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Inverse segregation and centreline shrinkage Minakawa, Sakae 1984

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INVERSE  SEGREGATION  AND  CENTRELINE  SHRINKAGE  by  SAKAE B.E.,  MINAKAWA  Tohoku U n i v e r s i t y ,  A THESIS THE  SUBMITTED  OF  1977.  IN PARTIAL FULFILMENT  REQUIREMENTS MASTER  Japan,  FOR  THE  APPLIED  DEGREE  OF  SCIENCE  in THE  FACULTY  Department  We  accept  OF  GRADUATE  of M e t a l l u r g i c a l  this  thesis  tp_ t h e r e q u i r e d  THE  UNIVERSITY  OF  June  ©  as  STUDIES Engineering  conforming  standard  BRITISH  COLUMBIA  1984  Sakae Minakawa,  1984  OF  In  presenting  this  thesis  in  partial  fulfilment  of  r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of Columbia,  I  available  for  permission  agree  for  p u r p o s e s may or  her  that  the  Library  shall  reference  and  study.  I  extensive  p u b l i c a t i o n of t h i s t h e s i s a l l o w e d w i t h o u t my  Department of  written  It for  is  permission.  Metallurgy  June 8,  1984  gain  that  freely  agree  that  scholarly  Department or  understood  financial  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  Date:  further  British  it  c o p y i n g of t h i s t h e s i s f o r  be g r a n t e d by the Head of my  representatives.  make  the  by  copying  shall  not  his or be  ii  Abstract One  of  shrinkage fluid flow  the major  which o c c u r s  results  from  solidification cast  during s o l i d i f i c a t i o n .  Interdendritic  a  cold  at  and  using a mathematical  The a l l o y s  c o n s i d e r e d were A l - C u ,  of  particular  The model  with  data  published  p r e v i o u s m o d e l s , t h e volume segregation the  present  significant from  did  role  the present  earlier  w i t h back  compared  should  and  show  model  differs  shrinkage  fluid  used  was  Regions  occur  were  of  expands  during  to  agreement In t h e inverse  have  expected. from  more  The  those  results of  the  alloys.  plate casting, associated was  combined the  examined u s i n g with a heat  plate  predicted  to published experimental  latter  c o n t r a c t i o n which, i n  found  liquid,  model  been  the  to c a l c u l a t e  significantly  in a steel  the  calculations.  inverse segregation.  i n the Al-Zn  flow  it  for  reasonable  i n c l u d e the thermal predictions,  segregation  f a c e have  Sb-Bi,  since  i n the s e g r e g a t i o n than  of the system.  porosity  face  flow of i n t e r d e n d r i t i c  interdendritic  Al-Zn  inverse  values  and computer  shrinkage  models, p a r t i c u l a r l y  Centreline  model  not  model  model  results  of c h i l l  Inverse  by  o f many  in  to the c h i l l  significance  solidification.  results  Quantitative  adjacent  left  i n the case  investigation,  determined  being  the v o i d s  which  chill,  theoretically.  both  to f i l l  shrinkage,  In t h e p r e s e n t  examined  segregation  interdendritic  movement o f l i q u i d  against  and  of  and t h e r m a l  segregation. h a s been  macrosegregation  p o r o s i t y i n c a s t i n g s i s the extent  flow  alloys  factors controlling  results  Where  w i t h t h e new  the  transfer  centreline model  of p o r o s i t y i n  and  steel  plates. and was  experimental used  channels. this  E x c e l l e n t agreement  to  results.  determine  The p r e s e n t  application.  the  In  was  obtained  the  fluid  extent  results  of  flow  flow  indicate  between  in  model, the  the  predicted  Darcy's  Law  interdendritic  D a r c y ' s Law i s  valid  in  Table of C o n t e n t s Abstract L i s t of T a b l e s L i s t of F i g u r e s L i s t of Nomenclature Acknowledgement  i i vi v i i x xiv  ....  PART-A INVERSE SEGREGATION  IN BINARY ALLOYS  1  Chapter A-I INTRODUCTION AND LITERATURE REVIEW 1.1 I n t r o d u c t i o n 1.2 P r e v i o u s Models f o r C h i l l Face I n v e r s e S e g r e g a t i o n 1.3 I n v e r s e S e g r e g a t i o n away from t h e C h i l l Face ( p o s i t i o n a l Segregation) 1.4 P r e s e n t O b j e c t i v e s  2 2 3  7 ,...11  Chapter A - I I MODELING PROCEDURE 2.1 M a t h e m a t i c a l F o r m u l a t i o n 2.1.1 A l l o y D e n s i t i e s 2.1.2 Shrinkage Along The C h i l l Face 2.1.3 S e g r e g a t i o n i n the F i r s t Column A d j a c e n t To The C h i l l Face 2.1.4 P o s i t i o n a l S e g r e g a t i o n 2.1.5 The Length of I n v e r s e S e g r e t a t i o n Zone 2.2 Computer Programming  19 20 23 24  Chapter A - I I I RESULTS AND DISCUSSION 3.1 The Aluminium Copper System 3.1.1 Volume Change d u r i n g S o l i d i f i c a t i o n 3.1.2 S e g r e g a t i o n a t t h e C h i l l Face 3.1.3 P o s i t i o n a l S e g r e g a t i o n 3.2 The Aluminium Zinc System 3.3 The Antimony Bismuth System 3.4 E x a m i n a t i o n Of P r e v i o u s Model P r e d i c t i o n s  25 25 25 27 28 30 33 36  Chapter A-IV CONCLUSIONS  37  PART-B CENTERLINE SHRINKAGE IN STEEL PLATE CASTINGS  68  Chapter B-I INTRODUCTION 1.1 Gross Shrinkage 1.2 C e n t e r l i n e Shrinkage 1.3 P r e v i o u s Models 1.4 P e r m e a b i l i t y  •  13 13 16 17  .69 ........69 ..70 72 ...76  V  1.5 P r e s e n t O b j e c t i v e s Chapter B - I I MODELLING PROCEDURE 2.1 M a t h e m a t i c a l F o r m u l a t i o n 2.1.1 Temperature C a l c u l a t i o n s 2.1.2 P r e s s u r e R e q u i r e d To Feed S h r i n k a g e 2.2 Computer Programming 2.3 V a l i d a t i o n Of The Heat T r a n s f e r Model Chapter B - I I I RESULTS AND DISCUSSION 3.1 S o l i d i f i c a t i o n Sequence .' 3.2 P r e d i c t i o n Of C e n t e r l i n e S h r i n k a g e 9.3 The E f f e c t of an End C h i l l on t h e Length of t h e P o r o s i t y Free Region 3.4 Proposed S o l i d i f i c a t i o n Parameters D e f i n i n g t h e T r a n s i t i o n from Porous t o Nonporous C a s t i n g s  79 80 80 80 82 84 84 86 86 87 89 90  Chapter B-IV CONCLUSIONS  92  BIBLIOGRAPHY  107  APPENDIX A - RECALCULATION OF THE SCHEIL-YOUDELIS MODEL PREDICTIONS  110  APPENDIX B - DERIVATION OF NODAL EQUATIONS  111  APPENDIX C - FORTRAN PROGRAM FOR THE PREDICTION OF CENTRELINE SHRINKAGE  113  vi  List  I.  of  Tables  The c o m p a r i s o n o f a l l o y d e n s i t i e s w i t h c a l c u l a t e d from d a t a f o r p u r e m e t a l s  II.  The t h e r m a l temperature  p r o p e r t i e s employed distribution  III.  Physical  IV.  Dimension of  V.  C o m p a r i s o n of t h e c r i t i c a l v a l u e s o f parameters f o r c e n t e r l i n e shrinkage  data  employed the  those  f o r t h e c a l c u l a t i o n of 40  in calculations  steel plate  39  casting  examined  93 94  solidfication 94  vii  L i s t of F i g u r e s  1. E q u i l i b r i u m phase diagram and s p e c i f i c volume f o r the Al-Cu a l l o y s 41 2. Schematic d e s c r i p t i o n of s o l i d i f y i n g zone; (a) d e n d r i t e morphology, (b) s o l i d and l i q u i d c o m p o s i t i o n s a l o n g c h i l l f a c e , and ( c ) c o m p o s i t i o n of i n t e r d e n d r i t i c l i q u i d 42 3. D e n s i t i e s of copper and aluminium vs temperature  43  4. D e n s i t i e s of Z i n c , Aluminium and Bismuth vs temperature 44 5. Schematic c o n f i g u r a t i o n of t h e model i n v e s t i g a t e d and the s u b d i v i s i o n f o r the n u m e r i c a l s i m u l a t i o n 45 6. Schematic c o n f i g u r a t i o n of t h e f e e d i n g sequence; (a) F l u i d f l o w , and (b) p r o f i l e of t h e l i q u i d c o m p o s i t i o n  46  7. Summary of the f l u i d f l o w induced by the s h r i n k a g e d u r i n g the s o l i d i f i c a t i o n of t h e model w i t h 10X10 subdivisions  47  8. Schematic d e s c r i p t i o n of t h e d i l u t i o n e f f e c t of t h e i n t e r d e n d r i t i c f l u i d flow during s o l i d i f i c a t i o n  48  9. Flow c h a r t f o r the c a l c u l a t i o n of i n v e r s e s e g r e g a t i o n 10. Flow c h a r t f o r the c a l c u l a t i o n of the l e n g t h of s o l i d / l i q u i d zone a d j a c e n t t o t h e c h i l l f a c e  49 50  11. C a l c u l a t e d volume change d u r i n g s o l i d i f i c a t i o n of A l 1 0 % C U . The number on t h e curve denotes each element solidified 51 12. C a l c u l a t e d volume change and l i q u i d c o m p o s i t i o n i n the r e p r e s e n t a t i v e volume f o r A l - 5 % C u . The number on the c u r v e denotes each element s o l i d f i e d 52 13. S o l i d i f i c a t i o n s h r i n k a g e r a t i o vs c o m p o s i t i o n of the A l Cu a l l o y s . The d o t t e d l i n e was d e r i v e d from F i g 1. ...53 14. P o s i t i o n a l s e g r e g a t i o n p r o f i l e s f o r v a r i o u s of t h e A l - C u a l l o y s  compositions 54  15. Comparison of the i n v e r s e s e g r e g a t i o n a t t h e c h i l l for the Al-Cu a l l o y s  face 55  vi ii  16. E f f e c t of a i r gap a t the metal/mold i n t e r f a c e on t h e l e n g t h of s o l i d / l i q u i d zone a d j a c e n t t o t h e c h i l l f a c e 17. Comparison of p o s i t i o n a l s e g r e g a t i o n  56  f o r A 1 - 1 0 % C U ....57  18. E q u i l i b r i u m phase diagram f o r A l - Z n a l l o y s  58  19. T o t a l s h r i n k a g e r a t i o vs c o m p o s i t i o n f o r t h e A l - Z n alloys  59  20. C a l c u l a t e d volume change and t e m p e r a t u r e i n t h e r e p r e s e n t a t i v e volume f o r A l - l O % Z n . The number on t h e c u r v e denotes each element s o l i f i e d 60 21. C a l c u l a t e d p r o f i l e s of temperature i n t h e r e p r e s e n t a t i v e volume f o r v a r i o u s c o m p o s i t i o n s of t h e A l - Z n a l l o y s indicated 61 22. Comparison of the i n v e r s e s e g r e g a t i o n for t h e A l - Z n a l l o y s  at the c h i l l  face 62  23. E q u i l i b r i u m phase diagram f o r Sb-Bi a l l o y s  63  24. T o t a l s h r i n k a g e r a t i o vs c o m p o s i t i o n f o r t h e Sb-Bi alloys  64  25. C a l c u l a t e d volume change and t e m p e r a t u r e i n t h e r e p r e s e n t a t i v e volume f o r S b - l O % B i . The number on t h e c u r v e denotes each element s o l i d i f i e d 65 26. Comparison of t h e i n v e r s e s e g r e g a t i o n for t h e Sb-Bi a l l o y s  a t the c h i l l  face 66  27. C a l c u l a t e d volume change and t e m p e r a t u r e i n t h e r e p r e s e n t a t i v e volume f o r S b - 2 0 % B i . The number on t h e c u r v e denotes each element s o l i d i f i e d 67 28. F e e d i n g r e l a t i o n s h i p determined e x p e r i m e n t a l l y i n t h e s t e e l c a s t i n g s ; (a) P l a t e s , and (b) Square b a r s ..95 3 0  3 1  29. Comparison o f measured p e r m e a b i l i t i e s vs volume f r a c t i o n liquid ..96 30. C o n f i g u r a t i o n  of t h e system i n v e s t i g a t e d  97  31. Flow c h a r t of the computer program f o r the p r e d i c t i o n of c e n t e r l i n e shrinkage 98 32. Temperature d i s t r i b u t i o n a l o n g t h e c e n t e r l i n e of t h e plate casting  99  33. S o l i d u s movement a l o n g t h e c e n t e r l i n e of p l a t e c a s t i n g 100  ix  34. The d i s t r i b u t i o n of s o l i d i f i c a t i o n contour l i n e s a t the ( a ) i n i t i a l , (b) m i d d l e and (c) l a s t stages of solidification 101 35. D i s t r i b u t i o n of t h e p r e s s u r e at the end of s o l i d i f i c a t i o n  r e q u i r e d t o feed (s=5cm)  shrinkage 102  36. D i s t r i b u t i o n of t h e p r e s s u r e a t the end of s o l i d i f i c a t i o n  r e q u i r e d t o feed (s=2.5cm)  shrinkage 103  37. D i s t r i b u t i o n of t h e p r e s s u r e r e q u i r e d t o feed a t the end of s o l i d i f i c a t i o n (s=0.5cm)  shrinkage 104  38. Soundness of s t e e l p l a t e c a s t i n g s  105  39. D i s t r i b u t i o n of t h e p r e s s u r e r e q u i r e d t o f e e d s h r i n k a g e at the end of s o l i d i f i c a t i o n -end c h i l l e d c a s t i n g ...106 40. Comparison of the i n v e r s e s e g r e g a t i o n a t t h e c h i l l f o r the A l - C u a l l o y s  face 126  41. Comparison of the i n v e r s e s e g r e g a t i o n a t t h e c h i l l f o r the A l - Z n a l l o y s  face 127  42. Comparison of t h e i n v e r s e s e g r e g a t i o n a t t h e c h i l l f o r the Sb-Bi a l l o y s  face 128  43. D i f f e r e n t  types of nodes i n t h e model i n v e s t i g a t e d ..129  List  of  Nomenclature  area,(cm ) 2  eutectic  contraction  Biot  number,*  mean  composition  amount  of  of volume  element,(wt%)  segregation,(wt%)  segregation of  coefficient,"  element  formed  during  the s o l i d i f i c a t i o n  (i,j),(wt%)  segregation  in j-th  compositions  column,(wt%)  of l i q u i d and s o l i d ,  respectively,(wt%) composition sum in  of l i q u i d at eutectic,(wt%)  of the composition j - t h column  total  when  dilution  dilution  effects  i - t h element  of composition  solidifies,(wt%)  in  j-th  column,(wt%) initial  composition,(wt%)  specific  heat,(cal/g°C)  local  average  composition  mean c o m p o s i t i o n length primary grey  of p l a t e  weight  shape  number, fraction  temperature  solid  at eutectic,(wt%)  casting,(cm)  dendrite  body  Fourier  of cored  of solid,(wt%)  arm  spacing,(cm)  factor,* +  solid  and  liquid,respectively,  gradient,(°C/cm)  +  xi  volume  fraction  of  eutectic,*  volume  fraction  of  solid  and  liquid,  respectively,* acceleration latent  heat  due  2  of s o l i d i f i c a t i o n , ( c a l / g )  heat  transfer  heat  transfer  metal  to gravity,981(cm/sec )  coefficient,(cal/cm sec°C) 2  coefficient  to ambient  from  sand mold  and  a i r , respectively,(cal/cm sec°C) 2  permeability,(cm ) 2  thermal  conductivity,(cal/cm.sec°C)  thermal  conductivity  of  air,(cal/cm.sec°C)  thermal  conductivity  of  sand  mold  and  metal,  respectively,(cal/cm.sec°C) equilibrium  partition  ratio,*  length  of  solid/liquid  zone,(cm)  length  of  solid/liquid  zone  length  of  fluid  at  feeding distance,(cm)  solid  liquid  representative m  g  and  state,(cm)  flow,(cm)  capillary and  steady  masses,  respectively,  in a  volume,(g)  ,respectively,  at  eutectic  temperature,(g) radius  of  capillary,(cm)  pressure,(dyn/cm ) 2  ambient  pressure,(dyn/cm ) 2  pressure  d r o p due  pressure  acting  to  due  fluid  flow,(dyn/cm )  to c a p i l l a r y  2  flow,(dyn/cm ) 2  1  X I  gradient  of  solid,(1/cm)  fraction  heat f l u x , ( c a l / c m s e c ) 2  rate,(°C/sec)  cooling  thickness bar  of  plate  casting  or  size  of  square  casting,(cm)  temperature,(°C) temperature interface  after  At,(°C)  temperature  of  ingot  and  mold,  respectively,(°C) temperature,(°C)  ambient  temperature  i n metal  and  sand  mold,  respectively,(°C) initial  temperature,(°C)  pouring  temperature,(°C)  time,(sec) representative  volume  or  fluid  volume  to  flow,(cm ) 3  volume-to-area  ratio  of  casting,(cm)  fluid  flow  velocity  in a  local  flow  velocity  of  capillary,(cm/sec)  interdendritic  liquid,(cm/sec) superficial  flow  velocity  of  interdendritic  fluid,(cm/sec) velocity  of  solidus,movement  centerline  of  local  velocity  liquid  flow  a  plate of  along  the  casting,(cm/sec) interdendritic  in x-direction,(cm/sec)  XI 1 1  element  size,(cm)  metallostatic thermal  head,(cm)  diffusivity,(cm /sec) 2  solidification thickness  of  emissivity,  shrinkage  air  ratio,*  gap,(cm)  +  viscosity,(poise) specific  volume  of  liquid  and  solid,  respectively,(cm /g) 3  density,(g/cm ) 3  local  mean  density  initial  density  density  of  solid  1 .37x10-  1 2  tortuosity dimensionless  of  liquid  density  of  of  solid  and  liquid,(g/cm ) 3  liquid,(g/cm ) 3  and  solid,  eutectic  (cal/cm sec°K 2  factor,*  f t  )  respectively,(g/cm  composition,(g/cm ) 3  3  xiv  Acknowledgement  I w i s h t o express my and  Dr.  Indira  guidance and  sincere gratitude to  V.  Samarasekera  Prof.  Fred  Weinberg  for their invaluable advice,  immense encouragement t h r o u g h o u t the course of  this  investigation. Thanks are a l s o extended t o my  f e l l o w graduate s t u d e n t s f o r  t h e i r v o l u n t a r y a s s i s t a n c e and c o - o p e r a t i o n . The faculty  assistance  members i n the Department of M e t a l l u r g i c a l  of  Engineering,  U n i v e r s i t y of B r i t i s h Columbia i s g r e a t l y a p p r e c i a t e d . I am g r a t e f u l  to  RIKEN  Corporation,  p r o v i d i n g me w i t h t h i s o p p o r t u n i t y and  financial  I w i l l not be d o i n g j u s t i c e t o myself hard  time  Tokyo,  Japan,  for  support.  without  acknowledging  and s a c r i f i c e s which my w i f e , Yuko, and my  Momoko, had t o s t a n d d u r i n g the c o u r s e of the p r e s e n t  daughter, study.  1  PART-A  INVERSE SEGREGATION IN BINARY ALLOYS  2  A-I.  INTRODUCTION AND  LITERATURE REVIEW  1 .1 I n t r o d u c t i o n  During  the  l o n g i n d u s t r i a l h i s t o r y of foundry  technology,  the c a s t i n g p r o c e s s has been c o n s i d e r e d from an e m p i r i c a l of  view  rather  than t h e o r e t i c a l l y .  the c o m p l e x i t y of the basic  casting  investigations  solidification occurs  under  of  of  multiphase  unsteady  state  forms  from  which  process.  systems  is  has For  discouraged example,  heterogeneous,  liquid  binary of  alloys,  solid  structure  and  properties.  Recently  m o d e l l i n g of the s o l i d i f i c a t i o n p r o c e s s has enabled t o be made which g i v e the temperature and  liquid  during  cheaply.  The  l a b o r a t o r y experiments  shrinkage.  major  one This  alloys  with  mathematical calculations solid  solidification.  done  reasonably,  With quickly  r e s u l t s of the c a l c u l a t i o n have been shown t o  agree w e l l w i t h the c o r r e s p o n d i n g  The  of  d i s t r i b u t i o n i n the  non-equilibrium  computers, t h e s e c a l c u l a t i o n s can be and  and  another c o m p o s i t i o n .  o c c u r r e n c e makes i t i m p o s s i b l e t o c a s t homogeneous uniform  the  and n o n - e q u i l i b r i u m c o n d i t i o n s .  D u r i n g s o l i d i f i c a t i o n of s i m p l e composition  T h i s can be a t t r i b u t e d t o  process the  point  temperature  and i n i n d u s t r i a l in  castings  in  situations. are m a c r o s e g r e g a t i o n  and  Both have a v e r y s t r o n g i n f l u e n c e on the q u a l i t y  of  the c a s t i n g s .  defects  measurements  I t i s g e n e r a l l y recognized that these d e f e c t s are  3  largely  related  particularly and  thermal The  to  fluid  interdendritic  occurence  equilibrium  flow r e s u l t i n g  phase  is  opposite  diagram  segregation  interdendritic thermal  liquid  is  to  during  Inverse  by a number of r e s e a r c h e r s .  and  solidification  due  solidification-,  to  long  that  this rich  solidification  Segregation  derive  for  at  face d u r i n g s o l i d i f i c a t i o n .  precise  analytical  the amount of i n v e r s e s e g r e g a t i o n which  system, the s e g r e g a t i o n  = c" - Co  the  solidification.  expressions  eutectic alloy  has  the back flow of s o l u t e  the v o i d c r e a t e d by  person  by  s e g r e g a t i o n has been examined  to  c o n t r a c t i o n s during  the c h i l l  predicted  I t i s g e n e r a l l y accepted  to f i l l  S c h e i l was the f i r s t  i n which the s o l u t e  that  1.2 Previous Models f o r C h i l l Face Inverse  AC  solidification,  from  of i n v e r s e s e g r e g a t i o n ,  interested metallurgists.  of  during  shrinkage.  gradient d i s t r i b u t i o n  type  flow  For a  typical  occurs binary  (AC) i s given by: (A-l)  with  "Wis - — — c --  Kirkaldy  +  m s E  c  S E  E  and  Youdelis  2  extended the S c h e i l equation (A-2)  4  to  determine  the solute  face  a n d i n t h e body  the  chill  face  representative volumes  On  =  V  a  volume,  S S m  V  +  profile  as a  V,  i s  both  function  unidirectionally  and s o l i d  the assumption  constant,  of the metal  for  of l i q u i d V  concentration  related  at the  of distance  solidified  to  chill  mass  ingot.  and  A  specific  as:  j A  (A-3)  that  equation  from  no c o n t r a c t i o n  (A-3)  void  forms,  and that  V  i s  reduces t o ,  = - a dm„  dnL. X  (A-4)  s  where a = l s V  The  solute  volume  ^  +  L  V  L  mass  i s given  d m  (A-5)  S  balance  f o r incremental  d(m C ) = - C d where change  L  s  the change of  into i s  solid/liquid  phase  change  neglected gives  d m  L  L  in  model.  m  s  T  _  V  Q  d  T  solid  plus  contraction.  and c o n t r a c t i o n Note  Combining  that  x ( A  in the liquid  o f two t e r m s :  differential  v  ( - ^ j -  the by  up  change,  d  mass  of the l i q u i d .  in this  the basic  mass  made  V  + | c  the volume  contribution  volume  m s  of s o l u t e  solute  transported  i n the  by: f  L  solidification  i s equated  6 )  the  the  solute  The  contraction  contraction due t o thermal  equations  to  _  due  the  mass  to the  specific  contraction  (A-4)  to  is  (A-6)  equation;  dc L  ^""X  (A-7)  5  where  r  A.I.-SL L  The  L  integration  (A-8)  of  (A-4)  s u b s t i t u t e d i n t o equation change  (A-7)  equation  gives  a  of t h e mass i n a c o r e d c r y s t a l  remaining  gives  m  value  which  when  for m.  The  L  s  i s r e l a t e d to that i n the  l i q u i d as: d ( m  S S C  )  =  C  S S ( A - 4 ) and ( A - 7 ) w i t h e q u a t i o n  Combining e q u a t i o n s  d ( m  Accordingly, integration  S S C  )  (A-9) yields:  ^ V ^ L  =  the of  (A-9)  d m  (A-10)  substitution  equation  of  ( A - 1 0 ) ,  m C ,  calculated  together with m  T  by  and m  IJ  equation chill  (A-2) w i l l  into S  g i v e t h e maximum i n v e r s e s e g r e g a t i o n a t t h e  face. For  t h e Al-Cu  system,  s p e c i f i c volumes of conditions  the  t h e temperature  solid  and  liquid  were c o m p i l e d by S a u e r w a l d .  dependence of t h e under  equilibrium  These v a l u e s a r e shown  3  i n F i g 1 a l o n g w i t h c o n s t i t u t i o n a l phase diagram of t h e Using  the  this  the c h i l l calculated  data, S c h e i l  face f o r a values  c a l c u l a t e d t h e maximum s e g r e g a t i o n a t  1  unidirectionally  were found  solidified  over t h e e n t i r e c o m p o s i t i o n above  theory  a l l o y s by Y o u d e l i s e t  ingot.  The  t o be i n e x c e l l e n t agreement w i t h  measured v a l u e s of t h e s o l u t e c o n c e n t r a t i o n a t  The  alloy.  the c h i l l  face  range examined.  was a l s o a p p l i e d t o t h e A l - Z n and Sb-Bi al"  j 5  using  r e p o r t e d by P r e z e l and S c h n e i d e r . 6  the  specific  volume  Again the t h e o r e t i c a l  data values  6  of  the  chill  face  concentration  e x p e r i m e n t a l measurements. the  experimental  data  agreed  In the A l - Z n a l l o y  well the  with scatter  l e s s d e f i n e d than i n  particular  i n comparing t h e o r y and experiment  r e s u l t s f o r the Sb-Bi a l l o y . accompanied  by  volume  inverse  from expansion  other  In t h i s system  expansion  and c o n t r a c t i o n a t low B i v a l u e s . negative  segregation,  systems.  the  solidification  is  a t the h i g h B i c o n c e n t r a t i o n s The change from  corresponding  positive  t o the  inverse  to  transition  t o c o n t r a c t i o n , o c c u r s at a p p r o x i m a t e l y  of  Of  are  In the S c h e i l and Y o u d e l i s development of the formulation  of  i s l a r g e , making the c o m p a r i s o n between  t h e o r y and experiment interest  the  30%Bi.  mathematical  s e g r e g a t i o n , the f o l l o w i n g assumptions  were made. 1)  The  s p e c i f i c volume and  composition  of  the  primary  s o l i d i f i e d metal are c o n s t a n t . 2)  The  thermal  contraction  during  solidification  is  negligible. These assumptions a r e i n v a l i d or o n l y the  f o l l o w i n g reasons.  approximately  valid  The model i s o n l y a p p l i c a b l e t o systems  w i t h n e g l i g i b l e s o l i d s o l u b i l i t y t o comply w i t h  assumption  and  to  for  alloys  assumption ( 2 ) .  with  for  short  freezing  ranges  Note t h a t , w i t h complete c o r i n g i n  comply w i t h the  solid,  the f i n a l s o l i d i f i c a t i o n o c c u r s a t the e u t e c t i c t e m p e r a t u r e , a t the s o l i d u s temperature.  (1)  not  7  1.3  Inverse Segregation  away from t h e C h i l l Face  (positional  Segregation)  Assuming  linear  mass  distribution  solid/liquid  mushy zone,  composition  i n the d i r e c t i o n  Youdelis  and  a  of  specific  liquid  gradient  of s o l i d i f i c a t i o n ,  i n the  of  liquid  K i r k a l d y and  have extended S c h e i l ' s a n a l y s i s t o the c a l c u l a t i o n  2  of  the p o s i t i o n a l v a r i a t i o n of i n v e r s e s e g r e g a t i o n . At liquid Away  the c h i l l  face, during s o l i d i f i c a t i o n ,  f l o w s i n t o t h e e l e m e n t a l volume w i t h from  the c h i l l  ho  face i n t e r d e n d r i t i c l i q u i d  and out of t h e e l e m e n t a l volume t o t a k e up account  to liquid  d(  where a C the inward  15%Cu  V  t  )  f l o w s both  volume  change.  into To  ( A - 6 ) , must be r e w r i t t e n by adding a term  - - C d. s  s  +  j C d^ ( )  _ i  -  L  1- ^  a  s  ^  A'L i s t h e mean l i q u i d c o n c e n t r a t i o n g r a d i e n t and S i s flow d i s t a n c e past p o i n t Equation  the p o s i t i o n a l  L  due  to  segregation  p o s i t i o n a l segregation  With gave  this  (A-11)  p r o f i l e s f o r A 1 - 1 0 % C U and A l -  of t h e maximum s e g r e g a t i o n a t t h e c h i l l section.  contraction i n  (A-6) was r e p l a c e d w i t h e q u a t i o n  were c a l c u l a t e d i n t h e s i m i l a r manner t o  previous  flow.  flow out a t t h e p o s i t i o n L:  inner regions. and  outward  f o r t h e outward f l u i d f l o w t h e e q u a t i o n f o r t h e s o l u t e  mass b a l a n c e , e q u a t i o n due  interdendritic  calculation  f a c e as d e s c r i b e d i n t h e  theory,  results  the  c a l c u l a t i o n s of t h e  which  agreed  well  with  8  experimentally  measured v a l u e s of t h e s e g r e g a t i o n as a f u n c t i o n ,  of p o s i t i o n from t h e c h i l l Prabhakar and W e i n b e r g at  by  Kirkaldy  and  t h e i r measurements. careful tracer  inverse  at  7  questioned  Youdelis  2  segregation and  the  that  face  copper c h i l l . thermal  the  drops  results  some  radioactive  maximum  inverse  of t h e melt when c a s t .  than  copper,  With  the concentration  has  at  for a  a  lower  the inverse segregation i s  when compared t o t h e copper  but a l s o drops w i t h i n c r e a s e d superheat.  the c h i l l  out  r a p i d l y w i t h i n c r e a s i n g superheat  higher a t higher superheats  of  using  W i t h a s t a i n l e s s s t e e l c h i l l which  conductivity  indication  carried  face i s h i g h l y dependent of c a s t i n g  temperature g r e a t e r than 40°C,  chill  7  measurements  showed  the c h i l l  arrived  on t h e b a s i s o f t h e a c c u r a c y of  c o n d i t i o n s , i n c l u d i n g t h e superheat superheat  the conclusions  Prabhakar and W e i n b e r g  techniques  segregation  face.  chill,  Moreover, t h e r e was no  i n v e r s e s e g r e g a t i o n i n t h e r e g i o n w e l l away from  face as r e p o r t e d by indicate  Kirkadly  and  Youdelis .  These  2  t h a t v a r i o u s c a s t i n g p a r a m e t e r s , not i n c l u d e d  i n t h e t h e o r e t i c a l model, can s t r o n g l y i n f l u e n c e t h e  extent  of  i n t e r d e n d r i t i c f l u i d f l o w d u r i n g s o l i d i f i c a t i o n and a c c o r d i n g l y , the i n v e r s e s e g r e g a t i o n . Based  on  the concept  that  a l l solute  macrosegregation  r e s u l t s from i n t e r d e n d r i t i c f l u i d f l o w of s o l u t e r i c h l i q u i d feed  solidification  shrinkage  S c h e i l - Y o u d e l i s model was account  and  extended  thermal by  to  c o n t r a c t i o n , the  Flemings  f o r a wide range o f s e g r e g a t i o n e f f e c t s .  et a l ' 8  1 1  to  This included  9  centerline  segretation,  segregation,  banding,  and t h e n e g a t i v e  inverse segregation.  under-riser  positive  cone of s e g r e g a t i o n as w e l l a s  C o n s e r v a t i o n o f s o l u t e mass i n t h e  volume  element d u r i n g s o l i d i f i c a t i o n r e q u i r e s t h a t : £CpC) = - n^ C v) L  (A-12)  L  C o n s e r v a t i o n of t o t a l mass i n t h e volume element g i v e s ,  Combining e q u a t i o n s The  change  (A-12) and (A-13),  i n s o l u t e mass o f t h e volume element i s t h e sum o f  the changes i n t h e l i q u i d and s o l i d phases; £(PC)  =^(C P g s  S i m i l a r l y , t h e change TT  TT  =  (  % s 8  s  s  +  C  L P L  (A-15)  g ) L  i n t o t a l mass of t h e volume element i s , PL L  +  8  (A-16)  )  Assuming l o c a l e q u i l i b r i u m a t  the s o l i d / l i q u i d  interface,  no  d i f f u s i o n i n t h e s o l i d , and a c o n s t a n t s o l i d d e n s i t y , g i v e s :  at V s ^ = o L s T T (  k  p  C  (A-17)  and dg  = - dg .  (A-18)  L  S u b s t i t u t i n g e q u a t i o n s (A-17) and (A-18) i n e q u a t i o n (A-15), and combining i t w i t h e q u a t i o n s (A-16) and (A-14) g i v e s t h e f i n a l partial differential  equation;  l^k = _ / lzg_i , vVT . L 8  3C  n  L  T  \l-k ) \  o  1  +  C~ L  For one d i m e n s i o n a l problem, t h a t i s u n i d i r e c t i o n a l  (A-19)  10  solidification, e  9  3  C  equation  L  1  L  i s rewritten to: g  V G  e  X  L  (A-20)  =  Macrosegregation  -  -  (A-19)  i s defined  S M S  C m  for a  +  p  eutectic  system as,  SE% LE C  P (l-g ) + s  binary  p  E  g E  g  •  E  (A-21)  AC = C - C  o which  i s analogous to S c h e i l ' s  (A-2).  Macrosegregation  21)  equation  and  Two  (A-19)  limiting  state,  integrating  calculated  with  and  equation  (A-  (A-20).  cases  can  be  easily  studied.  For  steady  (A-22)  into  equation  (A-20)  and  yields, l-k  L  Q  L (c;)  °  limiting  case  8  other  CA-22)  equation  C =  (A-23) can  be  defined  at c h i l l  face  where,  similarly, C . L  8  L  Assuming  and  be  (A-1)  ~ T-p"  Substituting  and  or  now  equation  _ R  The  can  expression,  e Q  IC "/  (A-24)  7  o  a  taking  Flemings  =  1-  l-k  constant  the value  el a l AC  =  8  partition  for  calculated  0.43%  ratio,  solidification t h e amount  at c h i l l  face  of  k =0.172,  o  f o r Al-4.5%Cu,  shrinkage segregation  as as:  /3=0.055  11  AC = 0 a t steady The  state  c a l c u l a t e d i n v e r s e s e g r e g a t i o n a t the c h i l l  the  experimental  v a l u e s r e p o r t e d by S c h e i l  s e g r e g a t i o n a t steady  1  s t a t e w i t h the r e s u l t s  face agrees w i t h  and  the c a l c u l a t e d  of  Prabhakar  and  Weinberg . 7  Note t h a t , i f the s o l i d i f i c a t i o n s h r i n k a g e  = —k  3 P  as  S 8  alloys  negative  of  containing  more  than  9%Cu.  When  0 i s adopted t o above t h e o r y , the face i s a l s o n e g a t i v e , which  does  w i t h p o s i t i v e i n v e r s e s e g r e g a t i o n r e p o r t e d by S c h e i l 2  always  8  not 1  Present  In  Objectives  the  this  segregation  models  thus  far  i s a c r i t i c a l variable.  reviewed,the It i s believed  i s not c l e a r l y d e f i n e d i n the models, and the  values  adopted t o c a l c u l a t e the i n v e r s e s e g r e g a t i o n are i n c o r r e c t . addition  other  satisfactory. inverse  /3  system.  s o l i d i f i c a t i o n shrinkage that  9  and  be p o s i t i v e , t h e r e f o r e , the d e f i n i t i o n of /3 i s not  v a l i d f o r the Al-Cu  1.4  a  resulting  K i r k a l d y et a l . To v a l i d a t e the model of F l e m i n g s e t a l ' , must  25)  L  s e g r e g a t i o n a t the c h i l l agree  _  (A  d i d , the v a l u e s of 0 o b t a i n e d from F i g 1 a r e  n e g a t i v e i n the value  by:  = V  Flemings et a l  i s defined  assumptions made i n the t h e o r y a r e not  In  entirely  In order t o c l a r i f y t h e s e problems and t o p r e d i c t  segregation  in  a  unidirectionally  solidified  ingot  12  p r e c i s e l y , a new model, w i t h minimum a s s u m p t i o n s , was  developed.  13  A-II.  2.1  Mathematical  In  a  MODELING PROCEDURE  Formulation  unidirectionally  solidified  ingot three  dimensional  c r y s t a l growth o c c u r s s i n c e a d e n d r i t e grows t h r e e d i m e n s i o n a l l y because of m i c r o s e g r e g a t i o n between d e n d r i t e s and t h e d i f f e r e n c e of t h e r m a l d i f f u s i v i t i e s complex  problem,  pyramid  shape  branches.  The  mathematical  i n s o l i d and l i q u i d .  To s i m p l i f y  this  i t i s assumed t h a t t h e p r i m a r y d e n d r i t e has a  without  secondary  following  or  higher  assumptions  order  are also  dendrite  made  i n the  formulation;  1)  No s u r f a c e e x u d a t i o n o c c u r s a t t h e c h i l l  2)  There  is  no  diffusion  in  the  face. solid  during  solidification. 3)  Residual  interdendritic  liquid  is  homogeneous i n t h e p l a n e p e r p e n d i c u l a r t o  completely the  growth  direction. 4)  Local e q u i l i b r i u m c o n d i t i o n s e x i s t a t the s o l i d / l i q u i d interface.  5)  The  dendrite  shape  does  not  change  during  solidification. 6)  The m i c r o s e g r e g a t i o n fluid  7)  i s not a f f e c t e d by i n t e r d e n d r i t i c  flow.  No s h r i n k a g e p o r o s i t y o c c u r s  during  solidification.  14  This  r e q u i r e s low gas l e v e l s and u n r e s t r i c t e d f l o w of  l i q u i d through the d e n d r i t e channels. Under these c o n d i t i o n s , t h e f i r s t form i s of c o m p o s i t i o n \ C }  subsequent  s m a l l amount of s o l i d  a t the l i q u i d u s temperature.  Q  solidification,  solute  During  i s enriched i n the r e s i d u a l  l i q u i d which l e a d s t o h i g h e r s o l u t e c o n c e n t r a t i o n s i n t h e as  solidification  progresses.  solid  Since d i f f u s i o n i n the s o l i d i s  assumed n e g l i g i b l e , t h e s o l u t e d i s t r i b u t i o n i n t h e s o l i d not change a f t e r s o l i d i f i c a t i o n A  quantitative  solute  e q u a t i n g t h e amount of s o l u t e  mass  balance  rejected  can be e x p r e s s e d by  from  the  solid/liquid  This balance i s  (C - C ) d f _ = f d C S L L T  since C  o  (A-26a)  and f = 1-f :  L  L  dC C  T  s  = k C  s  s  df„ V  u s  Integrating  l - f  equation  (A-26b)  s  (A-26b) from C  g  = k C Q  s o l i d c o m p o s i t i o n as a f u n c t i o n of f r a c t i o n C  or  S " o k  C  o  (  1  -  f  S  )  V  1  o  Equations called  L  =  C  o L ° f  k  at f =0 y i e l d s the g  solid  ,  CA-27a)  i n terms of l i q u i d c o m p o s i t i o n and f r a c t i o n C  does  occurs.  i n t e r f a c e t o the s o l u t e increase i n the l i q u i d . L  to  liquid, (A-27b)  1  (A-27) have been d e r i v e d by S c h e i l  the S c h e i l  equation  or  e q u a t i o n has been used t o p r e d i c t  Pfann  1 2  and P f a n n , 1 3  are  equation.  The S c h e i l  microsegregation  f o r normal  15  conditions  of c a s t i n g and  by Brody et a l * and  ingot s o l i d i f i c a t i o n  Bower  1  et a l  1  5  .  They  i n binary a l l o y s reported  c o r r e l a t i o n between the e q u a t i o n and e x p e r i m e n t a l At  the  region  be  projecting  represented  into  the  schematically  which  give  a l o n g the c h i l l reaches  L,  the c h i l l adjacent  the  solid  i s governed  fraction solid  pyramids  F o l l o w i n g the by  and l i q u i d c o m p o s i t i o n  f a c e as shown i n F i g 2b.  the  solid/liquid  solid  l i q u i d as shown i n F i g 2a.  above d i s c u s s i o n , the m i c r o s e g r e g a t i o n (A-27)  by  good  results.  i n i t i a l s t a g e s of s o l i d i f i c a t i o n , the  can  a  equation profiles  When the d e n d r i t e  tip  i n the volume element away from  face ( f ) can be r e l a t e d t o t h a t i n the volume element s  t o the c h i l l •P »  —  S "  f  l*~  f a c e (f ) by, s  c  x  ~L~  f  S  and the c o m p o s i t i o n  CA-28)  profile  i s g i v e n by the S c h e i l k  C  T  = k C (i-ir* S o o \ L  q  f)  °  SJ  equation,  -1 (A-29a)  and k T  L-x  L  Fig  2c  shows  composition (A-29b).  the in  which  1  distribution  the  of  interdendritic  liquid  growing d i r e c t i o n c a l c u l a t e d from e q u a t i o n  I t can be seen t h a t the f l o w of s o l u t e e n r i c h e d  t o feed s h r i n k a g e The  -  r e s u l t s i n inverse segregation.  next s t e p i s t o e s t a b l i s h the  is  related  liquid  to  the  amount  d e s c r i b e d i n the p r e v i o u s p a p e r s ' , 5  9  of  amount  of  fluid  flow,  volume c o n t r a c t i o n .  the volume change  As  occuring  16  i n the p r o c e s s of s o l i d i f i c a t i o n  i s made up of t h r e e mechanisms.  1)  The change due t o the l i q u i d / s o l i d  2)  The  change  phase change.  due t o the d e n s i t y change i n the r e s i d u a l  l i q u i d a s s o c i a t e d w i t h c o m p o s i t i o n changes. 3)  Thermal  contraction  associated  with  temperature  changes. The  first  two  volume  changes  can  be  calculated  equations for microsegregation, equations  (A-27)  from  and  the  (A-29).  The t h e r m a l c o n t r a c t i o n i s d e t e r m i n e d by the t e m p e r a t u r e d r o p .  2.1.1  Alloy Densities  The  calculation  solidification  and  of the t o t a l temperature  amount of c o n t r a c t i o n due t o changes  requires  accurate  i n f o r m a t i o n of the d e n s i t i e s of the l i q u i d and s o l i d a l l o y s as a f u n c t i o n of both c o m p o s i t i o n and t e m p e r a t u r e . by  Sauerwald  The d a t a r e p o r t e d  (see F i g 1) and P r e z e l et a l a r e not a p p l i c a b l e  3  6  i n the p r e s e n t a n a l y s i s s i n c e the  densities  they  gave  assume  t h a t t h e a l l o y c o m p o s i t i o n a t a g i v e n temperature a r e d e f i n e d by the the  equilibrium  phase diagram, which i s not the c a s e .  reason t h a t the previo'us m o d e l s ' ' 1  2  8  were f o r c e d t o  This i s neglect  thermal contraction during s o l i d i f i c a t i o n . The  only  reliable  data  which can be used i n the p r e s e n t  case a r e the p u b l i s h e d v a l u e s f o r the temperature dependence the  density  for  pure  metals.  Fig 3  shows  of  the temperature  dependence of the d e n s i t i e s of copper and aluminium c o m p i l e d  by  17  Smithells zinc,  and E l l i o t e t a l  1 6  antimony  1 7  .  The c o r r e s p o n d i n g d e n s i t i e s f o r  and bismuth a r e shown i n F i g 4.  Provided that a  b i n a r y e x i s t s as a s i m p l e m i x t u r e of each component, t h e d e n s i t y of t h e a l l o y can components  be  determined  from  the percentage  of t h e  i n t h e a l l o y and t h e components d e n s i t i e s a t a g i v e n  temperature.  C a l c u l a t e d v a l u e s of a l l o y d e n s i t i e s f o r the A l - C u  system u s i n g  t h e component  measured Smithels  directly 1 6  or  densites  compiled  a r e compared  by  Bornemann  i n T a b l e I . F o r both s o l i d s  agreement  is  obtained  between  d e t e r m i n e d from t h e c o n s t i t u e n t s  and  and  to  et a l  those 1  8  and  liquids  excellent  the c a l c u l a t e d  densities  the d i r e c t  measurements  w i t h d i f f e r e n c e s of l e s s than 5% a t most. Similar  comparisons  were  made  f o r t h e A l - Z n and Pb-Sn  systems u s i n g t h e data of Bornemann e t a l (Table I ) .  Very  good  2.1.2  Thresh  et a l  2 0  and  the Pb-Sn  systems,  Face  the representative  face ( F i g 2a).  The s o l i d / l i q u i d  solidification  advances  microsegregation calculation,  and  These e r r o r s a r e n e g l i g i b l e i n the p r e s e n t c a s e .  S h r i n k a g e Along The C h i l l  Consider  9  agreement i s a g a i n e v i d e n t w i t h e r r o r s ,  l e s s than 1.6% and 1.2% i n the A l - Z n respectively.  1  t h e i - t h element  interface  i s plane  and t h e  because  of t h e  unidirectionally  discussed  t h e volume  volume a d j a c e n t t o t h e c h i l l  previously.  In  the  numerical  was s u b d i v i d e d i n t o n e l e m e n t s .  i ssolidified,  composition  of  the  When  remaining  18  liquid i s : i C  L±  =  C  o  ( 1  ~ n  k  o  -  1  °  }  (A-30a)  and t h e average s o l i d c o m p o s i t i o n of t h e i - t h element i s : 5  until  C  S i - J L l V * • -=.{(1- ^ ) °  - (1- i)k°j  k  reaches t h e e u t e c t i c c o m p o s i t i o n .  L  i temperature  < -30 > A  b  The c o r r e s p o n d i n g  of t h e r e p r e s e n t a t i v e volume (T^ ) can  be  obtained " -  i  from  t h e phase  composition C  diagram  .  T  as  t h e l i q u i d u s temperature  Once t h e e u t e c t i c c o m p o s i t i o n i s reached,  s o l u t e r e j e c t i o n occurs a t the s o l i d / l i q u i d discussion  f o r the  interface.  no  From t h e  i n t h e p r e v i o u s s e c t i o n , t h e d e n s i t y of each element  i s c a l c u l a t e d f o r t h e average s o l i d c o m p o s i t i o n  (C  ) or  liquid  i  composition  (C  )  a t temperature  T.  T  .  To c a l c u l a t e t h e volume  L.  change i n each element, a c o n s t a n t mass w i t h i n t h e element assumed. Then t h e s h r i n k a g e r a t i o i n each element i s : l~ o = — - P  SH.  is  p  (  A  _  3  1  )  and t h e . t o t a l s h r i n k a g e of the whole volume i s : n e  The  i  = Z 1=1  shrinkage  S  H  i  (A-32)  formed  during  the s o l i d i f i c a t i o n  of the i - t h  element i s :  Ae  ±  = e. - e _ i  1  (A-33)  19 2.1.3  S e g r e g a t i o n i n the F i r s t Column A d j a c e n t To The C h i l l Face  C o n s i d e r the system subdivision  in  assumption element the  the  x  shown and  y  (3) d i c t a t e s t h a t  by  one  element  in  If  directions  the  in  F i g 5.  the  number  of  are e q u a l then  the  solidification  both  directions.  The  d e n d r i t e i n the x d i r e c t i o n r e s u l t s from the  heat  transfer  and  the  growth i n  y  m i c r o s e g r e g a t i o n e x p r e s s e d by e q u a t i o n The volume  column  growth of  from  the  (A-27). t o the  d i s c u s s e d i n the p r e v i o u s s e c t i o n .  The  second column.  one  unidirectional  direction  f i r s t column i n F i g 5 coresponds  i n the f i r s t column i s c o m p l e t e l y  proceeds  fed  by  the  representative s h r i n k a g e formed liquid  in  the  D u r i n g the s o l i d i f i c a t i o n of the i - t h element i n  1,  the  liquid  C  to C  .  in  column 2,  composition  i n column 2 i n c r e a s e s from  T a k i n g the average f o r the  the s e g r e g a t i o n caused  liquid  composition  by the f e e d i n g f l o w d u r i n g  this period i s : (A-34) and the f i n a l s e g r e g a t i o n a f t e r complete  solidification  of  the  f i r s t column i s : AC  1 "  n  2  i=l  A C  L  i,l  (A-35)  20  2.1.4  Positional  Based  Segregation  on  solidification residual  the and  thermal  liquid,  continuously during occurs  at  shrinkage  the  solidification.  The  face  since  The  system  in.  in  line  solid/liquid for  the  the  chill  that  exudations).  fluid  face  thin  of  flow across  the  dendrite  unchanged shape  liquid  amount  in  which  direction  profile is  l o c a t i o n of  basis change  of l i q u i d  shown  in  the  homogeneous.  the not  of  schematically.  actual  on  does  corresponding  solidifying  the  the  the  columns  denotes  liquid  that  dilutes  decreasing  of  Note t h a t the r a t i o of l e n g t h t o  remains  The  into  F i g 6a  interface.  solidification. in  segregation  than  i s assumed t o be c o m p l e t e l y  in  dendrite  assumption  higher  the  volume,  subdivided  occur  t h e r e i s no l i q u i d . f l o w  is  Accordingly,  the  with  should  maximum  F i g 6a shows the f e e d i n g sequence is  dotted  volume  the  to  fed  p o s i t i o n of the i n g o t , the c o m p o s i t i o n  interdendritic liquid The  flow  due  the l i q u i d r e q u i r e d t o f e e d  i n s o l u t e and  volume element away from  segregation.  completely  f a c e (note t h a t we n e g l e c t s u r f a c e  flowing  concentration  is  shrinkage  f l o w i n g out of the e l e m e n t a l liquid  shrinkage  fluid  i s concentrated  an i n t e r i o r  that  interdendritic  chill  a c r o s s the c h i l l At  assumption  width of  the  during  composition  F i g 6b.  Fig 7  summarizes the occurance of a l l the i n t e r d e n d r i t i c f l u i d f l o w i n the model w i t h 10X10 reaches  eutectic  s u b d i v i s i o n s by the time temperature,  volume has s o l i d i f i e d .  Each  that  dotted  is line  the  position  L  when the e n t i r e model denotes  the  actual  21  location  of  solid/liquid  interface,  and  the  digit  i n each  element the number of columns whose s h r i n k a g e s cause f l u i d across  a p a r t i c u l a r column a t the c o r r e s p o n d i n g  interface. that  flow  l o c a t i o n of the  The number 5 a p p e a r i n g a t ( b , 6 ) , f o r example,  means  the s h r i n k a g e s i n f i v e columns, 1-5, induce the f l u i d  flow  a c r o s s column 6 because of the f l u i d f l o w t o f e e d s h r i n k a g e s elements  ( 9 , 1 ) , ( 8 , 2 ) , ( 7 , 3 ) , ( 6 , 4 ) and ( 5 , 5 ) .  Thus the f l u i d i n - ~  f l o w t o f e e d i t s own s h r i n k a g e i s not i n c l u d e d i n t h i s When the element ( i , j ) s o l i d i f i e s , occur  in  (j-1) f l u i d  figure.  flows  will  a c r o s s the j - t h column, which a r e induced by the f l o w s t o  feed s h r i n k a g e s i n t h e column ( j - 1 ) , ( j - 2 ) , ( j - 3 ) , - - - - , which i s d e p i c t e d i n F i g 8 (Note t h i s d e s c r i p t i o n a p p l i e s 11).  Then,  the l i q u i d c o m p o s i t i o n s  for  i n c r e a s e from C  C L  column.  to  i-2  L _!  in  i-1  the  L  i  (j+l)-th  ±  T a k i n g the a v e r a g e s f o r the c o m p o s i t i o n s , the l i q u i d t o  flow out of the column j i s (C L  into  C  ^  to C L  i n the j - t h column and from  i+j  is  (C  L  i-2  +C L  )/2 and the l i q u i d t o f l o w L  i  )/2 so t h a t the d i f f e r e n c e between them i s  i-1  g i v e n by  (C  affected  by f l u i d f l o w , the s o l i d i f i c a t i o n s h r i n k a g e s formed i n  inner  L  -C i i-3  +C i-1  columns  )/2.  L  are  the  c a l c u l a t e d by e q u a t i o n flow  induced  by  Since  same  the  as  (A-33).  microsegregation  that  in  the  first  is  not  column  The d i l u t i o n e f f e c t of the above  the s h r i n k a g e i n the ( j - 1 ) - t h column i n which  ( i + l ) - t h element s o l i d i f i e s i s : c  —c  - (-V**  -  c  >«  22  The sum of d i l u t i o n e f f e c t s  AC  L.  i+j  ..  ( Z  i-i+1  ** )(  f o r ( j - 1 ) flow occurances i s :  . V  C  \  ±  '  Lj  ^  CA-37)  - o) C  2  When t h e element (5,4) s o l i d i f i e s i n F i g 7 f o r example, t h e shrinkages fluid  i n the f i r s t ,  flow  second and  a c r o s s the column 4.  third  columns - w i l l  The c o m p o s i t i o n  the l i q u i d f l o w i n g i n t o and out of t h e column 4  cause  d i f f e r e n c e of  i s ( C -C,  )/2  T  and t h e sum of d i l u t i o n e f f e c t s i s : C  A C  L  5 > 4  = <AB + AB 6  7+  T h i s k i n d of d i l u t i o n solidifies  L ~L C  AP )(-i—-cJ 8  f l o w c o n t i n u e s u n t i l t h e whole column  completely, therefore, the t o t a l d i l u t i o n e f f e c t f o r  the j - t h column i^s g i v e n by: AC  X  i=i  L  i  a  c  L  S i n c e t h e f e e d i n g flow i n t o shrinkage  increase  (  a  column  the c o m p o s i t i o n  t o compensate of  A  _  3  8  )  i t s own  t h e column by t h e an  amount e q u a l t o t h e i n c r e a s e i n t h e f i r s t column, t h e  resulting  s e g r e g a t i o n i n t h e j - t h column i s : AC.  -  A  C l  -  AC _ L  (  A  _  3  9  )  23  2.1.5  The Length of I n v e r s e S e g r e t a t i o n Zone  On  t h e assumption  that  e x i s t at  the l i q u i d / s o l i d  dendrite  t i p corresponds  alloy. state  the l o c a l e q u i l i b r i u m c o n d i t i o n s  interface,  at a  t o t h e l i q u i d u s temperature o f the  Hence u s i n g the d e n d r i t e length  the temperature  of s o l i d / l i q u i d  t i p temperature,  t h e steady  zone, L , can be d e f i n e d a s the s  d i s t a n c e between l i q u i d u s i s o t h e r m and the c h i l l metal  a t the c h i l l  face  when t h e  f a c e reaches the e u t e c t i c t e m p e r a t u r e . The  v a l u e of L„ can be c a l c u l a t e d n u m e r i c a l l y .  The heat  transfer  S  e q u a t i o n i s g i v e n by: 3 T t  k  =  9  for  a  p  3 T . 2  (A-40)  2  C p  3  x  s e m i - i n f i n i t e mold and i n g o t .  During s o l i d i f i c a t i o n  m e t a l s h r i n k s from the copper c h i l l p r o d u c i n g an a i r lowers  gap which  the heat t r a n s f e r between the m e t a l and mould.  The  heat  t r a n s f e r c o e f f i c i e n t t h r o u g h the a i r gap can be d e t e r m i n e d the  from  following expression: k h  = -f-  (T + 273) T  + OF  T  latent  heat  4  - (T + 273)  4  - T  I  The  the  ( " ^) A  4  II  of s o l i d i f i c a t i o n  i s taken  t o be r e l e a s e d  u n i f o r m l y between the l i q u i d u s and s o l i d u s t e m p e r a t u r e s .  Values  f o r the t h e r m a l p r o p e r t i e s used i n the n u m e r i c a l c a l c u l a t i o n a r e l i s t e d i n Table I I .  24  2 . 2 Computer Programming  Based discussed  on in  the  mathematical  chart  for  the  shows  the and  face. inverse  partial  f o r heat t r a n s f e r , e q u a t i o n  differential  equation  segregation  zone,  s o l v e d i m p l i c i t l y w i t h the f i n i t e d i f f e r e n c e method.  n o d a l e q u a t i o n s f o r i n t e r n a l nodes a r e : *  T.  - T.  At  and  Fig 9  segregation  To c a l c u l a t e the l e n g t h of the  ( A - 4 0 ) , was The  computer.  information  c a l c u l a t i o n of i n v e r s e s e g r e g a t i o n a t  a d j a c e n t t o the c h i l l  the  and  the p r e v i o u s c h a p t e r , c a l c u l a t i o n s of  were performed n u m e r i c a l l y w i t h a flow  formulation  *  . p  T.  N  -  p  2T.  *  (Ax)  + T.  *  .a  2  f o r s u r f a c e h a l f node:  At each time  step,  the  Gauss-Jordan  a p p l i e d t o s o l v e the t r i d i a g o n a l m a t r i x . sequence i s shown i n F i g  10.  elimination The  method  was  actual calculation  25  A-III.  The was  a p p l i e d t o the c a l c u l a t i o n of i n v e r s e s e g r e g a t i o n  The  The  Al-Cu  system  i s a t y p i c a l eutectic a l l o y with  Volume Change d u r i n g  at  is  solidification  phase of  the  where  the  change  appears  Al-Cu a l l o y .  solid  over  The  most  contraction stages  this  phenomena  l i n e denotes the sequence of volume the  dotted  line  in  Here the e q u i l i b r i u m p a r t i t i o n r a t i o was  t o have a c o n s t a n t v a l u e ,  of  c a l c u l a t e d r e s u l t s of  F i g 11, a l s o r e v e a l  change i n each s o l i d i f i e d element and residual liquid.  a  Solidification  volume change f o r A 1 - 1 0 % C U , clearly,  giving  solidification.  seen i n F i g 1 t h a t e x p a n s i o n i n s t e a d of  liquid/solid  nearly  s o l i d u s l i n e s i n the phase diagram ( F i g 1).  l i n e s a r e assumed t o be l i n e a r i n the c a l c u l a t i o n  It  Al-Cu,  Aluminium Copper System  constant p a r t i t i o n r a t i o during  3.1.1  in  Sb-Bi systems.  l i n e a r l i q u i d u s and The  DISCUSSION  m a t h e m a t i c a l model e s t a b l i s h e d i n the p r e v i o u s c h a p t e r  A l - Z n and  3.1  RESULTS AND  k =0.171.  the taken  Each element w i t h l e s s than  e u t e c t i c c o m p o s i t i o n shows expansion d u r i n g  solidification.  The  26  volume  expansion  ratio  increases  with  an  c o n c e n t r a t i o n because the volume change i s h i g h l y the d i f f e r e n c e i n copper c o n c e n t r a t i o n On  the  other  hand,  we  eutectic solidification place  during  reached.  The  of  dependent  on  i n the l i q u i d and  because  shrinkage  no  composition  change  i n F i g 11.  Of  here  i s t h a t , even i n the s o l i d i f i c a t i o n  the r e g i o n  with  composition  the whole system c o n t i n u e s  is  of the whole r e p r e s e n t a t i v e volume"  importance  expansion  takes  a f t e r the e u t e c t i c c o m p o s i t i o n  i s a l s o p l o t t e d as a dot dashed l i n e  solidification  solid.  can observe volume c o n t r a c t i o n d u r i n g  solidification total  increase  less  than  the  o c c u r s , the t o t a l to increase  particular process  eutectic,  shrinkage  almost  of  where  ratio  for  steadily.  This  phenomena can be a t t r i b u t e d t o the i n c r e a s e i n l i q u i d d e n s i t y as a  result  of  solute  enrichment i n the r e s i d u a l l i q u i d , and  t h e r m a l c o n t r a c t i o n which i s observed i n F i g 11 increase  of  the  solidification. volume  shrinkage  occurs  t r a n s f o r m a t i o n , the sum in  g r e a t e r than the Fig each  follows of  during  the  even  slight  after  its  though  the  liquid/solid  of the c o n t r a c t i o n  due  to  the  phase solute  the l i q u i d and the t e m p e r a t u r e drop i s a l w a y s expansion.  in  Al-5%Cu  of remaining  liquid.  alloy  of  volume  solidification, The  Note t h a t the  decreasing volume  change  t o g e t h e r w i t h the volume  the l i q u i d / s o l i d phase t r a n s f o r m a t i o n  composition.  the  element  that,  12 shows the c a l c u l a t e d r e s u l t s  element  profile  each  Hence we can c o n c l u d e  expansion  concentration  in  as  to  change  with  eventually  composition contraction  i n the  increase  of  beginning of  reverses  alloy from  27  contraction  to  corresponding  expansion  between  composition which  9%Cu,  Sauerwald .  The  4  i s less  than  calculated  the  critical  The  value  of  These can be compared w i t h the change  obtained  s p e c i f i c volume d a t a of F i g 1, a l s o shown i n F i g - 1 3 .  t o t a l shrinkage  change,  The  t o t a l shrinkage r a t i o s are p l o t t e d a g a i n s t  volume change r a t i o s a t the l i q u i d / s o l i d phase the  5.  3  the a l l o y c o m p o s i t i o n i n F i g 13.  from  and  of t h e r e v e r s a l p o i n t i s found t o be  a p p r o x i m a t e l y 7%Cu o b t a i n e d by  elements  ratio,  differing  from  the  local  volume  i s always p o s i t i v e w i t h v a l u e s r a n g i n g between 5.2%  A l - 2 5 % C u t o 7.2%  f o r n e a r l y pure A l .  t o t a l s h r i n k a g e r a t i o near pure  Al  The abrupt i n c r e a s e i n the results  from  the  contraction.  N e g l e c t i n g t h i s volume c o n t r a c t i o n as was  the  investigation ' '  previous  significant  1  error  in  the  2  would  8  for  calculated  clearly  result  thermal done i n in  v a l u e s of s h r i n k a g e  a and  segregation.  3.1.2  S e g r e g a t i o n a t the C h i l l  Face  S e g r e g a t i o n p r o f i l e s of the a l l o y s can using  the  complete  be  calculated  volume changes d i s c u s s e d above.  Positional  i n v e r s e s e g r e g a t i o n p r o f i l e s r e s u l t i n g from are  now  these  calculations  g i v e n i n F i g 14 f o r A l - C u a l l o y s as f u n c t i o n s of  d i s t a n c e from the c h i l l decrease  with  face.  increasing  segregation i s obtained  at  The  i n i t i a l s l o p e s of the  alloy the  fractional  composition.  chill  face.  The The  curves maximum  calculated  28  inverse  segregation  a t the c h i l l  f u n c t i o n of c o m p o s i t i o n  f a c e of t h e A l - C u a l l o y s as a  i s shown by t h e s o l i d l i n e  in  F i g 15.  As t h e c o n c e n t r a t i o n of copper i n the a l l o y i n c r e a s e s , the c h i l l face  inverse  about  segregation  Al-l2%Cu,  composition.  then  Also  by  decreases  plotted  measurements by S c h e i l predictions  i n c r e a s e s r a p i d l y t o a peak v a l u e a t  and  1  Scheil  in  to  zero  F i g 15  Kirkaldy  at  are et a l  the the  2  and  eutectic  experimental theoretical  and F l e m i n g s e t a l . The agreement of  1  8  the model p r e d i c t i o n s w i t h the e x p e r i m e n t a l v a l u e s i s e x c e l l e n t for  copper  above  15%  concentrations the  experimental  calculated  than  results  15%.  are  At c o n c e n t r a t i o n s  greater  than  Segregation  length  of the s o l i d / l i q u i d  zone a d j a c e n t  t o the c h i l l  f a c e i s dependent on the r a t e of heat e x t r a c t i o n from the at  the  chill  face.  between  the  metal  s u r f a c e s due t o the c o n t r a c t i o n of the m e t a l . region  metal  T h i s i n t u r n can be markedly i n f l u e n c e d by  the presence of an a i r gap  solid/liquid  the  values.  3.1.3 P o s i t i o n a l  The  less  i s defined  as  the  and  the  chill  The l e n g t h of the  d i s t a n c e between the  l i q u i d u s and the c h i l l a t the moment when the metal a t t h e c h i l l f a c e reaches the length  is  eutectic  plotted  temperature.  The  zone  zone  i n F i g 16 as a f u n c t i o n of t h e w i d t h of a i r  gap f o r t h r e e melt s u p e r h e a t s , as i n d i c a t e d . solid/liquid  calculated  is  observed  The l e n g t h of  the  to increase r a p i d l y with small  -  29  i n c r e a s e s i n the a i r gap. 15mm  The l e n g t h goes from n e a r l y  zero  w i t h the f o r m a t i o n of a s m a l l gap of lO^m w i t h a  of 4 0 ° C .  The r a t e of i n c r e a s e i s h i g h e s t w i t h the  to  superheat  lowest  melt  superheat. There  are  relatively  few  reliable  measurements  of the  l e n g t h of the s o l i d / l i q u i d r e g i o n , p a r t i c u l a r l y w i t h - t h e  length  b e i n g v e r y s e n s i t i v e t o the s o l i d i f i c a t i o n c o n d i t i o n s . measurements have been r e p o r t e d by Prabhakar techniques  and W e i n b e r g  segregation.  T h e i r r e s u l t s show t h a t the l e n g t h of s o l i d / l i q u i d  is  indicated  from  i s approximately  5mm  in their  et  F i g 17. et a l  7  For and  a l a r e a l s o shown.  the  face  (positional  c a l c u l a t e d from the p r e s e n t t h e o r y f o r A 1 - 1 0 % C U i s  segregation)  Prabhakar  for A1-10%CU.  measurements.  I n v e r s e s e g r e g a t i o n a d j a c e n t t o the c h i l l  in  inverse  F i g 16 t h a t t h e r e was an a i r gap of 3MITI  between the a l l o y and the c h i l l  shown  the  using  tracer  It  measure  7  radioactive  r e g i o n w i t h 4 1 ° C superheat  to  Reliable  2  present  theory  r e s u l t s of Prabhakar  comparison the  the  theoretical  results  reported  p r e d i c t i o n of K i r k a l d y  The i n v e r s e s e g r e g a t i o n c a l c u l a t e d is  a  little  et a l .  The  by  from  lower than t h e e x p e r i m e n t a l results  by  Kirkaldy  et a l  d e v i a t e f u r t h e r from t h e e x p e r i m e n t a l r e s u l t s near t h e c h i l l and maintain zero. away  a  p o s i t i v e v a l u e a f t e r 5mm when the measured v a l u e i s  I n v e r s e s e g r e g a t i o n i s r e p o r t e d by from  the  chill  face,  Kirkaldy  et a l  d i f f e r i n g from the r e s u l t s of the  p r e s e n t model and the e x p e r i m e n t a l r e s u l t s of Prabhakar Prabhakar  and W e i n b e r g  7  well  have  also  found  that  et a l .  with  melt  30  superheats  above 40°C the c o n c e n t r a t i o n at the c h i l l  r a p i d l y w i t h i n c r e a s i n g superheat occurance The  can be accounted  superheat  reduces  a g a i n s t a copper c h i l l .  is  quite  the  difficult  This  f o r from the r e s u l t s g i v e n i n F i g  to  16.  t h i c k n e s s of a i r gap a t mold/metal  i n t e r f a c e which lowers the l e n g t h of the It  face drops  solid/liquid  measure the c h i l l  region.  face s e g r e g a t i o n  p r o p e r l y w i t h i n a v e r y narrow i n v e r s e s e g r e g a t i o n zone.  3.2  The Aluminium Zinc System  The phase diagram of the A l - Z n a l l o y Fig  18.  In  the  is  shown  in  t h i s case the s o l i d u s and l i q u i d u s l i n e s cannot be  assumed t o be l i n e a r . system,  system  phase  To a p p l y the s e g r e g a t i o n diagram was  every 5% a l o n g l i q u i d u s .  The  model  to  this  d i v i d e d into nineteen p o r t i o n s ,  s o l i d u s and  liquidus  lines  assumed t o be l i n e a r i n each p o r t i o n g i v i n g a c o n s t a n t  were  partition  ratio. The function decreases then  calculated of  total  composition  shrinkage in  F i g 19.  w i t h i n c r e a s e of c o m p o s i t i o n  increases  composition.  slightly The  as  increase  it of  which i s much more prominent than t h a t  to  The  is  plotted  shrinkage  as a ratio  i n the low a l l o y r e g i o n ,  approaches  total  a l s o caused by the thermal s h r i n k a g e .  ratio  the  eutectic  s h r i n k a g e near pure A l , in  Al-Cu  ( F i g 12),  is  In the r e g i o n between  10%  40%Zn, the d e c l i n e of s h r i n k a g e i s markedly a c c e l e r a t e d .  31  Fig  20  shows  Al-lO%Zn.  the  Similar  shrinkage  to  c a l c u l a t e d r e s u l t s of volume change f o r  to  the  expansion  in  t r a n s f o r m a t i o n i s observed critcial  point  lies  Al-Cu  alloy,  volume  each  i n the low c o m p o s i t i o n  between  the  for  both,  each  raised.  This occurrence  thermal  shrinkage  element 10. from  The  and can  their be  This  However,  solidification  keeps i n c r e a s i n g s t e a d i l y  reaches the f i n a l p a r t of  element  region.  d u r i n g the  D u r i n g the s o l i d i f i c a t i o n of the f i n a l  from  liquid/solid  elements 7 and 8.  element, the t o t a l s h r i n k a g e  u n t i l the s o l i d / l i q u i d  reversal  change d u r i n g  independent of s h r i n k a g e or expansion of  the  solidification.  element, total,  accounted  the  are for  shrinkage  significantly by  the  large  r e s u l t i n g from the l o n g f r e e z i n g range of  l i q u i d composition  in  this  element  the  increases  31% t o 95%Zn because of the s o l u t e r e j e c t i o n away from the  solid/liquid  interface during  its  temperature  of  as  the  system,  r e q u i r e d t o drop from 600°C  to  solidification seen  in  382°C.  i n t e r d e n d r i t i c f l u i d t o feed s h r i n k a g e  solidification  makes  Considering  Fig  21  compositions  summarizes of A l - Z n .  that  ( e u t e c t i c ) at  of c o m p o s i t i o n  the  the  formed a t the l a t e stage of macrosegregation  stage. the  temperature  profiles  in  various  I t can be c l e a r l y seen t h a t the  c o n t r a c t i o n a t the end of s o l i d i f i c a t i o n r a p i d l y d e c r e a s e s increase  the  i s solute enriched during  f a r more c o n t r i b u t i o n t o  than t h a t a t the i n i t i a l  so  the same f i g u r e , i s  s o l i d i f i c a t i o n and has the h i g h e s t c o m p o s i t i o n end of s o l i d i f i c a t i o n , the s h r i n k a g e  and  i n the low a l l o y r e g i o n .  t h i s t h e r m a l c o n t r a c t i o n as Y o u d e l i s e t a l " d i d ,  I f we the  thermal with neglect  resulting  -  32  segregation  for  low  alloy  c a l c u l a t e d i n the p r e s e n t The r a p i d d e c r e a s e Fig  Al-Zn  must  be f a r l e s s than  that  work. i n ~ the - t o t a l  shrinkage  observed  in  19 can be a t t r i b u t e d t o the d e c r e a s e i n t h e r m a l c o n t r a c t i o n ,  proving  t h a t the t h e r m a l c o n t r a c t i o n p l a y s an i m p o r t a n t  s h r i n k a g e and m a c r o s e g r e g a t i o n s e g r e g a t i o n a t the c h i l l F i g 22.  in  Al-Zn  alloy.  -The  U n l i k e the smooth curve f o r Al-Cu a l l o y , the c u r v e f o r  followed eutectic.  by  a  i n composition  up  to  40%Zn,  slow p r o g r e s s i v e d e c r e a s e i n c o m p o s i t i o n  t o the  The d e c l i n e of the i n v e r s e s e g r e g a t i o n a f t e r t h e peak  a t 20%Zn i s due t o the a c c e l e r a t e d d e c r e a s e i n t o t a l The p r o g r e s s i v e slower decrease i n s e g r e g a t i o n  shrinkage.  after  due t o a n e a r l y c o n s t a n t and v e r y low s h r i n k a g e r a t i o  40%Zn  The p r e s e n t  model  is  ( F i g 19).  p l o t t e d i n F i g 22 a r e e x p e r i m e n t a l and t h e o r e t i c a l  by Y o u d e l i s and C o l t o n * . with  inverse  face, i s p l o t t e d against composition i n  A l - Z n c o n s i s t s of a r i s e and f a l l  Also  role i n  results  prediction  agrees  the r e p o r t e d r e s u l t s i n the r e g i o n up t o 23%Zn even though  thermal c o n t r a c t i o n was n e g l e c t e d i n the p r e v i o u s  investigation.  Beyond 23%Zn t h e r e i s a s t r o n g d i s c r e p a n c y between  the  results  I t s h o u l d be  and  the r e s u l t s o f ' Y o u d e l i s and C o l t o n " .  noted t h a t t h e r e i s c o n s i d e r a b l e s c a t t e r i n which  makes  the  comparison  models o n l y a p p r o x i m a t e .  the  present  reported  data  w i t h t h e measured r e s u l t s and the  As has been demonstrated e a r l i e r ,  the  i n v e r s e s e g r e g a t i o n i s v e r y s e n s i t i v e t o the c a s t i n g c o n d i t i o n s .  33  3.3 The Antimony  This  alloy  solidification system  Bismuth  is  System  is- unique  i n - t h a t both m e t a l s expand d u r i n g  -  (see F i g 4 ) .  The phase  shown i n F i g 23.  diagram  other  two  systems  range,  investigated.  lines,  the  phase  s e c t i o n s , each of 5 % B i ,  diagram  and  the  was  solidus  solubility  Following  from  the  same  solidus  divided and  Sb-Bi  differing  p r o c e d u r e used i n the A l - Z n system w i t h non l i n e a r liquidus  the  There i s complete s o l i d  of B i i n Sb over the e n i t r e c o m p o s i t i o n the  for  :  and  i n t o twenty  liquidus  assumed l i n e a r i n each s e c t i o n w i t h a c o n s t a n t p a r t i t i o n  lines ratio.  The t o t a l s h r i n k a g e r a t i o f o r S b - B i , a s a f u n c t i o n of a l l o y composition  is  given  in  F i g 24.  sequences a r e c l e a r l y o b s e r v e d , 30%Bi  where  the  shrinkage  one  Two region  contraction  during  between  shrinkage zero  and  r a t i o d e c r e a s e d r a p i d l y , and above  30%Bi where the s h r i n k a g e d e c r e a s e s s l o w l y . i n the s h r i n k a g e near pure  different  Sb  is  solidification,  The a b r u p t i n c r e a s e  attributed which  to  the  thermal  s h i f t s the n e g a t i v e  v a l u e of s h r i n k a g e f o r the pure Sb t o the p o s i t i v e  side.  The d e t a i l e d volume change and temperature p r o f i l e f o r lO%Bi  is  shown  in  F i g 25.  Sb-  The s o l i d l i n e denotes the volume  change i n each s o l i d i f i e d element and the d o t t e d l i n e the volume change i n r e s i d u a l l i q u i d .  E v e r y element shows e x t e n s i v e volume  e x p a n s i o n on s o l i d i f i c a t i o n e x c e p t f o r the volume Although  expansion the  is  compensated  remaining  liquid  last  element  whose  by the t h e r m a l c o n t r a c t i o n . continues  to  shrink  during  34  solidification  because  of  the  density  change  c o n c e n t r a t i o n and a s l i g h t temperature drop, expands  steadily  until  the s o l i d / l i q u i d  f i n a l element t o s o l i d i f y . system  must  cause  the  The  volume  due t o s o l u t e  t h e whole  i n t e r f a c e reaches the  expansion  as  On  a  total  i n t e r d e n d r i t i c f l o w of s o l u t e e n r i c h e d  f l u i d t o move away from t h e volume elememt r e s u l t i n g segregation.  volume  in  normal  t h e o t h e r hand, t h e c o n t r a c t i o n o c c u r s  during  the s o l i d i f i c a t i o n o f t h e l a s t element r e v e r s i n g t h e f l o w .  The  temperature  the  profile  shown  t e m p e r a t u r e drops from 530°C which  results  i n F i g 25 to  271°C  indicates  during  T h i s thermal c o n t r a c t i o n a t the  of s o l i d i f i c a t i o n r e v e r s e s t h e t o t a l s h r i n k a g e which  maximum c o m p o s i t i o n , calculated  which  feeds  the  and  inverse  segregation  theoretical  segregation  segregation  at  approximately with  the  since  has t h e  lOO%Bi, a t t h e end of s o l i d i f i c a t i o n .  results  Maximum s e g r e g a t i o n a t t h e c h i l l minimum  stage  from expansion t o  shrinkage  p r o f i l e a t the c h i l l  f a c e f o r t h e Sb-Bi a l l o y i s shown i n F i g 26, experimental  late  r e s u l t s i n extensive macrosegregation  the i n t e r d e n d r i t i c f l u i d  The  solidification,  i n a marked c o n t r a c t i o n , as c l e a r l y observed i n  the volume change.  contraction,  that  at  70%Bi.  as  as the  r e p o r t e d by Y o u d e l i s . 5  f a c e was observed The  well  reversal  a t 1 3 % B i , and from  inverse  t h e c h i l l f a c e t o normal s e g r e g a t i o n o c c u r s a t  30%Bi.  The p r e s e n t  experimental  r e s u l t s agree reasonably-  well  r e s u l t s , c o n s i d e r i n g t h e u n c e r t a i n i t y of  the measured d a t a . Comparing the s e g r e g a t i o n p r o f i l e w i t h t h e t o t a l  shrinkage  -  35  ratio  ( F i g 24) shows the c o m p o s i t i o n  a t which the s e g r e g a t i o n i s r e v e r s e d with  the  point  Between 20%  and  solidification anomalous.  i n the s e g r e g a t i o n ( 3 0 % B i ) does  a t which the s h r i n k a g e 30%Bi,  there  is  expansion  and yet i n v e r s e s e g r e g a t i o n  sequence  Sb-20%Bi  which  almost zero s o l i d i f i c a t i o n shrinkage  ratio.  distribution be observed  of  "for  associated  i s observed  negative  c o n t r a c t i o n due solidification.  change  shrinkage  is  In F i g 27,  similar  solidification.  made  up  t o the temperature drop a t The  same  s o l i d i f i c a t i o n of Sb-Bi a l l o y  thing up  to  and  i s found t o have a r f  a t f r e e z i n g of each element  s l i g h t decrease of t o t a l s h r i n k a g e d u r i n g  with  which i s  volume change t o t h a t of Sb-!0%Bi ( F i g 25)  w i t h l a r g e expansion  integrated  coincide  r a t i o i s zero (20%Bi).  T h i s can be a c c o u n t e d f o r by the volume  temperature  confirmed  not  profile  by  the  must  late  occur  30%Bi.  the  It  can and The  thermal stage  during can  be  of the now  t h a t , even though the t o t a l s h r i n k a g e r a t i o i s z e r o or  negative,  the  positive  s e g r e g a t i o n r e s u l t i n g from the  thermal  c o n t r a c t i o n a t the l a t e stage of s o l i d i f i c a t i o n always overcomes the n e g a t i v e s e g r e g a t i o n formed a t the p r i m a r y Bi  alloy  with  20%  to  30%Bi.  Again  residual l i q u i d i s enriched in solute solid/liquid  interface.  Of  the  i n the  i t i s reminded t h a t the  by  the  advance  of  change.  The  solidification behaviour  the  i n the a l l o y a c c u r a t e l y ,  i s e s s e n t i a l t o o b t a i n the i n f o r m a t i o n of volume change entire  Sb-  p a r t i c u l a r importance i s t h a t , i n  o r d e r t o c a l c u l a t e the m a c r o s e g r e g a t i o n it  stage  over  p r o c e s s as w e l l as the t o t a l volume  of i n t e r d e n d r i t i c f l u i d f l o w a t the  stage of s o l i d i f i c a t i o n c o n t r o l s the o v e r a l l  segregation.  late  36  3.4  Examination  In  the  Of P r e v i o u s Model P r e d i c t i o n s  discussion  of- t h e  r e s u l t s i t has been p o i n t e d out plays  a  :  present  that  the  thermal  calculated contraction  much more s i g n i f i c a n t r o l e i n the f o r m a t i o n of i n v e r s e  s e g r e g a t i o n than i s g e n e r a l l y a c c e p t e d . the  computer  case  for  This  is  the A l - Z n and Sb-Bi a l l o y s a t low  particularly compositions.  P r e v i o u s c a l c u l a t i o n s d i d not i n c l u d e t h e r m a l c o n t r a c t i o n i n the models ' ' " . 1  2  8  1 1  Without thermal  contraction  the  calculations  w i l l g i v e low v a l u e s f o r the i n v e r s e s e g r e g a t i o n . To  clarify  the  p r e s e n t work w i t h the chill  face  inverse  models and a s s u m p t i o n s calculations  for  c o n t r a d i c t i o n s found previous  i n the comparison of  calculations,  calculations  of  s e g r e g a t i o n v a l u e s were r e p e a t e d u s i n g the used  previously ' '"' . 1  2  5  The  repeated  the t h r e e a l l o y systems d i d not c o i n c i d e w i t h  the p u b l i s h e d c u r v e s , as shown i n Appendix A. what i s the cause of the d i s c r e p a n c y .  It is  not. clear  37  A-IV.  The  present  the c h i l l liquid  r e s u l t s show i n v e r s e s e g r e g a t i o n observed  f a c e i s governed by t h e back flow of through  dendrite  contraction during developed  channels  solidification.  which  quantitatively  s e g r e g a t i o n a t the c h i l l The  CONCLUSIONS  resulting A  enriched  from  mathematical  determines  volume  model  the  f a c e , and a d j a c e n t t o t h e c h i l l  is  inverse face.  model i n c l u d e s t h e e f f e c t of t h e r m a l c o n t r a c t i o n due t o t h e  temperature changes as w e l l as c o n t r a c t i o n changes. the  solute  near  Comparing  model  values  shows  p r e d i c t e d and observed  Al-Zn  and  reasonable  Sb-Bi  composition  alloys  agreement  with  between  the  values.  The f o l l o w i n g c o n c l u s i o n s have been 1)  to  t h e v a l u e s of s e g r e g a t i o n d e t e r m i n e d from  f o r the Al-Cu,  experimental  due  reached.  A b i n a r y a l l o y may be c o n s i d e r e d as a  simple  mixture  of t h e two componets i n o r d e r t o determine t h e d e n s i t y of both s o l i d and l i q u i d . 2)  The  volume  change  during  solidification  c o n s i t s of  three p a r t s . a) .  The volume change due  to  the  liquid/solid  phase t r a n s f o r m a t i o n . b) .  Contraction  i n t h e r e s i d u a l l i q u i d due t o  solute concentration. c) .  Thermal  changes.  contraction  due  to  temperature  38  3)  Volume  expansion  occurs  over  most  stages  of  s o l i d i f i c a t i o n i n both the A l - C u and A l - Z n systems  due  t o l i q u i d / s o l i d change but t h i s i s compensated f o r  by  the o t h e r two e f f e c t s p r o d u c i n g a net c o n t r a c t i o n . 4)  Thermal  contraction  plays  a s i g n i f i c a n t r o l e i n the  i n v e r s e s e g r e g a t i o n of A l - Z n and Sb-Bi composition  regions  since  occur i n the l a s t stage of 5)  large  volume  change  over  alloy  temperature  drops  i t i s e s s e n t i a l to  the  p r o c e s s s i n c e the b e h a v i o u r flow  low  solidification.  To c a l c u l a t e m a c r o s e g r e g a t i o n the  in  entire  of  know  solidification  interdendritic  fluid  a t the l a t e stage of s o l i d i f i c a t i o n c o n t r o l s the  overall segregation. 6)  The s o l i d i f i c a t i o n s h r i n k a g e r a t i o of an function  of  the  volume  change  c o m p o s i t i o n a l changes i n the changes,  a l l of  which  on  liquid  must  be  alloy  is  a  solidification, and  temperature  considered  i n any  s i g n f i c a n t model f o r s e g r e g a t i o n . 7)  The l e n g t h of chill  solid/liquid  region  adjacent  to  the  f a c e i s h i g h l y s e n s i t i v e t o the c o n t a c t t h e r m a l  r e s i s t a n c e a t mold/metal i n t e r f a c e a s s o c i a t e d w i t h t h e f o r m a t i o n of an a i r gap a t the c h i l l f a c e .  -  39  T a b l e I - The comparison of a l l o y d e n s i t i e s w i t h c a l c u l a t e d from d a t a f o r pure m e t a l s Alloy Al-4.5%Cu Al-4.5%Cu Al-4.5%Cu Al-8%Cu Al-10%Cu  Temp(°C) p - s o l v e n t  1  p-solute  those  p-mean pmeasured E r r o r ( % )  1  Al-12%Cu Al-20%Cu Al-40%Cu Al-80%Cu  20 800 1000 20 800 1000 20 1000 800 1200  2.70 2.34 2.29 2.70 2.34 2.29 2.70 2.29 2.34 2.25  8.93 8.15 7.99 8.93 8.15 7.99 8.93 7.99 8.15 7.84  2.82 2.45 2.40 2.92 2.60 2.53 3.04 2.84 3.62 5.76  2.75 2.45 2.40 2.83 2.55 2.50 2.93 2.75 3.45 5.60  Al-5%Zn Al-5%Zn Al-10%Zn Al-10%Zn Al-20%Zn Al-40%Zn  700 900 700 900 700 700  2.36 2.31 2.36 2.31 2.36 2.36  6.45 6.35 6.45 6.35 6.45 6.45  2.45 2.39 2.54 2.49 2.74 3.24  2.45" 2.40* 2.55" 2.45* 2.70* 3.20*  0 0.4 0.4 1.6 1 .5 1 .5  Pb-5%Sn Pb-10%Sn Pb-20%Sn Pb-40%Sn  315 300 275 235  10.68 10.70 10.73 10.78  6.91 6.92 6.94 6.97  10.36 10.09 9.58 8.73  10. 4 0 10. 1 5 9.70 8.80  0.4 0.6 1 .2 0.8  A1-10%CU  Note: 1. 2. 3. 4. 5.  Smithells and E l l i o t e t a l Smithells Bornemann e t a l Bornemann e t a l Thresh e t a l 1 6  1 6  2 0  1  8  1  9  2 3 3 2 3 3 2 3 3 3  5  5  5  5  1  7  2.5 0 0 3.2 2.0 1 .2 3.8 3.3 4.9 2.9  40  T a b l e I I - The t h e r m a l p r o p e r t i e s employed f o r t h e c a l c u l a t i o n of temperature d i s t r i b u t i o n  A1-10%CU  Cu mold  t h e r m a l c o n d u c t i v i t y , k(cal/cm.s.°C)  0.43  0 .94  s p e c i f i c heat,  Cp (cal/g.°C)  0.259  0 .093  density,  p  (g/cm )  l a t e n t heat of s o l i d i f i c a t i o n ,  3  Hs ( c a l / g )  emissivity, e  t h e r m a l c o n d u c t i v i t y of a i r gap, ka shape f a c t o r through a i r gap  variable  8 .89 -  93.0 0.1.8  0 .18  1.0X10-* c a l / c m .s.°C 0.10  41  SPECIFIC VOLUME, cmVqm 0.42 0.38 0.54 0.30  5401 0  Figure  5 10 15 20 25 30 33 CONCENTRATION, wt-<& Cu  1 - E q u i l i b r i u m phase diagram and s p e c i f i c volume for the Al-Cu a l l o y s  LI W-  4  *  CL  koC<  "  "  (b)  B  (a)  X  (c) F i g u r e 2 - Schematic d e s c r i p t i o n of s o l i d i f y i n g zone? (a) d e n d r i t e morphology, (b) s o l i d and l i q u i d c o m p o s i t i o n s a l o n g c h i l l f a c e , and (c) c o m p o s i t i o n of i n t e r d e n d r i t i c l i q u i d  43  T E M P E R A T U R E CC)  Figure  3 - D e n s i t i e s of copper and aluminium vs temperature  44  200  400  TEMPERATURE  Figure  600  800  CO  4 - D e n s i t i e s of Z i n c , Aluminium and Bismuth vs temperature  45  y F i g u r e 5 - Schematic c o n f i g u r a t i o n of the model i n v e s t i g a t e d and the s u b d i v i s i o n f o r the n u m e r i c a l s i m u l a t i o n  46  Figure 6 - Schematic c o n f i g u r a t i o n of the f e e d i n q sequence; (a) F l u i d f l o w , and (b) p r o f i l e of the l i q u i d composition  1  J  1  ^  2  -"1  y  3  ^  6  y'\ y  >' 1  y y  y  y  y\  ^2  y  9  y  ''1  ^ :  f  y -"1  —3  y  y  <'2  y „  y  y y  y  ''1  . ' „  y  1  y  ^ ^  ^  y  " 7  .  .  3  .  y  ^ ..f^  <- 5  ACTUAL SOLIDIFICATION FRONT  < 2 -" 1 ^  y y  6 -  y  y  y y  y  y  5 .  - "4  y  ^  y  ^ 1  . " 6 -  ' A 3— .  - '2  y y  y  „  y  y  y  " 6  y  y y  y  y  "V  y  ^ ^  -is .  N  y  7 •*  y y  , " 3 -  y  . "  y'7  -®.  y  y  " 2  —  - "5  y  y  y  y  y  y  y  y  y  y  10  y  y  y  y  .  " ^ 10. -"9  ."1  y  y  y  y  2  y  .  y  .^  y  y  „  y  . " 2  y  -3  '  y  y  ^1  8°  y  2  y  y  ,  y y  y y  y  " -^9  ""-.5 " -^6 v  y  y  y  LU  .  ''\  y  iii  y  y  y y  " ^ 3 -2 y  y  y •  \  y y y y y  y '  y  .y  i  ^  y' y  -  ^  Figure 7 - Summary of t h e f l u i d flow induced by t h e s h r i n k a g e d u r i n g t h e s o l i d i f i c a t i o n of t h e model w i t h 1 0 X 1 0 subdivi sions  F i g u r e 8 - Schematic d e s c r i p t i o n of the d i l u t i o n e f f e c t of the i n t e r d e n d r i t i c f l u i d flow d u r i n g s o l i d i f i c a t i o n  49  (START)  n r  ( R E A D INPUT DATA  1CALCULATE MICROSEGREGATION] CALCULATE MEAN SOLID COMPOSITION AND LIQUIDUS TEMPERATURE FOR EACH ELEMENT  I N I T I A L I Z E VARIABLES L E T SOLIDIFY ONE ELEMENT  ]  CALCULATE SHRINKAGE |  I  CALCULATE SEGREGATION I N THE F I R S T COLUMN  CALCULATE D I L U T I O N EFFECT DUE TO F L U I D FLOW  CALCULATE POSITIONAL SEGREGATION  (STOP)  Figure  9 - Flow c h a r t f o r t h e c a l c u l a t i o n of i n v e r s e segregation  50  (START)  READ INPUT DATA INITIALIZE VARIABLES TIME-TIME+At  CALCULATE TEMPERATURE FIELD USING IMPLICIT FTfl  F i g u r e 10 - Flow c h a r t f o r t h e c a l c u l a t i o n of t h e l e n g t h of s o l i d / l i q u i d zone a d j a c e n t t o t h e c h i l l f a c e  51  F i g u r e 11 - C a l c u l a t e d volume change d u r i n g s o l i d i f i c a t i o n of A l - 1 0 % C u . The number on the c u r v e denotes each element solidified.  52  F i g u r e 12 - C a l c u l a t e d volume change and l i q u i d c o m p o s i t i o n i n the r e p r e s e n t a t i v e volume f o r A l - 5 % C u . The number on the c u r v e denotes each element s o l i d f i e d .  53  0  5  10  15  20  C O N C E N T R A T I O N  25  30  (wrtCu)  F i g u r e 13 - S o l i d i f i c a t i o n s h r i n k a g e r a t i o v s c o m p o s i t i o n of the A l - C u a l l o y s . The d o t t e d l i n e was d e r i v e d from F i g 1.  54  FRACTIONAL DISTANCE FROM THE CHILL FACE, x/L  F i g u r e 14 - P o s i t i o n a l s e g r e g a t i o n p r o f i l e s f o r v a r i o u s c o m p o s i t i o n s of t h e A l - C u a l l o y s  F i g u r e 15 - Comparison of the i n v e r s e s e g r e g a t i o n a t the c h i l l face f o r the A l - C u a l l o y s  56  0  1  1  2  '  4  AIR  GAP  1  1  r  6  8  10  (/M)  F i g u r e 16 - E f f e c t of a i r gap a t the metal/mold i n t e r f a c e on the l e n g t h of s o l i d / l i q u i d zone a d j a c e n t t o the c h i l l face  57  Figure  17 - Comparison of p o s i t i o n a l s e g r e g a t i o n 10%CU  for A l -  58  0  10  20,  30  40  50  Concentration  Figure  60  70  CC Lt  s  80  90  (%Zn)  18 - E q u i l i b r i u m phase diagram f o r A l - Z n a l l o y s  100  59  10 1  H I  0  1  20  1  1  HO 60 C O N C E N T R A T I O N  1  80 (wtfZn)  F i g u r e 19 - T o t a l s h r i n k a g e r a t i o vs c o m p o s i t i o n f o r the Al-Zn a l l o y s  1  100  60  POSITION OF SOLID/LIQUID INTERFACE ( ELEMENT NO.)  F i g u r e 20 - C a l c u l a t e d volume change and temperature i n the r e p r e s e n t a t i v e volume f o r A l - l O % Z n . The number on the curve denotes each element s o l i f i e d .  61  650 •  600  f  550  =3  500  450 -  400 -  POSITION OF SOLID/LIQUID INTERFACE (ELEfOfT NO.)  F i g u r e 21 - C a l c u l a t e d p r o f i l e s of t e m p e r a t u r e i n the r e p r e s e n t a t i v e volume f o r v a r i o u s c o m p o s i t i o n s of the A l - Z n alloys indicated.  Figure  22 - Comparison of the i n v e r s e s e g r e g a t i o n a t c h i l l face f o r the A l - Z n a l l o y s  the  63  0  20  40  Concentration  60  80  (wt % Bi)  F i g u r e 23 - E q u i l i b r i u m phase diagram f o r Sb-Bi a l l o y s  100  64  C O N C E N T R A T I O N ( wrt Bi )  F i g u r e 24 - T o t a l s h r i n k a g e r a t i o vs c o m p o s i t i o n f o r the Sb-Bi a l l o y s  1  I  2  I  3  I  »  I  5  I  «  I  7  I  8  I  «  I  10  POSITION OF SOLID/LIQUID INTERFACE ( ELEMENT NO.) F i g u r e 25 - C a l c u l a t e d volume change and t e m p e r a t u r e i n t h e r e p r e s e n t a t i v e volume f o r S b - l O % B i . The number on the c u r v e denotes each element s o l i d i f i e d .  66  F i g u r e 26 - Comparison of the i n v e r s e s e g r e g a t i o n a t the c h i l l f a c e f o r the Sb-Bi a l l o y s  F i g u r e 27 - C a l c u l a t e d volume change and t e m p e r a t u r e i n the r e p r e s e n t a t i v e volume f o r Sb-20%Bi. The number on the c u r v e denotes each element s o l i d i f i e d .  68  PART - B  CENTERLINE SHRINKAGE IN STEEL PLATE CASTINGS  69  B-I.  INTRODUCTION  A unique advantage of m e t a l c a s t i n g s i s t h a t h i g h l y complex shapes can r e a d i l y be f a b r i c a t e d . castings,  however,  This  inherent  advantage  i s o f t e n l o s t because of t h e d i f f i c u l t y of  c o n t r o l l i n g c a s t i n g d e f e c t s , among which s h r i n k a g e one  of  of  t h e major d e f e c t s .  porosity  is  S h r i n k a g e p o r o s i t y i s c o n t r o l l e d by  proper d e s i g n of t h e c a s t i n g , u s i n g s u i t a b l e g a t i n g and r i s e r i n g t e c h n i q u e s and moulding  materials.  Since the c a s t i n g process i s  complex, the d e s i g n of these systems  i s only  approximate  and  e m p i r i c a l so t h a t p o r o s i t y does occur i n c a s t i n g s .  1 . 1 Gross  Shrinkage  Recent  development  casting p r o c e s s a  simple  2 1  "  to  be  a p p l i e d a mathematical prediction  of  insufficient calculations  gross mass  with  the  numerical  simulation  calculated.  model  to  shrinkage feeding.  experimental  sand  mold  casting  2 6  f o r the  i n a r a i l wheel c a s t i n g due t o Comparing  their  numerical  o b s e r v a t i o n s i t was found t h a t well  with  the  which p r e d i c t e d t h e shape and p o s i t i o n of t h e l a s t  liquid to solidify. shrinkage  of  J e y a r a j a n and P e h l k e  the observed g r o s s s h r i n k a g e c a v i t y agreed v e r y calculations  of t h e  has e n a b l e d t h e s o l i d i f i c a t i o n sequence  2 5  casting  of  The r e m a i n i n g problem  i n castings  i n predicting  gross  i s t o extend t h e P e h l k e a n a l y s i s t o t h e  70  h i g h l y complex shapes used i n normal foundry p r a c t i c e .  1.2 C e n t e r l i n e  Towards shrinkage  Shrinkage  t h e end  must  be  of  solidification,  small  scattered  I f complete back f l o w does  interdendritic  pores  c e n t e r l i n e p o r o s i t y or c e n t e r l i n e s h r i n k a g e . model g i v e s flow  the  In  the  P e l l i n i et a l  2  7  "  3  0  carried  i n which t h e f e e d i n g  green-sand moulds was d e t e r m i n e d .  transfer  to  In a d d i t i o n f l u i d flow of the considered.  out an  extensive  series  of  distance  of s t e e l c a s t i n g s i n  I t was  found  c a s t i n g , 2.5 t o 5.0cm t h i c k ( F i g 28a),  for a  plate  that:  t h e maximum p l a t e l e n g t h c a s t c o m p l e t e l y sound by one r i s e r i s 4.5s ( s = p l a t e  2)  heat  i s complex and t h e r e f o r e u n c e r t a i n when a p p l i e d  i n t e r d e n d r i t i c l i q u i d must a l s o be  1)  A  called  l a t e r p a r t of s o l i d i f i c a t i o n t h e heat  c e n t e r l i n e p o r o s i t y problem.  measurements  forms,  not  i s o c h r o n a l s o l i d i f i c a t i o n l i n e s assuming s i m p l e heat  patterns.  flow l o c a l l y  solidification  f e d by l i q u i d f l o w i n g down h i g h l y i r r e g u l a r  narrow i n t e r d e n d r i t i c c h a n n e l s . occur,  the  the length D  thickness),  of t h e end of t h e p l a t e f r e e o f p o r o s i t y  i s always 2.5s f o r t h e p l a t e c a s t i n g  more  than  4.5s  l o n g , and 3)  t h e sound a r e a due t o r i s e r e f f e c t i s 2 s .  S i m i l a r l y t h e maximum l e n g t h c a s t c o m p l e t e l y sound i s 9.56 s and the  sound r e g i o n  due  to  end e f f e c t s i s 1.5s-2s i n square b a r  71  castings with 5 to examined  the  effect  found t h a t c h i l l s 5cm  and  20cm of  end  contribute  1s i n p l a t e and  Johnson  section  and  size  ( F i g 28b).  c h i l l on the f e e d i n g  examined  3 1  flat  plates  and  2.5cm which i s s m a l l e r et a l  2  7  '  2  8  .  They  Niyama  Sections  than  the  range  considered  during  effective  critical  was  given  by  solidification.  parameter  parameter  G//R  Bishop  (16.1/s-14)cm  for  to  castings.  They  the  predict  porosity.  i n the  prediction  to  3 3  in  bases  isochronal  showed  of s o l i d i f i c a t i o n  taken by Niyama et a l  that  is  a  can  composition considered.  and  the  the  only Pellini  proposing  a  be 3 0  . new  v a l u e of and  T h i s would be. independent of the a l l o y  size  and  shape  T h i s has been c h a l l e n g e d  q u e s t i o n e d the g e n e r a l  the  simple  However  p r e d i c t p o r o s i t y w i t h the c r i t i c a l  porosities.  be  plate  /min.°C/cm f o r t h i s p a r a m e t e r , a p p l i c a b l e t o b o t h g r o s s  centerline  who  by  e m p i r i c a l l y f o r each c a s t i n g as shown by  A f u r t h e r s t e p was  1.0  at the end  v a l u e of the g r a d i e n t  determined  l e s s than  p r e d i c t e d c e n t e r l i n e s h r i n k a g e on the  3 2  temperature gradient and  of  studied  square bars of t h i c k n e s s e s  of a n u m e r i c a l s i m u l a t i o n of the p r o c e s s g i v i n g contours  by  the c e n t e r l i n e p o r o s i t y  (12.7/s-8.4)cm f o r square bar et a l  region  found t h a t the maximum l e n g t h which c o u l d  cast without porosity c a s t i n g s and  and  respectively.  t h i n s t e e l s e c t i o n s c a s t i n green-sand mould. included  also  distance  t o i n c r e a s i n g the sound  square bar c a s t i n g s  Loper  They  of  the  steel  by C h i j i i w a and  casting Imafuku *  a p p l i c a b i l i t y of t h i s p a r a m e t e r .  an a l t e r n a t i v e they proposed t h a t the g r a d i e n t  3  As  of f r a c t i o n s o l i d  72  at cm  the end of s o l i d i f i c a t i o n has the c r i t i c a l v a l u e of 0.25 for c e n t e r l i n e shrinkage.  predict  porosity,  G,  G//R  The  three parameters  shrinkage  industrially.  proposed  and the g r a d i e n t of f r a c t i o n  might be e f f e c t i v e i n p r e d i c t i n g  the  However,  formation they  are  per to  solid,  of  centerline  all  effectively  e m p i r i c a l p a r a m e t e r s l a c k i n g a c l e a r p h y s i c a l b a s i s i n which  to  e s t a b l i s h a q u a n t i t a t i v e measure of the p o r o s i t y i n a c a s t i n g .  1.3  P r e v i o u s Models  P o r o s i t y i s a s s o c i a t e d w i t h f e e d i n g d i s t a n c e i n the c a s t i n g as  well  through  as  solidification  i n t e r d e n d r i t i c channels  numerical given  the  by: dv  simulation.  The  structure.  Feeding  of  i s c o n s i d e r e d by D a v i e s  3 5  liquid using  c a p i l l a r y f l o w t h r o u g h a channel  is  nr P 4  =  dt  8y  1  (B-1)  The amount of l i q u i d which must f l o w through the compensate f o r s o l i d i f i c a t i o n c o n t r a c t i o n i s ;  capillary  "v *  "  r2  Combining e q u a t i o n s  (B 2)  (B-1) and  ( B - 2 ) , g i v e s the c a p i l l a r y  feeding  distance, l , f  <  Pr  to  2  s  F r i c t i o n r e d u c e s the e f f e c t i v e p r e s s u r e , P=P -AP, where, f o r Q  73  laminar  flow  (r<<l): 2  A P  The  (B-4)  f l o w v e l o c i t y i s g i v e n by, v = Bv  (B-5)  s  Combining e q u a t i o n s (P - A P ) r  h  =  2  P r  =  Pr  gives:  2  ~s 1  s  When s h r i n k a g e  The  (B-3),(B-4) and (B-5)  (B_6)  appears, l = l  and:  f  2  c a p i l l a r y f e e d i n g d i s t a n c e was c a l c u l a t e d u s i n g e q u a t i o n (B-  7),  incorporating  an  empirical  correction  factor  B  which  includes the f a c t o r s that are d i f f i c u l t to consider: 1  The  f  BYP r = - — i _  yBv  2  (B-8)  s  c o r r e c t i o n f a c t o r , B, was chosen t o be 0.22  the  i n order  that  c a l c u l a t e d f e e d i n g d i s t a n c e f o r a s t e e l p l a t e c a s t i n green  sand mould has a f e e d i n g  distance  equal  to that  empirically  d e t e r m i n e d v a l u e of 4 . 5 s . 3 0  Based  on  t h e type of a l l o y , assumptions were made f o r t h e  f r a c t i o n s o l i d a t which p o r o s i t y due t o incomplete occur. could  F o r example, D a v i e s occur  99.8%Al; 99%  3 5  estimated  f e e d i n g would  t h a t incomplete  above s o l i d f r a c t i o n s of 95%, f o r 0.6%C s t e e l s and  above s o l i d f r a c t i o n of 90% f o r nodular c a s t  f o r pure  feeding  Al.  iron  and  The above model (Davies') was a p p l i e d t o a  wide range of m a t e r i a l s and good agreement was o b t a i n e d  between  74  the  c a l c u l a t e d r e s u l t s and t h e e x p e r i m e n t a l l y d e t e r m i n e d v a l u e s . Shortcomings  i n the  Davies'  model a r e t h a t an e m p i r i c a l  c o r r e c t i o n f a c t o r B i s i n c l u d e d i n t h e model, and the a s s u m p t i o n of  when i n c o m p l e t e f e e d i n g s t a r t s which s t r o n g l y i n f l u e n c e s  presence  of  centerline  shrinkage.  The  value  f o r B used i n  f i t t i n g t h e model t o e x p e r i m e n t a l r e s u l t s was d e r i v e d experiments for  on  cast  steel.  the  from  the  One would expect d i f f e r e n t v a l u e s  B would be r e q u i r e d f o r o t h e r m a t e r i a l s s i n c e B i n e q u a t i o n  (B-8) i n v o l v e s a l o t of f a c t o r s not i n v o l v e d i n h i s model. Another Flemings flow.  3 6  theoretical  for porosity  i n c o r p o r a t i n g D a r c y ' s Law  Assuming  t h a t the mold/metal  o v e r r i d i n g importance and extent,  model  remains  at  that  the  was proposed by  for interdendritic  fluid  i n t e r f a c e r e s i s t a n c e i s of mold,  being  infinite  in  i t s i n i t i a l temperature ( T ) , the r a t e o f Q  heat f l o w a c r o s s t h i s i n t e r f a c e i s : q = -h(T-T )  (B-9)  Q  The heat e n t e r i n g the mold comes n e a r l y e n t i r e l y from of  f u s i o n of t h e s o l i d i f y i n g m e t a l . V  q = (pH  )  sV  3  G  the  heat  Thus:  L  - ±  A 8t  (B-10)  c  Combining e q u a t i o n (B-9) w i t h (B-10) g i v e s : 3t " ~ where A.T=T-T C  Q  and  c  c o n t r o l l e d by the heat  is a  constant.  transfer  (B-11) S i n c e t h e heat f l o w i s  coefficient  h,  there  i s no  s i g n i f i c a n t t h e r m a l g r a d i e n t i n t h e s o l i d i f y i n g m e t a l and a ^ / S t i s independent of p o s i t i o n i n t h e c a s t i n g .  D a r c y ' s law i s g i v e n  75  by: v = -  (VP +  (  B  _  1  2  )  For one d i m e n s i o n a l f l u i d f l o w , e q u a t i o n (B-12) reduces t o : „ v  _ x  K  dP .  ~ "  dx"  (  Similarly,  +  . p  L r g  t h e mass  (B-13)  }  balance  equation  i n t h e volume  element,  e q u a t i o n (A-13), reduces t o : 3 t  o~k~ L L x ^ ( p  g  (B-14)  V  where P  =  P  Equation  s s g  +  p  L L  (B-15)  g  (B-15) i s r e a r r a n g e d f o r c o n s t a n t d e n s i t i e s  in liquid  and s o l i d a s :  _ __ Equations 3x  _  ( B  _  1 6 )  (B-11), (B-14) and (B-16) y i e l d ,  (  Since g gives  -  8  L V'  " 1=3  (B-17)  C  i s independent  of p o s i t i o n , i n t e g r a t i n g e q u a t i o n  the interdendritic  flow  velocity  as a  (B-17)  function  of  position; vX  P  cx  " 1-3 g  Substituting  (B-18)  T  e q u a t i o n (B-18) i n t o e q u a t i o n  (B-13)  76  and  i n t e g r a t i n g y i e l d s t h e p r e s s u r e a t x; P ?  This  =  a AS +  ( D 2  expression  formation  of  - ) x 2  +  p  L r 8  (B-19)  Y  i s theoretically  centerline  shrinkage,  system w i t h homogeneous temperature without  superheat,  enough  to  describe  the  but i s v a l i d o n l y f o r t h e  a l l over  t h e c a s t i n g and  which i s not s a t i s f i e d under normal c a s t i n g  conditions.  1.4 P e r m e a b i l i t y  Towards t h e end of s o l i d i f i c a t i o n when t h e f r a c t i o n in  the a l l o y  i s high,  shrinkage  occurs  channels.  This  and, rate  fluid  through flow  flow  narrow  solid  t o compensate f o r volume tortuous  interdendritic  i s analogous t o f l o w through packed beds  i f t h e f l o w o c c u r s under low p r e s s u r e c o n d i t i o n s , t h e w i l l obey t h e "Darcy's Law" e x p r e s s e d  For one d i m e n s i o n a l f l u i d flow  without  a  flow  i n equation (B-12). metallostatic  head,  e q u a t i o n (B-12) reduces t o : v- = -  y  1  ^  (  B  -  2  0  )  where v' i s s u p e r f i c i a l f l o w v e l o c i t y and v* = g v L x T  (B-21)  Flow v e l o c i t i e s through a s t r a i g h t c a p i l l a r y tube a r e g i v e n by  77  Hargen-Poisulle' s equation: dt  _ ILL  =  8y  Introducing  a  (B-22)  1  "tortuosity  factor,  T"  in  the case where t h e  c a p i l l a r y i s not s t r a i g h t and f o r n c a p i l l a r i e s per u n i t a r e a : v' = -  8y  (B-23)  T l  Equating  e q u a t i o n s (B-20) and 4 K = 8T and from geometry: 2 g = mrr x  (B-23):  n 7 r r  (B-24) (B-25)  L  rearranging gives: 2 r  2 2 2  =  n  IT  (B-26)  x  S u b s t i t u t i n g equation  (B-26) i n t o (B-24) y i e l d s :  2 K = —  3  shows  that  Q  This  permeability  (B-27) for  a  given  i s directly  volume f r a c t i o n  liquid  3 7  "  proportional 3 9  lead.  and  Flemings  3 7  to  morphology, the  T  with  a  by  u s i n g Al-4.5%Cu porous media and molten  s l o p e of 2.  e x p e r i m e n t a l l y when the f r a c t i o n A p e l i a n , F l e m i n g s and M e h r a b i a n " equation  square of t h e  (B-27) was f i r s t a t t e m p t e d  The t h e o r y p r e d i c t s l n ( K ) i s p r o p o r t i o n a l t o  constant  the  .  The v e r i f i c a t i o n of e q u a t i o n Piwonka  dendrite  T h i s was found t o be t h e case liquid  0  ln(g ) for  was  less  than  0.3.  noted t h a t the p e r m e a b i l i t y i n  (B-27) i s i n v e r s e l y p r o p o r t i o n a l t o the number of flow  78  c h a n n e l s per u n i t a r e a . those  of  They compared K v a l u e s of A l - 4 % S i  Al-4%Si-0.25%Ti,  c o n s i s t e n t l y lower  f o r the  and  with  found t h a t the p e r m e a b i l i t y i s  grain  refined  alloy,  the  latter,  which has more c a p i l l a r i e s i n a u n i t a r e a . Interdendritic and W e i n b e r g casting. equal  in  3 9  f l u i d f l o w was c a r e f u l l y measured by S t r e a t a  partially  solidfied  columnar  dendritic  Assuming the number of f l o w c h a n n e l s i n the c a s t i n g i s  to  the  of c h a n n e l s between t h e columnar  primary  d e n d r i t e branches and the s p a c i n g between these c h a n n e l s  equals  the p r i m a r y  number  d e n d r i t e arm s p a c i n g , e q u a t i o n g (DAS) 2  2  38ITT  K  (B-28)  Measurements  of  spacing  and  a  constant  permeability  K  obeys  permeability  i n t e r d e n d r i t i c flow. changes  (B-27) i s reduced t o :  due  dendrite  which  channels. examined  to  f o r a wide range of d e n d r i t e arm  fraction  equation After  liquid,  showed  the  (B-28) d u r i n g the e a r l y p a r t of  flow  has  occured  the  flow  rate  the i n t e r a c t i o n of t h e f l o w i n g l i q u i d w i t h the changes  the  size  and  configuration  However d e v i a t i o n from e q u a t i o n occured  comparison  0.19,  with  after the  a  relatively  solidification  of  the  (B-28) i n the system  long  time  times  period,  in  associated  with a  fraction  liquid  normal c a s t i n g . V a l u e s of p e r m e a b i l i t y as a reported  in  the  literature  r e s u l t s show t h a t e q u a t i o n fluid  flow  under  steady  are  function  summarized  (B-27) i s state  of  valid  conditions  in  for  F i g 29.  The  interdendritic  f o r volume  liquid  79  f r a c t i o n s l e s s than about  1.5  0.3.  Present Objectives  Even though s h r i n k a g e p o r o s i t y quality  of  a  is a  a  casting.  This  centerline porosity flow. proper  factor  c a s t i n g , no a n a l y t i c a l method has  which c l e a r l y p r e d i c t s whether c e n t e r l i n e in  major  To p r e d i c t  is  primarily  mathematical  formation simulation  of is  the  been developed  shrinkage  will  occur  because the mechanism f o r  i s complex, governed by the  in  interdendritic  centerline required  t r a n s f e r model w i t h an i n t e r d e n d r i t i c f l u i d  fluid  shrinkage,  a  c o m b i n i n g a heat  f l o w model.  80  B-II.  The  formation  insufficient  MODELLING PROCEDURE  of c e n t e r l i n e s h r i n k a g e  flow  of  interdendritic  s o l i d i f i c a t i o n shrinkage. has  The  i s associated  with  l i q u i d t o compensate f o r  mathematical  t o i n v o l v e both heat t r a n s f e r and f l u i d  model,  therefore,  flow.  2.1 M a t h e m a t i c a l F o r m u l a t i o n 2.1.1  Temperature C a l c u l a t i o n s  The  c a s t i n g and sand mold geometry i n v e s t i g a t e d i s shown i n  F i g 30.  The  riser  was  chosen t o be the same s i z e as t h e one  used i n t h e experiments r e p o r t e d 3s  wide  and 5s h i g h .  by P e l l i n i e t a l  2  7  '  3  0  ,  that  is  T h i s r i s e r s i z e i s l a r g e enough t o a v o i d  gross shrinkages i n the p l a t e c a s t i n g . The  basic  temperature  mathematical  relation  used  to  calculate  d i s t r i b u t i o n i n t h e system was t h e g e n e r a l  the  equation  t h a t governs heat c o n d u c t i o n i n a s o l i d ; fCpf^) = V(kAT) For c o n s t a n t heat c o n d u c t i v i t y and two  (B-29) dimensional  heat  flow,  e q u a t i o n (B-29)reduces t o ;  |T 3  t  Several  (  +  3x  lily 3y  (B-30)  key assumptions c o n c e r n i n g t h e c a s t i n g p r o c e s s a r e  81  required to adjust  equation  (B-30)  to a  form  suitable for  numerical s o l u t i o n ; 1)  The at  mold  i s instantaneously f i l l e d  with l i q u i d  metal  the pouring temperature.  2)  Once t h e mold i s f i l l e d ,  3)  The  thermal  contact  the l i q u i d metal i s stagnant.  resistance  at  the  sand/metal  interface i s negligible. 4)  Segregation  i n t h e s o l i d and s o l u t e enrichment  i n the  l i q u i d are n e g l i g i b l e . 5)  The  latent  heat  uniformly  of  between  solidification  is  the  and  liquidus  released solidus  temperatures. 6)  Liquidus  and  temperature  solidus  are  linear  functions  of  so t h a t t h e weight f r a c t i o n s o l i d i f i e d i s  g i v e n by, T  s  f  The  L  "  T  £-Tir L s  =  (B-31)  i n i t i a l c o n d i t i o n s a t t=0 a r e , T = T M P  (B-32)  T  (B-33)  M  = T o  s  The boundary c o n d i t i o n a t t h e v e r t i c a l sand mold s u r f a c e i s :  =V  - s rr> k  (  T  s  -V  (B-34)  and a t t h e h o r i z o n t a l mold s u r f a c e i s : " s k  (  ^  )  =  h  s  (  T  s - a > T  < " > B  35  and a t m e t a l s u r f a c e i s : 3 T  M ' 3"T ^ " a > C o n v e c t i v e heat t r a n s f e r from t h e m e t a l t o ambient _k  (  )  =  (  T  (B-36) air is  82  neglected  giving (T  \ There  =  + 273) - (T + 273) A  ° ^  is  ^-T  4  8  ( B  a  continuity  of  heat  flux  across  i n t e r f a c e , which i s expressed at the v e r t i c a l 3T 3 T k (—- ) = k (— ) s 3x ' *M 3 x and at the h o r i z o n t a l i n t e r f a c e by;  the  i n t e r f a c e by  9 T  2.1.2  s  M  (B-39)  9 T  Pressure Required To Feed  It  has  been  interdendritic  fluid  shown  drop  due  c a s t i n g , Darcy's Law  Shrinkage  in  the  i s l e s s than 0.3.  3 8  '* '' 0  1  that  at steady s t a t e when To  calculate  the  to i n t e r d e n d r i t i c f l u i d  flow i n s o l i d i f y i n g  was  centerline  p l a t e on the assumptions  a p p l i e d along that the Law  through the media with higher l i q u i d temperature  literature  flow obeys Darcy's Law  the volume f r a c t i o n l i q u i d pressure  (B-38)  ;  V  3 7 )  sand/metal  M  l  "  gradient  present  in  the  i s also v a l i d  alloy.  the  f o r the flow  f r a c t i o n than 0.3 the  of  The  and f o r a governing  equation without a m e t a l l o s t a t i c head i s ,  K v  x  =  dP  ~ ^ 7 dx"  (B-40)  83  rearranging gives, dP = -  yg dx L  v  K  xx  (B-41)  where u i s c o n s t a n t , K calculated  i s given  i n F i g 29,  and  g  L  can  be  from,  - s L *L-f /p.(1-f.)/^ ( 1  f  8  Equation  ) / p  <-">  +  (B-41)  expresses  a  pressure  drop  within  an  appears  at  i n f i n i t e s i m a l element as a f u n c t i o n of f l u i d v e l o c i t y . The boundary solid/liquid  condition  interface  f o r equation  where,  (B-41)  i f t h e s h r i n k a g e i s f e d by the  l i q u i d , t h e r a t e of pore f o r m a t i o n i s e q u a l t o t h e r a t e of f l u i d flow.  This gives: (v Ag )6 = -vAg s  L  ,  L  ( B  _  4 3 )  yielding, V  x -  " V  (B-44)  and the c o n t i n u i t y of f l u i d f l o w g i v e s , V L  The  =  c o n  ^ant  ( B  _  4 5 )  t o t a l p r e s s u r e drop, c a l c u l a t e d by, AP =  denotes  required  Z AP j  the  shrinkage.  the  3  (B-46) 3  pressure  required  to  feed  the  solidification  S h r i n k a g e p o r o s i t y w i l l form when t h e t o t a l to  system,  pressure  feed  t h e s h r i n k a g e exceeds t h e p r e s s u r e a c t i n g on  that  i s the  m e t a l l o s t a t i c head p r e s s u r e .  atmospheric  pressure  plus  the  84  2.2 Computer Programming  The  explicit  finite  difference  method  is  used f o r the  s o l u t i o n of heat t r a n s f e r e q u a t i o n s s i n c e a s m a l l time required  step  t o c a l c u l a t e s m a l l changes d u r i n g s o l i d i f i c a t i o n .  d e r i v a t i o n of the n o d a l e q u a t i o n i s g i v e n calculation  was  first  performed  in  since  The  t o determine the temperature V a l u e s of  p e r m e a b i l i t y were t a k e n from the Piwonka and F l e m i n g s 29  The  Appendix B.  d i s t r i b u t i o n and then t o d e t e r m i n e the p r e s s u r e d r o p .  Fig  is  they c o v e r a wide range of l i q u i d  3 7  data  fraction,  in  which  i s r e q u i r e d i n the p r e s e n t c a s e . The a c t u a l c a l c u l a t i o n sequence chart,  F i g 31,  Appendix C. of at  used i s shown i n  flow  and a sample of the F o r t r a n program i s g i v e n i n  The computer program a l s o i n v o l v e s the  temperature g r a d i e n t and the s o l i d i f i c a t i o n the end of  the  calculation  parameter,  G/^R,  calculating  the  solidification.  2.3 V a l i d a t i o n Of The Heat T r a n s f e r Model  To  validate  temperature  the  present  model  d i s t r i b u t i o n i n the c a s t i n g , some c a l c u l a t i o n s were  performed and the r e s u l t s measurements  of  Bishop  were and  compared Pellini  c a l c u l a t e d v a l u e s f o r the temperature centerline  for  at  different  2 7  .  to  the  experimental  F i g 32 shows t y p i c a l  distributions  along  the  time i n t e r v a l s a f t e r p o u r i n g , f o r the  85  p l a t e c a s t i n g w i t h 5cm t h i c k and 33cm l o n g .  A l s o p l o t t e d i n the  f i g u r e s a r e the temperature p r o f i l e s d e t e r m i n e d e x p e r i m e n t a l l y . The measured l i q u i d u s carbon  steel,  and  having  solidus  the  temperatures  following  for  the  low  composition  Fe-0.34%C-  0.68%Si-0.89%Mn were 1507°C and 1463°C r e s p e c t i v e l y .  Reasonable  agreement can be observed between the model p r e d i c t i o n s and  the  e x p e r i m e n t a l r e s u l t s over the e n t i r e l e n g t h of the p l a t e , e x c e p t at  positions  close  to  the end s u r f a c e and t o the r i s e r .  p r e d i c t e d t e m p e r a t u r e s are region  of  the  quite  liquid/solid  accurate  co-existing  especially zone  noted  It  the  i s most will  be  from the c u r v e s i n F i g 32 t h a t the temperature g r a d i e n t s ,  e s t a b l i s h e d soon a f t e r s o l i d i f i c a t i o n s t a r t s by heat the  in  which  r e l e v a n t t o the f o r m a t i o n of c e n t e r l i n e s h r i n k a g e .  The  plate  from  the  riser  and  flow  into  heat l o s s e s a t the end of the  p l a t e , t e n d t o p r o g r e s s toward each o t h e r d u r i n g s o l i d i f i c a t i o n . The  movement  centerline  is  of  shown  the in  solidus F i g 33.  temperature  along  the  The c a l c u l a t e d v a l u e s a r e i n  good agreement w i t h the measured temperature p r o f i l e s except f o r a small deviation.  The f r e e z i n g r a t e , g i v e n by the s l o p e of the  c u r v e i n F i g 33 i s observed t o be  very  rapid  in  the  r e g i o n of the c a s t i n g and r e l a t i v e l y slow a t e i t h e r Values  of  calculations physical  data  the  are  physical  listed  for  interdendritic fluid  the flow.  in  properties Table I I I  calculation  of  used  end.  in  together pressure  central  the  with  above other  drop due t o  86  B-III.  The mathematical was  applied  to  RESULTS AND DISCUSSION  model d e s c r i b e d i n  a  the  previous  section  range of s t e e l p l a t e s of d i f f e r e n t s i z e , t o  p r e d i c t t h e f o r m a t i o n of c e n t e r l i n e s h r i n k a g e .  C a l c u a t i o n s were  a l s o c a r r i e d out t o e v a l u a t e the e f f e c t of t h e end c h i l l on c a s t i n g soundness.  Table IV l i s t s the p l a t e s s i z e s c o n s i d e r e d .  3.1 S o l i d i f i c a t i o n  The  Sequence  temperature  d i s t r i b u t i o n a l o n g t h e c e n t e r l i n e , F i g 32,  can be d i v i d e d i n t o t h r e e p a r t s . relatively  steep  temperature  zero  temperature  effectively the  temperature  shrinkage  must  the  gradient form  in  is the  These a r e t h e r i s e r end w i t h a gradient,  the  centre  with  g r a d i e n t and t h e p l a t e end where again  steep.  The  centerline  c e n t r a l r e g i o n where d i r e c t i o n a l  s o l i d i f i c a t i o n does not o c c u r . T y p i c a l c o n t o u r l i n e s d u r i n g s o l i d i f i c a t i o n determined the model a r e  shown  indicate  critical  the  s h r i n k a g e determined the  intial  stage  corresponding  in  F i g 34. points  solid  vertical  dotted  of  lines  f o r the f o r m a t i o n of c e n t e r l i n e  e x p e r i m e n t a l l y by B i s h o p and P e l l i n i solidification  ( F i g 34a)  2 8  .  contour  At line  t o 70% s o l i d p r o j e c t s out a l o n g the c e n t e r l i n e of  the p l a t e toward the s o l i d u s . 70%  The  from  region  will  The r e m a i n i n g  contain  liquid  within  i n t e r d e n d r i t i c channels  the broad  87  enough t o feed t h e s o l i d i f i c a t i o n c o n t r a c t i o n solid/liquid (Fig  region.  At  t h e middle  34b), t h e s o l i d u s s t i l l  of  the c a s t i n g .  The  solidus contains l i t t l e liquid  to  flow  liquid  through  the  between  making  i t very  the e x i s t i n g  solid/liquid  shrinkage  was  region i s located  found  solidification  solidification porosity  channel  to  t h e shape  lines occur.  and  indicate However  The  bottom  of  the  end  of  line  approach  position  of t h e  whether the  centerline  contour  line  cannot q u a n t i t a t i v e l y p r e d i c t t h e r e g i o n a l o n g t h e  c e n t e r l i n e where s h r i n k a g e p o r o s i t y w i l l  3.2 P r e d i c t i o n Of C e n t e r l i n e  The  t o f e e d volume  Near  2 7  Thus,  for  i n the area where c e n t e r l i n e  experimentally .  contour  i s likely  configuration  difficult  t h e s o l i d u s and 70% s o l i d c o n t o u r  each o t h e r ( F i g 3 4 c ) .  the r i s e r  90% s o l i d and t h e  shrinkage at the s o l i d u s along the c e n t e r l i n e . the  centerline  l i n e s move back toward  region  of  stage of s o l i d i f i c a t i o n  p r o j e c t s out a l o n g  whereas t h e 70% and 90% c o n t o u r end  a t t h e bottom  calculated  occur.  Shrinkage  pressure  required  to  feed the shrinkage  a l o n g t h e c e n t e r l i n e f o r a 5cm t h i c k p l a t e c a s t i n g i s shown i n Fig  35. The p r e s s u r e a c t i n g on t h e system i s d e f i n e d as the sum  of  atmospheric  pressure  When t h e p r e s s u r e r e q u i r e d end of s o l i d i f i c a t i o n  and t h e m e t a l l o s t a t i c head p r e s s u r e . t o feed t h e s h r i n k a g e formed  a t the  i s g r e a t e r than t h e p r e s s u r e a c t i n g on t h e  system, t h e s h r i n k a g e w i l l n o t be f e d c o m p l e t e l y .  88  The  pressure  r e q u i r e d t o f e e d a 4.6s(23cm) l o n g c a s t i n g i s  a l w a y s l e s s than required  to  that  feed  acting  long  on  castings,  g r e a t e r t h a n the a c t i n g p r e s s u r e . be  cast  completely  t h e system. greater  The  pressure  than 4.8s(24cm) i s  The maximum l e n g t h which  can  sound u s i n g one r i s e r , d e t e r m i n e d from t h e  model, i s 4.7s f o r 5cm t h i c k p l a t e .  This value agrees w e l l with  the e x p e r i m e n t a l l y d e t e r m i n e d v a l u e of 4 . 5 s .  In  sound  t o t h e f r e e end  3 0  r e g i o n determined e x p e r i m e n t a l l y a d j a c e n t  F i g 35 t h e  of t h e p l a t e i s shown f o r c a s t i n g s g r e a t e r than 4.8s i n l e n g t h . The l e n g t h f r e e of p o r o s i t y shown 2.35s  i n the f i g u r e  ranges  t o 2.7s, which a g r e e s w e l l w i t h t h e e x p e r i m e n t a l  2.5s .  Of p a r t i c u l a r i n t e r e s t i s t h a t  3 0  pressure  increases  casting  was  exponentially  subjected  solidification,  for  to  2  example,  from  v a l u e of  t h e maximum  required  with casting length.  I f the  atmosphere  pressure  during  the e f f e c t  would  small,  be  i n c r e a s i n g t h e r e g i o n f r e e of p o r o s i t y by 0.2s(1cm). Calculated shrinkage  profiles  along  of  the pressure  the c e n t e r l i n e  of  required  to  feed  2.5 and 0.5cm t h i c k p l a t e  castings  a r e shown  pressure  p r o f i l e s f o r both p l a t e t h i c k n e s s a r e s i m i l a r t o those  i n F i g 35. casting  i n F i g s 36 and 37,  respectively.  I t i s found f o r 2.5cm t h i c k p l a t e t h a t  length  which  values  agree  with  experimental  respectively.  For 0.5cm t h i c k  sound  i s also  length  t h e maximum  i s f r e e of p o r o s i t y i s 4.6s and t h a t t h e  sound r e g i o n a t the f r e e end o f t h e p l a t e These  The  4.6s.  casting  i s 2.5s data,  t o 2.7s.  4.5s  ( F i g 37),  and 2.5s  t h e maximum  There i s no i n d i c a t i o n t h a t t h e  maximum f e e d i n g d i s t a n c e t o t h i c k n e s s  ratio  decreases  with  a  89  decrease  i n plate  Johnson and Loper  thickness  thickness  3 1  i n F i g 38.  feeding  distances  plates  which  l e s s than 2.5cm, as  reported .  P r e d i c t e d maximum f e e d i n g plate  i n the region  distances  are plotted  E x p e r i m e n t a l r e s u l t s of t h e maximum  are a l s o  plotted  are thicker  than  i n the  figure  For  thinner  than  of  distances  a r e l e s s than those p r e d i c t e d .  for  2 7  '  3 1  .  For  2cm t h e agreement between t h e  c a l c u l a t e d and measured v a l u e s i s e x c e l l e n t . 2cm  against  the experimental  values  the the  T h i s can be  plates feeding  accounted  by t h e l a r g e drop i n temperature of t h e l i q u i d m e t a l a t t h e  f r e e end of t h e p l a t e as t h e l i q u i d m e t a l i s f i l l i n g during  pouring.  present  model.  effective feeding  t h e mold  T h i s temperature d r o p i s not c o n s i d e r e d The  superheat  rapid of  cooling  t h e melt  at  t h e end  i n the  lowers  the  and e f f e c t i v e l y reduces t h e  distance.  9.3 The E f f e c t of an End C h i l l on t h e L e n g t h of t h e P o r o s i t y Free Region  The region  c o n t r i b u t i o n of an end c h i l l t o  increasing  sound  i n a p l a t e c a s t i n g was a l s o examined by a p p l y i n g a s t e e l  c h i l l a t t h e c a s t i n g end of a 5cm t h i c k p l a t e . 5cm  t h i c k , was chosen s i n c e i t i s r e p o r t e d  that  I s c h i l l s a r e of s u f f i c i e n t t h i c k n e s s  when  the  i t s cross  (Fig 30).  section  The c h i l l  size,  by Myscowski e t a l for plate  2 9  castings  i s t h e same as t h a t o f t h e c a s t i n g s  90  The c a l c u l a t e d p r e s s u r e s as  f o r t h e c a s t i n g w i t h an end  chill  a f u n c t i o n of d i s t a n c e from t h e edge a l o n g t h e c e n t e r l i n e i s  shown i n F i g 39. pressure  Comparing t h i s f i g u r e t o F i g 35,  in this  t h e maximum  case i s s h i f t e d towards t h e r i s e r because of  the c h i l l .  The maximum l e n g t h of t h e p l a t e c a s t sound  and  sound  the  approximately  region  due  to  end  chill  effect  is  3.5s. T h i s shows t h a t t h e end c h i l l i n c r e a s e s t h e  sound r e g i o n by 1s (5cm) which c o i n c i d e s determined  and  i s 5.7s  experimentally.  i n d i c a t i n g t h a t end c h i l l s  with  the value,  -  5cm  This distance i s r e l a t i v e l y small, will  not  signficantly  improve  the  soundness of p l a t e s c a s t i n sand molds.  3.4 Proposed S o l i d i f i c a t i o n Parameters D e f i n i n g t h e T r a n s i t i o n from Porous t o Nonporous C a s t i n g s  As  described  i n the previous section three  solidification  p a r a m e t e r s have been proposed t o d e f i n e when c e n t e r l i n e p o r o s i t y w i l l occur (G) ,  in a casting.  the temperature  3 0  cooling (P^^) . 3 5  rate  (G/vfc)  These  33  These  are the temperature  gradient  g r a d i e n t d i v i d e d by t h e square r o o t o f and  parameters  the gradient were  of  calculated  fraction i n the  solid present  computer program  ( F i g 31) and t h e c r i t i c a l v a l u e s f o r c e n t e r l i n e  shrinkage  obtained  were  corresponding by  the  position  t o t h e boundary f o r c e n t e r l i n e s h r i n k a g e  predicted  the present  model.  as  the values  at  The c r i t i c a l v a l u e s f o r each parameter  a r e l i s t e d i n T a b l e V f o r each p l a t e  thickness.  The  critical  91  values  of  thickness.  both The  G  and  P  a r e observed  proposed v a l u e s a r e not  t o change w i t h p l a t e  applicable  generally.  On the o t h e r hand, the c r i t i c a l v a l u e s f o r (G/^R) a r e v e r y  close  t o the proposed v a l u e , u n i t y , independent of c a s t i n g s i z e i n the range examined. effective  A c c o r d i n g l y , the parameter G/ y/R i s a s i m p l e  parameter  to  predict  roughly  c e n t r e l i n e shrinkage i n a plate c a s t i n g .  the  location  and of  92  B-IV.  CONCLUSIONS  The l o c a t i o n of c e n t r e l i n e p o r o s i t y sand  molds  has  been  examined  i n the plates  s o l i d i f i c a t i o n and Darcy's  the  f l u i d flow.  1)  There  i s good  Law  define  i n t h e l i t e r a t u r e from  which  may be made. agreement  between  t h e c a l c u l a t e d and  o b s e r v e d v a l u e s of c e n t r e l i n e p o r o s i t y . indicates  to  The c a l c u l a t i o n s were compared  with experimental r e s u l t s reported the f o l l o w i n g c o n c l u s i o n s  This  clearly  t h e c e n t r e l i n e s h r i n k a g e can be a t t r i b u t e d  t o i n s u f f i c i e n t i n t e r d e n d r i t i c f l u i d flow t o f e e d volume c o n t r a c t i o n 2)  Darcy's  Law  during  can  be  i n t e r d e n d r i t i c f l u i d flow  The p r e s s u r e r e q u i r e d increases  used  to  increased. not  As  appreciably  a  calculate  the  i n a c a s t i n g even though t h e flow.  t o produce a sound p l a t e  exponentially  the  solidification.  system i s not a t s t e a d y s t a t e d u r i n g 3)  in  u s i n g c a l c u l a t e d v a l u e s of t h e  thermal f i e l d during interdendritic  cast  as  the c a s t i n g  casting  length  is  r e s u l t c a s t i n g under p r e s s u r e w i l l  reduce c e n t r e l i n e  porosity.  93  T a b l e I I I - P h y s i c a l d a t a employed i n c a l c u l a t i o n s  s p e c i f i c h e a t , Cp (cal/g°C)  Steel  Mold  0.20  0.25  d e n s i t y of l i q u i d , p i (g/cm )  7.10  ps (g/cm )  7.50  1 .65  0.074  0.0037  3  d e n s i t y of s o l i d , thermal  3  c o n d u c t i v i t y , k ( c a l / c m . s . °C)  heat t r a n s f e r c o e f f i c i e n t , hs (cal/cm .s.°C)  5.0X10-*  2  l a t e n t heat of s o l i d i f i c a t i o n , solidification liquidus  Hs ( c a l / g )  shrinkage r a t i o , 0  temperature, T l  65.0 0.03  (°C)  1507  solidus  temperature,  Ts (°C)  1463  pouring  temperature,  Tp _(°C)  1595  initial  temperature,  To (°C)  emissivity, e v i s c o s i t y of l i q u i d , M  ambient t e m p e r a t u r e ,  20 0.45  (poise)  Ta (°C)  0.05  20  94  Table IV- Dimenstion of the s t e e l p l a t e c a s t i n g examined l e n g t h (cm)  thickness(cm) 4. 6s, 4. 4S, 4. 4s, 4. 0s, 5. 6s,  s*5.0 2.5 1.25 0.5 5.0 1  4 .8s, 4 .6s, 4 .6s, 4 .4s, 5 .8s,  5.0s, 4.8s, 5.0s, 4.6s, 6.6s,  6 6s, 5 0s, 6 6s, 4 8s, 9 ,0s  9 0s, 6 6s  12.0s  5 8s  Mote: 1. end c h i l l e d by 5cm t h i c k s t e e l  Table v  ~ Comparison of the c r i t i c a l v a l u e s of s o l i d f i c a t i o n parameters f o r c e n t e r l i n e s h r i n k a g e  proposed  obtained v a l u e s f o r each p l a t e t h i c k n e s s  5cm  2.5cm  0.22-0.44"  1.8-2.2  3.6-4.4  Pj(/min°C/cm)  1 .0"  0.92-1.10  0.93-1.08  PJJ(1/cm)  0.25  parameters  G(°C/cm)  value  15  Kote: P . -  1.25cm  6.6-8.0  0.5cm  14.6-19.7  0.83-0.98 0.94-1.07  0.037-0.042 0.072-0.082 0.12-0.14 0.37-0.41  G/fR  95  rvi  MAXIMUM LENGTH CAST SOUND  V  4.5s s  r \V /i  LENGTH GREATER THAN MAXIMUM  2s *  2.5s al  u.  -J  s VARIABLE  rvi  MAXIMUM LENGTH CAST SOUND  Y  rvi Y  •>  LENGTH GREATER THAN MAXIMUM  2s  2s j  I,  '1  1'  -  1  VARIABLE  ...  c S  VARIABLE  F i g u r e 28 - F e e d i n g r e l a t i o n s h i p determined e x p e r i m e n t a l l y i n t h e s t e e l c a s t i n g s ; (a) P l a t e s , and (b) Square b a r s 3 0  3 1  96  Figure  29 - Comparison of measured p e r m e a b i l i t i e s vs volume fraction liquid  *  t>  *  .GREEN SAND MOLD  RISER  CO  *  • •  • • *#  .  a '  . :• SOUND * ; . * SOUND \ *'v CENTERLINE. * •••*•• DUE TO • DUE TO " -SHRINKAGE.'. 1 *. END EFFECT • . ' RISER EFFECT!  vo  1  * ••  •.  1.  CHILL  «  to  f  •la—I*-—  *  .  • •i  p  • n  3s  • ••  •.  • - •• » . •• »  i  4s-12s S = PLATE THICKNESS  F i g u r e 30 - C o n f i g u r a t i o n  X  of t h e system i n v e s t i g a t e d  CO  98  I START) READ INPUT DATA INITIALIZE VARIABLES ^TIMF=TlME+*t|*  CALCULATE TEMPERATURE FIELD USING EXPLICIT FDM  CALCULATA 9u*& AND Vx  I  CALCULATE THE REQUIRED PRESSURE TO FEED SHRINKAGE  YES CALCULATE -SOLIDIF ICATION PARAMETERS  (STOP) F i g u r e 31 - Flow c h a r t of t h e computer program f o r t h e p r e d i c t i o n of c e n t e r l i n e s h r i n k a g e  99  LIQUIDUS  0  5  10 15 20 25 DISTANCE FROM RISER ( CM )  30  F i g u r e 32 - Temperature d i s t r i b u t i o n a l o n g t h e c e n t e r l i n e of t h e p l a t e c a s t i n g  RISER " 30 c  .34 .89 .68  MN SI CASTING SIZE : 2" x 13" 20  -  o o  MPASII&FT) nv  10 •  BISHOP AND PELLINI  0  -r-  10 TIME  F i g u r e 33  •  CALCULATED RESULTS  15  2u"  ( MIN )  S o l i d u s movement a l o n g the c e n t e r l i n e of p l a t e casting  57  101  CASTING SIZE ! 5 X 33 CM  Z7 SOLIDUS  SOUND DUE TO END EFFECT'  7055  SOLID  SOUND DUE TO RISER EFFECT  (b)  (c)  F i g u r e 34 - The d i s t r i b u t i o n of s o l i d i f i c a t i o n c o n t o u r l i n e s a t t h e ( a ) i n i t i a l , (b) m i d d l e and (c) l a s t s t a g e s of solidification  102  F i g u r e 35 - D i s t r i b u t i o n of the p r e s s u r e r e q u i r e d t o feed s h r i n k a g e a t the end of s o l i d i f i c a t i o n (s=5cm)  F i g u r e 36 - D i s t r i b u t i o n of the p r e s s u r e r e q u i r e d t o feed s h r i n k a g e at the end of s o l i d i f i c a t i o n (s=2.5cm)  104  F i g u r e 37 - D i s t r i b u t i o n of the p r e s s u r e r e q u i r e d t o feed s h r i n k a g e a t the end of s o l i d i f i c a t i o n (s=0.5cm)  1 0 5  40-  35 -  °  •  EXPERIMENT BY BISHOP AND PELLINl'  °  •  EXPERIMENT BY JOHNSON AND LOPER  7  31  EXPERIMENTAL BOUNDARY PRESENT WORK  30 -  - 25 •  20 -  15 -  10 -  5 -  I  2  1  1  r  3 4 5 PLATE THICKNESS, s ( CM )  F i g u r e 38 - Soundness of s t e e l  T-  6  plate castings  106  F i g u r e 39 - D i s t r i b u t i o n of the p r e s s u r e r e q u i r e d t o feed s h r i n k a g e a t the end of s o l i d i f i c a t i o n -end c h i l l e d c a s t i n g  107  BIBLIOGRAPHY 1.  E.Scheil; Metallforschung,  2 .  J . K i r k a l d y and W.V.Youdelis; T r a n s . p 8 3 3 .  2  ( 1 9 4 7 ) ,  p 6 9 .  AIME,  2 1 2  ( 1 9 5 8 ) ,  -  3 .  F.Sauerwald; M e t a l l w i r s c h a f t ,  4 .  W.V.Youdelis and D.R.Colton; T r a n s .  22  ( 1 9 4 3 ) ,  p 5 4 3 .  AIME,  2 1 8  ( 1 9 6 0 ) ,  p 8 0 9 .  5.  W.V.Youdelis; "The S o l i d i f i c a t i o n of M e t a l s " ( B r i g h t o n C o n f e r e n c e ) , The I r o n and S t e e l I n s t . London ( 1 9 6 7 ) , p i  6 .  E . P l e z e l and Z . S c h n e i d e r ; Z. p i  7.  1 2 .  21  Metallkunde,  3 5  ( 1 9 4 3 ) ,  .  B.Prabhakar and F.Weinberg; Met. T r a n s .  ASM,  9 B  ( 1 9 7 8 ) ,  P 1 5 0 . 8 .  M.C.Flemings  and G.E.Nereo; T r a n s .  AIME,  2 3 9  ( 1 9 6 7 ) ,  p 1 4 4 9 . 9.  M.C.Flemings, R.Mehrabian and G.E.Nereo; i b i d ,  2 4 2  ( 1 9 6 8 ) ,  p 4 1 . 1 0 .  M.C.Flemings  11.  R.Mehrabian, M.A.Keane and M.C.Flemings; Met T r a n s .  1 2 .  E . S c h e i l ; Z.Metallkunde,  3 4  1 3 .  W.G.Pfann; T r a n s .  1 9 4  1 4 .  H.D.Brody and M.C.Flemings;  1 5 .  T.F.Bower, H.D.Brody and M.C.Flemings;  1  ( 1 9 7 0 ) ,  and G.E.Nereo; i b i d ,  2 4 2  ( 1 9 6 8 ) ,  p 5 0 .  ASM,  p 3 2 3 8  AIME,  ( 1 9 4 2 ) ,  p 7 0 .  ( 1 9 5 2 ) ,  ibid,  p 7 4 7 .  2 3 6  ( 1 9 6 6 ) ,  p 6 l 5 .  ibid,  2 3 6  ( 1 9 6 6 ) ,  p 6 2 4 .  16.  C . J . S m i t h e l l s ; " M e t a l s R e f e r e n c e Book" 4 t h Ed., B u t t e r w o r t h s , London ( 1 9 6 7 ) , p 6 8 5 . .  17.  J . F . E l l i o t and M . G l e i s e r ; "Thermochemistry of S t e e l m a k i n g " , Addison-Wesley Pub. Co. ( i 9 6 0  1 8 .  K.Bornemann and F.Sauerwald; Z . M e t a l l k u n d e ,  ) .  14  ( 1 9 2 2 ) ,  p l O .  1 9 .  K.Bornemann and F.Sauerwald; i b i d ,  14  ( 1 9 2 2 ) ,  p ! 5 4 .  108  20.  H.R.Thresh, A.F.Crawley and D.W.G.White; T r a n s . (1968), p 8 l 9 .  AIME, 242  21.  R.E.Morrone, J.O.Wilkes and R.D.Pehlke; AFS Cast M e t a l s Research J o u r n a l , Dec (1970), p i 8 4 .  22.  R.E.Morrone, J.O.Wilkes and R.D.Pehlke; i b i d , Dec (1970), pl88.  23.  G.Sciama;  24.  G.Sciama; i b i d , Dec (1972), p145  25.  I.Ohnaka, Y.Yashima and T.Fukusako; Imono, 52 (1980), No 1, p l O .  26.  A . J e y a r a j a n and R.D.Pehlke; AFS Trans."  86 ( 1 978), p457.  27.  H.F.Bishop and W . S . P e l l i n i ; AFS T r a n s .  58 (1950), p l 8 5 .  28.  H.F.Bishop, E.T.Myskowski and W . S . P e l l i n i ; i b i d , 59 (1951) , p171.  29.  E.T.Myskowski, H.F.Bishop and W . S . P e l l i n i ; i b i d , 60  i b i d , Mar (1972), p20.  (1952) , p389. 30.  W . S . P e l l i n i ; i b i d , 61 (1953), p 6 l .  31. 32.  S.B.Johnson and C.R.Loper,Jr; i b i d , 77 (1969), p360. E.Niyama, T.Uchida, M.Morioka and S . S a i t o ; AFS I n t e r n a t i o n a l Cast M e t a l s J o u r n a l ; 6 (1981), No.2, p16.  33.  E.Niyama, T.Uchida, M.Morioka and S . S a i t o ; i b i d , 7 (1982), No.3, p52.  34.  K . C h i j i i w a and I.Imafuku; Imono, 55 (1983), No.5, p 2 7 l .  35.  V . L . D a v i e s ; AFS Cast M e t a l s Research J o u r n a l , Jun (1975), p33.  36.  M.C.Flemings; " S o l i d i f i c a t i o n P r o c e s s i n g " , Inc. (1974), p234.  37.  T.S.Piwonka and M.C.Flemings; T r a n s . p1157.  38.  R.Mehrabian, M.Kreane and M.C.Flemings; Met. 1 (1970), p1209.  39.  N . S t r e a t and F.Weinberg; Met. P417.  40.  D . A p e l i a n , M.C.Flemings and R.Mehrabian; Met.  Trans.  McGraw-Hill  AIME, 236 (1966), Trans.  ASM, 7B (1976), Trans.  ASM,  109  ASM, 5 (1974), p2533.  110  APPENDIX A ~ RECALCULATION OF THE SCHEIL-YOUDELIS MODEL PREDICTIONS  The S c h e i l - Y o u d e l i s '  model p r e d i c t i o n s have been  r e c a l c u l a t e d f o r the A l - C u , A l - Z n and Sb-Bi systems and a r e shown i n F i g s 40, 41 and 42, r e s p e c t i v e l y .  C a l c u l a t i o n s from  t h i s i n v e s t i g a t i o n plus c a l c u l a t i o n s published experimental r e s u l t s reported  are included.  and t h e  For the Al-Cu  a l l o y s F i g 40 t h e r e c a l c u l a t e d v a l u e s a r e s i g n i f i c a n t l y than t h e c u r v e p u b l i s h e d  lower  by S c h e i l and K i r k a l d y e t a l , a l t h o u g h 2  the g e n e r a l shape of the c u r v e i s t h e same.  I t i s not c l e a r why  the r e c a l c u l a t e d curve d i f f e r s from t h e p u b l i s h e d  one.  A l - Z n and Sb-Bi a l l o y s t h e r e a r e v e r y l a r g e d i f f e r e n c e s the r e c a l c u l a t e d c u r v e s and t h o s e r e p o r t e d p a r t i c u l a r a t t h e low a l l o y  compositions.  In t h e between  i n the l i t e r a t u r e , i n  111  APPENDIX B ~ DERIVATION OF NODAL EQUATIONS  The p a r t i a l  differential  t r a n s i e n t problem  e q u a t i o n f o r two d i m e n s i o n a l  ( e q u a t i o n (B-30)) and boundary e q u a t i o n s  ( e q u a t i o n s B-34 t o B-36) were s o l v e d e x p l i c i t l y u s i n g f i n i t e d i f f e r e n c e method. model examined.  F i g 43 shows t y p e s o f nodes a p p e a r i n g i n t h e  The nodal d i f f e r e n c e e q u a t i o n s were d e r i v e d by  a p p l y i n g heat b a l a n c e f o r each element as f o l l o w i n g s ; I n t e r i o r nodes (node type 10) T. k  . ,-  T.  .  kL  Ax  + k  +  A y J  T. 1  +  (1  -  .  T. ...) i.J+1  2Fo  T.  Ax  . - T. >j Ay  1  T* . = Fo (T. . ,+ i,J x i,j-l +  T.  -JbJ±i  k  x  -  2Fo  .  i i i  T. A y  J  +  k  *  + Fo  y  . - T. Ay  *p  .  i  iLL ^  T . - T . . ^ At  Ax = AxAypC J  .  ^ l l t l  '  (1)  ( T . . .+ T . . . .) i-l,J i+l,J  )T. . y' i,j  (1)'  Corner node i n t h e mold (node type 1) T* i,j  = 2Fo T . . . . + x i,j+l +  (1  -  2Fo T . . . _ .+ 2(Fo B i + Fo B i )T y i+l,j x x y y' a 2F6  x  -  2Fo  y  -  2Fo  x  Bi x  2Fo  y  B i )T. . y' i,j  (2)  S u r f a c e nodes i n t h e mold (node type 2)  +  (1  -  2Fo  x  -  2Fo  y  -  2Fo  y  Bi  y  )T I,J  (3)  1 12  Boundary node i n t h e mold (node type 4) 1  T (T,  ^Ax  ±AX  ''I-  ^  T, O A y + k  ^ "  1  - T . ^ Ay + k  c  S  '  A X  T c  S  * T. + k  1  +  - T. >l id. Ay  1  c  S  T  . - T.  Ax = AxAyp C  ± > J  p  J  At  2  *  . = T - ~  1,  i,j  k  + k s  F6 T. . - + Fo T. ... + Fo (T, . .+ T, y »J  M  x  x  2k  (  +  M  ~ TTTT S M  1  F  k  ,)  i + 1  o  + k  X  ~ Fo - 2Fo ) T. y X  S u r f a c e boundary nodes i n t h e m e t a l (node type 6) -  (  _  T  1. 1. i , j - 1 yAx yAx •=r— + U  T  \*L+  ±,j 2 ' ;  v  T  *M  +  J  i>j+l"  Ax  T  l , j Ay_  _  (  2  "  U  a  1  S  T  *  2  k  ^d  *M  +  - T "Ay  T  id- & "  ^.1  C  =  *< * ^  -p  - T  At  9  T. . = T—-P- Fo T. , + Fo T. . + 2Fo T._,_. . + 2Fo B i T i,j kg+kj^ x i , j - l x i,j+l y l+l,j y y a A  s ( 1 - V — r f Fo s M 2 k  +  v  k  + k  x  - Fo - 2Fo - 2Fo B i ) T. . y y »J x  y  X  Corner nodes i n t h e m e t a l (node type 9)  T A  i,j  2 k  = k  c  + k  s M  (Fo T  ,+ Fo T  « i . j - i  .) .+ Fo T  J M  y i+i.j  2k  .,,+ Fo T. , .  3  y  x  i - < V 4 ' < V V 1 i.a 1  I +  f  F  T  113  APPENDIX C ~ FORTRAN PROGRAM FOR THE PREDICTION OF CENTRELINE SHRINKAGE C C r  C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C Q  c c  * * * * * * * * * * * * * * * * * * * * * * * * * * > * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *  F I L E NAME : CENTO91  CALC. OF SOLIDUS PRESSURE AT GL=  FEEDING DISTANCE DETERMINATION OF HORIZONTAL PLATE CA MOLD BY NUMERICAL SIMULATION.(APPLICATION OF 1 ~D DARC 2-D UNSTEADY-STATE PROBLEM. HANDLING OF LATENT HEAT = EQUIVALLENT SPECIFIC HEAT M ASSUMPTIONS; 1)CONST.SPECIFIC HEAT,CONDUCTIVITY A DEN 2)LINEAR SOLIDUS A LIQUIDUS 3)NO H.T.RESISTANCE"AT S/M INTERFACE 4)THE MOLD I S INSTANTANEOUSLY FILLED WIT METAL WHICH I S STAGNANT SYMBOLS; DX =GRID DISTANCE DY =GRID DISTANCE DT =TIME STEP H =HEAT TRANSF.COE TA =AMBIENT TEMP. TO =INITIAL TEMP.OF TP =POURING TEMP. N =NO.OF NODES(Y) M =NO.OF NODES(X) OTI=OUTPUT TIME INT OT =OUTPUT TIME CPM=SPECIFIC HEAT 0 CPS=SPECIFIC HEAT OF SAN DM =DENSITY OF META DS =DENSITY OF SAND KM =CONDUCTIVITY OF KS =CONDUCTIVITY OF SAND TL =LIQUIDUS TEMP. TS =SOLIDUS TEMP. HS =LATENT HEAT OF TM =TIME TMS=SOLIDN.START T l TM7=70% SOLIDN.TIME TM9=90% SOLIDN. TIM TMF=SOLIDN.COMPLETE TIME TMLS=LOCAL SOLIDN.T CR =MEAN COOLING RATE GR =TEMP.GRADIENT P =SOLIDN.PARAMETER AM,AS =THERMAL DIFFUSIVITY FMX,FMY=FOURIER NO.OF METAL FSX,FSY=FOURIER NO.OF SAND BSX,BSY=BIOT NO IN SAND DDX,DDY=1/DX**2,1/DY**2 A1,A2 =2*(KS,KM)/(KS+KM) CPME =EQUIVALLENT SPECIFIC HEAT OF METAL T,TB =NEW A OLD TEMP. VSX=SOLIDUS VELOCITY VX =FLUID VELOCITY VXI,VXO=IN- A OUT-FLOW VELOCITY GL,GLB =NEW A OLD VOLUME FRACTION LIQUID TA =ATOM. PRESSURE AK = PERMEABILITY DP =PRESSURE DROP PR =PRESSURE AT SOL B =CONTRACTION RATIO **********************************************************  REAL KM,KS DIMENSION T(100,100),TB(100,100),AM(100,100),FMX(100,100),  1 14  C C C  C  1 CP( 100, 100),TMS( 100, 100),PRR( 100, 100), 2TM98(100,100), GR(100,100),P(100,100), 3FMY(100,100),TMF(100,100),VSX(100,100),VX(100,100), 4GL(100,100),GLB(100,100),DP(100,100),PR(100,100) ' ***** DATA INPUT READ(5,1000) N,LC,DX,DY,DT,OTI 1000 FORMAT(//,2I12,4F12.5) READ(5,1020) CPM,DM,KM,HS,B READ(5,1020) TL,TS,TP,TA,TO 1020 FORMAT(/,5F12.5) READ(5,1030) CPS,DS,KS,H1,E,VIS 1030 FORMAT(/,6F12.5)  C c  C  C  C  C c  C C C C  C  M=25+LC MM5=M-5 MM6=M-6 ***** DETERMINATION OF INITIAL VALUES DO 20 J=1,M DO 10 1=1,N 10 TB(I,J)=T0 20 CONTINUE DO 40 J=6,20 DO 30 1=1,25 GLB(I,J)=1.0 30 TB(I,J)=TP 40 CONTINUE DO 60 J=21,MM5 DO 50 1=21,25 GLB(I,J)=1.0 50 TB(I,J)=TP 60 CONTINUE TM=0.0 OT=OTI JF=MM6 ***** ITERATION START 70 TM=TM+DT/60.0 ***** CHECK OF SPECIFIC HEAT OF METAL CALL CPCHK(MM5,TB,TL,TS,CP,CPM,CPME,HS) ***** CALCULATION OF CONSTANTS DDX=1/(DX**2) DDY=1/(DY**2) A1=2.0*KM/(KS+KM) A2=2.0*KS/(KS+KM) DO 130 J=6,20  115  C  C  C  c C C c  C C C  C C  C  DO '120 1 = 1 ,25 AM(I,J)=KM/(DM*CP(I,J)) FMX(I,J)=AM(I,J)*DT*DDX 120 FMY (I , J ) =AM(I , J ) *DT.*DDY 130 CONTINUE DO 150 J=21,MM5 DO 140 1=21,25 AM(I,J)=KM/(DM*CP(I,J)) FMX(I,J)=AM(I,J)*DT*DDX 140 FMY(I,J)=AM(I,J)*DT*DDY 150 CONTINUE  - .  AS=KS/(DS*CPS) FSX=AS*DT*DDX FSY=AS*DT*DDY BSX=H1*DX/KS BSY=H1*DY/KS PA=1.0l3E+06 G=981.0 ***** CALCULATION OF T'S AT TIME=TM CALL TCALC(N,M,T,TB,TA,FSX,FSY,FMX,FMY,BSX,BSY,E,DY,KM,A1,A ***** CALCULATION OF SOLIDIFICATION TIMES CALL TMSOL(TL,TS,MM5,T,TB,TM,TMS, TM98,TMF) ***** CALC. OF FLUID VELOCITY I F ( J F . L T . 2 1 ) GO TO 235 1=23 DO 165 J=21,JF I F ( T ( I , J ) . G T . T L ) GO TO 155 I F ( T ( I , J ) . L T . T S ) GO TO 160 FS=(TL-T(I,J))/(TL-TS) GL(I,J)=((1.0-FS)/7.1)/(FS/7.5+(1.0-FS)/7.1) GO TO 165 155 GL(I,J)=1.0 GO TO 165 160 GL(I,J)=0.0 165 CONTINUE 185 IF(GL(23,JF).GT.0.02) GO TO 228 DO 190 L=1,30 IF((TM98(23,JF)-TM98(23,JF+L)).LE.0.0) GOTO 190 VSX(23,JF)=DX*L/ ((TM98(23,JF)-TM98(23,JF+L))*60.0) GO TO 195 190 CONTINUE 195 VX(23,JF)=B*VSX(23,JF) I F ( J F . L E . 2 1 ) GO TO 220 DO 200 L=1,50 VX(23,JF-L)=VX(23,JF)*GLB(23,JF)/GLB(23,JF-L)  116  C  I F ( ( J F - L ) . E Q . 2 1 ) GO TO 220 200 CONTINUE  c  220  222 223 C  225  ***** APPLICATION OF DARCY'S LAW DPTX=0.0 1=23 - . I F ( G L ( I , 2 1 ) . L E . 0 . 0 2 ) GO TO 235 DO 225 J=21, JF IF(GLB(I,J).GT.0.3) GO TO 222 AK=6.89E-09*GLB(I,J)**2 GO TO 223 AK=1.0E-05*GLB(I,J)**8 DP(I,J)=VIS*GLB(I,J) *VX(I,J)*DX/AK DPTX=DPTX+DP(I , J ) CONTINUE  PR(23,JF)=DPTX PRR(23,JF)=PR(23,JF)/PA WRITE(6,1032) JF,TM 1 032 FORMAT(//,' (23, ,12,') IS SOLIDIFIED AT',F10.5,' (MIN)',/) WRITE(6,1033) (T(23,J),J=21,MM5) 1033 FORMAT(20F6.0) WRITE(6,1034) 1034 FORMAT(/,'NO',5X,'T',7X,'GL',9X,'Vf',7X,'P.DROP') WRITE(6,1035) 1035 FORMAT(7X,'C,16X,'cm/sec',5X,'dyn/cm2',/) WRITE(6,1036) ( J , T ( 2 3 , J ) , G L B ( 2 3 , J ) , V X ( 2 3 , J ) , D P ( 2 3 , J ) , J = 2 1 , J 1036 FORMAT(12,F2.1,F3.4,F11.5,E12.4) WRITE(6,1040) PRR(23,JF) 1040 FORMAT*/,5X,'PRESSURE REQUIRED =',F11.4,' atm',//) C WRITE(6,1032) JF,TM C1032 FORMAT(//,' (23, ' ,12,') IS SOLIDIFIED AT',F10.5,' (MIN)',/) C WRITE(6,1034) (VX(23,J),J=21,JF) C1034 FORMATO0E10.3,' cm/sec') C WRITE(6,1036) (DP(23,J),J=21,JF) C1036 FORMAT(10E10.3,' dyn/cm2') C WRITE(6, 1038) DPTX,PR(2 3,JF) C1038 FORMAT(/,5X, E12.4,' PRESSURE =',E12.4,' (dyn/cm2)') C JF=JF-1 I F ( J F . L T . 2 1 ) GO TO 235 C IF(GL(23,JF).LE.0.02) GO TO 185 1  C  C C C c  228 1=23 DO 229 J=21,JF 229 G L B ( I , J ) = G L ( I , J ) ***** CALCULATION OF TEMP.GRADIENTS 235 CALL TGR(MM5,T,TB,TS,DX,DY,GR) ***** CHECK OF COMPLETE SOLIDIFICATION IF(T(23,21).GT.TS) GO TO 239  1 17  C c  C c  C  C  ***** SOLIDN. PARAMETERS 1=23 DO 237 J=21,MM6 TMLS =TMF(I , J)-TMS(I , J ) CR =(TL-TS)/(TMLS *"-60..0) 237 P(I,J)=GR(I,J)/SQRT(CR) ***** OUTPUT OF T'S AT COMPLETE SOLIDIFICATION WRITE(6,1050) TM 1050 FORMATC////,'TEMP DISTRIBUTION AT',F10.4,' MIN WRITE(6,1080) ( ( T ( I , J ) , I = 1 6 , N ) , J= 1,M) 1080 FORMAT(15F5.0)  ',//)  WRITE(6,1170) 1170 FORMAT(////,'COMPLETE SOLIDN. TIME : TMF (MIN)',//) WRITE(6,1180) ((TMF(I,J),1=21,25),J=21,MM5) 1180 FORMAT(5F10.3) WRITE(6,1190) FORMAT(////,'NO',6X,'T.GRAD',8X,'P',7X,'V(SOLID.M)',2X, 1'PRESSURE',7X,'PR/PA', 2/,9X,'C/cm',19X,'cm/sec',5X,'dyn/cm',9X,'atm',/) 1=23 WRITE(6,1200) ( J , G R ( I , J ) , P ( I , J ) , V S X ( I , J ) , P R ( I , J ) , P R R ( I , J ) , 1J=21,MM6) 1200 F0RMAT(I2,3F12.4,E12.3,F12.4)  1190  C C C C c  C c  C C C C  C C C  STOP ***** OUTPUT TIME CHECK DURING ITERATION 239 IF(TM.LT.OT) GO TO 240 ***** OUTPUT OF T'S WRITE(6,1245) TM 1245 FORMAT(////,'TEMP. DISTRIBUTION AT',F10.4,'MIN WRITE(6,1246) ((T(I,J),I=16,N),J=1,M) 1246 FORMAT(15F5.0) OT=OT+OTI ***** SUBSTITUTION OF TEMP. A ITERATION 240 DO 260 J=1,M DO 250 1=1,N 250 T B ( I , J ) = T ( I , J ) 260 CONTINUE GO TO 70 END  ',//)  118  C C  c c Q  ************************************  C  . CPCHK(MM5,TB,TL,TS,CP,CPM,CPME,HS) :  SUBROUTINE  C C C  ASSIGN THE EQUIVALLENT SPECIFIC HEAT TO METAL IN THE RAN  Q  *********************************************************  c c  DIMENSION  C C  C C C C  C  C C C C  C C C C C C Q  c C  5 5 10 20 30  35 35 40 50 60  TB(100,100),CP(100,100)  TLS=(TL+TS)/2.0 DO 30 J=6,20 DO 20 1=1,25 IF (TB (I , J).GT.TL.OR.TB(I,J).LT.TS) GO TO 10 I F ( T B ( I , J ) . G T . T L S ) GO TO 5 CP(I,J)=CPM+(TB(I,J)-TS)*(4.0*HS/(TL-TS)**2) GO TO 20 CP(I,J)=CPM+(TL-TB(I,J))*(4.0*HS/(TL-TS)**2) CP(I,J)=CPM+HS/(TL-TS) GO TO 20 CP(I,J)=CPM CONTINUE CONTINUE DO 60 J=21,MM5 DO 50 1=21,25 IF(TB(I,J).GT.TL.OR.TB(I,J).LT.TS) GO TO 40 I F ( T B ( I , J ) . G T . T L S ) GO TO 35 CP(I,J)=CPM+(TB(I,J)-TS)*(4.0*HS/(TL-TS)**2) GO TO 50 CP(I,J)=CPM+(TL-TB(I,J))*(4.0*HS/(TL-TS)**2) CP(I,J)=CPM+HS/(TL-TS) GO TO 50 CP(I,J)=CPM CONTINUE CONTINUE RETURN END  ***********************************************************  SUBROUTINE TCALC(N,M,T,TB,TA,FSX,FSY,FMX,FMY,BSX,BSY,E,DY,K 1A1,A2)  119  C C  CALCULATION NODES' TEMPS. BY EXPLICIT FINITE DIFFERENCE  Q  *************************************  C  c  C C C C  C  C  C  C  C  C  REAL KM DIMENSION MM1=M-1 MM2=M-2 MM3=M-3 MM4=M-4 MM5=M-5 MM6=M-6  T(100,100),TB(100,100),FMX(100,100),FMY(100,100)  ***** TEMPS. IN SAND NODE 1 DO 15 J=2,4 DO 10 1=2,26 10 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * ( T B ( I - 1,J)+TB(I + 1,J)) 1 +(1.0-2.0*FSX-2.0*FSY)*TB(l,J) 15 CONTINUE 1 = 26 J=5 20 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * ( T B ( I - 1,J)+TB(I + 1 , J ) ) 1 +(1.0~2.0*FSX-2.0*FSY)*TB(I,J)  DO 40 1=27,29 DO 30 J=2,MM1 30 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * ( T B ( I - 1,J)+TB(I+1 ,J) ) 1 +(1.0-2.0*FSX~2.0*FSY)*TB(I,J) 40 CONTINUE DO 45 1=2,19 DO 42 J=22,MM1 42 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * ( T B ( I - 1 , J ) + T B ( I + 1 , J ) ) 1 +(1.0-2.0*FSX~2.0*FSY)*TB(I,J) 45 CONTINUE 1 = 20 J=MM4 50 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * ( T B ( I - 1 , J ) + T B ( I + 1 , J ) ) 1 + (1.0-2.0*FSX-2.0*FSY)*TB(I,J) 1 = 26 J=MM4 55 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * ( T B ( I - 1 , J ) + T B ( I + 1 , J ) ) 1 +(1.0-2.0*FSX-2.0*FSY)*TB(I,J)  DO 70 1=20,26 DO 60 J=MM3,MM1 60 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * ( T B ( I - 1 , J ) + T B ( I + 1 , J ) ) 1 +(1.0-2.0*FSX-2.0*FSY)*TB(I,J)  120  C C  C C  C  C C  C C  C C  C C  70 CONTINUE NODE 2 1 = 1 J=1 1 00 T ( l , J ) = F S X * T B ( I , J+1 )+FSY*TB-(l+J , J ) + (FSX*BSX+FSY*BSY) *TA 1 +(1 .0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(l ,.J) NODE 3  1 = 1 DO 110 J = 2,4 110 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB(1+1,J)+FSY*BSY*TA 1 +(1.0-2.0*FSX-FSY~FSY*BSY)*TB(I,J) 1 = 1 DO 115 J=22,MM1 115 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB(1+1,J)+FSY*BSY*TA 1 +(1.0-2.0*FSX-FSY-FSY*BSY)*TB(I,J)  NODE 4 1 = 1 J=5 120 T(I,J)=FSX*TB(I,J-1)+FSY*TB(l+1,J)+A1*FSX*TB(I,J+1)+FSY*BSY 1 +(1.0-FSX-FSY-A1*FSX-FSY*BSY)*TB(I,J) NODE 5 1 = 1 J=21 125 T(I,J)=FSX*TB(I ,J+1 )+FSY*TB(I + 1 , J)+A1 *FSX*TB('l ,J-1 )+FSY*BSY 1 +(1.0-FSX-FSY-A1*FSX-FSY*BSY)*TB(l,J) NODE 6 1 = 1 J=M 128 T(I,J)=FSX*TB(I,J-1)+FSY*TB(l+1,J)+(FSX*BSX+FSY*BSY)*TA 1 +(1.0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(I,J) NODE 7  J=1  DO 130 1=2,29  C C  1 30 T ( I , J ) = F S Y * ( T B ( I - 1 , J ) + T B ( I + 1 ,J))+FSX*TB(I,J+1 )+FSX*BSX*TA 1 +(1.O-FSX-2.0*FSY~FSX*BSX)*TB(I,J) NODE 8  J=5  DO 180 1=2,25  180 T(I,J)=FSY*(TB(I-1,J)+TB(l+1,J))+FSX*TB(I,J-1)+A1*FSX*TB(I, 1 +(1.0-2.0*FSY-FSX-A1*FSX)*TB(l,J) C C  NODE 9  J=21  DO 185 1=2,19  185 T(I,J)=FSY*(TB(I-1,J)+TB(I+1,J))+FSX*TB(I,J+1)+A1*FSX*TB(I, 1 +(1.0-2.0*FSY-FSX-A1*FSX)*TB(l,J)  121  C  J=MM4 DO 187 1=21,25 187 T(I,J)=FSY*(TB(I-1,J)+TB(I+1,J))+FSX*TB(I,J+1)+A1*FSX*TB(I, 1 +(1.0-2.0*FSY-FSX-A1*FSX)*TB(l,J) C C NODE 10 J=M 190 C C  C C  C C  C C  C C  C C  C C C  C C  DO 190 1=2,29  1  T(I,J)=FSY*(TB(I-1,J)+TB(I+1,J))+FSX*TB(I,J-1)+FSX*BSX*TA +(1.0-FSX-2.0*FSY-FSX*BSX)*TB(l,J)  NODE 11 1 = 20 J=21 220 T(I,J)=FSX*TB(I,J+1)+FSY*TB(I-1,J)+A1*FSX*TB(I,J-1) 1 +A1*FSY*TB(I + 1 ,J) + ( 1 .0-FSX-FSY-A1*"(FSX+FSY) )*TB(I ,J) NODE 12  1=20  DO 225 J=22,MM5 225 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * T B ( l - 1 , J ) + A 1 * F S Y * T B ( I + 1 +(1.0-2.0*FSX-FSY-A1*FSY)*TB(l,J) NODE 13  1 = 26 DO 230 J=6,MM5 230 T ( I , J ) = F S X * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F S Y * T B ( l + 1 , J ) + A 1 * F S Y * T B ( I 1 +(1.0-2.0*FSX-FSY-A1*FSY)*TB(I,J)  NODE 14 1 = 30 J=1 240 T(I,J)=FSX*TB(I,J+1)+FSY*TB(I-1,J)+(FSX*BSX+FSY*BSY)*TA 1 +(1,0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(l,J) NODE 15  1 = 30 DO 250 J=2,MM1 250 T(I,J)=FSX*(TB(I,J-1)+TB(l,J+1))+FSY*TB(1-1,J)+FSY*BSY*TA 1 +(1.0-2.0*FSX-FSY-FSY*BSY)*TB(I,J)  NODE 16 1 = 30 J=M 260 T(I,J)=FSX*TB(I,J-1)+FSY*TB(I-1,J)+(FSX*BSX+FSY*BSY)*TA 1 +(1.0-FSX-FSY-FSX*BSX-FSY*BSY)*TB(I,J) ***** TEMPS. IN METAL NODE 17  DO 280 1=2,24  DO 270 J=7,19  122  C  C  C C  C C  C C  C C  C C  270 T ( I , J ) = F M X ( I , J ) * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F M Y ( I , J ) * ( T B ( I - 1 , J ) 1 +TB(I+1,J))+(1.0-2.0*FMX(l,J)-2.0*FMY(I,J))*TB(I,J) 280 CONTINUE 1=21 J=20 - . 290 T ( I , J ) = F M X ( I , J ) * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F M Y ( I , J ) * ( T B ( I - 1 , J ) 1 +TB(I+1,J))+(1.0-2.0*FMX(l,J)-2.0*FMY(I,J))*TB(I,J) DO 310 1=22,24 DO 300 J=20,MM5 300 T ( I , J ) = F M X ( I , J ) * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F M Y ( l , J ) * ( T B ( I - 1 , J ) + T 1 (I+1,J))+(1.0-2.0*FMX(I,J)-2.0*FMY(I,J))*TB(I,J) 310 CONTINUE NODE 18 1= 1 J =6 H=1.37E-12*E*((TB(I,J)+273.0)**4-(TA+273.0)**4)/(TB(I,J)-TA BMY=H*DY/KM 320 T(I,J)=FMX(I,J)*TB(I,J+1)+FMY(I,J)*TB(I+1,J)+A2*FMX(I,J)*TB 1 (I,J-1)+FMY(l,J)*BMY*TA+(1.0-FMX(I,J)-FMY(I,J)-A2*FMX(I 2 -BMY*FMY(I,J))*TB(I,J) NODE 19  1= 1 DO 330 J=7,19 H=1.37E-12*E*((TB(I,J)+273.0)**4-(TA+273.0)**4)/(TB(I,J)-TA BMY=H*DY/KM 330 T ( I , J ) = F M X ( I , J ) * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F M Y ( l , J ) * T B ( I + 1 , J ) 1 +BMY*FMY(I,J)*TA 2 +(1.0-2.0*FMX(I,J)-(1.0+BMY)*FMY(I,J))*TB(I,J)  NODE 20 1= 1 J=20 H=1.37E-12*E*((TB(I,J)+273.0)**4-(TA+273.0)**4)/(TB(I,J)-TA BMY=H*DY/KM 340 T(I,J)=FMX(I,J)*TB(I,J-1)+FMY(I,J)*TB(1+1,J)+A2*FMX(I,J)*TB 1 (I,J+1)+FMY(l,J)*BMY*TA+(1.0-FMX(I,J)-FMY(I,J)-A2*FMX(I 2 -BMY*FMY(I,J))*TB(I,J) NODE 21 J=6 DO 350 1=2,24 350 T(I,J)=FMX(I,J)*TB(I,J+1)+FMY(I,J)*(TB(I-1,J)+TB(I+1,J)) 1 +A2*FMX(I,J)*TB(I,J-1) 2 +(1.0-2.0*FMY(I,J)-(1.0+A2)*FMX(I,J))*TB(I,J) NODE 22 J=20 DO 355 1=2,20 355 T ( I , J ) = F M X ( I , J ) * T B ( I , J - 1 ) + F M Y ( I , J ) * ( T B ( I - 1 , J ) + T B ( I + 1 , J ) ) 1 +A2*FMX(I,J)*TB(I,J+1)  123  C  C C  C C  2  +(1.0-2.0*FMY(l,J)-(1.0+A2)*FMX(l,J))*TB(I,J)  J=MM5 DO 360 1=22,24 360 T ( I , J)=FMX(I ,J)*TB(I , J-1 )+FMY(l , J ) * ( T B ( I - 1 , J ) + T B ( I + 1 , J.) ) 1 +A2*FMX(I,J)*TB(I,J+1)2 +(1.0-2.0*FMY(l,J)-(1.0+A2)*FMX(I,J))*TB(I,J) NODE 23 1=21 DO 370 J=21,MM6 370 T ( I , J ) = F M Y ( I , J ) * T B ( I + 1 , J ) + F M X ( I , J ) * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) 1 +A2*FMY(I,J)*TB(1-1,J) 2 +(1,0-2.0*FMX(l,J)-(1.0+A2)*FMY(I,J))*TB(I,J) NODE 24 1=21 J=MM5 380 T(I,J)=FMX(I,J)*TB(I,J 1)+FMY(I,J)*TB(1+1,J) 1 +A2*FMX(I,J)*TB(I,J+1)+A2*FMY(l,J)*TB(I-1,J) 2 +(1.0-(1.0+A2)*(FMX(l,J)+FMY(I,J)))*TB(I,J) -  C C  C C  C C  C C C  c c Q  c  NODE 25 1=25 J =6 390 T ( I , j ) = F M X ( l , J ) * T B ( I , J + 1 ) + F M Y ( l , J ) * T B ( I - 1 , J ) 1 +A2*FMX(I,J)*TB(I,J-1)+A2*FMY(l,J)*TB(I+1,J) 2 +(1.0-(1.0+A2)*(FMX(I,J)+FMY(I,J)))*TB(I,J) NODE 26  1 = 25 DO 400 J=7,MM6 400 T ( I , J ) = F M X ( I , J ) * ( T B ( I , J - 1 ) + T B ( l , J + 1 ) ) + F M Y ( I , J ) * T B ( I - 1,J) 1 +A2*FMY(I,J)*TB(I+1,J) 2 +(1.0-2.0*FMX(l,J)-(1.0+A2)*FMY(l,J))*TB(I,J)  NODE 27 1=25 J=MM5 410 T ( I , J ) = F M X ( I , J ) * T B ( I , J - 1 ) + F M Y ( I , J ) * T B ( 1 - 1 , J ) 1 +A2*FMX(I,J)*TB(I,J+1)+A2*FMY(l,J)*TB(I+1,J) 2 +(1.0-(1.0+A2)*(FMX(I,J)+FMY(I,J)))*TB(I,J) RETURN END  ****************************************  SUBROUTINE TMSOL(TL,TS,MM5,T,TB,TM,TMS, C C  CALCULATION OF SOLIDN. TIMES  TM98,TMF)  124  C C C C  SYMBOLS  C Q  *************  c c  *'jk  TM7=70% AT T7 TMF=COMPLETE AT TS TM98=98% AT T98  *******************************  DO 60 J=21,MM5 DO 50 1=21,25  C 34 40 50 60  I F ( T ( I , J ) . G T . T L . O R . T B ( I , J ) . L E . T L ) GO TO 34 TMS(I,J)=TM I F ( T ( I , J ) . G T . T 9 8 . O R . T B ( I , J ) . L E . T 9 8 ) GO TO 40 TM98(I,J)=TM IF(T(I,J).GT.TS.OR.TB(I,J).LE.TS) GO TO 50 TMF(I,J)=TM CONTINUE CONTINUE RETURN END  c c Q  ***********************************************************  c C C C C C  ***********  TL T9 T95 T99  T98=TL-0.98*(TL-TS)  C  C C  *  AT AT AT AT  DIMENSION T(100.100),TB(100,100),TMS(100,100), 1 TMF(100,100),TM98(100,100)  C  C  : TMS=START TM9=90% TM95=95% TM99=99%  SUBROUTINE  TGR(MM5,T,TB,TS,DX,DY,GR)  CALC. OF TEMP. GRADIENTS(GR) AT THE END OF SOLIDN. GR IS DEFINED AS THE MAXIMUM POSITIVE VALUE AMONG THE GR FROM THE CENTER NODE TO THE 8 SURROUNDING NODES.  Q  ***********************************************************  C  c C C  DIMENSION  T(100,100),TB(100,100),GR(100,100)  DO 20 J=21,MM5 DO 10 1=22,24 G1 =(T(I-1,J-1)-T(l,J))/SQRT(DX**2+DY**2) IF(T(I,J).GT.TS.OR.TB(I,J).LE.TS) GO TO 10 G2 =(T(I,J-1)-T(l,J))/DX G3 =(T(I+1,J-1)-T(l,J))/SQRT(DX**2+DY**2) G4 =(T(I-1,J)-T(I,J))/DY G5 =(T(I+1,J)-T(I,J))/DY G6 =(T(I-1,J+1)-T(l,J))/SQRT(DX**2+DY**2)  125  G7 =(T(I,J+1)-T(l,J))/DX G8 =(T(I+1,J+1)-T(l,J))/SQRT(DX**2+DY**2) 5 GR(I,J)=AMAX1(G1,G2,G3,G4,G5,G6,G7,G8) 10 CONTINUE 20 CONTINUE RETURN " END  C  O  N  C  E  N  T  R  A  T  I  O  N  • (.WTXCU)  F i g u r e 40 - Comparison o f t h e i n v e r s e s e g r e g a t i o n a t t h e c h i l l face f o r the A l - C u a l l o y s  F i g u r e 41  128  F i g u r e 42 - Comparison of the i n v e r s e s e g r e g a t i o n a t the c h i l l f a c e f o r the Sb-Bi a l l o y s  ©  ©  ©  © RISER  ®  ©  ® F i g u r e 43  © ©  ® GREEN SAND MOLD  CASTING  

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