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The effect of beam oscillation rate on Al evaporation behavior in the electron beam melting process Nakamura, Hideo 1989

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THE EFFECT OF BEAM OSCILLATION RATE ON AL EVAPORATION BEHAVIOR IN THE ELECTRON BEAM MELTING PROCESS  by  HIDEO NAKAMURA M.Eng., Tokyo I n s t i t u t e o f Technology, 1979  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of M e t a l s and M a t e r i a l s E n g i n e e r i n g  We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA March 1989 © Hideo Nakamura, 1989  In  presenting  degree at the  this  thesis  in  University of  partial  fulfilment  of  of  department  this thesis for or  by  his  or  scholarly purposes may be her  representatives.  permission.  of  Metals and""Materials Engineering  The University of British Columbia Vancouver, Canada  Date  DE-6  (2/88)  March 18, 1989  for  an advanced  Library shall make it  agree that permission for extensive  It  publication of this thesis for financial gain shall not  Department  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  Abstract  Electron producing  beam  hearth  melting process i s  s u p e r a l l o y s , T i and  under the h i g h l y reduced  very d i f f i c u l t  control.  In  reducing  of the  studies'  t o c a r r y out an a c c u r a t e  o r d e r t o prevent the excess  chemical  composition  evaporative  t e c h n i q u e i s w e l l known  empirically,  have been made t o d a t e .  r a t e on the e v a p o r a t i o n b e h a v i o r .  of  m e l t i n g f u r n a c e . The  quantitative  effect  Ti-6A1-4.V a l l o y were  temperature  few  loss  scanning  the  effect  study  is,  the  beam  of  melted  in  on the melt s u r f a c e was  an  also  heat  dimensional beam.)  The  and  model  mass t r a n s f e r model was was  model was  developed  used  i n the case  dimensional  developed. of  a  (A  one  stationary  t o i n v e s t i g a t e the e f f e c t of the  ii  The  investigated.  On the b a s i s of the e x p e r i m e n t a l r e s u l t s , a two unsteady  EB  measured  s i t u by an o p t i c a l pyrometer d u r i n g the m e l t i n g p e r i o d .  e v a p o r a t i o n l o s s of both T i and A l was  by  fundamental  purpose of t h i s  oscillation  amounts  the  The  to  Small  clarify  Although  therefore,  in  pressure  o p e r a t i n g p r e s s u r e . T h i s makes  i s employed i n normal o p e r a t i o n .  this  in  disadvantages  the s u p e r h e a t i n g of the molten p o o l , the beam  technique of  However, one  the p r o c e s s i s t h a t a l l o y i n g elements w i t h h i g h vapor  evaporate it  used  i t s a l l o y s because of i t s e x c e l l e n t  metallurgical characteristics. of  widely  beam  o s c i l l a t i o n r a t e on t h e e v a p o r a t i o n the  optimum beam s c a n n i n g r a t e .  evaporative increase rate  at  loss  b e h a v i o r and a l s o t o I t was c l e a r l y shown  more  observed scanning  evaporative  anymore.  technique  I t was a l s o  i s useful  wt%Al,  not  but a l s o  l o s s o f b o t h T i and A l .  iii  the  by t h e  W i t h t h e beam o s c i l l a t i o n  t h a n 1 . 0 Hz, however, t h i s e f f e c t  clearly  concentration,  that  o f b o t h T i and A l c o u l d be suppressed  o f t h e beam o s c i l l a t i o n r a t e .  discuss  only  found  could that  n o t be the  i n controlling  i n suppressing  the  beam Al total  Table o f Contents  Abstract  . . . i i  List  of Figures  v i i  List  of Tables  ix  List  o f Photos  x  List  of P r i n c i p a l  Symbols  xi  Acknowledegment  xiii  Section 1 INTRODUCTION 1-1 G e n e r a l  1  1-2 EB m e l t i n g p r o c e s s  1  1-2-1 D r i p m e l t i n g p r o c e s s  1  1-2-2 Hearth m e l t i n g p r o c e s s  3  1-3 L i t e r a t u r e  review  6  1-3-1 E v a p o r a t i o n phenomena i n vacuum m e t a l l u r g y 1-3-2 P r o c e s s modeling  6  o f EB h e a r t h m e l t i n g  process  15  1-3-3 Moving heat source problem 1- 4- O b j e c t i v e o f t h i s study  16 20  Section 2 EXPERIMENTAL PROCEDURE AND RESULTS 2- 1 E x p e r i m e n t a l apparatus  22  2-2 E x p e r i m e n t a l procedure  30  2-3 R e s u l t s  -41  iv  Section 3 MATHEMATICAL MODEL 3-1  Formulation  64  3-1-1  Basic equation  64  3-1-2  E f f e c t of metal flow  65  3-1-3  Heat i n p u t and heat l o s s  68  3-1-4- Boundary  condition  3-2  Solution  3-3  R e s u l t s and d i s c u s s i o n  71 71 73  3-3-1  E f f e c t o f t h e number o f nodes  73  3-3-2  E f f e c t o f FF and beam o s c i l l a t i o n r a t e  75  3-3-3  Comparison w i t h e x p e r i m e n t a l  3-3-4  Temperature c o n t o u r on m e t a l s u r f a c e  89  3-3-5  T o t a l weight l o s s  91  3-3-6  Decrease i n w t % A l d u r i n g m e l t i n g  93  results  85  3- 4 I m p l i c a t i o n s o f t h i s s t u d y f o r i n d u s t r i a l process  •  99  Section 4 SUMMARY AND RECOMMENDATIONS FOR THE FUTURE WORK 4- 1 Summary  101  4-2  102  Recommendations f o r t h e f u t u r e work  L i s t of References  104  ,  Appendix 1  v  ESTIMATION OF RATE CONTROLLING STEP A-1-1 Complete d i f f u s i o n c o n t r o l  case  A-1-2 E f f e c t o f t h e t e m p e r a t u r e and t h e m e t a l f l o w on t h e r a t e c o n t r o l l i n g s t e p  107  ...110  Appendix 2 PROGRAM LIST AND FLOW CHART A-2-1 Program l i s t o f 1-D model  115  A-2-2 Program l i s t o f 2-D model  121  A-2-3 Flow c h a r t o f 2-D model  129  vi  L i s t of F i g u r e s Figure  Page  1.1  Schematic diagram of EB d r i p m e l t i n g p r o c e s s  2  1.2  Schematic diagram of EB h e a r t h m e l t i n g p r o c e s s  4  1.3  Dual c o o r d i n a t e system f o r moving heat  2.1  s o u r c e s . ( E=x-vt )  18  Schematic diagram of e x p e r i m e n t a l apparatus  24.  2.2 W i r i n g diagram of EH 30/20  25  2.3 Water-cooled copper mold  28  2.4  Schematic diagram of i n - s i t u temperature measurement system  29  2.5  C a l i b r a t i o n curve of the pyrometer,  31  2.6  Schematic drawing of arrangement  IRCON 1100  of charged  material  34-  2.7  Standard heating-up p a t t e r n  35  2.8  Sketch of specimen d u r i n g experiment  36  2.9  T y p i c a l r e s u l t of SEM/EDX a n a l y s i s  38  2.10  R e l a t i o n between wt%AL and I ^ l ^ T i  39  2.11  C o o l i n g curve of CP-Ti  4-2  2.12  Temperature  43  2.13  Changes i n mV output and d u r i n g experiment  2.14  changes w i t h time  V a r i a t i o n s o f AT ,. time  (T  calculated  temperature 48  ,-T. ) of c o o l i n g water w i t h out in b  50 56  2.15  D i s t r i b u t i o n of wt%Al i n the p o o l  2.16  Changes i n t o t a l weight l o s s w i t h time  60  2.17  Changes i n wt%Al w i t h time  61  2.18  E f f e c t of beam o s c i l l a t i o n r a t e on wt%Al a t 5 min and 10 min  62  vii  3.1 Heat balance taken i n c o n t r o l volume  66  3-2 Coordinate system used f o r two dimensional model  67  3-3 Meshes used i n the c a l c u l a t i o n  74-  3.4 Comparison of the number of nodes taken i n the c a l c u l a t i o n  76  3.5 Calculated temperature changes with time i n the case of stationary beam  79  3.6 Temperature d i s t r i b u t i o n at 600 sec with various 80  FF values 3-7 Calculated temperature changes with time (0.1Hz)  82  3.8 Calculated temperature changes with time (1.0Hz)  83  3.9 Calculated temperature changes with time (10.0Hz)  84  3.10 Comparison with experimental r e s u l t s  86  3.11 R e l a t i o n between c a l c u l a t e d and observed temperature changes with time 3-12 Calculated temperature contours on the melt surface a t 600 sec  88 90  3.13 Calculated change i n the t o t a l weight l o s s with time  92  3.14 Calculated t o t a l weight l o s s at 5 min and at 10 min  94  3.15 Calculated changes i n wt%Al with time when i n i t i a l wt%Al i s taken as 6%  95  3.16 Calculated changes i n wt%Al with time when i n i t i a l wt%Al i s taken as 4-5% 3.17 Calculated wt%Al a t 5 min  97 98  A-1.1 Changes i n wt%Al with time c a l c u l a t e d from Machlin's 109  model A-1 usedc o i calculation A-1 .2 .3 Geometry Evaporation n tnr othe l map  viii  111 114  List  of Tables  Table 2.1  Page  Experimental  conditions  32  2.2 Wt%Al a n a l y z e d by e m i s s i o n s p e c t r o s c o p y and I . / I „ . o b t a i n e d by SEM/EDX ....^  40  2.3 A l l r e s u l t s o b t a i n e d i n t h i s experiment  46  3.1  72  T  Constants  used  i n the c a l c u l a t i o n  3.2 Examples of computer i n p u t s and outputs  ix  77  L i s t of Photos Photo  Page  2.1  Appearance of 30 kW EB furnace at UBC  23  2.2  Cross section of samples a f t e r experiment  51  x  L i s t of P r i n c i p a l Symbols  A  m  or cm  surface area  C  mol/cm  concentration  C P  J/kgK  heat capacity  D  cm /sec  D' FF  cm /sec  2  diffusivity  2  ff  modified d i f f u s i v i t y m u l t i p l y i n g f a c t o r of heat considering metal flow  conductivity.  m u l t i p l y i n g f a c t o r of d i f f u s i v i t y considering metal flow  AH  kJ/kg  heat of evaporation  h  cm  depth of molten metal  Erp  W/m K  heat t r a n s f e r c o e f f i c i e n t between mold and metal  j  mol/sec.cm  mass f l u x  k  W/mK  heat c o n d u c t i v i t y  k' W/mK k"Me 1 M kg/mol c m / / s e c  modified heat c o n d u c t i v i t y mass t r a n s f e r c o e f f i c i e n t i n l i q u i d phase molecular weight  m  kg/sec  evaporation rate  P°  Pa  vapor pressure of pure element 2  q'  W/m  B q  2  W/m EB  q  heat l o s s through mold 2  W/m EV  q  t o t a l power input of EB 2  W/m R  q  sum of heat input and output  evaporation heat l o s s 2  W/m  r a d i a t i o n heat l o s s xi  R  J/mol.K  gas c o n s t a n t ,  T  K  temperature  Ta  K  ambient temperature,  v  m/sec  velocity  x a  mole m /sec  y  8.315  of heat source  fraction  heat d i f f u s i v i t y Raoultian a c t i v i t y  •e  298  coefficient  emissivity  0  sec  life  £  m  moving c o o r d i n a t e  p  kg/m  density  a  W/m  2  .K  L.  time of a s m a l l element a t the s u r f a c e  Stefan-Boltzmann c o n s t a n t ,  xii  5-67x10  —8  ACKNOWLEDEGMENT  I would l i k e t o express g r a t i t u d e t o P r o f e s s o r A l e c M i t c h e l l f o r h i s k i n d guidance of  this  study.  contribution fellow  to  graduate  and courteous a d v i c e throughout  I a l s o thank Rudy Cardeno f o r t h e e x p e r i m e n t a l work.  his  I am v e r y  students and f a c u l t y members a t  the  course  significant grateful  to  Department  of  Metals and M a t e r i a l s E n g i n e e r i n g .  I s h a l l never f o r g e t my f a m i l y ' s p a r t i n a c c o m p l i s h i n g study.  My w i f e , H i t o m i , has supported me w i t h l o v e and p a t i e n c e ,  particularly little  i n t y p i n g out the d r a f t o f t h i s t h e s i s .  Also,  my  daughters, Emi and R i s a , have g i v e n i n e s t i m a b l e comfort t o  me d u r i n g t h i s  The  this  study.  financial  support  of  appreciated.  xiii  NKK  Corporation  is  deeply  S e c t i o n 1. INTRODUCTION  1-1  General  In r e c e n t y e a r s , t h e EB p r o c e s s has been w i d e l y adopted f o r 1 ) the p r o d u c t i o n o f s o - c a l l e d advanced metals  . The energy d e n s i t y 2 2  of t h e e l e c t r o n beam i s so h i g h ( i n t h e o r d e r o f 10 elevated  ' temperature  characteristics  can  be  obtained  easily.  of t h i s process i s the h i g h l y  in  t h e m e l t i n g chamber(in  is  required  f o r emission  t h e o r d e r o f 10  kW/cm ) t h a t  reduced  - 10  of the electron  Another pressure  Torr),  beam.  These  m e t a l l u r g i c a l f e a t u r e s a l l o w even r e f r a c t o r y and r e a c t i v e t o be melted r e l a t i v e l y  which two  metals  easily.  Among s e v e r a l v a r i a t i o n s o f t h e EB f u r n a c e t h e d r i p - m e l t i n g mode  and t h e h e a r t h - m e l t i n g mode a r e now t h e two main methods i n  commercial u s e .  1-2  EB m e l t i n g p r o c e s s  1-2-1  Drip melting process  Fig.1-1 process. from  shows  a  schematic  diagram o f t h e d r i p  I n t h i s process a piece of s o l i d  feedstock  the t o p i s s l o w l y lowered i n t o the path o f  1  one  melting supported o r more  EB gun  Feed stock  Water cooled copper mold Ingot  Fig.1-1  Schematic diagram o f EB d r i p m e l t i n g p r o c e s s .  2  electron  beams.  feedstock  and  typical  bottom  f a l l s i n t o t h e w a t e r - c o o l e d mold t o  use o f t h e d r i p m e l t i n g  refining  p o i n t s exceed 1900  because  environment  of which  metallurgical  the  process i s f o r the melting  are  refining  degC. P r e v i o u s workers  high  temperature  and  created i n t h i s reactions  r e a c t i o n s (CO, H^, N^) and  of the  solidify.  o f r e f r a c t o r y metals such as Nb, Ta, Hf,and  melting that  The molten metal d r i p s o f f t h e  can  have  the  process, take  Zr,  and whose  claimed  high a  A  vacuum  number  place.  of  Degassing  e v a p o r a t i o n r e a c t i o n s ( A l , NbO) from 2)  the  Nb  i n g o t a r e some o f these examples  melting  operations are  high-purity 1-2-2  . Successive  o f t e n c a r r i e d out i n o r d e r  multiple  to  produce  metals.  Hearth m e l t i n g process  The  hearth  superalloys, diagram  of  charged  into  Ti this  melting and  process  i t s alloys.  process. S o l i d  i s used Fig.1-2  or  for  shows  particulate  one end of a h e a r t h beneath one o r  producing  a  schematic  feedstock more  electron  beams. The molten m e t a l f l o w s i n t h e h e a r t h by g r a v i t y and i n t o a mold t o s o l i d i f y . M e t a l l u r g i c a l advantages o f t h i s are as f o l l o w s :  1) Vacuum r e f i n i n g 2) Removal o f n o n - m e t a l l i c i n c l u s i o n s 3)  F l e x i b i l i t y o f forms of f e e d s t o c k  3  is  pours process  EB  Fig.1-2  gun  Schematic diagram of EB h e a r t h m e l t i n g p r o c e s s .  The vacuum r e f i n i n g e f f e c t i s b a s i c a l l y the same as that of the d r i p melting process as described p r e v i o u s l y . However, since the  residence time and the r e a c t i o n area are much  larger,  this  e f f e c t i s more enhanced i n the hearth melting process.  Non-metallic when  inclusions  can be removed  very  effectively  the metal i s t r a v e r s i n g the hearth. High density i n c l u s i o n s 3)  (HDI)  such as TaC and WC sink to the bottom  inclusions  while low density  (LDI) such as A^O^ i n superalloys f l o a t up  to the  s u r f a c e ^ . In a d d i t i o n some kinds of i n c l u s i o n s such as TiN i n T i can  d i s s o l v e because the melt i n the hearth has r e l a t i v e l y  superheat.  As  a  r e s u l t u l t r a - c l e a n metal  and a l l o y s  high  can be  produced i n t h i s process. Another  advantage of t h i s process i s that r e t u r n scrap  can  be d i r e c t l y fed i n t o the hearth, u n l i k e i n conventional processes such as VAR.  Since the cost of scrap  i s much lower than that of  raw m a t e r i a l s , the production cost can be reduced d r a s t i c a l l y i n t h i s process.  Although metallurgical has  the  hearth  melting  process  has  both  and economic advantages described above,  disadvantages.  A l l o y i n g elements with high  vapor  the  i t also pressure  evaporate v i g o r o u s l y under h i g h l y reduced pressure. For example, the removal of A l from T i - 6 A 1 - 4 V a l l o y and the removal of Cr and Mn  from  observe  Ni-base superalloys take place. I t i s not unusual a 1-2% y i e l d l o s s a f t e r the evaporation of the  element. In the a c t u a l process, A l i s added a f t e r EB 5  to  primary  processing,  while  Cr i s i n t e n t i o n a l l y e n r i c h e d i n the s t a r t i n g  Another technique used  f o r p r e v e n t i n g the e v a p o r a t i o n l o s s i s the  beam scanning t e c h n i q u e . In t h i s technique scanned on the melt residence method,  time  oscillation 1000  Hz.  described  rate  The  the e l e c t r o n beam  surface at a very high frequency.  of the beam a t one  overheat  feedstock  of used  the melt can be  S i n c e the  short  minimized.  in  this  Typical  beam  i n the a c t u a l p r o c e s s ranges  d i f f i c u l t y i n the  above i s one  spot i s v e r y  precise  is  from  composition  0  to  adjustment  of the p r i n c i p a l problems i n  the  hearth  is  highly  e a s i l y from molten  metals  melting process.  1-3  Literature  1-3-1  review  E v a p o r a t i o n i n vacuum m e t a l l u r g y  In  vacuum  m e t a l l u r g y the e v a p o r a t i o n phenomena  important. V o l a t i l e elements evaporate under  the v e r y low o p e r a t i n g p r e s s u r e . T h i s phenomena  utilized  i n a number of m e t a l l u r g i c a l p r o c e s s e s  distillation,  and  such as  5)  coating processes . Among these 5) _17)  p r o c e s s e s r e l a t i v e l y many s t u d i e s  on  evaporation  metallurgical  1-3-1-1 Theory  6  the  p r o c e s s e s which  In t h i s s e c t i o n  phenomena i n the i n d u c t i o n m e l t i n g  reviewed.  been  refining,  have been done about  e v a p o r a t i o n i n the vacuum i n d u c t i o n m e l t i n g (VIM) are m e c h a n i s t i c a l l y q u i t e s i m i l a r t o EB.  has  studies  process  was  The  removal  by  v a p o r i z a t i o n o f an  element  from  a  melt  i n v o l v e s t h r e e b a s i c s t e p s . These b a s i c steps a r e :  Step  (A): Transport  o f atoms from the bulk o f metal  the metal/vacuum  to  interface.  Step  ( B ) : E v a p o r a t i o n a t the metal/vacuum  Step  (C): Transport  interface.  o f atoms or molecules  i n the gas  phase  away from the i n t e r f a c e .  Usually  the  condensation  s t e p can . be  neglected  in  VIM  processes.  T r a n s p o r t from the b u l k o f metal  t o the i n t e r f a c e  18) Machlin  g i v e s the m a s s - t r a n s f e r  coefficient f o r transport  to the i n t e r f a c e i n i n d u c t i v e l y s t i r r e d melt a s : J = K  L  (Co - Cs)  (1-1 )  where  J = Flux of solute,  given  by Eq.1-2, Cs = C o n c e n t r a t i o n o f s o l u t e a t the  and  = L i q u i d mass t r a n s f e r  Co = C o n c e n t r a t i o n of s o l u t e i n b u l k  K  L  =  ( 4D /  TTt'  coefficient interface,  metal.  )*  (1-2)  where D = D i f f u s i v i t y of s o l u t e and t ' = L i f e time o f the element moving a l o n g the i n t e r f a c e g i v e n by Eq.1-3-  7  t  where  1  = r /  V  (1-3)  r = Radius o f c r u c i b l e and V = Mean s u r f a c e  velocity  of  solute.  The  b a s i c i d e a o f t h i s model i s e x a c t l y the same as H i g b i e ' s 19)  "penetration theory" s t u d i e s and proved  . T h i s model has been used i n a number  t o be v e r y  of  reasonable.  Evaporation r e a c t i o n  The  maximum  r a t e o f e v a p o r a t i o n from the f r e e  surface  is  19) given  by  the Hertz-Knudsen-Langmuir e q u a t i o n  element i n d i l u t e  s o l u t i o n , take the form:  J = K  where  E  Cs  (1-4)  = E v a p o r a t i o n mass t r a n s f e r c o e f f i c i e n t g i v e n by Eq.1-5.  K-, =  where  which, f o r an  a =  solute, Y  =  a  P°y  Condensation  /  P  m  (2TTRTM)^  f a c t o r , P° =  (1-5)  Vapor  Raoultian a c t i v i t y constant, p  M o l e c u l a r weight of s o l u t e ,  m  pressure  of  = Molar d e n s i t y ,  pure M =  R = Gas c o n s t a n t , T = Temperature.  T h i s e q u a t i o n h o l d s t r u e when the p a r t i a l p r e s s u r e o f the  8  dilute  elements i n the gas phase i s n e g l i g i b l y s m a l l .  T r a n s p o r t i n the gas phase away from the i n t e r f a c e  The melt  r a t e o f mass t r a n s p o r t i n the gas phase away  surface  through  this  gas phase  boundary  from  layer  the  can  be  r e p r e s e n t e d by:  J = K  where  / RT (Ps - Pb)  Q  = Gas phase mass t r a n s f e r c o e f f i c i e n t ,  p r e s s u r e a t the i n t e r f a c e , and  ranging  will  be  from  unity  convection  in  unimportant,  K^  0  to  the  Q  Pb = P r e s s u r e  proportional to a fractional  tank.  If  can be expressed  K  where  (1-6)  the  power values  Ps  =  Partial  of gas phase.  of  the  diffusivity  depending  thermal  on  convection  the is  as Eq.1-7.  = K ° / Pb  (1-7)  Q  = Gas phase mass t r a n s f e r c o e f f i c i e n t  characteristics  of the system.  Another form o f  K  Q  i s r e p r e s e n t e d by  = D / 1 K  14.) :  (1-8)  H  where 1 = D i f f u s i o n d i s t a n c e from i n t e r f a c e t o condensation  9  place  and  = Henry's c o n s t a n t .  This  s t e p can be u s u a l l y n e g l e c t e d when o p e r a t i n g  pressure  i s v e r y low as i n t h e EB p r o c e s s .  O v e r - a l l mass t r a n s p o r t c o e f f i c i e n t  Eqs.1-1 , 1-4-»  Combining  and 1-6 and r e a r r a n g i n g , we c a n  o b t a i n o v e r - a l l f l u x of molecules  J = Co / ( 1/K + 1/K + 1/K ) L  E  Q  (1-9)  3  O v e r - a l l mass t r a n s f e r c o e f f i c i e n t , K , i s d e f i n e d as  1/K  S  = 1/K + 1/K + 1/K L  £  Q  (1-10)  The v a p o r i z a t i o n o f most o f elements i s known t o be a f i r s t o r d e r r e a c t i o n when mono atomic v a p o r i z a t i o n i s c o n s i d e r e d .  Then  the e q u a t i o n r e l a t i n g t h e c o n c e n t r a t i o n o f a v a p o r i z a t i o n element is In  where Initial  C^ =  (C - C ) / ( C - C ) = K ( A / V ) t S  f  ±  f  F i n a l l y a t t a i n a b l e concentration of  (1-11)  solute,  C^ =  c o n c e n t r a t i o n o f s o l u t e , A = M e l t s u r f a c e a r e a , and V =  Volume o f m e l t .  1-3-1-2 S t u d i e s on e v a p o r a t i o n k i n e t i c s i n VIM p r o c e s s e s 10  The  p i o n e e r i n g work on the e v a p o r a t i o n r e a c t i o n i n the VIM  6) 7) was  carried  out by Ward  temperature  '  . He s t u d i e d the p r e s s u r e  dependence o f t h e e v a p o r a t i o n c o e f f i c i e n t  and the in  iron-  based melt. H i s r e s u l t s a r e summarized as f o l l o w s :  3 (1) K  i s independent o f the p r e s s u r e o f the tank below  70  um; t h a t i s , the r a t e c o n t r o l l i n g s t e p then,  is  both s t e p (A) and s t e p ( B ) . (2)  At lower  is  temperature the predominant c o n t r o l  step  step (B), whereas a t h i g h e r temperature i t i s s t e p  (A) under the low p r e s s u r e .  His  approach f o r d e t e r m i n i n g  been used i n other r e s e a r c h e r ' s  . Ohno  , et  n  behavior  each  (10  , 8),9),11 ),12) al.  . investigated  - 10  element  Mn>Cu>Sn>S>Cr.  increased  with  follows f i r s t  In  Ohno  e f f e c t on K  S  8) 9) 11) 12) ' ' '  kinetics  of  and the  i n the f o l l o w i n g o r d e r o f  the increase of s i l i c o n  stirring  and  order  the Fe-Si-S  l i n e a r r e l a t i o n s h i p between logK  Metal  i r o n - b a s e a l l o y i n the  atm). They showed t h a t the e v a p o r a t i o n r a t e  e v a p o r a t i o n c o n s t a n t decreases  element:  system)  the  . . evaporation  the  -6  alloying  specific  has  literatures.  o f Mn, Cu, Sn, S, Cr from -5  VIM  the r a t e c o n t r o l l i n g s t e p  alloys,  the  constant  content and t h e r e was  a  and % S i .  was a l s o s t u d i e d by Ward (Fe and  11  Cu  system).  7)  In  ( Fe their  experiments graphite  the s t i r r i n g i n the melt was reduced by i n s e r t i n g  c r u c i b l e between the r e f r a c t o r y and the c o i l . 5  It  a was  c l e a r l y shown that K i n shielded charges i s smaller than that i n unshielded  one. However no q u a n t i t a t i v e d i s c u s s i o n  was c a r r i e d  out. "15)  Harris from  '  copper  empirical  "16) 2 2 )  '  studied the evaporation of B i , As, and Sb  melt.  I n order to c a l c u l a t e  equation  on the basis  he derived an  of Eq.1-2  considering the  temperature dependence on D and V. K  L  =  =C  (8DV/TTr)*  (A/V)*  r * T exp(-C /T) f *  (1-12)  2  where' C^ , C^ = Constant and f = C h a r a c t e r i s t i c frequency of induction  furnace. This equation was derived using  Eq.1-13 and  Eq.1-U.  = D  V=V where  Q  exp(-  E  Q  [T^ (A/V)  D  (1-13)  / RT)  f]*  L  2  (1-U)  = Temperature independent f a c t o r r e l a t i n g D to T, E^ =  Activation  energy  f o r d i f f u s i o n , V^ = Temperature  independent  f a c t o r of V to T and (A/V).  This i s an i n t e r e s t i n g r e s u l t since Eq.1-12 i n d i c a t e s that the melt mass t r a n s f e r c o e f f i c i e n t increases as the melts  12  radius  increases  and  t h a t melt temperature i s a s i g n i f i c a n t  i n d u c t i o n s t i r r e d melt phase mass t r a n s p o r t . u s i n g Ks, Kg and K^,  K  where  a,  b  Q  copper was  to propose an e x p e r i m e n t a l  = Constant  17)  by-  e x p r e s s i o n of  (1-15)  OPR  =  (total  initial  melt  vapor  pressure).  s t u d i e d the e v a p o r a t i o n of B i , Pb,  melt. K^ was  determined  and  in  calculated  = a + b (OPR)  pressure)/(chamber  Ozberk  He  factor  c a l c u l a t e d from Eqs.1-2 and  As, and 1-3>  Sb from  i n which  to have f i x e d v a l u e of 10 cm/s.  K^ was  and K^,  evaporation  V  calculated  g from  K  ,  Kg,  and  concluded  t h a t the  rates  were l a r g e l y c o n t r o l l e d by s t e p ( C ) .  Barnhurst iron.  K^,  Kg,  i n v e s t i g a t e d the e v a p o r a t i o n of Mn K^ v a l u e were c a l c u l a t e d from Eqs.1-2,  r e s p e c t i v e l y . These r e s u l t s were much s m a l l e r then obtained  K  .  from  He  assumed t h i s was  because  1-5»  cast 1-7  experimentally  condensation  factor,  _2 was  l e s s than 1, t h a t i s 10  f u r n a c e used i n h i s  o r d e r . T h i s can be t r u e i n a  large  experiments.  10)13) Irons  '  showed  a  comparison  vaporization  melt,  those p r e d i c t e d on the b a s i s of a n u m e r i c a l  the t u r b u l e n t (Navier-Stokes fluid  flow equation  from an  experimentally  measured and  r a t e s of Mg  between  equation)  stirred  s o l u t i o n of  electromagnetically driven  (Maxwell's e q u a t i o n ) .  13  inductively  He assumed t h a t  0  step  (A)  was  V in  a r a t e c o n t r o l l i n g s t e p i n the p a r t i c u l a r system. 20)  Eq.1-3  was  obtained  c a l c u l a t e d by the r e s u l t s of S z e k e l y ' s r e p o r t  from the experiment was  8.81  x 10  cm/s,  w e l l w i t h t h e o r e t i c a l l y c a l c u l a t e d r e s u l t , 7.1 Oxygen interfere  and the  example,  the  which  x 10  of the e v a p o r a t i o n  effect  e v a p o r a t i o n of l i q u i d  of  the  i r o n and  of  dissolved copper was  the  oxygen  agreed  cm/s.  s u l f u r are known as s u r f a c e a c t i v e rate  .  elements  to  elements. on  For  rates  of  s t u d i e d by Hayakawa  et  21 ) al  .  They showed t h a t the r a t e s of e v a p o r a t i o n of l i q u i d  and  copper  liquid  decreased  i n t e r f a c e . The  f u n c t i o n of [%0] J  F  e  = P  e v a p o r a t i o n r a t e of Fe was  at  expressed  the as  a  as f o l l o w s :  (M/2TrRT)*{l-0.57(K [%0])/(l+K [%0])} ad  (1-16)  ad  of  temperature.  Since much  w i t h i n c r e a s i n g oxygen  K ^ = A d s o r p t i o n c o e f f i c i e n t r e p r e s e n t e d as a f u n c t i o n  where the  remarkably  iron  the a c t i v i t y c o e f f i c i e n t of these elements i n  s m a l l e r than i n i r o n , the e f f e c t of these elements  rates  of  evaporation  in  the T i system  is  considered  Ti on  is the  to  be  correspondingly smaller.  As various  described induction  e s t a b l i s h e d and  above  the  melting  evaporation processes  model is  developed  in  relatively  well  the phenomena can be e x p l a i n e d r e a s o n a b l y .  Since  these t h e o r i e s d e a l w i t h the e v a p o r a t i o n phenomena i n the  U  molten  p o o l system  , b a s i c a l l y they c o u l d be a p p l i e d t o t h e  evaporation  phenomena i n the EB m e l t i n g p r o c e s s .  1-3-2 P r o c e s s modeling  Like that  other  m e t a l l u r g i c a l p r o c e s s e s , i t i s beyond  the development o f an e x c e l l e n t p r o c e s s model  intelligent melting  c o n t r o l of the chemical compositions  process.  process, H  o f EB h e a r t h m e l t i n g p r o c e s s  However,  because o f the  dispute  enables  the  i n t h e EB h e a r t h  complexity  o n l y a few works on t h i s matter have been  of  this  reported  to  ; 23)-30)  date  27) Herbertson melting  examined e v a p o r a t i o n r e a c t i o n s i n the EB h e a r t h  p r o c e s s by r e l a t i n g the e v a p o r a t i o n r a t e  constant,  kg,  c a l c u l a t e d by Langmuir e q u a t i o n t o the o v e r a l l r a t e c o n s t a n t , kg, obtained  from  between small  the a c t u a l o p e r a t i o n . A  kg and k^ was found  well-known  relationship  c l e a r l y i n h i s study, t h a t i s ,  k-g (as i n the example case o f Cr i n s t a i n l e s s  with  steels)  kg  becomes n e a r l y equal t o k^. As a r e s u l t k^ can be p r e d i c t e d by c a l c u l a t i n g kg o n l y . Although a  t h i s a n a l y s i s can be a p p l i e d  rough e s t i m a t i o n of the s o l u t e e v a p o r a t i o n l o s s ,  chemical  c o n t r o l cannot  balance was taken i n t h i s  Since  the  hearth  be made because o n l y a macroscopic  mass  treatment.  i n the a c t u a l  process  is  relatively  from a c o n v e c t i v e v i e w p o i n t , a v a l i d d e s c r i p t i o n  hearth  thermal  regime a l l o w s the e s t i m a t i o n  associated  to  the p r e c i s e  stable  variables  just  w i t h the p r o c e s s . 15  of  Kheshgi  o f the  the  mass-flow  and  Gresho  28)  tried  t o make an u n s t e a d y - s t a t e heat t r a n s f e r model,  but  their  29)  calculated  results  were n o t converged.  Tripp  successfully  d e v e l o p e d a 3-D s t e a d y s t a t e heat t r a n s f e r model i n t h e EB h e a r t h by  assuming t h e h e a r t h has a f i x e d  time-averaged  temperature.  The maximum e v a p o r a t i o n l o s s o f T i and A l were a l s o e s t i m a t e d i n h i s study. However,  since  an  e l e c t r o n beam i s scanned on  the  melt  s u r f a c e r a p i d l y i n a c t u a l p r o c e s s e s , t h e problem o f  an u n s t e a d y -  state  of  heat  scanning account  transfer,  on  i n o t h e r words,  the e f f e c t  t h e l o c a l s u r f a c e t e m p e r a t u r e , must  be  i n i n t e r p r e t a t i o n of the evaporation r e a c t i o n  t h e beam taken in  into this  process.  1-3-3  Moving heat source problem  The supplied  .subject  dealing  with the process  i n which  heat  is  from one o r more moving heat s o u r c e s as i n t h e p r e s e n t  case i s c a l l e d "Moving Heat Source Problem".  Many  studies  have been made t o d a t e  f i e l d of welding metallurgy, such as gas a r c , for were  applied  microstructual affected  i n the  where v a r i o u s k i n d s o f heat s o u r c e s  e l e c t r o n beam,  joining materials.  particularly  plasma gas, and l a s e r a r e used  Both a n a l y t i c a l and n u m e r i c a l t r e a t m e n t s  t o the p r e d i c t i o n of m e t a l l u r g i c a l transformation  reactions  i n t h e weld puddle and  zone.  16  the  and heat  The  exact a n a l y t i c a l t h e o r y was developed by  Rosenthal  in  31 ) the  early  were  194-0's  . The major assumptions  made i n h i s t h e o r y  i ) q u a s i - s t a t i o n a r y heat f l o w , i i ) a p o i n t heat  r e p r e s e n t the weld a r c , i i i ) effects,  n e g l e c t o f c o n v e c t i o n and  a  coordinate  to  radiation  i v ) two d i m e n s i o n a l heat f l o w , v) n e g l e c t of  t r a n s f o r m a t i o n , v i ) temperature Using  source  heat  of  independent m a t e r i a l p r o p e r t i e s .  system  shown  i n Fig.1-3,  the  basic  d i f f e r e n t i a l e q u a t i o n was r e p r e s e n t e d by Eq.1-17.  3 T/85 2  where  2  + 8 T/3y 2  2  2  = - (v/ct) 3T/3?  2  a = heat d i f f u s i v i t y ,  ^ = x - vt  Many  + 3 T/3z  researchers  (1-17)  v = v e l o c i t y o f heat  source  moving c o o r d i n a t e  applied  h i s theory  to  the  temperature  a n a l y s i s i n v a r i o u s w e l d i n g p r o c e s s e s and confirmed t h a t t h i s was an  excellent  though  mathematical  many assumptions  Since  Rosenthal's  model as a f i r s t  were i n c l u d e d  theory  approximation  even  32)  assumes a  point  or  line  heat  source, i t cannot p r o v i d e any i n f o r m a t i o n c o n c e r n i n g the shape o f the  weld p o o l . To improve  t h i s , recent studies presented a  more  g e n e r a l s o l u t i o n o f a t r a v e l i n g d i s t r i b u t e d heat source, assuming  33) the heat source has a G a u s s i a n d i s t r i b u t i o n . traveling  d i s t r i b u t e d heat source problem 17  Eagar  examined a  by s o l v i n g a  modified  z  18  Rosenthal's  e q u a t i o n a n a l y t i c a l l y and showed t h a t the t h e o r e t i c a l  predictions  were i n good agreement w i t h e x p e r i m e n t a l r e s u l t s  on  carbon s t e e l s , T i , and A l .  The r e c e n t p r o g r e s s of the n u m e r i c a l approaches f o r a  governing  equation  makes  i t p o s s i b l e to deal  solving  with  a  more  complex m e t a l l u r g i c a l p r o c e s s .  The the  e f f e c t of the c o n v e c t i o n and the r a d i a t i o n heat l o s s  in  weld p o o l can be t a k e n i n t o c o n s i d e r a t i o n s i m u l t a n e o u s l y  by  using  a  n u m e r i c a l method and more a c c u r a t e  prediction  becomes  34) possible.  For example, Kou  transfer the  a three-dimensional  model i n c l u d i n g t h e e f f e c t o f the c o n v e c t i o n a l f l o w  case  metals  developed  of  to  b o t h the moving l a s e r and the  f i n d out the good agreement of  w i t h observed  moving  arc  experimental  the  EB m e l t i n g p r o c e s s e s ,  between two p r o c e s s e s . the  welded results  t h e r e a r e some major  analysis  differences  F i r s t , the v e l o c i t y of the heat source i n  EB m e l t i n g p r o c e s s i s much h i g h e r than t h a t employed i n  welding processes.  the r a d i a t i o n and e v a p o r a t i o n  cannot be n e g l e c t e d and must be c o n s i d e r e d not o n l y i n  beam impinged  heat the  a r e a but a l s o i n the r e s t of sample s u r f a c e .  Because of the above r e a s o n s , numerical  the  Second, s i n c e the whole m e t a l i s molten s t a t e  i n the EB m e l t i n g p r o c e s s e s , loss  in  ones.  A l t h o u g h the b a s i c i d e a can be a p p l i e d t o the heat in  heat  treatment  developed  in 19  n e i t h e r the a n a l y t i c a l or the the  field  of  the  welding  p r o c e s s e s can be a p p l i e d d i r e c t l y t o t h e p r e s e n t  1-4- O b j e c t i v e o f t h i s  From  an  concerning However,  case.  study  operational  point  of  view,  the  information  t h e optimum beam o s c i l l a t i o n r a t e i s v e r y due  t o t h e l a c k o f fundamental  studies  important.  as  mentioned  above, v e r y l i t t l e i s known on t h e optimum beam o s c i l l a t i o n Consequently typically  m e l t e r s tend t o adopt h i g h e r beam o s c i l l a t i o n r a t e s (  around 100 Hz) w h i c h r e q u i r e an  beam c i r c u l a t i n g  The the  heat  and c o n t r o l l i n g  additional  expensive  system equipment.  o b j e c t i v e of t h i s study, t h e r e f o r e , i s t o i n v e s t i g a t e  effect  behavior  rate.  of  of  transfer  t h e beam o s c i l l a t i o n  alloying  rate  on  the  elements by d e v e l o p i n g an  model, which w i l l a s s i s t  evaporation  unsteady  i n constructing  state a  more  g e n e r a l model o f t h e EB h e a r t h m e l t i n g p r o c e s s i n t h e f u t u r e .  In alloy,  this which  investigated.  study,  A l evaporation  i s used as an aerospace  reactions engine  from  Ti-6A1-4V  component,  were  One o f t h e reasons why t h i s a l l o y was chosen i s  that the a c t i v i t y c o e f f i c i e n t  of A l i n Ti-Al-V,  Y^l' i ^ w n  c  a  very important constant i n the a n a l y s i s of evaporation r e a c t i o n s , 30) was  already  determined  e x p e r i m e n t a l procedure described.  by T a k a g i ' s  and r e s u l t s  study  .  In  section  o f t h e s m a l l EB m e l t i n g  I n s e c t i o n 3» a d i s c u s s i o n o f  tests  will  be  beam  o s c i l l a t i o n r a t e i n t h e EB m e l t i n g p r o c e s s w i l l be made 20  the  2,  optimum by  constructing a mathematical model on the basis of the fundamental r e s u l t s obtained i n section 2.  21  S e c t i o n 2. EXPERIMENTAL  2-1  PROCEDURE AND RESULTS  Experimental apparatus  The  UBC's  30kW EB f u r n a c e shown i n Photo.2-1  was used i n  t h i s s t u d y . Fig.2-1 shows a schematic diagram o f t h e e x p e r i m e n t a l apparatus.  This  apparatus  mainly consists  of the following  parts:  Power s u p p l y system EB gun Beam r o t a t i n g d e v i c e High vacuum  system  W a t e r - c o o l e d mold I n - s i t u temperature measurement system  The  power  ENTERPRISES voltage beam  supply  INC.).  system was s u p p l i e d by NHE  The maximum power o u t p u t  (and t h e c u r r e n t ) o f t h e f i l a m e n t ,  (NORTH  HILL  i s 37.5kw.  The  t h e e x c i t e r and t h e  can be c o n t r o l l e d by a manual adjustment o f d i a l s  on t h e  c o n t r o l l i n g panel.  The  EB gun made by Von Ardenne (EH-30/20) was used i n  this  s t u d y . The nominal power i s 30kW when t h e a c c e l e r a t i n g v o l t a g e i s 20kV.  35) A  wiring  diagram  i s given  22  i n Fig.2-2  . The  heated  Photo 2-1 Appearance of 3 0 kW EB f u r n a c e a t UBC.  23  Mechanical Pump  Power  Diffusion Pump  Supply  [Waveform  Chamber '  n  Flowmeter  To Mold  Thermometer  Fig.2-1  Schematic d i a g r a m o f e x p e r i m e n t a l a p p a r a t u s .  24  Generator]  -cr°v - "\  Anode Solid Cathode Filament Filament  Fig.2-2  I  J  o<  0 - -20KV,  DC DC  -1.5 A  -0.5 A J 0 - - 1 KV, I I o< AC J 0-- 5 V , 0 - -25 A III  Wiring diagram EH 30/20.  25  t u n g s t e n f i l a m e n t g i v e s o f f e l e c t r o n s , which bombard t h e t u n g s t e n solid  cathode. T h i s bombardment causes t h e s o l i d cathode  heated  so  t h a t h i g h d e n s i t y e l e c t r o n beams  Next,  the  electron  electrostatic  a  are  accelerated  &  Y  kinetic  an  the  X  of  in  d e f l e c t e d by magnetic l e n s e s . By c o n t r o l l i n g t h e s t r e n g t h o f f o r both  high l e v e l  generated.  and  field  to  emitted  be  be  energy  magnetic  field  beams  can  to  directions,  various  beam  t r a j e c t o r i e s , o r beam p a t t e r n s can be o b t a i n e d .  Signals  given  t o magnetic l e n s e s were produced by  a  wave  form s y n t h e s i z e r board (WSB-10 made by QUATECH INC.), which  were  i n s t a l l e d i n t h e IBM-PC. T h i s d e v i c e can g i v e output s i g n a l s w i t h desired  frequencies  according t o the user's  own  program.  boards were employed i n t h i s s t u d y ; one r e f e r r e d t o as a  Two  'Master'  board f o r X d i r e c t i o n i n a magnetic l e n s , another r e f e r r e d t o  as  a ' S l a v e ' board f o r Y d i r e c t i o n . A program g e n e r a t i n g a s i n e wave for  a 'Master'  board and a c o s i n e wave f o r a 'Slave'  made f o r each beam o s c i l l a t i o n r a t e (0.1, The s i g n a l s generated  1.0,  10.0  board  and 50.0  was Hz).  were m a g n i f i e d by t h e a m p l i f i e r and sent t o  magnetic l e n s e s . The a m p l i t u d e  o f these s i g n a l s can be c o n t r o l l e d  manually.  Two  independent  vacuum  were  employed  in  this  furnace;  one  chamber.  I n the gun chamber a vacuum environment i s r e q u i r e d f o r  beam  f o r the  systems  generation  environment excessive  in  and  gun chamber and t h e o t h e r f o r t h e  control.  The  maintenance  t h e melt chamber i s a l s o  scattering  of the  beam. 26  It  of  a  melt  vacuum  necessary  to  prevent  i s usual  t o use  two  separate  vacuum systems  chamber  without  evolution consist  sudden p r e s s u r e r i s e  from the melt chamber. of  vacuum  the  t o guarantee t h e h i g h vacuum i n t h e gun  an  Both o f  o i l d i f f u s i o n pump and a  achieved  by  caused  by  t h e vacuum  mechanical  t h i s system i s <1x10  t h e gas  Torr  systems  pump.  The  f o r t h e gun  _5 chamber and <5x10  T o r r f o r t h e melt chamber.  A w a t e r - c o o l e d copper mold was used as a shows  a drawing o f the mold.  approximately and  30 l / m i n .  Fig.2-3  crucible.  Water f l o w r a t e i n t h i s system was  Two thermometers m o n i t o r i n g t h e  inlet  o u t l e t water temperature were used i n o r d e r t o e s t i m a t e the  heat l o s s through t h e mold d u r i n g m e l t i n g .  Fig.Z-U set-up. the  shows  a schematic o f t h e  An IRCON 1100 i n f r a r e d  in-situ  special  temperature  set-up  temperature  o p t i c a l pyrometer was  measurement o f the  was i n s t a l l e d  measurement  molten  t o p r o t e c t the window  metal. glass  from  coated  during  shield  (20 cm  d i a m e t e r and 28 cm h i g h ) was p l a c e d on t h e mold t o  t h e metal vapor on an i n n e r w a l l . A  A  A  being  condense  evaporation of the metal.  used f o r  small  cylindrical  h o l e (6  mm  d i a m e t e r ) was d r i l l e d  i n o r d e r t o a l l o w o b s e r v a t i o n o f t h e molten  metal  the  surface  Furthermore,  from a  pyrometer  copper tube  s e t outside  (2 cm diameter)  was  the  installed  a v o i d g l a s s c o a t i n g . The o b s e r v a t i o n a n g l e and s i t e was to  be  chamber.  a p p r o x i m a t e l y 60 degree and 3 cm from t h e c e n t e r  to  adjusted of the  melt s u r f a c e . The mV output from the pyrometer was r e c o r d e d u s i n g a r e c o r d e r f o r temperature  analysis.  27  Fig.2-3  Water-cooled copper mold. ( u n i t : mm  28  )  29  Temperature Fig.2-5.  can be read o f f the c a l i b r a t i o n curve  The (Output%  of f u l l  scale) i s calculated  shown  using  in  Eq.2-  36)  1  , i f the e m i s s i v i t y of the molten metal  (measured  mV)  x 100 = (0utput% 50 x ( e m i s s i v i t y ) 2-2 E x p e r i m e n t a l procedure  The  of i n t e r e s t i s known,  of  full  scale)  (2-1)  e m i s s i v i t y i n t h i s p a r t i c u l a r f u r n a c e was estimated  the f o l l o w i n g way. emissivity  The e m i s s i v i t y mentioned here means the t o t a l  correction  observation  angle  in  which  and  the  i n c l u d e s any f a c t o r s such window  glass  as  absorption.  the The  rearrangement o f Eq.2-1 g i v e s Eq.2-2.  2 x (measured  mV)  (emissivity) =  (2-2) (Output%  To  of f u l l  scale)  o b t a i n the e m i s s i v i t y o f molten T i the  (1667  degC)  was  (0utput% of f u l l from  Fig.2-5.  allows  used as a 'known'  liquidus  standard  temperature  temperature.  The'  s c a l e ) a t l i q u i d u s temperature can be o b t a i n e d The  measurement of the mv  output  of  the c a l c u l a t i o n of the e m i s s i v i t y of the molten  pyrometer T i . The  c o o l i n g curve was o b t a i n e d by m e l t i n g pure T i ( C P - T i ) b l o c k s u s i n g the temperature measurement set-up d e s c r i b e d b e f o r e .  Table  2-1  shows the e x p e r i m e n t a l  30  c o n d i t i o n s used  in  this  100 90 80 70  60 50 40 30 Ill  _J  < o  20  (/)  _J _l D  Li.  LL  O  ^ D  0. r-  D O  10 9 8 7 6 1  5 4  /  /  /  /  T E • M P =:RA T U R F 1100  CA L U 3 R A 1 H O N  SERIES  ) 1 UVlI M 1 I O i» RAI M G E 2100°C  /  _  r"lr K u r  I R C ON,  INC  00 12 00 1300 14 00 1500 16 00 1700 18 00 1900 20 00 2 00 2200 2300 2400 25 Temp. (°C) Fig.2-5  C a l i b r a t i o n curve o f t h e pyrometer, IRCON 1100,  31  Table 2-1  Experimental  conditions.  EB power  15.5Kw (18KV-0.86A)  Beam travelling radius  1. 5cm  Frequency  0, 0.1, 1, 10Hz  Time  0', 1', 5', 10'  32  study. (1),  Four 60 mm  cut  from  shows and  pieces  x 60 mm  of Ti-6A1-4.V a l l o y  x 15 mm  (1), 10 mm  a l a r g e round i n g o t and  (50 mm  x 4-5  x 15 mm  placed i n  the  the arrangement of the charged m a t e r i a l s . the  gun  chamber  were evacuated  to  x 55  a  mm  mm  x 25  mm  (2) ) were Fig.2-6  mold. Both the  melt  determined  vacuum  pressure.  gas  Since r a p i d power i n p u t causes the v i g o r o u s  e v o l u t i o n of  from  beam  charged m a t e r i a l s which d e t e r i o r a t e s  conditions, a preheating The  standardized  spot  was  period.  Since to  generating  o p e r a t i o n was  necessary.  heating-up p a t t e r n i s shown i n F i g . 2 - 7 .  molten p o o l produced was  of  1.5-2cm and  the other  during  then observed  portion  The an  to  remained  another,  it  was  difficult  A f t e r t h i s premelting  center  position  and  to  keep  the  have  the power was  raised  to  a  unmelted.  melting  p e r i o d the beam was  beam 11kV  the amount of gas i n the charged m a t e r i a l v a r i e d from  constant. the  premelting  moved around on the charged m a t e r i a l The  diameter  and  the  one  pattern  s e t back  to  15-5  as  kW  q u i c k l y as p o s s i b l e .  Beam 50.0  Hz.  oscillation  r a t e s chosen were 0, 0.1,  In the case of the 0 Hz  1.0,  10.0  experiment the beam was  set  and to  the c e n t e r of the charged m a t e r i a l , whereas i n the case of moving beam experiments the beam t r a v e l i n g r a d i u s was cm. they  Although appeared  observed  beam spot s i z e s c o u l d not be to be 1-2mm  i n diameter. The  adjusted  measured molten  to be  1.5  precisely, surface  was  by s e e i n g an image r e f l e c t e d i n a g l a s s s e t i n s i d e  the  vacuum chamber. Fig.2-8 shows a s k e t c h of specimen d u r i n g  33  tests.  Fig.2-6  Schematic drawing of arrangement  34  of charged m a t e r i a l .  2 5  15.5KW  2 0  1 5  1 0  2.2KW  5  0  0 . 8 6 A '  0 . 8  0 . 6  0 . 4  0 . 1 A  0.15A  0 . 2 0  A  0 . 2  0  Beam Moved  2 0  Power Raised  7 K V 1 5  1 8 K V  11KV  9 K V  Experiment  1 0  5  0 1 0 0  2 0 0  3 0 0  4 0 0  5 0 0  Time (sec)  6 0 0  7 0 0  8 0 0  After power  a  determined  h o l d i n g time  ( 1 , 5 and 10  min.) the  was turned o f f . Using a c o o l i n g curve o b t a i n e d  experiment  from  , the e m i s s i v i t y was c a l c u l a t e d f o r a l l charges  each i n the  same manner as was d e s c r i b e d b e f o r e .  All  samples were p o l i s h e d u s i n g a 6um diamond p o l i s h e r  analyzed  by  seconds.  Fig.2-9  deconvolution  SEM/EDX, where X - r a d i a t i o n was  was  shows  a  typical  SEM/EDX  c a r r i e d out t o i n t e g r a t e  and  collected  f o r 200  result.  Gaussian  each  element's  Ka  count. A n a l y s i s was done t h r e e times a t the same p o i n t (6mm below the  surface  a l o n g the c e n t e r l i n e ) and the averaged  value  was  c a l c u l a t e d . To i n v e s t i g a t e A l d i s t r i b u t i o n i n the p o o l , the whole p o o l area was analyzed i n s e v e r a l samples.  S i x standard samples were a n a l y z e d by e m i s s i o n (carried between count  out  by  OREMET Co.).  F i g . 2-10  shows  spectroscopy the  %A1 by e m i s s i o n s p e c t r o s c o p y and the r a t i o , of  relation  I^l^Ti  A l t o Ka count o f T i ) o b t a i n e d by SEM/EDX. As  seen i n t h i s f i g u r e , good c o r r e l a t i o n was found. An  ^  a  can be  experimental  e q u a t i o n was o b t a i n e d by r e g r e s s i o n a n a l y s i s as f o l l o w s ;  wt%Al = 19590.6R R = I  A  -1034-2.1R + 351 .74-R + 0.01879  3  2  1  / I  Table 2-2 shows d e t a i l e d  Metal  which  T  (2-4)  ( 0 < A l < 6.5 % )  i  results.  d e p o s i t e d on the i n n e r w a l l o f the s h i e l d  37  was  00 o 111 w  © © © O.J O..J  •1 :  II  Ii  o  1  -H H : 1L1 W: (•'j Q. • U 'ti : i_ •—i : a.. U :  CO CD  m +-••  i~  © Ii")  o o  ©  U J  CO  cn  OJ  © © ©  in  trj I  -p CO  111 z>  Fig.2-9  T y p i c a l r e s u l t of SEM/EDX a n a l y s i s .  38  R  =  Ul/lTi  {% AO = 195903.6R -10342.1R + 351.74R + 0.01879 3  2  7. 6  / / / /  5 /  s  /  /  /  /  / /  <  i—«H  4  /£m—i  /  /  /  /  S  /  3  /  / /  /  /  2 /  /  /  /  / /  1  /  /  / J  0  I  1  2  L  3  lAl/lTi  Fig.2-10  R e l a t i o n between w t % A l and  39  1^/1^.  X10- 2  Table 2 - 2  Wt%Al a n a l y z e d by e m i s s i o n s p e c t r o s c o p y and o b t a i n e d by SEM/EDX.  Sample  w t % U l / l T i  #  Al  V  27  0.83  5.30  0.00235  23-2  2.87  4.80  0.0118  23-1  3.97  4.54  0.0181  21  4.00  4.51  0.0152  0-2  6.26  4.38  0.0291  30  6.28  4.40  0.0286  4-0  <DATA> 0.00170 0.00319 0.0112 0.00120 0.0203 0.00168 0.0160 0.00163 0.0295 0.00294 0.0291 0.00279  2 8 4 2 3 5 122 173 135 2 8 4 2 8 8  2 2 5 178  a l s o a n a l y z e d t o check the c o m p o s i t i o n o f the evaporant. case  "Standardless  corrections  EDX  v i a MAGIC  analysis  V),  one  technique" of  In t h i s  method  t h e software  (ZAF  routines  i n c o r p o r a t e d i n the EDX was used f o r c a l c u l a t i n g each wt%, Al  concentration  experimental  2-3  was l a r g e r than the upper l i m i t o f t h e above  equation.  Results  Fig.2-11 Ti.  A  shows a t y p i c a l r e s u l t o f a c o o l i n g curve o f  distinct  temperature be  0.116.  horizontal  was o b t a i n e d .  each  £  Fig.2-12 Hz  c o u l d be c a l c u l a t e d .  experiment, ±20  Hz experiments  a r e 10  50.0  Ti  liquidus  was c a l c u l a t e d t o  value  changes w i t h time. In the case o f temperature  fluctuated  within  i n 0.1  and  time c y c l e s . No temperature  Hz and 50.0  Hz experiments,  Hz experiments  b e h a v i o r o f the 10.0  chemical  Hz and  were n o t c a r r i e d o u t .  compositions  41  time  t o be no  r e s u l t s o b t a i n e d i n t h i s study a r e shown i n T a b l e  includes  1.0  cycles  since  s e c ) . S i n c e t h e r e appears  d i f f e r e n c e i n the temperature Hz t e s t s , 50.0  c  A typical  s e c . and 1 s e c . r e s p e c t i v e l y , which a r e i n  a r e v e r y s h o r t (< 0.1  All which  the  agreement w i t h expected  cycles  e  degC. Temperature c y c l e s found  were d e t e c t e d i n the 10.0  large  From t h i s r e s u l t  shows temperature  approximately  good  corresponding t o  0.11-0.13-  was i n the range o f  0  line  pure  S i n c e s i m i l a r c o o l i n g curves were o b t a i n e d a f t e r a l l  experiments,  a  since  o f raw  materials  2-3, and  0 Time  1.78mV  44mV Start  Stop  Time  0 Hz  Start  1.46mV  1.42mV  Stop  Time 0.1 Fig.2-12-a)  Hz  Temperature changes w i t h time  43  -  .. 0 Hz and 0.1  Hz.  2.44mV  4 1.46mV Start  Stop  Time  1 .0 Hz  Start  Stop  Time  10.0 Hz Fig.2-12-b)  Temperature  changes w i t h time  44  1 .-0 Hz and 10.0 Hz.  Start  Time  50.0 Hz  Fig.2-12-c)  Temperature changes w i t h time  45  50.0  C O  C M  CD  O  — 1 co  oo — 1 1 co r—  * eccentric  a a a^ CD CO  —  OQ  C O  C O  1 co  1 r—  1 C D  1  —  C M C O CM  C D C O «—  C D  1  CD  ^  ^  eo  CO  L O  C O  ^J-  C O  C M  CD  C D  C D  1  1  iO  C M  C O  C O  cn ^J.  O O  <D  1 1 cr> to  CO  CO  1  29.4 46.4  CD  30.2  CM  o  18  rt  L O  29.4  mt  r-^  32.4  o  CD  co  O  CO  —  CD  C M  co r-—  C M  —  C D  C O  C O  O  C O  C O  LO  co  r«-  C O  «—  r-» oo  «—  CD  C M  —  29.4  TIN  JCM  — r-~ r-»  C D  29.0  LO  CM CO  29.4  cz>  C O  —  co  29.4  C D C M  L O  29.4  •^r  29.4  C O  29.4  CO  iO  29.4  QCD  CO  LO  29.4  oco  TOUT  *  68  CO  0.97%  C O  3.82%  —  < WATER  -ci-  CD  5.26%  a *s  Ti W/o  (0)-8  DEPTH CENTER EDGE  <t  5.77%  4.77%  after  NOTE  /—\  V W/o SEMI QUANTITATVE ANALYSIS 4  experiment.  28  obtained i n t h i s  3' 1 5 . 5 k w  A l l results  Al W/o  Table 2-3  41.5  26  14.1  C O  57.0  co  C O  30  CO  —  oo  io  —  C D  UTJ  C D  W N  < N CNI  CO  L O  46  547.5 589  fi 0 1 z1 0  L O  C D  C D  CD  C D  O O  CD  oo r*-  CO  O O  f-—  «—•  L O  —  L O  O  C D  cn  CD  C M  eo  CD  1  0.88  U. of  0.90  f-*  CD  «—  C D L O  co r*-  Deposition  O  r-  0.90  n Q7 U. 0/  0.88  0.80 0.85  .n u i1  CO  ir^ cn co  O C D C O  Raw Material  -  CsJ  000 L O  CD  CO C M  746.8 732.7  r-— r—  LO  L O  0.85  co  L O  L O  o  —  598.4 541.4 t*- co  LO  LO  U. 0/  CM  L O  mm  co  713.1 697.4  640"  r-- co  CD LO  L O  0.80  645"  r--  C D  C D  640.2 585.4  591  CM CO CO  LO  0.86  KW  C D C D C O ^ J " C O C O  LO  KV  POWER TIME  CD CD CO  co r^.  co  HZ Mark  C D  co  <  CD  r-1"*—  605  CD CO  650"  693.4 661.0  C D C D  OUH 699.6 653.2  / 00 . L t>-  AAA  624.4 o CD co  Wf Wo  PREHEAT & MELT TIME  762.5  i  WEIGHT (g)  <  15.7  Lu  d e p o s i t e d metal on an i n n e r w a l l of t h e s h i e l d .  Fig.2-13 calculated  shows the mV output o b t a i n e d from the pyrometer and  temperature  temperature,  an  during  melting. In  order  e m i s s i v i t y , which was a l s o shown  t o estimate i n t h e same  f i g u r e , was c a l c u l a t e d u s i n g a r e s u l t of a c o o l i n g curve o b t a i n e d after in  each experiment.  While the mV output s c a t t e r i n g i s  t h e case o f 0 Hz and 10.0  minimum  mV  output  Hz experiments,  a r e shown i n the case o f  shown  t h e maximum and 0.1  and  1.0  Hz  experiments c o r r e s p o n d i n g t o time c y c l e s observed c l e a r l y i n F i g . 2-12. is  The output mV decreases w i t h time; the " d e c r e a s i n g  14-.20%  around  indicates  at  5 min,  20-26%  around  10  at  ratio"  min. T h i s  t h a t c o a t i n g o f t h e g l a s s window c o u l d n o t be  avoided  completely.  Fig.2-14 temperature  i n d i c a t e s changes i n  at  t h e o u t l e t and a t t h e i n l e t ) of  A f t e r 2 min a c o n s t a n t of heat this  water.  AT (= 6 degree C) was o b t a i n e d r e g a r d l e s s steady  state  t r a n s f e r c o n d i t i o n appeared t o be a t t a i n e d a f t e r 2 min  in  experiment.  clearly,  2-2  shows p o o l p r o f i l e s o f samples. As can be  seen  1 min samples have r e l a t i v e l y s h a l l o w p o o l depth,  that  t h e p o o l was n o t f u l l y developed. T h i s agrees w e l l w i t h t h e  data of  AT mentioned  was more than 15 the  cooling  t h e d i f f e r e n c e i n beam o s c i l l a t i o n r a t e s . The  Photo  is,  AT ( = d i f f e r e n c e between the  deepest  above. The p o o l depth o f t h e o t h e r  samples  mm. In some samples, however, the p o s i t i o n  pool  depth  was  47  not  centered  because  of the  0 Hz  0.1Hz  ^ >  2.52  •3.0  £  2.00  2.48  A * 1% 2  1.44 £ = 0.128  1.50  CD  °  1.00  2.1  J = 14% J« = 1 8%  1.46  1 80  1  2.0-  8>  E  ^*°^— ~\ Lj  1  60"  i  2  3  i  4  i  5  1823 1800  1800  1700  1700  1600  1600  6  H-11  H-18 -1900  1900  3.0  J' = 27% J = 10% !  2.0 -  1.76  =  0.133 1.20  1.0  i  0  H-15  H-11  H-18 3.00 -  1.0  0  1  2  3  4  5  6  7  8  9 H-15  19001-  1800 -  1740  1730 E v  10  1700  1—  1645  1  60' Time  -  (min)  1600 -  2  3  4  Time  0 1  5  2  3  4  5 Time  (min)  6  7  8  9  (min)  0.1 Hz .2-13-a)  Changes i n mV output and c a l c u l a t e d temperature d u r i n g experiment 0 Hz and 0.1 Hz.  48  10  1Hz  3.0 >  H-19  2.0 1.0  t  = 0.133  0 60" 1900  0  1900  H-19  ^ 1800 o  ' 2 3 4 5 6 0 1  7 8 9 10  1900  H-22  1800  1800  1700  .1700  1600  1600  1795  2 3 4 5 66  H-16  TTio  1705  I 1700  l-  1600 0 60" Time (min)  1710 1660  0 1 2 3 4 5 6 0 1 2 Time (min)  3  4 5 6 7 ( )  T i m e  8  9 10  min  1 .0 Hz  10Hz  H-20 , 3.0k  3.Oh> J. 2.0  H-13  H-17  3.0  2.0  2.0  1.0  .1.0  £ =  0.128  1.0  0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8  0 65" 1900  H-20  _ 1800 o % 1700 II  1900  H-13  1745  -  1700: 160065" Time (sec)  9 9'20"  1900  H-17  —^  —•  1800  18001-  1600r 0  :1.23 £ = 0.094  1750 17107528  :  1705. 1720  i690 ' 7 0 0 1600 i_  i  i  1  1  1  1  1  0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 Time (min) Time (min)  1 1 —  9 9'20"  10.0 Hz Fig.2-13-b)  Changes i n mV output and c a l c u l a t e d temperature d u r i n g experiment 1.0 Hz and 10.0 Hz.  49  OHz  0.1Hz -X  O  X 18 oil x 15  p  I-  -A 2  4  6  8  J2  10  4I  Time (min) 8  -ft  6  I-  ' 10  Time (min)  K  10Hz o  X • 19 o|2 x 16  2  *—x • 20 o!3 x17  p  0 2* 4.  -I  2  1  1  [_  4  6  8  Fig.2-U  9*  -I2  10  Time (min)  b  8'  1Hz X  4  O  6'  31  41  6I  8I  Time (min)  V a r i a t i o n s of AT (T . - T. ) o f c o o l i n g water w i t h , . out in time.  50  L. 10  ( ZH L'O ) sa-rdures j o uox^oes ggojrj  (q-Z"Z  nospitai Supply Canada Inc.  uxui OL (°  °1 {d 0l  hospitalier American du Canada  111  uxra c, ( q  Canlab  American Hospital Supply Canada Inc  ,  i|2  Materiel hospitalier American du Canada inc III  ,  Ii2  ,  I|3  Photo 2-2-c)  Cross s e c t i o n of samples  53  ( 1.0 Hz ).  Canlab®  Hospital Supply  Maienei' hospitaller  *  beam  c o u l d not be c e n t e r e d a c c u r a t e l y . T h e r e f o r e care was  taken  i n the i n t e r p r e t a t i o n o f these r e s u l t s .  Fig.2-15  indicates  the d i s t r i b u t i o n o f %wtAl i n  o b t a i n e d a f t e r 5 min experiment. In a l l samples  the  %wtAl was  almost  c o n s t a n t throughout i n the p o o l . In Fig.2-15-a) %wtAl on a surface  i s a l s o shown. There i s no l a r g e d i f f e r e n c e  concentration (3.96%). was  on  the  surface  (3.23%) and  These r e s u l t s a l l imply t h a t t h e  mainly  controlled  by  the  interface  that  pool  metal  between in  the  evaporation reaction  Al pool  reaction  under  the  Fig.2-16 shows the change i n the weight l o s s w i t h time.  The  experimental conditions.  weight  l o s s i n c r e a s e s almost l i n e a r l y w i t h time; 4-0-60g  samples was l o s t i n 10 min.  of  The e f f e c t o f beam o s c i l l a t i o n  the rate  i s a l s o observed i n the p r e s e n t experiment.  Fig.2-17 shows wt%Al changes w i t h time. Fig.2-18 shows wt%Al a t 5 min and 10 min. A f t e r 10 min experiments, A l was 1-2.5% large  i n a l l experimental conditions. difference  reduced t o  There appears t o  between the beam o s c i l l a t i o n r a t e  be  no  changes  as  expected.  A f t e r each experiment a r e l a t i v e l y l a r g e amount o f evaporant was d e p o s i t e d on the i n n e r w a l l o f the s h i e l d .  A deposit  after  method  a  0  previously.  Hz experiment was a n a l y z e d by the  described  As seen i n T a b l e 2-3 the c o m p o s i t i o n o f the  55  sampled  deposit  45mm  10,|J0.|X1 J Pool  Center  10mm f r o m C e n t e r  6  5 3.96%  3.96%  +-> 5  O-  3.23%  -O  (#)  3  0  3  6  9  Distance From  0  12 16  3  6  9  12  Distance From  Surface  16 Surface  (mm)  (mm)  20mm f r o m C e n t e r  30mm f r o m C e n t e r 6 -  <  <  4.11%  ON +->  5  -A  0  3  5  A-  6  9  Distance From  3  6 910  Distance From  Surface  Distribution  -x  0  12 14  Surface  (mm)  (mm)  Fig.2-15-a)  4.00%  4->  o f wt%Al i n t h e p o o l  56  ( 0 Hz),  45mm  H-11 (0.1HzPool  20mm from  Centerline  Center  C  5 5  5  3.55%  '0  3  6  9  j  12  3.29%  i  —O—  i_  15 17  0  Distance From Surface (mm)  3  6  9  12 14  Distance From Surface (mm)  40mm from Centerline  5-  "  3.37%  A 3,  i  i  3  6  Distance From Surface (mm)  Fig.2-15-b)  Distribution  o f wt%Al i n t h e p o o l .  57  ( 0.1 H z )  45mm  H-12 (1Hz~5) Pool  10mm from  Centerline  Centerline  < 5  2.38%  6  2.41%  •  r  _i  i_  9  12  6  Distance From Surface (mm)  9  13  Distance From Surface (mm)  20mm from  30mm from  Centerline  Centerline  < t; 4  2.54%  -A. A •0  3  6  2.38%  £  X  910  Distance From Surface (mm)  Fig.2-15-c)  Distribution  Distance From Surface (mm)  o f w t % A l i n the p o o l .  58  ( 1.0  Hz)  20mm  ,  20mm  20mm  H-13 (10Hz~5) Pool  6h  Center  20mm left  from  Center  <  < 5 5  5  3.79%  J  0  _]  I  3  6  L.  9  5 3.81%  0  12 15 18  3  9  12  Distance From Surface (mm)  Distance From Surface (mm)  40mm left  6  from  Center  < ON  3.91% 3.82%  J  0  0  3 5  Distribution  6  L  9  Distance From Surface (mm)  Distance From Surface (mm)  Fig.2-15-d)  I  3  o f wt%Al i n t h e p o o l .  59  ( 10.0 H z )  to o lap '(U  Time  Fig.2-16  (min)  Changes i n t o t a l weight l o s s w i t h time.  60  7 •  OHz  0.1Hz ® 1Hz x 10Hz A  A  3 ®  A  •premelt  0  Fig.2-17  1  J  2  I  L  4  6  Time  (min)  Changes i n wt%Al d u r i n g  61  8  experiment.  J  L_  10  0.1  1.0  10.0  Hz Fig.2-18  E f f e c t of beam o s c i l l a t i o n at 5 min and 10 min.  62  r a t e on wt%Al  was  Al=28%,  evaporate %  Ti=68%,  and  V=4%.  T h i s i n d i c a t e s T i and  h e a v i l y a l o n g w i t h A l i n t h i s experiment.  ( T i + V) to %A1 i s around 2.6.  A l was  also  The r a t i o of  h i g h l y concentrated  the d e p o s i t because of i t s h i g h vapor p r e s s u r e .  63  V  in  Section 3 MATHEMATICAL MODEL  3-1 F o r m u l a t i o n  3-1-1 B a s i c e q u a t i o n  As  described  observed  in  positioned regarded  this  at  i n S e c t i o n 2, the shape o f  the  molten  study was almost i d e n t i c a l when the  a  center,  o f the sample, and  hence  pool  beam  was  i t can  be  as a c y l i n d e r w i t h a p p r o x i m a t e l y 9 cm diameter and  1.5  cm depth.  Since experiment moving flow  the  molten  (the r a t i o  beam problem  pool  was  relatively  shallow 6.7)  of diameter t o depth i s around  can be d e s c r i b e d as a two  model and the s t a t i o n a r y beam problem  i n the  dimensional  as a one  the heat  dimensional  heat f l o w model.  Assuming m a t e r i a l p r o p e r t i e s such as heat c o n d u c t i v i t y , capacity,  and d e n s i t y  are constant,  c y l i n d r i c a l c o o r d i n a t e system  1 3 _  __(  r 3r  3T r  _ _  3r  q'  2  )  +  __  r  a governing equation i n a  can be expressed  1 3 T + _ 2  38  s  64  heat  as f o l l o w s :  1 3T =  k'  _ _ _  a 3t  (  3  _ U  Where k' = Heat c o n d u c t i v i t y o f T i i n c l u d i n g the e f f e c t of molten f l o w (W/mK), 2 q' = Sum o f heat i n p u t and output a  (W/m ),  = Heat d i f f u s i v i t y o f T i ( m / s e c ) , 2  (=k'/C p), p  Cp = Heat c a p a c i t y o f T i (J/kgK), and p  Fig.3-1 into  account  = D e n s i t y o f T i (kg/m ). 3  shows a schematic  diagram o f the heat balance  i n this calculation.  the 2-D model i s i l l u s t r a t e d  3-1-2 E f f e c t o f metal  A c o o r d i n a t e system used f o r  i n Fig.3-2.  flow  In the p r e s e n t experiment, a s l i g h t metal f l o w was during in  melting period.  taken  The  observed  d r i v i n g f o r c e of t h e f l o w  observed  t h i s study can be c o n s i d e r e d t o be a temperature g r a d i e n t  the  metal, which causes two flow t y p e s :  i)  Natural  in  convection  37) flow  and  i i ) Surface t e n s i o n d r i v e n flow  .  These  two  flow  types o r i g i n a t e from t h e change i n p h y s i c a l p r o p e r t i e s , t h a t i s , gravity  and  surface  tension  respectively  according  e x i s t e n c e o f the temperature g r a d i e n t on a metal be  seen i n t h i s The  s u r f a c e , as  the can  case.  molten metal  tends  t o expand w i t h temperature,  the metal f l o w takes p l a c e on a metal "hot s p o t " .  to  s u r f a c e r a d i a l l y from an EB  T h i s phenomenon has been observed  65  so t h a t  i n various casting  Heat Input  Heat Output  EB power, q EB  Radiation, q  0  r;  Evaporation,  <=£> Conduction, q Cond  Fig.3-1  Heat balance taken i n c o n t r o l  66  volume.  9Omm0  g.3-2  Coordinate  system used f o r two  d i m e n s i o n a l model.  processes  and many s t u d i e s have been r e p o r t e d  On  the  surface  from  other the  hand,  the f l u i d w i l l be  drawn  r e g i o n o f the lower s u r f a c e  r e g i o n o f the h i g h e r  surface tension.  tends t o decrease w i t h 'hot  to d a t e .  tension  S i n c e the s u r f a c e  temperature, f l u i d  flow.  the  t o the tension  f l o w outward from an EB  s p o t ' i s a n t i c i p a t e d a g a i n i n the case o f  driven  along  surface  tension  T h i s flow has been observed mainly i n the welding  process.  In  o r d e r t o take account o f the e f f e c t o f the metal f l o w  mentioned  above,  a m u l t i p l y i n g f a c t o r FF was c o n s i d e r e d  as  i n the  mathematical model as f o l l o w s .  k' =  where k r e p r e s e n t s  FF  FF  k  (3-2)  an ' o r d i n a r y ' heat c o n d u c t i v i t y o f T i (W/mK).  i s reported  t o be around 7 i n the case  of  a  natural  QQ \  convection natural  . However no r e p o r t s have been g i v e n y e t when  convection  flow and s u r f a c e t e n s i o n  place.  3-1-3  Heat i n p u t and heat  loss  q' i n Eq.3-2 i s r e p r e s e n t e d  by Eq.3-3-  68  driven  flow  both take  qi = q  Where  q  EB q  q  The reported  EB  _  R q  _  EV  , . (3-3)  q  = T o t a l power i n p u t of EB (W/m  R  = Radiation  EV  heat  loss  (W/m  = E v a p o r a t i o n heat l o s s  (W/m  2  2  2 ), ), and  ).  energy  efficiency  to  around 80%, when beam impinges  be  of the EB power  input  for  normal  Ti is to  the  39) metal  .  The  rest  of  the  input  energy  is  lost  by  b a c k s c a t t e r i n g of e l e c t r o n beams. In the p r e s e n t case, where  the the  beam impinged almost normal, 83 degrees, to the molten T i a l l o y s , the e f f e c t i v e power i n p u t can be c a l c u l a t e d t o be 12 kW  (l5-5kW  x 0.8). R Radiation  heat  loss,  q  2 (W/m  ) was  represented  by  following equation. q  R  e a ( T^ -  =  where  )  a  ( _4) 3  £ = E m i s s i v i t y of T i ^ ° \ 0.4-, O =  Stefan-Boltzmann c o n s t a n t , 5.67x10"  (W/in¥),  8  T^ = Ambient temperature,  E v a p o r a t i o n heat l o s s , q  EV  (W/m  2  298K.  ) was g i v e n as f o l l o w s .  FV q  =  m. T  AH . T  +  m  A1  69  AH  A 1  (3-5)  the  where  m^,^,m^ AH ,  AH  T1  In  this  included  = E v a p o r a t i o n r a t e of T i and A l (kg/m = Heat of e v a p o r a t i o n of T i and  A 1  study  the e v a p o r a t i o n b e h a v i o r  i s important  estimation Since, the  of  Vanadium  to know the r e a c t i o n c o n t r o l s t e p  appeared  t o p l a y a main r o l e  rate  each  of  Langmuir e q u a t i o n as f o l l o w s  m. l  = [y.  1  P.° M. 1  1  element  / (2 M.  be  interest.  reaction study,  expressed  at the  using  (3-6)  l  (-),  (kg/mol),  (-),  and  P° = Vapor p r e s s u r e of pure element  pressure  the  (=8.315J/Kmol),  X = Mole f r a c t i o n  vapor  the  of  for  Al)  M = M o l e c u l a r weight constant  can  in  7T R T ) * ] X.  1  where y = A c t i v i t y c o e f f i c i e n t  The  pressure  19)  ( i = T i and  R = Gas  was  the same.  as d e s c r i b e d i n Appendix 1, the e v a p o r a t i o n  evaporation  and  A l (kJ/kg).  of the e v a p o r a t i o n r a t e of the element  surface  follows  sec)  i n t h a t of T i f o r s i m p l i c i t y because the vapor  of both elements i s almost  It  2  of  pure T i and  41).  70  -Al  (Pa).  is  expressed  as  P  ° = 1.33 "o  p  =  1 > 3 3 x 1 0  o (  x 1  2 3 2 0 0 / T  - ' 0  6 6 l  °g  T + 1 3  - 4 7  )  (-l6380/T-logT 12.32) +  Constants used i n the c a l c u l a t i o n are summarized  i n Table 3-  1  3-1-4- Boundary Heat of  condition  l o s s through a mold,  Eq.3-1,  which g i v e s a boundary  condition  i s g i v e n by the f o l l o w i n g e q u a t i o n  q  where  B  = h  T  A  AT  (3-7)  - Heat t r a n s f e r c o e f f i c i e n t between mold and molten metal (W/m^) A  = I n t e r f a c e a r e a (m )  AT = Temperature  d i f f e r e n c e between mold  and  molten  metal (K), 1690 K (= 1693-283 )•  ~ h  2 T  increase  was  estimated  t o be 1140 W/m K  c o n s i d e r i n g the  i n the temperature d i f f e r e n c e o f c o o l i n g water  degC, f l o w r a t e = 29-4 1/min) measured 3-2  by  ( AT"=8  d u r i n g experiment.  Solution  Eq.3-1  was  solved  by  a  numerical  method.  The  finite  d i f f e r e n c e method ( e x p l i c i t method) was used f o r c a l c u l a t i o n .  71  In  T a b l e 3-1  Constants used i n the c a l c u l a t i o n .  Molten Pool Size  9cm<j>X 1. 5cm h (425g)  a  5.67X10"  e  0.4  JHT,  8 . 88X10  JHAI  1.08X 10  k-n  28.2 (wm/k)  P  4540 (kg/m )  Cp  8  (w/m -k ) 2  (J/kg)  6  7  (J/kg)  3  T i  690 ( J / k g - k )  7Ti  1.0  7AI  0.028  72  4  order  to  cope w i t h the  n o n - l i n e a r term,  q',  an  extrapolation L2)  method was  used i n order to c a l c u l a t e  Denoting the  T(t  Meshes indicates  The First,  p r e v i o u s step by  n + l / 2  )  = 3/2  used  are  T(t )  the  p l a c e d a t one  were c a l c u l a t e d .  taking  account of the  Third,  the  equations until  finite was  n - 1  )  (3-8)  In t h i s  figure  beam i s impinged a t a c e r t a i n  c a l c u l a t i o n was  of the  an a d j u s t e d temperature c a l c u l a t e d loss  T(t  shown i n F i g . 3 - 3 •  scheme used i n the  beam was  temperature  , then,  - 1/2  n  nodes where the  basic  t  an a d j u s t e d  Second,  'Beam' period.  as  follows.  'Beam' p o s i t i o n s .  by Eq.3-8,  Using  r a d i a t i o n and calculated  by  e v a p o r a t i o n weight l o s s of both T i and  Al.  difference  c a r r i e d out.  mass balance was  calculation  The  for  above c a l c u l a t i o n  heat  transfer  was  repeated  c a l c u l a t i o n time reaches beam r e s i d e n c e time a t one  Finally,  the  calculations  beam were  heat  was  moved to the  repeated  next node.  u n t i l a determined  A l l the  node. above  time  step  was  of 2D  model  are  achieved.  Both programs (1D shown i n Appendix  3-3  R e s u l t s and  3-3-1  and  2D)  and  the  2.  discussion  E f f e c t of the  number of nodes  73  flow chart  2D  —  ID  O 'Beam' • r = 2.7cm A r = 3. 6cm  Fig.3-3  Meshes used i n the  74  calculation.  Fig.3-4-  shows the e f f e c t o f the number o f  the d i f f e r e n c e i n the temperature the  temperature  distribution  nodes.  Although  was found w i t h i n a 1 cm r e g i o n ,  was  almost  the  same.  From  economical p o i n t o f view i n computer c a l c u l a t i o n , NR=5 was NS, in  the number o f nodes f o r 6 d i r e c t i o n was determined order  t o meet the s t a b i l i t y  Consequently  condition  in  the  an  used.  t o be  10  calculation.  the t o t a l number of nodes used was 51 i n the case o f  the 2D model and 6 f o r the 1D model.  Examples o f computer outputs a r e shown i n Table 3-2 t o g e t h e r w i t h the i n p u t d a t a .  3-3-2 E f f e c t o f FF and beam o s c i l l a t i o n  Fig.3-5  shows  case a t FF=10.0. time.  While  2750 degC,  t h e temperature the temperature  Fig.3-6 distribution  a c a l c u l a t e d r e s u l t o f the  A steady s t a t e  steep temperature  shows in  at  stationary  beam  can be a c h i e v e d i n a v e r y s h o r t  a t t h e beam hot spot reaches  around  a t 2.7 cm i s as low as 1670 degC.  A  g r a d i e n t i s observed.  the  effect  of  FF  an i n g o t a t 600 s e c .  g i v e s a f l a t t e r temperature temperature  rate  r=0  cm  gradient.  the  temperature  Obviously the  higher  at  r=  1200 degC i n the case o f  150 degC i n t h e case o f FF=100.  the metal f l o w appears t o be v e r y l a r g e .  75  FF  The d i f f e r e n c e between the  (EB hot spot) and t h a t  ( s u r f a c e o f the i n g o t ) i s around w h i l e i t i s around  on  4«5  cm  FF=10,  The e f f e c t o f  I  Distance from center (cm)  Fig.3-4  Comparison of the number of nodes taken i n the calculation.  76  T a b l e 3-2-a)  Examples of computer i n p u t s and outputs and 0.1 Hz.  0 Hz  •DIMENSION OF MOLTEN M E T A L " RA0IUS-O.O45M DEPTH-0.0I5M AREA-0.6362E-02M2  •EB  V0LUME-O.9543E-O4M3  WEIGHT-  0.423KG  CONDITIONS**  BEAM POWER- 12.000KW BEAM AREA-0.6362E-O4M2 FREQUENCY- CONOITIOS"* 0.0 HZ •CALCULATING RNODE : NUMBER5 S I Z E - 0.0090M  TIME S T E P END TIME •  POWER DENSITY-  O.189E*06KW/M2  0.050 SEC 6 0 0 . 0 0 0 SEC  FF20.00 HEAT CONDUCTIVITY OF T I - S64.00WM/K HEAT TRANSFER C O E F F I C I E N T 1140.0O0W/M2/K I N I T I A L AL WEIGHT PERCENT- G.00 •CONTROLLER*• TOTAL S T E P S - 1 2 0 0 0  PRINT FREQUENCY-  400  ••RESULTS-* •INITIAL  TEMPERATURE-1700.OODEGC  •TEMPERATURE 2471.60,  THROUGHOUT IN MELT  D I S T R I B U T I O N IDEGCI AT TIME2041.21,  1903.82,  600.OOO  1825.95.  SEC  1774.29.  1737.71,  •TOTAL T I LOSS-0.4 175E-OMKG) •TOTAL AL LOSS-0. 1353E-OMKG)  I  •FINAL AL CONCENTRATION- 3.22WTV.  i ) 0 Hz •DIMENSION OF MOLTEN METAL** RA0IUS-0.045M 0EPTH-0.015M AREA-O.G362E-02M2 VOLUME-0.9543E-04M3  WEIGHT-  •EB CONDITIONS** EB TRAVELLING RADIUS-0.OISM BEAM POWER- 13. COOKW BEAM AREA-0.101BE-03M2 FREQUENCY0.10HZ •CALCULATING CONOITIOS** RNODE : NUMBER5 THETA NODE: NUMBER10 TIME S T E P END TIME -  0.423KG  POWER DENSITY -  0.118E+06KW/M2  S I Z E - 0.0090M SIZE-0.628RAD  0.050 SEC 6 0 0 . 0 0 0 SEC  FF20.00 HEAT CONDUCTIVITY OF T I - 564.O0WM/K HEAT TRANSFER C O E F F I C I E N T 1140.OOOW/M2/K I N I T I A L AL WEIGHT PERCENT- 6.00 •CONTROLLER*• TOTAL STEPS-120OO  STEPS/NODE-  20  PRINT FREQUENCY-  20  ••RESULTS** •INITIAL  TEMPERATURE-1700.OOOEGC  •TEMPERATURE 1931 1973 1967 1853 1774 1729  . 10 .51 .89 .81 .61 .31  THROUGHOUT IN MELT  D I S T R I B U T I O N (DEGC) AT TIME-  1906 74 1849 97 1784 .76 1735 .59 1699 85  1874 1818 1775 1735 1703  . 15 .49 .72 .92 .97  1866 .31 1820 .28 1784 . 35 1752 . 81 1722. 19  1873 1836 1806 1777 1746  600.000 SEC  .97 .93 .37 . 15 .86  1B91 1861 1834 1805 1774  .85 .98 . 12 .52 .98  •TOTAL TI L O S S - 0 . 2 8 5 0 E - O K K G ) •TOTAL AL LOSS-0. 1 2 9 9 E - 0 K K G ) •FINAL  AL CONCENTRATION-  3.25WTX  i i ) 0.1 Hz  77  1918 1894 1865 1835 1604  .39 .32 .87 .82 .36  1955 40 1939 99 1901 . 40 1864 . 38 1830. S3  2006 2034 1946 1882 1840  .43 26 .54 .36 80  2059.23 2348.28 2019.36 1875.01 1813.09  T a b l e 3-2-b)  Examples o f computer i n p u t s and outputs and 10.0 Hz.  ••DIMENSION OF MOLTEN M E T A L * ' RA0IUS-O.O45M DEPTH=O.015M AREA-0.63S2E-02M2 VOLUME"O.954 3E-04M3  WEIGHT-  ••E8 CONDITIONS" EB TRAVELLING R A D I U S ' O . 0 1 5 M BEAM POWER- I2.0O0KW BEAM A R E A - 0 . 1 0 1 8 E - 0 3 M 2 FREQUENCY. tOOHZ ••CALCULATING CONDITIOS** RNOOE : NUMBER5 THETA NODE: NUMBER10 TIME S T E P END TIME «  1.0 Hz  0.423KG  POWER D E N S I T Y "  O.118E»06KW/M2  S I Z E - 0.0090M SIZE=0.628RA0  . 0 . 0 5 0 SEC 6 0 0 . 0 0 0 SEC  FF• 20.00 HEAT CONDUCTIVITY OF T I 564.00WM/K HEAT TRANSFER C O E F F I C I E N T 1140.0O0W/M2/K I N I T I A L AL WEIGHT PERCENT- 6 . 0 0 ••CONTROLLER** TOTAL S T E P S - 1 2 0 0 0  STEPS/NODE-  2  PRINT  FREQUENCY =  20  •RESULTS** •INITIAL  TEMPERATURE - 1700. OOOEGC  •TEMPERATURE D I S T R I B U T I O N 1950.23 1924.40 1903.79 1857. 14 1818.61  •TOTAL •TOTAL  1926 51 1899 78 1861 62 1820 63 1785 03  1933 1907 1866 1822 1785  THROUGHOUT IN  (OEGC) AT T I M E -  194 1 1916 187 1 1823 1785  63 71 49 35 61  87 87 63 67 67  MELT  S 0 0 . 0 0 0 SEC  1950 90 1927 61 1876 99 1824 49 1785 18  19E0 1940 1882 1824 1784  03 74 42 62 16  1968 1957 1887 1823 1782  62 64 44 82 82  1977 1981 1890 1821 1781  30 54 72 87 59  1974 2013 1889 1818 1781  1 1 88 73 73 28  196 1 21 19 1868 1816 1782  39 27 67 56 59  TI L O S S - 0 . 1 8 6 2 E - 0 K K G ) AL L O S S - 0 . 1 2 2 0 E - O H K G )  • F I N A L AL CONCENTRATION- 3 36WT%  i ) 1.0 Hz - • D I M E N S I O N OF MOLTEN M E T A L " RA0IUS-0.045M D E P T H - 0 015M AREA-0.6362E-02M2 VOLUME-0.9S43E-04M3  WEIGHT-  ••EB CONDITIONS" EB TRAVELLING R A O I U S - 0 015M BEAM POWER- 12.000KW BEAM A R E A - O . 1 0 I 8 E - 0 3 M 2 FREQUENCY* 10.00HZ "CALCULATING CONDITIOS" ANODE : NUMBER5 THETA NODE:  NUMBER-  10  SIZE-  0.423KG  !  POWER D E N S I T Y -  O.118E*06KW/M2  1  0.0090M  SIZE-O.628RAD  |  TIME S T E P 0 . 0 1 0 SEC ENO TIME • 6 0 O . 0 0 0 SEC FF20.OO HEAT CONDUCTIVITY OF T I 564.00WM/K HEAT TRANSFER C O E F F I C I E N T 1140.0O0W/M2/K I N I T I A L AL WEIGHT PERCENT- 6 . 0 0  I  ; ' !  "CONTROLLER" TOTAL S T E P S - 6 0 0 0 0  STEPS/NOOE-  1  PRINT  FREQUENCY- 2COO  "RESULTS" •INITIAL  TEMPERATURE-1700.OOOEGC  •TEMPERATURE DISTRIBUTION 1952.77 1954 05 1948 64 1877 45 1823 80 1785 99  1954 1950 1877 1823 1785  34 76 61 80 99  1954 1952 1877 1823 1785  57 97 74 80 99  THROUGHOUT IN MELT  (DEGC) AT T I M E 1954 1955 1877 1823 1785  73 28 82 80 99  1954 1957 1877 1823 1785  6 0 0 . 0 0 0 SEC 81 70 87 79 99  1954 I960 1877 1823 1785  80 23 86 79 99  •TOTAL TI L O S S - 0 . I 6 9 6 E - 0 K K G ) • T O T A L AL L O S S - 0 . 1 1 9 6 E - 0 K K G ) •FINAL  AL CONCENTRATION* 3.40WTX  ii)  10.0 Hz  78  1954 1962 1B77 1823 I78S  71 88 80 78 99  1954 1965 1877 1823 1785  51 66 69 78 99  1954 1968 1877 1823 1785  21 58 52 79 99  1953 1972 1877 1823 1785  71 06 26 79 99  12kW,FF=10.0 OHZ  2900 2700  Po-  Center  r = 0 cm  2500 O  2300  o  ri. E  K  r = 0.9cm •  -»—•-  2100 1900  r — 2. 7cm -*-X  1700 k . 1500 0  X-  100  200  300  400  X  X  500  600  Time (sec)  Fig.3-5  C a l c u l a t e d temperature changes w i t h time i n the case of s t a t i o n a r y beam.  79  80  Fig.3-7  shows  the temperature changes  beam impinged p o s i t i o n , r=2.7 the  beam  at three  points,  cm and r=3.6 cm r e s p e c t i v e l y ,  o s c i l l a t i o n r a t e i s 0.1 Hz.  As can be  seen  a  when  in  this  f i g u r e , a p e r i o d i c a l temperature change c o r r e s p o n d i n g t o the beam oscillation result  rate  also  stationary  clearly  movement  with  the  increase  of  FF.  This  unlike condition  . S i m i l a r t o the 0 Hz case the temperature  flatter  difference of FF.  shows t h a t the beam  obtained.  beam case g i v e s a l o c a l u n s t e a d y - s t a t e  each p o s i t i o n becomes  (time c y c l e = 10 second) was  The  the to  gradient  temperature  among t h r e e p o i n t s becomes s m a l l e r w i t h the  increase  E s p e c i a l l y i n the case o f FF=100, t h e r e i s o n l y a  slight  temperature g r a d i e n t i n t h e melt.  Fig.3-8 rate  is  1.0  shows c a l c u l a t e d r e s u l t s when the beam Hz.  A shaded a r e a i n t h i s  figure  oscillation  indicates  the  temperature range; an upper l i n e connects a l l maximum temperature points  o b t a i n e d i n one time c y c l e and a lower l i n e connects a l l  minimum  temperature p o i n t s .  More d e t a i l e d  temperature  changes  from 50 sec t o 51 sec a r e a l s o shown i n the same f i g u r e , i n which maximum  and  Similar  to  minimum the  points are  depicted  r e s u l t s on 0 Hz and 1.0  f o r each Hz  position.  calculations,  the  e f f e c t of FF i s v e r y l a r g e .  Fig.3-9 rate •to  shows c a l c u l a t e d r e s u l t s when the beam  i s 10.0 Hz. More d e t a i l e d temperature changes  oscillation from 9-0  9-1 sec a r e shown i n the same f i g u r e . The d i f f e r e n c e  the maximum and minimum temperature i s v e r y s m a l l Similar  to  three  other  cases, 81  the  effect  of  sec  between  even a t FF=10. FF  i s again  2900 12Kw 0.1Hz F F = 10.0 ° 'Beam' • r = 2.7cm «r = 3.6cm]  2700  2700 12kW  p  2500  2500  2300  2300  o'Beam'  0.1Hz  FF=20.0  •r=2.7cm  * r = 3.6cm  V 1500,  0  10  20  30  40  50  60  0  '0  20  Time (sec)  30 Time (sec)  FF = 10  FF = 20  2700 12kW 0.1Hz FF=100.0 ° Beam » r = 2 . 7 c m *r=3.6cm  2500  2300  O £  2100  ID  H 1900  1700  1500  0  10  20  30  40  50  60  Time (sec)  FF = 100  Fig.3-7  C a l c u l a t e d temperature changes w i t h time(0.1 Hz)  82  40  50  60  2700 12Kw  1Hz F F = 10.0  o'Beam'  « r = 2.7cm  x r = 3.6cm  FF = 10  Time (sec)  Time (sec)  2700 12kW 2500  1 Hz  o 'Beam'  FF=20.0  • r = 2.7cm x r = 3.6cm  2300  Max  1500  _i  10  20  Min  Min  i_  30  40  50  60  Max  15001—L 50  51  Time (sec)  FF = 20  Time (sec)  2700 12Kw 2500  d.  1Hz F F = 100.0  o 'Beam'  * r = 2.7cm  *r=3.6cm  2300  2300  2100  2100  E  Max(O) 1900  1900  1700  1700  1500 0  10  20  — 301  L . 40  50  60  Time (sec)  f Min(») \ Min(X)  Max(X)  1500'—^ 50  51 Time (sec)  FF = 100 i  Fig.3-8  C a l c u l a t e d temperature changes w i t h time.(l.OHz)  83  2700 !2Kw 2300  10Hz F F =  o'Beam'  10.0  « r = 2.7cm  * r  = 3.6cm  a E : 4) •  0  10  20  30 Time  40  50  60  (sec)  FF = 10 2700 12kW 2500  10Hz  o'Beam'  FF=20.0  «>r = 2 . 7 c m x r = 3 . 6 c m  2300  p  E H<D  20  30  40  9.0  9.1 T i m e (sec)  T i m e (sec)  FF = 20  2700 12Kw 2500  a E  10Hz F F =  o'Beam'  « r =  100.0  2.7cm  X r = 3.6cm  2300  2300  2100  2100  <D  r-  1900  M  ^ ^*  X(0)  )  Min(») j"'°) M  1700  1500  J 10  L. 20  30 Time  Fig.3-9  40  50  60  (sec)  1500  9.0  9.1 Time  (sec)  FF = 100  C a l c u l a t e d temperature changes w i t h time(10.0Hz)  8U  rather  l a r g e i n t h i s case.  t h r e e p o i n t s i s reduced  3-3-3  When  FF=100, the d i f f e r e n c e  t o be w i t h i n as s m a l l as around 60 degC.  Comparison w i t h e x p e r i m e n t a l  In o r d e r to determine  results  the FF i n t h i s experiment,  v a l u e s were compared w i t h e x p e r i m e n t a l  From  among  calculated  results.  an e x p e r i m e n t a l p o i n t of view t h a t i ) t h e beam may  not  have been c e n t e r e d p r e c i s e l y and i i ) t h e beam t r a v e l i n g r a d i u s  may  not have been a d j u s t e d p r e c i s e l y , both c a l c u l a t e d temperatures  at  r=2.7  cm  and  temperature  at in  r=3«6 cm  were used as  determining  o b s e r v a t i o n s i t e was  FF,  an  although  region  experiment test,  in  shows  whereas  Fig.3-10-b)  VV  reality  temperature  range  melted.  obtained the  difference  FF=10-20 seems t o  the  agree  well  results.  the  case of 0.1  Hz.  Although  more than i n the case of 0 Hz,  i n d i c a t e s the case of 1.0  between  in  A  preliminary  the e x p e r i m e n t a l and  85  Hz and  10.0  calculated  the  FF=10-20 can  f o r e x p e r i m e n t a l r e s u l t s e s p e c i a l l y i n the CP-Ti  Fig.3-10-c)  the  ( S t a t i o n a r y beam).  shows t h a t o b t a i n e d i n  indicates  d i s c r e p a n c y found was account  the  which CP-Ti was  w i t h the e x p e r i m e n t a l  in  site  s e t t o be r=3cm.  Fig.3-10-a) i n d i c a t e s the case of 0 Hz shaded  observation  test.  Hz.  The  values  is  Temp. ("C)  TO -  Temp. ('C)  Temp. ('C)  Temp. (*C)  FT  I  \ \ \ \ \  O  o 3  03 Hto O  Temp. (*C)  \  \  "n  11  0  X  H-  CO to  s;  oo  c+ ts"  cn  OO  \ \  CD X  3 3  \  t3  1>  CD  H-  3 CD  c+  CD 03 H c+ W •  •  II II  o  —' —^  —^ • o  •o  tn  N  o  tn  N  8°  a  1  o  O  —i  tn  tsi  relatively  l a r g e f o r Ti-6A1-4.V.  .However,  i n the case of CP-Ti  any FF v a l u e seems to agree w i t h e x p e r i m e n t a l r e s u l t s  relatively  well.  From  the  measurement, to  be  data  of view of  since  the  o b t a i n e d from CP-Ti experiment  temperature  are c o n s i d e r e d by  Ti-6A1-4.V  the degree of g l a s s c o a t i n g on the viewing  examining  consideration,  Fig.3-10  FF=20  T h i s i s supported  window  case  stage. weight  When FF i s s m a l l , f o r example FF=10, the becomes h i g h e r and the t o t a l  On  weight  the o t h e r hand, when FF i s l a r g e ,  the weight  the  example  l o s s p r e d i c t e d by t h i s model i s much  smaller  experiment.  v a l u e of FF=20 i s t w i c e as much as t h a t of  loss  for  than t h a t o b t a i n e d i n the  The  this  into  by the model i s much l a r g e r than t h a t o b t a i n e d i n  experiment. FF=100,  at  by the e x p e r i m e n t a l r e s u l t s on the t o t a l  spot temperature  predicted  and t a k i n g the above d i s c u s s i o n  seems to be reasonable  l o s s from the specimen.  the  in  much l e s s i n the case of C P - T i .  By  beam  accuracy  more r e l i a b l e compared w i t h those o b t a i n e d  tests, was  point  n a t u r a l convection.  Considering  estimated the  in  additional  e f f e c t of the s u r f a c e t e n s i o n d r i v e n f l o w i n the EB p r o c e s s ,  this  might be a gopd e s t i m a t i o n .  Fig.3-11 measured  in  shows  a  comparison  the experiment  of  the  temperature  changes  (CP-Ti) and those c a l c u l a t e d by  87  the  9 Fig.3-11  R e l a t i o n between c a l c u l a t e d and observed changes w i t h time.  88  temperature  model  a t FF=20.  As seen i n t h i s f i g u r e , the model i s  in  very  good agreement w i t h the e x p e r i m e n t a l r e s u l t s .  3-3-4- Temperature  Fig.3-12  contour on metal s u r f a c e  shows  temperature  contours  under  various  beam  o s c i l l a t i o n r a t e s a t 600 s e c , when the s t e a d y - s t a t e heat t r a n s f e r condition  ( i n terms o f "time averaged") appears t o be  achieved.  In t h i s ' c a l c u l a t i o n , FF=20 was used i n a l l c a s e s . As c l e a r l y in  t h i s f i g u r e , t h e r e i s a l a r g e d i f f e r e n c e i n each  contour.  seen  temperature  The EB hot spot temperatures reached a t 600 sec a r e  as  follows;  Beam O s c i l l a t i o n Rate 0  The steep in  Hot Spot  Hz  Temperature  2470 degC  0.1  2249  1.0  2121  10.0  1974  temperature g r a d i e n t on the metal s u r f a c e becomes  w i t h the i n c r e a s e o f beam o s c i l l a t i o n r a t e .  the  less  Particularly  case o f 10.0 Hz, c a l c u l a t e d temperatures i n  the  molten'  metal were r e l a t i v e l y low ( i n the range o f 1788 - 1974 degC) a  nearly  result than  concentric  temperature contour  was  obtained.  i m p l i e s t h a t w i t h a h i g h e r beam o s c i l l a t i o n r a t e o f 10.0  obtained  Hz,  the same k i n d o f temperature  and  contour  This more  would  and the e f f e c t of the r a p i d beam scanning would n o t  89  be be  0 Hz  0.1 Hz  anticipated  anymore.  From  this  overheating  of  calculation,  it  is  well  confirmed  molten metals can be avoided t o  a  that  considerable  e x t e n t by beam s c a n n i n g .  3-3-5  T o t a l weight  loss  Fig.3-13 shows the c a l c u l a t e d with The  time.  Again  total  weight  oscillation  rate.  oscillation  rate  difference  here, FF=20  increasing  The  this  calculation. in  From t h i s model i t i s c l e a r t h a t g i v e s the l a r g e r weight  T h i s r e s u l t suggests t h a t  the beam o s c i l l a t i o n r a t e  subtracting  loss.  Hz and t h a t  tendency  the  t o more than 10.0  Hz.  e x p e r i m e n t a l r e s u l t s o b t a i n e d are l o s s a t each time was  which corresponds t o a weight l o s s a t 0  the c a l c u l a t e d Fig.3-13,  r e s u l t s with it  Hz  there i s a l i m i t a t i o n i n  5.7g,  in  lower  i n 10.0  weight  indicated  beam  However,  total  comparing  as  any the  was  also  calculated  the weight l o s s o r i g i n a l l y measured a f t e r each  By ones  used i n  l o s s i n c r e a s e s almost l i n e a r l y  In the same f i g u r e ,  from  was  between the weight l o s s i n 1.0  i s very small.  shown.  t o t a l weight l o s s of T i and A l  the  by min,  experiment.  experimental  confirmed  that  the  o f the change i n the t o t a l weight l o s s o b t a i n e d i n  experiment Accordingly,  can  be  this  explained  reasonably  mathematical  model  r e a s o n a b l e f i r s t - s t e p model.  91  can  w e l l ' by be  the  regarded  the  model. as  a  100 too in CO  o  -J  +> _c S 50 CD  Experimental 0 Hz A 0.1Hz © 1.0Hz X 10Hz  •  Calculated line  1  "co +-> o  Time (Min) Fig.3-13  C a l c u l a t e d change i n the t o t a l weight l o s s w i t h time.  92  Fig.3-14weight  loss  shows  are  weight  r e l a t i o n between  the  calculated  and the beam o s c i l l a t i o n r a t e a t 5 min and  respectively. Ti  a  of  evaporates  10  In t h i s f i g u r e , the weight l o s s o f A l and t h a t  illustrated loss  total  i n d i v i d u a l l y t o g e t h e r w i t h the  these  two  elements.  s u b s t a n t i a l l y as w e l l as A l .  As  sum  clearly  This i s  of  seen,  attributed  min of the Ti to  the f a c t t h a t T i has l a r g e X and y i n Eq.3-6 i n s p i t e o f the lower vapor Al  pressure.  is  in  the  experimental  The r a t i o o f the weight l o s s o f T i and t h a t range  result,  of 1 - /+• (wt%Ti+wt%V)  This  agrees  /(wt%Al) =  well 2.6,  of  with  the  which  was  o b t a i n e d by the a n a l y s i s of the metal d e p o s i t e d on an i n n e r  wall  of the s h i e l d a f t e r 0 Hz t e s t s .  Another oscillation can  i n t e r e s t i n g . p o i n t i s t h a t the e f f e c t o f r a t e becomes s m a l l e r a t 1.0 - 10.0 Hz.  be supported by the r e s u l t s o f  contours  the  This  calculated  beam  result  temperature  as p r e v i o u s l y shown i n Fig.3-12, i n which a  homogenized temperature  the  relatively  d i s t r i b u t i o n was o b t a i n e d i n the case  of  1.0 Hz and 10.0 Hz as compared w i t h i n o t h e r c a s e s .  3-3-6 Decrease  Fig.3-15 time, results  where  i n wt%Al d u r i n g m e l t i n g  shows c a l c u l a t e d r e s u l t s o f changes i n w t % A l the i n i t i a l w t % A l i s taken as  a r e a l s o shown i n t h i s f i g u r e .  6.0%.  Experimental  The c a l c u l a t e d  r e v e a l t h a t wt%Al d e c r e a s e s almost l i n e a r l y w i t h time.  93  with  results Although  .55 10  _o -p  top  0  0.1  1  10  0  Frequency (Hz)  Fig.3-14-  0.1  1  Frequency (Hz)  C a l c u l a t e d t o t a l weight l o s s a t  94  5 min  and  10  min.  10  [Al]  0  =6%  ®  Experiment  •  0 Hz  A 0.1Hz  ©  1 Hz 10Hz  X  0• 0  1  1  I  1  L_  2  4  6  8  10  Time (sec)  Fig.3-15  C a l c u l a t e d changes i n wt%Al w i t h time when i n i t i a l wt%Al i s taken as 6%.  95  the  decrease  i n the wt%Al can be suppressed s l i g h t l y  i n c r e a s e o f the beam o s c i l l a t i o n r a t e , o s c i l l a t i o n rates i s very small. large  evaporative  with  the  the d i f f e r e n c e among beam  T h i s i s because  l o s s o f T i , as shown  the  relatively  previously,  makes  the  apparent change i n the A l c o n c e n t r a t i o n expressed by wt% s m a l l e r . This  appears t o be a main reason t h a t the d i f f e r e n c e among  oscillation presumably  r a t e s c o u l d n o t be found c l e a r l y i n the the  difference  i s s m a l l enough  to  beam  experiment:  be  beyond  the  a c c u r a c y o f the experiment.  The b e h a v i o r of the decrease i n w t % A l c o u l d n o t be e x p l a i n e d well At  when the i n i t i a l wt%Al i s 6.0% as i l l u s t r a t e d "time"=0.0,  wt%Al  concentration,  6.1%,  must due  have to  decreased  the  in  from  the  evaporative loss  p r e m e l t i n g p e r i o d p r i o r t o the a c t u a l experiment. true are  i n i t i a l wt%Al shown  taken  cannot be known e x a c t l y ,  i n Fig.3-16 when the i n i t i a l  as 4-.5%.  Fig.3-17 calculated almost  a comparison  of the  in  the  results.  96  can  well.  experimental  beam o s c i l l a t i o n r a t e .  model i s i n good agreement  the  results  the model  around 3«4-%, by t h i s model r e g a r d l e s s  the  a  arbitrarily  w t % A l a t 5 min. Wt%Al a t 5 min was c a l c u l a t e d  constant,  difference figure,  shows  Although  was  As c l e a r l y seen i n t h i s f i g u r e ,  initial  during  calculated  wt%Al  e x p l a i n the e x p e r i m e n t a l r e s u l t s r e l a t i v e l y  Fig.3-15.  with  As  shown the  and the to  be  o f the in  this  experimental  [AI]o=4.5%  ^9  -  1 "  0  Experiment  •  ®  OHz  0.1Hz ® 1 Hz 10Hz A  A  4  6  8  1 0  T i m e (min)  Fig.3-16  C a l c u l a t e d changes i n wt%Al w i t h time when wt%Al i s taken as 4--5%.  97  initial  [AI]o=4.5%  _ <  •  3  • •  -p  *  2  0  •  • Experiment — Calculated i  0  i  .  0.1  1.0  10.0  Frequency (Hz) Fig.3-17  C a l c u l a t e d w t % A l a t 5 min.  98  3-4  I m p l i c a t i o n s o f t h i s work f o r i n d u s t r i a l  process  As d e s c r i b e d p r e v i o u s l y , the mathematical  model developed i n  t h i s study was proven t o be r e a s o n a b l e f o r the i n t e r p r e t a t i o n evaporation 1.5  phenomena i n a s m a l l molten  cm d e p t h ) ,  a l t h o u g h many assumptions  metal  (9  cm diameter and  were i n c l u d e d .  s m a l l sample which was c o n s i d e r e d i n d e v e l o p i n g the model  can be regarded as a p a r t of a l a r g e h e a r t h ,  obtained  can  be  applied  to  an  actual  of  Since a  mathematical the  results  process  semi-  quantitatively.  A study  r e l a t i o n between the beam scanning v e l o c i t y used i n  this  and t h a t used t y p i c a l l y i n the a c t u a l p r o c e s s , i n which  beam scanning l e n g t h i s around  Frequency (Hz)  Beam Experiment  0.0 0.1 1.0 10.0 The fourth  4-0  cm, i s shown as f o l l o w s ;  V e l o c i t y (cm/sec) . A c t u a l process  0.0 0.94 9-4 94-0  0 4 40 400  beam v e l o c i t i e s used i n t h i s study a r e a p p r o x i m a t e l y of  oscillation  Although  those  in  a  the t y p i c a l a c t u a l process  at  each  a  beam  rate.  a p r e c i s e d e s c r i p t i o n i s necessary f o r a  u n d e r s t a n d i n g o f the the EB h e a r t h m e l t i n g p r o c e s s , the.  99  complete tendency  of  the  deduced  temperature  rate.  c o n t r o l , the e f f e c t  very l a r g e .  process  and the  evaporative  from the r e s u l t s o f t h i s model.  composition not  profile  is  Instead,  loss  can  Namely, i n terms  of  of the beam o s c i l l a t i o n r a t e  the t o t a l weight l o s s  suppressed by the i n c r e a s e o f the  From an o p e r a t i o n a l p o i n t o f view, t h i s  during beam  i s very  be a is  melting  oscillation important  t o prevent the d e p o s i t i o n o f the evaporant on a i n n e r w a l l o f the chamber, which makes the y i e l d energy e f f i c i e n c y . Hz the e f f e c t  low, and the decrease i n the  However, a t beam o s c i l l a t i o n r a t e s over  cannot be expected.  10.0  I n c o n c l u s i o n the optimum beam  o s c i l l a t i o n r a t e appears t o be 1.0 - 10.0 Hz.  100  beam  S e c t i o n 4SUMMARY AND  4.-1  RECOMMENDATIONS FOR  THE FUTURE WORK  Summary  In causes  o r d e r t o a v o i d o v e r h e a t i n g the a  substantial  metal,  which  of  valuable  evaporative loss  sometimes elements  d u r i n g the EB h e a r t h m e l t i n g p r o c e s s , the beam scanning t e c h n i q u e is  employed  on  i n the a c t u a l p r o c e s s .  In t h i s study, i n v e s t i g a t i o n  the optimum beam o s c i l l a t i o n r a t e was  mathematical  model  on  conducted by  the b a s i s of a s m a l l  scale  making EB  a  melting  experiment.  Small  amounts of Ti-6A1-4V  a l l o y were melted i n the  size  EB m e l t i n g f u r n a c e .  rate  on the e v a p o r a t i o n b e h a v i o r  beam  o s c i l l a t i o n r a t e s used i n t h i s study were 0, 0.1,  10.0  Hz.  The i n f l u e n c e of the beam o f A l was  small  oscillation  investigated.  The  1.0  The temperature c y c l e observed i n t h i s study was  and found  to be i n good agreement w i t h the g i v e n beam o s c i l l a t i o n r a t e . all  samples,  a s i g n i f i c a n t amount of the wt%Al decrease and  t o t a l weight l o s s oscillation but  could  present  On  ( A l + T i ) was  r a t e was not  observed.  In the  The e f f e c t of the beam  observed i n the t o t a l weight  be found c l e a r l y i n the wt%Al  loss  decrease  change, in  the  experiment.  the  basis  of  the  results  from  the  experiment,  p a r t i c u l a r the r e s u l t s of the temperature measurement, 2-D  101  and  in 1-  D  unsteady  moving  beam  problem. Al  and  heat t r a n s f e r models were made: problem  model f o r  the  model f o r  stationary  Ti  was  a l s o taken i n t o account  by  assuming  r e a c t i o n a t the i n t e r f a c e p l a y s a major  comparison of the c a l c u l a t e d and  explain  beam  that  containing the  That From a  e x p e r i m e n t a l r e s u l t s , FF=20.0, a  the e f f e c t of the molten metal,  experimental  the  role.  the e v a p o r a t i o n r e a c t i o n i s a r a t e - d e t e r m i n i n g s t e p .  factor  the  In these models, the mass balance w i t h r e s p e c t t o both  evaporation is,  and a 1-D  a 2-D  results  reasonably  appeared  well  to  under  the  that  the  present experimental c o n d i t i o n s .  According evaporative  to  loss  this of  model, i t was  both  A l and T i  clearly could  shown be  the  Hz.  This  result  o b t a i n e d because the o v e r h e a t i n g of the molten  metal  was  prevented  more than 1.0 anymore. of  Al,  substantially.  1.0  by  increasing was  beam o s c i l l a t i o n r a t e t o around  suppressed  With the beam o s c i l l a t i o n r a t e  Hz, however, t h i s e f f e c t c o u l d not be found  On the o t h e r hand, i n terms of the composition the  significant,  effect  of  the  since T i also  beam  oscillation  evaporated  rate  simultaneously  at  clearly control  was during  not a  m e l t i n g p e r i o d under the low o p e r a t i n g p r e s s u r e .  In  conclusion,  the optimum beam o s c i l l a t i o n  c o n s i d e r e d t o be i n the range of 1.0 o p e r a t i o n a l p o i n t of view.  4-2  Recommendations f o r f u t u r e work  102  - 10.0  rate  can  Hz e s p e c i a l l y from  be an  U n f o r t u n a t e l y the e f f e c t of the beam o s c i l l a t i o n r a t e not be d e t e c t e d c l e a r l y i n the p r e s e n t experiment.  could  In the f u t u r e ,  t h e r e f o r e , e x p e r i m e n t a l a c c u r a c y should be improved f u r t h e r .  For  example,  we  and  minimize  the  various  should effect  parameters  improve the  beam  of the p r e m e l t i n g  rotating period.  equipment In  such as EB power i n p u t , vacuum  addition,  degree,  and  s o l u t e elements h a v i n g d i f f e r e n t a c t i v i t y c o e f f i c i e n t s should taken i n t o account and  developed  phenomena, the will  i n order t o a l l o w the model t o be more p r e c i s e  reliable.  Furthermore, be  be  by  a t h r e e d i m e n s i o n a l mathematical c o n s i d e r i n g the e f f e c t of  the  ultimately  l e a d to the development of a  pool,  gravity.  complete  of the EB h e a r t h m e l t i n g p r o c e s s e s commercially  103  should  solidification  which w i l l g i v e a p r e d i c t i o n of the metal  metal f l o w d r i v e n by both s u r f a c e t e n s i o n and  modeling  model  and This  process used.  List  1.  2.  D . A p e l i a n , C.H.Entrekin (1986), pp.77-90.  of References  : I n t e r n a t i o n a l Metals Reviews, 31,  K.Schulze, O.Bach, D.Lupton, and F . S c h r e i b e r : "Niobium", P r o c . of the I n t e r n a t i o n a l Symposium, (1981), AIME, P i t t s b u r g h , PA, pp. 163-237.  3. C.H.Entrekin : " E l e c t r o n Beam Remelting and R e f i n i n g State of the A r t 1985 P t . I " , Proc.Conf., (1985), B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ , pp.4-0-47.. 4.  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G r u z l e s k i 5, (1979), pp.294-300.  15. R. H a r r i s and W.G. 581-588. 16. i b i d . ,  Davenport  : Ironmaking  & Steelmaking,  : Met. Trans. 13B, (1982), pp.  pp.589-591 -  17. E. Ozberk and R.I.L. G u t h r i e : Met. Trans. 17B, (1986), pp. 19-29.  104  18. E.S.Machlin  : Trans, of Met. Soc. of AIME, 218, ( i 9 6 0 ) , pp.  3U-326. 19- G.H.Geiger and D . R . P o i r i e r : " T r a n s p o r t phenomena i n m e t a l l u r g y " Readings, Mass, Addison-Wesley, (1973)20. J . S z e k e l y , C.W. (1977), p517.  Chang, and W.E.  Johnson  : Met. Trans. 8B,  21. S. Hayakawa, T. Choh, and M. Inoue : I . S . I . J . 22, (1982), pp. 637-645. 22. R. H a r r i s  : Met. Trans. 15B, (1984),  pp.251-257.  23- T . S a n t a l a and C.A.Adams : J . Vac. S c i . T e c h n o l . , 17, (1970), ss.22-2924. Y.Nakamura  and M.Kuwabara : I . S . I . J . , 15, (1975),  pp.103-108.  25. S . S c h i l l e r and H . F S r s t e r : " E l e c t r o n Beam Remelting and R e f i n i n g S t a t e of the A r t 1984" Proc.Conf., (1984), B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ , pp.49-6926. A . M i t c h e l l and K.Takagi (1984), pp.89-99.  : Proc. Vacuum M e t a l l u r g y conf.,  27. J . H e r b e r t s o n : " E l e c t r o n Beam Remelting and R e f i n i n g S t a t e o f the A r t 1986 P t . I I " Proc.Conf., (1986), B a k i s h Materials C o r p o r a t i o n , Englewood, NJ, pp.19-29.  28. H.S.Kheshgi and P.M.Gresho : " E l e c t r o n Beam Remelting and R e f i n i n g S t a t e of the A r t 1986 P t . I I " Proc.Conf., (1986), B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ, pp.68-7929. D.Tripp : Master's T h e s i s , U n i v e r s i t y of B r i t i s h (1987). 30. K.Takagi (1984).  : Master's T h e s i s , U n i v e r s i t y of B r i t i s h  31. D. R o s e n t h a l : Trans, of the ASME, 43,  (1946),  Columbia,  Columbia,  pp.849-866.  32. For example, N. C h r i s t e n s e n , V. Davies, K.Gjermundsen : B r i t i s h Welding J o u r n a l , 12, (1965), pp.54-7533-  T.W.Eager and N.S.Tsai  : Welding J o u r n a l , 62,  (1983),346s-  355s. 34-  S.Kou and Y.H.Wang : M e t a l l . Trans., 17A, 2270.  (1986), pp.2265-  35- O p e r a t i o n s Manual o f EH-30/20, Von Ardenne, E a s t Germany.  105  36. O p e r a t i o n s Manual o f Automatic O p t i c a l pyrometer IRCON Inc., I l l i n o i s .  S e r i e s 1100,  37. C.R.Heiple, J.R.Roper : Welding J o u r n a l , 61, (1982), 97s102s. 38. A . S . B a l l a n t y n e : Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, (1978). 39- S . S c h i l l e r , U . H e i s i g , and S.Panzer : " E l e c t r o n Beam Technology", (1982), John W i l e y & Sons, I n c . , New York, 4.0. "Thermal P r o p e r t i e s o f T i t a n i u m A l l o y s " , Defense I n f o r m a t i o n Center, B a t t e l l e Memorial I n s t i t u t e , Ohio. 4.1. O.Kubaschewski, C.B.Alcock : " M e t a l l u r g i c a l 5th ed., Pergamon P r e s s , Oxford, (1979).  NY.  Materials Columbus,  Thermochemistry",  42. P. C a c c i a t o r e : "Modeling o f C a s t i n g and Welding (1980), AIME.  Processes",  43- A . M i t c h e l l , H.Nakamura and D.Tripp : " E l e c t r o n Beam Remelting and R e f i n i n g S t a t e o f the A r t 1987" Proc.Conf., (1987), B a k i s h M a t e r i a l s C o r p o r a t i o n , Englewood, NJ , pp. 23-32.  106  Appendix 1 ESTIMATION OF RATE CONTROLLING STEP  A-1-1 Complete d i f f u s i o n c o n t r o l  The  case  p o s s i b i l i t y of a "complete"  d i f f u s i o n c o n t r o l case  was  18)  examined  by  using  M a c h l i n ' s model  Basic equation Assuming  that  r e a c t i o n system this  system,  Machlin's  model, which  was  developed  i n the i n d u c t i o n f u r n a c e , can be a l s o a p p l i e d the mass t r a n s f e r c o e f f i c i e n t ,  ^ (cm/sec),  for to is  g i v e n as f o l l o w s .  Me,l  where D  2 ( D / 7T  e )*  (A-1-1 )  D i f f u s i v i t y o f s o l u t e element  (cm / s e c ) ,  10 ^ cm^/sec 6  " L i f e time" o f a s m a l l element ( s e c )  On the o t h e r hand, o v e r a l l mass f l u x o f A l , j  Al  (mole/sec),  can be expressed by the f o l l o w i n g e q u a t i o n .  J  A1  = A k, Me,l :  C  (A-1-2)  A1  107  where A  = R e a c t i o n area (cm  ),  3 = A l c o n c e n t r a t i o n (mole/cm )  S u b s t i t u t i n g Eq.A-1-1 i n t o Eq.A-1-2, the f o l l o w i n g can be  obtained.  (wt%Al) = ( w t % A l )  0  exp(-(l/h) k  M g j l  t)  (A-1-3)  where h = Depth of molten p o o l (cm), 1.5 (wt%Al)  Q  = I n i t i a l wt%Al,  (wt%Al) - 4.5  exp(-7.52 x 1 0 "  t a k i n g 8 as a parameter,  cm  and  4..5%  F i n a l l y , the f o l l o w i n g e q u a t i o n can be  By  equation  3  obtained.  0 ~* t )  (A-1-4.)  the change i n wt%Al w i t h time  was  calculated.  Results  Fig.A-1.1 figure,  shows c a l c u l a t e d curves w i t h v a r i o u s 0.  the e x p e r i m e n t a l r e s u l t s are a l s o shown.  In  It is  this  clearly  shown t h a t the e x p e r i m e n t a l r e s u l t s can be e x p l a i n e d w e l l a t 8 10 the  - 4.0 s e c . However, from the o b s e r v a t i o n of the metal f l o w s u r f a c e d u r i n g experiment,  case larger  was  i n the o r d e r of 0.1  -  than the c a l c u l a t e d one.  the l i f e 1.0  sec.  time i n t h i s This i s  T h e r e f o r e , the  108  = on  particular  substantially  whole  reaction  5  U  \\ " A "  <3 o 5  A  e=  (sec)  N  N  2 Experiment^, •  1  0 A 0.1 ® 1 x10  0  2  JO  Hz Hz Hz Hz  ® "A  6 ^ 8  4  10  Time (min) Fig.A-1.1  Changes i n wt%Al w i t h time c a l c u l a t e d from M a c h l i n ' s model.  109  would  not have been c o n t r o l l e d by a d i f f u s i o n  controlling  step  only.  A-1-2  E f f e c t of the controlling  A  simple  e s t i m a t e the  temperature and  mathematical  e f f e c t of the  e q u a t i o n and  Fig.A-1.2 The  model was  temperature and  the  in  stirring  evaporation r e a c t i o n .  calculation.  taken i n t o account by i n t r o d u c i n g  a factor,  9*C  2.  A b a s i c e q u a t i o n can be g i v e n  9C  Al  follows.  Al  (A-1-5)  9t  D'  ff D  and  D  'Ordinary'  diffusivity  of s o l u t e element  The  reaction influence  t a k i n g mass balance i n t h i s system as  where  condition  The  i n Section  D  to  solution  removed by the  as d e s c r i b e d  order  follows.  i l l u s t r a t e s a geometry used i n t h i s  of the metal f l o w was ff,  developed  s o l u t e element d i f f u s e s from the bulk metal to the  i n t e r f a c e and  rate  step  on the r a t e c o n t r o l l i n g step as  Basic  the metal f l o w on the  boundary c o n d i t i o n s  are  given  110  as  (cm  2  /sec).  follows.  by  [metal surface] E  LO  9Omm0 g.A-1.2  Geometry used i n the c a l c u l a t i o n .  111  a t t = 0.0 sec  B.C.2  3C /3x =  B.C.3  - D » 3 C / 3 x = K„ C  condition  Kg,  %  C  A 1  A1  a t the bottom, x =  0.0  A1  The i n i t i a l  B.C.3  = A.5  B.C.1  a t the s u r f a c e , x=0.0mm  c o n d i t i o n i s expressed by B.C.1.  B.C.2  gives  t h a t t h e r e i s no mass f l u x a t the bottom of the  g i v e s a heterogeneous r e a c t i o n r a t e on the Langmuir r a t e c o n s t a n t ,  15mm  melt  a  pool.  surface.  i s expressed by Eq.A-1-6.  (Same  as  Eq.1-5)  K  = Y P  E  A 1  °  / P  m  (27TMRT)*  (A-1-6)  Thermochemical d a t a used i n the c a l c u l a t i o n a r e t a b u l a t e d  below.  Temperature(degC)  1600  1700  1800  1900  2000  2100  D  (cm /sec) x 10"  0.70  0.80  0.90  1.0  1.1  1.2  (cm/sec) 10"  0.15  0.4.0  0.93  2.00  4--01  7.56  2  A  ±  4  K_ E  *  x  3  D  =  D  e x p ( - E / R T ) ; D =constant,  Q  D  Q  E^  =  Activation  22) energy, 4-OkJ x10  4  Eq.A-1-2  .  a t 1900 degC was assumed t o be  1.0  cm^/sec. was  s o l v e d by t a k i n g f f (1 - 100)and  3100 degC) as parameters.  112  T  (1700  -  Results  R e s u l t s a r e summarized i n F i g . A-1.3, where numbers r e f e r t o U3) the r a t i o o f s u r f a c e c o n c e n t r a t i o n t o b u l k c o n c e n t r a t i o n this  figure,  i t i s seen t h a t t h e r e i s a s u b s t a n t i a l  both  f f and  temperature on t h e mode o f  e v a p o r a t i o n . A low temperature,  (<1900  t h e "average"  degC),  relatively natural  namely  strong convection  (corresponding experiment,  to  the  to d i f f u s i o n  overheat  and  was suppressed,  condition, the  ff=10 - 30),  effect  of  control  in  control.  molten temperature was  stirring  From  h i g h v e l o c i t y regime w i l l l e a d t o  e v a p o r a t i o n c o n t r o l ; the converse  Since  reaction  .  which  and  arises  surface-tension was  relatively  observed  low  also  the  from  the  driven i n the  flow present  c o n t r i b u t i o n o f t h e e v a p o r a t i o n s t e p would have  been v e r y l a r g e .  Therefore, the  rate  molten  approximation,  should  be r e a s o n a b l e  namely,  i n t h e molten p o o l was v i r t u a l l y f l a t .  h e a t - and  i n the p r e s e n t  t h i s assumption was a l s o supported  r e s u l t s d e s c r i b e d i n S e c t i o n 2;  question  t h e assumption  c o n t r o l l i n g step was t h e e v a p o r a t i o n r e a c t i o n  surface  addition,  as a f i r s t  about t h a t t h e simultaneous mass- t r a n s f e r e q u a t i o n s  interpretation  a t the  case.  In  by t h e e x p e r i m e n t a l  t h e d i s t r i b u t i o n o f %A1 ( Of course, t h e r e i s no  s o l u t i o n o f the momentum-,  i s necessary  f o r the p r e c i s e  o f t h e e v a p o r a t i o n phenomena i n the system  i s a f u t u r e work as d e s c r i b e d i n S e c t i o n 4«)  113  that  this  5-95%  95% Evaporation control  1700  •Mixed control  2100  Fig.A-1.3  2500 2900 Temp. ( °C)  E v a p o r a t i o n c o n t r o l map.  1U  <5% Diffusion control  3300  Appendix 2 PROGRAM LIST AND A-2-1  Program l i s t  Listing 1 2 3 4 5 G 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  of  o f 1-D  STSPOT a t  FLOW CHART  model  11:29:27  on  JAN 2 5 ,  198B  for  CC1d=NHER on G  C PROGRAM * S T S P O T * ( ( S T A T I O N A R Y BEAM)) C C CYLINDRICAL COORDINATE SYSTEM (1D MODEL) C DIMENSION T ( 1 0 0 ) , T C ( 1 0 0 ) , T N ( 1 0 0 ) , T A ( 100) , 1 T0(1OO),0R(1OO).0EB(1O0).STIJ(1OO). 2 SALJ(100),NTYPE(10),A 1(100),A2(100), 3 A3(10O).AREA(100) C C DEFINE STATEMENT FUNCTIONS C C * R A D I A T I O N HEAT L O S S * C TTA(TT)=TT*TT+TAMBT*TAMBT TTB(TT)=TT+TAMBT TTC(TT)=TT-TAMBT TTT(TT)=TTA(TT)*TTB(TT)*TTC(TT) ORA(TT)=EMISS*SIGMA*TTT(TT) C C *EVAPORATION HEAT L O S S * C P T I ( T T ) = 1 3 3 . 2 8 * 1 0 * * ( - 2 3 2 0 0 . O / T T - 0 . 6 6 * A L O G 1 0 ( T T ) + 1 1 .74) PAL(TT)=133.28*10**(-16380.0/TT-AL0G10(TT)+12.32) TIJ(TT)=GAMTI*PTI(TT)*TIM0L/SQRT(2.O*PI*TIM0L*R*TT)*XTI ALJ(TT)=GAMAL*PAL(TT)*ALMOL/SORT(2.0*PI*ALMOL*R*TT)*XAL QTI(TT)=TId(TT)*HTI . OAL(TT)=ALJ(TT)*HAL QEV(TT)=QTI(TT)+QAL(TT) C C *SUM OF HEAT LOSS TERM (W/M2) C QL(TT)=QRA(TT)+QEV(TT) C C READ INPUT DATA C READ ( 5 , 4 6 0 ) RAD.DEPTH READ ( 5 . 4 7 0 ) N R . I F R E Q RNR = F L O A T ( N R ) READ ( 5 . 4 6 0 ) POWERK, HZ POWERW = POWERK * 1 0 0 0 . 0 READ ( 5 , 4 6 0 ) WPCTA, TEMPIC TEMPIK = TEMPIC + 2 7 3 . 1 5 READ ( 5 , 4 6 0 ) D E L T , TIMEN READ ( 5 , 4 6 0 ) F F , HT IMAX = TIMEN / DELT + 0 . 5 C C AREA.VOLUME,AND WEIGHT OF LIQUID TI C PI = 3 . 1 4 1 5 9 2 6 5 4 RHO = 4 5 4 0 . 0 RHOAL = 2 7 0 0 . 0 WPCTT = 1 0 0 . 0 - WPCTA RHOM = (RHO*WPCTT + RHOAL*WPCTA) / 100.0 C AREAH •= RAD * RAD * PI VOLUME = AREAH * DEPTH WEIGHT = VOLUME * RHOM C  115  Listing 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116  o f STSPOT a t 11:29:27 on JAN 25. 1988 f o r CC1d=NHER C NODE AREA C DELR = RAO / RNR AREAO = ( D E L R * D E L R ) * PI / 4.0 DO 20 1 = 1, NR RI = F L O A T ( I ) I F ( I . E O . N R ) GO TO 10 C * INNER* AREA(I)=2.0*PI*RI*DELR*DELR GO TO 20 C *OUTER* 10 AREA(I)=PI*(RNR-O.25)*0ELR*DELR C 20 CONTINUE C C PHYSICAL PROPERTIES ( S I - U N I T ) C *HEAT CONDUCTION* C TIK = 28.2 CPTI = 6 9 0 . 0 FTIK = TIK * FF ALPHA = FTIK / ( C P T I * R H O ) FO = ALPHA * DELT / (DELR*DELR) BI = HT * DELR / F T I K C C *RADTATION* C EMISS = 0.4 SIGMA = 5.67E-8 TAMBT = 298 . 15 C C *EVAPORATION* C R = 8.3144 TIMOL = 0.0479 ALMOL = 0.027 GAMTI = 1.0 GAMAL = 0.028 HTI = 8.88E6 HAL = 1.08E7 C C EB POWER C OIK = POWERK/AREAO 01 = OIK * 1000.0 C C I N I T I A L CONCENTRATION OF ALLOY MASS BALANCE C WTI0=WEIGHT*WPCTT/100.0 WALO=WEIGHT-WTIO XPNAL = WPCTA / ALMOL XPNTI = WPCTT / TIMOL XAL = XPNAL / (XPNAL + X P N T I ) XTI = 1.0 - XAL C C WRITE LIQUID DIMENSIONS AND ETC, C WRITE ( 6 , 4 8 0 ) RAD, DEPTH  116  on G  Listing 1 17 1 18 1 19 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 13G 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  o f STSPOT a t 11:29:27 o n JAN 25. 1988 f o r CC1d=NHER o n G WRITE WRITE WRITE WRITE WRITE WRITE  (6.490) (6,500) (6,510) (6,520) (6,530) (6,540)  AREAH, VOLUME, WEIGHT POWERK, AREAO, OIK , HZ NR, DELR DELT, TIMEN F F , F T I K , HT, WPCTA IMAX.IFREO  C C SORT NODE TYPE C DO 70 I = 1, NR NTYPE(I) = 2 IF ( I .EQ. 1) N T Y P E ( I ) = 1 IF ( I .EQ. NR) N T Y P E ( I ) = 3 70 CONTINUE C C TERMS C DO 80 I = 1, NR RI = F L O A T ( I ) A1 ( I ) = (RI - 0.5) / RI A 2 ( I ) = ( R I + 0.5) / RI 8 0 CONTINUE B = DELR * DELR / ( D E P T H * F T I K ) D1 = 2.0 * (RNR - 0.5) / (RNR - 0. 25; D2 = 2.0 * RNR / (RNR - 0.25) C C SET I N I T I A L CONDITIONS C TO = TEMPIK TOO = TEMPIK TOC = TEMPIC S T I O J = 0.0 S A L O J = 0.0 QEB0=QI DD 100 I = 1, NR T ( I ) = TEMPIK T O ( I ) = TEMPIK T C ( I ) = TEMPIC S T I J ( I ) = 0.0 S A L J ( I ) = 0.0 QEB(I)=0.0 100 CONTINUE TIME = 0.0 ICOUNT = 0 C C WRITE I N I T I A L CONDITIONS C WRITE ( 6 . 5 5 0 ) TEMPIC WRITE ( 7 , 6 0 0 ) HZ WRITE ( 7 . 6 1 0 ) F F , FTIK,POWERK WRITE ( 7 , 6 2 0 ) WRITE ( 7 , 6 3 0 ) TIME, TOC C C START CALCULATION C 1 10 ICOUNT = ICOUNT + 1 IF ( I COUNT .GT. IMAX) GO TO 420 TIME = F L O A T ( I C O U N T ) * D E L T  117  Listing 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 •190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232  o f STSPOT a t 11:29:27 on JAN 25. 1988 f o r CC1d=NHER o n G C C TEMPERATURE ADJUSTMENT FOR CALCULATION OF C RADIATION 6 EVAPORATION HEAT LOSS C TOA = 1.5 * TO - 0.5 * TOO DO 140 I = 1, NR T A ( I ) = 1.5 * T ( I ) - 0.5 * T O ( I ) 140 CONTINUE C C RADIATION + EVAPORATION HEAT LOSS C QRO = O L ( T O A ) DO 160 I = 1 . NR TA1 = T A ( I ) OR(I) = 0L(TA1) 160 CONTINUE  c c c  c c c  c c c c c c  SUM  OF T I & AL LOSS AT EACH NODE(KG/M2)  180  S T I O J = S T I O J + T I J ( T O A ) * DELT SALOJ = S A L O J + A L J ( T O A ) * DELT DO 180 I = 1, NR TA1 = T A ( I ) T1J1 = T I J ( T A 1 ) ALJ1 = A L J ( T A 1) S T I J ( I ) = S T I J . ( I ) + T I J 1 * DELT S A L J ( I ) = S A L J ( I ) +.ALJ1 * DELT CONTINUE  TOTAL T I & AL LOSS  (KG)  A T I J = STIOJ*AREAO A A L J = SALOJ*AREAO DO 220 I = 1, NR IF ( I .EO. NR) GO TO 190 * INNER* AATIJ = S T I J ( I ) * AREA(I) AAALJ = S A L J ( I ) * A R E A ( I ) GO TO 200 •OUTER* 190 AATIJ = S T I J ( I ) * AREA(I) AAALJ = S A L J ( I ) * A R E A ( I ) *SUM* 200 ATIJ = ATIJ + AATIJ A A L J = A A L J + AAALJ 220 CONTINUE MASS BALANCE  : ADJUST TI & AL QUANTITY  WTI = WTIO WAL = WALO WT = WTI + WPCTA •= WAL XNTI = WTI XNAL = WAL XNT = XNTI XAL = XNAL XTI = 1 . 0 -  - ATIJ - AALJ WAL / WT* 100 .0 / TIMOL / ALMOL + XNAL / XNT XAL  118  IN MELT  Llstlng'of 233 234 235 236 237 238 239 240 241 242 243 244 245 24G 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  STSPOT at 11:29:27 on OAN 25, 198B f o r CC1d=NHER on G C C FINITE DIFFERENCE EQUATIONS C C •CENTER PART* C T0N=F0*(4.0*T(1)+B*(QEBO-QRO))+(1-4.0*F0)*T0 C C SORT C DO 360 I = 1, NR NTY = NTYPE(I) GO TO (260, 270. 280), NTY  c c c c c c c c c  c c c  c c c  c c c c  c  *TYPE1* 260  1  TN(I)=F0*(A1(I)*T0+A2(I)*T(1+1)+ B*(QEB(I)-QR(I)))+(1.0-2,0*FO)*T(I) GO TO 360  •TYPE2* 270  1  TN(I)=F0*(A1 (I )*T(1-1 )+A2(I)*T(1 + 1) + B*(QEB(I)-QR(I)))+(1.0-2.0*F0)*T(I) GO TO 360  *TYPE3* 280  1  TN(I) = F0*(D1*T(I-1 )+BI*D2*TAMBT+B*(QEB(I)-QR(I +(1.0-D1*F0-D2*F0*BI)*T(I)  360 CONTINUE TEMPERATURE MUST BE LESS THAN 3285.0 DEGC (3558.15 K) IF(T0N.GT.3558.15) T0N=3558.15 DO 364 I=1,NR T1=TN(I) IF(T1.GT.3558.15) TN(I)= 3558.15 364 CONTINUE SUBSTITUTE NEW TEMPERATURE IN OLD ONE TOO = TO TO = TON DO 380 I = 1, NR TO(I-) = T ( I ) T ( I ) = TN(I) 380 CONTINUE PRINT OUT CONTROL IF ( (ICOUNT/1FREQ)*IFREQ .NE. ICOUNT) GO TO 410 CHANGE TEMPERATURE UNIT: K TO C TOC = TO - 273.15 DO 400 I = 1, NR TC( I ) •= T( I ) - 273 . 15 400 CONTINUE  119  Listing 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347  o f STSPOT a t 11:29:27 on JAN 25. 1988 f o r CC1d=NHER o n G C WRITE TEMPERATURE & TI AND AL LOSS C WRITE ( 7 , 6 3 0 ) TIME, T O C , T C ( 1 ) . T C ( 3 ) , T C ( 5 ) , A T I J , A A L J . WPCTA C C NEXT STEP C 4 1 0 GO TO 110 C C WRITE RESULTS AT END TIME C 420 WRITE ( 6 , 5 6 0 ) TIME WRITE ( 6 , 5 7 0 ) T O C , ( T C ( I ) , I = 1 , N R ) WRITE ( 6 , 5 8 0 ) A T I J , A A L J , WPCTA GO TO 1000 C C READ FORMAT 4 6 0 FORMAT ( 5 F 1 0 . 5 ) 4 7 0 FORMAT ( 5 1 1 0 ) C C WRITE FORMAT C 480 FORMAT (5X,'**DIMENSION OF MOLTEN METAL**'/8X,'RADIUS=',F5.3, 1 'M',3X,'DEPTH='.F5.3,'M') 4 9 0 FORMAT (8X,'AREA=',E10.4,'M2',3X,'VOLUME=',E10.4,'M3',3X, 1 'WEIGHT=',F7.3,'KG'//) 500 FORMAT ( 5 X . ' * * E B CONDITIONS**'/8X, 1 'BEAM POWER=',F7.3,'KW',3X,'BEAM AREA='.E10.4, 2 'M2',3X,'POWER DENSITY'',E15.3,'KW/M2'/8X, 3 'FREOUENCY=',F7.2,'HZ'/) 510 FORMAT ( 5 X , ' * * C A L C U L A T I N G CONDITIOS**'/8X,'RNODE : NUMBER 1 I5.3X,'SIZE=',F7.4,'M'//) 520 FORMAT (8X,'TIME S T E P ' , F 1 0 . 3 , ' SEC'/8X,'END TIME =',F10.3, 1 ' SEC'/) 5 3 0 FORMAT (8X,'FF=',F7.2,3X,'HEAT CONDUCTIVITY OF T I = ' , F 7 . 2 , 1 'WM/K'/8X,'HEAT TRANSFER C O E F F I C I E N T = ' , F 1 0 . 3 , 2 'W/M2/K'/8X,'INITIAL AL WEIGHT P E R C E N T ' , F 5 . 2 / ) 540 FORMAT (5X, '*'CONTROLLER**'/8X, 'TOTAL STEPS=', 15, 3X. 1 'PRINT FREQUENCY ', 15///) 550 FORMAT (5X, '**RESULTS**'//8X, ' ' I N I T I A L TEMPERATURE ', F 7 . 2 , 1 ' D E G C , 3X, 'THROUGHOUT IN MELT'//) 560 FORMAT (8X,''TEMPERATURE DISTRIBUTION (DEGC) AT TIME=',F10.3, 1 ' SEC'/) 570 FORMAT (8X, 15( 1X.F7.2, ', ' , 3 X ) ) 580 FORMAT (//8X, ' *TOTAL TI L O S S ' , E10.4, ' ( K G ) ' / 8 X , 1 '*TOTAL AL L O S S ' , E10.4, ' ( K G ) ' / / 8 X , 2 '"FINAL AL CONCENTRATION ', F5.2,'WT%') 6 0 0 FORMAT (5X,'FREQUENCY=' . F 7 . 2 , 'HZ') 6 1 0 FORMAT ( 5 X , ' F F ' , F7.2,4X,'HEAT CONDUCTIVITY OF MOLTEN T I ' . 1 F 7 . 2 , 'WM/K'/5X, 'INPUT POWER ' ,F7 . 2, 'KW'/) 6 2 0 FORMAT (/5X.'DATA AT A CENTER POINT' 1 //6X,'TIME(S)',5X,'TEMP.(C)',4X, 2 ' * T I N ( C ) * ' , 3 X , '*TMID(C)*',3X, ' * T O B S ( C ) * ' .4X, 3 'TI L O S S ( K G ) ' , 4 X , ' A L LOSS(KG)',5X,'WT%AL'/) 6 3 0 FORMAT ( 5 X , F 8 . 2 , 4 ( 2 X , F 1 0 . 2 ) , 2 ( 5 X . E 1 0 . 4 ) , 4 X , F 6 . 2 ) C 1000 STOP END 3  3  3  3  3  3  3  3  3  3  120  A-2-2  Program l i s t  Listing 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 1G 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  of  MVSPOT  at  of 2-D  model  11:29:43  o n JAN 2 5 .  1988  for  CC1d=NHER on G  C PROGRAM *MVSPOT* C C CYLINDRICAL COORDINATE SYSTEM ( 2 - D MODEL) C DIMENSION T ( 1 0 0 , 1 0 0 ) , T C ( 1 0 0 , 1 0 0 ) , T N ( 1 0 0 , 1 0 0 ) . T A ( 1 0 0 , 1 0 0 ) , 1 T0(100,100),0R(100.100),OEB(100,100),STIJ(100.100). 2 SALJ(100,1O0),NTYPE(100,100),A1(100),A2(100). 3 A3(100),AREA(100) C C DEFINE STATEMENT FUNCTIONS C C ' R A D I A T I O N HEAT L O S S * C TTA(TT)=TT*TT+TAMBT*TAMBT TTB(TT)=TT+TAMBT TTC(TT)=TT-TAMBT TTT(TT)=TTA(TT)*TTB(TT)*TTC(TT) ORA(TT)=EMISS*SIGMA*TTT(TT) C C *EVAPORATION HEAT L O S S * C PTI(TT)=133.28*10**(-23200.0/TT-0.66*ALOG10(TT)+11.74) PAL(TT) = 1 3 3 . 2 8 * 1 0 * * ( - 16380.O/TT-ALOG10(TT)+12.32) TIJ(TT)=GAMTI*PTI(TT)*TIM0L/SQRT(2.O*PI*TIMOL*R*TT)*XTI ALJ(TT)=GAMAL*PAL(TT)* ALMOL/SORT(2.0*PI *ALMOL*R* TT)*XAL OTI(TT)=TIJ(TT)*HTI OAL(TT)=ALJ(TT)*HAL QEV(TT)=QTI(TT)+QAL(TT) C C *SUM OF HEAT LOSS TERM (W/M2) C QL(TT)=QRA(TT)+QEV(TT) C C READ INPUT DATA C READ ( 5 , 4 6 0 ) RAD, D E P T H , EBRAD READ ( 5 , 4 7 0 ) NR, NS. IFREO RNR = FLOAT(NR ) RNS = F L O A T ( N S ) READ ( 5 . 4 6 0 ) POWERK, HZ POWERW = POWERK * 1 0 0 0 . 0 READ ( 5 . 4 6 0 ) WPCTA, TEMPIC TEMPIK = TEMPIC + 2 7 3 . 1 5 READ ( 5 , 4 6 0 ) D E L T , TIMEN READ ( 5 . 4 6 0 ) F F , HT IMAX = TIMEN / DELT + 0 . 5 C C AREA,VOLUME.AND WEIGHT OF LIQUID TI C PI = 3 . 141592654 RHO = 4 5 4 0 . 0 RHOAL = 2 7 0 0 . 0 WPCTT = 1 0 0 . 0 - WPCTA RHOM = (RHO*WPCTT + RHOAL *WPCTA) / 100.0 C AREAH = RAD * RAD * PI VOLUME = AREAH * DEPTH WEIGHT * VOLUME * RHOM  121  Listing 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 1 10 11 1 1 12 1 13 1 14 1 15 1 16  o f MVSPOT a t 11:29:43 o n JAN 25,  c c  NODE  1988 f o r CC1d=NHER on G  AREA  C  c c c c c  DELR = RAD / RNR DELS " 2.0 * PI / RNS AREAO = ( D E L R * D E L R ) * PI / 4.0 DO 20 I = 1, NR RI = F L O A T ( I ) I F ( I . E O . N R ) GO TO 10 •INNER* AREA(I)=RI*DELR*DELR*DELS GO TO 20 •OUTER* 10 AREA(I)=0.5*(RNR-0.25)*DELR*DELR*DELS 20 NODE  CONTINUE ARRANGEMENTS  C  NNEI = EBRAD / DELR + 0.5 CYCLE = 1.0 / HZ TIM1 = CYCLE / RNS ISMAX = TIM1 / DELT + 0.5 IF (ISMAX .LT. 1) GO TO 440 C C PHYSICAL PROPERTIES ( S I - U N I T ) C •HEAT CONDUCTION* C T I K = 28.2 CPTI = 6 9 0 . 0 F T I K = T I K * FF ALPHA = F T I K / (CPTI*RHO) FO = ALPHA * DELT / (DELR*DELR) BI = HT * DELR / F T I K C C •RADIATION* C EMISS = 0.4 SIGMA = 5.67E-8 TAMBT = 298. 15 C C •EVAPORATION* C R = 8.3144 TIMOL = 0.0479 ALMOL = 0 . 0 2 7 GAMTI = 1.0 GAMAL = 0.028 HTI = 8.88E6 HAL = 1.08E7 C c EB POWER C AEB = F L O A T ( N N E I ) * DELS * DELR * DELR OIK •= POWERK / AEB 01 « OIK * 1000.0  c  C INITIAL  CONCENTRATION  OF  122  ALLOY  MASS  BALANCE  Listing 1 17 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174  o f MVSPOT a t 1 1:29:43 on JAN 25,  1988 f o r CC i d = NHER o n G  C WTIO = WEIGHT * WPCTT / 100.0 WALO = WEIGHT - WTIO XPNAL = WPCTA / ALMOL XPNTI = WPCTT / TIMOL XAL = XPNAL / (XPNAL + XPNTI) XTI = 1 . 0 - XAL C C WRITE LIQUID DIMENSIONS AND ETC, C WRITE ( 6 , 4 8 0 ) RAD, DEPTH WRITE ( 6 , 4 9 0 ) AREAH, VOLUME. WEIGHT WRITE ( 6 , 5 0 0 ) EBRAD, POWERK, AEB, QIK, HZ WRITE ( 6 , 5 1 0 ) NR, DELR, NS, DELS WRITE ( 6 , 5 2 0 ) DELT, TIMEN WRITE ( 6 , 5 3 0 ) F F , F T I K , HT, WPCTA WRITE ( 6 , 5 4 0 ) IMAX. ISMAX, IFREQ C C SORT NODE TYPE C DO 70 1 = 1 , NR DO 60 J = 1 . NS IF ( J .EQ. 1) GO TO 40 - IF ( J .EQ. NS) GO TO 50 C *MIDDLE* NTYPE(I.J) = 5 IF ( I .EQ. 1) N T Y P E ( I . J ) = 4 IF ( I .EQ. NR) N T Y P E ( I . J ) = 6 GO TO 6 0 C *INITIAL* 40 NTYPE(I.J) = 2 IF ( I .EQ. 1) N T Y P E ( I . J ) = 1 IF ( I .EQ. NR) N T Y P E ( I . J ) = 3 GO TO 60 C "LAST* 50 NTYPE(I.J) = 8 IF ( I .EQ. 1) N T Y P E ( I . J ) = 7 IF ( I .EQ. NR) N T Y P E ( I . J ) = 9 60 CONTINUE 70 CONTINUE C C TERMS C DO 80 I = 1, NR RI = F L O A T ( I ) A 1 ( I ) = (RI - 0.5) / RI A2( I ) = (RI + 0.5) / RI A 3 ( I ) = 1.0 / ( R I * R I * D E L S * D E L S ) 8 0 CONTINUE B = DELR * DELR / ( D E P T H * F T I K ) C = 2.0 * DELS / PI 01 = 2.0 * (RNR - 0.5) / (RNR - 0.25) D2 = 2.0 * RNR / (RNR - 0.25) D3 = 1.0 / (RNR *(RNR - 0.25 ) *DELS*DELS) C C SET I N I T I A L CONDITIONS C TO = TEMPIK  123  Listing 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 21 1 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232  o f MVSPOT a t 11:29:43 o n JAN 25,  90 100  1988 f o r CC1d=NHER on G  TOO « TEMPIK TOC = TEMPIC S T I O J = 0.0 SALOJ = 0.0 DO 100 I = 1, NR DO 9 0 J = 1, NS T ( I , J ) = TEMPIK T O ( I . J ) = TEMPIK T C ( I , J ) = TEMPIC S T I J ( I , J ) = 0.0 S A L J ( I , J ) = 0.0 CONTINUE CONTINUE TIME = 0.0 I COUNT = 0 JNODE = 0  C C WRITE I N I T I A L CONDITIONS C WRITE ( 6 , 5 5 0 ) TEMPIC WRITE ( 7 , 6 0 0 ) HZ WRITE ( 7 , 6 1 0 ) F F , F T I K , POWERK WRITE (.7,620) EBRAD, NNEI WRITE ( 7 , 6 3 0 ) TIME. T C ( N N E I , 1 ) C C START C A L C U L A T I O N C 1 10 NEB = 1 120 ICOUNT = ICOUNT + 1 IF (ICOUNT .GT. I MAX) GO TO 420 JNODE = JNODE + 1 TIME = F L O A T ( I C O U N T ) * D E L T C C TEMPERATURE ADJUSTMENT FOR CALCULATION OF RADIATION & EVAPORATION HEAT LOSS C C TOA = 1 . 5 * TO - 0 . 5 * TOO DO 140 I = 1, NR DO 130 J = 1, NS T A ( I , J ) = 1.5 * T ( I , J ) - 0.5 * T O ( I , J ) CONTINUE 130 140 CONTINUE C C RADIATION + EVAPORATION HEAT LOSS C QRO = O L ( T O A ) DO 160 I = 1, NR DO 150 J = 1, NS TA1 = T A ( I , J ) OR(I,J) = 0L(TA1) 150 CONTINUE 160 CONTINUE C C SUM OF TI & AL LOSS AT EACH NODE(KG/M2) C S T I O J • S T I O J + T I J ( T O A ) * DELT SALOJ = S A L O J + A L J ( T O A ) * DELT DO 180 I = 1, NR  124  Listing 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  o f MVSPOT a t 11:29:43 on JAN 25. 1988 f o r CC1d=NHER DO  170 J = 1. NS TA1 = T A ( I . J ) TIJ1 = TIJ(TA1) ALJ1 = ALJ(TA1 ) S T I J ( I . J ) = S T I J ( I . J ) + T I J 1 * DELT S A L J ( I . J ) = S A L J ( I . J ) + A L J 1 * DELT 170 CONTINUE 180 CONTINUE C C TOTAL TI 8 AL LOSS (KG) C A T I J = S T I O J * AREAO A A L J = S A L O J * AREAO DO 220 I = 1, NR DO 210 J = 1, NS AATIJ = S T I J ( I . J ) * AREA(I) AAALJ = S A L J ( I . J ) * AREA(I ) C *SUM* ATIJ = ATIJ + AATIJ A A L J = A A L J + AAALJ 210 CONTINUE 2 2 0 CONTINUE C C MASS BALANCE : ADJUST T I & AL QUANTITY IN MELT C WTI = WTIO - A T I J WAL = WALO - A A L J WT = WTI + WAL WPCTA = WAL / WT * 100.0 XNTI = WTI / TIMOL XNAL = WAL / ALMOL XNT = XNTI + XNAL XAL = XNAL / XNT XTI = 1 . 0 - XAL C C SET EB MOVEMENT C QEBO=0.0 DO 240 I = 1. NR DO 230 J = 1. NS Q E B ( I , J ) = 0.0 IF ( I .NE. NNEI) GO TO 230 IF ( J .EQ. NEB) Q E B ( I . J ) = QI 230 CONTINUE 240 CONTINUE C C F I N I T E D I F F E R E N C E EQUATIONS C C "CENTER PART* C SUMT = 0.0 DO 250 J = 1, NS SUMT = SUMT + T ( 1 , J ) 2 5 0 CONTINUE TON=C*F0*(SUMT-RNS*TO)+(QEBO-QRO)*B*F0+TO C C SORT C  125  on G  Listing 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348  o f MVSPOT  a t 11:29:43 o n JAN 25, 1988 f o r CC1d=NHER  on G  DO 3 6 0 1 = 1 , NR DO 350 J = 1. NS NTY = N T Y P E ( I . J ) GO TO ( 2 6 0 , 2 7 0 . 2 8 0 , 290, 3 0 0 . 310. 320. 330. 3 4 0 ) . NTY C C C  *TYPE1* 260 1 2  C C C  *TYPE2* 270 1 2  C C C  TN(I,J)=F0*(A1(I)*T(1-1,J)+A2(I)*T(1+1,J)+A3(I)*( T(I,NS)+T(I,J+1))+B*(QEB(I,J)-QR(I,J)))+(1.0-2.0* FO-2.0*FO*A3(I))*T(I.J) GO TO 350  *TYPE3* 280 1 2  C C C  T N ( I , J ) = F O * ( A 1 ( I ) * T 0 + A 2 ( I ) * T ( 1 + 1 ,J) + A3(I ) * ( T ( I , N S ) + T(I,J+1))+B*(OEB(I,J)-0R(I,J)))+(1.0-2.0*F0-2.0*FO* A3(I)) * T(I.J) GO TO 350  TN(I,J)=F0*(D1*T(I-1,J)+D2*BI*TAMBT+D3*(T(I,NS)+T( I.J+1))+B*(OEB(I,J)-OR(I,J)))+(1.0-D1*F0-D2*BI*F0 -2.0*D3*FO)*T(I,J) GO TO 350  *TYPE4* 290 1 2  TN(I,J)=FO*(A1(I)*T0+A2(I)*T(1+1,J)+A3(I)*(T(I,J1 ) + T ( I , J + 1 ) ) + B * ( O E B ( I , J ) - O R ( I , J ) ) ) + (1.0-2.0*F02.0*FO*A3(I))*T(I,J) GO TO 350  C C *TYPE5* C 300 T N ( I , J ) = F 0 * ( A 1 ( I ) * T ( I - 1 , J ) + A 2 ( I ) * T ( I + 1,J)+A3 ( I )* ( 1 T ( I , J - 1 )+T(I,J+1 ) ) + B * ( O E B ( I , J ) - O R ( I , J ) ) ) + (1.0-2. 2 0*FO-2.0*FO*A3(I))*T(I.J) GO TO 350 C C *TYPE6* C 310 T N ( I , J ) = F0*(D1*T(1-1,J)+D2*BI*TAMBT+D3*(T(I , J-1) + 1 T(I,J+1))+B*(OEB(I,J)-OR(I.J)))+(1.0-D1*FO-D2*BI* 2 FO-2.0*D3*FO)*T(I,J) GO TO 3 5 0 C C *TYPE7* C 320 T N ( I , J ) = FO*(A 1 ( I ) * T 0 + A 2 ( I ) * T ( 1 + 1,J) + A 3 ( I ) * ( T ( I , J 1 1 )+T(I , 1 ) ) + B * ( Q E B ( I , J ) - Q R ( I . J ) ) ) + ( 1 . 0 - 2 . 0 * F 0 - 2 . 0 * F 0 * 2 A3(I ) ) * T ( I , J ) GO TO 350 C C C  *TYPE8* 330  T N ( I , J ) = F O * ( A 1 ( I ) * T ( 1 - 1 , J ) + A 2 ( I ) * T ( 1 + 1 , J ) + A3( I )*(  126  L i s t i n g of MVSPOT at 11:29:43 on JAN 25. 1988 f o r CC1d=NHER on G 349 350 351 352 353 354 355 356' 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 38 1 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406  1 2  T ( I . J - 1 )+T(I,1))+B*(QEB(I,J)-QR<I.J))) + (1.0-2.0* FO - 2.0*FO*A3(I))* T ( I , J ) GO TO 350  C C *TYPE9* C 340 TN(I,J )=F0*(D1 * T ( I - 1 . J )+D2*BI*TAMBT+D3*(T(I,J-1) + 1 T ( I , 1 ) )+B*(OEB(I,J)-OR(I.J))) + (1.0-D1*FO-D2*BI*FO2 2.0*D3*FO)*T(I.J) C 350 CONTINUE 360 CONTINUE C C TEMPERATURE MUST BE LESS THAN 3285.0 DEGC (3558.15 K) C IF(TON.GT.3558.15) T0N=3558.15 DO 364 1=1,NR DO 362 J=1,NS T1=TN(I,J) IF (T1.GT.3558 .15) TN(I.J)= 3558.15 362 CONTINUE 364 CONTINUE C C SUBSTITUTE NEW TEMPERATURE IN OLD ONE C TOO = TO TO = TON DO 380 I = 1 , NR DO 370 J = 1, NS T0(I.J) = T(I,J) T(I , J ) = TN(I,J) 370 CONTINUE 380 CONTINUE C PRINT OUT CONTROL IF ((ICOUNT/IFREO)*IFREO .NE. ICOUNT) GO TO 410 C C CHANGE TEMPERATURE UNIT: K TO C C TOC = TO - 273.15 DO 400 I = 1, NR DO 390 J = 1. NS T C ( I , J ) = T ( I , J ) - 273.15 390 CONTINUE 400 CONTINUE C C WRITE TEMPERATURE & TI AND AL LOSS C WRITE(7,630)TIME.TC(NNEI.1),TC(3.2).TC(4.2),OEB( NNEI .1).ATIJ. 1 AALJ,WPCTA C C NEXT STEP & CONTROL C 410 IF (JNODE .LT. ISMAX) GO TO 120 JNODE = 0 NEB = NEB + 1 IF (NEB .LE. NS) GO TO 120 GO TO 110 C  127  Listing 407 408 409 410 411 412 4 13 414 415 416 4 17 418 419 . 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465  o f MVSPOT a t 11:29:43 on JAN 25. 1988 f o r CC1d=NHER  on G  C WRITE RESULTS AT END TIME C , 4 2 0 WRITE ( 6 , 5 6 0 ) TIME ' 1 WRITE ( 6 , 5 6 5 ) TOC DO 430 I » 1, NR WRITE ( 6 . 5 7 0 ) ( T C ( I . J ) . J = 1 . N S ) 4 3 0 CONTINUE WRITE ( 6 , 5 8 0 ) A T I J , A A L J , WPCTA GO TO 1000 4 4 0 WRITE ( 6 , 5 9 0 ) GO TO 1000 C C READ FORMAT C 460 FORMAT ( 5 F 1 0 . 5 ) 4 7 0 FORMAT ( 5 1 1 0 ) C C WRITE FORMAT C 4 8 0 FORMAT (5X, ' " D I M E N S I O N OF MOLTEN METAL**'/8X,' RADIUS''.F5.3, 1 'M' ,3X, 'DEPTH'' ,F5 . 3, 'M' ) 4 9 0 FORMAT (8X,'AREA=',E10.4,'M2',3X.'VOLUME=',E10.4,'M3',3X, 1 'WEIGHT=',F7.3,'KG'//) 500 FORMAT ( 5 X , ' * * E B CONDITIONS**'/8X,'EB T R A V E L L I N G RADIUS''.F5.3 1 'M'/8X,'BEAM POWER=',F7.3,'KW',3X,'BEAM AREA=',E10.4, 2 ' M2 ' , 3X, 'POWER DENSITY'' ,E15.3, 'KW/M2'/8X, 3 'FREQUENCY'',F7.2.'HZ'/) 510 FORMAT (5X, ' ""CALCULATING CONDITIOS**'/8X . 'RNODE : NUMBER' 1 I5.3X,'SIZE'',F7.4,'M'/8X.'THETA NODE: NUMBER'',15. 2 3X, ' S I Z E ' ' , F 5 . 3 . 'RAD'/) 5 2 0 FORMAT (SX.'TIME STEP'' ,F 10.3, ' SEC'/8X,'END TIME =',F10.3, 1 ' SEC'/) 530 FORMAT (8X, ' FF= ' . F7 . 2.3X, 'HEAT CONDUCTIVITY OF TI = ',F7.2, 1 'WM/K'/8X,'HEAT TRANSFER C O E F F I C I E N T = ' , F 1 0 . 3 , 2 'W/M2/K'/8X,'INITIAL AL WEIGHT P E R C E N T ' ' , F 5 . 2 / ) 540 FORMAT (5X. '*"CONTROLLER**'/8X, 'TOTAL S T E P S ' ' , 15, 3X, 1 'STEPS/NODE''. 15, 3X, 'PRINT FREQUENCY'', 15///) 550 FORMAT (5X, ' ""RESULTS**'//8X, '* I N I T I A L TEMPERATURE'', F7.2, 1 ' D E G C , 3X, 'THROUGHOUT IN MELT'//) 560 FORMAT (8X, '"TEMPERATURE D I S T R I B U T I O N (DEGC) AT TIME'' .F 10.3, 1 ' SEC'/) 565 FORMAT ( 9 X . F 7 . 2 ) 570 FORMAT (8X, 1 0 ( 1 X , F 7 . 2 , 1 X ) ) 580 FORMAT (//8X, '"TOTAL TI LOSS'', E10.4, ' ( K G ) " / 8 X , 1 '"TOTAL AL LOSS''. E10.4, ' ( K G ) ' / / 8 X , 2 '"FINAL AL CONCENTRATION'', F5 . 2 , ' WT"/.' ) 590 FORMAT (SX.'TIME STEP IS LARGER THAN TO BE SPENT IN ONE NODE') 6 0 0 FORMAT (5X,'FREQUENCY=' . F 7 . 2 . 'HZ') 610 FORMAT ( 5 X . ' F F = ' , F7.2.4X.'HEAT CONDUCTIVITY OF MOLTEN T I = ' , 1 F7.2.'WM/K'/5X.'POWER INPUT'',F7.2.'KW'/) 620 FORMAT (/5X,'DATA AT A POINT( R='. F 1 0 . 3 . '(M):'.1X, 1 'RNODE N O . » ' , 1 3 , ' )'//7X, ' T I M E ( S ) ' ,3X , ' T E M P . ( C ) ' , 3X , 2 ' " T M I D . ( C ) * ' , 2X , ' " T O B S . ( C ) " ' ,4X. 3 'EB(KW/M2)',6X . ' TI L O S S ( K G ) ' .4X, 'AL LOSS(KG ) ' ,4X. 4 ' WT*/.AL' ) 630 FORMAT ( 3 X . F 1 0 . 3 , 3 ( 2 X . F 1 0 . 2 ) . 3 ( 5 X . E 1 0 . 4 ) . 2 X , F 7 . 2 ) C 10O0 STOP ENO  128  A-2-3 Flow c h a r t of 2-D  (  START  model  j  >(p  I C O U N T = 0,  JNODE=0,  TIME =  0.0  —f~Z NEB=1  t ICOUNT = ICOUNT + l  JNODE = JNODE+l TIME = TIME + D E L T » Temperature Radiation & Heat  adjust.  Evaporation  Loss  t T i & Al loss Mass  FINITE  Balance  DIFFERENCE  CALCULATION  129  ( E N i [ )  

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