprays was the cause of cracks in Prime Western Grade z inc jumbos the design of a new spray system which would minimize t h i s phenomenon was under taken. A number of runs were made using the h e a t - f l o w model and the e f f e c t of h e a t - t r a n s f e r c o e f f i c i e n t at var ious po in ts along the s t rand on the reheat values was s t u d i e d . F i g . 6.10 shows the sur face temperature p r o f i l e s obta ined w i th d i f f e r e n t h e a t - t r a n s f e r c o e f f i c i e n t s in the sub-mould r e g i o n . The values of 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 were 20.93 2 2 kW/m K corresponding to the e x i s t i n g s e t - u p , 10.46 kW/m K 2 and 41.86 kW/m K. The value of heat t r a n s f e r c o e f f i c i e n t s used below the second spray have been kept the same f o r a l l the three runs . I t can be seen tha t decreasing the h e a t - t r a n s f e r c o e f f i c i e n t in the sub-mould reg ion r e s u l t s 151 500 4 0 0 h o o CU o CD Q. E CD H 300 2004-\00W F i g . 6.10 E f f e c t of the h e a t - t r a n s f e r c o e f f i c i e n t in the sub-mould reg ion on the sur face r e h e a t i n g at the bottom mid- face of a jumbo s e c t i o n . . ' . 1 5 2 in a decrease in the rehea t . I t was very c l e a r from these runs t h a t the reheat could be minimized by decreasing the i n t e n s i t y of the sprays in the sub-mould reg ion 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 w i t h a d d i t i o n a l sp rays . ^ . Thus the new coo l i ng arrangement was designed to ma in ta in un i form coo l i ng a l l the way down to the bottom of the l i q u i d pool to ensure t h a t rehea t ing of the sur face was minimized p r i o r to complete s o l i d i f i c a t i o n . This was accomplished by r e d i s t r i b u t i n g the t o t a l amount of water used over a wide area of the s u r f a c e . The arrangement of the sprays in the new coo l i ng assembly is shown in F i g . 6 . 1 1 . This corresponds to the bottom face o f the non-notched jumbo s e c t i o n . Other faces were a lso prov ided w i th s i m i l a r arrangement. As in the e x i s t i n g p r a c t i c e . a f l a t spray nozzle was used in the top r i n g imping ing on the mould to ensure adequate s o l i d i f i c a t i o n in the mould and a minimum of b r e a k - o u t s . However in order 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 se lec ted f o r the top r i n g had one -ha l f the capac i t y of the e x i s t i n g f l a t spray nozz le . For coo l i ng in the sub-mould reg ion a t o t a l of f ou r spray r ings were designed w i th 8 nozzles per spray r i n g (two per f a c e ) . The nozzles se lec ted f o r these f o u r 153 F i g . 6.11 Arrangement of spray nozzles in the new c o o l i n g assembly f o r the bottom sur face of a jumbo s e c t i o n . 154 r ings were d i f f e r e n t from the nozzles used in the o ld des ign . The new nozzles are of the wide angle type which prov ide a un i form water f l u x d i s t r i b u t i o n over a given face ( 8 8 ) . Thus the use of new spray nozzles r e s u l t s in a water f l u x o f 2 2 0.9 £/m s in comparison w i t h 4.8 £/m s. Al though there is a la rge d i f f e r e n c e between the two va lues , the t o t a l amount of water in the new design is only m a r g i n a l l y less than the o ld va lue . The new spray assembly was cons t ruc ted using 38 mm diameter ga lvan ized pipes w i t h threaded connect ions ' . A few minor m o d i f i c a t i o n s were requ i red to the c a s t i n g assembly to accommodate the new spray system. 6.7 Tes t ing of the New Cool ing System The new spray assembly was tes ted i n - p l a n t by making a t o t a l of fou r runs . In the f i r s t run the m a t e r i a l cast was high grade z inc at 1.27 mm/s. This run was c a r r i e d out e s s e n t i a l l y to check out the new assembly as w e l l as to gain conf idence of the operators f o r c a s t i n g the more d i f -f i c u l t Prime Western Grade z inc a l l o y . Since t h i s i n i t i a l run d id not pose any problem, Prime Western Grade z inc was cast at speeds of 1.27 mm/s and 1.48 mm/s in th ree runs . As before t ransverse sec t ions were inspected f o r the p re -sence of c racks . Of the twenty- two sec t ions inspec ted between two st rands in two runs no sec t ion showed any major cracks as seen in the previous campaigns. As before in 155 some s e c t i o n s , ext remely f i n e h a r i 1 i n e cracks were a lso observed. F i g . 6.11 shows a s e c t i o n from the new campaign. I t should be noted t h a t the f i n e h a i r l i n e - crack present was less v i s i b l e before e t ch ing w i t h h y d r o c h l o r i c a c i d . I t was the op in ion of the o p e r a t i n g people t h a t the q u a l i t y of the Prime Western Grade z inc cast 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 inc jumbos cast w i t h the o ld s e t - u p . 6.8 Summary The c rack ing problem encountered in c a s t i n g Prime Western Grade z inc has been i n v e s t i g a t e d , and shown to be. the r e s u l t of the shor t in tense spray c o o l i n g p r a c t i c e employed which generates sur face rehea t ing and t e n s i l e s t r a i n s at the s o l i d i f i c a t i o n f r o n t . Opening of cracks between ad jacent columnar dendr i tes under the i n f l u e n c e of these s t r a i n s is enhanced by the presence of l e a d - r i c h l i q u i d f i 1ms . A new spray system has been designed w i t h the a id o f the th ree-d imens iona l heat f low model to overcome the problem of sur face r e h e a t i n g by ma in ta in ing un i form coo l i ng to the bottom of the l i q u i d p o o l . The new assembly has been tes ted i n - p l a n t f o r the cas t i ng of Prime Western Grade z inc and shown to t o t a l l y e l i m i n a t e the severe c racks . 1 56 F i g . 6.12 A c r o s s - s e c t i o n of Prime Western Grade jumbo ingo t w i th the new coo l ing system. J Chapter 7 SUMMARY AND CONCLUSIONS A f u l l y th ree dimensional model has been developed to s imula te heat f low and s o l i d i f i c a t i o n in the D i r e c t C h i l l c a s t i n g of non- fe r rous metals w i th r e c t a n g u l a r as we l l as i r r e g u l a r notched c r o s s - s e c t i o n s . The model employs an a l t e r n a t i n g - d i r e c t i o n i m p l i c i t f i n i t e - d i f f e r e n c e method to solve the governing heat conduct ion equat ion and can take i n t o account both s t e a d y - s t a t e opera t ion as we l l as i n i t i a l t r a n s i e n t c o n d i t i o n s . The model has been tjested e x t e n s i v e l y f o r i n t e r n a l cons is tency and i t s v a l i d i t y has been checked by comparing p r e d i c t i o n s of pool p r o f i l e s and pool depths w i t h i n d u s t r i a l data f o r the D.C. c a s t i n g of aluminium and z i n c . The model has been used to i n v e s t i -gate the r e l a t i v e importance o f the i n d i v i d u a l components of heat conduct ion and has revealed the f o l l o w i n g : 1 . For t h i c k aluminium slabs e . g . 381 mm and, 457 mm t h i c k n e s s , sub jected to convent iona l D.C. c o o l i n g , a two-dimensional model in which,heat f low p a r a l l e l to the broad- face is neglected can be used when the aspect r a t i o exceeds 2 .5 . 2. For t h i c k z inc slabs of 381 mm th ickness the same 157 158 two-dimensional model i s adequate f o r aspect r a t i o s exceeding 2 . 0 . 3. For the case of reduced secondary c o o l i n g i t i s p r e f e r a b l e to neg lec t heat f low i n the a x i a l d i r e c t i o n , but cons ider both the two t ransve rse d i r e c t i o n s when f o r m u l a t i n g a two-d imensional model even though the aspect r a t i o exceeds 2 . 5 . 4 . The model has shown the f i r s t 25% o f the t o t a l c a s t i n g cyc le in the p roduc t ion of z inc jumbo sec t ions i s in the unsteady s t a t e . The t r a n -s i e n t pa r t of the c a s t i n g is also very impor tan t when reduced secondary coo l i ng is employed. For convent iona l D.C. coo l i ng o f aluminium the un-steady s t a t e i s impor tan t on ly in c a s t i n g l a rge s e c t i o n s , e . g . 457 x 1143 mm. A c rack ing problem encountered in the D.C. c a s t i n g of Prime Western Grade z inc has been i n v e s t i g a t e d w i t h the a id of the mathematical model. I t has shown t h a t the c rack ing is caused by the use of s h o r t - i n t e n s e spray c o o l i n g in the sub-mould r e g i o n . Temperature measurements and model c a l c u -l a t i o n s have revealed t h a t t h i s c o o l i n g p r a c t i c e r e s u l t s in the rehea t i ng o f the sur face below the spray r e g i o n , which in t u r n generates t e n s i l e s t r a i n s at the s o l i d i f i c a t i o n 159 f r o n t where f i l m s of l e a d - r i c h l i q u i d separate dend r i t es and f a c i l i t a t e the opening up o f c racks . Based on t h i s ana lys i s a new spray coo l i ng assembly has been designed f o r c a s t i n g Prime Western Grade z inc jumbo i n g o t s . This design at tempts to cool the sur face o f the jumbo more un i fo rm ly over the e n t i r e leng th o f the l i q u i d pool and also prevent sur face r e h e a t i n g . E x p e r i -ments have been undertaken i n - p l a n t to t e s t the new system, and r e s u l t s have shown t h a t the new coo l i ng assembly is e f f e c t i v e in p reven t ing c racks . 160 SYMBOLS Symbol Description Units c specific heat , Jg 1 K"1 -2 -1 h heat transfer coeff icient Wm K k thermal conductivity Wm~^  L latent heat of so l id i f i ca t ion J g~^ thickness of slab mm * ** *** T,T^,T 2 ,T,T , T temperature C t time s v casting speed mm s~^ 3 volume of a node mm x x-direct ion 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 fus iv i t y mm s Ax distance step in x-direct ion mm Ay distance step in y-direct ion mm Az distance step in z-direction mm At time step s e emissivity dimensionless -2 a Stefan-Boltzmann constant kWm _3 p density kgm e temperature Subscripts £ - liquidus s - solidus m - mushy n - present time interval n+1 - future time interval i - node ident i f icat ion in x-direction j - node ident i f icat ion in y-direction k - node ident i f icat ion in z-direction b ' - Base av - average BIBLIOGRAPHY 1 . 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T . , "A F i n i t e D i f fe rence Method o f High Order Accuracy f o r the So lu t i on of Three Dimensional T rans ien t Heat Conduction Problems", 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 in Conduction Heat  T r a n s f e r , McGraw-Hi l l , 1971. 79. Dus inber re , G.M., "Heat T rans fe r C a l c u l a t i o n s by  F i n i t e D i f f e r e n c e s " , I n t e r n a t i o n a l Text Book Company, Pennsy lvan ia , 1961. 80. H i l l s , A.W.D., "A General ised I n t e g r a l P r o f i l e Method f o r the Ana lys is 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 " , Transact ions of TMS AIME, v o l . 246, Ju ly 1969, pp. 1471-79. 8 1 . Ka r leka r , B.V. and Desmond, R.M., Engineer ing Heat . T r a n s f e r " , West Pub l i sh ing Co. , 1 977 . 82. Olson, F.C.W. and S c h u l t z , O.T.-, "Temperatures in So l ids During Heating and C o o l i n g " , I n d u s t r i a l Eng i -neer ing Chemist ry , v o l . 34, 1942, pp. 874-77. 83. Carslaw, H.S. and Jaeger, J . C , "Conduct ion o f Heat in S o l i d s " , Second E d i t i o n , Oxford Univ . Press 1959. 84. Su the r l and , J . G . , Personal Communications. 85. B e a t t i e , D.D., Personal Communications. 86. Van Drunen, G., Brimacombe, J .K. and Weinberg, F. , " I n t e r n a l Cracks in S t rand-cas t B i l l e t s , I ronmaking and Steelmaking ( Q u a r t e r l y ) , v o l . 2 , pp. 125-33, 1 975. 87. Agarwal , P.K., "Case Study of Spray Design f o r a Continuous B i l l e t Cas te r " , M.A. Sc. T h e s i s , Univ . of B r i t i s h Columbia, Dec. 1979. 88. Prabhakar, B., Personal Communications.. Hansen, M., " C o n s t i t u t i o n of Binary McGraw H i l l , 1958. A l T o y s " , APPENDIX 1 DEVELOPMENT OF FINITE DIFFERENCE EQUATIONS 1 70 171 APPENDIX 1 Al .1 A l t e r n a t i n g D i r e c t i o n F i n i t e D i f f e rence Equat ions f o r Three Dimensional Problems. The f i n i t e d i f f e r e n c e equat ions which rep lace the unsteady p a r t i a l d i f f e r e n t i a l equat ion have been ob ta ined using a heat balance approach. In t h i s method the m a t e r i a l being analysed i s d i v i d e d i n t o a number of d i s c r e t e elements of f i n i t e d imensions. In a three dimensional heat f l ow problem of a r e c t a n g u l a r para 11 el p iped , c a l c u l a t i o n s are performed only f o r one q u a r t e r o f the c a s t i n g as i n d i c a t e d in F i g . A l - 1 , because of the symmetry present at the mid-p lanes . In the z - d i r e c t i o n which is a lso the c a s t i n g d i r e c t i o n , the whole c a s t i n g had to be analysed because o f the d i f f e r e n t boundary c o n d i t i o n s invo lved at the top and bot tom. 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 is shown in F i g . A l - 2 . Ha l f nodes are present a t the sur face and centre w i t h respect to x and y d i r e c t i o n s and top and bottom w i th respect to z - d i r e c t i o n . I n F i g . A l -2 on ly the sur face nodes are v i s i b l e . In t h i s p a r t i -cu la r problem there are a l t o g e t h e r 27 d i f f e r e n t types of nodes depending on t h e i r l o c a t i o n in the c a s t i n g . 1 72 oo'x 'x, oyy'o' - zero heat f lux boundary condition yy 'd 'd , dd'x 'x, o ' y 'd ' x ' - heat-transfer coeff icient boundary condition oydx - constant temperature boundary condition F i g . Al .1 Dotted region is the volume over which c a l c u l a -t i o n s are per formed. Fig. Al . . 2 Discretization of the rectangular parallelpiped showing the surface nodes. 1 7 4 The genera t ion o f the simultaneous equat ions i s i l l u s t r a t e d below f o r the case of an i n t e r i o r node and a sur face node. I n t e r i o r Node: Let i , j , k be the ind ices of the node under i n v e s t i g a -t i o n and Ax, Ay, Az be i t s d imensions, as we l l as d i s tance between nodes . Stage I : I m p l i c i t i n x - D i r e c t i o n Rate o f Heat in by conduct ion j * T * = -kAyAz ( i , j ,k~ i - l , j , k ) in x - d i r e c t i o n Ax Rate of Heat out by conduct ion i n x - d i r e c t i o n -kAyAz ( T i + 1 , j , l ~ T i , ; i , k ) AX Rate of Heat in in the y - d i r e c t i o n = +kAxAz ( T i , j - l , k " T i , j , k ) Ay Rate of Heat out in the y - d i r e c t i o n = -kAxAz ( T i , j + l , k ~ T i , j , k ) Ay Rate of heat in in the z - d i r e c t i o n +kAxAy ( T i , j , k - l ~ T i , j , k ) AZ 175 Rate of Heat out in = - kAxAy (T. . k + 1 - T . . ) the z - d i r e c t i o n J I! ? J AZ Rate of heat consumption = 0 Rate of heat genera t ion = 0 Rate of heat accumulat ion = pc AxAyAz (T. . . -T. .' . ) A t / 2 ' ' ' ' * T i s the temperature a t the end of t ime Step At/2 From Energy Balance, (Rate of heat i n ) - (Rate of heat o u t ) + (Rate o f heat genera t ion ) - (Rate o f heat consumption) Rate-o f Accumulat ion k ^ ( T i - i , j , k - 2 T i , j , k + T i + i , j , k ) k - ^ ( T i , J - i v k - 2 T i , j , k + T i , : + 1 , k > + A y k ^ ( T i , j , k - l - Z T 1 . ] . k + T 1 J . k + l > - 2 axaytz pc ( T * . t - T , , k ) * Since only T s are unknown, they are kept on the l e f t hand side and e v e r y t h i n g e lse moved to the r i g h t Si mpli f y i ng we ge t , 176 ^ g p * T i . J , k + ^ f ( T i j - i ; k - 2 T i , j , k + T i , j + i , k > + ^ f ( T i , j , k - i - 2 T i . j . k + T i . j . k + i > • • • The form in which Eq. A l -1 is presented is very use fu l in cases where i t i s des i red to have a change in the node s izes in the d i f f e r e n t d i r e c t i o n s . I m p l i c i t in y - d i r e c t i o n : Fo l low ing on the same, l i n e s as before r e s u l t s i n , - T. /AXAZx + T. . . , 2 A x A Z + 2AxAyAZPCx i . j - l . k (-^y-) i . J . M - ^ - k i t ) 2 A X k y A f P C Ti>J»k + ^ f ( T i - l , j , k " 2 T i , j , k + T 1 + l , j , k > + ( T i , j \ k - l - 2 T i , j , k + T i , j , k + l ) . . . A l -2 where T are the unknown temperatures I m p l i c i t in the z - d i r e c t i o n : *** T. . *** AZ kAt "kick ( 2 " g f " C ' T l.J . k + ^ ' ( T l - i . j . k - 2 T i . j , k ** ** v i . j . k ) + ^ ( T i " ; j - i . k - 2 T i " j . k + T * ! j + i , k ^ Al-3 *** where T are the unknown temperatures E x p l i c i t formula f o r c a l c u l a t i n g the new temperature at the end of a t ime i n t e r v a l At i s given by Tn+1 i »jik » f T i - i , j , k - Z T i , j , k t V i , j , k > ** ** ** ^ ( T i , M , k - 2 T i , j , k + T i , j + l , k ' + ^ < T i , j , k - l - 2 T i , j , k + T i , j , k + l ' ) / /AXAyAZ PCX [ At ' + T i . j . k Al -4 .n + 1 where T. . u is the new temperature 1 » J » K I f the nodes were indexed i = 1,2 . . . L in the x D i r e c t i o n ^ 8 j = 1,2 . . . M in the y D i r e c t i o n k = 1,2 . . . N i n the z Di r e c t i on then the above equat ions are v a l i d f o r nodes having the ind ices i , j , k where i = 2,3 . . . L-1 j = 2,3 . . . M-1 k = 2,3 . . . N-l Equations f o r a node l y i n g on the boundary w i t h respect to  x d i r e c t i o n , but i n t e r i o r w i th respect to 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 ) I m p l i c i t i n the x - d i r e c t i o n : - T L - 1 , j , k O+ T L , j , k ( ^ + ^ p + h AyAZ\ = AxAyAzpe T . . + h_ T AyAz + AxAz k ' kAt L ' J ' K k M 2Ay ( T L , j - l , k " 2 T L , j , k + T L , j + l , k } + ^ ( T L , 3 , k - l - 2 T L , j , k + T L , j , k + l ) ' " I m p l i c i t in y - d i r e c t i o n : ** ** • T l j l b /AXAZ^ + T. . . /AXAZ + AxAyAZ PC\ ** T. . , , . ,AXAZN L , j + l , k ( ^ y - ) AXAyAZ T . + AZAV_ ( T , , , . - T. . . ) kAt ' J ' Ax ' J ' L ' J ' K h A Z A X (T, , . - JA + A X A X (T. . . , kAx L ' J ' K M 2 A Z L ' J J K 1 2 T L , j , k + T L , j , k + l ) I m p l i c i t i n z - d i r e c t i o n ~kieje "kick •T. k _ i f » + T , . . ,AXAy + AxAyAz PC ) * * * T, . L. j .k + 1 (§f4 AxAyAZ T l i k + ( T L -1 i k ~ TL i kAt ' J ' AX ' J > ' J ' h A Z A Z (Ti . k - T . ) + A X A Z (T. , , . Ak L , J ' K H 2Ay ' J ' ** ** 2 T L , j , k + T L , j + l , k } E x p l i c i t formula rn+l L,j ,k ^ ( T L - l , j , k " W + ** ** hPfL ( T L > j > k - T A) + Ax£z ( T L i j . , > k - 2 T u j i k + ** *** *** *** T L , J + l , k ) + § ^ ( T L , j , k - l - 2 T L s j , k + T L , j , k + l ) I^AxAyAz pc^ + T L,j ,k Al-8 where h is the heat t r a n s f e r c o e f f i c i e n t at the boundary i n the x d i r e c t i o n , and is the ambient temperature 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 fe rence Using Convect ive Type Boundary C o n d i t i o n s . The most s t r i n g e n t c r i t e r i o n w i l l be app l ied f o r a sur face node, which is on the bottom corner of the i n g o t . This p a r t i c u l a r node has three of i t s s i x faces sub jec ted to heat t r a n s f e r w i th the ou ts ide Let LMN stand f o r the node i d e n t i f i c a t i o n Doing a heat balance f o r t h i s node in e x p l i c i t f i n i t e d i f f e r e n c e form k ( T L -1 ,M,N' - TL,M,N } fgp. + k ( TL,M-1,N " \ ^ AXAZ + k (T 4 Ay L,M,N-1 L,M,N 4 Az ( T L > M j N " TA) A^AZ - h 0 ( T L > M ) N - TA) AXAZ h 3 ( TL,M,N " T A } If*. rn:+l P C A Z ( TL,M,N " TL,M,N } . Al-9 • 1 82 where h-j , h,, , h 3 a re t he hea t t r a n s f e r c o e f f i c i e n t s i n the x , y and z d i r e c t i o n . Here t he s t a b i l i t y c o n d i t i o n f o r e x p l i c i t method i s ( 1 - 2 At k - 2 At k - 2 At k , 2 2 2 AX pc Ay pc AZ PC 2 h 1 A t _ 2 h 2 A t _ 2 h 3 A t 0 pCAX pC Ay pC AZ s u b s t i t u t i n g k = -113 W/m K . P = 7140 k g / m 3 c = .3830 J / g K h 1 = 9210.9 W/m2K h 2 = 921 0.9 W/m2K h 3 = 209.34 W/m2K • Ax = Ay = 15 .24 mm Az = 20 mm we ge t At = < 0 .55 seconds APPENDIX 2 SOURCE LISTING OF THE COMPUTER PROGRAM 1 83 184 C A EBCGFAM W FITT EN IN FOBTEAN TO SIMULATE C THREE DIRENSIONAL HEAT TLOV AND S O I I D I F I C A T I O S C IH CASTING JUMBO SECTIONS OF ZINC. BECAUSE C OF THI SYBMETBY THE CALCULATIONS HAVE BEEN C PEfiFCEHED ONLY FOE ONE HALF OF THE CASTING C C c C NST AFT - 0 CORRESPONDS TO STARTING FBOB C THE BEGINNING C 1 FROM INTEBM E1ATE TIME C NITE - NOME EE OF TIRES CALCULATIONS ABE PEEFOBMED C HE1CT -NI7K EEB OF T1HES FLOTS ABE REQUIRED C RNOE - NUM EEB OF STEPS AFT EE HHICH A NEW S L I C E C IS ADDED TO THE Z-DIRECTION C Z - THICKNESS OF THE SLICE IN THE 2 DIBECTION C RKAY - THERMAL CONCDCT1VITY OF THE MAIEEIAL IN CGS C DT - TIRE INTEBVAL OVER WHICH CAICULATIONS ABE DONES C TAD - THE TOT AI CASTING TIME AT THE END OF EACH CALCtJLTION C L,R,N - THE NUMBER CF KCDES IN THE X Y Z DIRECTION - 1 C FOF JUMEC CALCD1TIONS L=9 AND M=15 . THE DI SCfi E H ZATION C I S CONE AS PEF EAT A SUBROUTINE AB EV CL C TECE1 - FOUEING TEMPERATDBE DEG C C TEME2 - SOME SBALLTEMPEEATOBE AS DUMMY VALUE C TO F I L L THE GBID IN THE NOTCH AREA C FOB THE PLOTTING SUBROUTINE C DEKI - DENSITY OF THE LIQUID G/CM3 C DENS - DENSITY OF THE SOLID G/CM3 C T I I C " LICUIEUS TEHPEBATUBE DEG C C TSCL - SOLIEUS TEMPEEATUPE DEG C C C E I - S F E C I F I C HEAT OF THE LIQUIE CAL/G C C ELHT - LATENT HEAT OF S O U CI FIC ATI C N CAL /G C H3 - EOTIOM BEAT TRANS FEB COEFFICIENT CGS UNITS C S P E C I F I C HEAT CI. SOLID INCORPORATED AS A C A FDNTICN OF TEHPEBATUBE IN THE FUNCTION C SUEBOUTINE CP C c c c DIBENSION T (10, 16,91),T 1 ( 10,16,91) ,T2(10,16,91) , 1 T3(10,16,91) ,18(10,16,91) DIfENSICN A (101) ,E (101) ,C (101) ,D (10 1) ,TPEIME (101) DI RENSICN AEX ( 16, 1 1,2) , AEY (11,16,2) ,AiiZ (1 1 , lb) DIMENSION TS(38,30,50) CORKON/C1/DX,EY,CZ,DT,BKAY COBMON/C2/S4(10,16,91) C0HB0N/C3/NTYPE(10, 16,9 1) ,LPS (10,16,91) 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 CCBRON/C12/BINPI(20) ,CPf (20) C0BB0N/C13/TA0 C0BB0N/C16/NNUH,NUB2 CCBBCN/C17/ABX,ABY,ABZ C O B B O N / C 1 8 / X 1 , T 1 , X 2 , Y 2 , Y 3 , Y 4 , T H C0I!E0N/C19/ABEA 1 , AB IA2 , AEEA3,ABEAU ,ABEA5,ABEA6 CCBECN/C21/TS N A K I L I S T / L I S T A / Z , B K A Y , D T H A E E L I S T / L I S T B / T S O L , T L I Q , E E N L , D E N S , C P L , B L f i I , d 3 C C BEADIKG INPUT DATA TO THE PBOGRAE C BEAT ( 5 , 2 0 0 ) NUBKUN BEAD ( 5 , 9 0 ) (BINPT ( I ) , 1=1 ,20 ) EEAD ( 5 , 9 5 ) NSTABT,NITE,NPLOT,NNUM 95 r o £ n A i ( i 4 ) * BEIE ( 5 , 1 0 0 ) Z , B K A Y , E T . T A U B E A E ( 5 , 2 C C ) L , « , N BEIE ( 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 B I A E ( 5 , 9 0 ) ( C P F ( I ) , 1 = 1 , 2 0 ) 90 FOEBAT(20A4) CALL GSET (TS, 1 0 4 0 , 5 0 , 1 0 4 0 , 0 . ) C C ECHO INPUT C WBI1E ( 6 . L I S T A ) H B 1 T E ( 6 , L I S T B ) 100 FCEPAT (SF 1 0 . 4 ) 200 FOBMAT (3 ( 1 4 , 1 X ) ) PLC ATL= L FLC ATI = H F L C AT N= N . DZ=2/FLOAIN CSE= EZ/(ET*NNUM) L = L + 1 B = B+1 H = N + 1 C C CALCULATION OF THEBflO PHYSICAL FBOPIBHES C FOB BUSHY ZCNE C DEKE= (DENS+DENL)/2 . CEB= (CPL+CP(TSOL) ) / 2 - + ( B L H T / ( I L I Q - I S O L ) ) FHYE=DFNM»CEH PHYL=CPL*DENL EHYS=DENS * CP(TSOI) C C I N I T I A L I S A T I O N PEOCEEUBE C I F (NSTABT.NE.0)GO 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 (T1 ,TEHP1,TEPP2) CAIL 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 ) C A I I I N I T I 2 ( T N , T E M P I , T E H P 2 ) DO 101 1 = 1 , L DO 101 J= 1,H DC 101 K=1,N L P S ( I , J , K ) = 1 101 CONTINUE GO TO 114 105 CONTINUE 1 86 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 ) 108 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 ) 113 CONTINUE NUB2 = 0 BCE=1 114 CONTINUE CA1L CUTPDT (T) CALL SUBFT C C C NCDE SOETING C CAIL NDSOET L L = L - 1 BB=B-1 NN=N-1 C C D ISCEIT IZAT ION OP THE CASTING C CALL ARIVOL 111 CChTINUE CALL ODTPT2 C C STAET Cf CALCULATIONS C DO 1001 K J I = 1,N PLOT DO 1002 KJJ = 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 ) C A I I T B I D A G ( 1 , L , A , E , C , E , T P R I S E ) DO 550 1 = 1 , L 11 » I , J , K ) = T P B I H E ( I ) 550 CONTINOE 600 CCMINOE C CALCOLATICN AT THE NCTCH C u=e J I = 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 ) C A I I T E I D A G ( 1 , 1 J , A , B . C , E . T P B I M E ) DO 630 1 = 1 , I J T l ( I , J I , K ) = T P E 1 « E ( I ) 630 CONTINUE 610 CCKTINUF J I = J I + 1 I J = I J + 1 615 CONTINOE C CALCOIATICN EELOS THE NOTCH C 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 ) 660 CC KTIND E 645 CONTINUE 640 CCKTINUE C C I H P L I C I T WITH BESPICT TO J DIBECTION C c 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 TFID8G ( 1 , R , A , fi.C, D. IPEIME) 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 ) 780 CONTINUE 770 COKTINOE 760 CONTINUE I J J = 7 J I I = 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 J I I = J I I + 1 I J J = I J J * 1 799 CONTINUE C C C I 8 E L I C I I WITH EESPECT TO Z DIBECTION C DO 900 1 = 1 , L DO SOO J = 1 , 4 C A I I C C E F F K ( 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 ) = TPEIBE(K) 850 CCKTINUE 900 CONTINUE C CALCUITICN AT THE NOTCH C I J K = 6 J I K = 5 DC 910. KKK=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) 920 CONTINOE 915 CONTINOE I J F = I J K + 1 J I K = J I K + 1 910 CONTINUE C C CAICDIATICN EELOW THE NOTCH DO 910 1 = 1 , L DO 950 J = 9 , H CAIL 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 TIHE INTEBVAL C CALL COHPUT ( 1 , L , 1 , « , N ) C A I L CCBPUT ( 1 , 6 , 5 , 5 , N ) CALL COSPDT(1 ,7 ,6 - ,6 ,N) C A I I C C B E U T ( 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 ) C A I I I A T H E T ( 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 ) C A I I L A T H E T ( 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 ) = I N ( I , J , K ) 1110 COMINUE CALL CHECK (MCH) v I F ( B C H . E Q . 0 ) G O TO 109 CALL AEENOD CALL COTPOT(T) CAIL SUBFT 109 CCKTINUE 1002 CONTINOE 189 C A I I GEAEfc C A I I F I L E I N ( 4 ) 1001 C C * I I N 0 E 901 CALL 7SARAY C A I I ELCTND STCP EN I C C I M P L I C I T CALCULTIONS I N THE X DIBECTION C CALCLLATINS OF THE IRIDIAGONAL 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 , 16 ,2 ) ,ABZ (1 1 , 16) E1EINSICN I ( 10 , 1 6 , 9 1) ,T 1 ( 1 0 , 1 6 , 9 1 ) 1 , 7 2 ( 1 0 , 1 6 , 9 1 ) , 3 3 ( 1 0 , 1 6 , 9 1 ) , T N ( 1 0 , 1 6 , 9 1 ) CCEECN/C1/EX,EY,E2,ET, f iKAY CCrECN/C17/ABX,ARY.ARZ C C E E C N / C 7 / T . T 1 , T 2 , T 3 , T N COBBON/C2/S4 ( 1 0 , 1 6 , 9 1 ) CCEEON/C3/NTYPE ( 1 0 , 16 ,91) ,LPS ( 10 , 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 rCN/C19/ABEA1,AREA2,ABEA3, AREA4 , ABEA5, AH EA6 EC 1000 I = I E , I I I J K = N T Y P E ( I , J , K ) GO TO ( 10 , 1 1 , 1 2 , 1 3 , 14, 15, 1 6 , 1 7 , 16 , 1 9 , 2 0 , 2 1 , 2 2 , 1 2 3 , 2 4 , 2 5 , 2 6 , 2 7 , 2 8 , 2 9 , 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 ) = A B X ( J , I , 2 ) + P H Y ( I , J , K ) * S 4 ( 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 • ( A F Y ( I , J , 2 ) ) * ( T ( I , J + 1 , K ) - T ( I , J , K ) ) 2 • P H Y ( I , J , K ) * S 4 ( I , J , K ) * 1 ( 1 , J , K ) 8 • ( A B E ( K ) - T ( I , J , K ) ) * (H4 (K) * ARY ( I , J , 2) *Y 2/BK A Y) GC TC 1000 11 A (1) = -AE X ( J , I , 1 ) B ( I ) = A B X ( J , I , 1 ) + A R X ( J , l , 2 ) + S 4 ( I , J , K ) * P H Y ( 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 ( * E Z ( I , J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) * T ( I , J , K * 1 ) ) • 2 E H Y ( I , J , K ) * S 4 ( I , J , K ) * T ( I , J , K ) 8 • (»BE(K) -T ( I , J , K ) ) » ( K 4 ( K ) * A E Y ( I , J , 2 ) * Y 2 / E K A i ) GC TO 1000 12 A ( I ) = - A B X ( 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 ( A E Y ( I , J , 2 ) ) * ( I ( I , J + 1 , K ) - T ( I , J , K ) ) t 2 ( S 4 ( I , J , K ) ) * P H Y ( I , J , K ) * T ( I , J , K ) * 3 (H1 (K) *AEX ( J , I , 1) *X1/BKAY*ABD (K) ) 8 • (ABE ( K ) - T ( 1 , J , K ) ) * (H4 (K) *ARY ( I , J , 2) *Y2/BKAY) GC TO 1000 13 E ( I ) = A B X ( J , I , 2 ) • S 4 ( I , J , K ) * P H Y ( I , J , K ) C ( I ) = - A B X ( J , I , 2 ) D ( I ) = A B Y ( I , J , 1 ) » ( T ( I , J - 1 , K ) - T ( I , J , K ) ) * A R Y ( I , J , 2 ) * 1 ( I ( I , J » 1 , K ) - T ( I , J , K ) ) • 1 90 3 ( A E Z ( I . J ) ) * ( T ( l , J , K - 1 ) - 2 . * T ( I , J , K ) • T ( I , J , K + 1) ) • 2 S I ( I , J . K ) * F f l Y ( I . J . K ) * T ( I . J . K ) GC TO 1000 14 A ( I ) = - » 2 X ( 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 ) = - A R X ( J , I , 2 ) D ( I ) = A B I ( I , J , 1 ) * ( T ( I , J - 1 , K ) - T ( I , J , K ) ) + A B Y ( I , J , 2 ) » 1 (T ( I , J + 1 ,K) - I ( I , J . K ) ) • 2 A E 2 ( I , J ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J . K ) + T ( 1 , J , K * 1 ) ) • 2 S « ( I , J , K ) * P E Y ( l . J . K ) » T ( I , J , K ) GC TO 1000 15 A ( I ) = - A B X ( J . I . I ) B ( l ) = ( A B X ( J . I . I ) ) • ( S 4 ( I , J , K ) * P H Y ( I , J , K ) ) • 1 (fc 1 (K) *AEX ( J . I , 1 ) +X1/BKAY) D ( I ) = (H1 (K) *AEX ( J . I , 1) *X1/RKAY*A»B (K) ) • > 1S4 ( J . J . K ) * P H Y ( I , J , K ) * T ( I , J , K ) • ' 2 ( A B Y ( I , J , 1 ) ) * ( T ( 1 , J - 1 , K ) - T ( I , J , K > ) • 2 ( A I Y ( 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 16 E ( I ) = ( A E X ( J , I , 2 ) ) • ( S U ( I . J . K ) ) * P H Y ( I . J . K ) C ( I ) = -ARX ( J . I , 2 ) D ( I ) = ( A R Y ( I , J , 1 ) ) * ( T ( 1 , J - 1 , K ) - T ( I . J . K ) ) • 1 ( H2 ( K ) * A B Y ( I , J , 1 ) *Y1/BKAY ) * (Af lE(K) - T ( l . J . K ) ) • 2 ( A F Z ( I . J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + T ( I , J , K * 1 ) ) 3 + | S 4 ( I , J , K ) ) * P E Y ( I . J . K ) • T ( I , J , K ) GC TO 1000 17 A ( I ) = - A E X ( 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 ) = - A B X ( 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 ( A I Z ( I , J ) ) * ( T ( 1 , J , K - 1 ) - 2 . * T ( 1 , J , K ) • T ( I , J , K * 1 ) ) -3 (K2 (K) *AEY ( I . J , 1) *Y1/BKAY) * ( 1 ( 1 , J . K ) - A (IB (K) ) GC TC 10CO 18 A ( I ) = -ABX ( J . I , 1) B ( I ) = ( A B X ( J , I , 1 ) ) • HI ( K ) * A B X ( J . I . 1 ) *X1 /BK AY * 1 (SO ( I , J , K ) ) * f H Y ( I , J . K ) D ( I ) = Hi ( K ) * A R X ( J , I , 1) *X1/EKAY*ARE(K) • 1 ( A E Y ( I . J . I ) ) * ( 1 ( 1 , 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 ) ) -3 (F.2 (K) *AEY ( I . J . 1 ) *Y1/EKAY) * (T ( I , J , K) - ABB (K) ) • 4 ( S 4 ( I , J , K ) ) • P H Y ( I . J . K ) * T ( I . J . K ) GC TC 1000 19 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 ) = ( A B Y ( I , J . 2 ) / 2 . ) * ( T ( I , J + 1 , K ) - T ( I , J , K ) ) • 1 ( A B Z ( I , J J ) * ( 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 20 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 ) = S 4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I , J , K ) • 1 (AFY ( I , J , 2) / 2 . ) * ( T ( I , J * 1 , K ) - T ( I . J . K ) ) + 2 ( A E Z ( 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 21 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 ) = (HI (K) *ABX ( J , l , 1) *X 1/BKAY*AM£ (K) /2. ) • 1 ( J H ( I r J , 2 ) / 2 . ) » ( T ( I , J * 1 , K ) - T ( I , J , K ) ) • 2 ( A F Z ( 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 ) *Ph 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 ( A F Z ( I , J ) ) * ( T ( I , J , K - 1 ) - T ( I , J , K ) ) 3 ( H 3 * A R Z ( I , J ) *D2/BKAY) * ( 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 ( A B Y ( I , J , 1 ) / 2 . ) * ( K I f J - 1 » K ) - I ( I , J , K ) ) + 1 (AFY ( I , J , 2 ) / 2 . ) * ( T ( I , J * 1 , K ) - T ( 1 , J , K ) ) • 2 | A F Z ( I , J ) ) * ( T ( I , J , K - 1 ) - T ( I , J , K ) ) -3 (B3*AFZ ( 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 ) = S 4 ( I , J , K ) * P h Y ( l . J . K ) * T ( I . J . K ) • 1 ( A F Y ( 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 ( A F Z ( I . J ) ) * ( T ( I , J . K - 1 ) - T ( I , J , K ) ) -3 (H3*ABZ ( I , J) *EZ/BKAY ) • ( T ( I , J , K ) - AHB (K) ) ' • 4 ( H 1 ( K ) * A E X ( J , I , 1 ) * X 1 / E K A Y * A B B ( 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 ( A E Z ( 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 ) = - A R X ( 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 ) = S 4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I , J , K ) -1 (F.2 (K) *ABY ( I , J , 1) *Y 1 /BKAY/2 . ) * (T ( I , J . K ) - A H B (K) ) -2 ( E 3 » A B Z ( I , J ) *EZ/BKAY ) * (T ( I , J , K ) - A B B (K) ) • 3 ( J F Y ( 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 /EKAY/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 . ) • 2 ( I B Y ( I , J . 1 ) / 2 . ) » (T ( I , J - 1 . K ) - T ( I , J . K ) ) • 3 ( A B Z ( I . J ) ) * ( T ( I . J . K - I ) - T ( I . J . K ) ) -4 (E2 (K) *ABY ( I , J . 1 ) * Y 1 / E K A Y / 2 . ) * (T ( I , J , K ) - A B u (K) ) -5 ( 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 ) = - A E X ( 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 ) = - A B X ( 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 S 4 ( I , J , K ) » PHY ( I . J . K ) * T ( 1 , J , K ) 3 • (H7 (K)*ABEA 1*AMB (K) ) 4 * ( H 5 ( K ) * A B E A 4 ) * (ABB ( K ) - T ( I , J . K ) ) GC TC 1000 29 » ( I ) = - A B X ( J , I , 1 ) / 2 . B ( I ) = (AFX ( J . I , 1) *ABX ( J , I , 2) ) / 2 . +S4 ( I . J . K ) •PaY ( I . J . K ) 1 • ( H 7 ( K ) * A B E A 1 / 2 . ) C ( l ) = -ABX ( J , I , 2 ) / 2 . D ( I ) = S 4 ( I , J , K ) * PHY ( I . J . K ) * T ( I , J , K ) + 1 ( A F Y ( 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 ( A B Z ( 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) *AEEA2*SIN (TB) ) D ( I ) = (H7 (K)*ABEA5*ABE ( K ) ) • 1S4 ( I . J . K ) * P H Y ( I . J . K ) * T ( I , J . K ) + 2 ( A E Y ( I , J , 1 ) ) * ( T ( I , J - 1 , K ) - T ( I , J , K ) ) • 2 ( A E Y ( I , J , 2 ) ) * ( T ( I , J » 1 , K ) - T ( I , J , K ) ) • 3 ( A E Z ( I , J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + T ( I , J r K * 1 ) ) 4 • (H6 (K) »AEEA2»SIN (TE)*AHB (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 ) ) » P H Y ( 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 ( I E Y ( I , J , 1 ) / 2 . ) * ( K 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 ( A E Z ( I , J ) ) * ( 1 ( I , J , K - 1 ) - T ( l . J . K ) ) -3 (H3*ABZ ( I . J ) *EZ/EKAY ) * ( T ( I , J , K ) - ABB (K) ) «• 4 ( H 7 ( K ) * A B E A 5 * A B B ( K ) / 2 . ) 5 + (H6 (K) *AEIA2*AHE(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 ( A F Z ( I , J ) ) * ( 1 ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + T ( 1 , J , K * 1 ) ) -3 (H5 (K) *AEY ( I , J , 1) *Y3/EKAY) • ( T ( I . J . K ) - AMB (K) ) GC TO 1000 33 M I ) = -AEX ( J , I , 1 ) / 2 . B ( I ) = ( A B X ( J , I , 1 ) + A R X ( J . I , 2 ) ) / 2 . • < S 4 ( I . J , K ) * P H Y ( I , J . K ) ) C ( I ) = -ABX ( J . l , 2 ) / 2 . E ( I ) = S 4 ( I , J , K ) * P H Y ( 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)-AMB (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 34 A ( I ) = - A B X ( J , I , 1 ) B ( I ) = ( A B X ( J , I , 1 ) ) • ( S4 ( I , J . K ) *PHY ( I , J . K ) ) • 1 (H6 (K) *ARX ( J . 1 , 1 ) *X2/BKAY) 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 ) ) * P H Y ( I , J , K ) * T ( I , J , K ) + 3 (B6(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 35 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)*ABX ( J . I , 1 ) * X 2 / B K A Y * A H E ( K ) / 2 . ) • 1 ( A I Y ( I . J . 2 ) / 2 . ) * ( T ( I , J * 1 , K ) - T ( 1 , J , K ) ) • 2 ( l E Z ( 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 36 A U ) = - A E X ( J . I . I ) E ( I ) = ( A R X ( J , I , 1 ) ) + ( S4 ( I , J , K) * PHY ( I , J , K) ) • 1 (B1(K) *AE£A6) 2 • (H6 ( K ) * A E E A 3 * S I N (TH)) D ( I ) = (ABZ ( 1 , J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) * T ( I , J , K * 1 ) ) + 1 ( A I Y ( I , J , 2 ) ) * ( T ( I , J * 1 , K ) - T ( I , J , K ) ) • 2 ( 5 4 ( 1 , J , K ) ) * P H Y ( 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 37 A ( I ) = - A B X ( 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) *ABEA3*SIN ( T H ) / 2 . ) E ( I ) = ( H l ( K ) * A B E A 6 * A M B ( K ) / 2 . ) • 1 ( » E Y ( I , J , 2 ) / 2 . ) * < T ( I , J + 1 , K ) - T ( I . J . K ) ) • 2 ( A E Z ( 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) *AEEA3*SIN (TB) *AMB (K) / 2 . ) 3 • (H6 (K) * ABE A3 *COS ( T H ) / 2 . ) * (A BB ( K ) - T ( I , J , K) ) GC TC 1000 38 A ( I ) = - A R X ( 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 ) ) * F H Y ( 1 , J , K ) D ( I ) = HI (K)*AEX ( J . I , 1 ) * X 1 / B K A Y * A B B ( K ) • 1 ( * E Y ( I , J , 1 ) ) * ( T ( I , J - 1 , K ) - 1 ( 1 , J . K ) ) • 2 ( A F Z ( I . J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) • T ( I , J , K * 1 ) ) -3 (H5 (K) *AFY ( I , J , 1 ) *Y3 /BKAY) * (T ( I . J . K ) - A M B (K) ) • 4 ( £ 4 ( 1 , J . K ) ) * P H Y ( I . J . K ) • T ( I , J , K ) 1 94 GC 30 1000 39 Ml) = -AEX ( J , I , 1 ) / 2 . B ( I ) = (ABX ( J , I , 1 ) / 2 . ) *S4 ( I , J , K ) * F H Y ( 1 , J , K) • 1 HI (K) * A F X ( J , I , 1) *X 1 /BKAY/2 . D ( I ) = (S4 ( I . J . K ) * F H Y ( I , J . K ) • T ( I , J , K ) ) + 1 ( E 1 ( K ) * A 6 X ( J , I , 1 ) * X 1 / E K A Y * A B E ( K ) / 2 . ) + 2 (AEY ( I , J , 1 ) / 2 . ) * ( T ( I . J - I . K ) - T ( I , J , K ) ) • 3 ( A F Z ( 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 ) -ABB (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 ( 16 , 1 1,2) ,ABY ( 1 1 , 16 ,2 ) ,ARZ ( 1 1 , 1 6 ) E IS INSION T ( 1 0 , 16 ,91) ,T 1 ( 1 0 , 16 ,91 ) 1 , 3 2 ( 1 0 , 1 6 , 9 1 ) , 1 3 ( 1 0 , 1 6 , 9 1 ) , T N ( 1 0 , 1 6 , 9 1 ) CCFRCN/C 1 / C X , I Y ,EZ,ET,EKAY CCrCCN/C17/ABX,AEY,AEZ C O K B O N / C 7 / T , I 1 , 3 2 , 1 3 , T N 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 C O B B O N / C 1 8 / X 1 , Y 1 , X 2 , Y 2 , Y 3 , Y 4 , T H CCBBCN/C8/PHY( 1 0 , 1 6 , 9 1 ) CCePCN/C19/ABEA1,AEEA2,AEEA3,ABEA4,ABEA5,ABEA6 EC 2000 J = I E , I E I J K = N T Y ? E ( I , J , K ) GOTO ( 1 1 0 , 1 1 1 , 1 1 2 , 1 1 3 , 1 1 4 , 1 1 5 , 1 1 6 , 1 1 7 , 1 1 8 , 1 1 1 5 , 1 2 0 , 1 2 1 , 1 2 2 , 1 2 3 , 1 2 4 , 1 2 5 , 1 2 6 , 1 2 7 , 1 2 8 , 1 2 9 , 1 3 0 , . 2 13 1 , 13 2 , 133 , 13 4 , 1 3 5 , 13 6 , 137, 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 ( A F Z ( I . J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) • 1 ( I , J , K + 1 ) ) • 2 ( A I X ( J , I , 2 ) ) • ( T 1 ( I « 1 , J , K ) - 1 1 ( 1 , J . K ) ) 8 • < A F Y ( I , J , 2 ) * B 4 ( K ) * Y 2 * A B B ( K ) / B K A Y ) 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) *Y2/EKAY C ( J ) = -AEY ( I , J , 2 ) D ( J ) = S4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I , J , K ) • 1 ( ! F Z ( 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 ( A E X ( J , I , 2 ) ) * ( T 1 ( I f 1 , J , K ) - T 1 ( I , J , K ) ) 8 • (AEY ( I , J , 2 ) » H 4 ( K ) * Y 2 * A E B ( K J / B K A Y ) GC TC 2000 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 ) • 1 ( A F Z ( I , J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + 1 ( I , J , K + 1 ) ) • 2 ( * B X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J , K ) - T 1 ( I , J , K ) ) -3 (E1 (K)*AEX ( 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) /BKAY) GO 30 2000 113 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 ) » P f l Y ( 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 A F X ( J , I , 2 ) » ( T 1 ( i + 1 , J , K ) - T 1 ( 1 , J , K ) ) GO 10 2000 114 A ( J ) = - A E Y ( 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 ) = - I B Y ( 1 , J . 2 ) C ( J ) = S 4 ( I , J , K ) * P H Y ( I . J . K ) * T ( I . J . K ) + 1 A B Z ( 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 ) * ( 1 1 (1 + 1 , J , K ) - H ( I . J . K ) ) GC IC 200C 115 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 ) * P H Y ( 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 ( A I X ( J . I . I ) ) * ( T 1 ( I - 1 , J , K ) - T 1 ( I , J , K ) ) -3 (E1 (K)*ARX ( J , 1 , 1 ) *X1/BKAY) • ( T 1 ( I , J , K ) - A3E (K) ) GC IC 2000 116 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)*AFY ( I , J , 1)*Y 1/BKAY) C ( J ) = £ 4 ( 1 , J , K ) * F H Y ( 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 ( A E Z ( I . J ) ) * ( T ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + I ( I , J , K + 1 ) ) • 3 ( E 2 ( K ) * A E Y ( 1 , J , 1 ) » Y 1 / B K A Y * A H E ( K ) ) GC I C 2000 117 A (J ) = -AEY ( I . J , 1) B ( J ) = ( A E Y ( I , J , 1 ) ) • PEY ( I . J . K ) » S 4 ( I . J . K ) • 1 ( E 2 ( K ) • A F Y ( I , J , 1 ) * Y 1 / E K A Y ) D ( J ) = S 4 ( X , J , K ) * P H Y ( I . J . K ) * T ( I , J , K ) • 1 ( A E X ( J , I , 1 ) ) • ( 1 1 ( 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 (B2(K) * A E Y ( I , J , 1) *Y1/EKAY*AMB (K) ) GO 10 2000 116 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 ( A E X ( J , I , 1 ) ) * ( T 1 ( I - 1 , J , K ) - T 1 ( I , J , K ) ) -2 (B1(K) * A E X ( J , I , 1 ) *X1/EKAY ) * ( T 1 ( I , J , K ) - ASE(K) ) + 3 ( * F Z ( 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 2000 119 B ( J ) = (AEY ( I . J , 2 ) / 2 . ) • S 4 ( I , J , K ) • P H Y ( I , J , K ) 1 + A E Y ( I . J . 2 ) * H 4 ( K ) * Y 2 / F K A Y / 2 . C ( J ) = -AEY ( I , J . 2 ) / 2 . D ( J ) = S 4 ( I , J , K ) * P H Y ( 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 ( A B Z ( I . J ) ) • ( T ( I , J , K - 1 ) - T ( I . J . K ) ) -3 (E3*ARZ ( I , J ) * E Z / B K A Y ) * (T (1, J . K)—ABB (K) ) 8 • (AFY ( I , J , 2) * E-4 (K)*Y2*ABE (K)/B KA Y/2. ) GC TO 2000 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 ) *Y2/BKAY/2. C (J) = -ARY ( I , J . 2 ) / 2 . D(J) = £4(1,J.K) * PHY(1,J,K) * T ( I , J , K ) • 1 ( A P X ( J , I , 1 ) / 2 . ) * ( 1 1 ( I - n J # K } - T 1 ( I , J , K ) ) • 1 ( I B X ( J , I , 2 ) / 2 . ) * (T1 (1 + 1 , J , K ) - T 1 ( I , J , K ) ) + 2 ( A F Z ( I , J ) ) * ( T ( I , J , K - 1 ) - T ( I . J . K ) ) -3 (6 3*ABZ ( I . J ) * C2/B KAY ) * ( T ( I . J . K ) -AH3(K) ) 8 • (AEY ( I , J.2) *H4 (K)*Y2«AI'.B (K)/RKAY/2. ) GO TC 2000 121 E ( J ) = (AEY ( I . J , 2 ) / 2 . ) + S 4 ( I , J , K ) * EHY(1,J,K) 1 • ABY ( I , J , 2 ) *B4 (K)*Y2/BKAY/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)*ABX (J.1,1)*X1/BKAY/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 (E3*AEZ ( 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)/EK*Y/2. ) GC TO 2000 122 A ( J ) = - A B Y ( I , J , 1 ) / 2 . E ( J ) = ( A E Y ( l . J . I ) • ABY ( I . J , 2 ) ) / 2 . • S 4 ( i , J , K ) *EHY(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 ( A E Z ( 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 2000 123 A ( J ) = -ABY ( I . J , 1) / 2 . E ( J ) = (ARY (I,J,1)+ABY ( I , J , 2 ) ) / 2 . *S4 ( I . J . K ) * P H Y ( I . J . K ) C (J) = -IEY ( 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.KJ-T1 ( I . J . K ) ) • 1 (AEX ( J , 1 , 2 ) / 2 - ) * (T 1 ( I t 1 , J , K ) - T 1 ( I . J . K ) ) • 2 ( A E Z ( I . J ) ) » ( T ( I , J , K - 1 ) - T ( I . J . K ) J 3 - (H3*AEZ ( I . J ) *EZ/EKAY ) * ( T ( l . J . K ) - ABB (K) ) GO TO 2000 124 A ( J ) = -ABY ( I , J , 1 )/2. B ( J ) = (ABY ( I . J , 1 ) + A R Y ( 1 , J , 2 ) )/2. + S 4 ( I . J . K ) * P H Y ( 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)*ABX(J,1,1) *X1/BKAY/2.) * ( T1 ( I . J . K ) - A f l d (K) ) • 3 ( A B Z ( 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 ) = S 4 ( I , J , K ) * P B Y ( 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 ( A F Z ( 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 ( K 2 ( K ) * A E Y ( I , J , 1 ) •Y1/BKAY*ABB(K)/2. ) GC TO 2000 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 1 ( E 2 (K) * A F Y ( I , J , 1 ) * Y 1 / B K A Y / 2 . ) E ( 0 ) = S 4 ( I , 0 , K ) * 1 ( I . 0 , K ) * P H Y ( I , 0 , K ) • 1 < A B X ( 0 , I , 1 ) / 2 . ) * ( T l ( 1 - 1 , 0 , K J - T 1 (1 , 0 ,K) ) + 1 ( 1 F X ( J , I , 2 ) / 2 . ) * ( T l ( I t 1 , 0 , K ) - T 1 ( I , J , K ) • ) • 2 ( J F Z ( I . J ) ) * ( T ( 1 , 0 , K - 1 ) - I ( I , J,K) ) -3 ( E 3 * A E Z ( I , J ) • E Z / E K A Y ) • (T ( 1 , J , K ) - A H B (K) ) • U ( K 2 ( K ) * » E Y ( 1 , 0 , 1 } * Y 1 / B K A Y * A M B ( K ) / 2 . ) GO TO 2 0 0 0 127 A ( J ) = - A E Y ( 1 , 0 , 1 ) / 2 . B ( J ) = ( A E Y ( I , 0 , 1 ) / 2 . ) • (SU ( I , 0 , K ) * P H Y ( I , 0 , K ) ) • 1 ( H 2 (K) * A E Y ( 1 , 0 , 1 ) *Y l / B K A Y / 2 . ) D(J) = S U ( 1 , 0 , K ) » P H Y ( I , 0 , K ) * T ( I , J , K ) • 1 ( A E X ( 0 , I , 1 ) / 2 . ) » (T1 (1-1 , 0 , K ) - T 1 ( I , J , K ) ) -2 ( B 1 (K) * A E X ( J , l , 1) * X 1 / R K A Y / 2 . ) * ( 1 1 ( 1 , 0 , K ) - AMB (K) ) • 3 ( * E Z ( I , J ) } * (1 ( 1 , 0 , K - 1 J - T ( 1 , 0 , K ) ) -U ( H 3 * A B Z (1 , 0 ) * E Z / E K AY ) * ( T ( I , 0 , K ) - A M B (K) ) • 5 (H2 (K) * A F Y ( 1 , 0 , 1) *Y 1 / E K A Y * A M B ( K ) / 2 . ) GO 1 0 2 0 0 0 128 A ( 0 ) = - A E Y ( I , J , 1 ) B ( 0 ) = A E Y ( 1 , 0 , 1) +AEY ( 1 , 0 , 2 ) • S I ( I , 0 , K ) * PHY ( I , 0 , K ) 1 • ( B 5 ( K ) • A E E A U ) C ( 0 ) = - A R Y ( 1 , 0 , 2 ) 0 ( 0 ) = S 4 ( 1 , J , K ) » PHY ( 1 , 0 , K ) * T ( I , 0 , K ) + 1 A F Z ( I , 0 ) » (1 ( I , J , K - 1 ) - 2 . * T ( I , 0 , K ) * T ( I , 0 , K + 1) ) • 2 A F X ( 0 , I , 1 ) * (T 1 ( I - 1 , 0 , K ) - T 1 ( 1 , 0 , K ) ) • 3 A I X ( 0 , 1 , 2 ) * (11 (1 + 1 , 0 , K ) - T 1 ( 1 , 0 , K ) ) *» - <H7 (K) * A E E A 1) * (T 1 ( 1 , 0 , K ) - A M B ( K ) ) 5 • ( K 5(K) * A R EAU *A ME (K) ) G C T C 2 0 0 0 129 A ( 0 ) = - A E Y ( 1 , 0 , 1 ) / 2 . B(J) = ( A E Y ( 1 , 0 , 1 ) + A K Y ( 1 , 0 , 2 ) ) / 2 . + S U ( I , J , K ) * P H Y ( I , J , I 1 • ( E 5 ( K ) * A E E A U / 2 . ) C ( 0 ) = - A E Y ( 1 , 0 , 2 ) / 2 . D ( 0 ) = S 4 ( I , 0 , K ) * I E Y ( 1 , J , K ) * T ( I , 0 , K ) • 1 ( i F X ( 0 , 1 , 1 ) / 2 . ) * ( T 1 (1 - 1 , 0 , K)-11 ( 1 , 0 , K ) ) • 1 ( A F X ( 0 , I , 2 ) / 2 . ) * (T 1 ( 1+ 1 , J , K ) - 1 1 ( I , 0 , K ) ) • 2 ( A E Z ( I . O ) ) • ( 1 ( I , J , K - 1 ) - T ( I , J , K ) ) 3 - < H 3 * A R Z ( I , J ) * E Z / R K A Y ) » ( T ( I , J , K ) - A M E ( K ) ) 4 - (B7 (K) * A E E A 1 / 2 . ) * (T 1 ( I , 0 , K ) - A H B (K) ) 1 • ( B 5 ( K ) * A R E A 4 * A M B ( K ) / 2 . ) GO TO 2 0 0 0 130 A ( 0 ) = - A B Y ( 1 , 0 , 1 ) B ( 0 ) = A R Y ( 1 , 0 , 1 ) + ARY ( 1 , 0 , 2 ) • S « ( I , 0 , K ) * P H Y ( I , J , K ) 1 • H6 ( K ) * A E E A 2 * C 0 S ( T F ) C ( 0 ) = - A R Y ( 1 , 0 , 2 ) D (0) = S 4 ( I , 0 , K ) * P H Y ( 1 , 0 , K ) * T ( I , 0 , K ) + 1 ( A E Z ( I , 0 ) ) * ( T ( 1 , 0 , K - 1 ) - 2 . *T ( I , J,K) • T ( I , J , K * 1 ) ) • 2 ( A F X ( 0 , I , 1 ) ) * ( T 1 ( I - 1 , 0 , K ) - T 1 ( I , J , K ) ) -3 ( H 7 (K) * A E I A 5 ) * ( T 1 ( I , 0 , K ) - A H B(K) ) U- ( H 6 ( K ) * A 8 E A 2 * S I N ( T H ) ) * ( T 1 ( I , 0 , K ) - A M B ( K ) ) 5 • R 6 ( K ) * A F E A 2 * C 0 S ( T H ) * A R E (K) G C 10 2 0 0 0 1 3 1 A ( 0 ) = - A R Y ( 1 , 0 , 1 ) / 2 . E ( 0 ) = ( A E Y (1, J, 1) +ARY (1, J , 2) ) / 2 . • SU ( 1 , 0 , K) * P H Y ( i , 0 , K) 1 • E 6 (K) * A E £ A 2 * C 0 S ( T H ) / 2 . C ( 0 ) = - A E Y ( 1 , 0 , 2 ) / 2 . D ( 0 ) = S 4 ( I , J , K ) * E H Y ( 1 , 0 , K ) * T ( I , J , K ) • 1 ( A F X ( J , I , 1 ) / 2 . ) * ( T 1 ( 1 - 1 , 0 , K ) - T 1 ( 1 , 0 , K ) ) -2 ( H 7 ( K ) * A F E A 5 / 2 . ) • ( 1 1 ( I , J , K) - A MB <K) ) • 3 ( A E Z ( I , 0 ) ) * ( T ( I , 0 , K - 1 ) - 1 ( 1 , 0 , K ) ) -198 4 ( H 3 * A B Z ( I . J ) * E Z / R K A Y ) • (T ( I , J , K) - A HE (K) ) 5 - (H6 ( K ) * A E E A 2 * S I N ( I f i ) / 2 . ) * ( T l ( I , J , K ) - A H B ( K ) ) 6 • H6 ( K ) * A E E A 2 * C 0 S ( T H ) * A H B ( K ) / 2 . GO TO 2 0 0 0 132 A { 0 ) = - A B Y ( I . J . I ) E ( J ) = ( A R Y ( I , J , 1 ) ) • F H Y ( I , J , K ) * S 4 ( I , J , K) • ' 1 ( B 5 (K) * A E Y ( I , J , 1) * Y 3 / B K A Y ) E ( J ) = S 4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I , J , K ) • 1 ( « F X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J , K ) - T 1 ( I , J , K ) ) + 1 ( A F X ( J , I , 2 ) ) * ( T 1 ( I * 1 , J , K ) - T 1 ( I , J , K ) ) • 2 ( I F Z ( I , J ) ) * (1 ( I , J , K - 1 ) - 2 . * I ( I , J , K ) * T ( I , J . K + 1 ) ) • 3 ( H E (K) * A F Y ( 1 , 0 , 1 ) » Y 3 / B K A Y * A M B ( K ) ) GO TO 2 0 0 0 1 3 3 A ( J ) = - A B Y ( 1 , J , 1 ) / 2 . B ( J ) = ( A B Y ( I , J , 1 ) / 2 . ) • ( S 4 ( I , J , K) * P H Y ( I , J , K) ) • 1 ( B £ ( K ) * A R Y ( 1 , J , 1 ) • Y 3 / B K A Y / 2 . ) D ( J ) = S 4 ( I , J , K ) * T ( 1 , J , K ) » P d Y ( I , J , K ) • 1 ( A F X ( J , I , 1 ) / 2 . ) » ( T 1 ( J - 1 , J , K ) - T 1 ( 1 , J , K ) ) • 1 ( A E X ( J , I , 2 ) / 2 . ) * ( T 1 (1 + 1 , 0 , K ) - 1 1 ( I , J , K) ) • 2 ( I I Z ( I . J ) ) * ( T ( 1 , J , K - 1 ) - T ( I , J , K ) ) -3 ( H 3 * * E Z ( I . J ) * E Z / 3 K A Y ) * (T ( I , J , K ) - A H 5 (K) ) • 4 ( H 5 ( K ) * A F Y ( I . J , 1 ) * Y 3 / B K A Y * A K B ( K ) / 2 . ) G C T C 2 0 0 0 1 3 4 E ( J ) = ( A E Y ( I , J , 2 ) ) • S 4 ( I , J , K ) * E H Y ( I . J . K ) 1 • AFY ( I , J , 2 ) *H6 ( K ) * Y 4 / E K A Y C ( J ) = - A B Y ( I , J , 2 ) D ( J ) = E 4 ( I , J , K ) » P F . Y ( 1 , J . K ) * T ( I , J , K ) + 1 ( A E Z ( I , J ) ) * ( 1 ( I , J , K - 1) - 2 . *T ( I . J , K) • T ( I , J . K + 1) ) • 2 ( A F X ( J , 1 , 1 ) ) * ( T 1 ( I - 1 , J , K ) - T 1 ( I , J , K ) ) -3 ( H E (K) * A B X ( J . I , 1) * X 2 / E K A Y ) * ( T 1 ( I , J , K ) - AHB (K) ) 8 • ( A E Y ( I , J , 2 ) * h 6 ( K ) * Y 4 * A H B ( K J / B K A Y ) GO TO 2 0 0 0 1 3 5 B ( J ) = ( A E Y ( I , J , 2 ) / 2 . ) + S 4 ( I , J , K) * FH Y ( I , J , K) 1 + A E Y ( I , J . 2 ) * K 6 ( K ) * Y 4 / B K A Y / 2 . C ( J ) = - A F Y ( I , J , 2 ) / 2 . D ( J ) = S 4 ( I , J , K ) * E E Y ( I . J . K ) * T ( l . J . K ) • 1 ( J I X ( J , I , 1 ) / 2 . ) * ( T 1 ( I - 1 , J , K ) - T 1 ( I . J . K ) ) -2 ( b 6 (K) * A F X ( J , I , 1 ) * X 2 / B K A Y / 2 . ) * ( 1 1 ( I . J . K ) - A H S ( K ) ) • 3 ( A B Z ( I . J ) ) * ( T ( I , J , K - 1 ) - T ( I , J , K ) ) -4 ( E 3 * A B Z ( I , J ) » E Z / R K A Y ) * (T ( I , J , K ) - A H B (K) ) 8 • ( A E Y ( I , J . 2 ) * E 6 ( K ) * Y 4 * A H B ( K ) / B K A Y / 2 . ) GO TO 2 0 0 0 1 3 6 E ( J ) = (ABY ( I . J , 2 ) ) • S 4 ( I , J , K ) * P U Y ( I . J . K ) 2 • ( H6 (K) * A E E A 3 * C O S (TH) ) C ( J ) = - A B Y ( I , J . 2 ) D ( J ) = S4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I , J , K ) • 1 ( A F Z ( I , J ) ) * ( 1 ( I , J , K - 1 ) - 2 . * T ( I , J , K ) + T ( I , J . K + 1 ) ') • 2 ( * I X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J . K ) - T 1 ( I . J . K ) ) -3 ( H I ( K ) * A B E A 6 ) * ( T I ( I . J . K ) - A H E (K) ) 3 • ( H6 ( K ) * A E E A 3 * C O S ( T H ) » A H B ( K ) ) 4 - (H6 ( K ) * A E E A 3 * S I N ( T H ) ) * (T 1 ( I , J , K ) - A H B ( K ) ) GO 1 0 2 0 0 0 1 3 7 B ( J ) = ( A E Y ( I , J , 2 ) / 2 . ) • S 4 ( I , J , K) * P h Y ( I , J , K) 1 • ( B 6 ( K ) * A B E A 3 * C O S ( I H ) / 2 « ) C ( O ) = - A B Y ( I . J , 2 ) / 2 . D ( J ) = S 4 ( I , J , K ) • E B Y ( I , J , K ) * T ( I . J . K ) • 1 ( H X ( J , 1 , 1 ) / 2 . ) * ( T 1 ( I - 1 , J , K ) - T 1 ( I . J . K ) ) -2 ( H I ( K ) * A F E A 6 / 2 . ) * ( T 1 ( l . J . K ) - A H B (K) ) • 3 ( A B Z ( I . J ) ) * ( T ( I . J . K - I ) - T ( I , J , K ) ) -4 ( H 3 * A E Z ( I , J ) * E Z / B K A Y ) * (T ( I , J , K ) - A H B ( K ) ) 1 99 1 • (f.6 (K) * A E E A3 * C O S ( T H ) / 2 . * A H B (K) ) 2 - (H6 ( K ) * A B E A 3 * £ I H ( T H ) / 2 . ) * ( T l ( I . J . K ) - A B S (K) ) GO 1C 2 0 0 C 1 3 8 A (J) = - A B Y ( I , J , 1 ) B ( J ) = < A E Y ( I , J , 1 ) ) + (SU ( I , J , K ) * P H Y ( I , J , K ) ) • 1 (H5 ( K ) * A E Y ( I , J , 1 ) * Y 3 / B K A Y ) E ( J ) = £"{!,J , K ) * ' P H Y ( I . J . K ) * T ( I . J . K ) • 1 ( 1 R X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J , K ) - T 1 ( I , J , K ) ) -2 ( 6 1 (K) * A E X ( J . I , 1) * X 1 / E K A Y ) * ( T I ( l . J . K ) - A R B (K) ) • 3 ( A E Z ( I . J ) ) * ( T ( 1 , J , K - 1 ) - 2 . * 1 ( 1 , J , K ) + T ' ( I , J . K + 1 ) ) + U ( H5 ( K ) * A B Y ( 1 , 0 , 1 ) * Y 3 / E K A Y * A H B ( K ) ) GO 1C 2 0 0 0 1 3 9 A ( J ) = - A B Y ( 1 , J , 1 ) / 2 . B ( J ) = ( A B Y ( 1 , 0 , 1 ) / 2 . ) • ( S U ( I , J , K ) * P H Y ( I , J , K ) ) • 1 (HE ( K ) * A E Y ( 1 , 0 , 1 ) * Y 3 / E F A Y / 2 . ) D ( 0 ) = S 4 ( I , 0 , K ) * P H Y ( I , 0 , K ) * 1(1, J , K ) • 1 ( 1 E X ( J , I , 1 ) / 2 . ) * ( T l ( 1 - 1 , 0 , K ) - T 1 ( I , 0 , K ) ) -2 ( t 1 ( K ) * A B X ( J . I , 1 ) * X 1 / E K A Y / 2 . ) * (11(1, J , K ) - AHB (K) ) + 3 ( A E Z ( I , 0 ) ) * ( T ( 1 , J , K - 1 ) - T ( I . O . K ) ) -4 ( H 3 * A B Z ( I , J ) * E Z / R K A Y ) * (T ( I , J , K ) - A H E (K) ) • 5 ( B 5 (K) * A E Y ( I , J , 1 ) * S 3 / B K A Y * A H E ( K ) / 2 . ) GO TO 2 0 0 0 2 0 0 0 C O N T I N U E B E T U E K EN E C C I f l l L I C I T C A L C U L A T I O N S I N T H E Z D I R E C T I O N C C C C S U E E C U T I K E C O E I E K ( I E , I E , 1 , J , A , B , C , D ) D I R I N S 1 C N A (1) , E ( 1 ) , C ( 1 ) , D ( 1 ) D I E E N S I C N ABX ( 1 6 , 1 1, 2) , ARY ( 1 1, 1 6 , 2) , AHZ ( 1 1 , 16) C I B E N S I C N 1 ( 1 0 , 1 6 , 9 1 ) , 1 1 ( 1 0 , 1 6 , 9 1 ) 1 , 1 2 ( 1 0 , 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 C O K E C N / C 1 7 / A R X , A B 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 C 0 B B C N / C 6 / H 3 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 = N T Y P E ( I , J . K ) G C T O ( 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 2 1 0 A ( K ) = - A B Z ( 1 , 0 ) B ( K ) = ( A E Z ( I , J ) * 2 . ) • S « ( I . J . K ) * P H Y ( I . J , K ) C ( K ) = A (K) D ( K) = S U ( 1 , 0 , K ) * P H Y ( 1 , J , K ) * T ( I . J . K ) • 1 ( A E X ( 0 , I , 2 ) ) * ( T l (1 + 1 , 0 , K ) - T 1 ( I , J . K ) ) • 2 ( A I Y ( I , J , 2 ) ) * ( T 2 ( 1 , J + 1 , K ) - 1 2 ( 1 , J . K ) ) 8 • ( A F Y ( I , J . 2 ) * K 4 ( K ) * Y 2 / R K A Y ) * ( A B B ( K ) - 1 2 ( I . J . K ) ) I F ( K . E Q . 2 ) D ( K ) = C ( K ) • ( A E Z ( I . J ) ) * T ( I . J , 1 ) GO 1 0 3 0 0 0 200 2 1 1 A ( K ) = - A B Z ( I , J ) B(K) = A E Z ( I , J ) * 2 . • S 4 ( I , J , K ) * F H Y ( I , J . K ) C ( K ) = A ( K ) D ( K ) = S 4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I . J . K ) •. 1 ( l F X ( J , l , 1 ) ) » ( I 1 ( I - 1 , J , K ) - - r l ( I . J , K ) ) • 1 ( A E X ( J , I , 2 ) ) * (T 1 ( 1 + 1 , J , K ) - T 1 ( I , J , K ) ) • 2 A F Y ( I , J , 2 ) * ( I 2 ( I , J + 1 , K ) - I 2 ( I , J . K ) ) 8 • ( A F Y ( I , J , 2 ) * f c 4 ( K ) * Y 2 / B K A Y ) * ( A A B ( K ) - T 2 ( I , J . K ) ) I F ( K . F . 0 . 2 ) D ( K ) = D ( K ) + ( A B Z ( I , J ) ) * T ( I . J , 1) GO T C 3 0 0 C 2 1 2 A ( K ) = - A F Z ( I . J ) B ( K ) = ( A E Z ( I , J ) * 2 . ) . • SU ( I . J . K ) * P H Y ( I . J . K ) C ( K ) = A (K) C ( K ) = S U ( I . J . K ) * P H Y ( I . J . K ) * T ( I . J . K ) + 1 ( A F X ( J » I , 1 ) ) * ( T 1 ( I - I . J . K ) - T 1 ( I . J . K ) ) -2 ( H i (K) * A B X ( J . I , 1) *X 1 / B K A I ) * ( T I ( I . J . K ) - A H B (K) ) • 3 ( * F Y ( I , J , 2 ) ) » ( T 2 ( I , J + 1 , K ) - T 2 ( I , J , K ) ) 8 • ( A E Y ( I , J , 2 ) * E U ( K ) » Y 2 / E K A Y ) » ( A B B ( K ) - 1 2 ( I , J , K ) ) I F ( K . F Q . 2 ) D (K) =D (K) • ( A E Z ( I . J ) ) * T ( I . J , 1) GO T C 3 0 0 C 2 1 3 A ( F ) = - A E Z ( I . J ) B ( K ) = A B Z ( I , J ) * 2 . + S4 ( I . J . K ) * P H Y ( I . J . K ) C ( K ) = A (K) C ( K ) = S U ( I . J . K ) * PHY ( I . J . K ) * T ( I , J , K ) • 1 A F X ( J , I , 2 ) * (T 1 ( 1 + 1 , J . K ) - T 1 ( I . J . K ) ) • 2 ( A E Y ( I , J , 1 ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) + 3 ( A F Y ( I , J , 2 ) ) * ( T 2 ( I , 0 + 1 , K ) - T 2 ( I , J , K ) ) I F ( K . E C . . 2 ) E (K) =E (K) • ( A E Z ( l . J ) ) * T ( I . J , 1) GO 1 0 3 0 0 0 2 1 4 A ( K ) = - A E Z ( I , J ) £ ( K ) = A E Z ( I , J ) * 2 . • S U ( I . J . K ) * P K Y ( I , J . K ) C ( K ) = A (K) D ( K ) = S U ( I , J , K ) * F H Y ( I . J . K ) * T ( I , J , K ) • 1 ( * F X ( J , I , 1 ) ) * (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 ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I . J . K ) ) • 2 ( I B Y ( I , J , 2 ) ) * ( T 2 ( I , J + 1 , K ) - T 2 ( I , J , K ) ) I F ( K . E C . 2 ) D ( K ) = E ( K ) + A B Z ( I , J ) * T ( I , J , 1 ) GO 1 0 3 0 0 0 2 1 5 A ( K ) = - A E Z ( I , J ) B(K) = A E Z ( I , J ) * 2 . • S 4 ( 1 , J . K ) * P H Y ( I , J . K ) C ( K ) = A ( K ) D ( K ) = S 4 ( I , J , K ) * E E Y ( I , J , K ) * T ( I , J , K ) • 1 ( A B X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J , K ) - T 1 ( I , J , K ) ) -2 ( E 1 ( K ) * A E X ( J . I , 1 ) * X 1 / B K A Y ) * ( T 1 ( I , J , K ) - A B B (K) ) • 2 ( A I Y ( I , J , 1 ) ) • ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) • 2 ( A E Y ( I , J , 2 ) ) • ( T 2 ( I , 0 + 1 , K ) - T 2 ( I , J , K ) ) I F ( K . E C . 2 ) D (K) = t ( K ) • ( A B Z ( I , J ) ) * T ( I , J , 1 ) GO 1 0 3 0 0 0 2 1 6 A ( K ) = - A B Z ( I , J ) B ( K ) = ( A B Z ( I , J ) * 2 . ) • S 4 ( I , J , K ) * P H Y ( I , J , K ) C ( K ) = A (K) C ( K ) = S 4 ( I , J , K ) * F H Y ( I , J , K ) * T ( I , J , K ) • 1 ( A E X ( J , I , 2 ) ) * ( T l (1 + 1 , J , K ) - T 1 ( I , J . K ) ) + 2 ( A F Y ( I . J . I ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( 1 , J , K ) ) -3 <B2 (K) * A F Y ( I , J , 1) *Y 1 / E K A Y ) * ( T 2 ( I , J , K ) - A H B (K) ) I F ( K . E Q . 2 ) D ( K ) = D ( K ) • ( A B Z ( 1 , J ) ) * T ( I , J , 1 ) G O 1 0 3 0 0 0 2 1 7 A ( K ) = - A B Z ( I , J ) B ( K ) = A B Z ( I , J ) * 2 . + S 4 ( I , J , K ) * P H Y ( 1 , J , K ) 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 ( A B Y ( I , J , 1 ) ) • ( T 2 ( I , 0 - 1 , K ) - T 2 ( 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 21E A (K ) = - A R Z ( I , J ) B(K) = (AEZ ( I , J ) * 2 . ) • S U ( I , J . K ) * £ H Y ( 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 ( A F X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J . K ) - T 1 ( 1 , J , K ) ) -2 (HI (K) * A B X ( J . I , 1 ) * X 1 / B K A Y ) * ( T I ( I . J . K ) -AHE(K) ) • 3 ( I F Y ( I , J , 1 ) ) • ( T 2 ( I , J - 1 , K ) - 1 2 ( 1 , J . K ) ) -1 (H2(K) *AEY ( 1 , 0 , 1 ) *Y1/EKAY) * ( T 2 ( I , J , K ) -AHB (K) ) I F ( K . E C . 2 ) D ( K ) = C ( K ) + ( A B Z < I , J ) ) * T ( I , J , 1 ) GC 10 3000 219 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 ) - T 1 ( I , J , K ) ) • 2 ( A B Y ( I , J , 2 ) / 2 . ) * { 1 2 ( I , J O , K ) - T 2 ( I , J , K ) ) • 3 ( B 3 * A B Z ( I , J ) * r Z / B K A Y * A H B ( 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 220 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 221 A (K) = - A E Z ( I . J ) £ (K) = ( A E Z ( I . J ) ) • ( S 4 ( I , J , K ) * E H Y ( 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 ( * F X ( J , I , 1 ) / 2 . ) * ( T 1 ( I - 1 , J , K ) - 1 1 ( 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 3000 222 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 ( A F Y ( I , J , 1 ) / 2 . ) * ( 1 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) * 2 (AEY ( I . J . 2 ) / 2 . ) * (12 ( I . J O , K ) - T 2 ( I . J . K ) ) • 3 (B3*AFZ ( I . J ) * rZ /BKAY*AHE (K) ) GO 10 3000 223 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 ) • 1 ( A E X ( J , I , 1 ) / 2 . ) * ( T 1 ( I - 1 , 0 , K ) - T 1 ( I , J , K ) ) • 1 ( A F X ( 0 , 1 , 2 ) / 2 . ) * ( H d * 1 , 0 , F ) - T 1 ( 1 , 0 , K ) ) • 2 ( AEY ( 1 , 0 , 1 ) / 2 . ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) • 2 ( I F Y ( I , J , 2 ) / 2 . ) * ( 1 2 ( I , J * 1 , K ) - 1 2 ( 1 , J , K ) ) + 3 H 3 * A E Z ( I , J ) * E Z / B K A Y * A B E < K ) GO I C 3 0 0 0 22<4 A ( K ) = - A B Z ( I , J ) B ( K ) = ( A E Z ( I , J ) ) * ( S 4 ( 1 , J , K ) * P H Y ( I , J , K ) ) • 1 H 3 * A E Z ( I , J ) * E Z / E K A Y r ( K ) = S U ( I , J , K ) * P H Y ( I , J , K ) * T ( I , J , K ) + 1 ( I E X ( J , I , 1 ) / 2 . ) * ( 1 1 ( 1 - 1 , J , K ) - T 1 ( I , J , K ) ) -2 ( H 1 (K) * AEX ( J , I , 1) * X 1 / B K A Y / 2 . ) * ( 11 ( I , J , K ) - A f l £ ( K ) ) • . 3 ( A E Y ( I , J , 1 ) / 2 . ) * ( 12 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) • 3 ( A E Y ( I , J , 2 ) / 2 . ) * ( T 2 ( I , J * 1 , K ) - T 2 ( I , J . K ) ) • « ( E 3 * A BZ ( I , J ) * E Z / B K A Y * A BB (K) ) GC 1 0 3 00 0 2 2 5 A ( K ) = - A B Z ( I , J ) B ( K ) = ( A E Z ( I , J ) ) • ( S U ( I , J , K) * P H Y ( I , J . K ) ) • 1 ( E 3 * A E Z ( I . J ) * E 2 / B K A Y ) E ( K ) = S U ( I . O . K ) * E H Y ( I , J , K ) * T ( I . J . K ) • 1 ( A B X ( 0 , I , 2 ) / 2 . ) • ( 1 1 ( 1 * 1 , J . K ) - T I ( I . J . K ) ) + 2 ( A I Y ( I , J , 1 ) / 2 . ) » ( T 2 ( I , J - 1 , K ) - I 2 ( I . J . K ) ) -3 ( E 2 ( K ) * A E Y ( I , 0 , 1 ) * Y 1 / E K A Y / 2 . ) * ( T 2 ( I , J , K ) - A B B (K) ) • 4 ( E 2 * A B Z ( 1 , 0 ) * D Z / B K A Y * A B B (K) ) GO 1 0 3 0 0 0 2 2 6 A ( K ) = - A E Z ( I , J ) B ( K ) = ( A E Z ( I . J ) ) + ( S U ( 1 , 0 , K ) * £ H Y ( I . J . K ) ) • 1 ( E 3 » A E Z ( 1 , J ) * E Z / B K A Y ) E ( K ) = £ 4 ( 1 , 0 , K ) * P H Y ( I . J . K ) * X ( I . J . K ) • 1 ( A E X ( J . I , 1 ) / 2 . ) » (11 ( 1 -1 , J , K ) - I 1 ( I , J . K ) ) * 1 ( A f X ( J . I . 2 ) / 2 . ) * ( I 1 ( 1 * 1 , J . K ) - T l ( l . J . K ) ) • 2 ( AEY ( 1 , J , 1) / 2 . ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) -3 ( H2 ( K ) * » B Y ( I . J , 1 ) * Y 1 / B K A Y / 2 . ) * ( T 2 ( I . J . K ) - A H B ( K ) ) 4 ( H 3 * A B Z ( I . J ) * E Z / B K A Y * A B E ( K ) ) GO TO 3 0 0 0 2 2 7 A (K) = - A E Z ( 1 , 0 ) B ( K ) = ( A B Z ( I , J ) ) • ( S 4 ( 1 , J , K ) * P H Y ( I , J , K ) ) * 1 ( K 3 * A E Z ( I , J ) * I Z / B K A Y ) D ( K ) = £ 4 ( 1 , J , K ) * F B Y ( 1 , J , K ) • T ( I , J , K ) + 1 ( A I X ( J , I , 1 ) / 2 . ) * ( T 1 ( I - 1 , 0 , K ) - T 1 ( I , J , K ) ) -2 (H 1 (K) * A B X ( 0 , 1 , 1 ) * X 1 / E K A Y / 2 . ) * ( I 1 ( 1 , J , K ) - A H B ( K ) ) • 3 ( A F Y ( I , J , 1 ) / 2 . ) * ( T 2 ( I , J - 1 , K ) - 1 2 ( I , J , K ) ) -4 (12 ( K ) * A F Y ( 1 , 0 , 1 ) * Y l / B K A Y / 2 . ) * ( T 2 ( 1 , 0 , K ) - A H B ( K ) ) • 5 ( E 3 * A F Z ( I , J ) * I Z / B K A Y * A B E ( K ) ) GO TO 3 0 0 0 2 2 8 A ( K ) = - A E Z ( I , J ) B ( K ) = A E Z ( I , J ) * 2 . • SU ( I , J , K ) * P H Y ( I , J , K ) C ( K ) = A (K) E ( K ) = S U ( I , J , K ) * F B I ( 1 , J , K ) * T ( I , J , K ) * 1 ( A F X ( J , I , 1 ) ) * (T 1 ( 1 - 1 , J , K ) - T 1 ( 1 , J , K ) ) • 1 ( A E X ( J , I , 2 ) ) * (T 1 ( I * 1 , 0 , K ) - T l ( 1 , 0 , K ) ) * 2 ( A F Y ( I , 0 , 1 ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I . J . K ) ) • 2 ( * F Y ( I , J , 2 ) ) * ( 1 2 ( 1 , 0 * 1 , K ) - T 2 ( I , 0 , K ) ) U - ( H 7 ( K ) * A B E A 1 ) * ( T 1 ( I , 0 , K ) - A K B ( K ) ) 5 • ( H 5 ( K ) * AE E AU) * ( A H B ( K ) - T 2 ( 1 , 0 , K) ) I E ( K . E C . 2 ) D ( K ) = E (K) + A B Z ( 1 , 0 ) * T ( I , 0 , 1) GO 10 3 0 0 0 2 2 9 A ( K ) = - A E Z ( 1 , 0 ) £ ( K ) = A E Z ( I , 0 ) • S U ( I , 0 , K ) * PHY ( I , J . K ) • 1 ( H 3 * A E Z ( I , J ) * E Z / B K A Y ) C ( K ) = S 4 ( I , J , K ) * PHY ( I . J , K ) * T ( I . J . K ) • 1 < A E X ( J , I , 1 ) / 2 . ) * ( I 1 ( I - 1 , J . K ) - T 1 ( I , J , K ) ) * 1 ( A I X ( J , I , 2 ) / 2 . ) * ( 1 1 ( I * 1 , J , K ) - T 1 ( l . J . K ) ) • 2 ( A R Y ( I . J , l ) / 2 . ) * ( T 2 ( I , 0 - 1 . K ) - T 2 ( I , J . K ) ) • 2 ( / I Y ( 1 , J , 2 ) / 2 . ) * { T 2 ( I , J + 1 , K ) - T 2 ( I , J , K ) ) • 3 H 3 * A B Z ( I , J ) » E Z / B K A Y * A H E ( K ) IJ - ( B 7 ( K ) * A R E A l / 2 . ) * ( 1 1 ( I . J . K ) - A K E (K) ) 5 • < E 5 ( K ) * A B E A 4 / 2 . ) * (AMB ( K ) - 1 2 ( I . J . K ) ) G C 1 C 3 0 0 0 2 3 0 A ( K ) = - A B Z ( I . J ) ^ -E ( K ) = A E Z ( I , J ) * 2 . • S 4 ( l . J . K ) * P H Y ( l . J . K ) C ( K ) = A ( K ) E ( K ) = S U ( I . J . K ) * P K Y ( I , J , K ) * 1 ( 1 , J . K ) • 1 ( A B X ( J , I , 1 ) ) * ( 11 ( 1-1 , J . K ) - T 1 ( I . J . K ) ) -2 ( H 7 ( K ) * A F E * 5 ) • ( 1 1 ( 1 , J . K ) - A E B ( K ) ) + 2 ( A E Y ( I . J . I ) ) * (12 ( I . J - 1 . K ) - 1 2 ( 1 , J . K ) ) • 2 ( A B Y ( I , J , 2 ) ) * ( 1 2 ( I , J * 1 , K ) - 1 2 ( I . J . K ) ) 3 - (H6 (K) * A B E A 2 * S I N ( T B ) ) * ( 1 1 ( I . J . K ) - A M B ( K ) ) 4 • . (H6 (K) * A R E A 2 * C O S ( T H ) ) * (AMB (K) - 1 2 ( I , J.K) ) I E ( K . E Q - 2 ) D ( K ) = C ( K ) • ( A B Z ( I , J ) ) * T ( I . J . I ) G C T C 3 0 0 0 2 3 1 A ( K ) = - A B Z ( I , J ) B ( K ) = ( A E Z ( I , J ) ) • < S 4 ( 3 , J , K ) * P H Y ( I , J , K ) ) • 1 H 3 * A B Z ( I . J ) * D Z / B K A Y D (K) = S 4 ( I , J , K ) * PHY ( 1 , J . K ) » T ( l . J . K ) • 1 ( A E X ( J . I , 1 ) / 2 . ) • ( 1 1 ( 1 - 1 , J . K ) - T I ( I . J . K ) ) -2 ( E 7 ( K ) * A E E A 5 / 2 . ) * ( 1 1 ( I . J . K ) - A ME (K) ) • 3 ( I E Y ( I , J , 1 ) / 2 . ) * ( T 2 ( I , J - 1 . K ) - T 2 ( 1 , J . K ) ) • 3 ( A F Y ( 1 , J , 2 ) / 2 . ) * ( I 2 ( I , J - H , K ) - T 2 ( I , J . K ) ) • 4 ( E 3 * A P Z ( 1 , J ) * f Z / F K A Y * A £ E (K) ) 3 - (H6 (K) * A R E A 2 * S I K ( T H ) / 2. ) * (1 1 ( I , J . K ) - AMB (K) ) 1 • ( H 6 ( K ) * A E E A 2 * C 0 S ( T b ) / 2 . ) * ( A B B (K) - 1 2 ( l . J . K ) ) G C 1 C 3 0 0 0 2 3 2 A ( K ) = - A B Z ( I . J ) B ( K ) = A F Z ( I , J ) * 2 . • S 4 ( , 1 , J , K ) * P H Y ( 1 , J , K ) C ( K ) = A (K) C ( K ) = S U ( I . J . K ) * E H Y ( 3 , J , K ) * T ( I , J , K ) • 1 ( A E X ( J , I , 1 ) ) • ( T l ( 1 - 1 , J , K ) - T 1 ( l . J . K ) ) • 1 ( I F X ( J , I , 2 ) ) * ( 1 1 ( I M . J . K ) - T I ( l . J . K ) ) + 2 ( A F Y ( 1 , J , 1 ) ) * ( 1 2 ( 1 , J - 1 . K ) - T 2 ( I , J , K ) ) -3 ( H 5 ( K ) * A B Y ( I , J , 1 ) * Y 3 / E K A Y ) * ( T 2 ( I . J . K ) - A B B ( K ) ) I F ( K . E C 2 ) E (K) = E (K) • ( A E Z ( I . J ) ) * 1 ( I . J . 1) G O 1 0 3 0 0 0 2 3 3 A ( K ) = - A E Z ( I . J ) B ( K ) = ( A E Z ( I . J ) ) • ( S U ( I . J . K ) * E h Y ( i , J , K ) ) + 1 ( h 3 * A E Z ( I . J ) * E Z / P K A Y ) 0 ( K ) = S 4 ( I , J , K ) * P H Y ( 3 , J . K ) * l ( I . J . K ) • 1 ( A B X ( J . I . I ) / 2 . ) * ( T 1 ( I - l . J . K ) - T l ( I , J . K ) ) • 1 ( I I X ( J , I , 2 ) / 2 . ) • ( 1 1 ( 1 + 1 . J . K ) - 1 1 ( I . J . K ) ) • 2 ( AEY ( I . J , 1 ) / 2 . ) * ( 1 2 ( 1 , J - 1 , K) - T 2 ( I , J , K ) ) -3 ( H 5 ( K ) * AB Y ( I . J , 1 ) * Y 3 / E K A Y / 2 . ) * ( 1 2 ( I . J . K ) - A B B ( K ) ) 4 ( B 3 * A B Z ( I . J ) * E Z / B K A Y * A H E (K) ) G C 1 C 3 0 0 0 2 3 4 A ( K ) = — A B Z ( I . J ) B ( K ) = ( A E Z ( I , J ) * 2 . ) • S 4 ( I , J , K ) * P H Y ( I . J . K ) C ( K ) = A (K) D ( K ) = £ 4 ( I , J , K ) * P H Y ( I , J , K ) * T ( I . J . K ) • 1 ( I E X ( J , I , 1 ) ) • ( T 1 ( 1 - l . J . K ) - T l ( I . J . K ) ) -2 ( H £ ( K ) » A B X ( J . I , 1) * X 2 / B K A Y ) * ( T 1 ( I , J , K ) - AMB (K) ) 3 ( A E Y ( I , J , 2 ) ) * ( T 2 ( I , J * 1 , K ) - T 2 ( I , J , K ) ) 8 • ( A B Y ( 1 , 0 , 2 ) * B 6 ( K ) * Y 4 / E K A Y ) * ( A H B ( K ) - T 2 ( 1 , 0 , K ) ) I F ( K . I C . 2 ) C ( K ) = E ( K ) * ( A E Z ( 1 , J ) ) * T ( I , J , 1 ) G C I C 3 0 0 0 2 3 5 A ( K ) = - A B Z ( I . O ) B ( K ) = ( A E Z ( I , J ) ) • ( S 4 ( 1 , J , K ) * P B Y ( I . J . K ) ) • 1 ( E 3* A E Z ( I . J ) * E 2 / E K A Y ) D ( K ) = S 4 ( I , J , K ) * PHY ( I . J . K ) * I | 1 , J , K ) • 1 ( A B X ( J , I , 1 ) / 2 . ) * ( T 1 ( 1 - 1 , J . K ) - 1 1 ( I , J . K ) ) 2 ( B 6 ( K ) » A B X ( J . I , 1) * X 2 / B K A Y / 2 . ) * ( 1 1 ( I . J . K ) - A B E ( K ) ) • 3 ( I F Y ( I , J , 2 ) / 2 . ) * ( T 2 ( I . J * 1 . K )-12 ( I . J . K ) ) • 3 ( H 3 * A R Z ( I , J ) * D 2 / B K A Y * A J E ( K ) ) 6 • ( A E Y ( I , J . 2 ) * B 6 ( K ) * Y 4 / E K A Y / 2 . ) * ( A B B ( K ) - 1 2 ( I , J . K ) ) G C I C 3 0 0 0 2 3 6 A ( K ) •= - A B Z ( I . J ) B ( K ) = ( A E Z ( I , J ) * 2 . ) • S 4 ( 1 , J , K ) * P H Y ( I . J . K ) C ( K ) = A (K) D (F ) = S 4 ( I , J , K ) * P E Y ( I , J , K ) * T ( I , J , K ) • 1 ( A E X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J . K ) - 1 1 ( I , J . K ) ) -2 ( H 1 ( K ) * A F E A 6 ) * ( 1 1 ( 1 , J . K ) - A B B (K) ) • 3 ( A F Y ( I . J , 2 ) ) * ( T 2 ( I , J * 1 , K ) - T 2 ( I , J , K ) ) 4 + ( B 6 ( K ) • A E E A 3 * C C S ( I B ) ) * (AHB ( K ) - T 2 ( I , J , K ) ) 5 - ( H 6 ( K ) * A E E A 3 * S 1 N ( T R ) ) * ( T 1 ( I , J , K ) - A H B ( K ) ) I F ( K . E C - 2 ) E (K) =E (K) + ( A E Z ( I . J ) ) * T ( I , J , 1) GO 10 3 0 0 0 2 3 7 A ( K ) = - A B Z ( I . J ) E ( K ) = ( A E Z ( I . J ) ) + ( S 4 ( I , J . K ) * F H Y ( I . J . K ) ) • 1 ( B 3 » A E Z ( I , J ) * E Z / F K A Y ) E ( K ) = £ 4 ( 1 , J , K ) * P B Y ( I , J , K ) * T ( I . J . K ) + 1 ( J E X ( J , I , 1 ) / 2 . ) * ( T 1 ( 1 - 1 , J . K J - 1 1 ( I , J . K ) ) -2 ( E l (K) * AE E A 6 / 2 . ) * ( 1 1 ( 1 , J . K ) - A B B (K) ) • 3 ( A E Y ( I . J , 2 ) /2. ) * { T 2 ( I , J + 1 , K ) - T 2 ( I , J , K ) ) • 3 ( H 3 * A E Z ( I . J ) * E Z / B K A Y * A B E ( K ) ) 4 • ( H 6 ( K ) * A B E A 3 * C O S ( T H ) / 2 . ) * ( A B E ( K ) - T 2 ( I , J , K ) ) 5 - ( H 6 ( K ) • A B E A 3 * S I S ( T H ) / 2 . ) * ( 1 1 ( I , J , K ) - A B B (K) ) GC TO 3 0 0 0 2 3 6 A ( K ) = - A K Z ( I . J ) B ( K ) = ( A E Z ( I . J ) * 2 . ) • S 4 ( I , J . K ) * F H Y ( I , J . K ) C ( K ) = A (K) D ( K ) = £ 4 ( 1 , J . K ) * E B Y ( 1 , J , K ) * T ( I , J , K ) • 1 ( A E X ( J , I , 1 ) ) * ( 1 1 ( 1 - 1 , J . K ) - T U l . J . K ) ) -2 ( E 1 ( K ) » A B X ( J , I , 1 ) * X 1 / B K A Y ) * ( T 1 ( I , J , K ) - A B B ( K ) ) + 3 ( A F I d . J . I ) ) • ( T 2 ( 1 , J - 1 , K ) - T 2 ( I , J , K ) ) -4 ( H 5 (K) * A B Y ( I , J , 1) * Y 3 / F K A Y ) * ( T 2 ( I , J , K ) - A H B ( K ) ) I F ( K . E C . 2 ) D ( K ) = E ( K ) • ( A E Z ( I , J ) ) * T ( I , J , 1) GC TO 3 0 0 0 2 3 9 A ( K ) = — A B Z ( I , J ) B ( K ) = ( A F Z ( I , J ) ) + ( S 4 ( 1 , J , K ) * P K Y ( I , J , K ) ) • 1 ( B 3 * A B Z ( I , J ) * D Z / B K A Y ) D ( K ) = S 4 ( I , J , K ) * PHY ( 1 , 0 , K ) * T ( I , J , K ) • 1 ( A E X ( J , I , 1 ) / 2 . ) • (T 1 ( I - 1 , 0 , K ) - I 1 ( 1 , 0 , K ) ) -2 ( E 1 ( K ) * A B X ( J , I , 1 ) * X 1 / E K A Y / 2 . ) * ( T 1 ( I , J , K ) - A B £ ( K ) ) • 3 ( A E Y ( I , J , 1 ) / 2 . ) • ( T 2 ( I , J - 1 , K ) - 1 2 ( I , J , K ) ) -4 ( h f ( K ) * A E Y ( I , 0 , 1) * Y 3 / E K A Y / 2 . ) * ( 1 2 ( I , 0 , K ) - A H E ( K ) ) • 5 ( E 3 * A R Z ( I , J ) * D Z / B K A Y * A f B (K) ) GO T C 3 0 0 0 3 0 0 0 C O K T I N O E F E T U E K EN E 205 c S U E E O 0 I I N E C 0 S P U T ( 1 1 , L 2 , B 1 , E 2 , K 1 ) C S O E B C U T I H E T O C A L C U L A T E T H E T E B P E B A T U E E S A T T H E C E K D CE A T I H E S T E P C c E I E E H S I O N T ( 1 0 , 1 6 , 9 1 ) , T 1 ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 , 1 6 , 9 1 ) , 1 T 3 ( 1 0 , 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 1 ) , C ( 1 0 1 ) , D ( 1 0 1) , T PR IM E (1 0 1) E I E E N S I G N A E X ( 1 6 , 1 1 , 2 ) , A B Y < 1 1 , 1 6 , 2 ) , A 8 2 ( 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 ) C O E E O N / C 3 / N T Y P E ( 1 0 , 1 6 , 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 C O B B O N / C 6 / H 3 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 ) C C E E C N / C 9 / E H Y H , E H Y S , P H Y L 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 ) , C P I ( 2 0 ) C O E E C K / C 1 3 / T 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 , A E Z 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 3 , A B E A 4 , A B E A 5 , A E E A 6 DO 1 0 0 0 I = L 1 , L 2 DO 1 0 0 0 J = B 1 , H 2 DO 1 0 C 0 K = 2 , K 1 I J K = K T Y E E ( I , J , K ) GO 1 0 ( 3 1 0 , 3 1 1 , 3 1 2 , 3 1 3 , 3 1 4 , 3 1 5 , 3 1 6 , 3 1 7 , 1 3 1 8 , 3 1 9 , 3 2 0 , 3 2 1 , 3 2 2 , 3 2 3 , 3 2 4 , 3 2 5 , 3 2 6 , 3 2 7 , 3 2 6 , 3 2 9 , 2 3 3 0 , 3 3 1 , 3 3 2 , 3 3 3 , 3 3 4 , 3 3 5 , 3 3 6 , 3 3 7 , 3 3 8 , 3 3 9 ) , I J K 3 1 0 I N ( I , J , K ) = ( ( A E X ( J , I , 2 ) ) * < T 1 ( 1 + 1 , J , K ) - T l ( I , J , K ) ) • 1 ( A B Y ( I , J , 2 ) ) * ( 1 2 ( I , J + 1 , K ) - T 2 ( I , J , K ) ) • 4 ( A B Y ( I . J . 2 ) * H 4 ( K ) » Y 2 / £ K A Y ) * (4MB ( K ) - 1 2 ( I . J . K ) ) + 2 ( A E Z ( I . J ) ) * ( 1 3 ( I , J . K - 1 ) - 2 . «1 3 ( I , J , K ) + T 3 (I , J , K * 1) ) ) / 3 ( < £ 4 ( I , J . K ) / 2 . ) * P H Y ( I , J , K ) ) • T ( I . J . K ) GO T O 1 0 0 0 3 1 1 T N ( I , 0 , K ) = ( ( A R X ( J , l , 1 ) ) * (T 1 ( 1 - 1 , J , K ) - T 1 ( I , J , K ) ) • 1 ( A E X ( J , 1 , 2 J ) * (T 1 ( I + 1 , J , K ) - T 1 ( I . J . K ) ) • 1 ( A B Y ( I , J , 2 ) ) * ( T 2 ( I , J * 1 , K ) - T 2 ( I , J , K ) ) + 8 ( A E Y ( I , J , 2 ) * H 4 ( K ) * Y 2 / E K A Y ) * ( A B B ( K ) - 1 2 ( I , J , K ) ) + 2 ( A E Z ( I . J ) ) * ( T 3 ( I , J , K - 1 ) - 2 . * T 3 ( I , J , K ) * T 3 ( I , J , K + 1) ) ) / 3 ( < S 4 ( I , O , K ) / 2 . ) * P H Y ( I , 0 , K ) ) + T ( I , J , K ) GO T O 1 0 0 0 3 1 2 T B ( I , J , K ) = ( ( A E X ( J , I , 1) ) * (T 1 ( I - 1 , J , K ) - T 1 ( I , J , K ) ) -1 ( E 1 ( K ) * A E X ( J , I , 1 ) * X 1 / E K A Y ) * ( T 1 ( I , J , K ) - A B E ( K ) ) • 2 ( A E Y ( I , J , 2 ) ) * ( T 2 ( I , J * 1 . K ) - T 2 ( I , J , K ) ) + 8 ( A E Y ( 1 , 0 , 2 ) * H 4 ( K ) * Y 2 / B K A Y ) * ( A B B ( K ) - 1 2 ( 1 , 0 , K ) ) + 3 ( A E Z ( I , J ) ) * ( 1 3 ( I , J , K - 1 ) - 2 . * T 3 ( I , 0 , K ) t T 3 ( I , J . K + 1) ) ) / 4 ( (S4 ( I , J , K ) / 2 . ) * E B Y ( I , J , K ) ) • T ( l . J . K ) GO TO 1 0 0 0 3 1 3 T N ( I , J , K ) = ( ( A B X ( J , l , 2 ) ) * (T 1 ( I * 1 , J , K ) - T 1 ( I , J . K ) ) • 1 ( A E Y ( 1 , J , 1 ) ) * ( T 2 ( I . J - 1 , K ) - T 2 ( I , J , K ) ) • 1 ( A E Y ( I , 0 , 2 ) ) * ( T 2 ( 1 , 0 * 1 , K ) - 1 2 ( 1 , 0 , K ) ) • 2 ( A E Z ( I , J ) ) * ( 1 3 ( I , J , K - 1 ) - 2 . » 1 3 ( I , J . K ) * T 3 ( I , 0 , K * 1 ) ) ) / 3 ( ( « 4 ( I , J , K ) / 2 . ) * P H Y ( I , 0 , K ) ) • T . ( I , J , K ) G C 1 C 1 0 0 0 3 1 4 T N d . J . K ) = ( ( A B X ( J . I , 1 ) ) * (T 1 ( 1 - 1 , J . K ) - T l ( l . J . K ) ) • 1 ( A E X ( J , I , 2 ) ) • (11 ( I + 1 , J , K ) - T 1 ( I . J . K ) ) • 2 ( I F Y ( I . J . l ) ) * ( T 2 ( I , J - 1 . K ) - T 2 ( I , J , K ) ) 2 < * E Y ( I , J , 2 ) ) * ( 1 2 ( 1 , J + 1 . K ) - I 2 ( I , J , K ) ) + 2 A E Z ( I , J ) * ( T 3 ( I . J . K - 1 ) - 2 . * 1 3 ( I , J . K ) + T 3 ( I , J . K + 1) ) ) / 3 ( < £ 4 ( I , J , K ) / 2 . ) * P b 5 f ( I # J r K ) ) + T ( I , J , K ) GO 10 1 0 0 0 3 1 5 1 N U , J , K ) = ( ( A B X ( J , I , 1) ) * (T 1 ( 1 - 1 , J , K ) - T l ( I , J , K ) ) -1 ( HI (K)* A E X ( J . I , 1 ) * X 1 / B K A Y ) * ( T I ( I . J . K ) - A H B (K) ) • 2 ( A E Y ( I , J , 1 ) ) » ( 1 2 ( I , J - 1 , K ) - 1 2 ( I , J , K ) ) • 2 ( A F Y ( I , J , 2 ) ) * ( T 2 ( I , J + 1 , K ) - T 2 ( 1 , J , K ) ) • 3 ( A E Z ( I . J ) ) * ( 1 3 ( I . J . K - 1 ) - 2 . * 1 3 ( I . J . K ) + T 3 ( I , J . K + 1) ) ) / 4 ( ( S 4 ( I , J , K ) / 2 . ) * P H Y ( 1 , J , K ) ) + 1 ( 1 , J , K ) GO 10 1 0 0 0 3 1 6 1 J J ( I , J , K ) = ( ( A B X ( J . I . 2 ) ) * (11 (1+ 1 , J . K ) - T 1 ( I , J . K ) ) • 1 ( A F Y ( I . J . I ) ) * ( T 2 ( I , J - 1 . K ) - T 2 ( I , J , K ) ) -2 ( H 2 (K) * A E Y ( I , J , 1) * Y 1 / E K A Y ) * ( T 2 ( I , J , K ) - A H E (K) ) + 3 ( A E Z ( I , J ) ) * ( T 3 ( I . J . K - 1 ) - 2 . * 1 3 ( I , J , K ) + 1 3 ( I , J . K + 1 ) ) ) / 4 ( < £ « ( I , J , K ) / 2 . ) * P H Y ( I , J , K ) ) + T ( I . J . K ) GC 1 0 1 0 0 0 3 1 7 T i l (I.J.K) = ( ( A B X ( J . I . I ) ) * ( T l ( 1 - 1 , J . K ) - T 1 ( i . J . K ) ) • 1 ( A B X ( J , I , 2 ) ) * ( T 1 (1+1 , J , K ) - T 1 ( l . J . K ) ) + 2 ( I F Y ( I . O . l ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) -3 <E2 (K) * A B Y ( I , J , 1) » Y 1 / E K A Y ) * (T 2 ( I , J, K) - AHB (K) ) • 4 ( A E Z ( I.J) ) * ( 1 3 ( I , J , K - 1 ) - 2 . * T 3 ( I , J . K ) + T 3 ( I , J . K + 1) ) ) / 5 ( ( £ 4 ( I , J , K ) / 2 . ) * P K Y ( I , J , K ) ) • T . ( I . J . K ) GO TO 1 0 0 0 3 1 8 T N I I . J , K ) = ( ( A E X ( J . I , 1) ) * (1 1 ( I - 1 , J . K ) - T 1 ( I . J . K ) ) -1 ( £ 1 ( K ) * A E X (J ,1,1) * X 1 / E K A Y ) * (1 1 ( I , J . K ) - A H E ( K ) ) + 2 ( J E Y ( I , J , 1 ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I . J . K ) ) -3 (B2 (K) * A £ Y ( I . J , 1) * Y 1 / R K A Y ) * ( T 2 ( l . J . K ) - AHB (K) ) • 4 ( A E Z ( I . J ) ) * ( 1 3 ( I . J . K - 1 ) - 2 - *T 3 ( I . J . K ) +T 3 ( I , J . K + 1 ) ) ) / 5 ( ( S 4 ( I , J , K ) / 2 . ) * P H Y ( I , J , K ) ) + T ( I , J , K ) GO 1 0 1 0 0 0 3 1 9 1 N ( 1 , J , K ) = ( ( A E X ( J . I , 2 ) / 2 . ) * ( T l (1+ 1 , J . K ) - T 1 ( I . J . K ) ) + 1 ( H Y ( I , J , 2 ) / 2 . ) * ( T 2 (I.J+ 1 . K ) - 1 2 ( I . J . K ) ) + 8 ( 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 ) ) • 2 ( A i Z ( I . J ) ) * ( T 3 ( I , J , K - 1 ) - T 3 ( I , J . K ) ) -3 ( B 3 * A R Z ( I.J) * C Z / S K A Y ) » (T 3 ( 1 , J . K ) - A f l B ( K ) ) ) / 4 ( ( E 4 ( I , J , K ) / 2 . ) * P E Y ( 1 , J , K ) ) + T ( I , J , K ) GC 1 C 1 0 0 0 3 2 0 T N ( I . J . K ) = ( ( A B X ( J . I , 1 ) / 2 . ) * ( T 1 ( 1 - 1 , J . K J - T 1 ( I . J . K ) ) • 1 ( 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 . ) * ( 1 2 (I,J+ 1 , K ) - T 2 ( I , J . K ) ) • 8 (ABY ( 1 , J . 2 ) * H 4 ( K ) * Y 2 / E K A Y / 2 . ) * ( A H B ( K ) - T 2 ( I . J . K ) ) • 3 ( I f Z ( I . J ) ) * ( T 3 J I . J . K - 1 ) - T 3 ( I , J , K ) ) -4 ( B 3 * A E Z ( I , J) * E Z / B K A Y ) * ( T 3 ( I . J . K ) - A H B (K) ) ) / 5 ( ( S 4 ( I , J , K ) / 2 . ) * P M ( I , J , K ) ) • T ( I . J . K ) G C 1 C 1 0 0 0 3 2 1 T N ( l . J . K ) = ( ( A B X ( J , I . 1 ) / 2 . ) * (T 1 ( 1 - 1 , J . K ) - T 1 ( I . J . K ) ) -1 ( E l ( K ) * A F X ( J . 1 , 1 ) * X 1 / B K A Y / 2 . ) * ( 1 1 ( 1 , J . K ) - AHB ( K) ) • 8 ( A E Y ( I . J , 2 ) *H 4 ( K ) * Y 2 / F K A Y / 2 . ) * ( A H B ( K ) - T 2 ( I , J , K ) ) + 2 ( A F Y ( I . J , 2 ) / 2 . ) * ( T 2 ( I . J + 1 . K J - T 2 ( I . J . K ) ) • 3 ( A F Z ( I . J ) ) * ( T 3 ( I , J , K - 1 ) - T 3 ( I , J , K ) ) -4 ( H 3 * A E Z ( I , J ) * E Z / R K A Y ) * ( T 3 ( I , J , R ) - A H B (K) ) ) / 5 ( ( £ 4 ( 1 , J . K ) / 2 . ) * P H Y ( I , J , K ) ) • T ( I . J . K ) G C T C 1 0 0 0 3 2 2 T N ( l . J . K ) = ( ( A R X ( J , I , 2 ) / 2 . ) * (11 (1 + 1 , J . K ) - T 1 ( I . J . K ) ) • 2 ( * 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 ( I , j + 1 , R ) - T 2 ( I , J , K ) ) * 2 ( A B Z ( I . J ) ) * ( T 3 ( I , J , K - 1 ) - T 3 ( I , J , K ) ) -3 < B 3 * A E Z ( I , J ) * E Z / B K A Y ) » (T 3 ( I , J , K) - A H B ( K ) ) ) / tl ( (£<4 ( I , J , K ) / 2 . ) * P H Y ( I . J . K ) ) • T ( I . J . K ) GO 10 1 0 0 0 3 2 3 m | I , J , K ) = ( ( A B X ( J . 1 , 1 ) / 2 . ) * (T 1 ( 1 - 1, J . K ) - T l ( I . J . K ) ) 1 ( A I X ( J , I , 2 ) / 2 . ) * ( 1 1 ( I * 1 . J , K ) - 1 U I , J , K ) ) + . 2 ( A F Y ( I , J , 1 ) / 2 . ) * ( 1 2 ( I , 0 - 1 , K ) - T 2 ( I , 0 , K ) ) * 2 ( I F Y ( I , 0 , 2 ) / 2 . ) * ( T 2 ( I , 0 + 1 , K ) - T 2 ( I , J , K ) ) • 3 ( A F Z ( I . J ) ) • ( 1 3 ( I , J , K - 1 ) - T 3 ( I , J , K ) ) - s 4 ( E 3 * A B Z ( I , J ) » C Z / E K A Y ) * ( T 2 ( I , J , K ) - A H E ( K ) ) ) / 5 ( | S U ( I , J . M / 2 . ) * P H ! ( I , J , K ) ) » I ( I , J , K | GO 1 0 1 0 0 0 3 2 4 T N ( I , J , K ) = ( ( A B X ( J , I , 1 ) / 2 . ) * (T 1 ( I - 1 , J , K ) - T 1 ( 1 , J , K) ) 1 ( E l (R) * A B X ( 0 , 1 , 1 ) * X 1 / H K A Y / 2 . ) * (1 1 ( I , J , K ) - A H B (K) ) • 2 ( * F Y ( I , J , 1 ) / 2 . ) * ( 1 2 ( I , J - 1 , K ) - T 2 ( 1 , J , K ) ) * 2 ( A I Y ( I , 0 , 2 ) / 2 . ) * ( 1 2 ( I , 0 * 1 , K ) - T 2 ( I , J , K ) ) • 3 ( A B Z ( I , 0 ) ) * ( 1 3 ( I , 0 , K - 1 ) - T 3 ( 1 , 0 , K ) ) -4 ( H 3 * A B Z ( 1 , 0 ) * E Z / B K A Y ) » ( T 3 ( I , J , K) - AHB ( K ) ) ) / 5 ( (£4 ( I , 0 , K ) / 2 . ) * P H Y ( I , J . K ) ) + T ( 1 , J , K ) GO 1 0 1 0 0 0 3 2 5 T N ( I , J , K) = ( ( A B X ( 0 , 1 , 2 ) / 2 . ) * (T 1 ( 1 + 1 , 0 , K ) - T l ( 1 , J , K ) ) • 1 ( 1 J Y ( 1 , 0 , 1) / 2 . ) * ( T 2 ( 1 , 0 - 1 , K ) - T 2 ( I , 0 , K ) ) -2 ( H 2 ( K ) * A E Y ( I , J , 1 ) * Y 1 / B K A Y / 2 . ) * ( 1 2 ( I , J , K) - A K 3 ( K ) ) • 3 ( A E Z ( I . J ) ) * ( 1 3 ( 1 , 0 , K - 1 ) - T 3 ( 1 , 0 , K ) ) -4 ( E 3 * A B Z ( 1 , J ) * E Z / F K A Y ) * ( T 3 ( 1 , 0 , K) - AHB (K) ) ) / 5 ( ( S 4 ( 1 , J , K ) / 2 . ) * P E Y ( I , J . K ) ) • T f l . J . K ) GO 1 0 1 0 0 0 3 2 6 T N ( I . 0 , K ) = ( ( A B X ( 0 , I , 1 ) / 2 . ) » |T 1 ( I - 1 , J , K ) - 1 1 ( I , J , K ) ) • 1 ( A FX ( 0 , 1 , 2 ) / 2 . ) * ( 1 1 ( 1 + l . J . K ) - T l ( 1 , 0 , K ) ) • 2 ( A F Y ( I , J , 1 ) / 2 . ) * ( T 2 ( I , 0 - 1 , R ) - T 2 ( I , 0 , K ) ) -3 ( B 2 (K) * A E Y ( I , J , 1) *Y 1 / B K A Y / 2 . ) * ( 1 2 ( I , J . K ) - A M B (K) ) • 4 ( A F Z ( I , J ) ) * ( T 3 ( 1 , 0 , K - 1) - T 3 ( 1 , 0 , R ) ) -5 ( K 3 » A F Z ( I , J ) * D Z / B F A Y ) * ( 1 3 ( 1 , 0 , R ) - AHB ( K ) ) ) / 6 ( (£4 ( I , J , K ) / 2 . ) * P B Y ( I , J , K ) ) • T ( I . J . K ) GO 1 0 1 0 0 0 3 2 7 T N ( 1 , 0 , K ) = ( ( A E X ( 0 , 1 , 1 ) / 2 . ) * (T 1 ( 1 - 1 , 0 , K ) - T l ( 1 , J , K ) ) 1 ( E 1 ( K ) * A B X ( 0 , 1 , 1 ) * X 1 / B K A Y / 2 . ) * ( 1 1 ( I , 0 , K ) - A B B (K) ) 2 ( * F Y ( I , J , 1 ) / 2 . ) * ( T 2 ( I , 0 - 1 , K ) - T 2 ( I , J , K ) ) -3 ( H 2 ( K ) * A F Y ( I , J , 1 ) * Y 1 / £ h A Y / 2 . ) * ( 1 2 ( 1 , 0 , K ) - A f l B ( K ) ) • 4 ( A E Z ( I , J ) ) * ( 1 3 ( 1 , 0 , R - 1 ) - I 3 ( 1 , 0 , K ) ) -5 ( B 3 * A B Z ( I , J ) * E Z / R K A Y ) » ( T 3 ( X , J , K ) - A M E ( K ) ) ) / 6 ( < S 4 ( I , J , K ) / 2 . ) * P H Y ( I , 0 , K ) ) • T ( I . J . K ) GO 1 0 1 0 0 0 3 2 8 I N ( I . J . K ) = ( (ABX ( J . I , 1 ) ) * ( T 1 ( I - 1 , J , K ) - T 1 ( I . . J , K ) ) • 1 ( A F X ( J , I , 2 ) ) * ( 1 1 ( 1 * 1 , 0 , R ) - T 1 ( 1 , 0 , K ) ) • 2 ( A F Y ( 1 , J , 1 ) ) * ( 1 2 ( 1 , 0 - 1 , K ) - T 2 ( I , J , K ) ) • 2 ( A F Y ( I , 0 , 2 ) ) * ( T 2 ( 1 , J * 1 , K ) - T 2 ( I , J , K ) ) 6 - (H7 (K) * A R E A 1) * ( T l ( I , J , K ) - A F B ( K ) ) 7 • (H5 ( K ) * A B E A U ) * ( AME ( K ) - T 2 ( I , 0 , K ) ) • 2 A B Z ( I , J ) * ( 1 3 ( I , J , K - 1 ) - 2 . * T 3 ( I , J , K ) + T 3 ( I , J , K • 1 ) ) ) / 3 ( ( £ 4 ( I , J , K ) / 2 . ) * P E Y ( I , J , K ) ) + 1 ( 1 , J . K ) G C 1 C 1 0 0 0 3 2 9 T N ( 1 , J , K ) = ( <ABX ( J . I . 1 ) / 2 . ) • ( 1 1 ( 1 - 1 , J . K ) - T 1 ( I . J . K ) ) • 1 ( J E X ( J . I , 2 ) / 2 . ) * (1 1 ( 1 * 1 , J . K ) - T l ( 1 , J . K ) ) • 2 ( A F Y ( I . J , 1 ) / 2 . ) * ( 1 2 ( I . J - 1 . K ) - T 2 ( I , J . K ) ) • 2 ( A E Y ( I . J , 2 ) / 2 . ) * ( 1 2 ( I , J + 1 , K ) - T 2 ( I , J . K ) ) 7 - (H7 ( K ) * A B I A 1 / 2 . ) * (T 1 ( I . J . K ) - A H E (K) ) e f (H5 (K) * A B E A 4 / 2 . ) * ( A B B ( K J - 1 2 ( I . J . K ) ) • 3 ( A E Z ( I . J ) ) * ( 1 3 ( I , J , K - 1 ) - T 3 ( I , J . K ) ) -4 ( H 3 » A B Z ( I , J ) * E Z / B K A Y ) * ( T 3 ( I , J , K ) - A H B (K) ) ) / 5 < < S M I , J , K ) / 2 . ) *Pfn > • T ( I , 0 , K ) GC 3 0 1 0 0 0 3 3 0 I N ( I , 0 , K ) = ( ( A E X ( 0 , 1 , 1) ) * <T 1 ( 1 - 1 . 0 , K ) - T l ( I . J . K ) ) -1 ( E 7 ( K ) * A B E A 5 ) * ( T I ( I . J . K ) - A H B (K) ) • 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 2 ( I , J + 1 , K ) - I 2 ( I , J , K ) ) 5 - (H6 ( K ) * A B E A 2 * S I N ( T E ) ) * ( T 1 ( I . J . K ) - A H B ( K ) ) 6 + (H6 (K) * A E E A 2 * C O S ( I B ) ) * ( A H B (K) - 1 2 ( I . J . K ) ) • 3 ( A B Z ( 1 , 0 ) ) * ( 3 3 ( I , J , K - 1 ) - 2 . * T 3 ( I , J . K ) * T 3 ( I , J . K * 1 ) ) ) / 4 ( (£4 ( I , J . K ) / 2 . ) * P B Y ( I , J . K ) ) • T ( 1 , J , K ) GO TO 1 0 0 0 3 3 1 I N ( I , J , K ) = ( ( A B X ( J , I , 1 ) / 2 . ) * ( T 1 ( 1 - 1 , J , K ) - T 1 ( Z . J . K ) ) -1 ( H 7 ( K ) * A E £ A 5 / 2 . ) • ( T 1 ( I , J , K ) - A H E ( K ) ) • 2 ( 1 1 Y ( I . J , 1 ) / 2 . ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I , J . K ) ) • 2 ( A E Y ( I , J , 2 ) / 2 . ) * ( T 2 ( I , J * 1 , K ) - T 2 ( l . J . K ) ) 6 - (H6 ( K ) * A B E A 2 * S I R ( T H J / 2 . ) * ( T l ( I . J . K ) - A H B ( K ) ) 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 ( A F Z ( I . J ) ) * ( T 3 ( I , J , K - 1 ) - T 3 ( 1 , J , K ) ) -4 ( E 3 * A B Z ( I , J ) * E Z / B K A Y ) * ( T 3 ( 1 , 0 , K) - A H B (K) ) ) / 5 ( ( S 4 ( I , 0 , K ) / 2 . ) * P S Y ( I , J , K ) ) • T ( I , 0 , K ) GC T C 1 0 0 0 3 3 2 T N ( I . J . K ) = ( ( A B X ( J , I , 1 ) ) * ( T l ( 1 - 1 , J . K ) - T l ( I , J , K ) ) + 1 ( A I X ( J , I , 2 ) ) * ( T 1 ( I t 1 , J , K ) - T 1 ( I , J , K ) ) + 2 ( A E Y ( I , J , 1 ) ) * ( T 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) -3 (EE (K) * A E Y ( I , J , 1 ) * Y 3 / E K A Y ) * ( T 2 ( I , J , K ) - A H E ( K ) ) • 4 ( A I Z ( I , J ) ) * ( T 3 ( I , J . K - 1 ) - 2 . * T 3 ( I , J , K ) + 1 3 ( I , J , K * 1 ) ) ) / 5 ( ( S 4 ( I , J , K ) / 2 . ) * P E Y ( I , J , K ) ) • T ( I . J . K ) G C I O 1 C 0 C 3 3 3 T N ( I , J , K ) = ( ( A f i X ( J , I , 1 ) / 2 . ) * ( " l ( I - l » 0 , K ) - T l ( 1 , 0 , K) ) • 1 ( A I X ( J , I , 2 ) / 2 . ) * ( I 1 ( I 4 1 , J , K ) - T 1 ( I , J , K ) ) + 2 ( A F Y ( I , J , 1 ) / 2 . ) * ( I 2 ( I , J - 1 , K ) - T 2 ( I , J , K ) ) -3 ( E E ( K ) * A E Y ( I , J , 1 ) * Y 3 / E K A X / 2 . ) * (12 ( I , J , K ) - A B £ (K) ' ) • 4 ( A I Z ( I . J ) ) » ( 1 3 ( I , J , K - 1 ) - T 3 ( I , J , K ) ) -5 ( H 3 * A E Z ( I , J ) * E Z / R K A Y ) * ( 1 3 ( I , J , K ) - A H B ( K ) ) ) / 6 ( ( £ « ( ! , 0 , K ) / 2 . ) * P H Y ( I , 0 , K ) ) + T ( I , J , K ) GO 3 0 1 0 0 0 3 3 4 T N ( I , J , K ) = ( ( A E X ( J . I , 1) ) » (T 1 ( 1 - 1 , 0 , K) - 3 1 ( I . J . K ) ) -1 ( B 6 ( K ) * A F X ( J , I , 1 ) * X 2 / B K A Y ) * ( T 1 ( I , 0 , K ) - AHD (K) ) 8 + ( A B Y ( 1 , 0 , 2 ) * H 6 ( K ) * Y 4 / E K A Y ) * ( A H E ( K ) - T 2 ( I , J . K ) ) + 2 ( i I Y ( I , J , 2 ) ) * ( T 2 ( I , J 4 1 , K ) - T 2 ( I , 0 , K ) ) + 3 ( A F Z ( 1 , 0 ) ) * ( I 3 ( I , J , K - 1 ) - 2 . * T 3 ( I , J , K ) + T 3 ( I , J . K + 1 ) ) ) / 4 ( ( £ 4 ( I , J , K ) / 2 . ) * P H Y ( I , J , K ) ) • T ( I . J . K ) GO T C 1 0 0 0 3 3 5 T N ( 1 , J , K ) = ( ( A E X ( J , I , 1 ) / 2 . ) * (T 1 ( 1 - 1 , J , K ) - T 1 ( I . J . K ) ) -1 ( B 6 ( K ) * A E X ( J . I , 1 ) * X 2 / E K A Y / 2 . ) * ( T 1 ( I . J . K ) - A B B ( K ) ) • 2 ( A F Y ( I , J , 2 ) / 2 . ) * ( T 2 ( I , 0 + 1 , K ) - I 2 ( I , 0 , K ) ) 8 * ( A E Y ( 1 , 0 , 2 ) *H6 (K) * Y 4 / B K A Y / 2 . ) • (AHB ( K ) - 1 2 { 1 , 0 , K ) ) • • 3 ( A E Z ( I , 0 ) ) * ( T 3 ( I , 0 , K - 1 ) - T 3 ( I , 0 , K ) ) -4 ( B 3 * A E Z ( 1 , 0 ) * r z / B K A Y ) * ( T 3 ( I . J . K ) - AHB ( K ) ) ) / 5 ( ( £ 4 ( I , J , K ) / 2 . ) • P B Y ( I . O . K ) ) • T ( I . J . K ) G C I C 1 0 0 0 3 3 6 T N ( I . J . K ) = ( ( A B X ( J , I , 1) ) * (T 1 ( I - I . J . K ) - T l ( I . J . K ) ) -1 ( H 1 (K) » AB E A6) * ( 1 1 ( 1 , J . K ) - A H B (K) ) • 2 ( A F Y ( I , J , 2 ) ) * ( T 2 ( I , J * 1 , K ) - T 2 ( I , J , K ) ) 5 • ( H 6 ( K ) * A B E A 3 * C O S ( T H ) ) * ( A E E ( K ) - T 2 ( I . J . K ) ) 6 - (H6 (K) * A E E A 3 * S I N ( T E ) ) * ( T 1 ( I , J , K ) - A H B (K) ) '* 3 ( A F Z ( I . J ) ) * ( T 3 ( I , J , K - 1 ) - 2 . « T 3 ( I , J , K ) * T 3 ( I , J . K * 1 ) ) ) / 4 ( ( £ 4 ( I , J , K ) / 2 . ) * F H Y ( I , 0 , K ) ) * T ( I . J . K ) GO TO 1 0 0 0 3 3 7 TU ( I , J , K ) = ( ( A E X ( J . I . 1 ) / 2 . ) * (T 1 ( 1 - 1, J.K) - T l ( I . J . K ) ) -1 ( B 1 ( K ) * A E E A 6 / 2 . ) * ( 1 1 ( I . J . K ) - A E B (K) ) • 2 ( A F Y ( I , J , 2 ) / 2 . ) * ( T 2 ( I , J * 1 , K ) - T 2 ( I , J , K ) ) 6 • (H6 ( K ) » A B E A 3 * C 0 S ( T H ) / 2 . ) * <*BB ( K ) - T 2 ( I , J . K ) ) 7 - (H6 (K) * A E E A 3 * S I N (T E) / 2 . ) * (11 ( I , J , K) - AHE(K) ) + 3 | A B 2 ( I , J ) ) * ( T 3 ( I , J , K - 1 ) - T 3 ( I , J , K ) ) -4 ( H 3 * A B 2 ( I , J) » E 2 / B K A Y ) » ( T 3 ( I , J , K ) - A H B (K) ) )/ 5 ( ( S 4 ( I , J , K ) / 2 . > * P B Y ( I , J , K ) ) • T ( I . J . K ) GO TO 1 0 0 0 3 3 8 T H ( I , J , K ) = ( ( A B X ( J . I , 1 ) ) * ( T 1 ( 1 - 1 . J . K ) - T 1 ( I , J . K ) ) -1 ( B 1 (K) * A E X ( J . J . 1) * X 1 / B K A Y ) * ( T l ( I . J . K ) - A H E ( K ) ) • 2 ( A E Y ( I , J , 1 ) ) * ( T 2 ( I . J - 1 , K ) - T 2 ( I . J . K ) ) -3 ( B 5 ( K ) * A E Y ( I , J , 1 ) * Y 3 / B K A Y ) * ( T 2 ( I , J . K ) - A H B (K) ) • 4 ( l I 2 ( l , J ) ) * ( T 3 ( I , J , K - 1 ) - 2 . • T 3 ( I , J , K ) + T 3 ( I , J , K + 1 ) ) ) / 5 ( ( S 4 ( I , J , K ) / 2 . ) * F B Y ( I , J . K ) ) +T ( I . J . K ) GO TO 1 0 0 0 3 3 9 TH ( I , J . K ) = ( ( A E X (J, 1 , 1 ) / 2 . ) * (T 1 ( 1 - 1 , J . K ) - T 1 ( 1 , J . K ) ) -1 ( B I ( K ) * A E X ( J . I , 1) * X 1 / B K A Y / 2 . ) * (T 1 (I ,J.K) - AHB(K) ) • 2 ( 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 ( I , J, 1) * Y 3 / E K A Y / 2 . ) * ( 1 2 ( I . J . K ) - A H B (K) ) • 4 ( A E Z ( I . J ) ) * ( T 3 ( I , J , K - 1 ) - T 3 ( I , J , K ) ) -5 ( B 3 * A B Z ( 1 , J) * E 2 / E K A Y ) * ( T 3 ( I , J , K ) - A H B (K) ) ) / 6 ( ( S 4 ( I , J , K ) / 2 . ) * P E Y ( I . J . K ) ) • T ( l . J . K ) GO 1 0 1 0 0 0 1 0 0 0 C O K T I N O E B E I U R N E N E C C C S D E E C C T I N E F O B S O L V I N G A S Y S T E H C F I I N E A B C S I B U I 1 A N E O U S E Q U A T I O N S H A V I N G A T B I E l AGON AL C C C I E I J C I E N T H ATI! I X S D E E O 0 1 I N E T B I D A G ( I F . L . A . B . C . D . V ) D I E E N S I C N A ( 1 ) , E ( 1 ) , C ( 1 ) , D ( 1 ) , V ( 1 ) , B E T A ( 1 0 1 ) , G AHH A ( 1 0 1 ) C C C C C H P O I E I N I E F H E D I I T E A E R B A Y S O F B E T A AND G A H H A C E E I A ( I F ) = E ( I E ) G A B E A ( I f ) = D ( I E ) / B E T A ( I F ) I F H = I F +1 DO 1 I = I F P 1 , L B E T A ( I ) * E ( I ) - A (3) * C ( 3 - 1 ) / B E T A ( I - 1 ) 1 G A E E A ( l ) = ( E ( I ) - A ( I ) * G A E E A ( 1 - 1 ) ) / E E T A ( I ) C C C O B E C T E F I N A L S C L D T I C N C F V E C T C E V C V ( I ) =G AHB A ( L ) L A S T = L - I F DO 2 K = 1 , L A S T I = L - K 2 V ( I ) = G A M H A ( I ) - C ( I ) * V ( 1 + 1 ) / B E T A ( I ) B E T U I N E N E C C S U E B C D T I N E TO C A L C U L A T E T E E D I F F E B E K T A B E C A K E V C I U E E T E B B S I O B T H E E L E M E N T S C D I S C B E T I Z A T I O N O F T H E C A S T I N G I S A L E E A D Y B U I L T I N C I H E O U G H T B I E A T A S T A T E M E N T S C 210 S U E F O D T I N E A E E V C L D I B E N S I O N ARX ( 1 6 , 1 1 , 2 ) , A E Y ( 1 1 , 1 6 , 2 ) ( A B Z ( 1 1 , 1 6 ) E I E I N S I C N XY ( 1 0 , 2 ) , Y X ( 1 6 , 2 ) D I B E N S I O N XX ( 1 6 ) , Y Y ( 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 C O B B O N / C 2 / S 4 ( 1 0 , 1 6 , 9 1 ) C C E E C N / C 3 / N T Y P E ( 1 0 , 1 6 , 9 1 ) , L F S ( 1 0 , 1 6 , 9 1 ) C O E E C N / C 4 / 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 E A 2 , A B E A 3 , A F E A 4 , A R E A 5 , A B E A 6 D A T A X Y / 0 . , 1 . 9 7 5 , 8 * 1 . 4 2 5 , 2 * 1 . 9 7 5 , 7 * 1 . 4 2 5 , 0 . / E A T A 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 , 1 9 * 1 . 7 5 , 2 * 1 . 1 9 , 0 . / T H - . 8 8 7 4 1 L L = L - 1 flfl=E-1 N N = N - 1 C C A I C O I A T I N G T H E C O C E E I N A T ES O F T H E X AND Y C G F I C E O I N T S X X ( 1 ) = 0 . DO 1 J = 2 , B X X ( J ) = Y X ( J - 1 , 2 ) + Y X ( J , 1 ) + X X ( J - 1 ) 1 C O N T I N U E YY ( 1 ) = 0 . CO 2 I = 2 , L YY <I) = XY ( 1 - 1 , 2 ) +XY ( I , 1) + Y Y ( I - 1 ) 2 C O N T I N U E DC 1 3 0 0 1 = 1 , L L DO 1 4 C 0 J = 1 , « A E X ( J , I , 2 ) = (YX ( J , 1) +YX ( 0 , 2 ) ) * C Z / ( X Y ( I , 2 ) + XY ( 1 + 1 , 1 ) ) 1 4 0 0 C O N T I N U E 1 3 0 0 C O M I N U E DO 1 5 0 0 1= 2 , L CO 1 6 0 0 0 = 1 , B A B X ( 0 , I , 1 ) = A B X ( 0 , 1 - 1 , 2 ) 1 6 0 0 C O M I N U E 1 5 0 0 C O N T I N U E DC 1 7 0 0 J = 1 , B B CC 1 8 0 0 1= 1 , L A B Y ( 1 , 0 , 2 ) = ( X Y ( I , 1 ) + X Y ( 1 , 2 ) ) * D Z / ( Y X ( J , 2 ) • Y X ( J + 1 , 1 ) ) 1 8 0 C C O N T I N U E 1 7 0 0 C O M I N U E CC 1 9 C 0 0 = 2 , M DO 2 0 0 0 1 = 1 , L A B Y ( I , J , 1 ) = A E Y ( I , J - 1 , 2 ) 2 0 0 0 C O N 1 I N U E 1 9 0 C C O N T I N U E 4 D C 2 1 0 0 1 = 6 , L L A B X ( 4 , 1 , 2 ) = A B X ( 4 , 1 , 2 ) / 2 . 2 1 0 0 C C M I N U E CC 2 2 0 0 1= 7 , L A F X ( 4,1, 1 ) = A B X ( 4 , 1 - 1 , 2 ) 2 2 0 0 C O N T I N U E 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 ( 4 , 2 ) » E Z / R K A Y AB E A 4 = X Y ( 6 , 2 ) * D Z / R K A Y 211 A E E A 5 = Y X ( 5 , 1 ) * C Z / R K A Y A B 1 A 2 = Y X ( 5 , 2 ) • D Z / E K A Y / S I K ( T H ) A B 1 1 3 = Y X ( 9 , 1 ) * D Z / B K A Y / S I * ( T H ) A B I A 6 = Y X ( 9 , 2) * E Z / B K A Y X1 = X Y ( L - 1 , 2 ) + X Y (1 , 1 ) X 2 = XY ( 7 , 2 ) + XY ( 8 , 1 ) Y 1 = I X ( B - 1 , 2 ) + Y X ( f i , 1 ) Y 2 = Y X ( 1 , 2 ) + Y X ( 2 , 1) Y 4 = Y X ( 7 , 2 ) + Y X ( 8 , 1 ) Y 3 = Y X ( 3 , 2 ) • Y X ( 4 , 1 ) y . DO 2 3 0 0 1 = 1 , L CO 2 4 0 0 0 = 1 , f l A B Z ( I , J ) = ( X I ( 1 , 1) + X Y (1 , 2 ) ) * (YX ( J , 1) + Y X ( 0 , 2 ) ) 2 4 0 0 C O N T I N U E 2 3 0 0 C O N T I N U E DO 2 5 0 0 1 = 7 , L A B Z ( I , 4 ) = A B Z (1,1)/ 2 . 2 5 0 0 C C S 1 I N U E A E Z ( 6 , 1 ) = A B Z ( 6 , 4 ) - ( Y X ( 4 , 2 ) * X Y ( 6 , 2 ) ) A E Z ( 6 , 5 ) = ( X Y (6 , 1) * Y X ( 5 , 1) ) • ( Y X ( 6 , 2 ) / 2 . ) * (XY ( 6 , 1 ) + 1 XY ( 6 , 1 ) • XY ( 6 , 2 ) ) A B Z ( 7 , 6 ) = ( Y X ( 6 , 1) +YX ( 6 , 2 ) ) * (XY (7 , 1) +XY ( 7 , 2 ) ) / 2 . A B Z ( 8 , 7 ) = ( Y X ( 7 , 1) +YX ( 7 , 2 ) ) * (XY (8, 1) +XY ( 6 , 2 ) ) /2. A E Z ( 9 , 8 ) = ( Y X ( 8 , 1) + YX ( 6 , 2 ) ) * ( X Y ( 9 , 1) + XY ( 9 , 2 ) ) / 2 . A E Z ( 1 C . 9 ) = ( Y X ( 9 , 1 ) + Y X ( 9 , 2 ) ) * X Y ( 1 0 , 1 ) - ( 0 . 5 * X Y ( 1 0 , 1 ) * Y X ( 9 , 1) ) DO 2 6 0 0 1=1,1 CC 2 7 0 C J = 1 , H A B Z ( 1 , 0 ) = A E Z ( 1 , 0 ) / I Z 2 7 0 0 C O N T I N U E 2 6 0 0 C C F T I N O E CO 1 1 C 0 1=1,1 DC 1 1 0 0 J = 1 , H DO 1 1 C 0 K = 2 , N N S 4 < I , 0 , K ) = A B Z ( I , J ) * E Z * C Z » 2 . / ( D T * B K A Y ) 1 1 0 0 C O N T I N U E D C 1 2 0 0 1= 1 , L DO 1 2 0 0 0 = 1 , B S 4 ( 1 , 0 , N ) = S 4 ( 1 , 0 , N N ) / 2 . 1 2 0 0 C O N T I N O E F . E T D E K E N D C C C C S O E B C C T I N E 1 0 P E I K T T H E B A T E I X O f T E B P E B A T O E E S C C c S O E E O C T I N E O U 1 P U T ( 1 ) D I E I N S I C N T ( 1 0 , 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 C O E H 0 N / C 1 3 / T A U 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 1 0 0 F O F E A T ( ' 1 ' , 1 0 X , ' T E B P E B A T U B E S AT T E E E N D O F T I M i ' , 1 F 8 . 1 , 1 X , ' S E C O N D S ' ) Z T = E Z * F 1 C AT ( N - 1) H B I I E ( 6 , 1 4 5 ) Z T 1 4 5 F O F E A 1 ( 1 0 X , • S I Z E O F T H E I N G C T I N Z C I E E C T I O N • , F 1 0 . 2 , • C B S •) 1 5 0 F O E E A T ( 5 X , * K = • , 2X , 1 3 ) B B I 1 E ( 6 , 1 8 5 ) 1 8 5 F O E E A T ( 2 X , ' C E N T R E I I N E 1 E E E E £ A T U F E S •) S B I T I ( 6 , 1 8 0 ) (T ( 1 , 9 , K ) , K = 1 , N ) W E I 1 E ( 6 , 1 9 0 ) 1 9 0 P O I E A T ( 1 X , / ) 1 8 0 F O E E A T ( 1 X , 1 1 ( F 1 0 . 1 , 1 X ) ) K1 = ( N / 2 ) * 1 W B I 1 E ( 6 , 1 5 0 ) K 1 E C 2 1 0 J = 1 , H K B I 1 E ( 6 , 1 8 0 ) (1 ( I , J . M ) , 1 = 1 , 1 ) 2 1 0 C O M I N D E W B I 1 E ( 6 , 1 9 0 ) R 2 = F CO 1C9 K = 1 , N K K = K - K + 1 I F < l ( 1 , 9 , N K ) . L E . I S O L ) K 2 = N K 1 0 9 C C M 1 N 0 E I F ( K 2 . E C . N ) K 2 = K 2 - 1 K 3 = K 2 + 1 C C 3 C 0 K = K 2 , K 3 W R I T E ( 6 , 1 5 0 ) K DO 3 1 C J = 1 , H WR H E ( 6 , 1 8 0 ) (T ( 3 , J . K ) , 1 = 1 , L ) 3 1 0 C O N 1 I N E E 3 0 0 C O M I N U E B E I C R N Et<T c C C H E A T 1 E A N S E E 5 C O E E F I C I E N 1 S U B E C U 1 I N E S . T H E C V A B I C O S S U E E C U T I N E S H I T H B C O G H b 7 A B E F O B C D E S C E I E I N G E T C A l I K E V A R I O U S F C R 1 I O K C F C T H E S U F F A C E C F T H E C A S T I N G . T H E Y A L L H A V E C B E E S S I I F O E S Y B K E 1 E I C A I C C C 1 I N G C C c F U N C 1 I O N H 1 ( 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 T C 10 H 1 = . 5 B E 1 0 R N 10 H 1 = 0 . 3 4 1 7 8 * E X P ( - ( ( ( K - 1 ) * D Z - ( D Z / 2 . ) ) * . 0 2 4 3 5 ) ) E E 1 C E N E N I F D S C 1 1 0 H H 2 ( 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 T O 10 H2 = . 5 B E I C E N 10 H2 = 0 . 3 4 1 7 8 * E X P ( - ( ( ( K - 1 ) * D Z - ( D Z / 2 . ) ) * . 0 2 4 3 5 ) ) E E 1 C E N I N C F U N C T I O N 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 T C 10 H 4 = . 5 B E T CBN 1 0 H 4 = 0 . 3 4 1 7 6 * E X P ( - ( ( ( K - 1 ) * D Z - ( D Z / 2 . ) ) * . 0 2 4 3 5 ) ) B E T U B N 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 I F ( K . G T . 1 3 ) G O T O 10 H 5 = . 5 E E 1 U F N 10 H 5 = C . 3 4 1 7 6 * E X P ( - ( ( ( K - 1 ) * D Z - ( D Z / 2 . ) ) * . 0 2 4 3 5 ) ) E E T O E N EN E F U K C T I C N 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 T O 10 ' H 6 = . 5 E E T O E N 10 H 6 = C . 3 4 1 7 8 * E X P ( - ( ( ( K - 1 ) * D Z - ( D Z / 2 . ) ) * . 0 2 4 3 b ) ) E E T O E N ' E N E 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 T O 10 B 7 = . 5 E E T O E N 10 H 7 = C . 3 4 1 7 8 * E X P ( - ( { ( K - 1 ) * E Z - ( D Z / 2 . ) ) * . 0 2 4 3 5 ) ) E E I D E N E N E C C c F U N C T I O N A H B ( K ) A B E = 5 . B E T D F N 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 S U E F O O T I N E I N I T I A (T , T 1 , T 2 ) E I BE NS I O N T ( 1 0 , 1 6 , 9 1) C O B B C K / C 4 / L , H , N I t = I - 1 HB = fi- 1 K N = N - 1 DO 1 0 0 1 = 1 , L L CO 100 J = 1 , H M DO 1 C 0 K = 2 , N N T ( I , J , K ) = T 1 1 0 0 C O N T I N U E DC 1 5 0 1 = 1 , L L DO 1 5 0 J = 1 , H B T ( I , J , 1) = T 1 T ( I , J , N ) = T 1 1 5 0 C C K T I N U E E C 1 6 0 K = 2 , N N E C 160 J = 1 , B B 1 ( 1 . J , K ) = T1 160 C O M I N U E EO 1 7 0 1 = 1 , L L E C 170 K = 2 , K N T ( I , B , K ) = T 2 1 7 0 C O I I I N U E DO 1 9 0 J = 1 , H H T ( I , J , 1 ) = T 1 I ( I , J , N ) = T 1 190 C O I T I F O E DO 2 1 0 1 = 1 , L L T ( I , E , 1 ) = T 2 T ( I , H , N ) = T 2 2 1 0 C O N T I N U E DO 2 2 0 K = 2 , N N T ( 1 , f l , K ) = T 2 T ( L , 1 , K ) = T 1 T ( I , E , K ) = T 2 2 2 0 C O B T I N O E X ( I . 1 , 1 ) = I 1 T ( L , H , 1) = T 2 T ( 1 , B , 1 ) = T 2 T ( L , 1 , N ) = T 1 I ( I , f l . N ) = T 2 T { 1 , H , N ) = T 2 E E T U F K E N E C c c c S O E B C O T I N E L A T H E T (L 1 , L 2 , B 1 , B 2 , K 1 ) C C S U B R C U T I N E TO E E L E AS E T H E L A T E N T H E A T O F S O L I D I F 1 C A H O N C A T T E E E I F F E E E N T N C E E S C T H I S A L S O C H A R A C T E R I Z E S T E E P H Y S I C A L S T A T E O F E A C H NODE C E E 1 N C L I Q U I D , B U S H Y C E S C U D E E G 1 0 N C D I B E N S I O N T ( 1 0 , 1 6 , 9 1) , T 1 ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 , 1 6 , 9 1 ) D I E E N S I C N T 3 ( 1 0 , 1 6 , 9 1 ) , T N ( 1 C , 1 6 , 9 1 ) C O E B O N / C 7 / T , T 1 , T 2 , T 2 , T N C O B E O N / C 2 / N T Y P E ( 1 0 , 1 6 , 9 1 ) , L E S ( 1 0 , 1 6 , 9 1 ) C C E H C N / C 8 / P H Y ( 1 0 , 1 6 , 9 1 ) 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 1 0 0 0 I = L 1 , L 2 CO 1 0 C 0 J = B 1 , H 2 DO 1 0 0 0 K = 2 , K 1 L N E = L P S ( I , 0 , K ) G C T O ( 1 0 , 2 0 , 1 0 0 0 ) , L N D C C H C D E I N I T I A L L Y I N L I C U I E C 10 I F (TN ( I , 3 , K ) . G E . T I I C ) GO T O 1 0 0 0 C C C H A N G E C E S T A T E C C D C E S F I N A l I E B P E N E DP I N BUSfcY Z O N E ? 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 T O C C B O D E E N D S DP I N H O S E Y B E G J C N C T N ( I , J , K ) = T L I C - ( I L I Q - T N ( I , J , K ) ) * E H Y L / P H Y f l 215 L P S ( I , J , K ) = 2 GO 1 0 1 0 0 0 c C R O D E H O V E S I N T O S O L I D C 1 0 T N ( I , J , K ) = T S C L - ( ( T L I Q - T N < I , 0 , K ) ) * E H Y L - ( T L I C - T S O L ) 1 * E E Y B ) / E H Y S L P S < I , 0 , K ) = 3 G C TO 1 0 0 0 C / • • C H C D I I K I T I A L L Y I K T H E B U S H Y E E G I O N C 2 0 I P ( TK ( I , 0 , K ) . G I . T S C L ) GC T C 1 0 0 0 C C N O D E B O V E S I N T O S C L I D C T K ( I , J , K ) = T S O L - ( T S O L - T K ( 1 , 0 , K ) ) * F H Y H / P H Y S I E S ( I , 0 , K ) = 3 1 0 0 0 C O N T I N U E F E T U F N EN I C C c S U E E C O T I R I O U T I N T ( I I , N N ) C C S U E B C O T I K E TO P E I N T C U T T H E T KB EE D I B E N S I O N A L C A E E A Y C O N T A I N I N G I NT E G E E N U H E E E S C D I B F N S I C N I I ( 1 0 , 1 6 , S 1) C O B K O N / C 4 / L , B , N E C 1 0 0 0 K= 1 , N , N N S B I T E ( 6 , 9 0 0 ) K 9 0 0 F O E B A T ( 5 X , • K = » , 2 X , 1 4 / / ) DO 6 C 0 0 = 1 , B H B I T I ( 6 , 8 5 0 ) ( I I ( I , J , K ) , 1 = 1 , L ) £ 5 0 F O E B A T ( 5 X , 11 ( J K , 2 X ) ) 8 0 0 C C M I N U E 1 0 0 0 C O N T I N U E E E 1 U B N ENE C C F U N C T I O N C P ( T ) C C F O K T I C N B C U T I N E C A L C U L A T E S T H I S P E C I F I C H E A T C C i Z I N C AT ANY E A E T I C U L A E T E H E E E A T U E E I N C O N I T S C F C A L / G . C C T K = I + 2 7 3 . C P = 0 . 0 8 1 8 4 + 0 . 0 3 6 7 * 1 . E - 0 3 * T K E E 1 U F N EN C C C S O E F C O T I N E P H Y P E P C C C A L C O I A T I S T H E P B C E U C T O F D E N S I T Y AND C S P E C I F I C H E A T F O E A l l T H E N O D E S AND C S T C B E S I T I N T H E ABB AY P K Y D I E E N S I O N T ( 1 0 , 1 6 , S 1) , 1 1 ( 1 0 , 1 6 , 9 1) , 7 2 ( 1 0 , 1 6 , 9 1 ) D I B E N S I O N 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 ) , L F S ( 1 0 , 1 6 , 9 1 ) C C E E C N / C 8 / P H Y ( 1 0 , 1 6 , 9 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 1 C C 0 1 = 1 , 1 DO 1 0 0 0 J = 1 , H E C 1 0 C 0 K = 2 , N I N C = 1 E S ( I , J , K ) GO 1 0 ( 1 0 , 2 0 , 3 0 ) , I . ND C C B C E E S A E O V E L I Q U I D U S C 10 P H Y (I , J , K ) = F H Y L GO 1 0 1 0 0 0 C C S C E E S I N T H E B U S H Y E E G I O N C 2 0 E H Y ( I , J , K ) = E H Y H GO 1 0 1 0 0 0 C C N O D E S I N T H E S O L I D R E G I C K C 3 0 P H Y ( I , J , K ) = C E ( I ( I , J , K ) ) * E E N S 1 0 0 0 C O K 1 I N U I R E T U R N END CC c C S D E E C U T I N E A I C N C E C S 0 E B C U 1 I N E F O E A D D I N G O N E X T E A S E T C F K O D E S C B E S U I 1 I N G I B O H T H E G E O W T H O F I E E I N G O I I N C T H E Z D I R E C T I O N . A L S C S O B E I N I T I A L I S A T I O N C c D I B E N S I O N T ( 1 0 , 1 6 , 9 1 ) , 1 1 ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 , 1 6 , 9 1 ) D I H E N S I O N 1 3 ( 1 0 , 1 6 , 9 1 ) , 1N ( 1 0 , 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 C O B H O N / C 3 / N 1 Y P E ( 1 0 , 1 6 , 9 1 ) , L F S ( 1 0 , 1 6 , 9 1 ) C O E E C R / C 5 / N C B , T E B P 1 C O B B O N / C 2 / S 4 ( 1 0 , 1 6 , 9 1 ) C C H A K E T H E T O P S L I C E I N T H E E E E V I C O S I N T E S V A L C AS T E E S E C O N D S L I C E E E O H I E E T O P F O E C I B I S 1 I H I S T E P C CO 10 K = 1 , N CO 10 J = 1 , B CO 10 1 = 1 ,L S C = » * 2 - K T ( I , J , N C ) = 1 ( I , J . N C - 1 ) L F S ( I , J , N C ) = L E S ( I , J , N C - 1 ) 10 C O N T I N U E KN=*-1 21 7 DO 15 K= 1 , 2 DO 15 1= 1 , L C C 15 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 ( 1 , J , N C - 1) 15 C C K T I N U E C C I N I T I A L I S A T I O N O F T H E N E W L Y A D D E D S L I C E C <• CO 20 1 = 1 , L DC 20 J = 1 , B 1 ( 1 , J , 1 ) = 1 ( 1 , 0 , 2 ) L E S ( 1 , 0 , 1 ) = L P S ( I . J , 2 ) 2 0 C 0 N 1 I N U E N= K • 1 B E 1 CBN E N E C C C O F Y I K G C E T E E 1 EM F E B A1UB E F I E L D IN 1 H E C C A S 1 I K G I N E I N A B Y F O E S U E S E C U E N T U S E I K C S T A F T I K G T H E P B O G E A H C S U E F C U T 1 N E F I L E I N ( N I ) D I M E N S I O N T ( 1 0 , 1 6 , 9 1 ) , 1 1 ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 , 1 6 , 9 1 ) D I B F N S I C N 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 C O I ! E O N / ' C « / L , M , ^ • C O ! ! E O N / C 3 / N T Y P E ( 1 0 , 1 6 , 9 1 ) , L P S ( 1 0 , 1 6 , 9 1 ) CO 1C K=1,N DC 10 J = 1 , B K B I I E ( N I ) ( T ( I , 0 , K ) , 1 = 1 , 1 ) , ( 1 F S ( I , 0 , K ) , I = 1 , L ) 10 C 0 K 1 I N U E B E 1 C F N E N E C c c S U E F C 0 T 1 N E G B A P H C C T H I S S D E E C U T I N E F 1 0 T S T E E C C N I O U E P B O E I L E S O F C 1 B E T I B P E E A T U R E E 1 E L E C D I B E N S I O N T ( 1 0 , 1 6 , y i ) , T 1 ( 1 0 , 1 6 , 9 1 ) , 1 2 ( 1 0 , 1 6 , 9 1 ) C I E E K S I O N XX ( 1 6 ) , Y Y ( 1 0 ) , Z 2 ( 1 6 , 1 0 ) D I H E N S I O N 1 3 ( 1 0 , 1 6 , 9 1 ) , 1 N ( 1 0 , 1 6 , 9 1 ) C I B E N S I O N Z P 1 ( 1 C 1 , 10) , Z E 2 ( 1 0 1 , 1 6 ) , X E ( 1 0 1 ) , Y E 1 ( 1 1 ) , 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) = 1 ( 1 , 9 , 0 ) 10 C O K T I N O E DO 20 1=1,H DC 20 0 = 1 , N ZE2 < J , I ) = T ( 1 , I , J ) 20 CONTINUE C C SCALIKG THE X AXIS (COBEESFONDS TO ORIGINAL C X AN I Y AXES ) C DYY1P=4. DYY2F=«. SY 1 = YY( 10) /EYY1E SY2 = XX ( 16) /CYY2E CO 15 I = 1 , L YF1 ( I ) = YY ( I ) /EYY IP 15 CONTINOE DC 16 J = 1 , f l YP2 (J) =XX ( J ) / D Y Y 2 P 16 CCMINOE SCALING THE X AXIS (ORIGINALLY 2 AXIS) DXX=CZ/4. SX=rXX*FLOAT (N-1) DXXI=U. XP(1) = 0 . NX = N - 1 DO 4C 1=1,NX XP ( I + l ) = XP ( I ) • CXX 40 CONTINUE CCNTCl'E OETAINEE EEOK SECTION FER PE KCICUL AE TC Y AXIS XZ ELANE C A I I FLCTEI ( ' X S I Z E• , 128) C A I I E L C 1 E L ( ' Y S 1 Z E , , 6 6 ) C A I I FFAME1 C A I I CNTCUB (XF,N,YP 1 ,L ,ZE 1, 10 1, T L I C , 3. , T L I C ) C A I I CNTCOF. (XP,N , YP 1 , 1 ,Z P1 , 1 01 , T S C L , 3 . ,TSOL) 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. C A I I E L C T ( X B , 0 . , - 3 ) C A I I XYFLAN C A I I CtJTLNE XH 1 = SY2 + 4 . C A I I E L C T ( X M 1 , 0 . , - 3 ) BETCRN EN C SUEFCDTINI OUTPT2 C C r K O N / C 1 C / T L I C , T S C l , D £ K S , D E K I 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 C O E E O N / C 6 / B 3 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 ) ( R I N F T ( I ) , 1 = 1 , 2 0 ) 10 F C E E AT (* 1 * , 2 6 X , 2 0 4 4 / / ) S E I T E ( 6 , 2 0 ) N O H B U N : • • / 20 F O I E A T ( 5 5 X , • E U N NO • , 2 X , I 4 / / / ) B B I T E ( 6 , 3 0 ) 3 0 F O E E A T ( 5 X , ' T H E B B O P H Y S I C A L P E C P E B T I E S ' / ) B B I T E ( 6 , 4 0 ) T L I C 4 0 F O B H A T ( 1 0 X , ' L I Q O I D O S T E E I E E A T U B E = • , I X , F 7 . 1 , 2 X , ' D E G C ) B B I T E ( 6 , 5 0 ) 7 S C L 5 0 F 0 F E A 1 ( 1 C X , ' S O L I D U S T E H F E B A T O E E = • , 1 X , F 7 . 1 , • D E G C ) BB H E ( 6 , 6 0 ) D E N I 6 0 F O E B 4 T ( 1 0 X , • C E N S I T Y O E T E E L I Q U I D = • , 1 X , F 5 . 1 , 1 X , • G / C H 3 • ) B B I T I ( 6 , 7 0 ) C E N S 7 0 F O F B A T ( 1 0 X , ' D E N S I T Y O F T H E S O L I D = ' , 1 X , F 5 . 1 , • G / C M 3 ' ) B B I T E ( 6 , 8 0 ) C E L 8 0 F O F B A T ( 1 0 X , ' S P E C I F I C H E A T OF T H E L I Q U I D = ' , I X , F 5 . 2 , • C A L / G B • ) B B I T E ( 6 , S 0 ) ( C P F ( I ) , 1 = 1 , 2 0 ) 9 0 F O E B S T ( I O X , ' S P E C I F I C H E A T OF T H E S O L I D ' , 1 X , 2 0 ( A 4 ) / ) B £ I T E ( 6 , 1 0 0 ) B L E T 1 0 0 F O F B A T ( I O X , ' L A T E N T B E A T C F S O L I D I F I C A T I O N = • , F 6 . 1 , ' C A L / G f l ' ) N B I T E ( 6 , 1 2 0 ) B K A Y 120 F C E B A T ( 1 0 X , ' T H E B B A L C O N I U C I 1 V I T I O F T H E L I Q U I D = ' , F 5 . 2 , 1 « C A L / C B . D E G . C . S E C ' ) B B I T E ( 6 , 1 3 0 ) B K A Y 1 3 0 F C F E A T ( 1 0 X , ' I H E B H A i C O N D U C T I V I T Y O F T H E S O L I D = ' , F 5 . 2 , I ' C I I / C B . C E G . C . S E C . • ) B E I 1 E ( 6 , 1 4 0 ) 1 4 0 F O B B A T ( 1 0 X , / / / / , 5 X , ' C A S T I N G C O N D I T I O N S * , F 5 . 2 , ' C H S / S E C ) B E I T I ( 6 , 1 5 0 ) C S P 1 5 0 F O F E A I ( I O X , ' C A S T I N G S P E E D = ' , F 5 . 2 , ' C B S / S E C • ) B B I T E ( 6 , 1 6 0 ) 1 6 0 F O E E A T ( I O X , ' H E A T T B A N S F E E 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 1 7 0 E O E B A T ( 1 5 X , ' B C 1 T C B H E A T 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 ) E E T U E N E N E C C C S O E E O U T I N E F B A B E 1 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 , D Y Y 2 P 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 AU C A L L F L C T E L ( ' B E T E I C ' , 1 ) C A I I A X C I F L ( ' S I D E ' , - 1 ) C A I I A X C I B L ( ' D I G I T S ' , 1 ) C A L L AX E L C T ( • D I S T A L O N G Z - A X I S ( C H S ) ; » , 0 . , S i , 0 . , D X X P ) C A I I A X C T F L (• S I D E ' , 1) C A L L A X P L O T ( ' D I S T A L O N G X - A X I S ( C B S ) ; ' , 9 0 . , S Y 1 , 0 . , D Y Y 1 P ) C A I I 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 A L O N G Z - A X I S ( C B S ) ; ' , 0 . , S X , 0 . , D X X P ) C A I I F L C T ( 0 . , 0 . , 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 ) C A I I A X C T E L ( ' S I E E ' , - 1 ) C A I I A X F L C I (• D I S T A L O N G X - A X I S ( C E S ) ; • , 9 0 . , S Y 1 , 0 . , DY X 1P) 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 . ( , I O B I G I N ' , 0 . ) F L C A T = N O f l E O N C A I I S Y B E C L ( 1 . , 1 . , 0 . 4 , ' E U N ' , 9 0 . , 4 ) C A I I N U B E E E ( 1 . , 2 . 5 , 0 . 4 , E L O A T , 9 0 . , - 1) C A I I S Y B E O L ( 1 . 5 , 1 . 0 , 0 . 4 , , T I E E = * , S 0 . , 5 ) C A I I N U B E E E ( 1 . 5 , 3 . 0 , 0 . 4 , T A U , 9 0 . , 0 ) E E T O E N EN I S 0 E E O 0 T I N E I B A B E 2 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 C C E B O N / C 1 3 / T AO C A I L E L C T E L ( C A I I A X C T E L ( C A L L A X C T E L ( C A I I A X F L C T ( C A I I A X C T E L ( C A L L A X P L C T ( C A I I F L O T ( 0 . C A I I A X C T E L ( C A L L A X C T E L ( C A I I AX E L C T ( C A I I E L C T ( 0 . C A L L A X C T E L ( C A I I A X C T E L ( C A L L A X C T E L ( C A I I AX E L C T ( C A I I A X C T E L ( C A I I A X C T E L ( F L C AT= NUB EON C A L L S Y H E C L ( C A I I N U E E E E ( C A L 1 S Y B E O L ( C A L L N U f l E E E ( E E T U E N EN t ( C B S ) ; ' , 0 . , S X , 0 . , D X X P ) ( C B S ) ; • , 9 0 . , S Y 2 , 0 . , D Y Y 2 F ) ( C B S ) ; ' , 0 . , S X , 0 . , D X X P ) B E T B I C • , 1) S I t E ' , - 1) D I G I T S • , 1) D I S T A L O N G Z - A X 1 S S H E ' , 1) D I S T A L O N G Y - A X I S 0 . , 3 ) Y C E 1 G I N « , S Y 2 ) S I D E * , 1 ) D I S T A L O N G Z - A X I S 0 . , 3 ) Y O E I G I N • , 0 . ) X C E I G I N ' , S X ) S I C E ' , - 1 ) D I S T A L O N G Y - A X I S ( C B S ) ; • , 9 0 . , S Y 2 , 0 . , D Y Y 2 P ) X O E I G I N ^ O . ) Y C E I G I N ' , 0 . ) . , 1 . , 0 . 4 , « E U K • , 9 0 . , 4 ) . , 2 . 5 , 0 . 4 , E L O A T , 9 0 . , - I ) • 5 , 1 . 0 , 0 . 4 , ' T I E E ^ , 9 0 . ,5) . 5 , 3 . 0 , 0 . 4 , T A U , 9 0 . , 0 ) S U E F C D T I N E C H E C K ( B C E) C O E E C N / C 1 6 / K N U E , N U E 2 B U E 2 = N D B 2 + 1 I F ( F U B 2 . E C . K N U B ) G O TO 10 R C E = 0 B E T GEN B C B = 1 N U B 2 = C B E T O E K E N D 221 C S 0 B B C 0 1 I N E T O P E I H T T E E H A T B I X C S U E E C U T I N E O U T P T 1 ( T ) E I B E N S I O N T ( 1 0 , 1 6 , 9 1 ) C O e E C » / C « / L , H , H C C 8 B O N / C 1 / E X , E X , D Z , D T , B K A Y C O B B C N / C 1 3 / I A C C O E B C N / C 1 0 / 1 L I Q , T S C I , D E S S , D E N I B B I 1 I ( 6 , 1 0 0 ) T A U 1 0 0 to ft. 11 ( ' 0 ' , 1 0 X , ' T E H F E R A T O R E S AT T B E E H D O F T I H E ' , 1 F 6 . 1 , I X , ' S E C C N E S ' ) Z T = E Z * F I C A T ( N - f ) H B I T E ( 6 . 1 4 5 ) 2 1 1 4 5 F O E B A T ( I O X , ' S I Z E O f T B E J H G O T I N Z D I B E C T I O N , , F 1 0 . 2 , * C B S ' ) 1 5 0 F O B B A T ( 5 X , , K = , , 2 X , I 3 ) DO 5 0 1 K = 1 , H H B I T E ( 6 , 150)K E C 5 0 2 J = 1 , f l • H B I 1 E ( 6 , 1 8 0 ) ( T ( 1 , J , K ) , 1 = 1 , L ) 5 0 2 C C K T I N U E 5 0 1 C O N T I N O E 1 6 0 F C I E A T ( 1 X , 1 1 ( E 1 0 . 1 , 1 X ) ) I F ( K . E C . 1 0 0 ) G C 1 0 1 B E I O F N 1 K = 1 1 B B I I E ( 6 , 1 5 0 ) K DO 5 0 3 J = 1 , 1 1 W B I I £ ( 6 , 1 8 0 ) ( T ( I , J , K ) , 1 = 1 , L ) 5 0 3 C O N T I N O E DC 5 0 4 K = 2 0 , 2 1 H B I 1 E ( 6 , 1 5 0 )K DO 5 0 5 0 = 1 , 1 1 H B I 1 E ( 6 , 1 8 0 ) ( 1 ( 1 , 0 , K ) , 1 = 1 , 1 ) 5 0 5 C C K T I N U E S O U C 0 N 3 I N D E B E I O F N E B C C C C c S O E I C O T I N E X X F I A N C S O E B O C I I N E I C P I C T C O H T O O B S I N T H E XX P L A N E C B C I E T E A T T H E C O N T O D f i P B O G E A B t O E S NOT HOB EC I F C T E E G E 1 E I S 1 B B E G D L A B L Y S H A P E D . T H I S C A N B E C O V E B C C E E B I G I V I N G D O B B I V A L U E S TO NON E X I S T I N G C G £ I D F C I N T S D I B E N S I O N T ( 1 0 , 1 6 , 9 1 ) , T 1 ( 1 0 , 1 6 , 9 1 ) , I 2 ( 1 0 , 1 6 , 9 1 ) E I B E N S I O N 1 3 ( 1 0 , 1 6 , 9 1 ) , 1 8 ( 1 0 , 1 6 , 9 1 ) D I E E K S I C N XX ( 1 6 ) , Y Y ( 1 0 ) , Z Z ( 1 6 , 1 0 ) D I B E N S I C N X X I ( 1 6 ) , Y Y 1 ( 10) , 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 C O H F C N / C 1 3 / T 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 C C G B I C F C I B T S ON X A X I S ( O R I G I N A L L Y I A X I S ) 222 C S C A I I N G DO 1C0 J = 1 , f l XX 1 ( J ) = XX ( J ) / E Y Y 2 P 100 C O N T I N U E DC 2 0 0 1 = 1 , L Y Y 1 ( I ) = Y Y ( I ) / D Y Y 1 P 2 0 0 C C M I N O I C D E A N I KG A C E O S S S E C T I O N O F T H E J O E B C I N G O T C C A I L P L C T E L ( * H E T B I C ' , 1 ) C A I I A X C T F L ( ' S H E ' , 0) C A I I A X E L C 3 C C 1 S T A L O N G Y A X I S ( C B S ) ; ' , 0 . , X X I ( 1 6 ) , 0 . , D Y Y 2 P ) C A I I A X E L C T ( ' D I S T A L O N G X A X I S ( C B S ) ; ' , 9 0 . , Y Y 1 ( 1 0 ) , 0 . , D Y Y 1 P ) C A I I E L C T (XX 1 ( 1 ) , Y Y 1 ( 1 0 ) , 3 ) C A L L F L O T ( X X 1 ( 4 ) , Y Y 1 ( 1 0 ) , 2 ) C A I I F L C T ( X X I ( 4 ) , Y Y 1 ( 6 ) , 2 ) C A L L P L O T ( X X 1 ( 5 ) , Y Y 1 ( 6 ) , 2 ) 4 C A I I E L C T ( X X 1 ( 9 ) , Y Y 1 ( 1 0 ) , 2 ) C A L L F L O T ( X X I ( 1 6 ) , Y Y 1 ( 1 0 ) , 2 ) C A I I F L C T (XX 1 ( 16) , YY 1 ( 1) , 2 ) 1 0 C A L L P L O T ( X X 1 ( 1 ) , Y Y 1 ( 1 ) , 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 ' , 9 0 . , 4 ) C A I I K U B E I E ( 1 . , 2 . 5 , 0 . 4 , E L O A T , 9 0 . , - 1 ) C A I I 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 ) E E T O E N C F I L L I N G T H E G R I D K I T H T E B P E E A T U E E V A L U E S C E N T I Y C U T L N E K 2 = ( N / 2 ) • 1 DO 3 C 0 1 = 1 , L DO 3 5 0 J = 1 , 4 Z Z 1 ( J , I ) = T ( I , J , K 2 ) 3 5 0 C C M I N U E 3 C 0 C O N T I N U E CO 4 0 0 1= 1 , 6 Z Z 1 ( 5 , I ) = T ( I , J , K 2 ) 4 0 0 C C K T I N U E DO 4 5 0 1 = 1 , 7 Z Z 1 ( 6 , I ) = I ( I , 6 , K 2 ) 4 5 0 C O N T I N U E DO 5 0 0 1 = 1 , 8 Z Z 1 ( 7 , I ) = 1 ( 1 , 7 , K 2 ) 5 0 0 C O N T I N U E CO £ C 0 1 = 1 , 9 Z Z 1 ( 8 , I ) = T ( 1 , 8 , K 2 ) 6 0 0 C O N T I N O E DC 7 0 0 1 = 1 , L CO 8 0 0 J = 1 , 1 6 Z Z 1 ( J , I ) = T ( I , J , K 2) 8 0 0 C O N T I N U E 7 0 0 C O N T I N U E C A I L 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 C N T O U E ( X X I , 1 6 . Y Y 1 , 1 C , 2 Z 1 , 1 6 , I S O L , 3 . , I S O L ) T- B n E E T U B N EN C C C C 223 S O E E C O T I N I I N I T I 2 ( 1 , T E B E , D U K Y ) C S U E B O C 1 I S E T C I N I T I A L I Z E T E E T E M E E B A T U S E F I E L E C I N T H I C A S E O F A J U B E C I N G O T . N O T E T H A T T H E C E I I E E I T S O U T S I D E T E I J D H B C E A V I B E E N I N I T I A L I S E D C T C A rORHY V A L O E T O E N A B L E T H E U S E O F C O N T O U B C P B O G B A E D I E E N S I O N T ( 1 0 , 1 6 , S 1) C O E E O N / C 4 / L , H , N C A I I G S E T ( T , 1 6 0 , 9 1 , 1 6 0 , D U B Y ) DO S 9 9 9 K = 1 , N / DO 150 1= 1, L DC 2 C 0 J = 1 , 4 T ( 1 , J , K ) = T E H E 2 0 0 C O N T I N U E 1 5 0 C C F . 1 I N U E CO 2 5 C 1 = 1 , 6 T ( 1 , 5 , K ) = T E B F 2 5 0 C O N T I N U E DO 3 0 0 1 = 1 , 7 T ( I , 6 , K ) = T E M F 3 0 0 C O N T I N U E DO 4 C 0 1 = 1 , 8 T ( I , 7 , K ) = T E B P 1 0 0 C O N T I N U E DC 5 0 0 1 = 1 , 9 I ( I , 8 , K ) = 1 E H P 5 0 0 C O K 1 I N U E CO 6 C 0 1 = 1 , L DC 6 5 0 J = 9 , B I ( I , J , K ) = 1 E B P 6 5 0 C O K 1 I N U E 6 0 0 C O H 1 I N U I 9 9 9 9 C O S 1 I N U E B E 1 U B N EN I C c c C S D B I C U T 1 N E T C S C B T C U T T E E E 1 F F E B E N T T Y P E S O F N O D E S C c S U E E C U T I N I N E S O B T 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 B H = B - 1 N N = 5 - 1 CO S 9 9 9 1 = 1 , L DC 9 9 9 9 J = 1 , B DO S 9 9 9 K = 1 , N I F ( 1 . E C . 1) GO T O 1 0 0 I F ( I . E C . L ) GO TO 2 0 0 I F ( J . E C . 1) GO T O 3 0 0 I F ( J . E C . K ) GO T C 4 0 0 I F ( K . E C - N ) GO T O 5 0 0 N T Y I I ( I , J , K ) = 5 G C I C 9 9 9 9 5 0 0 N I Y P E ( I , J , K ) = 14 GO 1 0 9 9 9 9 1 0 0 IF ( J . E C . . 1) GO TO 6 0 0 I F ( J . E C . f l ) GO TO 7 0 0 I F ( K . E C . N) GO T O 8 0 0 N T Y E E ( I , J , K ) = 4 G C T C 9 9 9 9 8 0 0 N T Y P E ( I , J , K ) = 13 GC T C 9 9 9 9 2 0 0 I F ( J . E C . 1 ) GO TO 9 0 0 I F ( J . F C . M ) GO TO 1 0 0 0 I F ( K . E C - N ) GO T C 1 1 0 0 N T Y I E ( I , J , K ) = 6 GO TO 9 9 9 9 1 1 0 0 N T Y E E ( I , J . K ) = 15 GO TO 9 9 9 9 3 0 0 I F ( K . E C - N ) GO T O 1 2 0 0 N T Y E f ( I , J , K ) = 2 G C T C 9 9 S 9 1 2 0 0 N T Y F E ( I , J , K ) = 11 GO T C 9 9 9 9 t O O I F ( K . I C . N ) GO TO 1 3 0 0 N T Y I E ( I , J , K ) = 8 GO TO 9 9 9 9 1 3 0 0 N T Y E E ( 1 , J , K ) = 17 GC TO 9 9 9 9 6 0 0 I F ( K . F C . N ) GO T O 1 4 0 0 NT Y F E ( I , J , K ) = 1 GO T C 9 9 9 9 1 4 0 0 N T Y F I ( I , J , K ) = 10 GO T C 9 9 9 9 7 0 0 I F ( K . E C - N ) GC TO 1 5 0 0 N T Y E E ( 1 , J , K ) = 7 GO TO 9 9 9 9 1 5 0 0 N I Y F E ( I , J , K ) = 16 GO TO 9 9 9 S 9 0 0 I F ( K . E C . N ) GO T C 1 6 0 0 N T Y F E ( I , J , K ) = 3 G C T C 9 9 9 9 1 6 0 0 N T Y F E ( I , J , K ) = 12 GO T C 9 9 9 9 1 0 0 0 I f ( K . E C . N ) GO TO 1 7 0 0 N T Y F E ( I , J , K ) = 9 GO TO 9 9 9 9 1 7 0 0 N I Y F F ( I , J , K ) = 18 9 9 9 S C O N T I N O E DC 1 6 0 0 K = 1 , N N HT Y F E ( 6 , 4 , K ) = 1 9 1 8 0 0 C O N T I N O E E C 1 3 5 0 K = 1 , N N H T Y E E { 6 , 5 , K ) = 2 1 1 3 5 0 C O N T I N C E N T Y E E ( 6 , 4 , N ) = 2 0 N T Y F E ( 6 , 5 , N ) = 2 2 B C 1 9 0 0 K = 1 , N N DO 2 C C 0 1 = 7 , I I N T Y I E ( I , 4 , K ) = 2 3 2 0 0 0 C O N T I N O E 1 9 0 0 C O M I N U E DO 2 1 C 0 1 = 7 , L L N T Y E E ( I , 4 , N ) = 2 4 2 1 0 0 C O N T I N U E DC 2 2 0 0 K = 1 , N N 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 B E T CBN END C C C SOEECUTINE SUBIT C SDEECOT1KI TC S10EI THE SU B E AC E TEKFEEATUBES0 CP DIFTEfiENT C ELEEEK1S IN AN ABE AY. TE1S IS FCB STUDYING THE UKSXEADY C STATE ECEIION OF THE CASTING. C C 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 10 K2 = 30 20 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 21 CCKTINUE DO 25 J=2,4 IS (I,KK,K 1) = T (10> J,K3) 1=1*1 25 CONTINUE CO 30 J=1,4 I1=10-J TS (I,KK ,K1)=1 (11 ,4, K3) 1= I • 1 30 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 DC 35 J = 9 , 1 6 IS ( I , K K , K 1 ) = T ( 1 0 , J , K 3 ) 1=1+1 35 C O N T I N U E DO 40 J = 1 , 9 12 = 1 C - J TS ( 1 , K K , K 1 ) = T (12, 1 6 . K 3 ) 1 = 1 + 1 4 0 C O M I N U E 100 C O N T I N U E B E T U B N EN I C C SUE ECUTINE TSAEAY C C S U E R C D I I N E T C C O E Y T H E S U E F A C E T E H P E B A T U R E A R & A Y C O N T O * F I L E I N B I N A R Y C D I H E N S I O N T S ( 3 8 , 3 0 , 5 0 ) C O F . r C K / C 4 / L , H , N C0BH0N/C21/T S DC 10 K=1,N DO 1C J = 1 , 3 0 W E I T F ( 7 ) ( T S ( I , J , K ) , 1 = 1 , 3 6 ) 1 0 C O N T I N U E B E T U B N E N L APPENDIX 3 THREE DIMENSIONAL TEMPERATURE DISTRIBUTION IN THE CASTING FOR THE DIFFERENT RUNS 227 228 DIST. FBOH THE HOOLD 0.0 CKS 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 70C. 7 0 C . 7CC. 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 C . 7CC. 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 C . 7 0 C . 7 0 C . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 70 0. 7 0 C . 70C. 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 70C. 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . . 7 0 0 . 7 0C. 7 0 C . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 70 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 C . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . ' 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 70 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 70 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 C 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 70 0 . 7 0 0 . 7 0 J . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . D IST . FECH THE BOOLD 6 . 3 CBS 6 9 1 . 6 9 1 . 6 8 9 . 6 8 7 . 6 8 3 . 6 7 7 . 6 S 1 . 6 9 1 . 6 8 9 . 6 8 7 . 6 8 3 . 6 7 7 . 6 S 1 . 6 9 1 . 6 8 9 . 6 8 7 . 6 8 3 . 6 7 7 . 6 S 1 . 6 9 1 . 6 8 9 . 6 8 7 . 6 8 3 . 6 7 7 . 6 S 1 . 6 9 0 . 6 8 9 . 6 8 7 . 6 8 3 . 6 7 7 . 6 SC. 6 9 0 . 6 e s . 6 3 6 . 6 8 2 . 6 7 7 . 6 £ S . 6 3 8 . 6 8 7 . 6 8 5 . 6 8 1 . 6 7 6 . 6 £ 4 . 6 8 4 . 6 3 3 . 6 8 1 . 6 7 8 . 6 7 3 . 6 7 3 . 6 7 3 . 6 7 2 . 67 1 . 6 6 8 . 6 6 5 . 64C. 64C. 6 3 9 . 6 3 9 . 6 3 9 . 6 3 8 . 4 7 6 . 4 7 8 . 4 7 8 . 4 7 8 . 4 7 7 . 4 7 5 . 6 6 9 . 6 5 8 . 6 4 4 . 6 3 0 . 5 2 1 . 6 6 9 . 6 5 8 . 6 4 4 . 6 3 0 . 5 2 1 . 6 6 9 . 6 5 8 . 644 . 6 3 0 . 52 1 . 6 6 9 . 6 5 8 . 6 4 4 . 6 3 0 . 5 2 1 . 6 6 9 . 6 5 8 . 6 4 4 . . 6 3 0 . 5 2 1 . 6 6 9 . 6 5 7 . 6 4 4 . 6 3 0 . 5 2 1 . 6 6 8 . 6 5 7 . 64<4 . 6 3 0 . 52 1. 6 6 6 . 6 5 6 . 6 4 3 . 6 3 0 . 52 1 . 6 5 9 . 6 5 1 . 6 4 1 . ' 6 3 0 . 52 0 . 6 3 6 . 6 3 4 . 6 3 1 . 4 3 0 . 51 1 . 4 7 2 . 4 6 6 . 4 5 3 . 4 1 3 . 3 7 3 . DIST. FBCH THE HOOLD 12 .7 CHS 6 7 6 . 6 7 5 . 6 7 2 . 6 6 7 . 6 5 9 . 6 4 E . 6 3 1 . 5 7 1 . 4 6 1 . 3 2 0 . 1 4 9 . 6 7 £ . 6 7 5 . 67 2 . 6 6 7 . 6 5 9 . 6 4 3 . 6 3 1 . 5 7 1 . 4 6 1 . 3 2 0 . 1 4 9 . 6 7 6 . 6 7 5 . 6 7 2 . 6 6 7 . 6 5 9 . 6 4 £ . 6 3 1 . 5 7 1 . 461 . 3 2 0 . 1 4 9 . 6 7 6 . 6 7 5 . 6 7 2 . 6 6 7 . 6 5 9 . 6 4 8 . 6 3 1 . 5 7 1 . 4 6 1 . 3 2 0 . 1 4 9 . 6 7 5 . 6 7 4 . 6 7 1 . 6 6 6 . 6 5 9 . 6 4 8 . 6 3 1 . 5 7 1 . 4 6 0 . . 3 1 9 . 149 . 6 7 4 . 6 7 3 . 6 7 0 . 6 6 5 . 6 5 8 . 6 4 7 . 6 3 1 . 5 7 0 . 4 6 0 . 3 1 9 . 1 4 9 . 6 7 0 . 6 7 0 . 6 6 7 . 6 6 3 . 6 5 6 . 6 4 6 . 6 3 1 . 5 6 9 . 4 5 7 . 3 1 7 . 1 4 8 . 6 6 1 . 6 6 1 . 6 5 9 . 6 5 5 . 6 5 0 . 6 4 2 . 6 3 1 . 5 6 3 . 4 5 0 . •3 1 1 . 14 5 . 6 3 7 . 6 3 6 . 6 3 6 . 6 3 4 . 6 3 1 . 6 3 1 . 6 3 0 . 5 2 4 . 4 2 0 . 2 9 1 . 136. 5 4 6 . 5 4 6 . 5 4 3 . 5 2 1 . 5 0 2 . 4 8 1 . 4 5 1 . 3 9 6 . 3 2 1 . 2 2 6 . 1 0 8 . 1 4 7 . 147. 146. 144 . 139. 1 3 4 . 1 2 6 . 1 1 3 . 9 5 . 7 1 . 4 0 . Table A3.1 Three-dimensional temperature d i s t r i b u t i o n in 381 x 991 mm. aluminium i n g o t cast at 1.775 mm/s (convent iona l c o o l i n g ) . 229 D I S T . F E C H T H E H O U 1 D 1 9 . 0 C B S 6 5 6 . 6 5 7 . 6 5 4 . 6 4 8 . 6 4 0 . 6 3 1 . 6 5 8 . 6 5 7 . 6 5 4 . 6 4 8 . 6 4 0 . 6 3 1. 6 5 6 . 6 5 7 . 6 5 4 . 6 4 3 . 6 4 0 . 6 3 1 . 6 5 6 . 6 5 7 . 6 5 3 . 6 4 7 . 6 4 0 . 6 3 1. 6 5 7 . 6 5 6 . 6 5 3 . 6 4 7 . 6 3 9 . 6 3 1 . 6 5 5 . 6 5 4 . 6 5 1 . 6 4 6 . 6 3 9 . 6 3 1 . 6 5 1 . 6 5 C . 6 4 8 . 6 4 3 . 6 3 5 . 6 3 0 . 6<42. 6 4 2 . 6 4 0 . 6 3 5 . 6 3 1 . 5 8 9 . 6 3 1 . 6 3 1 . 6 3 1 . 6 3 0 . 5 7 4 . 5 1 3 . 4 0 2 . 4 0 0 . 3 9 4 . 3 8 2 . 3 6 2 . 3 3 4 . 1 C 4 . 1 0 4 . 1 0 2 . 9 9 . 9 5 . 8 9 . D I S T . F B C H T H E H O U 1 D 2 5 . 4 C M S 6 4 3 . 6 4 2 . 6 3 9 . 6 3 4 . 6 3 0 . 5 5 4 . 6 4 3 . 6 4 2 . 6 3 9 . 6 3 4 . 6 3 0 . 5 5 5 . 6 4 3 . 6 4 2 . 6 3 9 . 6 3 4 . 6 3 0 . 5 5 5 . 6 4 3 . 6 4 2 . 6 3 9 . 6 3 4 . 6 3 0 . 5 5 4 . 6 4 2 . 6 4 1 . 6 3 6 . 6 3 4 . 6 3 0 . 5 5 3 . 6 4 1. 6 4 0 . 6 3 7 . 6 3 3 . . 6 3 0 . 5 4 9 . 6 3 6 . 6 3 7 . 6 3 4 . 6 3 1 . 5 9 1 . 5 3 1 . 6 3 1. 63 1. 6 3 1. 6 3 0 . 5 5 7 . 4 9 1 . 5 1 7 . 5 1 2 . 4 9 9 . 4 7 4 . 4 3 9 . 3 9 4 . 3 2 0 . 3 1 7 . 3 0 9 . 2 9 5 . 2 7 5 . 2 4 9 . 6 4 . 8 4 . 8 2 . 7 9 . 7 4 . 6 8 . D I S T . F B C H T H E B O U 1 D 3 1 . 7 C H S 6 3 4 . 6 3 3 . 6 3 2 . 6 3 0 . 5 6 2 . 4 9 7 . 6 3 4 . 6 3 3 . 6 3 2 . 6 3 0 . 5 6 2 . 4 9 7 . 6 3 4 . 6 3 3 . 6 3 2 . 6 3 0 . 5 6 2 . 4 9 6 . 6 3 4 . ' 6 3 3 . 6 3 2 . 6 3 0 . 5 6 1. 4 9 5 . 6 3 4 . 6 3 3 . 6 3 2 . 6 3 0 . 5 5 9 . 4 9 2 . 6 3 3 . 6 3 2 . 6 3 1. 6 0 0 . 5 4 6 . 4 8 3 . 6 3 1 . 6 3 1 . 6 3 1 . 5 7 7 . 5 2 1 . 4 5 9 . 5 7 6 . 5 8 7 . 5 5 1 . 5 0 8 . 4 6 0 . 4 0 6 . 4 2 9 . 4 2 3 . 4 0 8 . 3 8 4 . 3 5 3 . 3 1 5 . 2 5 8 . 2 5 5 . 2 4 7 . 2 3 5 . 2 1 7 . 1 9 6 . 7 C . 6 9 . 6 8 . 6 5 . 6 1 . 5 6 . 5 5 4 . 4 6 3 . 3 5 8 . 2 4 C . 1 1 2 . 5 5 4 . ' 4 6 3 . . 3 5 8 . 2 4 0 . 1 1 2 . 5 5 4 . 4 6 3 . 3 5 8 . 2 4 0 . 1 1 2 . 5 5 4 . 4 6 3 . 3 5 6 . 2 4 0 . 1 1 2 . 5 5 4 . 4 6 2 . 3 5 7 . 2 3 9 . 1 1 2 . 5 5 2 . 4 6 0 . 3 5 5 . 2 3 8 . 11 1 . 5 4 6 . 4 5 4 . 3 5 0 . 2 3 4 . 1 1 0 . 5 1 8 . 4 3 2 . 3 3 3 . 2 2 3 . 10 5 . 4 4 8 . 3 7 4 . 2 8 9 . 1 9 4 . 9 3 . 2 9 8 . 2 5 2 . 1 9 7 . 1 3 5 . 6 7 . 8 0 . 7 0 . 5 7 . 4 3 . 2 7 . 4 7 8 . 3 9 4 . 3 0 1 . 2 0 0 . 9 5 . 4 7 9 . 3 9 4 . 3 0 1 . 2 0 0 . 9 5 . 4 7 8 . 3 9 4 . 3 0 0 . 2 0 0 . 9 5 . 4 7 7 . 3 9 3 . 3 0 0 . 2 0 0 . 9 5 . 4 7 6 . 3 9 1 . 2 9 9 . 1 9 9 . 9 4 . 4 7 1 . 3 8 7 . 2 9 5 . 1 9 6 . 9 3 . 4 5 8 . 3 7 5 . 2 8 6 . 1 9 0 . 9 0 . 4 2 1 . 3 4 5 . 2 6 3 . 1 7 5 . 8 4 . 3 4 1 . 2 8 2 . 2 1 6 . 1 4 5 . 7 1 . 2 1 8 . 1 6 2 . 1 4 1 . 9 7 . 5 0 . 6 1. 5 3 . 4 4 . 3 4 . 2 3 . 4 2 5 . 3 4 7 . 2 6 4 . 1 7 5 . 8 4 . 4 2 5 . 3 4 7 . 2 6 4 . 1 7 5 . 8 4 . 4 2 5 . 3 4 7 . 2 6 3 . 1 7 5 . 8 4 . 4 2 4 . 3 4 6 . 2 6 2 . 1 7 5 . 8 4 . 4 2 0 . 3 4 3 . 2 6 0 . 1 7 3 . 8 3 . 4 1 2 . 3 3 6 . 2 5 4 . 1 6 9 . 8 1 . 3 9 1 . 3 1 9 . 2 4 2 . 1 6 1 . 7 8 . 3 4 7 . 2 8 3 . 2 1 5 . 1 4 4 . 7 1 . 2 7 2 . 2 2 4 . 1 7 1 . 1 1 6 . 5 6 . 1 7 0 . 1 4 2 . 1 1 0 . 7 7 . 4 1 . 5 0 . 4 4 . 3 7 . 2 9 . 2 1 . 230 DIST. FECH TBI HOULD 38. 1 CHS 63 1. 631. 630. 57 1. 514. 452. 63 1. 631. 63C. 571. 514. 452. 631. 63 1. 630. 570. 513. 45 1. 631. 631. 630. 568. 510. 449. 6 31. 631. 6 19. 564. 505. 442. 631. 631. 5S6. 545. 489. 428. 630. 589. 549. 503. 452. 396. lilt. 470. 450. ' 420. 382. 339. 350. 346. 334. 314. 289. 256. 2 1C. 208. 201. 191. 176. 159. 5S. 58. 57. 55. 51. 47. DIST. FECH THE HOOID 44. 4 CHS 631. 630. 578. 527. 473. 4 14. 631. 63C. 577. 527. 472. 414. 631. 630. 576. 525. 470. 412. 631. £3C. 571. 520. 465. 407. 631. 60 1. 556. 507. 454. 397. 574. 555. 520. 477. 429. 376. 484. 476. 454. 422. 383. 338. 3SC. 385. 371. 34 8. 319. 283. 287. 284. 274. 259. 236. 212. 173. 171. 166. 157. 145. 131. 51. 50. 49. 47. 44. 41. DIST. FBCfi T BE HOULD 50.8 CHS 566. 544. 512. 472. 426. 374. 554. 541. 510. 471. 425. 373. 542. 530. 504. 466. 421. 370. 52S. 516. 492. 455. 412. 362. 4SS. 491. 468. 435. 394. 346. 454. 447. 429i 401. 365. 323. 393. 388. 374. 352. 322. 286. 32C. 317. 306. 289. 266. 237. 237. 234. 227. 214. 198. 178. 143. 142. 138. 131. 121. 110. 44. 44. 43. 4 1. 39. 36. 385. 313. 237. 158. 76. 385. 313. 237. 156.' 76. 384. 312. 236. 157. 76. 382. 310. 235. 156. 76. 376. 306. 231. 154. 75. 364. 295. 223. 149. 73. 337. 2 74. 206. 139. 6 8. 290. 237. 180. 121. 61. 222. 183. 141 . 96. 50. 138. 115. 90. 63. 3 6. 43. 38. 32. 26. 20. 352. 286. 216. 1 44. 70. 351. 285. 21o. 1HH. 70. 350. 284. 215. 143. 70. 346. 230. 212. 142. 69. 337. 273. 207. 138. 66. 319. 259. 197. 132. 65. 269. 236. 179. 1^0. 6 0. 243. 200. 153. 104. 53. 184. 152. 118. 61. 4 3. 114. 96. 75. 54. 32. 37. 33. 29. 24. 19. 319. 259. 196. 131. 65. 318. 258. 196. 131. 65. 315. 25b. J94. 130. 64. 309. 251. 190. 126. 63. 297. 242. 183. 123. 61. 276. 226. 172. 116. 56. 246. 2 02. 154. • 104. 53. 205. 169. 130. 89. 4 7. 154. 128. 100. 69. 3 8. 96. 81. 64. 47. 29. 33. 30. 26. 22. 18. 231 D I S T . F R C H T H F H 0 U 1 D 5 7 . 1 . C H S 4 5 5 . 4 4 9 . 4 3 2 . 1 0 5 . 3 7 0 . 3 2 9 . 4 5 3 . 4 4 7 . 4 3 0 . 4 0 4 . 3 6 9 . 321. 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 . 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 4 1 . 2 6 5 . 2 6 3 . 2 5 4 . 2 4 1 . 2 2 2 . 1 9 S . 1 S 7 . 1 9 5 . 1 8 9 . 1 7 9 . 1 6 6 . 1 4 9 . 1 2 0 . 1 1 9 . 1 1 5 . 1 1 0 . 1 0 2 . 9 3 . 3 S . 3 8 . 3 8 . 3 6 . 3 5 . 3 2 . D I S T . F E C H T H E H O U I D 6 3 . 5 C H S 3 8 4 . 3 8 C . 3 6 7 . 3 4 6 . 3 1 9 . 2 8 5 . 3 6 2 . 3 7 7 . 3 6 5 . 3 4 4 . 3 1 7 . 2 6 3 . 3 7 5 . 3 7 C . 3 5 6 . 3 3 8 . 3 1 1 . 2 7 6 . 3 6 1. 3 5 6 . 3 4 6 . 3 2 7 . 3 0 1. 2 6 9 . 3 4 1. 3 3 7 . 3 2 6 . 3 0 6 . 2 8 4 . 2 5 4 . 3 1 0 . 3 0 7 . 2 9 7 . 28 1. 2 5 9 . 2 3 2 . 2 7 1. 2 6 6 . 2 5 9 . 2 4 6 . 2 2 7 . 2 0 4 . 2 2 2 . 2 1 9 . 2 1 3 . 2 0 2 . 1 8 7 . 1 6 6 . 1 6 5 . 1 6 3 . 1 5 8 . 1 5 0 . 1 3 9 . 1 2 6 . 10 1. 1 0 1 . 9 8 . 9 3 . 8 7 . 7 9 . 3 4 . 3 4 . 3 4 . 3 3 . 3 1 . 2 9 . D I S T . F E O H T H E H O U I D 6 9 . 8 C H S 3 2 6 . 3 2 4 . 3 1 4 . 2 9 7 . 2 7 4 . 2 4 6 . 3 2 5 . 3 2 2 . 3 1 2 . 2 9 5 . 2 7 2 . 2 4 4 . 3 I S . 3 1 5 . 3 0 5 . 2 8 9 . 2 6 7 . 2 3 9 . 3 0 7 . 3 0 3 . 2 9 4 . 2 7 8 . 2 5 7 . 2 3 1. 2 6 6 . 2 8 5 . 2 7 6 . 2 6 1 . 2 4 2 . 2 1 7 . 2 6 2 . 2 5 9 . 2 5 1. 2 3 8 . 2 2 0 . 1 9 8 . 2 2 6 . 2 2 6 . 2 1 9 . 2 0 8 . 1 9 2 . 1 7 3 . 1 8 7 . 1 8 5 . 1 8 0 . 1 7 1 . 1 5 6 . 1 4 3 . 1 2 S . 1 3 6 . 1 3 4 . 1 2 8 . 1 1 9 . 1 0 7 . 8 7 . 8 6 . 8 4 . 8 0 . 7 5 . 6 8 . 3 1 . 3 1 . 3 0 . 3 0 . 2 8 . 2 7 . 2 8 2 . 2 3 0 . 1 7 5 . 1 1 6 . 5 9 . 2 8 1 . 2 2 9 . 1 7 5 . 1 1 6 . 5 9 . 2 7 7 . 2 2 6 . 1 7 2 . 1 1 6 . 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 9 4 . 1 4 9 . 10 1. 5 2 . 2 0 8 . 1 7 2 . 1 3 2 . 9 0 . 4 7 . 1 7 3 . 1 4 3 . 1 1 1 . 7 7 . 4 2 . 1 3 0 . 1 0 3 . 6 5 . 6 0 . 3 4 . 8 2 . 6 9 . 5 6 . 4 1 . 2 6 . 3 0 . 2 7 . 2 4 . 2 1 . 1 6 . 2 4 5 . 2 0 2 . 1 5 5 . 1 0 5 . 5 4 . 2 4 4 . 201. 1 5 4 . 1 0 4 . 5 3 . 2 4 0 . 1 9 7 . 1 5 1 . 1 0 3 . 5 3 . 2 3 2 . I S I - 1 4 7 . 1 0 0 . 5 1. 2 1 9 . 1 8 1 . 1 3 9 . 9 5 . 4 9 . 2 0 1 . 1 6 6 . 12a. 8 6 . 4 6 . 1 7 7 . 1 4 6 . 1 3 . 7 8 . 4 2 . 1 4 6 . 121 . 9 5 . 6 6 . 3 7 . 1 1 0 . 9 2 . 7 3 . 5 2 . 3 1. 7 0 . 6 0 . 4 9 . 3 7 . 2 4 . 2 7 . 2 5 . 2 3 . 2 0 . 1 7 . 2 1 3 . 1 7 6 . 1 3 5 . 9 3 . 4 8 . 2 1 1 . 1 7 5 . 1 3 5 . 9 2 . 4 6 . 2 0 7 . 1 7 1 . 1 3 2 . S O . 4 7 . 2 0 0 . 1 6 5 . 1 2 7 . 8 6 . 4 6 . 1 8 8 . 1 5 6 . 1 2 0 . 8 3 . 4 4 . 1 7 2 . 1 4 2 . 1 1 0 . 7 7 . 4 2 . 1 5 1 . 1 2 5 . 9 6 . 6 8 . 3 8 . 1 2 5 . 1 0 4 . 8 2 . 5 8 . 3 4 . 9 4 . 7 9 . 6 3 . 4 6 . 2 8 . 6 1 . 5 2 . 4 3 . 3 3 . 2 3 . 2 5 . 2 3 . 2 1 . 1 9 . 1 7 . D I S T . F E C H T H I H O O L D 2 8 3 . 2 8 0 . 2 7 1. 2 5 7 . 2 E 1 . 2 7 6 . 2 6 9 . 2 5 5 . 2 7 4 . 2 7 2 . 2 6 3 . 2 4 9 . 2 6 3 . 2 6 1 . 2 5 3 . 2 3 9 . 2 4 7 . 2 4 4 . 2 3 7 . 2 2 4 . 2 2 4 . 2 2 2 . 2 1 5 . 2 0 4 . 1 9 5 . 1 9 3 . 1 8 7 . 1 7 8 . 1 6 C . 1 5 6 . 1 5 4 . 1 4 6 . 1 2 0 . 1 1 9 . 1 1 5 . 1 1 0 . 7 5 . 7 5 . 7 3 . 7 0 . 2 9 . 2 8 . 2 8 . 2 7 . D I S T . F E C H T H E H O O L D 2 4 8 . 2 4 5 . 2 3 8 . 2 2 5 . 2 4 £ . 2 4 3 . 2 3 6 . 2 2 4 . 2 4 0 . 2 3 7 . 2 3 0 . 2 1 8 . 2 3 C . 2 2 7 . 2 2 1. 2 0 9 . 2 1 5 . 2 1 3 . 2 0 6 . 1 9 6 . 1 S S . 1 9 3 . 1 8 7 . 1 7 8 . 1 7 0 . 1 6 8 . 1 6 3 . 1 5 5 . 1 3 S . 1 3 8 . 1 3 4 . 1 2 7 . 1 0 5 . 1 0 4 . 1 0 1 . 9 6 . 6 7 . 6 6 . 6 4 . 6 2 . 2 7 . 2 6 . 2 6 . 2 5 . D I S T . F E C B T H I M O O L E I 2 2 1 . 2 1 9 . 2 1 3 . 2 0 2 . 2 2 C . 2 1 7 . 2 1 1 . 2 0 0 . 2 1 4 . 2 1 2 . 2 0 6 . 1 9 5 . 2 C 5 . 2 0 3 . 1 9 7 . 1 8 7 . 1 S 1 . 1 9 0 . 1 8 4 . 1 7 5 . 1 7 4 . 1 7 2 . 1 6 7 . 1 5 9 . 1 5 1 . 1 5 0 . 1 4 5 . 1 3 8 . 1 2 4 . 1 2 3 . 1 2 0 . 1 1 4 . 9 4 . 9 3 . 9 0 . 8 6 . £ C . 6 0 . 5 8 . 5 6 . 2 5 . 2 5 . 2 5 . 2 4 . 3 . 2 C H S 2 3 6 . 2 1 4 . 1 8 5 . 1 5 4 . 2 3 6 . 2 1 2 . 1 8 4 . 1 5 2 . 2 3 1. 2 0 7 . 1 8 0 . 1 4 9 . 2 2 2 . 1 9 9 . 1 7 3 . 1 4 4 . 2 0 6 . 1 8 7 . 1 6 3 . 1 3 5 . 1 8 9 . 1 7 0 . 1 4 8 . 1 2 3 . 1 6 5 . 1 4 S . 1 3 0 . 1 0 8 . 1 3 6 . 1 2 3 . 1 0 7 . 9 0 . 1 0 2 . 9 3 . 8 2 . 6 9 . 6 5 . 6 0 . 5 3 . 4 6 . 2 6 . 2 5 . 2 4 . 2 2 . ! . 5 i C H S 2 0 9 . 1 8 6 . 1 6 3 . 1 3 6 . 2 0 7 . 1 8 6 . 16 2 . 1 3 5 . 2 0 2 . 1 8 2 . 1 5 6 . 1 3 2 . 1 9 4 . 1 7 5 . 1 5 2 . 1 2 6 . 1 6 1 . 1 6 4 . 1 4 3 . 1 1 9 . 1 6 5 . 1 4 S . 1 3 0 . 1 0 9 . 1 4 4 . 1 3 0 . 1 1 4 . 9 5 . 1 1 9 . 1 0 7 . 9 4 . 8 0 . 9 0 . 8 2 . 7 2 . 6 2 . 5 8 . 5 3 . 4 8 . 4 2 . 2 5 . 2 4 . 2 2 . 21 . . 9 C H S 1 8 7 . 1 6 8 . 1 4 7 . 1 2 2 . 1 8 5 . 1 6 7 . 1 4 6 . 1 2 1 . 18 1. 1 6 3 . 1 4 2 . 1 1 9 . 1 7 3 . 1 5 6 . 1 3 6 . 1 1 4 . 1 6 2 . 1 4 6 . 1 2 8 . 1 0 7 . 1 4 7 . 1 3 3 . 1 1 6 . 9 7 . 1 2 6 . 1 1 6 . 1 0 2 . 6 6 . 1 0 6 . 9 6 . 8 5 . 7 2 . 8 1. 7 4 . 6 5 . 5 6 . 5 3 . 4 9 . 4 4 . 3 9 . 2 3 . 2 3 . 2 2 . 2 0 . 232 1 1 9 . 8 2 . 4 4 . 1 1 8 . 8 2 . . 4 4 . 1 1 6 . 6 0 . 4 3 . 1 1 1 . 7 7 . 4 2 . 1 0 5 . 7 3 . 4 0 . 9 6 . 6 7 . 3 8 . 8 5 . 6 0 . 3 4 . 7 1 . 5 1 . 3 1 . 5 6 . 4 1 . 2 6 . 3 8 . 3 0 . 2 2 . 2 0 . 1 6 . 1 6 . 1 0 6 . 7 3 . 4 0 . 1 0 5 . 7 3 . 4 0 . 1 0 3 . 7 2 . 3 9 . 9 9 . 6 9 . 3 8 . 9 3 . 6 5 . 3 7 . 8 5 . 6 0 . 3 4 . 7 5 . 5 4 . 3 2 . 6 3 . 4 6 . 2 8 . 5 0 . 3 8 . 2 5 . 3 5 . 2 8 . 2 1 . 2 0 . 1 6 . 1 6 . 9 5 . 6 7 . 3 7 . 9 5 . 6 7 . 3 7 . 9 3 . 6 5 . 3 7 . 8 9 . 6 3 . 3 6 . 8 4 . 5 9 . 3 4 . 7 7 . 5 5 . 3 2 . 6 3 . . 4 9 . 3 0 . 5 8 . 4 3 . 2 7 . 4 6 . 3 5 . 2 4 . 3 3 . 2 6 . 2 0 . 1 9 . 1 8 . 1 6 . J 233 D I S T . F E C H T H E H O U L D 9 5 . 2 C H S 2 0 0 . 2 0 2 . 1 9 6 . 1 8 6 . 1 7 2 . 1 5 5 . 1 3 5 . 1 1 3 . 8 9 . 6 3 . 3 5 . 2 C 2 . 2 0 0 . 1 9 4 . 1 8 4 . 17 1. 1 5 4 . 1 3 4 . 1 1 2 . . 8 8 . 6 2 . 3 5 . 1 S 7 . 1 9 5 . 1 8 S . 1 8 0 . 1 6 7 . 1 5 0 . 1 3 1 . 1 1 0 . 8 6 . 6 1 . 3 5 . 1 6 6 . 1 8 7 . 18 1. 1 7 2 . 1 6 0 . 1 4 4 . 1 2 6 . 1 0 5 . 8 3 . 5 9 . 3 4 . 1 7 6 . 1 7 4 . 1 6 9 . 1 6 1 . 1 4 9 . 1 3 5 . 1 1 8 . 9 9 . 7 8 . 5 6 . 3 2 . 1 5 9 . 1 5 8 . 1 5 3 . 1 4 6 . 1 3 5 . 1 2 3 . 1 0 7 . 9 0 . 7 1 . 5 1 . 3 1 . 1 3 9 . 1 3 6 . 1 3 4 . 1 2 7 . 1 1 8 . 1 0 7 . 9 4 . 7 9 . 6 3 . 4 6 . 2 8 . 1 1 4 . 1 1 3 . 1 1 0 . 1 0 5 . 9 6 . 8 9 . 7 9 . 6 7 . 5 4 . 4 0 . 2 6 . 6 7 . 8 6 . 8 4 . 8 0 . 7 5 . 6 8 . 6 1 . 5 2 . 4 3 . 3 3 . 2 3 . 5 6 . 5 6 . 5 4 . 5 2 . 4 9 . 4 6 . 4 1 . 3 6 . 3 1 . 2 5 . 1 9 . 2 4 . 2 4 . 2 4 . 2 3 . 2 3 . 2 2 . 2 1 . 2 0 . 1 9 . 1 7 . 1 6 . D I S T . FBOrt I H E HCULD 0.0 CfiS 234 7C0. 700. 700. 700. 7C0. 700. 700. 700. 700. 700. 700. C I S T . 6 7 8 . 6 7 8 . 6 7 8 . 6 7 8 . 6 7 8 . 6 7 7 . 6 7 5 . 6 7 1 . 6 6 2 . 6 4 7 . 5 5 7 . D I S T . 6 6 2 . 6 6 2 . 6 6 2 . 66 1 . 6 6 1 . 6 6 0 . 6 5 9 . 6 5 7 . 6 4 1 . 5 6 9 . 5 1 6 . 700. 700. 700. 700. 700. 700. 7C0. 700. 700. 700. 700. 6 7 7 . 6 7 7 . 6 7 7 . 6 7 7 . 6 7 6 . 6 7 6 . 6 7 4 . 6 7 0 . 6 6 2 . 6 4 6 . 5 5 5 . 6 6 1 . 6 6 1 . 6 6 1 . 6 6 1 . 6 6 0 . 6 6 0 . 6 5 9 . 6 5 7 . 6 3 5 . 5 6 5 . 5 1 2 . 700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 6 7 3 . 6 7 3 . 6 7 3 . 6 7 3 . 6 7 3 . 6 7 2 . 6 7 0 . 6 6 7 . 6 5 9 . 64 1 . 5 4 7 . 6 5 9 . 6 5 9 . 6 5 8 . 6 5 8 . 6 5 8 . 6 5 8 . 6 5 7 . 6 5 1 . 6 1 0 . 5 5 3 . 5 0 3 . 700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 6 6 6 . 6 6 6 . 6 6 6 . 6 6 6 . 6 6 6 . 6 6 5 . 6 6 4 . 6 6 1 . 6 5 6 . 6 1 1 . 5 3 1 . 6 5 4 . 6 5 4 . 6 5 4 . 6 5 4 . 6 5 4 . 6 5 2 . 6 4 8 . 6 2 2 . 5 8 4 . 5 3 6 . 4 8 9 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 C 0 . 7 0 0 . 7 0 0 . 7 0 0 . 6 5 o . 6 5 6 . 6 5 6 . 6 5 6 . 6 5 6 . 6 5 6 . 6 5 5 . 6 5 3 . 6 4 2 . 56 1 . 5 0 7 . 6 3 6 . 6 3 6 . 6 3 6 . 6 3 5 . 6 2 8 . 6 2 3 . 6 1 3 . 5 S 1 . 5 5 7 . 5 1 5 . 4 7 1 . 700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 700. 6 2 0 . 6 2 0 . 6 2 0 . 6 2 0 . 6 2 0 . 6 1 9 . 6 1 7 . 6 1 0 . 5 8 9 . 5 4 3 . 4 7 7 . 6 0 3 . 6 0 3 . 6 0 3 . 6 0 2 . 5 * 8 . 5 9 2 . 5 8 0 . 5 6 0 . 5 2 9 . 491 . 4 5 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 0 . 7 0 U . 7 0 U . 7 0 0 . 7 0 0 . 7 0 U . 7 0 0 . 70 0 . 57 1 . 5 7 1 . 5 7 1 . 5 7 1 . 5 7 0 . 5 6 S . 5 6 7 . 5 6 0 . 5 3 9 . 4 9 9 . 4 3 9 . 57 1 . 5 7 1 . 57 0 . 5 6 S . 5 6 6 . 5 6 0 . 5 4 9 . 5 3 0 . 5 0 1 . 4 6 5 . 4 2 6 . I B O a THE RCULD 1 0 . 4 CHS F E C K T H * KCULD 2 0 . 8 CMS Table A 3 . I I Three-dimensional temperature d i s t r i b u t i o n in 254 x 690 mm aluminium i n g o t cast a t 0.833 mm/s (Reduced Secondary C o o l i n g ) . DIST. FBOM THE MOULD 31.2 CMS 235 658. 657. 653. 637. 6C7. 577. 54 9. 658. 657. 653. 637. 6 06. 577. 549. 658. 657. 653. 636. 6C4. 575. . 54 b., 658. 6 57. 652. 627. 599. 570. 54 2. 658. 657. 650. 62 1. 59 1. 562. 53 a. 657. 655- 642. 607. 577. 548. 52 1. 649. 64 1. 6 1 1. 582. 554. 528. 502. 586. 580. 566. 546. 524. 500. 476. 535. 532. 522. 507. 4 89. 458. 446. 491. 488. 480. 468. 453. 434. 414. 451. 449. 442. 431. 417. 401. 382. CIST. FEOM THE MOULD 41.6 CMS 655. 650. 622. 592. 560. 525. 486. 655. 649. 621. 590. 558. 524. 485. 654. 647. 6 16. 584. 552. 518. 480. 649. 638. 603. 572. 542. 509. 47 1. 605. 5S8. 577. 553. 525. 494. 458. "566. 561. 547. 528. 504. 475. 44 0. 529. 525. 515. 499. 478. 451 . 41 y. 4 92. 489. 481. 467. 449. 425. 394. 455. 452. 445. 433. 4 17. 395. 367. 415. 413. 407. 397. 382. 363. 337. 370. 368. 363. 354. 341. 324. 301. EI ST. IEOK THE MOULD 52.0 CM S 558. 554. 546. 535. 522. 509. 496. 554. 551. 543. 532. 520. 507. 494. 544. 541. 534. 524. 512. 500. 487. 529. 527. 521. 511. 500. 489. 476. 510. 508. 503. 495. 485. 4 74. 4b2. 489. 487. 483. 475. 466. 456. 445. 466. 464. 460. 454. 445. 436. 425. 442. 44 1. 437. 431. 423. 415. 405. 4 19. . 4 18. 4 14. 409. 402. 3S4. 384. 398. 397. 39a. 386. 382. 374. 365. 380. 379. 376. 371. 365. 358. 3*9. 1 r i s i . F R O M THE ECU LB 6 2 . 4 CMS 236 4 3 9 . 4 8 8 . 4 8 6 . 4 8 2 . 4 7 7 . 4 7 0 . 4 6 4 . 4 8 7 . 4 8 6 . 4 8 4 . 4 6 0 . 4 7 5 . 4 6 9 . 4 6 2 . 4 8 2 . 4 8 1 . 4 7 8 . 4 7 4 . 4 6 9 . 4 6 3 . 4 5 7 . 4 7 3 . 4 7 2 . 4 7 0 . 4 6 6 . 46 1 . 4 5 5 . 44>3. 4 6 1 . 4 6 0 . 4 5 8 . 4 5 4 . 4 5 0 . 4 4 4 . 4 3 8 . 4 4 7 . 4 4 6 . 4 4 4 . 44 1 . 4 3 6 . 4 3 1 . 4 2 5 . 4 3 2 . 4 3 1 . 4 2 9 . 4 2 6 . 4 2 2 . 4 1 6 . 4 1 1 . 4 1 6 . 4 1 6 . 4 1 4 . 4 1 1 . 4 0 7 . 4 0 2 . 3 i i 6 . 4 0 2 . 4 0 1 . 3 9 9 . 3 9 6 . 3 9 2 . 3 8 8 . 3 8 2 . 3 8 8 . 3 8 8 . 3 8 6 . 3 8 3 . 3 7 9 . 3 7 5 . 3 7 0 . 3 7 6 . 3 7 7 . 3 7 5 . 3 7 3 . 3 6 9 . 3 6 4 . 3 5 9 . D I S T . FBCM THE KCULD 7 2 . 8 CMS 4 4 3 . 4 4 3 . 4 4 1 . 4 3 8 . 4 3 4 . 4 2 9 . 4 2 3 . 4 4 2 . 44 1. 4 4 0 . 4 3 7 . 4 3 3 . 4 2 8 . 4 2 2 . 4 3 8 . 4 3 8 . 4 3 6 . 4 3 3 . 4 2 y . 4 2 4 . 4 1 8 . 4 3 2 . 4 3 1 . 4 3 0 . 4 2 7 . 4 2 3 . 4 1 8 . 4 1 3 . 4 2 4 . 4 2 3 . 4 2 2 . 4 1 9 . 4 1 5 . 4 1 1 . 4 0 5 . 4 14. 4 1 4 . 4 1 2 . 4 1 0 . 4 0 6 . 401 . 3 9 6 . 4C4. 4 0 3 . 4 0 2 . 3 9 9 . 3 9 6 . 3 9 1 . 3 8 6 . 3 9 3 . 3 9 2 . 39 1 . 3 8 8 . 3 6 5 . 3 8 0 . 3 7 5 . 3 8 2 . 3 8 1 . 3 8 0 . 3 7 7 . 3 7 4 . 3 7 0 . 3 6 5 . 3 7 1 . 3 7 1 . 3 7 0 . 3 6 7 . 3 6 4 . 3 6 0 . 3 5 5 . 3 6 2 . 3 6 2 . 3 6 0 . 3 5 8 . 3 5 5 . 3 5 1 . 3 4 6 . D I S T . FECM THE MOULD 8 3 . 2 CMS 4 0 5 . 4 0 4 . 4 0 3 . 4 0 0 . 3 9 7 . 3 9 2 . 3 3 7 . 4 0 4 . 4 0 3 . 4 0 2 . 3 9 9 . 3 9 6 . 3 9 1 . 3 8 6 . 4 C 1 . 4 0 0 . 3 9 9 . 3 9 6 . 3 9 3 . 3 8 9 . 3 8 3 . 3 9 6 . 3 S 6 . 3 9 4 . 3 9 2 . 3 8 9 . 3 6 4 . 3 7 9 . 3 9 1 . 3 9 0 . 3 8 9 . 3 8 6 . 3 8 3 . , 3 7 9 . 3 7 3 . 3 8 3 . 3 8 3 . 3 8 2 . 3 7 9 . 3 7 6 . 3 7 2 . 3 6 7 . 3 7 6 . 3 7 5 . 3 7 4 . 3 7 1 . 3 6 8 . 3 6 4 . 3 5 9 . 3 6 7 . 3 6 7 . 3 6 5 . 3 6 3 . 3 6 0 . 3 5 6 . 3 5 1 . 3 5 8 . 3 5 8 . 3 5 7 . 3 5 4 . 3 5 1 . 3 4 7 . 3 4 3 . 3 5 0 . 3 4 9 . 3 4 8 . 3 4 6 . 3 4 3 . 3 3 9 . 3 3 4 . 3 4 1 . 34 1 . 3 3 9 . 3 3 7 . 3 3 4 . 3 3 1 . 3 2 6 . 237 F B C H . THE MODLC 9 3 . 6 CHS 3 5 8 . , 3 5 4 . 3 5 8 . 3 5 3 . 3 5 6 . 3 5 1 . 3 5 2 . 3 4 8 . 3 4 8 . 3 4 3 . 3 4 3 . 3 3 a . 3 3 6 . , 3 3 2 . 3 3 0 . 3 2 5 . 3 2 3 . 3 1 6 . 3 1 5 . 3 1 1 . 3 0 8 . 3 0 4 . D I S T . 3 7 0 . 3 6 9 . 3 6 7 . 3 6 3 . 3 5 9 . 3 5 3 . 3 4 7 . 3 4 0 . 3 3 3 . 3 2 5 . 3 1 7 . D I S T . 3 3 7 . 3 3 7 . 3 3 5 . 3 3 2 . 3 2 8 . 3 2 4 . 3 1 8 . 3 1 2 . 3 0 6 . 2 9 9 . 2 9 2 . 3 6 9 . 3 6 9 . 3 6 6 . 3 6 3 . 3 5 8 . 3 5 3 . 3 4 7 . 3 4 0 . 3 3 2 . 3 2 5 . 3 17. 3 3 7 . 3 3 6 . 3 3 5 . 3 3 2 . 3 2 8 . 3 2 3 . 3 1 8 . 3 1 2 . 3 0 5 . 2 9 8 . 29 1 . 3 6 8 . 3 6 7 . 3 6 5 . 3 6 2 . 3 5 7 . 3 5 2 . 3 4 5 . 3 3 8 . 3 3 1 . 3 2 4 . 3 1 6 . 3 3 6 . 3 3 5 . 3 3 3 . 3 3 1 . 3 2 7 . 3 2 2 . 3 17. 3 1 1 . 3 0 4 . 2 S 7 . 2 9 0 . 3 6 6 . 3 6 5 . 3 6 3 . 3 5 9 . 3 5 5 . 3 4 9 . 3 4 3 . 3 3 6 . 3 2 9 . 3 2 2 . 3 1 4 . 3 3 4 . 3 3 3 . 32 1 . 3 2 9 . 3 2 5 . 3 2 0 . 3 1 5 . 3 0 9 . 3 0 2 . 2 9 6 . 2 8 9 . 3 6 3 . 3 6 2 . 3 6 0 . 3 5 6 . 3 5 2 . 3 4 6 . 3 4 0 . 3 3 3 . 3 2 6 . 3 1 9 . 3 1 1 . 3 3 1 . 3 3 0 . 3 2 S . 3 2 6 . 3 2 2 . 3 1 7 . 3 1 2 . 3 0 6 . 3 0 0 . 2 9 3 . 2 6 6 . 3 2 7 . 3 2 7 . 3 2 5 . 3 2 2 . 3 1 8 . 3 1 4 . 3 0 8 . 3 0 3 . 2 9 6 . 2 9 0 . 2 6 3 . 3 2 3 . 3 2 2 . 3 2 0 . 3 1 8 . 3 1 4 . 3 0 9 . 3 0 4 . 2 9 9 . 2 9 2 . 2 8 6 . 2 7 9 . FROM T i i E M O O L D 1 0 4 . 0 CMS CIST FSOM THE MOULD 0 . 0 CMS 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 „ 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . 4 3 5 . r i s i JfrCK THE MOULD 1 4 . 2 CMS 3 2 . 3 1. 3 0 . 2 7 . 2 4 . 2 0 . 1 7 . 1 4 . 1 0 . 6 . 2 2 C . 2 1 5 . 2 0 1 . 1 8 2 . 1 5 4 . 1 2 3 . 9 6 . 7 1 . 4 3 . 1 1 . 3 7 8 . 3 7 6 . 3 4 4 . 3 0 1 . 2 3 2 . 1 6 0 . 1 1 4 . 8 0 . 4 7 . 1 1 . 4 2 C . 4 2 0 . 4 1 6 . 3 3 4 . 2 1 3 . 4 b . 2 0 . 1 4 . 1 0 . 6 . 4 2 5 . 4 2 3 . 4 1 8 . 3 4 9 . 2 2 4 . 3 9 . 4 2 7 . 4 2 5 . 4 2 C . 3 9 6 . 2 9 2 . 1 6 1 . 2 6 . 4 2 9 . 4 2 8 . 4 2 4 . 4 1 8 . 3 5 7 . 2 6 1 . 1 4 7 . 2 4 . 4 3 0 . 4 2 9 . 4 2 6 . 4 2 3 . 4 1 7 . 3 4 7 . 2 5 1 . 1 3 9 . 22. 4 3 1 . 4 3 0 . 4 2 8 . 4 2 5 . 42 1 . 4 1 5 . 3 3 4 . 2 2 8 . 1 1 5 . 1 8 . 4 3 0 . 4 3 0 . 4 2 8 . 4 2 6 . 4 2 4 . 4 2 0 . 3 8 4 . 2 8 1 . 1 5 7 . 2 8 . 4 2 7 . 4 2 7 . 4 2 6 . 4 2 5 . 4 2 3 . 4 2 0 . 4 1 3 . 2 9 6 . 1 7 0 . 3 0 . 4 2 2 . 4 2 2 . 4 2 2 . 4 2 1 . 4 2 0 . 4 1 9 . 4 1 0 . 2 8 8 . 1 6 6 . 2 9 . 4 1 2 . 4 1 2 . 4 1 2 . 4 1 2 . 4 1 1 . 3 6 6 . 3 2 4 . 2 4 2 . 1 4 1 . 2 5 . 2 6 C . 2 7 9 . 2 7 9 . 2 7 7 . 2 7 2 . 2 5 5 . 2 2 4 . 1 7 2 . 1 0 2 . 2 0 . 1 6 2 . 1 6 2 . 16 1 . 1 6 0 . 1 5 6 . 1 4 8 . 1 3 0 . 1 0 1 . 6 1 . 1 3 . 3 2 . 3 2 . 3 2 . 3 2 . 3 1 . 3 0 . 2 7 . 2 2 . 1 5 . 6 . Table A 3 . I l l Three-dimensional temperature d i s t r i b u t i o n in z inc jumbo ingo t cast at 1.27 mm/s. 2 3 9 CI £ 3 F E C K T H I HCULD 2 8 . 3 CMS 4 6 . 4 4 . 3 9 . 3 3 . 2 7 . 2 0 . 1 5 . 1 1 . 8 . 6 . 15 1 . 1 4 3 . 1 2 5 . 1 0 4 . 8 0 . 5 7 . 3 6 . 2 5 . 1 6 . fa. 26 1 . 2 4 6 . 2 1 C . 1 7 0 . 1 2 3 . 7 8 . 4 7 . 2 9 . 1 7 . 9 . 3 6 5 . 3 4 4 . 2 8 5 . 2 2 2 . 1 4 6 . 5 6 . 2 2 . 1 3 . 9 . 6 . 4 1 9 . 4 1 5 . 3 3 7 . 2 6 4 . 1 7 5 . 6 1 . 4 2 0 . 4 1 9 . 3 7 3 . 3 0 8 . 2 2 7 . 1 3 5 . 4 3 . 4 2 1 . 4 2 0 . 4 1 4 . 3 5 5 . 2 8 3 . 2 0 2 . 1 1 9 . 3 9 . 4 2 3 . 4 2 2 . 4 2 0 . 4 1 0 . 3 3 6 . 2 6 2 , 1 8 3 . 1 0 6 . 3 4 . 4 2 3 . 4 2 2 . 4 2 1 . 4 1 8 . 3 7 5 . 3 0 8 . 2 3 3 . 1 5 6 . 8 5 . 2 7 . 4 2 2 . 4 2 2 . 4 2 1. 4 2 0 . 4 14. 3 3 9 . 2 6 3 . 1 6 5 . 1 0 9 . 3 8 . 4 2 C . 4 2 0 . 4 2 0 . 4 1 9 . 4 1 4 . 3 3 9 . 2 6 7 . 1 9 1 . 1 1 5 . 4 1 . 4 1 2 . 4 1 2 . 4 1 0 . 3 7 9 . 3 5 0 . 3 0 0 . 2 4 1 . 1 7 5 . 1 C 7 . 3 8 . 3 0 7 . 3 C 6 . 3 0 C . 2 8 7 . 2 6 6 . 2 3 3 . 1 9 0 . 1 4 0 . 8 6 . 3 2 . 2 1 3 . 2 1 2 . 2 0 7 . 1 9 9 . 1 8 4 . 1 6 3 . 1 3 4 . 1 0 0 . 6 2 . 2 4 . 1 3 3 . 1 3 3 . 1 2 9 . 1 2 4 . 1 1 5 . 1 0 2 . 8 5 . 6 4 . 4 1 . 1 7 . 5 2 . 5 2 . 5 1. 4 9 . 4 6 . 4 1 . 3 5 . 2 7 . 1 8 . 9 . C I S T F E C K THE MOU1E 4 2 . 5 CMS 4 4 . 4 2 . 3 7 . 3 1 . 2 5 . 1 9 . 1 4 . 1 0 . 8 . 6 . 1 1 5 . 1 0 9 . 9 4 . 7 8 . 6 0 . 4 3 . 2 9 . 1 9 . 1 3 . 8 . 1 9 C . 1 7 9 . 1 5 3 . 1 2 4 . 9 1 . 5 9 . 3 6 . 2 2 . 1 4 . 8 . 2 6 9 . 2 5 2 . 2 1 3 . 1 6 9 . 1 1 5 . 5 2 . 2 2 . 1 3 . 9 . 6 . 3 4 2 . 3 1 7 . 2 6 8 . 2 1 3 . 1 4 7 . 6 3 . 4 1 5 . 3 7 4 . 3 1 6 . 2 5 8 . 1 9 2 . 1 2 0 . 4 7 . 4 2 C . 4 1 5 . 3 5 5 . 3 0 0 . 2 3 8 . 1 7 1 . 1 0 4 . 4 1 . 4 2 0 . 4 1 9 . 3 8 5 . 3 3 6 . 2 7 8 . 2 14. 1 5 1 . 9 0 . 3 5 . 4 2 C . 4 2 0 . 4 1 6 . 3 6 3 . 3 0 6 . 2 4 5 . 1 8 3 . 1 2 4 . 7 1 . 2 6 . 4 2 0 . 4 2 0 . 4 1 8 . 3 7 2 . 3 1 6 . 2 5 6 . 1 9 6 . 141 . 6 7 . 3 8 . 4 1 4 . 4 1 3 . 3 8 5 . 3 4 7 . 2 9 9 . 2 4 8 . 1 9 4 . 1 4 1 . 8 9 . 3 9 . 3 3 7 . 3 3 3 . 3 1 6 . 2 9 1 . 2 5 6 . 2 1 6 . 1 7 3 . 1 2 7 . 81 . 3 6 . 2 5 5 . 2 5 1 . 2 3 9 . 2 2 2 . 1 9 8 . 1 7 0 . 1 3 7 . 1 0 2 . 6 6 . 3 0 . 1 8 1 . 1 7 8 . 1 7 0 . 1 5 8 . 1 4 2 . 1 2 3 . 1 0 0 . 7 5 . 4 9 . 2 3 . 1 2 G . 1 1 8 . 1 1 2 . 1 0 5 . 9 5 . 8 2 . 6 7 . 5 1 . 3 4 . 1 7 . 5 7 . 5 6 . 5 4 . 5 0 . 4 6 . 4 0 . 3 3 . 2 6 . 1 8 . 1 0 . E I S T F B O f l T B I MOULD 5 6 . 6 C M S 4 4 . 4 2 . 3 7 . 3 2 . 2 6 . 2 0 . 1 5 . S 3 . 8 8 . 7 7 . 6 5 . 5 1 . 3 7 . 2 6 . 1 4 7 . 1 3 9 . 1 2 1 . 1 0 0 . 7 6 . 5 1 . 3 2 . 2 C 7 . 1 9 6 . 1 6 9 . 1 3 7 . 9 7 . 5 0 . 2 3 . 2 6 5 . 2 5 0 . 2 1 5 . 1 7 6 . 1 2 6 . 6 5 . 3 2 2 . 3 0 2 . 2 6 C . 2 1 6 . 1 6 4 . 1 0 8 . 5 1 . 3 8 0 . 3 5 4 . 3 0 3 . 2 5 4 . 2 0 2 . 1 4 8 . 9 4 . 4 1 9 . 4 1 0 . 3 3 8 . 2 8 5 . 2 3 2 . 1 7 9 . 1 2 8 . 4 2 0 . 4 1 6 . 3 5 1. 3 0 0 . 2 4 9 . 1 9 9 . 1 5 0 . 4 1 6 . 4 1 1 . 3 4 3 . 2 9 6 . 2 5 0 . 2 0 3 . 1 5 8 . 3 4 9 . 3 3 9 . 3 0 4 . 2 7 0 . 2 3 2 . 1 9 3 . 1 5 2 . 2 8 5 . 2 7 6 . 2 5 4 . 2 2 9 . 2 0 U . 1 6 8 . 1 3 5 . 2 1 7 . 2 1 1 . 1 9 6 . 1 7 9 . 1 5 8 . 1 3 4 . 1 0 6 . 1 5 8 . 1 5 4 . 1 4 4 . 1 3 2 . 1 1 7 . 1 0 0 . 8 1 . 1 0 9 . 1 0 7 . 1 0 0 . 9 2 . 8 2 . 7 1 . 5 6 . 6 1 . 6 0 . 5 6 . 5 2 . 4 7 . 4 0 . 3 4 . E I S T F B C M Ttil MOULD 7 0 . 8 CM c 4 4 . 4 2 . 3 8 . 3 3 . 2 7 . 2 1 . 1 6 . 8 C . 7 6 . 6 7 . 5 8 . 4 6 . 3 5 . 2 5 . 1 2 0 . 1 1 4 . 1 0 0 . 8 5 . 6 6 . 4 6 . 3 0 . 1 6 5 . 1 5 7 . 1 3 0 . 1 1 4 . 8 5 . 4 9 . 2 5 . 2 1 0 . 2 0 0 . 1 7 5 . 1 4 6 . 1 1 0 . 6 5 . 2 5 1 . 2 3 9 . 2 1 0 . 1 7 8 . 1 4 0 . 9 6 . 5 4 . 2 9 0 . 2 7 5 . 2 4 3 . 2 0 8 . 1 6 9 . 1 2 7 . 6 6 . 3 2 3 . 3 0 3 . 2 6 7 . 2 3 1 . 1 9 1 . 1 5 0 . 1 1 0 . 3 3 8 . 3 1 4 . 2 7 7 . 2 4 1 . 2 0 3 . 1 6 4 . 1 2 6 . 3 1 6 . 2 9 9 . 2 6 7 . 2 3 6 . 2 0 1 . 1 6 5 . 1 3 0 . 2 7 8 . 2 6 6 . 2 4 2 . 2 1 6 . 1 8 7 . 1 5 6 . 1 2 5 . 2 3 2 . 2 2 4 . 2 C 6 . 1 8 6 . 1t>3. 1 3 7 . 1 1 1 . 1 8 1 . 1 7 6 . 1 6 3 . 1 4 8 . 1 3 1 . 1 1 1 . 9 1 . 1 3 7 . 1 3 3 . 1 2 4 . 1 1 3 . 1 0 0 . 8 6 . 7 1 . 1 0 0 . 9 8 . 9 1. 8 4 . 7 4 . 6 4 . 5 3 . 6 4 . 6 3 . 5 9 . 5 4 . 4 8 . 4 2 . 3 5 . 1 1 . 8 . 7 . 1 8 . 1 2 . 8 . 2 0 . 1 3 . 9 . 1 4 . 9 . 7 . 4 5 . 8 0 . 3 7 . 1 0 4 . 6 3 . 3 0 . 1 1 4 . 7 4 o 3 8 . 1 1 2 . 7 5 . 3 9 . 1 0 1 . 6 8 . 3 6 . 8 2 . 5 6 . 3 0 . 6 2 . 4 3 . 2 4 . 4 5 . 3 1 „ 1 8 . 2 7 . 1 9 . 1 2 . 1 2 . 9 . 7 . 1 8 . 1 3 . 9 . 2 0 . 1 4 . 1 0 . 1 5 . 1 0 . 6 . 4 8 . 7 3 . 4 0 . S O . 5 8 . 3 3 . 9 7 . 6 6 . 4 0 . 9 5 . 6 6 . 4 0 . 8 5 . 6 1 . 3 7 . 7 1 . 5 1 . 3 1 . 5 5 . 4 0 . 2 5 . 4 2 . 3 1 . 2 0 . 2 8 . 2 1 . 1 4 . C I S T FfiOM THE MOULD 8 5 . 0 CMS 241 4 5 . 4 3 . 3 9 . 3 4 . 2 9 . 7 0 . 6 8 . 6 1 . 5 3 . 4 3 . 1 0 0 . 9 6 . 8 5 . 7 3 . 5 9 . 1 3 3 . 1 2 8 . 1 1 3 . 9 6 . 7 4 . 1 6 5 . 1 5 8 . 14 1 . 1 2 0 . 9 5 . 1 9 2 . 1 8 4 . 1 6 5 . 1 4 4 . 1 1 7 . 2 1 4 . 2 0 6 . 1 8 7 . 1 6 4 . 1 3 7 . 2 2 9 . 2 2 1 . 2 0 2 . 1 7 9 . 1 5 3 . 2 3 4 . 2 2 6 . 2 0 7 . 1 8 6 . 1 6 0 . 2 2 7 . 2 2 0 . 2 0 2 . 1 8 3 . 1 5 9 . 2 0 9 . 2 0 2 . 1 8 7 . 1 7 0 . 1 5 0 . 1 8 1 . 1 7 6 . 1 6 4 . 1 5 0 . 1 3 3 . 1 4 8 . 1 4 4 . 1 3 4 . 1 2 3 . 1 1 0 . 1 1 6 . 1 1 4 . 1 0 6 . 9 8 . 8 7 . 9 1 . 8 6 . 8 3 . 7 7 . 6 9 . 6 5 . 6 3 . 6 0 . 5 5 . 5 0 . E I S T FBOM THE MOULD 99 4 5 . 4 3 . 3 9 . 3 5 . 3 0 . 6 3 . 6 1 . 5 5 . 4 9 . 4 1 . 8 4 . 8 1 . 7 3 . 6 4 . 5 3 . 1 0 8 . 1 0 4 . 9 4 . 8 1 . 6 5 . 1 3 0 . 1 2 5 . 1 1 3 . 9 9 . 8 2 . 1 4 8 . 1 4 3 . 1 3 1 . 1 1 6 . S 7 . 1 6 3 . 1 5 8 . 1 4 5 . 1 3 0 . 1 1 2 . 1 7 3 . 1 6 8 . 1 5 5 . 1 4 0 . 1 2 2 . 1 7 6 . 1 7 1 . 1 5 9 . 1 4 5 . 1 2 8 . 1 7 2 . 1 6 7 . 1 5 6 . 1 4 3 . 1 2 7 . 1 6 1 . 1 5 6 . 1 4 6 . 1 3 5 . 1 2 0 . 1 4 3 . 1 3 9 . 13 1. 1 2 1 . 1 0 9 . 12 0 . 1 1 8 . 1 1 1 . 1 0 2 . 9 2 . 9 9 . 9 7 . 9 1 . 8 5 . 7 7 . 8 1 . 8 0 . 7 5 . 7 0 . 6 3 . 6 4 . 6 2 . 5 9 . 5 5 . 5 0 . 2 3 . 1 8 . 1 4 . 1 1 . 9 . 3 4 . 2 5 . 1 6 . 1 4 . 1 0 . 4 3 . 3 0 . 2 1 . 1 5 . 1 1 . 4 8 . 2 6 . 1 7 . 1 2 . 9 . 6 4 . 6 7 . 5 5 . 1 0 8 . 7 8 . 4 9 . 1 2 4 . S 5 . 6 7 . ' 4 2 . 1 3 3 . 1 0 5 . 7 9 . 5 5 . 3 5 . 1 3 4 . 1 0 9 . 8 4 . 61 . 4 1 . 1 2 7 . 1 0 4 . 8 2 . 6 1 . 4 1 . 1 1 4 . S 4 . 7 5 . 5 6 . 3 6 . 9 5 . 7 9 . 6 3 . 4 6 . 3 3 . 7 6 . 6 4 . 5 1 . 3 9 . 2 7 . 6 0 . 5 0 . 4 1 . 3 1 . 2 2 . 4 3 . 3 7 . 3 0 . 2 4 . 1 7 . 1 CMS 2 4 . 1 9 . 1 5 . 1 2 . 1 0 . 3 3 . 2 5 . 1 9 . 1 5 . 1 2 . 4 1 . 2 9 . 2 1 . 1 6 . 1 3 . 4 6 . 2 7 . 1 8 . 1 4 . 1 1 . 6 0 . 7 7 . «; e ^ w . 9 1 . 7 0 . 4 9 . 1 0 2 . 6 2 . 6 2 . 4 3 . 1 0 9 . 8 9 . 7 0 . 5 2 . 3 7 . 1 0 9 . 9 1 . 7 3 . 5 6 „ 4 2 . 1 0 5 . 8 8 . 7 2 . 5 6 . 4 2 . 9 5 . 8 0 . 6 6 . 5 2 . 3 9 . 8 1 . 6 9 . 5 7 . 4 6 . 3 4 . 6 8 . 5 8 . 4 8 . 3 9 . 2 9 . 5 6 . 4 8 . 4 0 . 3 2 . 2 5 . 4 4 . 3 £ . 3 2 . 2 6 . 2 0 . C I S T F F C M T H E M O U L D 1 1 3 . 3 C M S 242 4 4 . 4 2 . 3 9 . 3 5 . 3 1. 2 6 . \ 2 1 . 5 7 . 5 5 . 5 0 . 4 5 . 3 9 . 3 2 . 2 6 . 7 2 . 6 9 . 6 3 . 5 6 . 4 8 . 3 8 . 2 9 . 8 9 . 8 6 . 7 8 . 6 9 . 5 8 . 4 3 . 2 8 . 1 0 4 . 1 0 1 . 9 3 . 6 3 . 7 0 . 5 6 . 1 1 7 . 1 1 3 . 1 C 5 . 9 5 . 8 2 . 6 8 . 5 3 . 1 2 7 . 1 2 4 . 1 1 5 . 1 0 5 . 9 2 . 7 8 . 6 3 . 1 3 4 . 1 3 0 . 1 2 2 . 1 1 2 . 9 9 . 8 6 . 7 1 . 1 3 6 . 1 3 3 . 1 2 5 . 1 1 5 . 1 0 3 . 9 0 . 7 6 . 1 3 4 . 1 3 1 . 1 2 3 . 1 1 4 . 1 0 3 . S O . 7 7 . 1 2 6 . 1 2 3 . 1 1 6 . 1 0 6 . 9 6 . 8 7 . 7 5 . 1 1 5 . 1 1 2 . 1 0 6 . 9 9 . 9 0 . 8 0 . 7 0 . 9 9 . 9 7 . 9 2 . 8 6 . 7 9 . 7 0 . 6 2 . 8 5 . 6 3 . 7 9 . 7 4 . 6 6 . 6 1 . 5 3 . 7 3 . 7 2 . 6 6 . 6 4 . 5 9 . 5 3 . 4 6 . 6 1 . 6 0 . 5 7 . 5 3 . 4 9 . 4 4 . 3 9 . L I S T F F C M T H E MOULD 1 2 7 . 4 CM c 4 2 . 4 1 . 3 8 . 3 4 . 3 0 . 2 6 . 2 2 . 1 8 . 1 6 . 1 4 . 5 C . 4 9 . 4 6 . 4 1 . 3 6 . 3 1 . 2 6 . 2 1 . 1 3 . 1 5 . 6 1 . 5 9 . 5 5 . 5 0 . 4 3 . 3 6 . 2 6 . 2 2 . 1 8 . 1 6 . 7 3 . 7 1 . "66. 5 9 . 5 1 . 4 0 . 2 6 . 2 1 . 1 7 . 1 4 . 8 4 . 8 2 . 7 6 . 6 9 . 6 1. 5 2 . 9 3 . 9 1 . 8 5 . 7 8 . 6 9 . 6 0 . 5 0 . 1 0 0 . 9 8 . 9 2 . 8 5 . 7 6 . 6 7 . 5 7 . 4 7 . 1 0 5 . 1 C 3 . 9 7 . 9 0 . 8 1 . 7 2 . 6 2 . 5 2 . 4 3 . 1 0 7 . 1 0 5 . 9 9 . 9 2 . 8 4 . 7 5 . 6 5 . 5 6 . 4 7 . 3 9 . 1 0 5 . 1 G 3 . 9 8 . 9 1 . 8 4 . 7 5 . 6 6 . 5 7 . 4 9 . 4 1. 1 0 0 . s e . 9 4 . 8 8 . 8 1 . 7 3 . 6 4 . 5 6 , 4 6 . 4 1 . 9 3 . 9 1 . 8 7 . 8 1 . 7 5 . 6 8 . 6 0 . 5 3 . 4 6 . 3 S . 8 3 . 8 1 . 7 7 . 7 3 . 6 7 . 6 1 . 5 5 . 4 8 . 4 2 . 3 6 . 7 3 . 7 2 . 6 9 . 6 5 . 6 0 . 5 5 . 4 9 . 4 3 . 3 7 . 3 2 . 6 5 . 6 4 . 6 1 . 5 8 . 5 4 . 4 9 . 4 4 . 3 9 . 3 4 . 2 9 . 5 7 . 5 6 . 5 4 . 5 1 . 4 7 . 4 3 . 3 9 . 3 4 . 3 0 . 2 6 . 1 7 . 1 4 . 1 2 . 2 0 . 1 6 . 1 4 . 2 2 . 1 7 . 1 4 . 2 0 . 1 5 . 1 3 . 4 9 . 5 7 . 4 3 . 6 2 . 5 0 . 3 9 . 6 5 . 5 3 . 4 2 . 6 3 . 5 2 . 4 2 . 5 9 . 4 9 . 4 0 . 5 2 . 4 4 . 3 6 . 4 6 . 3 8 . 3 1 . 4 0 . 3 3 . 2 7 . 3 4 . 2 d . 2 4 . EIS1 FEOM THE HCU1D 141.6 C B S J 8 . 37 . 34. 32. 28. 25. 21. 18. 16. 14. 43. 42. 39. 36. 32. 28. 24. 2 0 . 17. 15. 51. 4 9 . 4 6 . 42. 37. 32. 26. 21. 18. 16. 5 9 . 58. 54. 49. 43. 36. 26. 2 0 . 17. 15. 6 7 . 6 5 . 61 . 57. 51. 4 5 . / 7 3. 72. 68. 63. 57. 51. 45. 79. 77. 73. 68. 62. 56. 4 9 . 4 3 . / 82. £ 1 . 76. 72. 66. 59. 52. 4 6 . 3 9 . 84. 8 2 . 78. 73. 68. 61. 54. 4 8 . 4 2 . 37. 8 3 . 61. 77. 73. 67. 61. 55. 4 9 . 4 3 . 3 8 . 7 9 . 7 8 . 74. 70. 65. 60. 54. 4 8 . 4 2 . 38. 74. 73. 70. 66. . 61. 56. 51. 45. 4 0 . 36. 6 7 . 66. 6 3 . 60. 56. 52. 47. 42. 37. 33. 61 . 60. 56. 55. 51. 47. 43. 38. 3 4 . 31. 56. 5 5 . 53. 50. 47. 43. 39. 35. 3 2 . 26. 51. 5 0 . 48. 46. 43. 39. 36. 32. 2 9 . 2 b . 

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