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Aspects of mould design in electroslag casting Sathaye, Jayant Moreshwar 1983

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ASPECTS OF MOULD DESIGN IN ELECTROSLAG CASTING by JAYANT MORESHWAR SATHAYE B.Tech., Indian I n s t i t u t e Of Technology, Madras, 1977  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of M e t a l l u r g i c a l  Engineering  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA February 1983  ©  Jayant Moreshwar Sathaye, 1983  In  presenting  this  thesis  in  partial  fulfilment  of  the  requirements f o r an advanced degree at the U n i v e r s i t y  of  British  Columbia,  I  it  freely  available  for  permission  agree  her  the  Library  shall  reference  and  study.  I  for  extensive  be  granted by  purposes may or  that  the Head of my It  of t h i s t h e s i s  allowed without my  Department of  written  further  agree  copying of t h i s t h e s i s f o r  representatives.  publication  make  for  is  financial  permission.  Metallurgical  Engineering  The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  scholarly  Department or  understood gain  that  that  by  his  copying or  shall  not  be  Abstract Electroslag  casting  i s a process technique used to produce  castings using e l e c t r o s l a g melting. cooled  It i s l i k e l y  Aluminium  moulds  w i l l be employed.  f o r the design of  moulds  to  be  used  that  channel  Thus, the c r i t e r i a  in  this  process  were  examined. Temperatures  i n a mould s e c t i o n were measured and compared  to those c a l c u l a t e d from a model developed steady  state  the model  the  for  The  two  major  boundary  conditions  heat t r a n s f e r  sensor was  found. the  In heat  to the c o o l a n t and the heat f l u x on the hot correlations  i n tubes such as the Petukhov and the S i e d e r -  The l a t t e r had to be measured and a  heat  were made to q u a l i t a t i v e l y e x p l a i n the heat  v a r i a t i o n along the mould h e i g h t , but i t the  simple manner. proposed  quasi  flux  developed f o r t h i s .  Proposals  estimate  are  former can be estimated from e s t a b l i s h e d  Tate r e l a t i o n s .  3-D  heat flow and reasonable agreement was  transfer coefficient face.  assuming  the  c a s t i n g was  heat  flux  was  not  possible  found to be  to  f o r a c a s t i n g of a r b i t r a r y shape i n a  According to the s l a g c r y s t a l l i s a t i o n influence  flux  mechanism  of mould on the s u r f a c e q u a l i t y of the negligible.  Table of Contents Abstract L i s t of Tables L i s t of F i g u r e s Acknowledgements  i i v vi ix  Chapter I INTRODUCTION 1.1 The E l e c t r o s l a g Process 1 .2 The Mould 1.2.1 Heat Flow In And To The Mould 1.2.2 Heat T r a n s f e r To The Mould In Mathematical Models 1.2.3 Design Of Moulds 1.2.4 Water Cooled Moulds Used Elsewhere 1.2.5 Probable Mould Type In ESC 1.3 Surface Q u a l i t y 1.3.1 Slag Skin C r y s t a l l i s a t i o n 1.3.2 Slag Skin E f f e c t s On Surface Q u a l i t y 1 . 4 Objectives  1 1 2 2 7 8 11 11 12 13 15 16  Chapter II EXPERIMENTAL 17 2.1 Experiments On The Furnace 17 2.1.1 The E l e c t r o s l a g Furnace At UBC 17 2.1.2 Temperature Measurements 18 2.1.3 The Heat Flow Sensor 22 2.1.4 V e r i f i c a t i o n Of Sensor Data 22 2.1.5 Experimental R e s u l t s 24 2.2 O b s e r v a t i o n s On Slag Skins 36 2.2.1 M i c r o s t r u c t u r e 36 2.2.2 O b s e r v a t i o n s On The SEM 42 2.2.3 The 70:30 S l a g Cap Near The L i q u i d Metal Meniscus 48 Chapter I I I CALCULATIONS 3.1 C a l c u l a t i o n s In 2 Dimensions 3.2 C a l c u l a t i o n s In 3 Dimensions 3.3 S p e c u l a t i o n On The E f f e c t Of Grooves On Surface Quality  50 .50 52 60  Chapter IV DISCUSSION 62 4.1 F a i l u r e Of 2-D Model 62 4.2 Apparent Success Of 3-D Quasi Steady S t a t e Model ..65 4.3 F a c t o r s A f f e c t i n g The Heat Flux To The Mould 67 4.4 C r y s t a l l i s a t i o n In Slag Skins 72 4.5 S l a g Composition E f f e c t s On The Heat F l u x 74 4.6 F a c t o r s A f f e c t i n g Surface Q u a l i t y 76  iv  Chapter V SUMMARY  78  REFERENCES  79  APPENDIX A - SLAG NOTATION  83  APPENDIX B - MISC. DATA AND CORRELATIONS  84  APPENDIX C - FORTRAN PROGRAM TO CALCULATE TEMPERATURES IN CHANNEL COOLED MOULDS 86 APPENDIX D - FORTRAN PROGRAM REFERRED TO IN SECTION 3.3 .101 APPENDIX E - F.E. FORMULATION PLANE'  FOR STEADY HEAT TRANSFER IN A 117  V  List  of T a b l e s  I.  Thermal R e s i s t a n c e - from Kusamichi, et a l  II.  Record of e x p e r i m e n t a l runs  25  III.  Comparison:  35  IV.  Chemical a n a l y s i s of some s k i n s  48  V.  Possible Error  65  VI.  Temp, drop through s l a g s k i n  67  VII.  Heat  68  Heat  flow and Power input  i n heat flow measurement  flow through a i r gap  .....6  vi  L i s t of F i g u r e s  1. Heat Loss v s . Parameter F  4  2. Heat Flux V a r i a t i o n  4  3. Sketch of Mould Assembly  19  4. Views of Mould Assembly  19  5. Dimensions of Mould S e c t i o n used  20  6. Dimensions of Copper S e c t i o n  21  7. Sketch of Heat Flux Sensor  23  8. For V e r i f i c a t i o n  23  of Sensor  9. E l e c t r o d e used i n Run 10 10. Temp. H i s t o r y of RUN 3  26 •  26  11. Smoothened Temp. H i s t o r y of Run 3  27  12. C a l c u l a t e d Temp, f o r Run 3  27  13. Temp.History of Run 2  28  14. C a l c u l a t e d Temp, f o r Run 2  28  15. Temp.History of Run 6  29  16. C a l c u l a t e d Temp, f o r Run 6  29  17. Temp.History of Run 10  30  18. C a l c u l a t e d Temp, f o r Run 10  30  19. Temp.History of Run 8  31  20. Heat Flux Measurement-Run 2  31  21. Heat Flux Measurement-Run 3  32  22. Heat Flux Measurement-Run 4  ....32  23. Heat Flux Measurement-Run 6  33  24. Heat Flux Measurement-Run 11  33  25. Sample comparison between C a l c u l a t e d temperature and  vi i  measured  34  26.  SEM p i c t u r e of 70:30 s k i n  ...37  27.  Spike on 70:30 s k i n  37  28.  70:15:15 s l a g s k i n  38  29.  50:30:20 s l a g s k i n  38  30.  Skin  31.  Skins of complex s l a g  39  32.  I n d u s t r i a l 70:30 s l a g s k i n  40  33.  I n d u s t r i a l 50:30:20 s l a g s k i n  40  34.  70:30 s l a g cap near metal meniscus  41  35.  70:30 s l a g cap near top  41  36.  EDX a n a l y s i s of 70:30 s k i n  44  37.  EDX a n a l y s i s of 70:15:15 s k i n  45  38.  EDX a n a l y s i s of 50:30:20 s k i n  46  39.  EDX a n a l y s i s of complex s l a g s k i n  47  40.  Temp, f i e l d c a l c . with 2-D assumptions  51  41.  The s l i c e which was d i s c r e t i s e d  51  42.  Calculated  Temp, f i e l d s  56  43.  Calculated  Temp, f i e l d s  57  44.  Calculated  Temp, f i e l d s  58  45.  Section  46.  Calculated  47.  Error  48.  Hypothetical  49.  Phase diagram f o r C a F - A I 2 O 3  69  50.  Phase diagram f o r C a F - A l 0 -CaO  73  51.  Shape f u n c t i o n s - R e c t .  of S i l i c a , Magnesia c o n t a i n i n g  slag  taken f o r F.E. c a l c u l a t i o n  58  r e s u l t s of s p e c u l a t i o n  59  i n heat f l u x measurement  64  heat f l u x v a r i a t i o n  64  2  2  2  3  isoparametric  39  element  118  vi i i  52.  Shape f u n c t i o n s - T r i a n g u l a r  element  ix  Acknowledgement  I would l i k e t o thank P r o f . A l e c M i t c h e l l time  he spared  f o r a l l the  f o r me and e s p e c i a l l y f o r h i s s t y l e of  guidance. Gus S i d l a ' s a s s i s t a n c e was i n v a l u a b l e and i s very much a p p r e c i a t e d . Ed B a r r y and Rudy Cardeno were h e l p f u l d u r i n g the e x p e r i m e n t a l The  pleasant  nature  department and of f e l l o w  very  stages. of  graduate  the  staff  students  in made  the my  s t a y very e n j o y a b l e . I  a l s o wish to  express my  C o r p o r a t i o n , Rancocas, New J e r s e y support.  gratitude  t o Consarc  for their  financial  1  I.  INTRODUCTION  1.1 The E l e c t r o s l a q Process The early  u t i l i s a t i o n of the e l e c t r o s l a g  1940's  with  the  Kellogg  electric  process was o r i g i n a l l y developed industry  d i d not  Investigators the  weld  involve  by  itself  i n the Soviet Union,  metal  electric  aim  furnace  castings  of  i n the  ingot p r o c e s s .  R.K.Hopkins,  but  seriously t i l l meanwhile,  The  western  the 1960's.  discovered  that  steel.  countries  They then developed the process  improving Since  too. Following  many other  began  i n an e l e c t r o s l a g weld was purer and had fewer  i n c l u s i o n s than the base metal. with the prime  process  the  quality  of  air-melted  then, i t has been used t o produce  t h i s l e a d the process has been used i n  i n the manufacture of i n g o t s .  A  Japanese  firm  uses  t h i s process to manufacture reformer tubes but apart  from  this  there  applications  of  are  almost  electroslag  no  significant  engineering  c a s t metal, as i s , i n the western  world. The  e l e c t r o s l a g process i n v o l v e s the  electrode  through  the  significant within  electrode  and  An e l e c t r i c p o t e n t i a l the mould;  r e s i s t a n c e and almost  i t . The e l e c t r o d e  solidifies  of  a  solid  a l i q u i d e l e c t r i c a l l y conducting s l a g i n t o a  water cooled m e t a l l i c mould. across  melting  is  applied  The s l a g forms the only  a l l the  heat  is  generated  i s melted p r o g r e s s i v e l y and the ingot  in s i m i l a r fashion,  i.e.  progressively  and  to  a  2  certain  extent  directionally.  number of i n c l u s i o n s and can  I t reduces,  control  the  significantly,  solidification  thus e l i m i n a t i n g shrinkage p o r o s i t y and minimising The  produced  metal  rate  segregation.  i s c o n s i d e r e d more r e l i a b l e .  Attempts are  being made to e s t a b l i s h e l e c t r o s l a g c a s t i n g s i n e n g i n e e r i n g The  be  an  c a s t i n g and  1.2  The  important  This s i g n i f i e s  aspect  that  mould  i n d e l i b e r a t i o n s about  design  electroslag  f r e s h designs w i l l be r e q u i r e d f o r every new  shape.  Mould  The mould, i n g e n e r a l , i s a c o n t a i n e r f o r the l i q u i d and  serves  to  shape  the  casting.  Additionally  minimise the p r o b a b i l i t y of such d e f e c t s as principles  of mould design u t i l i s e d  a p p l i c a b l e here. able  use.  term ' c a s t i n g ' i m p l i e s that complex shapes, r e l a t i v e to  i n g o t s , w i l l be c o n s i d e r e d . will  the  to  The  abstract  hot  e l e c t r o s l a g process and  heat  it  should  tears.  Basic  i n c o n v e n t i o n a l c a s t i n g are  e x t r a f e a t u r e i s that the  the  metal  fluxes  liable  mould to  must  be  occur  i n the  should f o s t e r the formation of a  smooth  surface.  1.2.1  Heat Flow In And To The Most  published  work  Mould  on heat  t r a n s f e r to the mould i n the  e l e c t r o s l a g process has c o n c e n t r a t e d on the c o n d i t i o n s o c c u r r i n g in ingot r e m e l t i n g . in  the  slag  passes  Nearly 35-50 percent of the heat directly  to  generated  the mould and another  20-25  3  percent passes i n d i r e c t l y to the m o u l d . ' ' ' " The heat f l u x  has  been found to r e f l e c t  the  1  the p a t t e r n of heat  2  3  generation  with  peak heat f l u x being i n the range of 1.6 MW/m. 2  Early  work  on the heat balance of the process was  S o v i e t workers at Kiev, the essence of which book  edited  configuration metal  slag  by  Medovar  (bifilar,  et  al.  interfacial  area;  developed f o r t h i s as shown i n with  on  between  the above.  the  electrode  3-phase, e t c . ) , the s l a g depth  input goes d i r e c t l y to the s l a g .  varies  i s reviewed i n  Depending  1  done by  and  the  46 to 65% of the power  An e m p i r i c a l r e l a t i o n has been fig.  In a d d i t i o n ,  1.  The  heat  flux  too  i t v a r i e s with the h e i g h t  of the ingot c a s t i n g , the power input and the mould Many measurements of heat f l u x are r e p o r t e d .  shape. In some cases  two peaks appear, one a t the a i r s l a g i n t e r f a c e and the other a t the s l a g metal i n t e r f a c e and i n other occur,  but  there  is  a  with  a  peak  does  not  broad band of high heat f l u x from the  l i q u i d s l a g , as i s shown in f i g 2. reduces  cases  The heat f l u x from the  slag  i n c r e a s i n g l i q u i d metal head, i n c r e a s i n g d i s t a n c e  from the e l e c t r o d e and d e c r e a s i n g power input.  As the s l a g  t h i c k n e s s i n c r e a s e s , the mid s l a g region heat f l u x /  metal  skin head  f l u x r a t i o tends to come c l o s e r to 1 . Much Joshi.  5  of  this  has  been  confirmed  by H o l z g r u b e r  S i m i l a r energy balances and heat flows  can  be  3  and by deduced  from the work d e s c r i b e d below. Pocklington slab production. was  and  Patrick  6  report  on heat flow i n 3 phase  The moulds were j a c k e t e d and  1 D approximation  used to c a l c u l a t e heat f l u x e s from thermocouples  inserted in  90 80 H  40 •  /  X  » 20-1  2 0 F  Figure  =  4 0  °S-mould  .nlerfoce  area  interface  area"  1 - Heat Loss v s .  Figure  3 0  slag-metal  s l  2 - Heat  Flux  5 0  Parameter  Variation  5  the w a l l .  Heat f l u x v a r i a t i o n s were found  composition  and the r e m e l t i n g c o n d i t i o n s .  were found i n what they c a l l e d  t o depend  Peaks of temperature  'Form' 4 curves, but t h i s d i d not  t r a n s l a t e i n t o peaks of heat f l u x .  Maximum heat f l u x e s were of  the order of 1 - 1.2MW/m f o r most s l a g s and about  1.8 MW/m f o r  2  70:15:15  slag.  p e r i p h e r y was suggested  Greater found  that  variation  compared  small  heat  on s l a g  of heat  to conticast flux  2  flow  around the  moulds.  differentials  It  was  may minimise  d i s t o r t i o n of the mould. Some experiments Research  Centre.  7  A  laboratory furnace. curve  with  were c a r r i e d out by the Albany 3 piece  jacketed  mould  They observed 3 peaks  maximum  flux  of about  d i f f e r e n t i a l of the c o o l i n g water o u t l e t was used to determine  was used i n a  i n t h e heat  3 MW/m.  flux  The temperature  2  between  Metallurgy  the i n l e t  and  the heat f l u x and i t i s l i k e l y  the  that  very high p r e c i s i o n measurements were r e q u i r e d . Kusamichi  et a l  8  conducted  i n t e r e s t i n g experiments  i n which  s l a g was f r o z e n onto a water c o o l e d copper pipe i n a c r u c i b l e of molten so  slag.  that  Two thermocouples  the slag  temperatures  were  conductivities  solidified used  were kept very c l o s e t o the pipe around  t o estimate  were  and t h i s i s shown i n Table I .  for different  crystallisation  the e f f e c t i v e  thermal  The r e s i s t a n c e of the s l a g s k i n and  that of the s l a g s k i n - mould i n t e r f a c e  noted  and the measured  of s l a g s k i n s , the thermal r e s i s t a n c e s , and the  heat t r a n s f e r c o e f f i c i e n t s .  major importance  them  slags,  i s different  found  t o be  of  D i f f e r e n c e s were  but i t must be p o i n t e d out that i n the r e a l p r o c e s s .  6  Table I - Thermal R e s i s t a n c e - from Kusamichi  et a l  a. i n C r u c i b l e CaF 0 50 100  Slag A1 0 2  2  3  50 25 0  Cao  Thermal R e s i s t a n c e (s.deg./Cal) Total BathCrust C r u s t - Cu PipeCrust Pipe Pipe water  50 25 0  35.4 30.3 21 .7  4.6 3.6 2.9  28.5 14.3 5.4  0.19 0.09  0.48 0.21  0.7 10.9 12.0  0.01 0.01 0.01  2.2 1.7 1 .3  0.007 0.007  0.04 0.02  b. i n ESR 0 50  50 25  50 25  0.81 0.43  0.09 0.10  In (b) read mould f o r p i p e . Kondo, Kodama et a l , measured heat 9  MW/m i n 800 mm 2  distribution  i n d u s t r i a l moulds.  was  seen  at  the  A  f l u x e s of up to 0.8 - 1  peak  s l a g metal  in  the  heat  flux  interface level for  70:15:15 s l a g and i t was claimed that r e m e l t i n g of the s l a g of a 70:30 s l a g was  skin  observed.  The d i f f e r e n c e between S o v i e t i n v e s t i g a t o r s and the  others  i s noteworthy.  The S o v i e t s developed  was  i n s u l a t e d from the r e s t of the mould, the s i g n a l  thermally  being generated by a thermopile.  a d i r e c t heat sensor which  The  others  merely  inserted  thermocouples i n t o the mould w a l l and^ j u s t i f i e d the technique by suggesting that 1-D approximation  was a c c e p t a b l e .  7  1.2.2  Heat T r a n s f e r To The Mould In Mathematical Models It  i s of i n t e r e s t to examine the boundary c o n d i t i o n s at the  s l a g mould i n t e r f a c e employed around.  Many  models  did  compare the c o n d i t i o n s  at  Pridgeon  the  determined  1 0  by the v a r i o u s mathematical models not  include  the  top  of  value  for  cals./cm sK. did  not  v a l u e s from workers.  model  .01  to  Joshi  13  to  the .02  advanced  Choudhary heat  by  1 7  '  '  1 9  2  did  Paton it  to  fluid  Schwerdtfeger These workers  and  Szekely  1 9  coefficient  used  wall  functions  recent to  to the s l a g s k i n boundary  transfer  situation  a  is  coefficient attributed  will to  a  used co-  be  .0115  flow  have  been  15  and by D i l a w a r i ,  did  not  consider  t r a n s f e r c o e f f i c i e n t s but f i x e d the s l a g boundary In  1 2  and  .02.  including  1 8  cals./cm sK  estimated  used  and  i t s l i q u i d u s temperature.  heat  1 0  models  and S z e k e l y .  and  rates.  and M a u l v a u l t  as  2  Kreyenberg  heating  0.4  Elliott  cals./cm sK  2  developed  used  1 1  slag.  cals./cm sK w h i l e B a l l a n t y n e More  Sun  h, by p l u n g i n g Copper  measuring  experimentally  5  ingot.  may  the mould v a r i e d from .006 to .00745  C a r v a j a l and G e i g e r  2  they  "h"  the  conductance,  rods i n t o molten s l a g and metal and Their  the s l a g , but one  paper  to be a t  Choudhary  and  e s t i m a t e the heat t r a n s f e r and  predicted  remain nearly  near  that  the  constant.  This  invariant  temperature  g r a d i e n t c l o s e to the mould w a l l i n the s l a g along the l e n g t h of the  mould f a c e .  In g e n e r a l , these m o d e l l e r s were i n t e r e s t e d i n  the p o o l shape and the l o c a l s o l i d i f i c a t i o n of the i n g o t .  time at  the  centre  8  1.2.3  Design Of Moulds In the case of ingot r e m e l t i n g most moulds are j a c k e t e d and  water  cooled  as  a  non-boiling  d e t a i l e d by M i t c h e l l and S m a i l e r . with h o l e s d r i l l e d in  the  system 2  using  principles  Channel-cooled  moulds (moulds  i n the w a l l s ) have a l s o been used  moving-mould  and  ingot-withdrawal  as  especially  techniques.  Early  moulds were made of s t e e l ( with spray c o o l i n g ) but p r a c t i c e soon switched to copper. for was  good  axial  heightened.  longer  and  S t e e l moulds had to be  relatively  heat conduction and the danger of F u r t h e r , copper  so were cheaper  thinner  burn-through  moulds were found to l a s t  i n the long run.  much  Larger moulds are  made from s l a b s of t h i c k e r s e c t i o n s i n c e the s t r e s s e s the  mould  must take are much h i g h e r . Cremisio  and  moulds i n d e t a i l . velocity wall  Zak  2 0  '  have  2 1  d i s c u s s e d the design of  Mould design s t a r t s with  the  r e q u i r e d to maintain n o n - b o i l i n g c o n d i t i o n s .  thickness  is  generally  view that f l a t curved  thermal  surfaces  as  restraint  in  round moulds.  horizontal  and  i n the t h i r d plane.  the the  The design should planes  while  Creep r e s u l t i n g  from  s t r e s s e s i s the c h i e f problem i n such moulds. restraints  a r e s u l t of t h e i r p h y s i c a l c o n s t r u c t i o n .  higher  They are of  vertical  Channel c o o l e d moulds have s u f f i c i e n t as  The mould  s u r f a c e s i n moulds face more demanding c o n d i t i o n s  permit movement i n the maintaining  water  s e l e c t e d from experience and  r e s t of the d e t a i l e d design f o l l o w s from t h i s .  than  minimum  slab  temperatures  frequently  not  be  on used  the in  hot  face.  built  in  However, they face  Such  a  design  f u l l - l e n g t h moulds s i n c e t h i s  may may  9  entail  i n o r d i n a t e l y high water p r e s s u r e s to ensure  high water-flow Soviet and  work has Highly and  rates.  researchers  have  sufficiently  done  were the e a r l y developers of the  extensive  work i n the  field.  The  process  g i s t of t h i s  been summarised i n the book e d i t e d by Medovar  et  al.  1  i n v o l v e d e m p i r i c a l design techniques have been developed  i t appears that t h e i r work i s the most d e t a i l e d p u b l i s h e d  so  far. V a r i o u s methods of water c o o l i n g were jacket  c o o l i n g , channel c o o l i n g and  c o o l i n g with l i q u i d metals and but  it  is  not  evaporative  known  and  forced  l a t t e r being g e n e r a l l y The  if  Soviet  empirical  into practice.  Both  with  the  d i r e c t i o n s ; from  to the mould.  methods mentioned i n s e c t i o n 1 . 2 . 1  chosen  " e f f i c i e n c y " ) and  formulated  considered  from the c o o l a n t  velocity  Taking a s a f e t y  depending  from t h i s the  on  the  detailed  the heat f l u x  factor  mould  Using  the  type  water  ( i . e . the  hydraulic  design  is  out.  The coolant  expected  maximum  interface  is  the  coolant  coolant,  coefficient. using  were  Schemes f o r  were  approach to design i s from two  i s estimated.  carried  put  e.g.,  preferred.  to mould w a l l is  liquids  was  convection  the s l a g to the mould and the  " r i b " cooling.  organic  this  investigated,  The  convection  and  estimated  heat  Borishansky's  mould  temperature from  the  heat  temperature  and  the  transfer  coefficient  correlation "  modified  at  5  the flux  heat is  f o r nucleate  by an e f f i c i e n c y f a c t o r to  mouldto  the  transfer  approximated b o i l i n g forced reflect  the  10  geometry.  The  temperature  the v e r t i c a l and h o r i z o n t a l  distribution sections  i s then worked out. i n  using  carbon  conducting  paper, i . e . a 2-D t e c h n i q u e . For  the v e r t i c a l  s e c t i o n through the c o o l i n g channel, the  heat f l u x estimate i s used as the boundary is  raised  30%  but  f o r the h o r i z o n t a l s e c t i o n a r b i t r a r i l y .  channel the boundary  c o n d i t i o n i s a f i x e d temperature  The maximum temperatures c a l c u l a t e d a r e maximum  condition,  temperature  tolerable  then  this  At the  condition.  compared  to the  and the design accepted on t h i s  basis. The design pays l e s s a t t e n t i o n t o the heat  flow.  to  the  coolant.  In  a  the maximum mould face temperature reached  The t o l e r a b l e temperature was given as 500°C. mentioned  of  axial  The most important parameters are the heat f l u x and  the heat t r a n s f e r c o e f f i c i e n t calculation  influence  No  sample 350°C.  material  was  but from the value of the thermal c o n d u c t i v i t y used i t  i s assumed that the m a t e r i a l was copper. This  r e p r e s e n t s very high mould temperatures when compared  with western d e s i g n .  I t must be p o i n t e d out, however, that most  of t h i s work was done more than 10-15 years  ago  with  probably  l i m i t e d computing r e s o u r c e s . The  approach  p r a c t i c e , mould m a t e r i a l employed to-day.  to  e v a p o r a t i v e mould design i s s i m i l a r .  thicknesses i s steel.  are  lower  (about  5mm)  In  and the  Such moulds a p p a r e n t l y a r e not used  11  1.2.4  Water Cooled Moulds Used Elsewhere Water  c o o l e d moulds are a l s o employed i n other  Vacuum arc m e l t i n g moulds techniques. of  h i g h heat  However, flux  are  also  the heat  using  the  f l u x e s are lower and  same  the region  i s narrow which, p r a c t i c a l l y , means that  water flow v e l o c i t i e s are used. the number of  heats  amortise  cost  the  designed  processes.  obtained of  lower  There i s creep deformation, is  r e p a i r and  sufficient  to  but  economically  the c o s t of the m o u l d .  same remarks are probably a p p l i c a b l e f o r e l e c t r o n  beam  21  The  melting  moulds. Moulds similar  in  heat  study. ' 2 3  2 4  continuous fluxes  casting  and  Earlier,  have  the  been  problem  attempted to be s o l v e d by lowering wall."  Attempts are now  1  intermittent  boiling  of  steel subjected  of  mould  s u b j e c t to  to  intensive  distortion  temperatures  in  the  and  to  gauge  the  2 4  Probable Mould Type In  effect  of  tall.  section.  One  rejection.  ESC  Further there w i l l probably  will  be  channels mould  have c a s t i n g d e f e c t s ( e.g.  d i f f i c u l t to e i t h e r integrally cast.  will  be  made  of  to  be  be frequent changes i n  p i e c e moulds, i f employed, w i l l have to be  Such moulds may  mould  thermal  In e l e c t r o s l a g c a s t i n g , the c a s t i n g i s not expected very  was  being made to reduce the p o s s i b i l i t y of  s t r e s s e s ; and so design t o reduce product  1.2.5  are  cast.  blow-holes ) and i t  j a c k e t them or to have the c o o l i n g  Therefore, short  it  sections  is  likely  joined  that  together  the by  12  mechanical means.  T h i s system  s i n c e i t i s thought  f a v o r s channel c o o l i n g of  moulds  that w a t e r - j a c k e t i n g of each s e c t i o n w i l l  be  too expensive. It  is  unlikely  that  e l e c t r o s l a g c a s t i n g p r o d u c t i o n runs  w i l l be very l a r g e meaning that make  this  commonly  process  noncompetitive.  available,  conductivity temperatures  is  copper  with  a  aluminium.  2 5  The  would  ,  probably  only other m a t e r i a l ,  sufficiently  This  i n the mould w a l l  moulds  change  high  will  cause  thermal higher  but there i s no r e p o r t of  any  untoward e f f e c t on the process or the c a s t metal.  1.3  Surface Q u a l i t y Electroslag  off  i n g o t s have a good s u r f a c e and t h i s was  i n t u i t i v e l y as being due  Etienne  2 6  t h i s was  suggested due  to a t h i n s l a g s k i n .  passed  Mitchell  and  t h a t , as i n other s o l i d i f i c a t i o n p r o c e s s e s ,  to the presence of a c y l i n d r i c a l  l i q u i d metal head.  T h i s i s g e n e r a l l y , accepted with v a r i a t i o n s  ,  existence  f i l m between the s l a g  or  non-existence  of  s k i n and the s o l i d i f y i n g metal. only  as  a  intermittent  result rippling.  of  a liquid  A bad s u r f a c e  meniscus  primarily  can  solidification,  come  on  the  about  leading  to  13  1.3.1 S l a g Skin C r y s t a l l i s a t i o n The use of water c o o l e d moulds makes solid  slag  layer,  inevitable. formation  called  the  formation  of  the s l a g s k i n , on the mould s u r f a c e  There has been l i t t l e p u b l i s h e d i n f o r m a t i o n on of  slag  a  skins  by  the  the  s o l i d i f i c a t i o n of d i f f e r e n t  slags. Sharapov  et a l ,  examined the r o l e of the s l a g s k i n .  6 0  d i v i d e d s l a g s i n t o two major groups, those with two and those with many components. layers  components,  Both types of s l a g caused three  t o form i n the s l a g s k i n with the two l a y e r s nearest the  metal h e l d to form on r e h e a t i n g of the o r i g i n a l s k i n . l a y e r c l o s e s t to the metal was s a i d to be l i q u i d t i l l solidified.  Multi-component  mould processes because Kamensky et a l , slag  2 7  structure  thickness  was  governed  of  a  were  the  preferred  of by  layer. of the  i t s boundary, the s l a g s k i n . heat  transfer.  On  70:30 s l a g s k i n they found f i v e l a y e r s .  l a y e r had c a l c i u m aluminate c r y s t a l s with the gaps  with  f l u o r i t e with the former as high as 70%.  has columnar corundum c r y s t a l s  flow.  in  the  third  filled layer layer  4 i s the same as l a y e r 3 with a small q u a n t i t y of  needles of corundum.  direction.  (upto 80%) and  The second  The  (92%, r e s t g l a s s ) o r i e n t e d i n the d i r e c t i o n of heat  Layer  corundum  The  petrographic  first  has f l u o r i t e  metal  i n moving  c o n s i d e r e d the thermal s t a b i l i t y  the  examination,  slags  The t h i r d  they p e r m i t t e d a t h i c k e r t h i r d  and  found  They  needles  Layer  (12mm  in  5  has  length)  These needles were much granulated  slag.  extremely  long  and  large  o r i e n t e d i n the heat flow longer  than  those  They observed that t h i s  to  be  structure  14  meant that the ocurring.  phenomenon  The  higher  of  fractional  alumina  content.of the s k i n  high ) was a t t r i b u t e d t o t h e r m o d i f f u s i o n enrichment by ions of the type Korousic  and  Osterc  crystallisation  of  ( twice as  fluoride  ions  and  (Al„0 ) . - 2  7  a l s o made a m i n e r a l o g i c  2 8  was  analysis.  They found that the phases c r y s t a l l i s i n g out d i d not f o l l o w phase  diagram.  metal  i o n exchange  The s l a g s k i n was s t a t e d t o form under impeded  crystallisation. vertical  sections  Medovar  et  with  It  consequent  unbalanced  fractional  i s unfortunate that they f a i l e d t o study  c l o s e to the metal meniscus. a l , mention  three  1  boundaries.  The  composition.  The f i r s t  last  layer  layers  with  proportions  no  abrupt  i s s a i d to have a near e u t e c t i c  l a y e r has an e q u i l i b r i u m  forms a t very h i g h c o o l i n g equilibrium  the  rates.  composition and  The middle l a y e r c o n t a i n s non  of phases and multicomponent  slags  are  more prone t o t h i s . B e l l and M i t c h e l l the  2 9  a l s o observed the s l a g s k i n of s l a g s i n  calcium fluoride-alumina  be a near e q u i l i b r i u m  system.  The s l a g s k i n was s a i d to  phase and they deduced the d i s s o l u t i o n and  m e l t i n g of the s k i n , as the metal meniscus approached, ray  maps.  s l a g s using and  CA  6  Mitchell  3 0  has  X-ray d i f f r a c t i o n .  were found.  from  X-  a n a l y s e d the s l a g s k i n s of v a r i o u s In a 70:30 s l a g s k i n C a F , 2  A 40:30:30 s l a g caused C  c r y s t a l l i s e and a 55:35:10 s l a g s k i n had C a F  2  1 2  A  7  and C^k F 7  and CA . 2  Al 0 2  3  to  15  1.3.2  Slag Skin E f f e c t s On In  the  West, the suggestion  been g e n e r a l l y accepted. if  a  The  of M i t c h e l l and E t i e n n e  s l a g composition  surface.  If  h e l d that  some  redissolved  is  the  conducive liquid  of  into  the  Russian  liquid  the  and  formation  leaving  high a  of  are  The  9  film  between  the s o l i d i f i e d s l a g s k i n .  They  envisage  and  rippled  attributed  to  meniscus 3 1  solidification.  tend to agree with the  surface  of  Russian  the ingot i s s a i d to be c o n t r o l l e d by  f a v o r s the formation  Speculation  of the s l a g .  A  the mould warmer.  of a smooth  However, M e l l b e r g  3 2  out that the heat f l u x i n the molten head region of  ingot i s very high and liquid.  This  t h i s may  would  a c t to c o n t r o l the superheat  mean t h a t running  the high  i n i n d u s t r y suggests that smooth skins may  encouraged by running  have no  Some  favor the M i t c h e l l viewpoint.  m e l t i n g phase p r e c i p i t a t i n g out  the  is  v i s u a l i s e a thin l i q u i d  1  the nature  pointed  phase  smooth surface f o r the  f r e e z i n g p o i n t of the a l l o y and  skin.  good  against.  Japanese r e s e a r c h e r s ' o p i n i o n s view, o t h e r s  a  then i t i s  melting  some r e m e l t i n g and/or d i s s o l u t i o n of the s l a g s k i n surfaces  and  solidification  head of metal i s present,  slag  workers  metal  the  precipitated  l i q u i d metal to s o l i d i f y The  to  has  2 6  i s important  high m e l t i n g phase p r e c i p i t a t e s out d u r i n g  then such a s l a g  the  Surface Q u a l i t y  a mould warmer  be has the in may  effect.  In i n d u s t r i a l p r a c t i c e , the problem of a poor been  generally  this  is  limited  dealt due  with to  by  surface  i n c r e a s i n g the power input,  adverse  effects  on  has but  solidification  16  structure.  It  i s a l s o known that a higher f i l l - r a t i o  b e t t e r s u r f a c e i n f i x e d mould p r a c t i c e . depth i s a l s o f a v o r a b l e but t h i s , The  mould,  An  increase  no  i t appears, has no e f f e c t .  mould  can  be  of  Shevtsov et a l , skin  skin  33  3 4  formation  made that has a face temperature i n  excess of the s l a g m e l t i n g p o i n t based on the premise slag  slag  i t i s stated, i s unexpected.  suggest that a mould can have no e f f e c t on s l a g because  favors a  thickness adjusts i t s e l f  that  the  to match the heat f l u x  from  the bulk of the bath t o the mould.  1.4 O b j e c t i v e s The f o r e g o i n g d i s c u s s i o n suggests s t r o n g l y that be  used  in  will  to  e l e c t r o s l a g c a s t i n g w i l l c o n s i s t of c h a n n e l - c o o l e d  s e c t i o n s made from aluminium. used  moulds  normally  investigations  were  c o u l d be designed  be  I t i s noted that e l e c t r o d e s t o be  round  carried  to  bars. out  ensure  With  to  smooth  this  in  mind,  determine i f such moulds surfaces  in  electroslag  casting. It  was  of  interest  to  design  moulds  l i m i t i n g temperature a t the mould s u r f a c e .  to  have a g i v e n  Besides the  obvious  n e c e s s i t y of keeping the mould c o o l enough to prevent d i s a s t r o u s melting, criterion  there  were  suggestions  f o r mould d e s i g n .  I t was  that  temperature  intended  to  i s the  determine i f  t h i s c r i t e r i o n was c o r r e c t or i f t h e r e were other more important criteria  i n mould d e s i g n .  17  II.  EXPERIMENTAL  T h i s chapter i s d i v i d e d i n t o two s e c t i o n s .  The  with experiments made on the 1 tonne e l e c t r o s l a g second  is  f i r s t deals  furnace and the  concerned with o b s e r v a t i o n s of the s l a g s k i n and the  s l a g cap as o b t a i n e d a f t e r m e l t i n g . 2.1  Experiments On The It  at  was  important to measure temperatures i n the mould  different  locations,  model p r e d i c t i o n s . since  it  Furnace  was  s i n c e data was  needed  A heat f l u x sensor a l s o had to be  geometries and v a l u e s of heat f l u x  as input t o mathematical models of heat t r a n s f e r  The E l e c t r o s l a g Furnace At A  detailed  earlier. castings  3 5  '  2 5  up  r a t e d 250 KVA second steps of  developed  description  were  needed  i n the mould.  UBC  of the furnace has been presented  The furnace  is  capable  of  melting  to 1 tonne.  There are two t r a n s f o r m e r s , the  ingots  with v o l t a g e dropping from 12.5KV to 600V and  delivering  electrical  power  at  and first the  between 25V to 60V i n  2.5V.  The e l e c t r o d e holder i s water c o o l e d and up  to compare with  known that the heat f l u x v a r i e d w i t h p o s i t i o n i n  non-symmetrical  2.1.1  wall  and down between aluminium  i s suspended  r a i l guides.  from a c h a i n , the c h a i n d r i v e n  moves  vertically  The h o l d e r c a r r i a g e by  a  motor  whose  18  speed i s c o n t i n u o u s l y v a r i a b l e up to 163mm/min. The  melting  rate  descent v e l o c i t y .  was c o n t r o l l e d by v a r y i n g the e l e c t r o d e  The. c u r r e n t was  i n d i r e c t l y c o n t r o l l e d by  the  e l e c t r o d e v e l o c i t y and a s a t u r a b l e c o r e r e a c t o r was not used. Vertical  channels were d r i l l e d  i n the mould and  horizontal  h o l e s connected the channels to brass n i p p l e s at the ends of the mould.  The open h o l e s a t the e x t r e m i t i e s of  sealed  by  welding.  the  The moulds were m e c h a n i c a l l y assembled  c o o l i n g water brought to the moulds through thermometers  m o n i t o r i n g the i n l e t  A  picture  of  and o u t l e t temperature.  Both  the  2.1.2  Temperature  course  i s shown i n f i g .  Sketches of the mould s e c t i o n s i n which the embedded are given i n f i g s .  and with  the assembly  rubber  were  hoses  aluminium and copper s e c t i o n s were used i n work.  channel  5(a,b,c) and  of 3 and  thermocouples  the 4. were  6.  Measurements  Thermocouples  are the most convenient means of temperature  measurement i n moulds and have been used e x t e n s i v e l y by p r e v i o u s workers. about  The maximum temperature expected to 300°C.  chosen.  Hence,  copper-constantan  These were a v a i l a b l e i n  thicknesses  be  measured  thermocouples of  10  thou  was were and  sheathed i n f i b r e g l a s s , which a l s o served as i n s u l a t i o n with the diameter of the assembly being about Several section  as  thermocouples  holes shown were  of diameter 1.25 in  figures  1mm. mm  5(b,c)  were d r i l l e d and  6.  The  i n a mould type  A  expected to g i v e an idea of the temperature  19  Figure  Figure  3 - Sketch of Mould Assembly  4 - Views of Mould Assembly  20  a  Hot F o c e Typ« A Pos. No.  Type A  X  Pos. No.  (in 3.57 4.61 3.11  95 .95 1.43  7 9 11  3.58 460 4.66  1.91 1.51 2.05  X cm  3 57 461 311 152297 356 1652 4.10 17 78 4-66 11-46 12 67  13  Type B  b  Figure  Y cms.)  1 3 5  o  1 3 5 7 9 11  Z cm  c  5 - Dimensions of Mould S e c t i o n used  21  13-5 20 20 10 dimensions in mms. 1  1  section height = 254 mm T/C No. U  X z y ( in mm ) 5-1 9-5 101-6  15  5-8  3 6  127  16  3-5  3-6  1524  COPPER  Figure distribution confirm  SECTION  USED  6 - Dimensions of Copper  i n the mould, while type B  the  assumption  Section  thermocouples  were  to  of steady s t a t e heat flow, i . e . type B  thermocouples were expected t o g i v e the same temperature h i s t o r y under steady s t a t e c o n d i t i o n s . The thermocouples were welded o u t s i d e and soldered bottom  of  the  h o l e s using STRONGSET No. 509 s o l d e r .  t o the The c o l d  j u n c t i o n s were e n c a p s u l a t e d i n i n s u l a t i o n  tape  and  test-tubes  them  to promote heat  which  had  some  mercury  in  t r a n s f e r and were kept at 0°C by p a r t i a l water  mixture.  The  measuring  immersion  junctions  placed  in  i n an i c e -  were connected t o a  Texas Instruments FM6WB m u l t i - c h a n n e l p o t e n t i o m e t e r - r e c o r d e r .  22  2.1.3  The Heat Flow Sensor Considerable d i f f i c u l t y  values  of  heat  flux  to  evident that heat flow was  was  found  the  in  measured  fitting  temperatures.  I t was  d e f i n i t e l y three dimensional and  c r e a t e d the need to develop a heat flow sensor. sensor  published  The  this  heat  flow  had t o be t h e r m a l l y i n s u l a t e d from the r e s t of the mould  and had to be as t h i n as p r a c t i c a l to permit a c c u r a t e heat measurement. o u t l e t was  Temperature measurement of water at the i n l e t  i n a v a i l a b l e measuring  The heat f l u x sensor 7.  sensor was  The  was  have  were  designed  shown  in  copper-constantan and  the  the  water  at  was  the  expected outlet.  i n FIBREFRAX to t h e r m a l l y  The  insulate  between  it. 1  The expected  temperature to  be  thermal  Biot  number  was  about  sensors  resistance  not matter.  between the thermocouples  proportional  to  the  P u b l i s h e d v a l u e s of the heat t r a n s f e r c o e f f i c i e n t the  wrapped  Data  difference  directly  Such  inlet  I t i s noted that  and so the c h o i c e of m a t e r i a l f o r the sensor may  V e r i f i c a t i o n Of Sensor  degrees  the  sensor was  been widely used by Russian r e s e a r c h e r s .  the metal probably c o n s t i t u t e s a n e g l i g i b l e  2.1.4  as  h e a v i l y water c o o l e d so that no more than 0.5  and  tightly  instruments.  finally  thermocouples  C e l s i u s change i n temperature water  and  c o n s i d e r e d and r e j e c t e d s i n c e i t r e q u i r e d too great a  sensitivity  figure  flux  .02 to .04.  heat  was  flux.  suggested  that  Here i t i s p o i n t e d out  that the B i o t number i s normally used i n cases where there i s no  23  o  G  - _ 50 mm-—  0  —»  . . . Thermocouples  Heat Flux Sensor  Figure  7 - Sketch of Heat Flux  Sensor  100  80  60  I  20'  0  0  10  20  30  40  50  60  70  Diff. in Temperature. *C  Figure  8 - For V e r i f i c a t i o n  of Sensor  24  sink as i s present i n the sensor, but an e x t e n s i o n of may  give  an  approximate line  insight  into  as i t may  variation  of  be.  the  effectiveness  to e x i s t  idea  of the sensor,  I t i s suggested that f o r  temperature  the  a  straight  i n the sensor the B i o t  number at the source should be small and that at the sink should be l a r g e .  S i n c e the heat t r a n s f e r c o e f f i c i e n t  interface hot  i s about  face,  it  temperature  c a l c u l a t e d and is  about  appears  The  response time, L /A, 2  and  A  the  found to be about  3 s.  plot  temperature  temperatures  should  of  The  the  be  difference  ( p r o p o r t i o n a l to the heat f l u x )  a  coolant  Provided that the sensor  straight at  the  line inlet.  I t was  used a f t e r  i t had been normalised u s i n g f i g u r e  c o n s i d e r e d that the data was  starting  at  was was the  T h i s i s shown i n adequate  to  be  8.  Experimental R e s u l t s A  summary  Temperatures  of the 15 melts made i s presented i n t a b l e I I .  were measured at d i f f e r e n t p o s i t i o n s i n the mould.  L a t e r , when i t became apparent predicted and  was  r a t e of r i s e of ingot  f i g u r e 8.  2.1.5  diffusivity,  To v e r i f y t h i s estimate the  a g a i n s t the temperatures.  good the  thermal  where L i s the  which suggests that the sensor w i l l probably  respond q u i c k l y enough.  plotted  coolant  l i k e l y that a s t r a i g h t l i n e v a r i a t i o n of  length  .05 cm/s  between the two  the  4 o r d e r s of magnitude higher than that at the  w i l l occur.  characteristic  at  reasonably  11 were recorded.  well, Run  that  the  temperature  only the temperature  8 had the mould s e c t i o n  could  be  of p o s i t i o n 3 reversed  so  TABLE I I  CaF„ wt%  CaO wt%  S l a g 8 kg. Al 0 wt %  1  70  —  30  _  _  75  Trial  2  70  —  30  _  _  75  Mould Temperatures and Heat F l u x Measured  _  113  T h i c k e r e l e c t r o d e , temp., heat f l u x measured  •_  113  II  113  M  Run  #  SIO wt %  MgO wt %  Electrode Dia. (mm)  Special  Features  3  70  15  15  _  4  70  _  30  _  5  50  20  30  6  49  16  17  12  6  113  Slag leak, melt  7  49  16  17  12  6  113  Thick electrode:  8  70  —  30  _  _  113  Mould s e c t i o n r e v e r s e d , power f a c t o r =  9  70  —  30  _  _  113  Melt aborted.  10  70  —  30  —  _  113  Copper s e c t i o n r e p l a c e s aluminum  11  70  -  30  -  -  113  Run #10 r e p e a t e d fluctuations  12  70  —  30  —  —  113  Grooved s e c t i o n used w i t h T/Cs  13  70  —  30  -  —  113  F l u x sensor a t c o r n e r of  box  14  70  —  30  —  —  113  F l u x sensor  box  15  70  -  30  -  -  113  E l e c t r o d e as i n F i g . 9  aborted temp., heat f l u x measured 0.96  Cable o v e r h e a t e d t o a v o i d i n f l u e n c e of power  a t c o r n e r of  and  flux  sensor  26  - Holding Stub  50 mm f  400 mm  Starter Stub  Figure  9 - E l e c t r o d e used i n Run 10  RUN 3(exp.|  Figure  10 - Temp.  H i s t o r y of RUN 3  27  £  s. 0  Figure  16  8  24  32 Time. mins.  40  11 - Smoothened Temp.  94=—, 0  Figure  1 8  ,  ,  16  .  . 24  .  ,  32 Time. mins.  ,  48  56  H i s t o r y of Run 3  . 40  12 - C a l c u l a t e d Temp.  ,  , 48  , r56  f o r Run 3  28  Figure  13 - Temp.History of Run 2  RUN 2lcalc>  0  8  16  21  32  40  48  56  Time, mins  Figure  14 - C a l c u l a t e d Temp.  f o r Run 2  29  Figure  Figure  15 - Temp.History  16 - C a l c u l a t e d Temp.  of Run 6  f o r Run  6  RUN  lOlexpl 14 • 15 — 16  0.  !0  20  30  40  50  •  60  Time. mins.  Figure 17 - Temp.History  gure 18 - C a l c u l a t e d Temp.  of Run 10  f o r Run 10  31  Figure  19 - Temp.History  of Run 8  F i g u r e 20 - Heat Flux Measurement-Run 2  32  0  I  .  0  1  .  ,  2  Heat Flux. MW.m"  3  r4  2  F i g u r e 21 - Heat Flux  Measurement-Run  F i g u r e 22 - Heat Flux  Measurement-Run  33  Time (mins)  i  i  r  Heot F l u » . MW.m"  F i g u r e 23 - Heat Flux Measurement-Run 6  RUN  0  1 2 Heot Flux. MW.m"  II  3  4  2  F i g u r e 24 - Heat Flux Measurement-Run 11  34  0  8  16  24  32  40  48  56  Time. mins.  F i g u r e 25 - Sample comparison between C a l c u l a t e d temperature and measured that  thermocouples  previously  now at a lower p o s i t i o n . determined  to  be  In  0.96.  p l a c e d higher i n the mould were  this  run  the  the  box s e c t i o n .  factor  Run 12 was made with a grooved  Runs 13 and 14 were made to measure the heat of  power  was  face.  f l u x a t the  corner  Run 15 had an e l e c t r o d e as shown i n f i g .  9. F i g u r e s 10,11,13,15,17,19 obtained.  The  the temperature Fig.10  shows  fluctuation.  detail  the  temperature  curves have been moved on the p l o t s t o simulate fields the  on  horizontal  variation  planes  obtained  in with  the  mould.  heavy  power  In the other runs the f l u c t u a t i o n was not so h i g h ,  but the curves have been smoothened t o reduce power  history  fluctuation  and  facilitate  better  the  influence  comparision  of  with  35  computed r e s u l t s . electrodes  I t i s c l e a r from runs 2 and  that  thinner  have lower m e l t i n g r a t e s and cause lower heat  to the mould than t h i c k e r e l e c t r o d e s . mould  4  rose  and  fell  smoothly  The  as  temperature  fluxes i n the  the s l a g passed a c r o s s the  measuring p l a n e , except when there was a power i n s t a b i l i t y . heat f l u x curve showed two peaks, one at the s l a g - a i r and  The  interface  the other at the l i q u i d metal head l e v e l with the peak f l u x  l e v e l a t the l i q u i d metal head being much higher than p r e v i o u s l y reported.  (see f i g s .  20 t o 24 )  The heat f l u x a t the  corner  of  determined w i t h some d i f f i c u l t y .  the  square  section  In the f i r s t attempt, the f l u x  was found t o be much h i g h e r than expected from mould measurements.  In  the  second  attempt,  d i r e c t e d a t e n s u r i n g that t h e r e was no s l a g the  sensor.  special  temperature e f f o r t s were  penetration  Apparently,  a i r gap between the s l a g and the mould p l a y e d a major  in c o n t r o l l i n g the heat f l u x .  T h e r e f o r e the v a l u e s of heat  determined at t h i s p o s i t i o n may not be very a c c u r a t e . clear  around  I t was thought that such p e n e t r a t i o n would prevent  c o n t r a c t i o n of the s l a g s k i n away from the sensor. the  is  was  that  the  What  role flux is  average heat f l u x and the magnitude of the  peak f l u x a r e both lower.  Based on these r e s u l t s the t o t a l heat  f l u x t o the mould was c a l c u l a t e d and compared to the power input and the match (shown i n t a b l e I I I ) was found t o be s a t i s f a c t o r y . Run transfer. found  8 was made to v e r i f y that there was some t r a n s i e n t heat The maximum temperature  reached  by  position  to be higher than that reached by p o s i t i o n  was i n the o p p o s i t e sense i n e a r l i e r  3  was  11, whereas i t  runs thus demonstrating the  36  Table III - Comparison: Heat flow and Power Run#  Area  Rate of Heat flow ingot r i s e (int.) cm/min. kW  cm 2 3 4 5 6  248.7 365 215 194 236.8  existence  .37825 .22325 .38372 .40959 .28069  127.8 1 10.7 112.1 107.98 90.02  135.0 1 35.0 1 35.0 135.0 126.0  formation  controlling surface.  Skins  transfer necessary  i n order  and to  in  the formation  examine  skins  formed  to a s c e r t a i n i f i t c o u l d be m o d i f i e d  with v a r i o u s  i t s phase diagram i s r e l a t i v e l y w e l l  2.2.1  of  the  T h i s s l a g was chosen s i n c e known.  Microstructure  35  three  skin  The s l a g  Some micrographs of the s l a g s k i n s are shown i n f i g u r e s to  in  so that  With t h i s view,  s l a g s were examined.  of 70:30 s l a g was a l s o examined.  part  of the metal  the process  a smooth c a s t i n g surface c o u l d be ensured.  cap  17.02 13.17 16.11 13.92 1 5.43  of the s l a g s k i n p l a y s an important  heat  I t was  formation  slag  Mould h t . considered cm  of t r a n s i e n t heat t r a n s f e r .  2.2 Observations On Slag The  Power input kW  input  from  layers.  which  particular  i t i s c l e a r that the s t r u c t u r e c o n s i s t s of  The f i r s t  some areas and l i g h t  26  seems to be a  i n others.  mixture  being  dark  in  There does not appear to be any  r e g u l a r s t r u c t u r e or any predominant component i n the  37  Figure 27 - Spike on 70:30 s k i n  38  F i g u r e 29 - 50:30:20 s l a g  skin  Figure 31 - Skins of complex s l a g  40  F i g u r e 32 - I n d u s t r i a l  F i g u r e 33 - I n d u s t r i a l  70:30 s l a g  skin  50:30:20 s l a g  skin  41  Figure 35 - 70:30 s l a g cap near top  42  layer. The  second, or the middle, l a y e r d i s p l a y s r e l a t i v e l y  c r y s t a l s o r i e n t e d toward the voids  but  most  of  ingot c a s t i n g .  phase  while  The  used  it  crystals  have  i s pertinent  t h i r d l a y e r , the one  some  retain  in  the  word  that procedure.  T h i s l a y e r was caused  there  some spikes of g l a s s y transparent  especially  the  at  random  detector  on  scanning  i d e n t i f y the elements present Quite but  obviously the  In  the  light  considerable lightly  70:30  slag,  material sticking the  other very  electron i n the  microscope,  microscope three  d i f f e r e n t s l a g s would present  general  to  slags  thin  and  the  EDX  SEM  the work under the  difficult  visible.  Observations On The Following  intervals.  of  70:15:15 s l a g , the t h i r d l a y e r was  at some p l a c e s not  2.2.2  In the p a r t i c u l a r case  does  the p h y s i c a l  Apparently i t i s only  the  to the s l a g s k i n  has  ingot c a s t i n g  held against were  the  'crystal'  s u r p r i s i n g l y , has  sample during p o l i s h i n g and  skin.  be  l i k e a g l a s s y porous mass.  c l o s e s t to the  secondary phase.  d i f f i c u l t y during  to  to note that i n some of the complex  primary c r y s t a l s , but  appearance of the  appear  Though the  s l a g s the middle l a y e r a c t u a l l y looks The  several  the darker phase i s probably the e u t e c t i c  l i q u i d which s o l i d i f i e d l a t e r . been  are  the space i n between the t h i c k c r y s t a l s i s  f i l l e d with a darker phase. primary  There  thick  pattern  was  different  used  to  layers.  different results,  became c l e a r e r as seen i n f i g s .  36  to  43  39. In the case of 70:30 s l a g the outer l'ayer was i n some areas A l would be present and i n This  other  inconsistent;  areas  absent.  was not the case with Ca; i t was always present and the Ca  peak was higher than the A l peak. was  In the  middle  layer,  there  a s t r o n g peak f o r A l , even l a r g e r than that f o r Ca.  In the  inner l a y e r , c l o s e to the i n g o t , A l was n o t i c e a b l y absent  except  in the c r y s t a l s that p i e r c e d t h e i r way through  middle  layer.  The  spikes  of  glassy  material  from  showed  s u r p r i s i n g composition; apart from the presence Si  the  an even more  of A l there  was  i n s i g n i f i c a n t content and n o t i c e a b l e q u a n t i t i e s of S,Fe and  Mn.  T h i s was s u r p r i s i n g because n e i t h e r  nor  the l i q u i d  the  remelting  f i l m theory c o u l d e x p l a i n the composition  theory of the  t h i r d l a y e r or that of the s p i k e s . The outer  s t r u c t u r e of the  layer  showed  slag  A l , Ca and S i .  and the peak due to A l layer,  70:15:15  had  was  different.  The middle  increased  in  l a y e r had no S i  height.  The  no s p i k e s on t h i s s l a g The the  50:30:20 Ca  distinguishable  peak  but  There  slag  skin  stronger.  had i n i t s outer l a y e r A l and Ca The  middle  layer  phases, one darker than the o t h e r .  smaller  were  skin.  Ca were present with the Ca peak s l i g h t l y phase  inner  c l o s e s t t o the i n g o t , showed an i n c o n s i s t e n t presence of  A l and S i e s p e c i a l l y a t the very edge of the s k i n .  with  The  i n the l i g h t phase.  larger  had  two  Both A l and in  the  dark  The inner l a y e r showed  very l i t t l e A l but S i , S and Mn were i n evidence. An i n d u s t r i a l s l a g of 70:30 type  used  to  melt  stainless  44  ••••'X • outer layer '  middle layer  '  • ^•?v>V.'^v-cvv.. o  .':  o>  .  •• . :  .  .  */...•  • •  - • -'".v. -f . "•' **'  ' V.Vt/'.V'v'.'  ...  inner layer  :  A. •  • * • ' V .** . • • • base of spike  •  .  .'.  • • • .v. •. •/ spike A| Si  '' t Figure  Ca Ca  f  Mn  t  .-. Fe  f '  k  36 - EDX a n a l y s i s of 70:30 s k i n  .. .••»."'•"*  45  •A  outer layer  • .  A •• •  o u o  middle  . 4  2 J> • •.  inner layer  Al Si  II Figure  Ca Ca  LL 37 - EDX a n a l y s i s of 70:15:15 s k i n  46  inner^ layer'  \~ \~y  light phase (middle) % o  o  •  tn  _i  -  •• •*  •V -.V*.- .'• dark phase (middle)  outer layer  Mg Al Si  S  ILL! Figure  Ca Ca  Mn  tt  T  38 - EDX a n a l y s i s of 50:30:20 s k i n  47  .  .  "  outer layer  ' ' ' * -V^'rvVV/V *\  I  light phase (middle)  dark phase (middle)  .'•  -.\Vv.  V:v^. V - . A . - . . . . . inner layer  Mq Al Si  ftt  Ca Ca  tt  Mn  *  Fe  t t  F i g u r e 3 9 - EDX a n a l y s i s of complex  slag  skin  rv.V.,^;..^**.  48  steel  was  examined and d i s p l a y e d  and T i appearing on the inner  much the same p a t t e r n  layers r e f l e c t i n g  the  with Cr  effect  of  the metal melted. A s l a g having 12% S i 0 , 2  rest  CaF  showed  2  the  6% MgO,  16% A l 0 , 2  presence of Mg, S i , A l , Ca and Mn, (the  l a s t a p i c k u p from the metal) i n the outer layer the  had  17% Cao, and the  3  layer.  The  middle  two phases, the darker of which had A l , S i , and Ca ;  l i g h t e r had Mg, A l t r a c e s of Ca and small amounts of Mn  Fe.  The  inner  l a y e r had a l l the elements mentioned  and  though the  peak of Ca was dominant.  2.2.3  The 70:30 Slag Cap Near The L i q u i d Metal  Meniscus  It was apparent that the middle l a y e r c o n s i s t e d primary  crystals.  It  was a l s o c l e a r that  mainly  of  these c r y s t a l s were  Table IV - Chemical a n a l y s i s of some skins Sample  CaF wt%  70:30(bulk) 70:30 s k i n 70:15:15 s k i n 50:30:20 s k i n much  thicker  than  precipitated  (those  Cao wt%  2  80.7 60 to 70 38.4 43.4 the  — — 17.9 29.7  crystals  observed  in  that the  A1 0 wt% 2  3  15 30 to 40 17.5 29.9 would  have  normally  s l a g bulk) and d e f i n i t e  macrosegregation was observed.  Micrographs were taken  proportion  i n the s k i n and i n the s l a g bulk  determined.  of  primary  phase  The p r o p o r t i o n  of  primary  phase  was  and the  4-5  times  49  higher i n the s k i n than i n the bulk. break  a t the s k i n and no evidence that r e m e l t i n g or d i s s o l u t i o n  took p l a c e a t the meniscus. about  F u r t h e r , there was a c l e a n  The middle  layer  started  1cm below the s l a g top and i n c r e a s e d i n t h i c k n e s s t i l l the  metal meniscus An  in a nearly l i n e a r  effort  was  p r o p o r t i o n of elements electron  probe  made  fashion.  to  present  quantitatively in  microanalysis.  the  The  necessary. presence  presence  sensitivity  voids so  in that  the  specimen  reliable  phases  of  spectrometer  was  coupled with the reduced  became  A n a l y s i s by atomic a b s o r p t i o n was p o s s i b l e and the in  Table  IV.  Apparently  much  using and  probably  analysis  the  oxygen  The f r a g i l e nature of the specimen of  determine  various  f l u o r i n e meant that the use of a l i g h t element  given  growing  the  difficult. analysis  is  of the alumina d i d not  d i s s o l v e i n the s l a g and the composition of the  70:30  slag  is  a c t u a l l y c l o s e t o 80:20, but the i n c r e a s e of alumina i n the s k i n is evident.  The l a t t e r a n a l y s i s cannot, of course, i n d i c a t e the  i n d i v i d u a l composition of the three l a y e r s .  50  III. Several  simple  models  e x p l a i n the o b s e r v a t i o n s . used,  CALCULATIONS were  formulated  The f i n i t e d i f f e r e n c e  In a  grooves  in  element  C a l c u l a t i o n s In 2 Dimensions  s o l v e the d i f f e r e n t i a l Good  explanations  K r e i t h and B l a c k , Wilkes.  equation f o r steady s t a t e heat t r a n s f e r .  are 3 8  a  Therefore,  horizontal slice finite  rough  i n many textbooks, e.g. those by and by  3 6  Carnahan,  Luther  are  cals/cm K). 2  surface  estimate of the temperature f i e l d positioning  a 2-D  of a f l a t  difference  results  given  by P a t a n k a r  needed to determine the mould.  a standard method to  and  3 7  Initially,  shown  thermocouples  in  steady s t a t e heat flow s i t u a t i o n mould  section  discretisation in  of  figure  40  was  was  made,  (for  a  examined. solved  heat  the i n a. The  and  flux  was  of  the 25  I t i s c l e a r that the temperature g r a d i e n t s near the are  very  high  and  any  e r r o r i n placement of a  thermocouple c o u l d lead t o l a r g e e r r o r s i n them  of  was  used.  The f i n i t e d i f f e r e n c e technique i s now  hot  technique  mould face on heat t r a n s f e r to the mould, the f i n i t e  method was  3.1  an e f f o r t to  with some success, to c a l c u l a t e mould temperatures.  s p e c u l a t i v e c a l c u l a t i o n to determine the e f f e c t the  in  unreliable.  measurement,  making  I t i s a l s o seen that the region marked B l i e s  in a area of low temperature g r a d i e n t which means  that  d r i l l s a hole from the rear face as shown there i s l i t t l e  if  one  chance  51  ure  40 - Temp.  field calc.  with 2-D assumptions  Hot Face  Using symmetry, only portion bounded by discretised. Arrows  heavy lines  show heat flow.  F i g u r e 41 - The s l i c e which was d i s c r e t i s e d  52  of  inaccuracy  positions,  i n temperature measurement  it  Type B  i t seems, may be used f o r v e r i f i c a t i o n .  Measured values and  i n the r e g i o n .  was  of heat f l u x were used i n  quickly discovered  calculation  that a 2-D model was  s i n c e i t p r e d i c t e d extremely high I t was evident  the  inadequate  temperatures at the hot f a c e .  that computation had  to  be  extended  to  three  dimensions.  3.2 C a l c u l a t i o n s In 3 Dimensions The  next  model c o u l d reasonably  step  was  calculate  to  determine i f a quasi  temperatures  representative.  suggests that an accuracy  in  the  steady  state  that  were  mould  The r e s u l t s of the previous of  no  more  than  about  section  5  Celsius  degrees i n temperature measurement may be expected. Following  the  sensor a s i m i l a r section  such  type  line  as  of  of  assessment used f o r the heat f l u x  reasoning  considered  assumption of quasi  was  s t a t e may e x i s t only  and  thickness so there  steady  condition  that  assumptions.  r i s e of the s l a g across  inherent  error  However, there  the mould f a c e .  mould f o r the  C a l c u l a t i o n s of of  quasi  in  steady  than 4-5  of the mould was 5cms and the height an  a  suitable  i n a body of s i z e no g r e a t e r  i s probably  state  very  steady s t a t e heat t r a n s f e r .  a " B i o t " number i n d i c a t e d that the  The  not  indicated  about 25 cms making  quasi  is a relatively  Further  cms.  other  slow  workers  have shown that the heat f l u x to the mould i n the s l a g region i s relatively  constant  1  and  it  is  possible  that  the  maximum  53  temperatures computed on the mould face may not i.e.  be  inaccurate,  a steady s t a t e c a l c u l a t i o n may p r e d i c t maximum  temperatures  with  acceptable  accuracy.  dimensional unsteady  It  i s a l s o to be noted that three  s t a t e computing  i s potentially  extremely  c o s t l y and hence u n s u i t a b l e i n the f i e l d . A  three  made. in  dimensional  steady  s t a t e c a l c u l a t i o n was  The technique used matrix s o l u t i o n s as w e l l as  an  effort  requirements. the  quasi  AMDAHL  were  computing  computer  costs  against  20 seconds  CPU  i n FORTRAN i s given i n Appendix  as  the  w r i t t e n assuming  memory time  at the U n i v e r s i t y of B r i t i s h Columbia.  r e s u l t s of the program w r i t t e n  used  The  balance  A s i n g l e run took about  program l i s t i n g The  to  iteration  initial  solution  on A  C.  f o r 2-D  approximation  estimate f o r the program  3-D steady s t a t e heat flow.  input r e q u i r e d i s the heat t r a n s f e r c o e f f i c i e n t  to the  c o o l a n t , the heat f l u x d i s t r i b u t i o n , the thermal c o n d u c t i v i t y of the the  mould,  the p h y s i c a l dimensions, the c o o l a n t flow r a t e , and  c o o l a n t temperature.  coefficient Sieder-Tate  to  the  In the c a l c u l a t i o n , the heat  cooling  c o r r e l a t i o n as  5 5  Petukhov. (App.B) 56  water well  was as  by these v a r i a t i o n s .  was  deemed s u f f i c i e n t  was no g r e a t e r than t h a t . constant  and  the  relation  given  by  were  not  significantly  The heat f l u x i n t o the mould face  was taken from the experimental d a t a . degree  estimated by both the  There was a s m a l l d i f f e r e n c e but i t was seen  l a t e r that the mould face temperatures changed  transfer  I t e r a t i o n t o the nearest  s i n c e the accuracy of measurement  The thermal c o n d u c t i v i t y was  equal t o the value a t 200°C f o r aluminium  assumed but f o r  54  copper a l i n e a r v a r i a t i o n with temperature was p e r m i t t e d . r e l a t i v e l y good agreement with experimental data the  program  was  was  Since  observed,  not r e f i n e d to a l l o w a b e t t e r estimate of the  thermal c o n d u c t i v i t y . The r e s u l t s a r e given i n f i g u r e s  12, 14, 16, 18 (p.27-30).  The temperatures were c a l c u l a t e d u s i n g 3 s p a t i a l dimensions the  vertical  dimension  but  (z) can be converted t o time s i n c e the  r a t e of r i s e of the ingot i s r e l a t i v e l y c o n s t a n t .  The  have  comparison.  It  been g i v e n with respect to time t o f a c i l i t a t e  i s to be noted that p o i n t s  position  within  mould  not  quite  the  same  as  The  the  temperatures  measured  comparison  i s visually available  The temperature  correct.  It  is  gradient  felt  doubt  that  e f f e c t on temperature that  steady  state  with  that  p r e d i c t maximum temperatures little  same  such  reasonable  in figures  time  calculated  temperatures  expected) but a r e c l o s e enough to be c o n s i d e r e d  23.  the  T h i s i s true except f o r the 2-3 cms at the  base and at the top of the mould.  this  in  with r e f e r e n c e t o the s l a g have approximately the same  temperature h i s t o r y .  are  the  results  i s also  reasonably  c a l c u l a t i o n s may be used t o  heat  fields.  there  is  t r a n s f e r has a d i s c e r n a b l e  T h i s i s confirmed  calculations  and  11 t o 18 and  i n mould s e c t i o n s though  transient  (as  by  the  fact  u n d e r - p r e d i c t temperatures at  type B p o s i t i o n s and a l s o by the r e s u l t s of run 8. F o l l o w i n g the a c c e p t a b l e success of the model the e f f e c t of v a r i a t i o n of mould m a t e r i a l , spacing  were  made.  channel  positioning  and  channel  As expected, the use of copper lowered hot  face temperatures d r a m a t i c a l l y .  T h i s i s seen i n f i g .  17 and 18  55  (p.30). Moving the channels c l o s e r reduced the face temperature made the temperature g r a d i e n t more uniform ( f i g . 4 3 a,b). channel i s brought c l o s e r to the hot f a c e , the face is  reduced  but the g r a d i e n t  (fig.43 c,d). channel  and  conditions.  A  coefficient  to  may  small the  I f the  temperature  i n c r e a s e s and becomes l e s s uniform  There i s a l s o a h i g h temperature at this  the  cooling  c r e a t e b o i l i n g thus changing the c o o l i n g change  (10%)  coolant  of  brings  the  only  heat small  transfer changes  temperature d i s t r i b u t i o n s u g g e s t i n g that the approximation to  estimate  the c o e f f i c i e n t  i s acceptable (fig.42 b,c).  to be noted that the heat t r a n s f e r c o e f f i c i e n t mean  temperature  of  the  with  the  length  that  used It i s  on  the  channel.  A  an a r b i t r a r y v a r i a t i o n was a l s o performed  and  showed i n s i g n i f i c a n t changes ( f i g . 4 2 water  depends  in  c o o l a n t f i l m a c c o r d i n g to the method  used and so should vary over calculation  and  of  d).  the  Variation  of  inlet  temperature a l s o had only a small e f f e c t , but i t i s noted i f the temperature r i s e s above 25-30°C at the same  velocities, occurring  then  there  ( f i g . 4 4 a,b).  is  a  strong  The h o r i z o n t a l  where the hot face reaches i t s highest  possibility plane  chosen  temperature.  cooling  of b o i l i n g is  that  c) h r a i s e d  10%  Figure 42 - C a l c u l a t e d  d)  arbitrary  Temp.  h  fields  57  a,b)  reduced i n t e r - c h a n n e l d i s t a n c e  c,d) channel c l o s e r to hot F i g u r e 43 - C a l c u l a t e d Temp.  face fields  58  a) water a t 25°C  b) water at 10°C  F i g u r e 44 - C a l c u l a t e d Temp.  fields  sketch of section token for finite element calculations  F i g u r e 45 - S e c t i o n taken f o r F.E.  calculation  59  90° V-groove 304  (cals/cm^s) 10-  + —  0  —  10  ,  ,  r-  20  30  40  °/o Relative Surface  U - groove  Heat Flux tcals/cm^s)  0  10 %>  20  30  Relative  Surface  LO  F i g u r e 46 - C a l c u l a t e d r e s u l t s of s p e c u l a t i o n  60  3.3 S p e c u l a t i o n The  On The E f f e c t Of Grooves On Surface  assumption  knowledge  of  the  conductivity  of steady s t a t e heat t r a n s f e r coupled with slag  skin  indicated  that  thickness  and  its  thermal  the a i r gap had a smaller  thermal  r e s i s t a n c e , though not i n s i g n i f i c a n t , compared t o slag  skin  that  of  (see Table I, VI and VII the l a t t e r on p.  Thus, a s l i g h t flux  Quality  increase of the a i r gap  significantly.  It  could  the  67-68).  reduce  the  heat  i s known that rough mould s u r f a c e s i n  metal ingot c a s t i n g lead to smoother s u r f a c e s , presumably due t o lower heat f l u x e s .  I t was thought that machining grooves i n the  mould s u r f a c e might have the same e f f e c t . There were two assumptions made.  The f i r s t  was  that  the  s l a g would not enter  the grooves s i n c e  slag  the space between two mould s e c t i o n s , which  d i d not enter  c o u l d be as l a r g e as 2mms. thickness To  The other  was  that  the  slag  skin  would not change as the molten metal head approached. determine  the e f f e c t of groove spacing  f i n i t e element model of a sketch  i t had been observed that  of  slice  of  mould  which i s shown i n f i g u r e 45.  the program  i s given  speculation  are  in  given  f l u x c o u l d be e f f e c t i v e l y  in  Appendix  The  a  results  of  the  I t seemed that the heat  reduced by up t o 25%  covered 50% of the hot f a c e .  formulated,  The FORTRAN l i s t i n g of  D.  f i g u r e 46.  was  on heat f l u x , a  i f the  grooves  Apparently, there was need t o make  experimental i n v e s t i g a t i o n s s i n c e i t seemed p o s s i b l e to design a mould  to  permit a molten head of metal, to e x i s t a t lower power  input than h i t h e r t o and t h i s was the b a s i s f o r run 12. The  experiment was u n s u c c e s s f u l  since  i t was  discovered  61  that  t o prevent  s l a g from e n t e r i n g the g r o o v e s ,  t o quench the s l a g e x t r e m e l y r a p i d l y .  it  i s necessary  The g e n e r a l h e a t i n g up of  the mould w i t h the r i s e of the s l a g makes t h i s  impossible.  No  e x p e r i m e n t s w i t h a groove w i d t h s m a l l e r than 1 mm were attempted since  it  was thought t h a t such f i n e grooves would be e x p e n s i v e  t o machine and so would be i m p r a c t i c a l .  62  IV.  4.1  F a i l u r e Of 2-D  Model  Russian workers  have c a l c u l a t e d temperature  1  carbon conducting paper. adequately  DISCUSSION  described  T h i s presupposes  as  two  fields  using  that heat flow can be  dimensional.  While t h i s may  be  p e r m i s s i b l e f o r a v e r t i c a l s e c t i o n e x a c t l y between two  channels  it  This  is  not  r e a l i s e d and flux  be  sufficient i t was  a  horizontal  suggested by these  arbitrarily  temperature  for  raised  by  section.  workers  30%.  that  the  heat  No other p r e d i c t i o n s of  f i e l d s are a v a i l a b l e f o r channel c o o l e d moulds.  Channel c o o l e d moulds are used i n continuous c a s t i n g . this  case t h e r e are some l i t e r a t u r e s t u d i e s .  and Simonov et a l Young,  Sunaryo  vertical conditions  and  section for  accepted 2-D  3 9  Cross  exactly  al  4 2  4 1  steady  state  used temperature  between  two  assumed  measurements of heat  negligible  vertical  4 0  measurements i n a as  resources. heat  al  approximations.  channels  of l i m i t e d computing  For  C h i z h i k o v et  the m o d e l l i n g of h o r i z o n t a l s e c t i o n s .  probably done because et  was  flow  boundary This  was  Alberny in  their  flux.  V e r t i c a l heat flow has o f t e n been n e g l e c t e d i n the past  in  measurements of heat f l u x i n both the e l e c t r o s l a g process and i n continuous  casting.  T h i s approach  heat f l u x g r a d i e n t s e x i s t . by  considering  a  i s flawed s i n c e a p p r e c i a b l e  The extent of e r r o r can be  rectangle  in  the  X-Y  examined  plane with boundary  63  c o n d i t i o n s as shown i n f i g .  47.  The e r r o r  is negligible  i n the  c e n t r a l region of the r e c t a n g l e but v a r i e s from 0 to 10% i n extreme r e g i o n s . (fig.48)  and  r e s u l t s are  A h y p o t h e t i c a l heat f l u x v a r i a t i o n was  calculations shown  thermocouples  in  performed  Table  V.  to check  It  is  the  assumed  the e r r o r .  clear  that  The  if  the  used to measure heat f l u x are p l a c e d c l o s e to the  hot face the e r r o r  i s low  error.  i t i s l i k e l y that the sharp peaks observed i n  However,  (about 10%)  and  t h i s study would not have been uncovered  within  s i n c e the type of e r r o r  a s s o c i a t e d with the measurements would have It  is  experimental  obliterated  them.  a l s o c l e a r the heat f l u x i n the X - d i r e c t i o n reduces with  i n c r e a s i n g t h i c k n e s s f a l l i n g about value.  T h i s i n d i c a t e s why  15-20%  temperature  from  the  hot  face  f i e l d s must be c a l c u l a t e d  i n 2 dimensions even i f heat f l u x measurements are made u s i n g 1D  approximation.  reduction  of  In  symmetry  the  case  of channel c o o l e d moulds, the  necessitates  an  extension  to  three  dimensions. In  the  early  stages  approximations of temperature The  heat  flux  from  of  t h i s work, i t was  f i e l d s were  the s l a g was  than that from the ingot or the caused  appreciable  vertical  grossly  above  flow  approximation would not be s a t i s f a c t o r y . s e c t i o n s are a c c e p t a b l e f o r cooling  channels  are  thin  probably  not  be  and 2-D  jacketed  acceptable  inaccurate.  the  slag  which  meant that a  2-D  models of v e r t i c a l moulds  extremely c l o s e l y spaced,  isotherms are f l a t and p a r a l l e l to the hot will  2-D  an order of magnitude higher  region heat  seen that  face.  or  if  the  i . e . where the Such  moulds  in electroslag casting  from  64  Figure 47 - E r r o r  i n heat f l u x measurement  F i g u r e 48 - H y p o t h e t i c a l heat f l u x  variation  65  Table V - P o s s i b l e E r r o r i n heat flow measurement a. Measuring T/C's 3 and 11 mm Position (cm)  A c t u a l Flux Cals/cm C  'measured' heat f l u x i n mould of t h i c k n e s s , cm.  20  1.5  5 10 35 35 21 9  0 6 12 17 24 30  from hot face  5.46 11.5 32.01 33.63 21 .08 1 1 .47  2.0  2.5  5.62 11.51 31 .99 33.55 21.11 11.51  4.0  5.74 1 1 .54 31 .93 33.46 21.13 1 1 .58  6.02 11.61 31.81 33.26 21.18 1 1 .74  7.33 14.16 28.01 31 .21 21.41 1 4.99  9.73 15.16 25.94 29.2 21 .82 17.12  b. 'Measurement' at c o l d f a c e . 0 6 12 17 24 30  5.92 12.86 29.84 32.59 21.16 13.27  5 10 35 35 21 9  reasoning given i n s e c t i o n  6.58 13.56 28.87 31.9 21 .28 14.17  1.2.5 and thus the need t o examine 3-  D models a r o s e .  4.2 Apparent  Success Of 3-D Quasi Steady State. Model  Three dimensional approximation explained  is partially  sucessful,  hereunder.  The c o r r e l a t i o n s developed f o r e s t i m a t i n g the heat coefficient  conditions  by  flow i n the channel and they are  where  - a l o n g the c h a n n e l . developed  transfer  t o the c o o l i n g channel a r e r e l a t i v e l y a c c u r a t e only  f o r f u l l y developed to  as  the heat f l u x does not vary  When the heat f l u x  Sleicher  of l o c a l heat t r a n s f e r  4 3  and o t h e r s  coefficients  does  4 4  '  4 5  '  6 2  should  vary  restricted  substantially the  methods  f o r the c a l c u l a t i o n be  employed.  In  66  theory,  even t h i s i s not good enough since the heat f l u x v a r i e s  with p o s i t i o n even flow  is  in a horizontal  not f u l l y developed.  s e c t i o n and i n any  I t i s fortunate  that  a x i a l conduction o c c u r s , so that the temperatures near  case  the  substantial  in  the  mould  the hot face are not very s e n s i t i v e to changes i n the heat  transfer c o e f f i c i e n t . Petukhov  relation  5 6  T h i s means may  be  that  used.  correlations  such  the  However, i t i s pointed  that b o i l i n g can cause d r a s t i c changes and  in  this  out  case  more  complex c a l c u l a t i o n s are necessary. The  other  question  regarding  c a l c u l a t i o n comes from the f a c t that time dependent. arguments  the  the heat flow  brought  f o r t h e a r l i e r are j u s t i f i e d ,  i s to l i m i t impetus  thermal s t r e s s e s is  However, i t i s gradients  may  the maximum hot face for  3-D  compare  unsteady  deformation  by  noted  the  that  be p r e d i c t e d  the  results  the  if  s t a t e c a l c u l a t i o n s may  cause  temperatures  and  f a i r l y w e l l using quasi  obtained.  to  come  A knowledge of for  c r e e p caused by such  be worthwhile  the  temperature.  make  mould  stresses. temperature  steady  state  measurements  I f the r e s u l t i s p r e c i s e  enough, no refinement of the assumptions may The most important v a r i a b l e , over which little  actually  especially  i s necessary s i n c e the major  c a l c u l a t i o n s and so i t may and  3-D  to some e x t e n t .  from the the need to c a l c u l a t e thermal s t r e s s e s .  retirement  is  a  i t would seem unnecessary to go to the  expense of making more complex c a l c u l a t i o n s  The  of  However, the experimental r e s u l t s show that  In i n d u s t r i a l c o n d i t i o n s ,  motive  utility  be necessary. the  c o n t r o l , i s the heat f l u x to the mould.  designer  has  There have been  67  many measurements of the heat f l u x under v a r i o u s c o n d i t i o n s , but as yet i t does not seem p o s s i b l e to p r e d i c t the heat f l u x f o r an a r b i t r a r y shape. reliable  There  estimates  i s need t o develop a  using  simple  techniques  approach may be used i n day to day work. the  method  of  before  making  such  I t i s suggested  an that  sharp f l u x peak observed may be n e g l e c t e d s i n c e the peak i s  very narrow and the heat f l u x approximated h i g h heat  as a  broad  band  of  flux.  4.3 F a c t o r s A f f e c t i n g The Heat Flux To The Mould  Table VI - Temp,  drop through s l a g  skin  a. Heat Flux = 65 c a l s / c m s 2  slag skin thickness(cm) 0.15 0. 12 0.10 0.08  Drop i n Temperature °C .008* .010* .006* 1 625 1 300 1 083 866  1218 975 812 650  975 780 650 520  b. Heat Flux = 35 c a l s / c m s 2  0.5 0.4 0.3 0.2 0.1  2916 2333 1750 1 1 66 583  * Thermal c o n d u c t i v i t y The  2187 1750 1312 875 437  1750 1400 1050 700 350  i n cals./cm.s.°C  thermal r e s i s t a n c e s to heat flow to the mould from the  s l a g or the ingot are the s l a g s k i n and the a i r gap, presumed to e x i s t between the s l a g s k i n and the mould.  Simple  fitting  of  68  Table VII - Heat flow through a i r gap Drop i n Temperature, °C for heat f l u x ( C a l s / c m s ) of 65 35  A i r Gap (cm)  2  0.1 0.05 0.001 0.0005 0.0001  65 000 32 500 650 325 65  35 000 17 500 350 175 35  A l l u n i t s i n the CGS system. Thermal C o n d u c t i v i t y of a i r taken t o be 0.0001 cals./cm.s.°C the  known  physical properties  observed heat f l u x e s  of a i r and the s o l i d s l a g to the  (as i n T a b l e s VI and  VII)  indicates  that  both the s l a g s k i n and the a i r gap are s i g n i f i c a n t r e s i s t a n c e s . It  appears  that  the  a i r gap t h i c k n e s s  microns and the temperature drop a c r o s s rest  cannot n e g l e c t  skin  about i s only  either  from p o i n t t o p o i n t .  a  simple  primary c r y s t a l s ( A l 0  bulk.  In  2  physically  any  case  resistance.  frozen  layer  d i s s o l u t i o n as has been p r e v i o u s l y  were  It  an average value and i n f a c t  examination of a 70:30 s l a g cap i n d i c a t e d that  i s not  skin  the  out that t h i s i s a very approximate a n a l y s i s and that  " a i r gap" w r i t t e n  The  The  400°C  o  probably v a r i e s d r a m a t i c a l l y one  i t roughly  of the drop of roughly 700~800 C a c r o s s the s l a g s k i n .  i s pointed the  i s of the order of 10  3  much  or CA ) 6  and  no  the s l a g  remelting  or  s t a t e d t o occur was observed. precipitated  in  the  slag  t h i c k e r than they were i n the s l a g  In f a c t , the average content of  aluminium  in  the  slag  s k i n i s much higher than i n the nominal composition of the s l a g . It appears that phase  from  the constant washing o f f of the secondary molten  the  slag  s k i n and the supply of f r e s h s l a g to the  69  region  allows continuous p r e c i p i t a t i o n of the primary phase  the passage between the c r y s t a l s flow  of  liquid  temperature of liquidus  is  no  the  slag  temperature  "equilibrium" temperature  layer (see  1800  longer  becomes possible.  skin-molten  and  the  constricted  is  49).  slag  interface  equal  Using  "that  In t h i s s c e n a r i o , is  temperature of the c h i l l e d  interface  fig.  so  till  the  to  the  values  the the  layer-  eutectic of  thermal  J  1700  1600  1500J  H00  1300, CoF  2  • Al 0  3  ICAg)  m o s s •/.  Al 0  2  1200. 20  •CoF,'  Figure c o n d u c t i v i t y given the  "equilibrium"  w e l l with  30 2  40  50  3  49 - Phase diagram f o r CaF^-A^O^ by T a y l o r layer  observations.  of  and  M i l l s et a l  0.3  mm  6  1  a  thickness  i s o b t a i n e d , which f i t s  of in  70  The high heat f l u x from  the  molten  explained. steel  ( n e a r l y double that  head  of  metal  to  i n the s l a g  the  region)  mould must s t i l l  From the measurements of other workers,  the  be  molten  i s g e n e r a l l y at a temperature of 1600-1650°C at the ingot  top s u r f a c e p e r i p h e r y . 1500°C.  If  For a 70:30 s l a g the l i q u i d u s  conduction  is  hot  face  sufficient  caused  about  c o n s i d e r e d to be the major mode of  heat t r a n s f e r then the. d i f f e r e n c e i n skin  is  by  the  temperature  rise  of  at  molten  the  slag  metal i s not  to e x p l a i n the d o u b l i n g of the heat f l u x .  Radiation  too i s not a s a t i s f a c t o r y answer s i n c e the s t r u c t u r e of the s l a g skin  ensures that s u b s t a n t i a l s c a t t e r i n g takes p l a c e , i . e . only  an i n c r e a s e i n e f f e c t i v e thermal c o n d u c t i v i t y may The  only  expected.  other reason c o u l d be a r e d u c t i o n of the a i r gap.  the face of i t , even t h i s appears d o u b t f u l reason  be  why  there  is  no  a couple of m i l l i m e t e r s of molten metal head should  do what 500 mm supposed  since  On  of s l a g  that  the  head  could  higher  metal some  not.  However,  temperature  if  will  cause  crystals,  the  s l a g s k i n w i l l weaken s t r u c t u r a l l y without subsequent  will  permit  gap.  I f the l i q u i d metal head  fracture  deformation  and  observed. '" 1  cause  re-solidification.  secondary  primary  This  then,  of the s l a g s k i n and decrease the a i r  runout  i s e x c e s s i v e l y b i g , the s k i n metal  fins  such  as  have  may been  7  As the ingot s o l i d i f i e s and withdraws contraction,  the  the  and  apparent change on  of  melt  is  secondary phase then  dissolution  it  the  phase  reduced and  pressure  from the s l a g s k i n on  created  draws  t h i s s o l i d i f i e s on the s o l i d  out ingot  the wall.  71  T h i s e x p l a i n s why  the t h i r d l a y e r c o n s i s t i n g  phase  i n the s l a g s k i n .  is  found  The  of  the  s p i k e s observed on  s u r f a c e mainly have e a s i l y o x i d i s a b l e elements. that  corrosion  products  collect  secondary  It i s  this  possible  near the metal meniscus s i n c e  they are not r e a d i l y s o l u b l e i n the s l a g which may  get  trapped  and s t i c k to the s l a g s k i n thus c r e a t i n g the s p i k e s . In  the  other  slags  examined,  for  example the 70:15:15  composition, t h e r e i s some melt back of the this  seems  to  the  and  Of course, some  deformation  s l a g s k i n i s a l s o p o s s i b l e but t h i s does not change the  situation  qualitatively.  T h i s e x p l a n a t i o n of the v a r i a t i o n of the reasonable, flux.  expected  be a reasonable e x p l a n a t i o n f o r the r i s e of the  heat f l u x at the metal head l e v e l . of  skin  but  there  is  still  no way  heat  coefficient  to  the  depths, t h i s i n turn depending field.  The  air  gap  Some  work  Medovar  et  al  showing  coefficient  of  of  the  heat  from the molten  slag  in  this that  on  this  nature 70:30  the  thermo-physical  area, has slags  flow  it  seems,  is  been r e p o r t e d by have  a  lower  thermal expansion than the complex s l a g s having  MgO  and S i 0 .  Al 0  at  the expense  of  2  depend  Work  necessary.  skin  on  on the magneto-hydrodynamic  will  p r o p e r t i e s of s l a g s k i n s .  4 6  slag  seems  of e s t i m a t i n g the heat  The t h i c k n e s s of the s l a g s k i n w i l l depend  transfer  flux  2  3  and CaO reduced  d u c t i l i t y of C a F - C a O - A l 0 2  2  3  were found to i n c r e a s e the s t r e n g t h ductility system  while  slags.  Si0  2  raised  the  72  4.4  C r y s t a l l i s a t i o n In Slag It  is  useful,  crystallisation diagram Keene"  8  for and  at  this  the  CaO-Al 0 -CaF 2  3  2  i s shown i n f i g . 5 0 .  show  that  assumed in the  juncture,  sequence a c c o r d i n g  s l a g s are marked out. others  Skins  The  there  phase  system  i s given  The  works of  2  this  by moisture and slag  p o i n t s are unknown but work  shown.  two  or A l 0 ,  6  2  9  5 0  '  5 1  '  5 2  '  and  s o l i d s o l u b i l i t y and  and  5 3  this is  is  up to  5%  evaporation  50  CaO  of A1F .  high  The  slag  plait  skins  and  be p l a c e d at p o i n t s A and  B as  l i q u i d u s the primary phase  will  C and  6  would  be  the  primary  D.  possibly  T h i s i s a l s o supported by the work of  determined that CA  to  At  3  i n the 2 - l i q u i d r e g i o n .  they may  due  both of which have been observed by M i t c h e l l  3  the s l a g s k i n . who  by M i l l s  l i q u i d s w i l l then have the composition  If C has a higher CA  The  compositions of four common  from a knowledge of the  of Z h m o i d i n  The  diagram.  Zhmoidin" '  is l i t t l e  the  following discussion.  h y d r o l y s i s of CaF  the  examine  to the  A 70:30 s l a g , in p r a c t i c e , c o n t a i n s  temperatures  to  be in  3 0  Zhmoidin  phase  in  50  this  system. A  70:15:15  though not and  CaO  will  i n i t . The crystallising  change  Therefore,  slag  the it  is  lies  at the edge of the 2 l i q u i d  primary phase should later.  However, small  crystallisation likely  be C A F  sequence  3  region  with  3  CaF  i n c r e a s e s of  2  CaO  significantly.  that C ^ A y F w i l l come out  i n s t e a d of  C A F. 3  3  A 55:35:10 s l a g i s i n the 2 l i q u i d region and  f o l l o w i n g the  arguments made e a r l i e r , the primary phase c o u l d be CA  2  or  C A F 3  3  73  mass V. CoO X  70:30  JL  70:15:15  m  55=35:10  rr  33:33:33  Figure  50 - P h a s e d i a g r a m  f o r CaF -Al^Og-CaO 2  74  depending  on  the  tie-lines.  s m a l l changes i n the  CaO  Here, as  content can  f o r the  change  70:15:15 s l a g ,  the  precipitation  sequence. 33:33:33 s l a g gave C^ i A F with C a F  A the  7  eutectic.  s k i n of t h i s  Mitchell  3 0  has  found C  A  1 2  diagram.  Kamensky et a l , the  and  Fractional 2 7  will  slag  one  expects  structure  plate  and  As  plates  as may  be  so one  of  metal ingot melting point  i n the  crystalline  the  thicken,  one  2  i n the  of  by in  T h i s means that  the  s l a g s k i n are  slag skins  of  The that  top p e r i p h e r y i s between This  2  low  primary phase i n  6  has  This  3  skin  grows  large  2  the  a closely  i n between the  l e s s of C a F  as  a  plates.  i n the  skin  ingots.  Heat Flux the metal temperature at 1600-1650°C  the  under  normal  temperature i s above the  melting  primary phases f o r a l l the with the  (CA  Al 0 ).  should f i n d  suggested e a r l i e r  preceding section  spoken  70:30 s l a g owing to  corundum  of alpha  finds CaF  circumstances.  of the  follows  skin.  to that  seen i n the  was  slag  motion  secondary phases i n the  Slag Composition E f f e c t s On It  i n the  7  slag  to f i n d , b a s i c a l l y , only the  slag  characteristics  the  A F  as  take p l a c e owing to the  f i n d i n g the  bulk of the  similar  1 1  constituting  skin  crystallisation  There i s a great d i f f e r e n c e  4.5  C  e l e c t r o s l a g process as d e s c r i b e d above.  chances of  the  and  7  CaO  slag.  Apparently c r y s t a l l i s a t i o n of the phase  and  2  s l a g s mentioned  e x c e p t i o n of the  70:30 s l a g .  in  the  75  The  implication  i s that the s l a g s k i n w i l l be melted back  in a l l cases except that of the 70:30 light  striations  surface 70:30  and  the  of a s l a g s k i n slag  further,  dealt  the  slag  manufacture),  absence  of  with skin  in is  i n cases where the s l a g  where  does  not  different. is  to  produced.  In the  fracture  Paton  plastic  to  addition  of MgO  probably  due  moving  reduce  the  and S i 0  skin  melts  This  the l a t t e r mould  thus  back  and  is  seen  in  s i t u a t i o n favors a  case,  2  2  more  defects,  that  chances  the  the  of  system  is  with  efficient.  as f i n s , are  skin  should  f r a c t u r e and suggests the This  "plasticity"  5 7  low This  fluoride is  due  contents  reduced  Kusamichi et a l , report  velocities.  there i s a d i s c e r n a b l e fluoride  content  r a d i a l temperature which  are  to t h e i r higher  viscosity  flow  is  this.  e l e c t r i c a l r e s i s t i v i t y and h i g h e r slag  be  of g l a s s present i n the s l a g and  probably f a v o r s slags  such  slag  to achieve t h i s .  to the s o f t e n i n g  i s known that  operationally  and  indicates  5 8  the a d d i t i o n of S i 0  low  ingot  I t has been observed that the s k i n of the 70:30 s l a g  prone  It  large,  9  mould c o n d i t i o n s ,  smoother s u r f a c e .  in  not even appear.  the r e s u l t s of Kondo et a l . static  Continuing  melt, as f o r the 70:30 s l a g , the heat f l u x  peak w i l l be d i f f u s e and may  In  the  be observed at the molten  level  the  The case f o r  (as  of  on the i n s i d e  earlier.  thicker  a peak of heat f l u x may  presence  "spikes"  detail  metal head it  The  i s i n d i c a t i v e of t h i s .  was  if  slag.  which  also  means  8  gradient  in  slags  that of  i s absent i n h i g h f l u o r i d e s l a g s .  The net e f f e c t s are t h i c k e r s l a g s k i n s and lower  heat  flux  to  76  the  mould f o r low  fluoride slags.  above, the s l a g s k i n s of low a  higher  melting  p o i n t and  Further,  discussion  f l u o r i d e (about 20-40%) s l a g s so r e m e l t i n g  be l e s s .  These s k i n s should  be  surfaces  in  operations.  moving  from the  mould  of the s l a g s k i n  stronger  and  so  have should  give  better  A 70:30 s l a g a l s o  give a t h i c k s k i n but owing to the p l a t e s t r u c t u r e  of  will  corundum  this skin fractures e a s i l y . The  best  s k i n , i t appears, must have a high melting  must c r y s t a l l i s e with no d i r e c t i o n a l b i a s and appropriate  4.6  degree of  the  regime has clear;  the  i s the only  by  metal and  by  appears  that the mould c o o l i n g  surface q u a l i t y .  mechanism  of  formation  a good surface  metal  Such  reducing  head.  resistance  thing  a  of  condition  to  a  is  bad  i s ensured by  the r e s i s t a n c e to heat flow  i n c r e a s i n g the  One  ingot means that meniscus  Therefore,  presence of a l i q u i d improved  it  r a t e of r i s e of the  surface.  an  Quality  to do with the  slow  solidification rippled  discussion,  little  possess  "plasticity".  F a c t o r s A f f e c t i n g Surface From  should  point,  heat  can  i n the flow  the be  liquid to  the  mould. The  former i s c o n t r o l l e d by geometry and  the system. slag skin. or  by  a  The  latter  i s favoured by a t h i c k e r and/or  A t h i c k e r s k i n can thinner  the power input  stronger  be generated by lower power  e l e c t r o d e , but  that the thermal r e s i s t a n c e i n the  input  t h i s i s counter p r o d u c t i v e liquid  metal  is  to  in  adversely  77  affected. The  other method i s t o s e l e c t the c o r r e c t  slag.  The use of  pure compounds or e u t e c t i c compositions i s not recommended in t h i s case remelting a  reduction  of  since  of the s l a g s k i n i s p o s s i b l e which causes  the thermal r e s i s t a n c e .  I t i s necessary f o r a  high m e l t i n g phase to p r e c i p i t a t e out and i t i s p r e f e r a b l e the  proportion  of  the  primary phase be as h i g h as p o s s i b l e .  Thus the tendency of the s l a g s k i n t o combated.  An  increased  dissolve  and  weaken  is  s l a g depth a t the same power input  may  a l s o be b e n e f i c i a l but t h i s i s a l i m i t e d option  i n that a  increase  penalty  losses.  of  slag  depth  that  has  a  substantial  large  i n power  78  V.  SUMMARY  In c o n c l u s i o n , 1.  Quasi s t e a d y - s t a t e three dimensional models may be used  calculate  temperatures  electroslag casting.  i n mould  Temperature  sections  to  be  used  to in  at the hot face may be changed  by changing the spacing between channels but d r a s t i c changes can be  made  only  production  by  runs,  changing temperature  the  mould  material.  For  appears to be a v a l i d  short  criterion,  but f o r long runs design t o minimise s t r e s s e s may be necessary.  2.  I t appears that temperature d i s t r i b u t i o n s generated i n t h i s  way may be used to c a l c u l a t e thermal s t r e s s e s , but  verification  i s necessary.  3.  A  proposal  to  e x p l a i n the v a r i a t i o n of heat f l u x to the  mould has been made.  Work t o determine  slag  used t o estimate heat f l u x e s f o r a r b i t r a r y  skins  can  be  i f the  properties  of  shapes i n a simple manner seems necessary.  4.  The s u r f a c e q u a l i t y of a c a s t i n g  design but by the s l a g composition, geometrical  relationship  and the c a s t i n g  shape.  i s not i n f l u e n c e d by mould the  power  input  and the  between the e l e c t r o d e , the s l a g depth  79  REFERENCES 1.  B.I.Medovar, V.L.Shevtsov, G.S.Marinsky, V.F.Demchenko, V.E.Machenko, Thermal Processes i n E l e c t r o s l a g Flows , Naukova Dumka, Kiev, 1978.  2.  A . M i t c h e l l , R.M.Smailer, p.231 .  3.  W.Holzgruber, Ref. 4.  4.  W.Duckworth, G.Hoyle, E l e c t r o s l a g R e f i n i n g H a l l , London, 1969.  5.  S . J o s h i , Ph.D. 1 971 .  6.  D.N.Pocklington, B . P a t r i c k , Met. p.359-366.  7.  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P a l e e v , A.A.Andreevsky,  , Harper and Row,  12266, I z v . Vuz.  Rev.  Report de Met.,  ASME, 79, 1957, p.789-797. ASME, 82, Aug 1960, p. 170.  S c i . Res., 4A, 1953, p.61-75.  Int.  Met.  Akad.  on S p e c i a l  Rev., ( 1 ) ,  Nauk.  1981, p.21of Mat.  Mat., Proc.  USSR, 6, 1969, B.S.Fokin,  82  M.E.Lavzentiev, K.P.Malyus-Malitsky, V.N.Fromzel, G.P.Danilova, I n t . J . Heat Mass Trans., ( 5 ) , 1973, p. 1073-1085. 55.  E.N.Sieder, G.N.Tate, Ind. Eng. p.1429.  Chem., 28, (12), 1936,  56.  B.S.Petukhov, Adv.  57.  J.D.W.Rawson, G.Jeszensky, A.W.Bryant, Proc. S i x t h I n t . Vac. Met. Conf. on S p e c i a l M e l t i n g , San Diego, 1979, p.848-863.  58.  B.E.Paton, B.I.Medovar, V.L.Artamonov, A.G.Bogachenko, Y.P.Shtanko, Henry Brutcher Trans. 9726, R e f i n i n g Remelting , Naukova Dumka, Kiev, 1972, p.49-53.  59.  K . C . M i l l s , J.S.Powell, J.W.Bryant, B.J.Keene, Can. Met. Quart., 20, ( 1 ) , 1981, p.93-99.  60.  A.A.Sharapov, C.E.Volkov, C.V.Lebedev, T e o r i y a M e t a l . , 3, 1957, p.131-136.  61.  R.Taylor, K . C . M i l l s , Arch. 63.  62.  R.N.Noyes, Trans.  i n Heat T r a n s f e r , 6, 1970, p.504-564.  E i s e n . , 53, ( 2 ) , 1982, p.55-  ASME, 83, 1961, p.96-97.  83  APPENDIX A - SLAG NOTATION  i. ii. iii. iv.  70:30 = 70% C a F  2  , 30% A l 0 2  3  70:15:15 = 70% C a F , 15% A l 0 , 2  2  3  50:30:20 = 50% C a F , 30% A l 0 , 2  33:33:33 = 1/3 C a F , 2  2  3  1/3 A l 0 , 2  3  15% CaO. 20% CaO. 1/3 CaO.  84  APPENDIX B ~ MISC. i.  DATA AND  CORRELATIONS  Water flow measurements through A l s e c t i o n 30 l i t r e s / m i n . through Cu s e c t i o n 30 l i t r e s / m i n . through heat f l u x sensor  14 l i t r e s / m i n .  These measurements were made by n o t i n g the time needed to f i l l a 100 l i t r e drum. i i.  Some p h y s i c a l p r o p e r t i e s  Material at Aluminium Copper  Thermal C o n d u c t i v i t y (W/m.K) 273K at 400K at 600K 236 401  Water a t s a t . Pr. and at  240 392  Density  Absolute Prandtl ViscosityNo. X 1 0 m /s  kg/m  3  273 313 353 373  iii.  K K K K  6  999.3 999.2 971.8 958.4  Sieder-Tate  232 383  2  1794 658 352 278  13.7 4.3 2.25 1.75  correlation  Nu = .027 Re  8  Pr  33  (n /n b  s  ) • • 1  where Nu = Nusselt  number  Re = Reynolds number Pr = P r a n d t l number Mi  = v i s c o s i t y of f l u i d at bulk  Mj  = v i s c o s i t y of f l u i d at s u r f a c e of tube  temperature  85  A l l p r o p e r t i e s except v i s c o s i t y taken at average bulk temperature. iv.  Petukhov  correlation  Nu  =  U  /8) Re Pr K,U)  + K (Pr)  U/8) -  2  where £ = (1.82 Log Re - 1.64)"  5  (Pr '  6 7  - 1)  2  K, = 1 + 3.4 £ K  2  = 11.7  + 1.8  Pr"•  3 3  A l l p r o p e r t i e s taken at the mean f i l m  temperature  86  APPENDIX C ~ FORTRAN PROGRAM TO CALCULATE TEMPERATURES IN CHANNEL COOLED MOULDS  C  c  C C C C C C C C Q  THIS IS A PROGRAMME TO CALCULATE TEMPERATURES IN CHANNEL COOLED MOULDS USED IN ELECTROSLAG CASTING THE METHOD IS TO USE MATRIX METHODS TO CALCULATE THE TEMPERATURES FOR EACH SLICE AND ITERATE FROM SLICE TO SLICE TILL CONVERGENCE IS REACHED. THE REQUIRED INPUT ON UNIT 5 IS AS GIVEN BELOW. THE INITIAL TEMP. ESTIMATE MUST BE GIVEN ON UNIT 4 AND THE OUTPUT IS GIVEN ON UNIT 7. *******************************  C C C C C C C C C C C C C C C C C C C  M = NUMBER OF 'X' DIVISIONS N = NUMBER OF Y NODES 0 = NUMBER OF Z NODES L = MAX. NO. OF ITERATIONS ALLOWED TOLER = TOLERANCE TW = INLET TEMPERATURE OF COOLANT H = HEAT TRANSFER COEFFICIENT TO THE COOLANT DELX = DISTANCE BETWEEN 2 X NODES DELZ = DISTANCE BETWEEN 2 Z NODES RH = RADIUS OF ROUND CHANNEL IN TERMS OF DELX UNITS CH = POSITION OF CHANNEL IN TERMS OF DELX UNITS COTC = COEFFICIENT OF THERMAL CONDUCTIVITY AT 0 C VELA = MASS FLOW VELOCITY OF COOLANT CEE = GRADIENT OF THERMAL CONDUCTIVITY WITH TEMP. QA(I) = HEAT FLUX INTO THE HOT FACE I N I T = 0 I F THERE IS NO GUESS SOLUTION, OTHERWISE ANY NO. IT IS SUGGESTED THAT ALL UNITS BE FED IN IN CGS UNITS TO AVERT FORMAT PROBLEMS. **********************************************************  C C C C C  [TE] {S} = {B} IS THE MATRIX EQUATION TO BE SOLVED AT EACH SLICE. X,Y,Z ARE SPECIAL VARIABLES. T — TEMPERATURE TT -- REFERENCE TEMPERATURE FIELD I PERM, TS — MEMORY REQUIRED BY SOLUTION ROUTINE  c  Q  C C C C C C C C  ******************************************************  THE ROUTINE FSLE IS A ROUTINE IN THE UBC LIBRARY AND USES GAUSSIAN ELIMINATION TO SOLVE SIMULTANEOUS EQUATIONS. ************************************************************** THE ROUTINES MATDAT, BOTDAT AND TOPDAT SERVE TO CALCULATE THE MATRIX TERMS FOR THE MIDDLE SLICES, THE BOTTOM SLICE AND THE TOP SLICE RESPECTIVELY  c***************************************************************  REAL*4 TT,B,T,DET DIMENSION TT(20,20,30),TS(400,400),IPERM(800),S(400),X(20)  87  1,Y(20),HT(30) COMMON T(20, 20,30),TE(400,400),B(400),QA(30),Z(20 , 20) , 1TH(30),DELT C C C  READ IN MATRIX DATA  INTEGER 0 READ(5,1) M,N,0,L,TOLER 1 FORMAT(2X,3I3,I4,F3.1) READ(5,2) TW,H 2 FORMAT(2X,F4.1,F7.3) READ(5,3) DELX,RH,CH,DEL Z 3 FORMAT(2X,F6.4,2F4.1,F6.4) READ(5,4) COTC,VELA,CEE 4 FORMAT(2X,F5.4,F7.3,F8.6) READ(5,6) (QA(I),1=1,0) 6 FORMAT(1X,10F6.2) READ(5,7) INIT 7 FORMAT(1X,I 2) IF (INIT .EQ. 0) GO TO 9 CONTINUE READ(4,8) (((T(I,J,K),J=1,N),I=1,M),K=1,0) 8 FORMAT(11F6.2) C ********************************************* C THE LINES BELOW TO BE USED TO VARY H ALONG THE HEIGHT OF C THE MOULD. Q  C C C C C C C C C C C C C C C C C C Q  C C C  ***********************************************************  HT(1)=H-0.18 HT(2)=H-0.16 HT(3) = H- 0.16 HT(4) = H-0.14 HT(5) = HT(4) HT(6)=H-0.10 HT(7) = H - 0.08 HT(8) = H HT(9) = H+.02 HT(10) = H+.04 HT(11) = H +.02 HT(12) = H- .06 HT(13) = H - 0.08 HT(14) = H-0.10 HT(15) = H-0.12 HT(16) = H - 0.14 HT(17) = H- 0.16 HT(18) = HT(17)  ***********************************************************  MN=M*N GIVE X, Y COORDINATES TO THE NODES DO 10 1=1,M DO 10 J=1,N DO 10 K=1,0 10 T(I,J,K) = 0.0  88  SUM =-1.0 DO 11 1 = 1 ,M SUM=SUM+1. 11 X ( I ) = SUM SCUM =-1.0 DO 12 J=1 ,N SCUM=SCUM+1. 12 Y(J) = SCUM C C C  CODE DIFFERENT TYPES OF NODES DO 100 1 = 1 ,M DO 100 J=1 ,N  C C THE FOLLOWING IS BASED ON COORDINATE GEOMETRY C AB =ABS(Y(J)-CH) A = (X(I)**2 + (Y(J) - CH)**2) IF ((RH**2) .LE. A) GO TO 20 C C THE NODE IS IN THE COOLING CHANNEL C Z(I,J) = 1 GO TO 100 20 IF (X(I) .GE. (RH+1.)) GO TO 50 IF (ABS(Y(J)-CH) .GE. (RH+1.)) GO TO 50 D = ABS(SQRT(ABS(RH**2-X(I)**2))-AB) C = ABS(X(I) - SQRT(ABS(RH**2 - (AB)**2))) IF (D .GE. 1.) GO TO 28 IF (X(I).GT. 0.) GO T022 IF (Y(J) .GT.CH) GO TO 21 C C CODES 2 TO 9 REFER TO NODES BOUNDING THE COOLING CHANNEL C Z(I,J) = 2 GO TO 100 21 Z(I,J) =3 GO TO 100 22 IF (C.GE.1.) GO TO 26 IF (Y(J)-CH) 23,24,25 23 Z(I,J) = 4 GO TO 100 24 Z ( I , J ) = 5 GO TO 100 25 Z ( I , J ) = 6 GO TO 100 26 IF (Y(J).GT.CH) GO TO 27 Z(I,J) =7 GO TO 100 27 Z(I,J) = 8 GO TO 100 28 IF (C.GE.1.) GO TO 50 Z(I,J) = 9 GO TO 100  89  C C C C C  50 IF (X(I).LT.(M-1)) GO TO 65 IF (Y(J).GT.O.) GO TO 55 " CODES 10 TO 12 REFER TO THE LEFT SIDE OF THE SECTION INCLUDING THE CORNER NODES BUT EXCLUDING THE NODES INSIDE THE COOLING CHANNEL AND THOSE BOUNDING IT. Z(I,J) = 10 GO TO 100 55 IF (Y(J).LT.(N-1)) GO TO 60 Z ( I , J ) = 11 GO TO 100 60 Z ( I , J ) = 12 GO TO 100 65 IF (X(I).GT.O.) GO TO 80 IF (Y(J).GT.O.) GO TO 70  C C CODES 13 TO 15 REFER TO THE NODES ON THE RIGHT SIDE OF C THE SECTION INCLUDING THE CORNER NODES C Z ( I , J ) = 13 GO TO 100 70 IF (Y(J).LT.(N-1)) GO TO 75 Z ( I , J ) =14 GO TO 100 75 Z(I,J) =15 GO TO 100 80 IF (Y(J).GT.O.) GO TO 85 C C CODE 16 REFERS TO THE BOUNDARY OPPOSITE THE HOT FACE C Z ( I , J ) =16 GO TO 100 85 IF (Y(J).LT.(N-1)) GO TO 90 C C CODE 17 REFERS TO THE HOT FACE BOUNDARY C Z ( I , J ) =17 GO TO 100 C C CODE 18 REFERS TO INTERIOR NODES C 90 Z(I,J) =18 100 CONTINUE C C SET THE TEMP. OF THE CHANNEL TO THAT OF THE COOLANT C AT THE INLET C DO 101 IC=1,0 101 TH(IC) = TW C C SET ASSUMED TEMP. FIELD = REFERENCE C DO 105 K=1,0  90  C C C  DO 105 J=1,N DO 105 1 = 1 ,M 105 T T ( l , J , K ) = T(I,J,K) ITERATE L TIMES  DO 310 LL=1,L C C RESET INCREASE OF TEMP. OF THE COOLANT TO ZERO C FOR THE NEXT SLICE C DELT =0.0 Q  ***************************************  C TO BE USED ONLY IF H IS TO BE VARIED C C DO 130 K=1,0 C H = HT(K)  c*********************************************************  C C C C C C C  BOTTOM SLICE IF  (K.EQ.1) GO TO 110  TOP SLICE IF (K.EQ.O) GO TO 120 CALL MIDDAT(M,N,O,K,COTC,DELX,DELZ,VELA,H,CEE) CALL FSLE(MN,400,TE., 1 , 400 , B, S , I PERM, 400 , TS , DET, JEXP)  C C RE-WRITE SOLUTION IN TERMS OF X,Y,Z COORDINATES C DO 106 11=1,MN I = (H-O/N+1 J= II-N*(I-1) 106 T(I,J,K) = S ( I I ) GO TO 130 110 CALL BOTDAT(M,N,0,K,COTC,DELX,DELZ,VELA,H,CEE) CALL FSLE(MN,400,TE, 1 ,400,B,S,I PERM,400,TS,DET,JEXP) DO 115 11=1,MN I=(II-1)/N+1 J= II-N*(I-1) 115 T(I,J,K) = S ( I I ) GO TO 130 120 CALL TOPDAT(M,N,0,K,COTC,DELX,DELZ,VELA,H,CEE) CALL FSLE(MN,400,TE,1,400,B,S,I PERM,400,TS,DET,JEXP) DO 125 11=1,MN I = (II-1)/N+1 J = II-N*(I-1) 125 T(I,J,K) = S ( I I ) 130 CONTINUE C C COMPARE REFERENCE AND CALCULATED FIELDS C  91  DO 200 1=1,M DO 200 J=1,N DO 200 K=1,0 IF ( A B S ( T T ( I , J K ) - T ( l , J , K ) ) .GT. TOLER) GO TO 300 200 CONTINUE f  C C WRITE OUT RESULTS. NOTE IF N > 20 THEN PRINTOUT WILL C LOOK FUNNY C DO 218 K=1,O DO 216 1 = 1 ,M WRITE(7,210) (T(I,J,K),J=1,N) 210 FORMAT(20F7.2) 216 CONTINUE WRITE(7,217) 217 FORMAT(/,/) 218 CONTINUE STOP C C RESET REFERENCE FIELDS C 300 DO 310 K=1,0 DO 310 J=1,N DO 310 1=1,M 310 TT(I,J,K) = T(I,J,K) C C NO. OF ITERARIONS WERE INSUFFICIENT. WRITE OUT FAILURE C AND TEMPERATURES. C WRITE(7,320) TOLER,L 320 FORMAT(1H ,'THE TEMPERATURES DO NOT CONVERGE TO WITHIN' 1' DEGREES IN',14,' ITERATION STEPS.') DO 328 K=1,0 DO 326 1=1,M WRITE(7,321) (T(I,J,K),J=1,N) 321 FORMAT(20F7.2) 326 CONTINUE WRITE(7,327) 327 FORMAT(/,/) 328 CONTINUE STOP END SUBROUTINE MIDDAT(M,N,O,K,COTC,DELX,DELZ,VELA,H,CEE) C C THIS ROUTINE CALCULATES MATRIX TERMS FOR THE MIDDLE SLICES C COMMON T(20,20,30),TT(400,400),B(400),QA(30),Z(20,20), 1TH(30),DELT INTEGER CODE,0 MN=M*N C C THE FOLLOWING CALCULATIONS ARE EQUIVALENT TO THE VARIOUS C COEFFICIENTS. THEY ARE MADE HERE TO AVOID REATING AGAIN C  92  H12 = H*DELX*DELZ*.5 H1 = H12*2. H2 = H12*4. H12TH = H12*TH(K) H1TH= H12TH*2. H2TH = H12TH*4. QA12 = QA(K)*DELX*DELZ*.5 QA1 = QA12*2.  C C INITIALISE MATRICES TO ZERO C CALL GSET(TT,400,400,400,0.) CALL GSET(B,400,1 ,400,0. ) THEAT =0.0 DDH = DELZ * DELX*H C C DO FOR EACH NODE DEPENDING ON CODE C DO 100 J=1,N DO 100 1=1,M C C SET THERMAL CONDUCTIVITY ACCORDING TO TEMP. OF NODE C AS IN THE PREVIOUS ITERATION C CT = CEE*T(I,J,K) + COTC COT12 = CT*.5*DELZ COT1 = COT12*2. COT2 = COT1*2. COT3 = COT1*3. COT4 =COT1*4. C14 = CT*(DELX**2)*.25/DELZ C12 = C14*2. C = C14*4. CODE = Z ( I , J ) KK = N*(I-1)+J GO TO (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),CODE 1 TT(KK,KK) = 1. B(KK) = TH(K) GO TO 100 2 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT1+ H12 + C12) TT(KK,KK+N) = COT12 B(KK) = -(H12TH+ C14*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT*.5 GO TO 100 3 TT(KK,KK) = -(COT1+ H12+ C12) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT12 B(KK) = -(H12TH+ C14*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) HEAT = DDH*( T ( I , J , K ) - T H ( K ) ) THEAT = THEAT + HEAT * .5 GO TO 100 4 TT(KK,KK-1) = COT1  93  5  6  7  8  9  10  11  TT(KK,KK) = -(C0T2+ H2+ C*2.) TT(KK,KK+N) = C0T1 B(KK) = -(H2TH+ C*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT+ HEAT *2. GO TO 100 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ H1+ C) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(H1TH + C12*(T(I,J,K+1)+T(l,J,K-1))) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 1. GO TO 100 TT(KK,KK) = -(COT2+ H2+ C*2.) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT1 B(KK) = -(H2TH+ C*(T(I,J,K+1)+T(I,J,K~1))) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT*2. GO TO 100 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ H1+ C) TT(KK,KK-N) = COT12 TT(KK,KK+N) = COT12 B(KK) = ~(H1TH+C12*(T(I,J,K+1)+T(l,J,K-1))) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT*1. GO TO 100 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT2+ H1 + C) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT12 B(KK) = -(H1TH + C12*(T(I,J,K+1) + T( I,J,K-1))) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT+ HEAT*1. GO TO 100 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ H1+ C) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(H1TH+ C12*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 1. GO TO 100 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT1+ C12) TT(KK,KK+1) = COT12 B(KK) = -(C14*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) GO TO 100 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT1+ C12) TT(KK,KK-1) = COT1 2 B(KK) = -(QA12+ C14*(T(I,J,K+1) + T ( I , J , K - 1 ) ) )  94  GO TO 100 12 TT(KK,KK-N) = COT1 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ C) TT(KK,KK+1) = COT1 2 B(KK) = -(C12*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) GO TO 100 13 TT(KK,KK) = -(COT1+ C12) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT12 B(KK) = -(C14*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) GO TO 100 14 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT1+ C12) TT(KK,KK+N) = COT12 B(KK) = -(QA12+ C14*(T(l,J,K+1) + T ( I , J , K - 1 ) ) ) GO TO 100 15 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ C) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(C12*(T(I J,K+1) + T ( I , J , K - 1 ) ) ) GO TO 100 16 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT2+ C) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT12 B(KK) = -(C12*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) GO TO 100 17 TT(KK,KK-N) = COT12 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ C) TT(KK,KK+N) = COT12 B(KK) = -(QA1+ C12*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) GO TO 100 18 TT(KK,KK-N) = COT1 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT4+ C*2.) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT1 B(KK) = -(C*(T(I,J,K+1) + T ( I , J , K - 1 ) ) ) 100 CONTINUE f  C C CALCULATE INCREASE OF TEMPERATURE AND ADD TO TEMP. C OF COOLANT C DELT = DELT + THEAT/VELA IF (DELT .LT. 0.001) GO TO 120 K01=K+1 DO 110 KO=K01,0 110 TH(KO) = TH(1) + DELT 120 CONTINUE RETURN END  95  SUBROUTINE BOTDAT(M,N,0,K,COTC,DELX,DEL Z,VELA,H,CEE) C C THIS ROUTINE SETS MATRIX TERMS FOR THE BOTTOM SLICE C ALL COMMENTS GIVEN IN THE ROUTINE MIDDAT APPLY HERE TOO C COMMON T(20,20,30),TT(400,400),B(400),QA(30),Z(20,20), 1TH(30),DELT INTEGER CODE,0 MN=M*N H12 = H*DELX*DELZ*.25 H1 = H12*2. H2 = H12*4. H12TH = H12*TH(K) H1TH= H12TH*2. H2TH = H12TH*4. QA12 = QA(K)*DELX*DELZ*.25 QA1 = QA12*2. CALL GSET(TT,400,400,400,0.) CALL GSET(B,400,1,400,0.) THEAT = 0.0 DDH = DELZ*DELX*H DO 100 J=1,N DO 100 1 = 1 ,M CT= CEE * T(I,J,K) +COTC COT12 = CT*.25*DELZ COT1 = COT12*2. COT2 = COT1*2. COT3 = COT1*3. COT4 =COT1*4. C14 = CT*(DELX**2)*.25/DELZ C12 = C14*2. C = C14*4. CODE = Z(I,J) KK = N*(I-1)+J GO TO (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),CODE 1 TT(KK,KK) = 1. B(KK) = TH(K) GO TO 100 2 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT1+ H12 + C14) TT(KK,KK+N) = COT12 B(KK) = -(H12TH+ C14*(T(I,J,K+1) + 0.)) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT *.25 GO TO 100 3 TT(KK,KK) = -(COT1+ H12+ C14) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT12 B(KK) = -(H12TH+ C14*(T(I,J,K+1) + 0.)) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 0.25 GO TO 100 4 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ H2+ C)  96  TT(KK,KK+N) = C0T1 B(KK) = -(H2TH+ C*(T(I,J,K+1) + 0.)) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 1. GO TO 100 5 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ H1+ C12) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(H1TH + C12*(T(I,J,K+1)+0.)) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 0.5 GO TO 100 6 TT(KK,KK) = -(COT2+ H2+ C) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT1 B(KK) = -(H2TH+ C*(T(I,J,K+1)+0.)) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 1. GO TO 100 7 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ H1+ C12) TT(KK,KK-N) = COT12 TT(KK,KK+N) = COT 12 B(KK) = -(H1TH+C12*(T(I,J,K+1)+0.)) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT *.5 GO TO 100 8 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT2+ H1 + C12) TT(KK,KK+1) = COT1 TT(KK,KK+N) - COT12 B(KK) = -(H1TH + C12*(T(I,J,K+1) + 0.)) HEAT = DDH*(T(I,J,K)-TH(K)) THEAT = THEAT + HEAT * 0.5 GO TO 100 9 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ H1+ C12) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(H1TH+ C12*(T(I,J,K+1) + 0.)) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT *0.5 GO TO 100 10 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT1+ C14) TT(KK,KK+1) = COT12 B(KK) = -(C14*(T(I,J,K+1) + 0.)) GO TO 100 1 1 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT1+ C14) TT(KK,KK-1) = COT12 B(KK) = -(QA12+ C14*(T(I,J,K+1) + 0.)) GO TO 100  97  12 TT(KK,KK-N) = C0T1 TT(KK,KK-1) = C0T12 TT(KK,KK) = -(C0T2+ C12) TT(KK,KK+1) = C0T12 B(KK) = -(C12*(T(I,J,K+1) + 0.)) GO TO 100 13 TT(KK,KK) = -(COT1+ C14) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT12 B(KK) = -(C14*(T(I,J,K+1) + 0.)) GO TO 100 14 TT(KK,KK-1) = COT1 2 TT(KK,KK) = -(COT1+ C14) TT(KK,KK+N) = COT12 B(KK) = -(QA12+ C14*(T(I,J,K+1) + 0.)) GO TO 100 15 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ C12) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(C12*(T(I,J,K+1) + 0.)) GO TO 100 16 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT2+ C12) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT12 B(KK) = -(C12*(T(I,J,K+1) + 0.)) GO TO 100 17 TT(KK,KK-N) = COT12 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ C12) TT(KK,KK+N) = COT12 B(KK) = -(QA1+ C12*(T(I,J,K+1) + 0.)) GO TO 100 18 TT(KK,KK-N) = COT1 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT4+ C) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT1 B(KK) = -(C*(T(I,J,K+1) + 0.)) 100 CONTINUE DELT = DELT + THEAT/VELA IF (DELT .LT. 0.001) GO TO 120 K01=K+1 DO 110 KO=K01,0 110 TH(KO) = TH(1) + DELT 120 CONTINUE RETURN END SUBROUTINE TOPDAT(M,N,O,K,COTC,DELX,DEL Z,VELA,H,CEE) C C THIS ROUTINE CALCULATES MATRIX TERMS FOR THE TOP SLICE C ALL OTHER COMMENTS GIVEN IN THE ROUTINE MIDDAT APPLY C  98  COMMON T(20,20,30),TT(400,400),B(400),QA(30),Z(20,20), 1TH(30),DELT INTEGER CODE,0 MN=M*N H12 = H*DELX*DELZ*.25 H1 = H12*2. H2 = H12*4. H12TH = H12*TH(K) H1TH= H12TH*2. H2TH = H12TH*4. QA12 = QA(K)*DELX*DELZ*.25 QA1 = QA12*2. CALL GSET(TT,400,400,400,0.) CALL GSET(B,400,1,400,0.) THEAT = 0.0 DDH = 0.0 DO 100 J=1 ,N DO 100 1 = 1 ,M CT = CEE * T(I,J,K) +COTC COT12 = CT*.25*DELZ COT1 = COT12*2. COT2 = COT1*2. COT3 = COT1*3. COT4 =COT1*4. C14 = CT*(DELX**2)*.25/DELZ C12 = C14*2. C = C14*4. CODE = Z ( I , J ) KK = N*(I-1)+J GO TO (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),CODE 1 TT(KK,KK) = 1. B(KK) = TH(K) GO TO 100 2 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT1+ H12 + C14) TT(KK,KK+N) =.COT12 B(KK) = -(H12TH+ C14*(0. + T ( I , J , K - 1 ) ) ) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 0.25 GO TO 100 3 TT(KK,KK) = -(COT1+ H12+ C14) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT12 B(KK) = -(H12TH+ C14*(0. + T ( I , J , K - 1 ) ) ) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT *0.25 GO TO 100 4 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ H2+ C) TT(KK,KK+N) = COT1 B(KK) = -(H2TH+ C*(0. + T ( I , J , K ~ 1 ) ) ) HEAT = DDH*(T(I,J,K) -TH(K)) THEAT =THEAT + HEAT * 1 . GO TO 100  99  5 TT(KK,KK-1) = C0T12 TT(KK,KK) = -(C0T2+ H1+ C12) TT(KK,KK+1) = C0T12 TT(KK,KK+N) = C0T1 B(KK) = -(H1TH + C12*(0.+T(I,J,K-1))) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 0.5 GO TO 100 6 TT(KK,KK) = -(COT2+ H2+ C) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT1 B(KK) = -(H2TH+ C*(0.+T(I,J,K-1))) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 1 . GO TO 100 7 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ H1+ C12) TT(KK,KK-N) = COT12 TT(KK,KK+N) = COT12 B(KK) = -(H1TH+C12*(0.+T(I,J,K-1))) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 0.5 GO TO 100 8 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT2+ H1 + C12) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT12 B(KK) = -(H1TH + C12*(0. + T ( I , J , K - 1 ) ) ) HEAT = DDH * ( T ( l , J , K ) - TH(K)) THEAT = THEAT + HEAT * 0.5 GO TO 100 9 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ H1+ C12) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(H1TH+ C12*(0. + T ( I , J , K - 1 ) ) ) HEAT = DDH*(T(I,J,K) - TH(K)) THEAT = THEAT + HEAT * 0.5 GO TO 100 10 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT1+ C14) TT(KK,KK+1) = COT12 B(KK) = -(C14*(0. + T ( I , J , K - 1 ) ) ) GO TO 100 11 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT1+ C14) TT(KK,KK-1) = COT12 B(KK) = -(QA12+ C14*(0. + T ( I , J , K - 1 ) ) ) GO TO 100 12 TT(KK,KK-N) = COT1 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ C12) TT(KK,KK+1) = COT12 B(KK) = -(C12*(0. + T ( I , J , K - 1 ) ) )  100  GO TO 100 13 TT(KK,KK) = -(COT1+ C14) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT12 B(KK) = -(C14*(0. + T ( I , J , K - 1 ) ) ) GO TO 100 14 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT1+ C14) TT(KK,KK+N) = COT12 B(KK) = -(QA12+ C14*(0. + T ( I , J , K - 1 ) ) ) GO TO 100 15 TT(KK,KK-1) = COT12 TT(KK,KK) = -(COT2+ C12) TT(KK,KK+1) = COT12 TT(KK,KK+N) = COT1 B(KK) = -(C12*(0. + T ( I , J , K - 1 ) ) ) GO TO 100 16 TT(KK,KK-N) = COT12 TT(KK,KK) = -(COT2+ C12) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT12 B(KK) = -(C12*(0. + T ( I , J , K - 1 ) ) ) GO TO 100 17 TT(KK,KK-N) = COT12 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT2+ C12) TT(KK,KK+N) = COT12 B(KK) = -(QA1+ C12*(0. + T ( I , J K - 1 ) ) ) GO TO 100 18 TT(KK,KK-N) = COT1 TT(KK,KK-1) = COT1 TT(KK,KK) = -(COT4+ C) TT(KK,KK+1) = COT1 TT(KK,KK+N) = COT1 B(KK) = - ( C * ( 0 . + T ( I , J , K - 1 ) ) ) 100 CONTINUE DELT = THEAT/VELA TH(K) = TH(K) RETURN END f  101  APPENDIX D - FORTRAN PROGRAM REFERRED TO IN SECTION 3.3  Q  C C C C C C C C C C C C C C C C C C C C C C C  C C C  **********************************************  IMPLICIT REAL*8(A-H,0-Z) INTEGER BCC THIS IS A PROGRAM USING TRIANGULAR OR RECTANGULAR FINITE ELEMENTS TO SOLVE STEADY STATE HEAT TRANSFER PROBLEMS. SPECIFICALLY IT IS BEING USED TO MODEL SECTIONS IN AN ELECTROSLAG CASTING MOULD. ALL VARIABLES BEGINNING WITH THE LETTERS A~H AND 0~Z ARE DECLARED DOUBLE PRECISION. DEFINE VARIABLES C = CONDUCTANCE MATRIX T = TEMPERATURE MATRIX B = [C]*[T] MATRIX ICO = ELEMENT CORNER"NODE NUMBERS X,Y = GLOBAL CO-ORDINATES D = CONDUCTIVITY MATRIX H = HEAT TRANSFER CO-EFFICIENT( ITYPE = TYPE OF ELEMENT IMAT = MATERIAL CODE BCC = BOUNDARY CODE(CONVECTION) BCT = BOUNDARY CODE (TEMPERATURE) IHF = CODE TO CALCULATE HEAT FLUX XX,YY,A,S ELEMENT CO-ORDINATES AND MATRICES  DIMENSION C(200,200),T(200),B(200),X(200),Y(200),ITYPE(200) 1BCT(200),IHF(200),IMAT(200),IPERM(400),TT(200,200),DA(2,2), 2XX(8),BCC(200),YY(8),NO(8),ICO(200,8),S4(4,4),S3(3,3),A4(4) 3S8(8,8),A8(8),A3(3),DC(2,2),RZ(200) COMMON /ZD/ DET,JEXP COMMON /DSLMP$/ NITER CALL SUBROUTINE TO READ IN NECESSARY DATA FOR PROBLEM CALL LAYOUT(X,Y,ICO,2 0 0,1TYPE,NE,NNODES,BCT,BCC,H,TW,COTC, 1IHF,COTA,IMAT)  C C INITIALISE ARRAYS C CALL DGSET(DC,2,2,2,0.D0) DC(1,1) = COTC DC(2,2) = DC(1,1) CALL DGSET(DA,2,2,2,0.D0) DA(1,1) = COTA DA(2,2) = DA(1,1) DO 3 1=1,200 B(I) = 0.D0 DO 3 J=1,200  102  C C C  3 C ( I , J ) = 0.D0 BEGIN DO-LOOP TO GO THROUGH ALL ELEMENTS DO 8 IEL=1,NE NN = ITYPE(IEL) BC = BCC(IEL) DO 4 I=1,NN NO(I) = IC0(IEL,I) XX(I) = X(NO(D) 4 YY(I) = Y(NO(l))  C C C  C C C C  CALL SUBROUTINE TO CALCULATE I-TH CONDUCTANCE MATRIX IM = IMAT(IEL) IF (IM .EQ. 0) GO TO 20 CALL COND(IEL,NN,XX,YY,S 3,S 4,S 8,DC,A3,A4,A8,H,TW,BC) GO TO 21 20 CALL C0ND(IEL,NN,XX,YY,S3,S4,S8,DA,A3,A4,A8,H,TW,BC) 21 CONTINUE CALL SUBROUTINE TO INSERT ELEMENT MATRIX INTO STRUCTURE MATRIX  5  C C C C  6 7 8  CALL SUBROUTINE TO ADJUST STRUCTURE MATRIX FOR BOUNDARY CONDITIONS  40 50 55 56  57 C C C  IF (NN .EQ. 3) GO TO 5 IF (NN .EQ. 8) GO TO 6 CALL SETUP(NO,NN,C,200,S4,NN,B,A4) GO TO 7 CALL SETUP(NO,NN,C,200,S3,NN,B,A3) GO TO 7 CALL SETUP(NO,NN,C,200,S8,NN,B,A8) CONTINUE CONTINUE  DO 50 1=1,NNODES IF (BCT(I)) 50,40,40 CALL BOUND(l,BCT,C,B,200,NNODES) CONTINUE WRITE(7,55) ((C(I,J),1=1,NNODES),J=1,NNODES) FORMAT(1X,2D30.22) WRITE(7,56) (B(I),1=1,NNODES) FORMAT(/,/,/,IX,(2D30.12)) DO 57 1=1,NNODES B(I) = B(I)*1.D45 DO 57 J=1,NNODES C ( I , J ) = C ( I , J ) * 1.D45 CALL MATRIX SOLVER FROM UBC LIBRARY DEPS = 1.D-16 CALL DSLIMP(C,TT,B,T,RZ,I PERM,NNODES,200,DEPS,1,14)  103  C C C  WRITE(6,25) DET,JEXP 25 FORMAT(1X,' DETERM. = ',G20.12,'* 10**' ,I 5,/) WRITE(6,26) ((IJK,T(IJK)),IJK=1,NNODES) 26 F0RMAT(1X,4('T(',13,') = ',F6.2,3X)) 27 CONTINUE CALCULATE HEAT FLUX  90  95 96 100 Q  Q  C C C  C C  '  AT MOLD INSIDE SURFACE AND PRINT  DO 100 IEL=1,NE IK = IHF(IEL) IF (IK .EQ. 0) GO TO 100 NN = ITYPE(IEL) DO 90 1=1,NN NO(I) = IC0(IEL,I) XX(I) = X(NO(l)) YY(I) = Y(NO(l)) T ( I ) = T(NO(l)) CONTINUE IM = IMAT(IEL) IF (IM .EQ. 0) GO TO 95 CALL HFLUX(IEL,NN,XX,YY,T,DC,NO) GO TO 96 CALL HFLUX(lEL,NN,XX,YY,T,DA,NO) CONTINUE CONTINUE STOP END  **************************************  SUBROUTINE COND(l,NN,X,Y,S3,S4,S8,D,A3,A4,A8,H,TW,BC) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(1),Y(1),S3(3,3),S4(4,4),D(2,2) 1,S8(8,8),A8(8),A3(3),A4(4) IF (NN.EQ. 4) GO TO 10 IF (NN .EQ. 8) GO TO 15 CALL TRICON(X,Y,S3,D,A3) GO TO 20 10 CALL QUACON(X,Y,S4,D,A4,H,TW,BC) GO TO 20 15 CALL LQCON(X,Y,S8,D,A8,H,TW,BC) 20 CONTINUE RETURN END  ***************************************************************  SUBROUTINE  QUACON(X,Y,S,D,A4,H,TW,BC)  CALCULATES CONDUCTANCE MATRIX FOR RECTANGULAR ELEMENT IMPLICIT REAL*8(A-H,0-Z) DIMENSION S(4,4),X(4),Y(4),D(2,2),B(2,4),BT(4,2),BTD(4,2) 1,AJ(2,2),XI(2),AI(2,2),ANU(4),ANV(4),ST(4,4) 2,SP(4,4),A4(4),AN(1,4),ANT(4,1) DATA XI/-0.577350269l89626D0,+0.577350269189626D0/ BEGIN DO-LOOPS FOR NUMERICAL INTEGRATION  104  CALL DGSET(S,4,4,4,0.D0) DO 26 1=1,2 DO 26 J=1,2 CALL DGSET(AJ,2,2,2,0.D0) U = XI(I) V = XI(J) CALCULATES DERIVATIVES OF SHAPE FUNCTIONS ANU(1) ANU(2) ANU(3) ANU(4) ANV(1) ANV(2) ANV(3) ANV(4)  = = = = = = = =  -(1.D0-V)*0.25D0 -ANU(1) (1.D0+V)*0.25D0 - ANU(3) - (1.D0 - U) *0.25D0 - (1.D0 +U) * 0.25D0 -ANV(2) -ANV(1)  CALCULATE JACOBIAN DO 6 K=1,4 AJ(1 , 1 ) = AJ(1 , 1) + ANU(K)*X(K) A J ( 1 , 2) = A J (1 , + 2)ANU(K) * Y(K) A J (2 , 1 ) = A J (2 , 1+ )ANV(K) * X(K) AJ(2,2) = AJ(2,2) + ANV(K) * Y(K) 6 CONTINUE - CALCULATE DETERMINANT AND INVERT AJ DET = AJ(1,1)*AJ(2,2) - AJ(1,2)*AJ(2,1) AI (1,1) = AJ(2,2)/DET AI(1,2) = - AJ(1,2)/DET AI(2,1) = -AJ(2,1)/DET AI(2,2) = AJ(1,1)/DET CALCULATE B DO 8 K=1,4 B(1,K) = AI(1,1)*ANU(K) +AI(1,2)*ANV(K) 8 B(2,K) = AI(2,1)*ANU(K) + AI(2,2)*ANV(K) CALCULATE BT*D*B CALL CALL CALL  DGTRAN(B,BT,2,4,2,4) DGMULT(BT,D,BTD,4,2,2,4,2,4) DGMULT(BTD,B,ST,4,2,4,4,2,4)  MULTIPLY BY DET AND WEIGHT DO 10 K=1,4 DO 10 L=1,4 10 S(K,L) = S(K,L) + ST(K,L)*DET 26 CONTINUE  105  30  31  32  33  34 35  C C C  ICODE= BC CALL DGSET(SP,4,4,4,0.DO) DO 30 K=1,4 A4(K) = 0.D0 DO 50 1=1,2 CALL DGSET(AJ,2,2,2,0.D0) GO TO (34,33,32,31,50), ICODE U = -1.DO V = XI(I ) ALINE = DABS(DSQRT((X(4)~X(1))**2 +(Y(4)-Y(1))**2)) GO TO 35 V = 1.DO U= X I ( I ) ALINE = DABS(DSQRT((X(3)~X(4))**2 + ( Y ( 3 ) - Y ( 4 ) ) * * 2 ) ) GO TO 35 U = 1.DO V = XI (I ) ALINE = DABS(DSQRT((X(2)-X(3))**2 + ( Y ( 2 ) - Y ( 3 ) ) * * 2 ) ) GO TO 35 V = -1.DO U = XI(I ) ALINE = DABS(DSQRT((X(1)-X(2))**2 + ( Y ( 1 ) - Y ( 2 ) ) * * 2 ) ) AN(1,1) = (1.D0-U)*(1,D0-V)*.25D0 AN(1,2) = (1.D0+U)*(1.D0-V) *.25D0 AN(1,3) = (1.D0+U)*(1.D0+V)*.25D0 AN(1,4) = (1.D0-U)*(1.D0+V)*.25D0  CALCULATE NT*N CALL DGTRAN(AN,ANT,1,4,1,4) CALL DGMULT(ANT,AN,SP,4,1,4,4,1,4) DO 40 K=1,4 DO 40 L=1,4 40 S(K,L) = S(K,L) + SP(K,L)*H*ALINE*.5D0 DO 45 K=1,4 45 A4(K) =A4(K) + AN(1,K)*H*TW*ALINE*.5D0 50 CONTINUE RETURN END  Q ***********  C C C  **  **************************************************  SUBROUTINE TRICON(X,Y,S,D,A3) CALCULATES CONDUCTANCE MATRIX FOR TRIANGULAR ELEMENT IMPLICIT REAL*8(A-H,0-Z) DIMENSION S(3,3),A3(3),X(3),Y(3),D(2,2),B(2,3),BT(3,2), 1BTD(3,2) A= X(2)*Y(3) - Y(2)*X(3) + X ( 1 ) * ( Y ( 2 ) - Y ( 3 ) ) + 1Y(1)*(X(3)-X(2)) AINV = 1.DO/A B(1,1) = (Y(2) -Y(3)) *AINV B(1 ,2) = (Y(3)-Y(1)) * AINV B( 1 ,3) = (Y(1)-Y(2)) * AINV B(2, 1) = (X(3)-X(2)) * AINV  106  B(2,2) B ( 2 , 3 ) C C C  CALCULATE  = =  20  Q  * *  AINV AINV  BT*D*B  CALL CALL  D G T R A N ( B , B T , 2 , 3 , 2 , 3 ) D G M U L T ( B T , D , B T D , 3 , 2 , 2 , 3 , 2 , 3 )  CALL  D G M U L T ( B T D , B , S , 3 , 2 , 3 , 3 , 2 , 3 )  DO 10 DO 10 S ( I , J ) DO 20 A3(I) = RETURN END  10  ( X ( 1 ) - X ( 3 ) ) ( X ( 2 ) - X ( 1 ) )  I=1,3 J=1,3 = S ( I , J ) * 0 . 5 D 0 * A 1=1,3 0.D0  *************************************  SUBROUTINE  LAYOUT(X,Y,ICO,NDIM,ITYPE,NE,NNODES,BCT,BCC,H,TW  1,COTC,IHF,COTA,IMAT) C C C  THIS ROUTINE PROBLEM  READS  IN  DATA  FOR  FINITE  ELEMENT  HEAT  CONDUCTION  C  28 29 60 30 32  40 41 44  C C C  IMPLICIT R E A L * 8 ( A - H , 0 - Z ) INTEGER BCC D I M E N S I O NX ( N D I M ) , I H F ( N D I M ) , Y ( N D I M ) , I C O ( N D I M , 8 ) , I T Y P E ( N D I M ) 1BCT(NDIM),BCC(NDIM),IMAT(NDIM) READ(5,28) COTC,COTA FORMAT(F5.3,F8.6) WRITE(6,29) COTC F O R M A T ( 1 X , ' C O - E F F . OF THERMAL C O N D U C T I V I T Y ( A L ) = ' , F 5 . 3 , / ) WRITE(6,60) COTA F O R M A T ( 1 X , ' C O - E F F . OF THERMAL C O N D U C T I V I T Y ( A I R ) = ' , F 8 . 6 , / ) READ(5,30) H,TW FORMAT(F7.3,F5.1) WRITE(6,32) H,TW FORMAT(1X,'HEAT TRAN. CO-EFF. = ' , F 7 . 3 , 3 X , ' W A T E R TEMP. = ' , 1 F 5 . 1 , / ) READ(5,40) NE,NNODES FORMAT(215) WRITE(6,41) NE,NNODES F O R M A T ( 1 X , ' N O . OF ELEMENTS= ' , I 5 , 2 X , ' N O . OF NODES= ' , 1 5 , / ) WRITE(6,44) FORMAT(6X,' X C0-0RD ' , 2 X , ' Y CO-ORD ',2X,'BOUNDARY 1CONDITIONS' ,/) DO 10 1=1,NNODES  IF  43 45 10  BCT  DOES  NOT  EXIST  ,  IT  SHOULD  BE  MADE  READ(5,43) X ( I ) , Y ( I ) , B C T ( I ) FORMAT(3F7.3) WRITE(6,45) I , X ( I ) , Y ( I ) , B C T ( I ) F O R M A T ( 1 X , I 3 , 2 X , F 8 . 3 , 2 X , F 1 0 . 3 , 5 X , F 8 . 3 ) CONTINUE  NEGATIVE  107  C ITYPE IS 4 IF RECTANGULAR ; 3 IF TRIANGULAR C READ(5,51) (ITYPE(I),I=1,NE) 51 FORMAT(20I2) WRITE(6,52) 52 FORMAT(/,/,1X, ELEMENT ',3X,'BOUNDARY',3X,'MATERIAL' 1,' NODE ') WRITE(6,53) 53 FORMAT(1X,' NO. ',3X,' CODE ',3X,' CODE ', 13X,' NUMBERS ',/) DO 11 1=1,NE IT = ITYPE(I) C C BCC > 1,2,3,4,5,: 1,2,3,4, REFER TO SIDES SUFFERING C CONVECTION C 5 IF THE ELEMENT SUFFERS NO CONVECTION C IHF > 0 IF THERE IS NO NEED TO CALCULATE HEAT FLUX C 1 OTHERWISE 1  c c c c  I MAT  •> 0 FOR AIR 1 FOR COPPER  READ(5,49) BCC(I),IHF(I),IMAT(I),(ICO(I,J),J=1,IT) 49 FORMAT(1515) WRITE(6,47) I,BCC(I),IMAT(I),(ICO(I,J),J=1,IT) 47 FORMAT(1X,I6,6X,I6,5X,I6,3X,8(I5, ,')) 11 CONTINUE RETURN END ****************************************** SUBROUTINE SETUP(NODES,NNODEL,A,NDIMA,B,NDIMB,PMAST,PELEM) 1  C PROGRAM TO SETUP MASTER STIFFNESS MATRIX FROM INDIVIDUAL C C FINITE ELEMENT STIFFNESS MATRICES. ALSO SETS UP LOAD VECTOR. C C NODES INTEGER VECTOR CONTAINING ELEMENT NODE NUMBERS C IN ORDER. C NNODEL NUMBER OF NODES PER ELEMENT C NVAR NUMBER OF VARIABLES PER NODE C MASTER STIFFNESS MATRIX WHICH MUST BE ZEROED A C BEFORE FIRST ENTRY TO SETUP C FIRST DIMENSION OF A NDIMA C ELEMENT STIFFNESS MATRIX B C FIRST DIMENSION OF B NDIMB C MASTER LOAD VECTOR WHICH MUST BE ZEROED BEFORE PMAST C FIRST ENTRY. C ELEMENT LOAD VECTOR PELEM C C IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(NDIMA,NDIMA),B(NDIMB,NDIMB),NODES(20), 1PMAST(NDIMA),PELEM(20) DO 1 1=1,NNODEL MM = NODES(I)  108  Q  PMAST (MM) = PMAST (MM) +PELEM (I ) DO 1 J=1,NNODEL NN = NODES(J) 1 A(MM,NN)=A(MM,NN)+B(I,J) RETURN END  *************************************  SUBROUTINE C C C C  Q  C C C  BOUND(I,BCT,AK,B,NDIM,NN)  THIS SUBROUTINE MODIFIES MATRIX FOR FIXED TEMPERATURE BOUNDARY CONDITIONS IMPLICIT REAL*8(A-H,0-Z) DIMENSION BCT(NDIM),AK(NDIM,NDIM),B(NDIM) PERM = AK(I,I) DO 10 J=1,NN B(J) = B(J) -AK(J,I)*BCT(I) 10 AK(I,J) = 0.D0 DO 11 J=1,NN 11 AK(J,I) = 0.D0 AK(I,I) = PERM B(I) = AK(I,1)*BCT(I) RETURN END  ***************************************************************  SUBROUTINE HFLUX(IEL,NN,X,Y,T,D,NO) THIS ROUTINE CALCULATES HEAT FLUX IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(NN),NO(8),HF(4),Y(NN),T(NN),HF4(4),D(2,2) 1,HF8(8) IF (NN.EQ. 3) GO TO 5 IF (NN .EQ. 8) GO TO 10 CALL FLUX4(X, Y, T,HF 4,D,HFI) WRITE(6,51) 51 FORMAT(1X,' ELEMENT NO. ',5X,' NODES ',' 1 HEAT FLOW ) WRITE(6,52) (NO(I),I=1,NN) 52 FORMAT( 1X, 1 6X, 4 (16 , 2X)) WRITE(6,53) IEL,(HF4(I),1=1,NN),HFI 53 FORMAT(1X,I6,1 OX,4(F6.2,2X),3X,F7.3) GO TO 20 5 CALL FLUX3(X,Y,T,HF 3,D) WRITE(6,54) 54 FORMAT(1X,' ELEMENT NO. ',5X, HRAT FLUX ') WRITE(6,55) IEL,HF3 55 FORMAT(1X,16,12X,F7.3) GO TO 20 10 CALL FLUX8(X,Y,T,HF8,D,HFI) WRITE(6,56) 56 FORMAT(1X,' ELEMENT NO. ',5X,' NODES ',32X,' 1 HEAT FLOW' ) WRITE(6,57) (NO(l),1=1,NN) T  109  Q  C C C C  C C C  57 FORMAT(1X,16X,8(I6,2X)) WRITE(6,58) IEL,(HF8(l),1=1,8),HFI 58 FORMAT(1X,I 6,10X,8(F6.2,2X),3X,F7.3) 20 CONTINUE RETURN END  ********************************************  SUBROUTINE FLUX3(X,Y,T,HF,D) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(3),Y(3),B(2,3),BT(2),T(3),BTD(2) A = X(2)*Y(3) - Y(2)*X(3) + X( 1 ) * (Y( 2)-Y ( 3) ) +Y(D* 1(X(3)-X(2)) AINV= 1.DO/A B(1 , 1) = (Y(2) -Y(3))*AINV B(1,2) = (Y(3) - Y(1))*AINV B(1,3) = (Y(1) - Y(2))*AINV B(2,1) = (X(3) -X(2))*AINV B(2,2) = (X(1) - X(3))*AINV B(2,3) = (X(2) - X(1))*AINV CALL DGMATV(B,T,BT,2,3,2) CALL DGMATV(D,BT,BTD,2,2,2) HF = BTD(1) RETURN END *************************************************************** SUBROUTINE FLUX4(X,Y,T,HF,D,HFI) CALCULATES HEAT FLUX IN RECTANGULAR ELEMENT IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(4),Y(4),T(4),AJ(2,2),AI(2,2),ANU(4),ANV(4) 1,XY(8),YX(8),YI(2),IC(4),B(2,4),XI(2),HF(4),BTD(2),BT(2), 2D(2 2) DATA YI/-.577350269189626D0, + .577350269189626D0/ DATA XI/-1.D0,+1.DO/ DATA IC/1,2,4,3/ IP = 0 DO 200 1=1,2 DO 200 J=1,2 CALL DGSET(AJ,2,2,2,0.D0) U = XI(I ) V = XI(J) ANU(1) = -(1.D0-V)*0.25D0 ANU(2) = -ANU(1) ANU(3) = (1.D0+V)*0.25D0 ANU(4) = - ANU(3) ANV(1) = - (1.D0 - U) *0.25D0 ANV(2) = - (1.D0 +U) * 0.25D0 ANV(3) = -ANV(2) ANV(4) = -ANV(1) CALCULATE JACOBIAN DO 6 K=1,4  110  IA = IC(K) XY(K) = X(IA) YX(K) = Y(IA) AJ(1,1) = AJ(1,1) AJ(1,2) = AJ(1,2) AJ(2,1) = AJ(2,1) AJ(2,2) = AJ(2,2) 6 CONTINUE C C C  C C C  C C C  + + + +  ANU(K)*X(K) ANU(K) * Y(K) ANV(K) * X(K) ANV(K) * Y(K)  CALCULATE DETERMINANT AND INVERT AJ DET = AJ(1,1)*AJ(2,2) - AJ(1,2)*AJ(2,1) AI ( 1 , 1 ) = AJ(2,2)/DET AI (1 ,2) = - AJ(1,2)/DET AI (2, 1 ) = -AJ(2,1)/DET AI (2,2) = AJ(1,1)/DET DO 100 K=1,4 B(1,K) = AI(1,1)*ANU(K) + AI(1,2)*ANV(K) 100 B(2,K) = AI(2,1)*ANU(K) + AI(2,2)*ANV(K) CALL DGMATV(B,T,BT,2,4,2) CALL DGMATV(D,BT,BTD,2,2,2) IP = IP + 1 IK = IC(IP) HF(IK) = BTD(1) 200 CONTINUE HFI = 0.D0 ALINE2 = DABS(DSQRT((X(2) - X(3))**2 + (Y(2) - Y ( 3 ) ) * * 2 ) ) DO 350 1=1,2 CALL DGSET(AJ,2,2,2,0.D0) U = 1.DO V = YI(I ) ANU(1) = -(1.D0-V)*0.25D0 ANU(2) = -ANU(1) ANU(3) = (1.D0+V)*0.25D0 ANU(4) = " ANU(3) ANV(1) = - (1.D0 - U) *0.25D0 ANV(2) = - (1.D0 +U) * 0.25D0 ANV(3) = -ANV(2) ANV(4) = -ANV(1) CALCULATE JACOBIAN DO 250 K=1,4 AJ(1,1) = AJ(1,1) AJ(1,2) = AJ(1,2) AJ(2,1) = AJ(2,1) AJ(2,2) = AJ(2,2) 250 CONTINUE  + + + +  ANU(K)*X(K) ANU(K) * Y(K) ANV(K) * X(K) ANV(K) * Y(K)  CALCULATE DETERMINANT AND INVERT AJ DET = AJ(1,1)*AJ(2,2) - AJ(1,2)*AJ(2,1) AI ( 1 , 1 ) = AJ(2,2)/DET AI (1 ,2) = - AJ(1,2)/DET  111  A l ( 2 , 1 ) = -AJ(2,1)/DET A l ( 2 , 2) = AJ(1 ,1)/DET DO 300 K=1,4 B(1,K) = Al(1,1)*ANU(K) + Al(1,2)*ANV(K) 300 B(2,K) = Al(2,1)*ANU(K) + Al(2,2)*ANV(K) CALL DGMATV(B,T,BT,2,4,2) CALL DGMATV(D,BT,BTD,2,2,2) HFI = HFI + BTD(1)*ALINE2*.5D0 350 CONTINUE RETURN END C ******************************************** SUBROUTINE LQCON(X,Y,S,D,A8,H,TW,BC) C C CALCULATES CONDUCTANCE MATRIX FOR 8 NODE C ISOPARAMETRIC ELEMENT. C IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(8),Y(8),S(8,8),D(2,2),A8(8),W(3),XI(3), 1AJ(2,2),AI(2,2),ANU(8),ANV(8),AN(1,8),ANT(8,1),ST(8,8), 2SP(8,8),BT(8,2),BTD(8,2),B(2,8) DATA W/0.5555555555556D0,0.8888888888889D0,0.5555555555556 1D0/ DATA XI/-0.774596669241483D0,0.DO,0.774596669241483D0/ C C ZERO ARRAYS C CALL DGSET(S,8,8,8,0.D0) C C BEGIN DO-LOOPS FOR NUMERICAL INTEGRATION C DO 26 1=1,3 DO 26 J=1,3 CALL DGSET(AJ,2,2,2,0.D0) U=XI(I) V= X I ( J ) C C CALCULATE DERIVATIVES FOR SHAPE FUNCTIONS C ANU(1) = (1.D0-V)*(2.D0*U+V)*.25D0 ANU(2) = (1.D0-V)*(2.D0*U-V)*.25D0 ANU(3) = (1.D0+V)*(2.D0*U+V)*.25D0 ANU(4) = (1.D0+V)*(2.D0*U-V)*.25D0 ANU(5) = -U*(1.D0-V) ANU(6) = (1.D0-V*V)*.5D0 ANU(7) = -U*(1.D0+V) ANU(8) = -(1.D0-V*V)*.5D0 ANV(1) = (1.D0-U)*(U+2.D0*V)*.25D0 ANV(2) = (1.D0+U)*(2.D0*V-U)*.25D0 • ANV(3) = (1.D0+U)*(2.D0*V+U)*.25D0 ANV(4) = (1.D0-U)*(2.D0*V-U)*.25D0 ANV(5) = -(1.D0-U*U)*.5D0 ANV(6) = -V*(1.D0+U) ANV(7) = (1.D0-U*U)*.5D0  1 12  C C C  - ANV(8) = -V*(1.DO-U) CALCULATE JACOBIAN DO 6 K=1,8 AJ(1,1) =AJ(1,1) +ANU(K)*X(K) AJ(1,2) = AJ(1,2) + ANU(K)*Y(K) AJ(2,1) = AJ(2,1) + ANV(K) * X(K) 6 AJ(2,2) = AJ(2,2) + ANV(K)*Y(K)  C C C  CALCULATE DET AND INVERT AJ  C C C  CASLCULATE CONDUCTANCE MATRIX  C C C  MULTIPLY BT*D*B  C C C C  MULTIPLY RESULT BY WEIGTHS AND DET. OF J AND ADD INTO CONDUCTANCE MATRIX  DET = AJ(1,1)*AJ(2,2) - AJ(1,2)*AJ(2,1) Al ( 1 , 1 ) = AJ(2,2)/DET Al(1 ,2) = -AJ(1,2)/DET Al(2,1) = -AJ(2,1)/DET AI(2,2) =AJ(1,1)/DET  DO 8 K=1,8 B(1,K) = A l ( 1 , 1 ) * ANU(K) +AI(1,2)* ANV(K) 8 B(2,K) = AI(2,1) *ANU(K) + AI(2,2) * ANV(K)  CALL DGTRAN(B,BT,2,8,2,8) CALL DGMULT(BT,D,BTD,8,2,2,8,2,8) CALL DGMULT(BTD,B,ST,8,2,8,8,2,8)  11 26  30  31 32 33 34  DO 11 K=1,8 DO 11 L=1,8 S(K,L) = S(K,L) +W(I)*W(J)*ST(K,L)*DET CONTINUE ICODE = BC CALL DGSET(SP,8,8,8,0.D0) DO 30 K=1,8 A8(K) = 0.D0 DO 50 1=1,3 CALL DGSET(AJ,2,2,2,0.D0) GO TO (34,33,32,31,50), ICODE U= -1.DO V= X I ( I ) GO TO 35 V= 1.DO U= X I ( I ) GO TO 35 U = 1.D0 V= X I ( I ) GO TO 35 V= -1.DO  1 13  U= XI(I ) 35 AN(1,1) = AN(1,2) = AN(1,3) = AN(1,4) = AN(1,5) = AN(1,6) = AN(1,7) = AN(1,8) = C C C  C C C  C C C  -(1.DO-U)*(1.DO-V)*(1.DO+U+V)*.25D0 -(1.DO+U)*(1.DO-V)*(1.DO-U+V)*.25D0 -(1.DO+U)*(1.DO+V)*(1.D0-U-V)*.25D0 -(1.DO-U)*(1.DO+V)*(1.DO+U-V)*.25D0 (1.D0-U**2)*(1.D0-V)*.5D0 (1.D0-V**2)*(1.D0+U)*.5D0 (1.DO-U**2)*(1.D0+V)*.5D0 (1.D0-V**2)*(1,D0-U)*.5D0  CALCULATE DERIVATIVES FOR SHAPE FUNCTIONS ANU(1) ANU(2) ANU(3) ANU(4) ANU(5) ANU(6) ANU(7) ANU(8) ANV(1) ANV(2) ANV(3) ANV(4) ANV(5) ANV(6) ANV(7) ANV(8)  = = = = = = = = = = = = = = = =  (1.D0-V)*(2.D0*U+V)*.25D0 (1.D0-V)*(2.D0*U-V)*.25D0 (1.D0+V)*(2.D0*U+V)*.25D0 (1.D0+V)*(2.D0*U-V)*.25D0 -U*(1.DO-V) (1.D0-V*V)*.5D0 -U*(1.DO+V) -(1.D0-V*V)*.5D0 (1.D0-U)*(U+2.D0*V)*.25D0 (1.D0+U)*(2.D0*V-U)*.25D0 (1.D0+U)*(2.D0*V+U)*.25D0 (1.D0-U)*(2.D0*V-U)*.25D0 -(1.D0-U*U)*.5D0 -V*(1.DO+U) (1.D0-U*U)*.5D0 -V*(1.DO-U)  CALCULATE JACOBIAN DO 37 K=1,8 AJ(1,1) =AJ(1,1) +ANU(K)*X(K) AJ(1,2) = AJ(1,2) + ANU(K)*Y(K) AJ(2,1) = AJ(2,1) + ANV(K) * X(K) 37 AJ(2,2) = AJ(2,2) + ANV(K)*Y(K) IF (ICODE .EQ. 1) GO TO 38 IF (ICODE .EQ. 3) GO TO 38 ALINE = Aj(2,1) + AJ(2,2) GO TO 39 38 ALINE = AJ(1,1) +AJ(1,2) 39 CONTINUE CALCULATE NT*N CALL DGTRAN(AN,ANT,1,8,1,8) CALL DGMULT(ANT,AN,SP,8,1,8,8,1,8) DO 40 K=1,8 DO 40 L=1,8 40 S(K,L) = S(K,L) +SP(K,L)*H*W(I)*ALINE DO 45 K=1,8 45 A8(K) = A8(K) + AN(1,K)*H*TW*W(I)*ALINE 50 CONTINUE RETURN  1 14  END C***************************************************** SUBROUTINE FLUX8(X,Y,T,HF,D,HFI) IMPLICIT REAL*8(A-H,0-Z) DIMENSION X(8),Y(8),T(8),AJ(2,2),AI(2,2),ANU(8),ANV(8) 1, XY(8),YX(8),YI(3),IC(8),B(2,8),XI(3),HF(8),BTD(2),BT(2) 2, D(2,2),W(3) DATA YI/-0.774596669241483D0,0.DO,0.774596669241483D0/ DATA W/0.5555555555556DO,0.8888888888889DO,0.5555555555556 1D0/ DATA XI/-1.D0,0.D0,+1.DO/ DATA IC/1,5,3,8,6,4,7,3/ IP=0 DO 200 1=1,3 DO 200 J=1,3 CALL DGSET(AJ,2,2,2,0.D0) U= X I ( I ) V = XI(J) IF (U .EQ. 0) GO TO 2 GO TO 3 2 IF (V .EQ. 0) GO TO 200 3 CONTINUE C C CALCULATE DERIVATIVES FOR SHAPE FUNCTIONS C ANU(1) = (1.D0-V)*(2.D0*U+V)*.25D0 ANU(2) = (1.D0-V)*(2.D0*U-V)*.25D0 ANU(3) = (1.D0+V)*(2.D0*U+V)*.25D0 ANU(4) = (1.D0+V)*(2.D0*U-V)*.25D0 ANU(5) = -U*(1.D0-V) ANU(6) = (1.D0-V*V)*.5D0 ANU(7) = -U*(1.D0+V) ANU(8) = -(1.D0-V*V)*.5D0 ANV(1) = (1.D0-U)*(U+2.D0*V)*.25D0 ANV(2) = (1.D0+U)*(2.D0*V-U)*.25D0 ANV(3) = (1.D0+U)*(2.D0*V+U)*.25D0 ANV(4) = (1.D0-U)*(2.D0*V-U)*.25D0 ANV(5) = -(1.D0-U*U)*.5D0 ANV(6) = -V*(1.D0+U) ANV(7) = (1.D0-U*U)*.5D0 ANV(8) = -V*(1.D0-U) C C CALCULATE JACOBIAN C DO 6 K=1,8 IA = IC(K) XY(K) = X(IA) YX(K) = Y(IA) AJ(1,1) =AJ(1,1) +ANU(K)*X(K) AJ(1,2) = AJ(1,2) + ANU(K)*Y(K) AJ(2,1) = AJ(2,1) + ANV(K) * X(K) 6 AJ(2,2) = AJ(2,2) + ANV(K)*Y(K) C C CALCULATE DET AND INVERT AJ  115  C  C C C  c  C C  C C C  DET = AJ(1,1)*AJ(2,2) - AJ(1,2)*AJ(2,1) Al(1,1) = AJ(2,2)/DET A l ( 1 , 2 ) = -AJ(1,2)/DET Al(2,1) = -AJ(2,1)/DET AI(2,2) =AJ(1,1)/DET DO 8 K=1,8 B(1,K) = AI(1,1)* ANU(K) +AI(1,2)* ANV(K) 8 B(2,K) = AI(2,1) *ANU(K) + AI(2,2) * ANV(K) CALL DGMATV(B,T,BT,2,8,2) CALL DGMATV(D,BT,BTD,2,2,2) IP = IP + 1 IK = IC(IP) HF(IK) = BTD(1) 200 CONTINUE HFI = 0.D0 DO 350 1=1,3 CALL DGSET(AJ,2,2,2,0.D0) U = 1.DO V= YI(I ) CALCULATE DERIVATIVES FOR SHAPE FUNCTIONS ANU(1) ANU(2) ANU(3) ANU(4) ANU(5) ANU(6) ANU(7) ANU(8) ANV(1) ANV(2) ANV(3) ANV(4) ANV(5) ANV(6) ANV(7) ANV(8)  == == == == == == == == == == == == == == == ==  (1.D0-V)*(2 .D0*U+V)* .25D0 (1.D0-V)*(2 .D0*U-V)* .25D0 (1.D0+V)*(2 .D0*U+V)* .25D0 (1.D0+V)*(2 .D0*U-V)* . 25D0 -U*(1.DO-V) (1.D0-V*V)* .5D0 -U*(1.DO+V) -(1.D0-V*V)*.5D0 (1.D0-U)*(U+2.D0*V)* .25D0 (1.D0+U)*(2 .D0*V-U)* .25D0 (1.D0+U)*(2 .D0*V+U)* .25D0 (1.D0-U)*(2 .D0*V-U)* .25D0 -(1.D0-U*U)*.5D0 -V*(1.DO+U) (1.D0-U*U)* .5D0 -V*(1.DO-U)  CALCULATE JACOBIAN DO 250 AJ(1,1) AJ(1,2) AJ(2,1) 250 AJ(2,2)  K=1,8 =AJ(1,1) +ANU(K)*X(K) = AJ(1,2) + ANU(K)*Y(K) = AJ(2,1) + ANV(K) * X(K) = AJ(2,2) + ANV(K)*Y(K)  CALCULATE DET AND INVERT AJ DET = AJ(1,1)*AJ(2,2) - AJ(1,2)*AJ(2,1) A l ( 1 , 1 ) = AJ(2,2)/DET Al(1,2) = -AJ(1,2)/DET Al(2,1) = -AJ(2,1)/DET  116  AI(2,2) =AJ(1,1)/DET DO 300 K=1,8 B(1,K) = AI(1,1)* ANU(K) +AI(1,2)* ANV(K) 300 B(2,K) = AI(2,1) *ANU(K) + Al(2,2) * ANV(K) CALL DGMATV(B,T,BT,2,8,2) CALL DGMATV(D,BT,BTD,2,2,2) HFI = HFI + BTD(1)*W(I)*(AJ(1,1)+AJ(1,2)) 350 CONTINUE RETURN END  1 17  APPENDIX E ~ F.E.  a.  FORMULATION FOR STEADY HEAT TRANSFER IN A PLANE  The F o u r i e r heat conduction equation f o r steady heat t r a n s f e r i n a plane may be w r i t t e n as  [Q where [k] =  k  x  k For homogeneous  x  k  q l  T  y  = " E k ] [dT/dx  9T/9y]  T  xy  k isotropic materials k  = 0 and k  x y  x  = k,  Then k ( 9 T / 9 x ) + k ( 9 T / 9 y ) = 0. 2  2  2  2  A f u n c t i o n a l f o r t h i s problem i s  n  = 0.5  f f ( { 9 T / 9 x 9T/9y} [k] {9T/9x 9T/9y} + fq* T dS + J ( h / 2 ) ( T  2  T  dA  - 2TT»)dS  where q* - p r e s c r i b e d heat f l u x on boundary S . N  h - heat t r a n s f e r c o e f f i c i e n t on boundary S . c  and T» - the attendant bulk temperature  of the surrounding  medium. On m i n i m i s a t i o n , the matrix equation [K ] {T} = {P^} + {p } r  i s obtained where {T} i s the s o l u t i o n f o r the d i s t r i b u t i o n and [K ] i s the conductance /[B  ]  T  [k] [B] dA ( The s p a t i a l f i e l d  temperature  matrix given by  f o r temperature  i n an  element i s [N ] {T } where [N] i s the shape f u n c t i o n matrix. Then {9T/9x 9T/9y} = [B ] {T } where [B ] = {9 /9x 9 /9y} [N ]) {P^} = /[N ]  T  q* dS  118  t  1  I  N =  N,  =-1  N  N  2  N ]  3  4  where N, = 1/4 ( 1 - s ) ( 1 - t ) N2=  1/4 (1 * s » 1-t)  N =  1/4 (1*s)(1*t)  N =  1/4 (1- s)(1*t|  3  t  F i g u r e 51 - Shape f u n c t i o n s - R e c t .  N =  2To7io-/ 1 N  N  2  isoparametric  element  N J 3  where  1  X Y -Y X 2  X  3  2  3  3 r 3 1 Y  Y  X  L 1 2" 1 2 X  Y  Y  X  Y -Y 2  V V  Y  Y  X -X  3  1 2  3  X  1 V  - X  X  2  3 1  F i g u r e 52 - Shape f u n c t i o n s - T r i a n g u l a r element  119  (P } = J[N ] r  the l a s t two equations b.  T  h T«, dS r e p r e s e n t i n g the boundary c o n d i t i o n s ,  The shape f u n c t i o n s f o r a r e c t a n g u l a r element and those in f i g .  Ref.  51 and  isoparametric  f o r a t r i a n g u l a r element are given  52.  R.D.Cook, Concepts and A p p l i c a t i o n s of F i n i t e  Analysis  , J.Wiley  and Sons,  1981.  Element  

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