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The thermal regime during electron beam hearth remelting Tripp, David William 1987

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T H E T H E R M A L REGIME DURING E L E C T R O N B E A M H E A R T H REMELTING by  DAVID WILLIAM TRIPP  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR T H E DEGREE OF MASTER OF APPLIED SCIENCE  in Metals and Materials Engineering THE FACULTY OF GRADUATE STUDIES APRIL 1987  We accept this thesis as conforming to the required standard  THE UNIVERSITY  OF BRITISH COLUMBIA  23 April 1987  © DAVID WILLIAM TRIPP,  1987  In  presenting  degree  at  this  the  thesis  in  University of  partial  fulfilment  of  of  department publication  this or of  thesis for by  his  or  requirements  British Columbia, I agree  freely available for reference and study. I further copying  the  that the  representatives.  an advanced  Library shall make it  agree that permission for extensive  scholarly purposes may be her  for  It  this thesis for financial- gain shall not  is  granted  by the  understood  that  head  of  copying  my or  be allowed without my written  permission.  Department of  M&T7*Z&> hjK>D  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6G/81)  ffT&tZ,, &t$>  (sJ&ZtL/'  Abstract  E l e c t r o n beam h e a r t h r e m e l t i n g i s e x t e n s i v e l y u s e d in  refining  of  superalloys,  titanium  alloys  and  the  r e c y c l i n g o f t h e s e m a t e r i a l s . The r e m o v a l o f i m p u r i t i e s a n d exhogenous p a r t i c l e s  during  depends p r i m a r i l y  the time  developed  on  the hearth  melting  a t temperature  w i t h i n a pool of molten metal.  m e l t e r s have a c t e d l a r g e l y such parameters as melt  In the past  hearth  to  specify  r a t e s , power l e v e l s a n d s k u l l  to p r e d i c t c e r t a i n parameters  i n p u t and melt  relationship  on e m p i r i c a l e v i d e n c e  T h i s work d e s c r i b e s a m a t h e m a t i c a l  element e v a p o r a t i o n  operation  sizes.  model w h i c h c o u l d be u s e d  (such as p o o l volume o r  alloy  r a t e s ) when g i v e n s k u l l g e o m e t r y ,  power  rate.  A three  dimensional  steady  state  heat  transfer  model o f b o t h t h e s k u l l and water c o o l e d c o p p e r mould d u r i n g electron has  been  beam h e a r t h r e m e l t i n g h a s been d e v e l o p e d . used  temperature, presence  to  liquid  of the  investigate motion,  power  h e a r t h mould  operating refining the  general  parameters  the  input,  choice  depends  skull  r a t e on  on  of a  any  surface geometry, parameters  combination  balance  of  between  the  ( i . e . l i q u i d volume)  and  e l e m e n t s by e v a p o r a t i o n . i i  of  melting.  c a p a c i t i y of the process  l o s s of a l l o y  effects  and melt  such a s p o o l volume d u r i n g s k u l l In  the  The m o d e l  In the case  of  m e l t i n g pure between  materials  refining It  l i q u i d pool  found  more e f f e c t i v e  alloy  increase elements  changes motion.  the pool within  the balance  energy  forced  is  use.  convection  is  i n i n c r e a s i n g the volume of the  s i n g l e parameter. Increasing  the s k u l l ,  removing the water c o o l e d also  that  t h a n any o t h e r  power i n p u t t o  CP t i t a n i u m )  c a p a c i t y and e f f i c i e n t  was  significantly  (e.g.  i n c r e a s i n g the copper mould  volume. the  skull  i n t h e power d i s t r i b u t i o n  The  s k u l l width  from around the evaporation  were  the  most  and skull  rates  of  effected  by  and t h e d e g r e e o f  liquid  Table of Contents  Abstract  i i  Table of Contents  iv  List  of Tables  ix  List  of F i g u r e s  x  List  of Symbols  xvi  Acknowledgements  xviii  1 .0 I n t r o d u c t i o n  1  1.1.  E l e c t r o n Beam S y s t e m s  2  1.1.1 Beam G e n e r a t i o n a n d C o n t r o l  2  1.1.2 Vacuum S y s t e m s  3  1.2. E l e c t r o n Beam R e m e l t i n g P r o c e s s e s  4  1.2.1 D r i p a n d P o o l M e l t i n g  4  1.2.2 H e a r t h M e l t i n g  6  1.2.2.1 P r o c e s s A d v a n t a g e s  8  1.2.2.2 P r o c e s s D i s a d v a n t a g e s  10  1.2.2.3 A p p l i c a t i o n s o f EBCHR  11  1.2.2.3.1 R e c y c l i n g o f T i t a n i u m S c r a p 1.2.2.3.2 Removal o f O x i d e  Inclusions  11 from  Superalloys  17  1.2.2.3.3 I n - S p e c  1.3.  18  1.2.2.4 O p e r a t i n g H e a r t h F u r n a c e s  18  The T h e r m a l Regime D u r i n g EBCHR  19  2.0 M o d e l l i n g H e a t F l o w 2.1.  Titanium Electrodes  i n t h e E l e c t r o n Beam H e a r t h  The Heat F l o w E q u a t i o n  22 22  iv  V  2.2. S k u l l  Boundary C o n d i t i o n s  23  2.2.1 Top B o u n d a r y C o n d i t i o n  23  2.2.2 S i d e  26  Boundary C o n d i t i o n s  2.2.3 B o t t o m B o u n d a r y C o n d i t i o n  27  2.2.4 C e n t e r l i n e B o u n d a r y C o n d i t i o n  27  2.3. H e a r t h B o u n d a r y C o n d i t i o n s  28  2.3.1 E x t e r i o r  Surface  Boundary C o n d i t i o n  28  2.3.2 I n t e r i o r  Surface  Boundary C o n d i t i o n s  28  2.3.3 C e n t e r l i n e B o u n d a r y C o n d i t i o n  29  2.3.4 W a t e r C h a n n e l B o u n d a r y C o n d i t i o n  29  2.4. B l o c k  Model  30  2.5. Summary  31  3.0 M o d e l F o r m u l a t i o n  33  3.1. I n t r o d u c t i o n  33  3.2. C o l d H e a r t h M o d e l  33  3.2.1 S k u l l  Boundary C o n d i t i o n s  34  3.2.1.1 Top B o u n d a r y C o n d i t i o n  34  3.2.1.2 S i d e B o u n d a r y C o n d i t i o n  37  3.2.1.3 B o t t o m B o u n d a r y C o n d i t i o n  39  3.2.1.4 T h e r m a l C o n d u c t i v i t y  39  3.2.2 H e a r t h B o u n d a r y C o n d i t i o n s  42  3.2.2.1 I n t e r i o r  Boundary C o n d i t i o n  45  3.2.2.2 E x t e r i o r  Boundary C o n d i t i o n s  45  3.2.2.3 W a t e r C h a n n e l B o u n d a r y C o n d i t i o n s  46  3.2.2.4 T h e r m a l C o n d u c t i v i t y  49  3.3. B l o c k  Model  3.4. N u m e r i c a l T e c h n i q u e  49 49  vi 3.4.1 Non L i n e a r i t i e s  50  3.4.1.1 T h e r m a l C o n d u c t i v i t y  51  3.4.1.2 R a d i a t i o n  52  3.4.1.3 I t e r a t i v e S o l u t i o n  53  4.0 M o d e l R e s u l t s 4.1.  55  Introduction  4.2. T e m p e r a t u r e  55 Distribution  Model  55  4.2.1 E f f e c t  of S u r f a c e Temperature  64  4.2.2 E f f e c t  of L i q u i d  66  4.3.  Power D i s t r i b u t i o n  Movement Model  74  4.3.1 The Power D i s t r i b u t i o n  78  4.3.1.1 Beam L o s s e s  78  4.3.1.2 M e l t R a t e A d j u s t m e n t s  80  4.3.2 R e s u l t s  88  4.3.2.1 E f f e c t  o f Power D e n s i t y  4.3.2.2 I n f l u e n c e o f L i q u i d  Motion  4.3.3 The H e a t B a l a n c e  88 94 94  4.4. Beam S p o t T e m p e r a t u r e  104  4.5. E x p e r i m e n t a l V e r i f i c a t i o n  109  5.0 H e a r t h D e s i g n a n d O p e r a t i o n 5.1. F a c t o r s A f f e c t i n g  H e a r t h D e s i g n and O p e r a t i o n  116 ...116  5.2. H e a r t h D e s i g n  116  5.2.1 G e o m e t r y  117  5.2.2 The E f f e c t 5.3.  of the H e a r t h Mould  Furnace Operations  5.3.1 E f f e c t  of M e l t Rate  5.3.2 Power D i s t r i b u t i o n  127 132 133 136  vi i 5.4. Summary  150  6.0 Summary a n d R e c o m m e n d a t i o n s f o r F u t u r e Work  152  6.1. Summary  152  6.2. R e c o m m e n d a t i o n s f o r F u t u r e Work  153  6.2.1 V e r i f i c a t i o n 6.2.2 Power D i s t r i b u t i o n 6.2.3 E v a p o r a t i o n R a t e s 6.3. C o n c l u d i n g Remarks List  of References  APPENDIX  1 Beam S p o t M o d e l  154 a n d Power L e v e l  154 155 155 156 159  List  of Tables  Table  Page  1.1 S p e c i f i c a t i o n s f o r I n t e r s t i t a l Titanium Alloys  Concentrations i n 16  4.1 H e a t B a l a n c e C a l c u l a t i o n s f o r T e m p e r a t u r e D i s t r i b u t i o n Boundary C o n d i t i o n  63  4.2 T h e o r e t i c a l Maximum E v a p o r a t i o n Temperature D i s t r i b u t i o n s  65  Rates f o r Various  4.3 Summary o f P o o l D a t a a n d H e a t B a l a n c e f o r T e m p e r a t u r e D i s t r i b u t i o n Runs  Calculations 73  4.4 Summary o f H e a t B a l a n c e C a l c u l a t i o n s , P o o l D a t a and E v a p o r a t i o n R a t e s f o r Power D i s t r i b u t i o n M o d e l . ..99 4.5 T h e r m a l E f f i c i e n c y o f E l e c t r o n Various Conditions  Beam M e l t i n g  Under  105  5.1 R e s u l t s o f C a l c u l a t i o n s on t h e EB S k u l l a s a Function of Thickness  121  5.2 R e s u l t s o f C a l c u l a t i o n s Function of Width  125  on t h e EB S k u l l a s a  5.3 E v a p o r a t i o n R a t e s , P o o l D a t a a n d H e a t B a l a n c e C a l c u l a t i o n s f o r an EB S k u l l W i t h a n d W i t h o u t a Hearth Mould  131  5.4 Power L e v e l s U s e d a s a F u n c t i o n o f M e l t R a t e  135  5.5 P o o l D a t a , E v a p o r a t i o n R a t e s a n d H e a t B a l a n c e C a l c u l a t i o n s as a F u n c t i o n of M e l t Rate  137  vi i i  IX  5.6 P o o l D a t a , H e a t B a l a n c e C a l c u l a t i o n s a n d E v a p o r a t i o n R a t e s f o r Power D i s t r i b u t i o n Runs  149  L i s t of F i g u r e s Page  Figure  1.1  E l e c t r o n Beam D r i p M e l t i n g  6  1.2  E l e c t r o n Beam P o o l M e l t i n g  7  1.3  Electron  9  1.4  T i t a n i u m - N i t r o g e n Phase Diagram  13  1.5  T i t a n i u m - Oxygen P h a s e D i a g r a m  14  1.6  T i t a n i u m - Carbon Phase Diagram  15  2.1  H e a t F l o w S c h e m a t i c o f t h e EBCHR P r o c e s s  25  3.1  G e o m e t r y o f t h e S k u l l a n d H e a r t h U s e d by V i k i n g Metallurgical  35  3.2  T h e r m a l C o n d u c t i v i t y o f T i t a n i u m 6A1-4V a s a F u n c t i o n of Temperature  41  3.3  T h e r m a l C o n d u c t i v i t y o f T i t a n i u m 6A1-4V a s a F u n c t i o n of Temperature F o l l o w i n g M o d i f i c a t i o n f o r L i q u i d Motion  43  3.4  X-Ray P h o t o g r a p h o f a J o h n s o n S k u l l F r a c t i o n of V o i d s  44  4.1  T y p i c a l Temperature D i s t r i b u t i o n Used. Superheat T e m p e r a t u r e = 150 °C  Beam C o l d H e a r t h R e m e l t i n g  x  Showing  Large  57  2  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 °C, Superheat = 1 1 0 °C, LTCMF = 1  3  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, Superheat = 1 1 0 °C, LTCMF = 1  4  Pool P r o f i l e s at the Centerline, Contouring I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF = 1  5  X - r a y P h o t o g r a p h o f an A. J o h n s o n S k u l l the L o c a t i o n of t h e S o l i d u s  6  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 °C, S u p e r h e a t = 150 °C, LTCMF = 1  7  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, S u p e r h e a t = 150 °C, LTCMF = 1  8  Pool P r o f i l e s at the Centerline, Contouring I n t e r v a l = 100 °C, S u p e r h e a t = 150 °C, LTCMF = 1  9  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 °C, S u p e r h e a t = 200 °C, LTCMF = 1  Showing  10 C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, S u p e r h e a t = 200 °C, LTCMF = 1 11 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 200 °C, LTCMF = 1 12 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF = 1  4.13 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF = 2  76  4.14 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF = 5  77  4.15 S c h e m a t i c D i a g r a m o f I n t e r a c t i o n s Between an Beam a n d an I r r a d i a t e d S u r f a c e I n d i c a t i n g E n e r g y L o s s Mechanisms  79  4.16 Number o f B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A t o m i c Number o f t h e I r r a d i a t e d M a t e r i a l f o r a N o r m a l Beam, A c c e l e r a t i n g V o l t a g e = 10 K V .  81  4.17 Number o f B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A n g l e o f I n c i d e n c e o f t h e Beam, A c c e l e r a t i n g V o l t a g e = 10 K V .  82  4.18 E n e r g y D i s t r i b u t i o n o f B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A t o m i c Number f o r a N o r m a l Beam, A c c e l e r a t i n g V o l t a g e = 10 K V .  83  4.19 Power L o s s e s Due t o B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A t o m i c Number f o r a N o r m a l Beam, A c c e l e r a t i n g V o l t a g e = 10 K V .  84  4.20 S c h e m a t i c Heat F l o w D i a g r a m o f t h e E l e c t r o n Beam S k u l l  86  2  2  2  2  4.21  A T y p i c a l Power D i s t r i b u t i o n U s e d Metallurgical  By V i k i n g  . .89  4.22 C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 °C, T o t a l Power = 33 KW, LTCMF = 1  90  4.23 C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, T o t a l Power = 33 KW, LTCMF = 1  91  xi i i 4.24 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 33 KW, LTCMF = 1  92  4.25 S u r f a c e T e m p e r a t u r e D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 1  93  Contouring 33 KW,  4.26 C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 °C, T o t a l Power = 36 KW, LTCMF = 1  95  4.27 C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, T o t a l Power = 36 KW, LTCMF = 1  96  4.28 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 36 KW, LTCMF = 1  97  4.29 S u r f a c e T e m p e r a t u r e D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 1  98  Contouring 36 KW,  4.30 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 33 KW, LTCMF = 2 4.31  S u r f a c e Temperature D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 2  Contouring 33 KW,  100  101  4.32 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 33 KW, LTCMF = 5  102  4.33 S u r f a c e T e m p e r a t u r e D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 5  103  Contouring 33 KW,  4.34 U n s t e a d y S t a t e R e s p o n s e o f t h e T e m p e r a t u r e Under t h e Beam S p o t a s a F u n c t i o n o f Power I n p u t  107  xiv 4.35  U n s t e a d y S t a t e Response o f t h e Temperature Under t h e Beam S p o t A f t e r 20 m i l l i s e c o n d s a t 200 KW  108  4.36 P h o t o m a c r o g r a p h o f t h e B l o c k U s e d D u r i n g t h e Experiment a t V i k i n g Showing t h e Boundary of t h e Liquid Pool. A - Liquid Pool, B - S o l i d Block, M a g n i f i c a t i o n : 0.375X , Beam Power : 80 KW  111  4.37 P h o t o m a c r o g r a p h o f t h e B l o c k U s e d D u r i n g t h e Experiment a t Johnson Showing t h e Boundary of t h e Liquid Pool. A - Liquid Pool, B - Solid Block, M a g n i f i c a t i o n : 0.580X  112  4.38 P o o l P r o f i l e s o f t h e B l o c k M o d e l U s i n g t h e C o n d i t i o n s o f t h e V i k i n g E x p e r i m e n t . LTCMF = 1. Power = 56 KW, 5 cm i n d i a  113  4.39  P o o l P r o f i l e s of t h e B l o c k Model f o r t h e V i k i n g E x p e r i m e n t . LTCMF = 1. Power = 50 KW, G a u s s i a n D i s t . a = 10 cm  114  5.1  Power D i s t r i b u t i o n U s e d t o E v a l u a t e S k u l l Thickness  119  5.2  The V a r i a t i o n Thickness  5.3  Variation  o f P o o l Volume w i t h  the E f f e c t of  Skull 120  of Average Heat F l u x w i t h  Skull  Thickness  122  5.4  V a r i a t i o n o f P o o l Volume w i t h S k u l l W i d t h  124  5.5  V a r i a t i o n of Pool Parameters w i t h Dimensionless S k u l l Width Power D i s t r i b u t i o n U s e d t o E v a l u a t e t h e E f f e c t s of t h e H e a r t h M o u l d  1 26 128  Pool P r o f i l e s Copper H e a r t h  129  5.6  5.7  f o r EB S k u l l U s i n g  a Water  Cooled  XV  5.8  P o o l P r o f i l e s f o r EB S k u l l W i t h o u t a W a t e r Copper H e a r t h  5.9  Shape o f t h e Power D i s t r i b u t i o n the E f f e c t  Cooled  Used t o E v a l u a t e  of Melt Rate.  5.10 T h e r m a l E f f i c i e n c y  130  134  as a F u n c t i o n of Melt Rate  138  5.11  P o o l Volume a n d E v a p o r a t i o n R a t e o f A l a s a F u n c t i o n of Melt Rate 5.12 B a s e C a s e Power D i s t r i b u t i o n f o r E v a l u a t i o n o f t h e E f f e c t o f Power D i s t r i b u t i o n on t h e T h e r m a l Regime  141  5.13 Power D i s t r i b u t i o n t o t h e Base Case  142  1. 3.15 KW C i r c l e i n A d d i t i o n  5.14 Power D i s t r i b u t i o n 2. Base C a s e Power D i s t r i b u t i o n 1/2 a s W i d e , Same T o t a l Power 5.15 Power D i s t r i b u t i o n T o t a l Area  139  3. R e c t a n g u l a r  143  Shape o f Same  a s t h e Base Case  144  5.16 P o o l P r o f i l e s f o r t h e Base C a s e  145  5.17 P o o l P r o f i l e s f o r Power D i s t r i b u t i o n  1  146  5.18 P o o l P r o f i l e s f o r Power D i s t r i b u t i o n  2  147  5.19 P o o l P r o f i l e s f o r Power D i s t r i b u t i o n  3  148  L i s t of Symbols Symbol  Definition  Tg  -  Temperature of t h e S k u l l  T  -  Temperature of t h e H e a r t h  u  hi  T  A  -  Ambient  T  w  -  Temperature of t h e C o o l i n g  kg  -  Thermal C o n d u c t i v i t y of the S k u l l  k  -  Thermal C o n d u c t i v i t y of the H e a r t h  eg  -  Emissivity  of t h e S k u l l  e  -  Emissivity  of t h e H e a r t h  -  Bandwidth of the Gaussian D i s t r i b u t i o n or the Stefan Boltzman Constant  ~  S k u l l / H e a r t h Heat T r a n s f e r  Coefficient  w  -  Copper/Water Heat T r a n s f e r  Coefficient  R e  -  R e y n o l d s Number  N„ Pr  -  Prandtl  Q.  -  Heat  V"  -  Water  5  -  Mesh S i z e  A  -  Area  I  -  Backscattered  I,  -  Beam C u r r e n t  H  H  a n  S/H h  N  w  Temperature  Number  Flux Velocity  Current  xvi  Water  xvi i  a  -  A n g l e o f Beam  E  -  Energy  e  -  Fundamental Charge of t h e E l e c t r o n  _  Beam A c c e l e r a t i n g  P P  r  -  Backscattering  -  Beam  Power  Incidence  Voltage  Power  Acknowledgements I would l i k e provided  by  S t e v e and  t o acknowledge the  the boys i n the o f f i c e , B a r r y as  d e p a r t m e n t who matter  how  g u i d a n c e and  contributed  General also  financial also the  encouragement  specifically  a l l the to t h i s  Bob,  other people  Mike, in  the  any  way,  no  Alec M i t c h e l l  for  his  thesis in  s i n c e r e s t thanks t o Dr. insights  t h a n k s a l s o t o A.  Thanks  w e l l as  and  small. My  and  support  i n t o the  Johnson  Electric go  Co.  out  support.  industry.  Metals, Viking Metallurgical for their  to  Tom  special melting  El  support  Dorado  d u r i n g the  Resources  N i c h o l of the  for  problems  Co. work. their  computing center  i n s t r u m e n t a l i n s o l v i n g some n u m e r i c a l  My  was  during  work. Finally,  interest, help, which aided  I would l i k e  support  and  i n the completion  t o t h a n k my  sometimes of t h i s  xvi i i  family for  constant  work.  their  badgering  CHAPTER 1 Introduction In recent process and  years  routes  alloys  i t has  become a p p a r e n t t h a t t h e  f o r the  production  f o r the aerospace  induction melting  (VIM)  (ESR)  arc  or  vacuum  material with the  specification  vacuum a r c density  of  (VAR)  s i z e s and the engine  processing  does  i n c l u s i o n s or the  materials  inadequate.  by e l e c t r o s l a g  remelting  inclusion  of super c l e a n  i n d u s t r y are  followed  traditional  not  does  Vacuum  remelting  not  produce  d i s t r i b u t i o n s that manufacturers.  As  remove  the  hard-alpha  either  defects  from  meet well, high  titanium  alloys. I n a d d i t i o n , buy i n d u s t r y are greater  -  extremely  jet  s i g n i f i c a n t amount available  t o be  the  in  of b o t h  on  the  engines.  i n the  the order production  This  means  aerospace  of  10:1  of  that there  s u p e r a l l o y and  particles  the manufacturing  f i n i s h e d product.  (HDI's) - u s u a l l y t o o l  introduced  processes  For  may  f i n d t h e i r way  example, h i g h d e n s i t y  remelting  1  treatment.  a  scrap  of  the  both scrap into  inclusions  b i t chips - in titanium w i l l  e v e n a t r i p l e vacuum a r c  is  titanium  into  or  critical  r e c y c l e d . In t r a d i t i o n a l p r o c e s s i n g  m a t e r i a l s , exogenous during  high -  especially  components f o r  to f l y r a t i o s  survive  2 The aircraft  aerospace industry  engine  manufacturers  process o f f e r i n g clean  processing  a n d more s p e c i f i c a l l y h a v e been  interested  to the  traditional  an a l t e r n a t i v e routes.  In general,  cold hearth  the  e l e c t r o n beam c o l d h e a r t h r e m e l t i n g f o r the  materials quality 1.1.  of  forging  Electron  integrated  a n d more  beam  melting  and t h e  have e x t e n s i v e  scrap to  high  systems  require  two  that  the various  1.1.1  Beam G e n e r a t i o n a n d C o n t r o l  these  i n order  valves  to  open  and  o f e l e c t r o n beams  are  sequence.  generation  and c o n t r o l  b a s e d on t h e p h y s i c s  of  o p t i c s and a l t h o u g h  many d i f f e r e n t t y p e s  high  manufactured ranging  megawatt l e v e l s ,  generation  Both of  computer c o n t r o l  i n the correct  energy p a r t i c l e s and  electron  of e l e c t r o n  guns  i n power f r o m t h e m i c r o w a t t t o t h e  t h e f u n d a m e n t a l s r e m a i n t h e same.  Electrons  directly  superalloy  titanium  vacuum s y s t e m .  close  emitter  clean  has  Beam S y s t e m s  g u i d e t h e beam a n d i n s u r e  are  that  stock.  c o n t r o l systems  The  super  (EBCHR)  s y s t e m s t o o p e r a t e . T h e s e a r e t h e beam  systems a l s o  a  specifically  process  super  and f o r t h e r e c y c l i n g of  Electron  and  processes  production  in  i t h a s been f o u n d  the  allowed  remelting  the  are  generated  (usually tungsten or i n d i r e c t l y .  by h e a t i n g  f o r melting  A high  an  applications)  electrical  electron either  p o t e n t i a l (20 - 50  3  KV f o r m e l t i n g )  is  then a p p l i e d  between t h e  hot  emitter  c a t h o d e a n d an a d d i t i o n a l p o s i t i v e e l e c t r o d e o r a n o d e . causes  electrons  to  be  emitted  a c c e l e r a t e d t o h i g h energy The  choice  levels  from  the  This  cathode  i n an e l e c t r o s t a t i c  and field.  o f h e a t i n g method, c a t h o d e s i z e and shape as w e l l  as e l e c t r o d e c o n f i g u r a t i o n d e p e n d s  on t h e a p p l i c a t i o n , t h e  operating parameters  conditions i n  of the  chamber a n d d e s i r e d c a t h o d e  gun,  the  melt  life . 1  I n a d d i t i o n t o t h e two r e q u i r e d e l e c t r o d e s and c a t h o d e ) a beam g e n e r a t i o n for  f o c u s s i n g and  shaping.  s y s t e m may c o n t a i n  Electrode  such as c u r r e n t  aperture  spot.  Once  generated  and  electrodes  s e l e c t i o n and  d e t e r m i n e t h e m a i n beam p a r a m e t e r s and l o c a t i o n of t h e f o c a l  focussed  the  control  electron  optics  electron  schemes  are  and  provide  beam  can  on  excellent  the  be  lenses.  principles  control  of  of  the  beam. At t h e present  melting  based  design density,  c o n t r o l l e d as t o p o s i t i o n u s i n g a s e r i e s of magnetic These  (anode  operations.  time, a x i a l  Further  guns a r e p r e f e r r e d  discussion  of r i n g ,  for  line  and  2-4 transverse  guns c a n be f o u n d i n t h e l i t e r a t u r e .  1.1.2 Vacuum S y s t e m s E l e c t r o n beam s y s t e m s  f o r melting  require  rather -4  extensive  vacuum s y s t e m s . A p r e s s u r e  on t h e o r d e r  o f 10  Pa  (10 ^ t o r r ) i n t h e gun chamber i s r e q u i r e d t o a l l o w t h e beam  4 g e n e r a t i o n system c l o s e l y designed  to  operate. With  passages  differential  between t h e  gun  and  pumping, the  melt  chamber a n d modern r o b u s t guns i t i s p o s s i b l e t o o p e r a t e p r e s s u r e s as h i g h as  13 Pa  (10  -1  torr)  Chamber p r e s s u r e s a r e more l i k e l y torr)  for conventional metal The  for both  i n the melt  t o be a r o u n d 0.13  at  5 chamber. Pa  (10  processing.  p r e s e n c e o f t h i s r e a s o n a b l y g o o d vacuum a l l o w s  advantages  electron  beam  the e l e c t r o n  beam  r e m e l t i n g p r o c e s s e s a r e a b l e t o remove d i s s o l v e d g a s e s  from  r e m e l t i n g . Because  and  disadvantages  of the  vacuum l e v e l  the f e e d s t o c k m a t e r i a l . the removal  of v o l a t i l e  manganese f r o m a l l o y environment alloying  1.2.  i m p u r i t i e s such  t i t a n i u m . On  also  t h e vacuum p r o v i d e s  cause  and  t h e o t h e r h a n d , t h e vacuum desired  e l e m e n t s s u c h a s a l u m i n u m f r o m t i t a n i u m 6%  aluminum  and c h r o m i u m f r o m  the  a s magnesium  for  of  4% v a n a d i u m  will  In a d d i t i o n  in  evaporation  superalloys.  E l e c t r o n Beam R e m e l t i n g P r o c e s s e s E l e c t r o n beam r e m e l t i n g on an  industrial  scale  has  5  been c o m m e r c i a l l y time e l e c t r o n  viable  beams  s i n c e t h e mid  have p r o v i d e d  m e l t i n g and r e f i n i n g  1950's. S i n c e  the  power  source  r e f r a c t o r y m e t a l s f o r use i n  1.2.1  of super c l e a n  D r i p and P o o l  Ta).  In  Melting  (see the  r e f i n i n g of the  t h i s process  m e l t i n g chamber  for  superalloy material.  E l e c t r o n beam d r i p m e l t i n g t h e m e l t i n g and  for  aerospace  components, f o r t h e r e c y c l e o f r e a c t i v e m e t a l s c r a p and the p r o d u c t i o n  that  i s used p r i m a r i l y  r e f r a c t o r y metals  bar feedstock  in either  (Nb,  i s introduced  to  a h o r i z o n t a l or v e r t i c a l  for V, the  fashion  f i g u r e 1 . 1 ) . One o r more e l e c t r o n beams t h e n i m p i n g e on s u r f a c e of  i n t o a water solidified  the bar  and d r o p s o f  c o o l e d copper c r u c i b l e  i n a c o n t r o l l e d way  molten m a t e r i a l where t h e m a t e r i a l  t o p r o d u c e an i n g o t .  i s t h e p r e f e r r e d mode s i n c e t h i s a v o i d s  a region  shadow  in  melting  electrode  is  crucible.  In  usually rotated  vertical  drip  i n order  is  Vertical  feeding  the  fall  to facilitate  of the even  melting. Pool melting  melting  is  virtually  except that the feed stock  small pieces t h i s case,  ( i . e . sponge, s c r a p )  remelting  Pool  melting  the r e s u l t i n g  using is  the  to  drip  i s u s u a l l y p a r t i c u l a t e or a s shown  m a t e r i a l i s fed into the l i q u i d  c r u c i b l e where i t m e l t s crucible.  identical  i f f i g u r e 1.2. I n pool  i n the ingot  heat p r e s e n t  generally  i n g o t one o r more  followed times.  within  the  by  drip  6  REMELT STICK ELECTON  GUN  CRUCIBLE  Figure  1.1  E l e c t r o n Beam D r i p  Melting.  7  ELECTRON GUNS  SCRAP HOPPER  INGOT CRUCIBLE  Figure  1.2  E l e c t r o n Beam P o o l  Melting.  8 1.2.2 H e a r t h  Melting  E l e c t r o n beam processes defects being  hearth  which i s able t o  from aerospace  remelting  guarantee the  one  of  the  absence of  quality materials.  some  I t i s currently  u s e d a s a c o m p l i m e n t a r y s t e p t o one o r two vacuum a r c  remelts  in  titanium processing  removal of oxide  and i s  general  of a  EBCHR i n v o l v e s  a t one  maintaining  the metal i n the l i q u i d  the other  end  level,  feed stock  trough  a l s o used  f o r the  i n c l u s i o n s from s u p e r a l l o y s .  On a v e r y  the  is  and a l l o w i n g  water c o o l e d  remelting  copper  trough,  s t a t e over the l e n g t h of  i t t o flow  into  an i n g o t m o u l d  at  end (see f i g u r e 1.3).  1.2.2.1 P r o c e s s The beam h e a r t h  Advantages  primary  metallurgical  remelting are twofold.  good vacuum a l l o w s  advantages of First  various v o l a t i l e  electron  the presence of  impurities  (Mn, Mg  a in  T i ) t o evaporate thus r e f i n i n g  the m a t e r i a l . Secondly,  the  presence  liquid  for  of  gravimetric inclusions  a  quantity  separation from t h e  of  of  bulk  high  as  metal  density  w e l l as  and  e l e c t r o n beam  the  power s o u r c e  a d v a n t a g e s . Because t h e accuracy,  i t i s possible  nature  and  provides  beam c a n be to cast a  low  providing a  volume f o r t h e d i s s o l u t i o n o f i n t e r - m e t a l l i c In a d d i t i o n  allows  density reaction  particles.  flexibility a number  of  c o n t r o l l e d with wide v a r i e t y of  of  the  process great shapes  9  ELECTRON GUNS  SCRAP FEEDER  INGOT  CRUCIBLE  Figure  1.3  E l e c t r o n Beam C o l d H e a r t h  Remelting.  10 (i.e.  squares,  disconnection (heat  s l a b s , rounds, of  t h e power  c a n be a p p l i e d  reduces the  amount o f  instabilities quality  of  product  lost  during  final  found  in  due t o  from  VAR  ingots^  the  reduces t h e  formation  feeding give  systems a  great  t o input feedstock,  1.2.2.2 P r o c e s s  of  rate  surface  amount  shrinkage  commonly degree  of  cavities  elements a l s o  in  hearth  flexibility  with  Disadvantages  i s that although  from t h e input  of  used  c h e m i s t r i e s and m e l t p r a c t i c e .  The b i g g e s t d i s a d v a n t a g e  feedstock during 7 evaporate.  o f e l e c t r o n beam  hearth  v o l a t i l e c o n t a m i n e n t s a r e removed melting, desired  Chemistry  a l l o y m a t e r i a l s becomes v e r y alloying  melt  solidification.  remelting also  remelting  feedstock)  due t o  improves  The  material  melting additional  i n g o t and  the  etc.).  the feed  macrosegregation  the f i n i s h e d  The  respect  source  without  30 ingots,  hollow  c o n t r o l when  alloying remelting  d i f f i c u l t , e s p e c i a l l y when  the  e l e m e n t c a n n o t be a d d e d t o t h e c h a r g e i n s u f f i c i e n t  quantities  to  associated with  make  up  alloy  the  remelting  homogeneity that r e s u l t s  deficit. is  Another  the l a c k  from poor mixing  problem  of  chemical  w i t h i n the  liquid  metal. Even understood,  i t  though is  also  electron very  beam complex  technology requiring  is a  well broad  11  knowledge  of  Therefore  a  required. and  very  vacuum  high  In addition,  vacuum  capital  physics,  systems  systems  level  of  and  electronics.  operator  training  t h e t r a n s f o r m e r s , computer  are  costly  making  the  is  controls  process  very  intensive.  1.2.2.3 A p p l i c a t i o n s o f EBCHR  1.2.2.3.1 R e c y c l i n g o f T i t a n i u m  Scrap  R e c y c l i n g of t i t a n i u m scrap reasons.  First,  expensive  and second a r e a s o n a b l y  scrap  virgin  (particularly  tungsten  sponge  carbide tool  t u r n i n g s and sources  inclusions  bits - are f o r type  relatively titanium  i s available.  problems a s s o c i a t e d  High d e n s i t y  is  l a r g e amount o f  machine t u r n i n g s )  a r e two s i g n i f i c a n t scrap recycle.  titanium  i s d e s i r a b l e f o r two  with the  There  titanium  (HDI's)  -  usually  often present  in  machine  I , hard alpha d e f e c t s are a l s o  common. It  i s possible to  from s c r a p w i t h out  remove h i g h d e n s i t y  making use of  inclusions  an e l e c t r o n beam  hearth  8 furnace. In a d d i t i o n oil,  i tis  passed  to cleaning  over  the scrap  a magnetic  of grease  and  separator removing  the  b u l k o f t h e H D I ' s . I n non-EB i n s t a l l a t i o n s x-rayed  and  particles work. When  visually  a r e removed.  inspected  and  This process  using electron  beam  the s c r a p i s then  any  high  i s expensive  remelting, the  density but  does  x-ray  and  12 visual  inspection  s t e p s a r e e l i m i n a t e d . Any  i n c l u s i o n s which enter bottom of t h e l i q u i d solid  s k u l l and  the hearth  been e x p e r i m e n t a l l y v e r i f i e d g a t A. J o h n s o n M e t a l s . Hard a l p h a from p a r t i c l e s of the l i q u i d and The  strongest  furnace w i l l  p o o l , where t h e y  d i s s o l v e over  a p e r i o d of  I)  through  a - s t a b i l i z e r s are  the a l p h a phase v e r y  titanium  burnt  sponge a n d  arise  entering  the f i n i s h e d  product.  the i n t e r s t i t i a l  elements  substantially  presence  i s not  properties.  elements  stabilize melting  (see f i g u r e s  1.4  flame c u t s c r a p a r e  of  the  interstitial  necessarily detrimental  Oxygen  up  to  a  level  c o n s i d e r e d an a l l o y i n g e l e m e n t a n d p.p.m. l e v e l 1.1).  from hard  will  alpha producing  material.  to  prime  a l l alloy  nitrogen  non-destructive  and  t o the 2000  in  mechanical p.p.m.  specifications ^ 1  p a r t i c l e s not  Unfortunately  elements  is  n i t r o g e n a r o u n d t h e 100  detrimental to the  c o n c e n t r a t i o n s of d e t e c t by  exceed  of  L o c a l c o n c e n t r a t i o n s of these  d i s s o l v e d may be the  Entrekin  defects i n titanium  to  has  of hard alpha d e f e c t s . The  table  by  s t r o n g l y and a l s o i n c r e a s e t h e  or l i q u i d u s temperature  sources  by t h e  strongly a-stabilized particles  remaining  Therefore  sink t o the  time. This  using plant t r i a l s  ( o r type  density  become t r a p p e d  (C, 0, N ) . S m a l l c o n c e n t r a t i o n s o f t h e s e  1.6).  high  the  elements being  (see  resulting completely  fatigue properties presence  oxygen  means. U s u a l l y  of  these  i s impossible the hard  of  to  alpha  13  Figure  1.4  Titanium  - N i t r o g e n Phase Diagram.  °c  Ti  5  10  15  20  Weight Percent Oxygen Figure  1.5  T i t a n i u m - Oxygen  Phase Diagram.  15  °C 3100 '  I  2700  /  L  /  /  1900 1500  1100 Ti  /  v-  ]  X  :  I  \  1650°  V"  /  1  /  /  -  /  1 1 1 \ \  /  2300  s  /  --* y / 31380 \ / 1 6.5% \  6  I  y  U  8  12  L.  16  20  Weight Percent Carbon  Figure  1.6  T i t a n i u m - Carbon Phase Diagram.  16  Alloy  Impurity Limits p.p.m. N (max)  0 (max)  C (max)  H (max)  Commercial P u r i t y (CP)  300  2000  1000  150  Ti-5Al-2.5Sn  500  2000  800  200  700  1200  800  125  Ti-6A1-4V  500  2000  1000  125  Ti-6A1-4V ELI  500  1300  800  125  Ti-10V-2Fe-3Al  500  1600  500  150  Ti-3Al-8V-6Cr-4Mo-4Zr (Beta-C)  300  1200  500  200  Ti-5Al-2Sn-2Zr-4Mo-4Cr (Titanium-17)  400  1300  500  125  Ti-5Al-2.5Sn  Table  ELI  1.1 S p e c i f i c a t i o n s f o r I n t e r s t i t a l Titanium A l l o y s .  Concentrations i n  17 defect  forms  surrounding  a  continuous  matrix  which  grain  is  boundary  invisible  with  to  the  ultrasonics.  M e t a l l o g r a p h i c t e c h n i q u e s c a n be u s e d t o d e t e c t t h e p r e s e n c e of  a-stabilization  defect  but  t h i s method  useful  i s p r e s e n t on t h e s u r f a c e o f t h e p a r t b e i n g The  accompanies  increase  in  required  liquidus  the i n t e r s t i t i a l  h i g h temperatures or long  the  i s only  if  the  examined.  temperature  e l e m e n t s means t h a t  which  extremely  times at higher temperatures  t o melt or d i s s o l v e  are  the p a r t i c l e s r e s p o n s i b l e  d e f e c t s . Vacuum a r c r e m e l t i n g  i s i n c a p a b l e of  for  supplying  e i t h e r c o n d i t i o n . EBCHR, on t h e o t h e r h a n d , c a n manage d e p e n d i n g on t h e d e s i r e d m e l t i n g The high  presence of  temperature  during  i n s u r e s t h a t most, will  conditions.  a liquid  volume  at a  electron  beam  hearth  i f not  both  a l l , of the  reasonably remelting  hard alpha  defects  be r e m o v e d . F u r t h e r w o r k , b o t h i n t h e l a b o r a t o r y a n d  an i n d u s t r i a l  s c a l e , must  t i m e s and t e m p e r a t u r e s complete  be done i n  order to specify  that are required  on the  to guarantee  the  removal of h a r d a l p h a d e f e c t s .  1.2.2.3.2 R e m o v a l o f O x i d e I n c l u s i o n s The  removal of  s u p e r a l l o y s h a s been  oxygen and  demonstrated 11  and on an  industrial  scale.  inclusions  from the f i n i s h e d  from  Superalloys  oxide inclusions  b o t h on  a small  from  scale  1 1  12 '  The  removal  of the  oxide  product i s accomplished  using  one o r more w a t e r c o o l e d c o p p e r dams o r m a k i n g e f f e c t i v e  use  18 o f beam sweep p a t t e r n s . I t improvements i n t o t a l and  mechanical  1.2.2.3.3  h a s been shown t h a t  o x y g e n c o n t e n t , maximum p a r t i c l e  size  p r o p e r t i e s c a n be a c h i e v e d .  In-Spec T i t a n i u m E l e c t r o d e s Current p r a c t i c e  for  r e c y c l i n g of t i t a n i u m  s c r a p i n v o l v e s c o n s o l i d a t i o n and p u r i f i c a t i o n t o p r o d u c e an  electrode depleted i n  a d d i t i o n s a r e then and  significant  alloy  made a d d i n g m a s t e r  and s t r u c t u r e .  elements.  alloy  Alloy ingot  chemistry,  Obviously there are  advantages t o removing the remelt  melting  a l l o y s t o the  vacuum a r c r e m e l t i n g t o g i v e t h e p r o p e r  homogenization  by EB  alloy  economic  s t e p u s i n g t h e vacuum  arc  furnace. The remelting  alloy  are  losses  unavoidable.  a c h i e v e d by  adding  element t o  the feed  met w i t h l i m i t e d  homogenous. T a k a g i analysis  as  an  in  alloy  13  the the  date t h i s i s not  have s u g g e s t e d  quantitative is  chemistry  but the a l l o y 14  and M i t c h e l l  beam  to achieve  t h e i n g o t . To  success  on-line  electron  q u a n t i t i e s of  stock hoping  c o m p o s i t i o n c o n t r o l . Work of t h i s  The  additional  chemical compositions has  during  proceeding  tool  can  be  alloying desired approach usually  using  for  on t h e  hearth  x-ray  chemical  development  technique.  1.2.2.4 O p e r a t i n g H e a r t h In N o r t h America operating  hearth  Furnaces there  remelting  a r e c u r r e n t l y a number  furnaces.  The  two  of  biggest  19 installations and A.  are at V i k i n g  Johnson  Metals  V i k i n g , t h e r e a r e two 2.4  MW  installation  Both  these  titanium which  hearth  foundry  4 of  600  is  KW  DeGussa  remelting  facility  f u r n a c e s , a 1.2  MW  u s e s 2 1.2  are  used  used  Von  MW  installation Von  guns.  Electronics) in  also  Vallejo,  for remelting  i s in  additional the  a  guns.  r.ecycling MW  furnace  ElectroMetals operates  a  (a  small  California.  superalloy  the S o v i e t Union  The  hearth  hearth  w o r l d . The  r a t e d m e l t i n g power o f 9  This  materials  and  remelting  for  largest  furnaces  single  i s reported  and  skull  design  in  the  2.4  f u r n a c e the s k u l l c o n s i s t s of a n e a r l y  area w i t h a narrow t r o u g h i s u s e d as t h e m e l t  p o u r l i p . A t A.  a  MW.  a r e by no means s t a n d a r d . F o r  Viking  hearth  t o have  installations  area  and  Ardenne for  At  applications.  o p e r a t i n g throughout  MW  Nevada  Pennsylvannia.  primarily  Ardenne  furnace  There are  furnace  Morgantown,  in Verdi,  s c r a p . A. J o h n s o n o p e r a t e s a s i n g l e 2.4  has  division  in  which  furnaces  Metallurgical  exiting end  and  f r o m one  various  i n s t a n c e , i n the  s i d e . The  square large  t h e n a r r o w t r o u g h a c t s as a  Johnson, the hearth i s a r e c t a n g u l a r  trough  20  with  a  semi-circular  cross  section.  This  variation  furnace d e s i g n a l l o w s f o r a l a r g e degree of f l e x i b l i t y respect t o melt ability 1.3.  r a t e s , melt  of a p a r t i c u l a r  practice  The process fields  beam h e a r t h  i s l a r g e l y d e p e n d e n t on c o n t r o l l i n g the s k u l l .  volume, r e s i d e n c e  islittle  generated  or  of  type  t o have a d e q u a t e temperature.  no a v a i l a b l e d a t a  addition  to finding  p r o c e s s more e f f i c i e n t , some method  data  To  date,  on t h e t h e r m a l  field  ways  t o make  the  i t w o u l d a l s o be b e n e f i c i a l  of designing  Current hearth  moulds  time and  mass  d u r i n g EBCHR. In  Although  temperature  to the dissolution  I defects i n t i t a n i u m , i t i s necessary on p o o l  the  remelting  F o r e x a m p l e , i n o r d e r t o make  transfer calculations pertaining  there  efficiently.  EBCHR  success of the e l e c t r o n  within  with  a n d may i n f l u e n c e t h e  furnace t o melt  The T h e r m a l Regime D u r i n g  in  hearths  d e s i g n seems  to  t o be  give better  a hit  EBCHR  to  have  results.  or miss  affair.  h e a r t h m e l t e r s have been u s i n g t h e i r c u r r e n t h e a r t h effectively  guarantee  (some  f o r many  years)  t h a t t h e s e d e s i g n s a r e t h e most  there  i s no  efficient.  As w i t h most o t h e r p y r o m e t a l l u r g i c a l p r o c e s s e s , i t is extremely furnaces.  d i f f i c u l t to  This  temperatures, nature  of  experiment  i s primarily  on  because  operating of  the  hearth elevated  t h e vacuum t h a t i s p r e s e n t a n d t h e s u p e r  the  materials  being  melted.  As  clean  previously  21 mentioned, e l e c t r o n  beam  i n t e n s i v e making p i l o t  melting  plant  in  general  ( a n d t o some e x t e n t  scale) c o n s t r u c t i o n u n r e a l i s t i c . Therefore which t o  study  beam h e a r t h model.  the  heat flow  remelting  i s by  is  laboratory  t h e o n l y way  c h a r a c t e r i s t i c s of constructing  capital  a  in  electron  mathematical  22 CHAPTER 2 M o d e l l i n g H e a t F l o w i n t h e E l e c t r o n Beam  2.1.  The H e a t F l o w  Equation  Because of a p p l y i n g power geometries  to  that  Hearth  the control  the  can  skull  and  of t h e  in  an  are being  e l e c t r o n beam  EBCHR used  furnace,  are  the surface of  t h e s k u l l . These  i n t o c o n s i d e r a t i o n when d e s i g n i n g t h e EBCHR  i t s most g e n e r a l  a s shown i n e q u a t i o n  3 3T — (k—) 3x 3x  2.1.  flow i n three  3 3T + — ( k — ) 3y 3y  During  form t h e heat  taken  model  for  the s k u l l  the  e l e c t r o n beam  which  moves a b o u t  i n d i c a t e t h a t t h e EBCHR  process  22  ...(2.1)  hearth melting  some s o r t o f p a t t e r n .  r e g i o n of very h i g h temperature in diameter)  unsteady  dimensions.  3 3T 3T — ( k — ) = pC ( — ) . 3z 3z P 3t  +  in  flow equation i s  This equation describes  i n p u t power i s c o n t r o l l e d by p r o g r a m m i n g over  manner  p o i n t s must be  a mathematical  In  process. In  s t a t e heat  the  varied.  a d d i t i o n , e n e r g y must be d e l i v e r e d i n an n o n - u n i f o r m over  in  t h e beam t o This results  a t t h e beam s p o t with the  process move in  (up t o  a  2cm  beam. T h i s  would  p r e s e n t s an u n s t e a d y  state  23 heat t r a n s f e r integrating  p r o b l e m . The  the  power  problem  input  c a n be  over time  to  d i s t r i b u t i o n . When t h e power d i s t r i b u t i o n the  s k u l l as boundary  simplified give  a  by  power  i s t h e n i m p o s e d on  c o n d i t i o n , the unsteady s t a t e  nature  of t h e p r o b l e m i s removed. As p r e v i o u s l y i n d i c a t e d  t h e o p e r a t i n g EB  hearths  a t A. J o h n s o n M e t a l s  and V i k i n g M e t a l l u r g i c a l I n c . , a r e  simple  boxes.  rectangular  T-shaped h e a r t h  section  Johnson use a r e c t a n g u l a r  neglect  any o f t h e  9 S — (k_ 9x 9x  geometries,  9  9 T  — (k 9x 9x H  H  a the A.  with a semi-circular  i t w o u l d be i n a c c u r a t e  to  d i r e c t i o n s as they a l l  regime.  we c a n w r i t e t h e h e a t f l o w e q u a t i o n s f o r  and s k u l l as :  9 T  b  and pour l i p form t h e s t i c k .  three heat flow  Therefore  uses  melt b a s i n forms  shaped t r o u g h  c o n t r i b u t e t o the thermal  the h e a r t h  Metallurgical  i n which the  c r o s s and t h e runout t r o u g h  bottom. In both these  Viking  not  ) +  ) +  9  9 T  —(k_  9y  S  ) +  ^ y  9  9 T  — (k 9y 9y H  T  H  9  9 T  —(k.  9z  b  S  9T ) +  ) = 0.  9z  — ( k 9z 9z H  H  ) = 0  ...(2.2)  ...(2.3)  24  2.2. S k u l l B o u n d a r y  2.2.1  Top B o u n d a r y A heat  Conditions  Condition flow schematic  f o r the  EBCHR p r o c e s s  is  shown i n F i g u r e 2.1. At t h e t r a n s f e r r e d by impinging  top surface  two p r o c e s s  electron  vacuum t o  the  beam  and  surrounding  of t h e : energy heat i s furnace.  s k u l l heat  is  i s delivered radiated Using  being  by  the  through  the  a  radiation  b o u n d a r y c o n d i t i o n t h e t o p b o u n d a r y c o n d i t i o n i s g i v e n by 9T -k  = P _ ( x , y ) - a e ( T ^ - T ? ) where z=0.  c b  S  c  3z  ...(2.4)  b  The b o u n d a r y c o n d i t i o n  2.4  is  complete. I t i s sometimes e a s i e r t o o b t a i n r e s u l t s u s i n g  an  a l t e r n a t i v e boundary c o n d i t i o n . distribution is  Instead of  equation  giving a  on t h e t o p s u r f a c e , a t e m p e r a t u r e  specified. This  distribution attributed  g i v e n by  to  under  power  distribution  i s a v a l i d a p p r o a c h s i n c e any t e m p e r a t u r e a  o n l y one  given  set  of  conditions  power d i s t r i b u t i o n .  More  can  be  formally  s t a t e d the boundary c o n d i t i o n i s :  T  Q  = T ( x , y ) where z=0.  . . . (2.5)  25  7-  /  in  \  A  Q  surf  \  /  \  <>EB  M  ^,^---*^r  * out  11  ^  ^nould  k s  ^bottom  z > z  Mnould  * ^m^Ti-V  c  z < z  Q - f(x.y) EB  F i g u r e 2.1  Heat  Flow Schematic  o f t h e EBCHR P r o c e s s .  26 An  added b e n e f i t of  u s i n g t h e power  a p p r o a c h i s t h a t the model then source  interactions with  becomes i n d e p e n d e n t of power  the s k u l l .  m o d e l i n d e p e n d e n t o f power s o u r c e  In f a c t  r e m e l t i n g power  heating  thermal  the  remelting are adequately 2.2.2  Side Boundary At the  transferred  exterior  t h e h e a r t h and of  surfaces  R s  S  3x  3 T  k s  arc  in  of  plasma  the s k u l l  i n t o the surrounding  arc  t h e power d i s t r i b u t i o n skull  the  may  skull  heat  water  s t a r t u p procedure,  In  cooled  i m p o s e d on  this  is  method of  form a s o l i d / s o l i d  surface.  boundary c o n d i t i o n i s w r i t t e n as 9 T  s o u r c e , plasma  known.  from the s k u l l  region  the  Conditions  s k u l l c o n s t r u c t i o n and  in the  the  with  conditions  c o p p e r h e a r t h . D e p e n d i n g on t h e  skull,  i t makes  a l l o w i n g i t s use  other p o t e n t i a l hearth provided  distribution  the  contact area  the  :  S  3y  h  S/H  ( T  S  T  H  ...(2.6)  )  a t t h e a p p r o p r i a t e l o c a t i o n s a l o n g t h e x and Below t h i s  r e g i o n of s o l i d / s o l i d  y  directions.  contact a  vacuum  27  gap w i l l  form. Since  the  only  way  a c r o s s a vacuum g a p i s by r a d i a t i o n , in this  "k  b  = ~k S  Q  bg  region  It contact  may  b  = -o(e T c  4  c  9 y  t h e boundary  transfer condition  4  - e T ) . H H u  i s also possible develop i n  scrap materials degree t o  to  is:  c  x  f o r heat  used  which t h i s  average between  the  u  ...(2.7)  that  t h e vacuum  some d e g r e e gap r e g i o n  in constructing point contact two  modes o f  the  point  due t o t h e  skull.  occurs i s heat  of  I f the  large  transfer  some  may  be  required. 2.2.3  Bottom Boundary The  s i d e boundary be a r e g i o n  Condition  bottom boundary  condition  c o n d i t i o n except that of s o l i d / s o l i d  t h a t a l a r g e number  of point  is  there  contact.  It  contacts  similar  to the  i s not l i k e l y i s quite  will  possible  d e v e l o p on  bottom n e c e s s i t a t i n g t h e use of t h e a v e r a g e mentioned The  only  other  mode o f h e a t  gap i s r a d i a t i o n g i v i n g t h e b o u n d a r y 3T_  -k  b  Q S  3z  4  to  the  above.  transfer across  the  c o n d i t i o n as :  4  = -a(e„T„ - e T )  ...(2.8)  28  2.2.4 C e n t e r l i n e B o u n d a r y If  we  assume  about the c e n t e r l i n e final  surface.  9  -k  S  T  field  the c e n t e r l i n e  will  heat  be no  is  flow  symmetrical  to define  the  across  this  :  = 0 where y=0  c S  the thermal  we p e r m i t  There  surface giving  Condition  3y  ...(2.9)  2.3. H e a r t h B o u n d a r y  2.3.1 E x t e r i o r  Conditions  Surface  Boundary  The e x t e r i o r  s u r f a c e s of t h e water c o o o l e d  hearth  a r e those  skull.  Since a reasonably  possible  9 T  -k„ Hg  H  exposed t o the furnace  copper  w a l l and not t o  vacuum i s p r e s e n t ,  i s r a d i a t i o n , thus the  the  the only  boundary  is  3 T  = -k H  H  u  x  good  mode o f h e a t f l o w  condition  Condition  9 y  at the appropriate  H = -k„ H g 9 T  ,4 4, = - a e ( T „ - T*) H H A H  z  locations.  , ...(2.10) x  29  2.3.2  Interior  Surface  Boundary  The i n t e r i o r  Conditions  surfaces  of the  hearth  are  those  which a r e d i r e c t l y exposed t o t h e s k u l l . For c o n s i s t e n c y the heat the  input to the hearth skull.  Therefore  magnitude t o those opposite  must be e q u a l  the  heat  to  fluxes  stated i n sections  i n s i g n . Thus t h e  the heat out  are  identical  2.2.2 a n d 2.2.3  f o l l o w i n g equations  hold at  of in but the  appropriate locations.  _ !!u . _ !!5 k  k  H  Hg  9 x  9 T  H  9 T  " I-U 9x k  =  H  ( -) T  T  S/H  y  H  H  H ~ R~T~ 9z  S  •  9 T  " H^ 9y k  =  H  k  "  =  using  thermal 9 T  field  and  H  ^ q q 4  e s  T  )  ...(2.12)  s  the  assumption  due t o  of  longitudinal  by t h e c e n t e r l i n e becomes  the  symmetrical  nature  of  the the  t h e boundary c o n d i t i o n i s :  H  -k  = 0 where H  T  Condition  symmetry, t h e s u r f a c e d e f i n e d surface  4  H H " H  2.3.3 C e n t e r l i n e B o u n d a r y  final  a ( e  H  Again  ...(2.11)  9Y  y=0 ...(2.13)  30  2.3.4 W a t e r C h a n n e l B o u n d a r y C o n d i t i o n The  f i n a l heat t r a n s f e r s u r f a c e  that of the hearth/water f l o w s from  i n t e r f a c e . At these  the copper  hearth  boundary c o n d i t i o n i s best 3T  3T  H  ""HIT " V =  3T  H  =  " H^ k  at the appropriate h  to  Block  This  by :  "VH " V  ...,.,  T  2  the nature  4)  coefficient  of t h e i n t e r f a c e  with  of the c o o l i n g water.  Model  been c o n s t r u c t e d  block of  water.  l o c a t i o n s . The h e a t t r a n s f e r  In a d d i t i o n t o has  heat  H  r e s p e c t t o b o i l i n g and t h e v e l o c i t y 2.4.  is  surfaces  the cooling  represented  =  i s d e t e r m i n e d b a s e d on  w  i n the problem  to describe  material freely  T h i s model p r e s e n t s hearth process.  t h e EBCHR p r o c e s s  one  the thermal  radiating  to  case  i tis  cases  to  cold infer  conditions  a t a l l b u t t h e t o p s u r f a c e c a n be w r i t t e n a s : 9 T  -k  R B  B  9 T  B  = -k 9y R  9x  B  at the appropriate  9 T  B  = -k B  9z  locations.  4  4  = -ae ( T ? - T l ) B B A  a  walls.  i n the  not necessary  the e x i s t e n c e of p o i n t c o n t a c t s and t h e boundary  model  regime i n  the furnace  of t h e l i m i t i n g  In t h i s  model, a  ...(2.15)  31 At in section 9  -k  n B  the top surface,  2.2.1 w i t h  t h e boundary c o n d i t i o n  t h e power d i s t r i b u t i o n w i l l  = P-,(x,y) - a e ( T B B n  9z  n  B  - T, ) where z = 0 A  ...(2.16)  Summary I n summary, t h e  three  dimensional  conditions. generally variety  The  of  This  model i n v o l v e s flow  boundary  the steady  equation  conditions  with  are  melting  requires  materials  expressed  and  the s p e c i f i c a t i o n  f o r the boundary  conditions  as  t o a wide heat  of heat rather  state  boundary  t o make t h e m o d e l a p p l i c a b l e  geometries,  coefficients heat  heat  as p o s s i b l e  conditions.  flow  transfer than  the  using  the  fluxes. All  of  the  computer runs  f o l l o w i n g e q u a t i o n s and boundary 9 9T — (k—) 9x 9x 9  9 T  9 9T + —(k—) 9y 9y  S  — (k 9x 9x c  9  9 T  H  — (k„ 9x 9x H  9T 9z  +  ) +  9 S ) + — ( k _ 9z 9 z  9 H — (k„ 9y 9 y  9  ) +  H  = P (x,y) - ae (T c  ) = 0 ...(2.2)  b  9 T  S  ...(2.1)  9 T  C  S  made  conditions.  9 S — ( k _ 9y 9 y b  ) +  were  9 9T 9T — ( k — ) = pC — 9z 9z P9t  9 T  b  c S  apply.  R  T  2.5.  -k  stated  c  (k 9z 9 z —  H  H  4  c  9 T  H  ) = 0  ...(2.3)  4  - T.) where z=0  ...(2.4)  32  T  = T ( x , y ) where z=0  Q  ...(2.5)  so  9 T  s  9x  s  s  s  ( T  S  - T ) ...(2.6)  • -*  9y  ( e  s s "" H H T  e  T  }  ...(2.7)  S  9 T  -k  " H^H ^ e  s  9z S  9 T  b  9y  H  9x  " H k  " H k  9x 9T  9z  _ h  S/H  9T  k  "  9y  ( T  H  H  « 9z  k  (2.11)  "  -  a U  H H T  -  e  T ) 4  ..(2.12)  ...(2.13)  9y  H  ..(2.10)  = 0 where y=0 9 T  H  9 T =  " H H  B  " R  k  9x 9 T  B  ~ R k  9z  9y  = P (x,y) B  9 T  =  H  » 9z  k  B  " R B  9 T  "  9y 9 T  =  k  H  k  9 x  9 T  B  T  H  " H  B  = - « < T * ' A> H  H  " H H  k  H  "  9T =  H  H  H  9  H  _ k  H  H y  k  9 T  9y 9 T  "  3x 9 T  H  H =  H  9 T  H  9 T  = -k  " H H  = 0 where y=0  H  9 T  ...(2.8)  ...(2.9)  k  k  h  S  9 T  = -k  9x  " S/H  =  9y  S  9 T  s  9 T  = -k  h  k  9z  "  -^B  T ) A 4  " -B  ( T  ( T  V  ...(2.14)  B  " R B  " W H "  =  B "  ( T  B - 2> T  where z = 0  ...(2.15)  . .(2.16)  33 CHAPTER 3 Model F o r m u l a t i o n  3.1.  Introduction By  f a r t h e most c r i t i c a l  work i s  the s p e c i f i c a t i o n  case of  Cold  rather  difficult  i n d u s t r y and industrial It  Hearth  i s also quite  f u r n a c e s due incurred.  to  Cold  conditions.  proprietary  large c a p i t a l fields  In  specification  costs  of  of  of  critical  be the  operating furnaces.  scale  that  ingot  the  can  nature  expensive t o b u i l d laboratory the  modelling  e x p e r i m e n t a t i o n on t h e i r  provide  hearth  would  remelting  and  into  the  h e a r t h model i s d i f f e r e n t from t h e b l o c k  model  the titanium  water  o f some o f t h e n e c e s s a r y  insight  be  parameters.  H e a r t h Model  i n that  cooled  to allow  casting  The only  o f any  understood reluctance  the  determination  this  the  Therefore  continuous  3.2.  to  the well  firms  of boundary  Remelting  due  portion  copper  hearth.  follows also applies As  mentioned  T-shape  Thus  i s surrounded  much  to the block  geometries are possible exaggerated  skull  above  of t h e  discussion  model d i s c u s s e d a  by a  number  which  later.  of  different  f o r t h e EBCHR p r o c e s s . Of t h e s e currently 33  being  used  by  the  Viking  34 M e t a l l u r g i c a l was c h o s e n f o r s t u d y . A d i a g r a m o f t h e assumed g e o m e t r y i s shown  i n Figure  c r o s s on t h e T) i s t h e m e l t a d d e d . The m o l t e n  3.1. The  larger portion  b a s i n t o which t h e feed stock i s  t i t a n i u m (or other  down t h e r u n o u t t r o u g h  (the  (the stick  m a t e r i a l ) then  on t h e  flows  T) a n d i n t o  the  i n g o t m o u l d a t t h e p o u r l i p . The h e a r t h g e o m e t r y u s e d d u r i n g the  initial  the  stages of t h i s  Viking  Although  hearth  work i s a s c a l e d down v e r s i o n  (approximately  the w a l l s of the hearth  are tapered  to  provide  7.5  i n the V i k i n g  f o r easy removal of  s m a l l e r gaps a l l o w i n g f o r enhanced heat in the study this  that the  configuration  times  smaller). installation  the s k u l l  and  f l o w , i t was assumed  hearth walls are  h a s no  of  vertical  appreciable effect  and  that  on t h e  heat  transfer conditions i n the hearth. The rather  water channels  complex  arrangment.  u s e d by V i k i n g a r e p l a c e d i n This  arrangement  difficult  to  describe mathematically.  alternate  s e t of c o o l i n g water channels  s h o u l d be a r e a s o n a b l e in  industrial  practice.  approximation  For this  a  would  be  reason  an  h a s been u s e d  of t h e c o n d i t i o n s  which found  35  520  MELT  BASIN  190  ft  RUN THROUGH TROUGH  POUR LIP  HEARTH  T 280  ft  v  SKULL  200 *_  •1200  T " 1gO  +*— 350  230  350  1900  A L L DIMENSIONS IN MILLIMETERS  Figure 3.1  Geometry o f t h e S k u l l Metallurgical.  NOT TO S C A L E  a n d H e a r t h U s e d by V i k i n g  4  36  3.2.1  Skull  Boundary  Conditions  3.2.1.1 Top  Boundary  Condition  As can  -k  be  w r i t t e n as  9T_ 5 c b  discussed  c  c  4  4 - T.)  c  b  two  adjustable  emissivity  the  top  boundary  condition  :  = P (x,y) - ae (T  3z  The  previously  of the  where  z=0  parameters in t h i s skull  e  and  g  2)  ...(3.1)  equation are  the  power  1)  the  distribution  P (x,y) . s  The  e m i s s i v i t y of  titanium  in a  vacuum has  been  15 reported = 0.4 1600  by  the Defence M a t e r i a l s  b e t w e e n 1 200 °C.  emissivity  °C and  Although i t  the  would  of t i t a n i u m w i t h  stated  v a l u e of  range  expected. Since  distribution  0.4  was  the  supplied  Information  Center  approximate melting be more  correct  t e m p e r a t u r e i t was  to  vary  the  that  as  b o u n d a r y c o n d i t i o n must a l s o power  the  temperature  is  s t e a d y s t a t e . In p r a c t i c e , of c o u r s e , the  state,  of  model the  steady  be  point  felt  r e a s o n a b l y good o v e r t h e  to  the  power be  distribution  37  is anything  but  steady  moving a t h i g h speed delivery  s t a t e . With  a point  across the s u r f a c e  o f power t o t h e  skull  power  source  of the s k u l l ,  surface varies greatly  the with  time. The time  to  a c t u a l power d i s t r i b u t i o n was  produce a  time  a v e r a g e d power  d i s t r i b u t i o n s used i n the c o u r s e  integrated  over  distribution.  The  o f t h e work were  provided  by V i k i n g M e t a l l u r g i c a l . 3.2.1.2 S i d e B o u n d a r y C o n d i t i o n In both c o n t i n u o u s processes,  a  r e g i o n of  c o p p e r m o u l d and exists  near  c o p p e r due gap  surface  progresses  to thermal  d e p e n d i n g on  skull  contact  a solidifying  the  solidification  c a s t i n g and  this  the  to the nature  the  s h r i n k s away f r o m  the  vacuum  t h e EBCHR p r o c e s s e s ,  the  scrap i n t o the hearth.  Due  be a s e r i e s  of  d i s c r e t e point contacts. V i s u a l examinations  o f a number  of  electron  beam  indicate that region  of  skulls t h i s was  solid/solid  approximately  3 t o 4 cm  more l i k e l y  provided  by  Viking  the  process  to  but  sections)  i n these  is  not c o n t i n u o u s  tubular  material  contact  probably  and  cooled  As  of t h e s c r a p g e n e r a l l y m e l t e d  (machine t u r n i n g s  metal.  l e a v i n g an a i r o r  In  i s c o n s t r u c t e d by m e l t i n g  water  of the c a s t liquid  shell  contraction  the p r o c e s s .  ingot remelting  between the shell  of  the  Metallurgical  the case.  There appeared  t o be  or  contact  extended  point  below the  that  s u r f a c e of t h e s k u l l .  a  This  38 r e g i o n was  followed  by an  b e t w e e n t h e h e a r t h and vacuum  w r i t t e n as  -k  S  S  -k  S  contact  i n d i c a t i n g the e x i s t e n c e of  a  boundary c o n d i t i o n p r e v i o u s l y d e s c r i b e d can  be  9 T  = -k  S  o  S  = -h  9 y  3T b  C  = -k  heat  s  3y  g i v e s the heat _2 0.02 c a l cm  - e T*) H H  coefficient  the contact  region in  -1  °C  50  sec c a l  mould and  1-2  expected  by  Ballantyne ^ 1  f o r t h e VAR  process  as  which  represents  °C  a  thermal  —1  cm  sec c a l  .  Of  this  resistance  -1  interface.  thermal  evaluated  —1  sec  of  be  c a n  more d e t a i l .  transfer coefficient  '2 1 °C cm  included  ...(3.3)  hg/j_j  2 resistance  ...(3.2)  4.  = - o ( e TZ  transfer  examining  / (T - T ) S/H S H , 4  C  9x  The  point  :  9 x  3T  skull  of s p a r s e  gap. The  9 T  area  belongs 2  °C  cm  Since  i n the  sec  these  t o t h e copper used as the —1 cal  belongs  two  resistances  s k u l l / h e a r t h heat  r e s i s t a n c e of a b o u t 48 for  the  resistance  in  s k u l l and  presence  imperfections  these  mould/water  are  not  to  transfer coefficient, 2 —1  °C cm  c o n t a c t between the of  to the  ingot  sec c a l  question.  should However,  tend  to  raise  a be the  the h e a r t h i s not p e r f e c t . will  be  The the  thermal 30%,  r e s i s t a n c e . I f the value i s a r b i t r a i l y i n c r e a s e d by 2 -1 a r e s i s t a n c e of 60 °C cm sec c a l i s obtained which  corresponds 0.016  c a l cm  to —2  a —  sec  1  heat °C  -1  .  transfer This value  coefficient falls  i n the  of range  39 o f 0.011  - 0.03 c a l cm  -2  sec  0  „-1 C  found  by F e n e c h  and  by E i s e n  and  17 Rohsenow  and a l s o  falls  in  t h e range used  •J o Campagna i n t h e i r work i n vacuum a r c m e l t i n g . C a r v a j a l a n d 19 -o - 1 - 1 Geiger u s e d 0.015 c a l cm sec °C i n work t h e y d i d in modelling  t h e ESR  process.  In a d d i t i o n the e m i s i v i t y of copper  i n a vacuum  e  H  20 has been o b t a i n e d Since  t h e two  from K r i e t h  who q u o t e s a v a l u e  surfaces are e s s e n t i a l l y  c l o s e p r o x i m i t y no g e o m e t r i c a l 3.2.1.3 B o t t o m B o u n d a r y At  parallel  of  0.1.  plates  in  view f a c t o r s a r e r e q u i r e d .  Condition  t h e bottom of t h e s k u l l  the boundary  condition  c a n be w r i t t e n a s :  4  s  -k  = -a(e T„  q  S  Q  3z  This  S  S  4,  - e T ) H  ...(3.4)  H  i s e x a c t l y t h e same c o n d i t i o n a s t h a t o f t h e vacuum g a p  on t h e s i d e s . The f o r m a t i o n surface  of p o i n t c o n t a c t s  i s also a p o s s i b i l i l t y .  this possibility reasonably  free  c o n t a c t s . Areas around beyond t h e l e v e l  of t h e  extrusions  the bottom s u r f a c e s which  t h e edges of bulk  regions t o the o v e r a l l heat  bottom  The s k u l l s were e x a m i n e d f o r  and, i n g e n e r a l , from  on t h e  result  the s k u l l  in  were point  d i d extrude  but t h e c o n t r i b u t i o n of transfer surface at the  these bottom  40  of t h e s k u l l  was deemed n e g l i g i b l e .  3.2.1.4 T h e r m a l C o n d u c t i v i t y The f u n c t i o n of  thermal  conductivity  temperature  has  M a t e r i a l s Information Center Researchers  of  been r e p o r t e d  15  i n the i n g o t r e m e l t i n g and  fields  have f o u n d  results  from n a t u r a l o r f o r c e d c o n v e c t i o n  setting the  thermal  that the l i q u i d  c o n d u c t i v i t y of  f a c t o r m u l t i p l i e d by t h e t h e r m a l temperature.  by t h e  B a l l a n t y n e ^ found 1  as  a  Defence  a n d i s shown i n f i g u r e 3.2.  casting  by  Ti-6Al-4V  metal  continuous  motion  i s best  that  simulated  the l i q u i d  to  some  c o n d u c t i v i t y at the s o l i d u s that the m u l t i p l y i n g  factor  s h o u l d be 2-3 f o r ESR m e l t s a n d s o m e t i m e s a s h i g h a s 10 f o r VAR m e l t s agreement 1 ft 0 1  using high with  by a  observations well,  in liquid  factor  other  result  which  is  in  Harrison  found  3  that  good this  not g r e a t e r  than  natural  conductivity  7-10.  l o w momentum, l i t t l e  Since  o r no  from i n t e r a c t i o n s of t h e l i q u i d m e t a l  power s o u r c e . The l i q u i d c o n f i g u r a t i o n and s m a l l amount  workers  t i n increases the thermal  e l e c t r o n beam h a s a v e r y  process  by  values are i n  0Q  As  should  r a t e s . These  0?  field. convection  melt  of  metal  liquid  stirring and  i s a l s o i n a thermal  therefore there motion  s h o u l d be  when  a  compared t o  factor  i n an  enhancement o f t h e f l u i d  the  stable  relatively the  f o r example. These o b s e r v a t i o n s would i n d i c a t e  the m u l t i p l y i n g  an  e l e c t r o n beam f u r n a c e w i t h  ESR that no  f l o w s h o u l d be on t h e o r d e r o f 1-2.  0.5-1  £  0.4 J  0.1-1 1000  1 1300  1 1600  r1900  T e m p e r a t u r e °C  Figure  3.2  Thermal C o n d u c t i v i t y of T i t a n i u m Function of Temperature.  6A1-4V a s a  42 Initially  the  s p e c i f i e d using  a step  instability  the  in  in  a  thermal c o n d u c t i v i t y  function. This  solution  solution to o s c i l l a t e . raised  increased  which  I f the  step-wise was  increased  over  (i.e.  l i q u i d u s ) then  the  d i s a p p e a r e d and  one  solution  thermal c o n d u c t i v i t y a be  function  of  noted that  caused  (i.e.  t e m p e r a t u r e ) but s o l i d u s to  procedure produced the  was  a  at  was  the  not  solidus  temperature  numerical  obtained.  an  numerical  thermal c o n d u c t i v i t y  manner  was  range  instability  The  resulting  ( w i t h enhancement f o r l i q u i d m o t i o n )  t e m p e r a t u r e i s shown i n f i g u r e 3.3.  It  when no  used,  e n h a n c e d l i q u i d m o t i o n was  as  should the  numerical s o l u t i o n always converged. During the received  f r o m A.  through the region voids  of and  cross  the  J o h n s o n and sections.  skull  therefore  These v o i d s  could  research  conductivity  of t h e  a d e q u a t e way  to q u a n t i f y  extensive  program of  b o t t o m was  usually  porosity  (see  s i g n i f i c a n t e f f e c t on  skull  the  skulls  V i k i n g M e t a l l u r g i c a l were  a high  have a  some o f  These s k u l l s r e v e a l e d  near the had  programme  in t h i s area.  cut  that  the  filled  with  figure the  3.4). thermal  Since there  is  no  t h i s e f f e c t without engaging  in  an  skull  s e c t i o n i n g or x - r a y  evaluation,  Figure  3.3  T h e r m a l C o n d u c t i v i t y o f T i t a n i u m 6A1-4V a s a Function of Temperature F o l l o w i n g M o d i f i c a t i o n for L i q u i d Motion.  F i g u r e 3.4  X-Ray P h o t o g r a p h o f a J o h n s o n S k u l l Large F r a c t i o n of V o i d s .  Showing  45  it  was i g n o r e d  w i t h some r e s e r v a t i o n s .  3.2.2 H e a r t h B o u n d a r y  Conditions  The b o u n d a r y divided  into 3 parts  3) t h e w a t e r  conditions  : 1) t h e i n t e r i o r ,  Boundary  already  s e c t i o n s on  hearth  hearth  been the  to  described  skull  2) t h e e x t e r i o r  on t h e i n t e r i o r  and  insure that  3.2.2.2 E x t e r i o r B o u n d a r y  the w a l l s of t h e furnace radiate heat.  the  It  is  only  the  boundary c o n d i t i o n s  on  the  conditions.  of the s k u l l .  Conditions  There  a r e some a s s u m p t i o n s  between t h e s e  should  t h a t c a n be  surfaces  to  the  the water channel  low  and t h e f u r n a c e  heat  flow  made  Firstly,  the temperatures at  be r e l a t i v e l y  Secondly, the temperature  into  gradients  and t h e heat  the  the flow  wall negligible  in  in  cooling  water.  the region  between  and t h e e x t e r i o r s u r f a c e of t h e h e a r t h a r e  e x p e c t e d t o be q u i t e s m a l l . T h e r e f o r e flow  to freely  t h e magnitude of t h i s heat f l u x , however.  surface  comparison  a r e exposed t o  and a r e t h e r e f o r e a l l o w e d  the water c o o l i n g i s s u f f i c i e n t ,  hearth  and  surfaces  characterized  The e x t e r i o r s u r f a c e s o f t h e h e a r t h  if  be  in  boundary  a r e c o n s i s t e n t w i t h those  regarding  can  Condition  The b o u n d a r y c o n d i t i o n s  necessary  the  channels.  3.2.2.1 I n t e r i o r  have  in  i n t o t h e water  from t h e s i d e  very  little  opposite  heat  the s k u l l .  will For  46 this  reason  i t is  between t h e  cooling  unnecessary c h a n n e l s and  Therefore the e x t e r i o r  H  3x  H  3y  boundary  the  the  exterior  generate s i g n i f i c a n t  material surfaces.  c o n d i t i o n s become :  ...(3.5) statement  temperatures are  furnace w a l l s .  include  9z  H  The o n l y e x c e p t i o n t o t h i s Here the  to  expected to  heat  flow  Therefore  i s at the top  a  from  be h i g h the  radiation  surface.  enough  hearth  boundary  to  to the  condition  exists :  H  9z  H  H  A  ...(3.6)  3.2.2.3 W a t e r C h a n n e l B o u n d a r y C o n d i t i o n s In are  operating  constructed  locations  by  i n d u s t r i a l h e a r t h s the water  either  drilling  holes  in  channels  the  proper  i n a copper b l o c k or c a s t i n g copper around  tubing  23 arranged  i n the desired  methods  produce  difficult  to  fashion.  cylindrical  handle  in  a  Both of these water  channels  rectangular  construction which  coordinate  d i f f e r e n c e system. T h e r e f o r e t h e model uses c o o l i n g of the  square c r o s s - s e c t i o n . P r o v i d e d same w e t t e d p e r i m e t e r a s  replace, the d i f f e r e n c e  i n the heat t r a n s f e r  b e t w e e n t h e two a r e n e g l i g i b l e .  finite  channels  the square channels  the c y l i n d r i c a l  are  channels  have they  characteristics  47  The  network  Metallurgical difficult  i s  o f c o o l i n g c h a n n e l s employed  complex  and  consequently  t o model a c c u r a t e l y . F o r t h i s  arrangement  o f c o o l i n g water  h e a r t h model.  The  at  extremely  reason an  arbitrary  c h a n n e l s was c h o s e n  c o o l i n g water  c h a n n e l s used  Viking  for the  i n the  model  are spaced e v e n l y over the whole o f the h e a r t h mould. The  boundary  condition  for  the c o o l i n g  channels  h a s been m a t h e m a t i c a l l y s t a t e d p r e v i o u s l y a s :  9 T  " H  H  3 T  K  " H  =  K  A X  Thus  H  9 T  = 9 Y  H  " H-^ K  the specification  =  V H  of  ...(3.7)  " V  T  h  specifies  w  the  c o n d i t i o n . Two e f f e c t s g o v e r n  t h e magnitude  transfer coefficient  are  h  w  >  These  interface with respect to b o i l i n g at the  o f t h e heat  : 1) t h e n a t u r e o f  and  2) t h e w a t e r  operate  i n either  regime.  I n the case  c o l d mould p r o c e s s e s  the non-boiling  velocity  of nucleate b o i l i n g ,  t e m p e r a t u r e and the s a t u r a t i o n - 100 °C. T h e r e f o r e ,  boiling  the water/copper  cold hearth i n t e r f a c e are  processes much l e s s  boiling  surface  excess surface  temperature - i s i n the  i n order t o achieve  s u r f a c e t e m p e r a t u r e s must be  °C a t  generally  or nucleate  t e m p e r a t u r e - d e f i n e d a s the d i f f e r e n c e between the  200  the  interface. C o l d c r u c i b l e and  of 5  boundary  I n the  t h e temperatures indicating  nucleate  i n the range o f  interface.  that  range  105  case o f  expected  at  non-boiling  the the heat  48 transfer  i s the operative In  non-boiling  convection, results  the  heat  Seider-Tate  transfer  0  8  Re  M / 1  N  3  Pr  (  /  B  ~>  \  °-  where N_, > 1 0000 a n d 0.7 « Re  produces  good  coefficient.  The  1 4  ,, ..  N„ Pr  T h i s r e l a t i o n s h i p has been used s u c c e s s f u l l y and  Brimacombe,  Joshi,  and  Applying the and  1.25 cm  r e l a t i o n s h i p using  square c o o l i n g  channels the  c o e f f i c i e n t becomes :  h„ = 0.01257  (V  W  r 7  )  is  200 cm s e c  a t 20  equation  in  = 0.871  cm s e c .  Using  1  watts  calculation  the  is  cm  typical  water  hearth/water  interface  total  transfer  transfer  resistance  velocity  velocity  of  gives  —1 °C  then  a  . I f a thermal resistance  used  to  determine  magnitude of t h i s thermal r e s i s t a n c e ,  heat  f o r the  ...(3.9)  for  1  °C  0 , 8  —2  of  water  W  where V  w  by S a m a r a s e k e r a  Ballantyne.  heat t r a n s f e r  h  forced  is: KI -  - n no-i 0 2 3  Nu " ° '  under  relationship  f o r t h e value of t h e heat t r a n s f e r  relationship u  mode.  The  relative  i t i s found that  accounts f o r less resistance.  the  than  bulk  i s a c c o u n t e d f o r by  1%  of  the  type  of  the the  the  heat  vacuum  gap  49 (---94%) w i t h t h e o t h e r for the remaining the v a l u e s  heat t r a n s f e r r e s i s t a n c e s  6%. From t h i s r e s u l t  obtained  accounting  i t would appear  using equation  3.9  are  that  sufficiently  accurate.  3.2.2.4 T h e r m a l C o n d u c t i v i t y Over t h e t e m p e r a t u r e range cooled copper h e a r t h , the thermal essentially  constant  the o v e r a l l  heat  a t 3.9 w a t t s  flow equation  expected  i n the  water  c o n d u c t i v i t y of copper cm  1  °C . ^ T h i s 1  allows  2  i n the hearth  is  t o be  written  as : 2 3 T  2 3 T  Z  H  + 3x  Block  = 0 3z  2  ...(3.10)  2  Model The  e l e c t r o n beam without  H  + 3y  2  3.3.  2 9^T  Z  H  b l o c k model w h i c h i s used under c o n d i t i o n s power i n p u t  i s essentially  the p r o v i s i o n f o r s o l i d / s o l i d  i s t o say the block  the s k u l l  or point contact.  i s free t o r a d i a t e t o the furnace  of  model That walls  50  in a l l directions. 3.4, N u m e r i c a l  Technique  A f t e r examining i s reasonably difficult  i f  clear not  that  p o s s i b l e t o choose  either  solution  the s o l u t i o n  using the f i n i t e  difference  of  numerical  be  solution  d i f f e r e n c e or  system  in  method h a s  i t s simplicity  in  finite  Both of of  the system  technique  advantage t h i s is  s o l u t i o n would  technique.  of a  no r e d u c t i o n  element approach  a  the f i n i t e  equations. Since  case, the only  Thus  s t a t e d model i t  As w i t h o t h e r n u m e r i c a l p r o b l e m s i t i s  f o r the  methods i n v o l v e  an a n a l y t i c a l  impossible.  t e c h n i q u e was c h o s e n .  e l e m e n t method  the mathematically  these  simultaneous i s possible  the steady over  state  the  finite  f o r m u l a t i o n and  the  c o m p a r a t i v e l y lower c o s t . Using  the  finite  difference  sparse m a t r i x r e p r e s e n t i n g t h e system generated. system  approach  of l i n e a r  In order t o s o l v e f o r the temperature  o f e q u a t i o n s must be r e d u c e d  After testing  a variety  found  successive  that  large  equations i s field,  to the s o l u t i o n  of the t e c h n i q u e s a v a i l a b l e , symmetric  a  this  vector. i t  overrelaxation  was with  26 conjugate  gradient acceleration  was t h e  best  i n terms  of  51  b o t h s p e e d and 3.4.1  Non  cost.  Linearities Unfortunately  technique requires the  current  using  the  the  system of  p r o b l e m two  finite  e q u a t i o n s t o be  non-linearities  1) v a r i a t i o n o f t h e r m a l c o n d u c t i v i t y  with  the  the  f o u r t h power  of  difference  temperature  in  linear.  In  e x i s t . These a r e  :  t e m p e r a t u r e and  2)  radiation  boundary  condition.  3.4.1.1 T h e r m a l The the  Conductivity  finite  s o l v i n g of  the  difference  temperature at the initial  g e n e r a t e d and  solved  and  other  conductivity with evaluating  previous  guess).  One  the  The the  evaluated iteration  the  (or  on  of  process  repeated.  condition  basis the  conductivity  at  i f the  used i s  not  For  average value.  a d j a c e n t nodes at t e m p e r a t u r e s T nodes have t h e r m a l  1  and  a  example, T  2  c o n d u c t i v i t i e s k,  of then  taken  node.  thermal  the  thermal  t e m p e r a t u r e i s t h a t c a r e must be thermal  of  is  varying  to  thermal  basis  equations  of  to produce l a r g e e r r o r s  two  on  system  possible  an  i s condusive  t h e r m a l c o n d u c t i v i t y p r o b l e m . The  c o n d u c t i v i t y a t e a c h node i s  some  technique  It  in is  conductivity consider  two  r e s p e c t i v e l y . These and  k  0  associated  52  w i t h them. G i v e n t h a t entering Q (-). A  2  i s greater  than  T  then the  1  heat  node 1 i s (  2"V  T  = k  1  T  A  1  ...(3.12)  and t h e h e a t l e a v i n g node two i s Q (-)_  (  2"V  T  = k,  A  ...(3.13)  A  If  k  * k  1  which i s  (as i s  2  physically  the  case here)  impossible.  average thermal c o n d u c t i v i t y k ( k  ^  =  =  1  + k  2  A V  To  then  (Q/A)  solve  the  i s defined  1  *  (Q/A)  problem  2  an  as  }  *  ...(3.14)  then  (-) A  1  = - ( - ) ,= k A  —2 A  L_ ...(3.15)  5 3  Radiation  3 . 4 . 1 . 2  The associated  second  with  the  n o n - l i n e a r i t y which f o u r t h power  of  develops i s  temperature  that  i n the  radiation equation :  " "8 The  < T  S  " J '  ...  T  f i n i t e d i f f e r e n c e method i s n o t a s a m e n a b l e t o  t h i s n o n - l i n e a r i t y . In t h i s case  t h e boundary  l i n e a r i z e d by d e f i n i n g a p s e u d o - h e a t a e h  S C  ( T  4  S  -  ( 3  .,  6 )  solving  condition  is  transfer coefficient :  TJ)  A  R = ( T - T . )  S  . . . ( 3 . 1 7 )  A  or a ( e  h  R "  S S T  - H H> T  (T -T ) g  Using  e  this  solution,  . . . ( 3 . 1 7 )  H  method  also  requires  i n which t h e pseudo-heat  e v a l u a t e d based  on  the  an  iterative  type  of  transfer coefficient i s  previous iteration  and  then  the  system of e q u a t i o n s i s s o l v e d . 3 . 4 . 1 . 3  Iterative Solution B o t h n o n - l i n e a r i t i e s i n t h e model r e q u i r e  solution  be  obtained  s o l u t i o n , an i n i t i a l  iteratively.  g u e s s must  To  that the  accomplish  be p r o v i d e d t o t h e  this system.  54 This i n i t i a l allow  for  guess  i s then  quicker  used t o generate a s o l u t i o n .  convergence  to the  s h o o t i n g technique which combines one  o r more p r e v i o u s s o l u t i o n s A t some p o i n t  must be r e a c h e d . I t was number o f d i f f e r e n t the d i f f e r e n c e  in  i t e r a t i o n s when differs  by  a  and  i s used.  t e m p e r a t u r e between  endpoint  to the  the heat l o s t power and  d i f f e r e n c e between small  to define  value  stop  i s s m a l l o r 2)  these d i f f e r e n c e s  input  sufficiently  was  i t e r a t i o n s and  of t h e s e d i f f e r e n c e s  the  a  p o s s i b l e to define the endpoint i n a  a l s o possible to c a l c u l a t e this  the c u r r e n t s o l u t i o n  i n t h e i t e r a t i v e p r o c e s s an  maximum ( a b s o l u t e v a l u e ) o f  and c o m p a r e  solution,  w a y s . The most o b v i o u s way  when 1) t h e a v e r a g e  was  final  To  the  i s small.  It  from the  skull  then  cease  these  to two  over a  values  period  of  iterat ions. In g e n e r a l  i t  was  found t h a t  p e r m i s s i b l e d i f f e r e n c e of about error  in  average  the  power c a l c u l a t i o n  d i f f e r e n c e of l e s s  d i f f e r e n c e and  °C p r o d u c e d of  t h a n 0.01  percentage error  been u s e d t o p r o d u c e  1  results.  using a  l e s s than  a  percentage 5%  °C. O n l y t h e  convergence  maximum  and  an  maximum  criteria  have  55 CHAPTER 4 Model  4.1.  Results  Introduction The  temperature  EBCHR or  model  power  boundary c o n d i t i o n . particular  power  can  be  run  distribution  as  distribution  will  ( i n other  maintain  temperature  distribution  clear.  It is  i s required), the link advantageous t o  variables using possible to  a  the  the  power  a  surface  a in  a  specific order  to  power  t h e two i s n o t effects  distribution  melting  top  specific  between  examine  temperature  ignore  words, a  either  of c o n d i t i o n s  produce  distribution  distribution  the  A l t h o u g h f o r any s e t  temperature a  using  and  of  some  where  i t is  liquid  metal  superheat a t t h e pour l i p . E v e n t h o u g h no c o m p r e h e n s i v e p r o g r a m model  was c a r r i e d  EBCHR s k u l l s  out, certain  and of  f u r n a c e s have a l l o w e d  t o v e r i f y the  o b s e r v a t i o n s made d i r e c t l y on  melting operations a s m a l l degree  i n the  industrial  of confidence  i n the  t h e r m a l f i e l d s c a l c u l a t e d u s i n g t h e model. 4.2.  Temperature D i s t r i b u t i o n A. J o h n s o n m e t a l s  the  surface  temperature  Model have c o n d u c t e d m e a s u r e m e n t s  in 55  their  hearth  furnace  of  using  56 26 i n f r a r e d pyrometry. They have r e p o r t e d t e m p e r a t u r e s  i n the  m e l t b a s i n a s h i g h a s 80 - 140  °C (150 - 250 °F) a b o v e  liquidus  meltstock.  temperature  distributions  of  were  the  constructed  to  Top  temperature  reflect  m e a s u r e m e n t s . To k e e p t h e t e m p e r a t u r e d i s t r i b u t i o n as p o s s i b l e ,  t e m p e r a t u r e s were  lower value a t t h e  varied  value of the l i q u i d u s temperature plateau at a  linearly  perimeter of t h e s k u l l  fixed distance  the  these as s i m p l e  from  some  t o a maximum  of  p l u s some s u p e r h e a t a t  form t h e  skull  perimeter.  example of t h i s t y p e of t e m p e r a t u r e d i s t r i b u t i o n  a An  i s shown i n  f i g u r e 4.1. The model aluminum 4 weight % stock. Using a l i q u i d  was  run  using  titanium  vanadium (Ti6A14V o r  6  6-4) a s t h e  thermal c o n d u c t i v i t y multipying  (LTCMF) o f 1 ( i n d i c a t i n g  n a t u r a l c o n v e c t i o n ) and  s u p e r h e a t , t h e b o u n d a r i e s o f t h e l i q u i d p o o l were T h e s e b o u n d a r i e s a r e shown i n f i g u r e  figure  4.4 shows  %  melt factor  a 110  °C  generated.  4.2 t h r o u g h f i g u r e 4.4.  F i g u r e 4.2 a n d 4.3 a r e c o n t o u r maps o f t h e i s o t h e r m s and  weight  the  liquidus/solidus  pool p r o f i l e  at the  c e n t e r l i n e o f t h e s k u l l . U s i n g f i g u r e s 4.2 a n d 4.3 i t c a n be s e e n t h a t t h e l i q u i d p o o l e x t e n d s o v e r more t h a t h a l f  of t h e  s u r f a c e area of t h e s k u l l and t o a depth of a p p r o x i m a t e l y cm. The l i q u i d / s o l i d  gap i s a p p r o x i m a t e l y 0.75 cm i n  and  i s present  cm.  therefore  liquid  t o a maximum d e p t h o f  1  length 1.75  57 o U~>  r -  o  —  o  I  s"ti F i g u r e 4.1  I  1 's 6 -  I  I  s'z.  I  I  s*s  I  1 9'£  Sd3l3NI1N33  I  I  s't  I  TV  eo-  T y p i c a l Temperature D i s t r i b u t i o n Used. Superheat Temperature = 1 5 0 °C.  58 o  in  —  o  ro  CM  — — —  o CM  o CD  —  o  -o in CO  —  •LU  1 1 1 u 1  -  cn  o  CD  in o  cn  i S"M F i g u r e 4.2  i  r~ i 9*6  —i—i—i  i—r~ i  i  r  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 Superheat =110 °C, LTCMF = 1.  (  C,  59  m  ' o . c n ' o  . r' o  LU  ' CD  o  o  ' LT) O  i—i—r 9'll 9/6 F i g u r e 4.3  i S'£  i  i 9'9  m S"E  i—r 9'1  S"0-  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, Superheat =110 °C, LTCMF = 1.  60  i—i—i—r O'SZMO/QtSIO'SZW  dU31 Figure  4.4  • JdHS  i  i I i 0'6 0'L  i  i n i 0'£ OT  i  i—r . 0"I O'l  ScJ313WIlN33  Pool P r o f i l e s a t the Centerline, Contouring I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF = 1.  61 This  pool  observed during Viking  depth  furnace  is  reasonably  operation.  Both  close A.  to  that  Johnson  and  i n d i c a t e t h a t t h e m e l t p o o l i s f r o m 2.5 - 5 cm (1 - 2  i n ) i n d e p t h . The x - r a y e x a m i n a t i o n o f a s i m i l a r f i g u r e 4.5) a l s o  shows t h e  p o o l d e p t h i s m a r k e d by This bright  pool depth t o  the bright  skull  (see  be s h a l l o w .  s p o t on t h e  The  photograph.  spot i s a h i g h d e n s i t y tungsten c a r b i d e  particle  w h i c h was d e l i b e r a t e l y p l a c e d i n t h e f e e d s t o c k t h a t  produced  this skull.  Since tungsten  than t i t a n i u m , the liquid/solid difficult validity  carbide  l o c a t i o n of the  interface at  to  i s significantly  make  any  o f t h e model  WC p a r t i c l e , m a r k s  this particular definite  It the s k u l l . 4.1.  of t h e temperature  i s informative to  power  temperature f i e l d 6.7 KW o r  evaporation  the to  distribution  i n the geometries of  of t h i s c a l c u l a t i o n required  to  a r e shown i n  produce  i s on t h e o r d e r  s u r f a c e of t h e s k u l l .  Also  about  c o n s t r u c t a heat balance  the  to the  The b a l a n c e  rate of  interest alloy  is  KW,  f u r n a c e from  the  ( o r 12.4 KW) i s  the  elements  the l i q u i d  .7 U s i n g d a t a p r o v i d e d by T a g a k i and assuming  removed  water.  calculation  from  table  19  t h r o u g h t h e h e a r t h by t h e c o o l i n g of  on  calculated  o f 19 KW. Of t h i s  <*35% i s r a d i a t e d b a c k  from t h e s k u l l  It is  s k u l l and t h e m o d e l l e d one.  Results  The  the  e v i d e n c e a l o n e due  i m p o s e d on t h e s k u l l a n d t h e d i f f e r e n c e t h e A. J o h n s o n  location.  conclusions  b a s e d on t h i s  the r a t h e r a b s t r a c t nature  denser  of  the  metal.  that evaporation  62  F i g u r e 4.5  X - r a y P h o t o g r a p h of an A. J o h n s o n the L o c a t i o n of the S o l i d u s .  Skull  Showing  63  Superheat Temperature T °C S H  Surface Losses KW (%)  Hearth Heat KW (%)  Total Power KW (%)  110  6.7  (34.9)  12.5  (65.1)  19.2  150  7.0  (35.6)  12.6  (64.4)  19.5  200  7.4  (36.5)  12.8  (63.5)  20.1  T a b l e 4.1  Heat B a l a n c e C a l c u l a t i o n s f o r Temperature D i s t r i b u t i o n Boundary C o n d i t i o n .  64 i s the rate l i m i t i n g vanadium and equation  m  s t e p , the e v a p o r a t i o n r a t e of aluminum,  t i t a n i u m can  be o b t a i n e d  u s i n g the  Langmuir  :  p°7 C A *x 'x X  =  p/27rRM T  x  ...(4.1)  x  where, p°  (Pa) i s t h e  vapour  the temperature T 7  i s the  M  the pure element  at  (°K),  activity  temperature  p r e s s u r e of  coefficient  o f component  x at  the  T,  i s the molecular weight  p i s the d e n s i t y  o f component  x,  i n moles/cm , 3  C  x  i s t h e c o n c e n t r a t i o n o f x i n (g/cm  ) and  A i s the area. Making  the  constant  gross assumption over  the  that the  entire  liquid  maximum e v a p o r a t i o n r a t e w i l l t o t a l mass f l u x  from the  i n t e g r a t i n g e q u a t i o n 4.1  concentration C  pool,  the  is  x  theoretical  be g i v e n by e q u a t i o n 4.1.  skull  can then  over the  The  be c a l c u l a t e d  a r e a of the l i q u i d  pool.  A p p l y i n g t h i s equation to the temperature d i s t r i b u t i o n to  generate  figures  4.2  through  4.4  results  by  in  used the  e v a p o r a t i o n r a t e s shown i n t a b l e 4.2. 4.2.1 E f f e c t o f S u r f a c e T e m p e r a t u r e I n c r e a s i n g the 150  °C has t h e  s u r f a c e superheat  e f f e c t of i n c r e a s i n g  temperature  both the s u r f a c e  to area  65  Superheat Temperature T °C S H  Table  Ti  Evaporation kg/hr Al  Rates v  110  0.056  1.40  0.00  150  0.096  2.19  0.00  200  0.158  3.03  0.00  4.2 T h e o r e t i c a l Maximum E v a p o r a t i o n R a t e s Temperature D i s t r i b u t i o n s .  f o r Various  66 and  the depth  4.8).  of t h e l i q u i d p o o l  The p o o l d e p t h  increases t o approximately  the l i q u i d / s o l i d  gap r e m a i n i n g  200 °C s u p e r h e a t  temperature  cm d e p t h  ( a s shown i n f i g u r e s 4.6 t o 1.5 cm  at approximately produces a  with  0.75 cm.  l i q u i d pool of  w i t h t h e same 0.75 cm l i q u i d / s o l i d  A =2  gap ( s e e f i g u r e s  4.9 t o 4 . 1 1 ) . These r e s u l t s of t h e heat w e l l as  the  temperatures  evaporation  also  These  lost  evaporation  rate.  the pool  as  superheat  t o the  there  representing i s apparent  isa  are  furnace  pool by  r a d i a t e d t o the furnace  and  On  is  the  two  extremes,  6.7 KW  data  and t h e  66  o f 135  cm , 7.4  at cm,  aluminum  On t h e o t h e r h a n d , a t 200 3  i n c r e a s e s of 105%,  when  of the l i q u i d offset  a n d an a l u m i n u m e v a p o r a t i o n  v o l u m e c a n be e x p e c t e d  from  however  p o o l volume  from t h i s  rises  volume i s a p p r o x i m a t e l y  r a t e i s 1.4 k g / h r .  r a d i a t e d heat  It  three  temperature  and volume  increases  evaporation  110 °C s u p e r h e a t  superheat  depth  i n t h e amount o f h e a t  t h e maximum  the heat  f o r the  surface superheat  200 °C t h e  increase.  increases  calculations  a r e s u m m a r i z e d i n t a b l e 4.3.  As t h e 110 °C t o  rates  balance  KW  °C of  r a t e o f 3.03 k g / h r  10% a n d 116% t h a t some  the temperature  respectively.  increase i n at the  pool  surface  67  i £' II F i g u r e 4.6  i  i S"6  i  i i i S'Z. S'S  I  i S'E  Sd313WIiN33  i  i i r i S" I • S'O-  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 Superheat =150 °C, LTCMF = 1.  °C,  68  F i g u r e 4.7  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, Superheat =150 °C, LTCMF = 1.  o  Figure  4.8  Pool P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 150 °C, LTCMF = 1.  70  in CM  n o r\j  o .  en  o . r ' o  . in •LU  - <rn  L  LU  o ro  in o m  Figure  i  i 9'£  i— 9'  9"I1  " i —S"B i i — iL— r  4.9  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 S u p e r h e a t = 200 °C, LTCMF = 1.  9"9  9'0-  'C,  71  . in C M  m  C M  o C M  O . CD  ' O  . r' o . in  • a  •UJ  -  m .  UJ  C D  o r -  a m  CD m  i—i—i S'll S'6 F i g u r e 4.10  i—i—i—i—i i S'S £"£  Sd313HIlN33  i  r £ ' I S" o-  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, S u p e r h e a t = 200 °C, LTCMF = 1.  Figure  4.11  Pool P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 200 °C, LTCMF = 1.  73  Superheat Temperature °C  LTCMF  110  Surface Losses KW  Hearth Heat KW  1  6.7  12.5  19.2  1 .75  66  150  1  7.0  12.6  19.5  2.25  105  200  1  7.4  12.8  20.1  2.75  1 35  110  2  6.7  12.8  19.5  2.75  95  110  5  6.7  13.6  20.3  4.25  Table  Total Power KW  Pool Depth cm  4.3 Summary o f P o o l D a t a a n d H e a t B a l a n c e C a l c u l a t i o n s f o r Temperature D i s t r i b u t i o n Runs.  Pool Volume cm 3  171  74  of t h e s k u l l  i s increased.  s m a l l expense  i n terms  This  of  increase i s obtained at  evaporation  rates  of  no  alloying  elements. 4.2.2 E f f e c t o f L i q u i d Movement The  e f f e c t s of l i q u i d  artificially  increasing  liquid  by  metal  the  c a n have  r e s u l t s of the heat  of  effect  on  the  by the very  thermal  m e t a l . The r e s u l t s o f v a r y i n g t h e  conductivity  shown i n f i g u r e s 4.12  conductivity  previously discussed  a major  c o n d u c t i v i t y of t h e l i q u i d thermal  thermal  some f a c t o r . As  small v e l o c i t i e s  liquid  movement c a n be o b t a i n e d  f a c t o r LTCMF  t o 4.14 f o r a  balance  from 1  to 5  150 °C s u p e r h e a t .  calculations are also  are The  tabulated  i n t a b l e 4.3. It table  4.3  i s q u i t e obvious that  small  significant  e f f e c t on  pool. This  effect  increasing of heat  liquid  i s much  greater without  by r a d i a t i o n a n d a l l o y e l e m e n t  in  temperature  increase the  in liquid  total  distribution.  a l s o apparent  power This  the data  velocities  t h e e x t e n t and  surface temperature  The increase  from  compiled also  have  volume o f t h e than  the  a  liquid  effect  the increased  in  of  losses  evaporation.  movement a l s o  necessary  to  increase i n  when t h e s u r f a c e t e m p e r a t u r e  causes  maintain  t o t a l power  i n increased  an the is and  Figure  4.12  Pool P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF = 1.  i  Figure  4.13  P o o l P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF =2.  Figure  4.14  Pool P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, S u p e r h e a t = 110 °C, LTCMF = 5.  78  i s o f t h e same o r d e r  of magnitude.  4.3. Power D i s t r i b u t i o n M o d e l  4.3.1  The Power D i s t r i b u t i o n Obtaining  a  power d i s t r i b u t i o n  power  model  o b t a i n i n g a temperature  distribution  is  somewhat  information  the  information  or  guns a n d  d u r a t i o n of constraint  a  more  use  particular  in  the  complex  distribution. Typically  beam t r a v e r s e s c o n t a i n gun  for  than  programmed  a s t o t h e power l e v e l s on about the  p a t t e r n . In  location  order  to  of t h e s t e a d y s t a t e heat f l o w e q u a t i o n  and  f i t the the  power  d i s t r i b u t i o n must be t i m e a v e r a g e d a n d m a g n i t u d e a d j u s t m e n t s for melting 4.3.1.1  power and beam l o s s e s must be made.  Beam L o s s e s Energy  various  losses  in  electron  beam i n t e r a c t i o n s a s  secondary  total  account the  e l e c t r o n s amount  incident  beam  f o r 10 -  m a t e r i a l being  to  the  from  thermionic  to less  t h a n 0.5%  Backscattered  of  the  electrons  can  i n c i d e n t energy depending 2 m e l t e d and t h e a c c e l e r a t i n g v o l t a g e .  atomic  accelerating voltage  of  backscattered  number and  the  of  the  electrons  is  irradiated  a n g l e of  The  emmisions  40 % o f t h e  The number only  energy.  result  i n d i c a t e d i n f i g u r e 4.15.  e n e r g y l o s s e s due t o x - r a y g e n e r a t i o n , and  beams  incidence  on  related  material, of  the  79  Electron Beam  090000  ©  Backscattered Electrons  ©  Secondary Electrons  © X-Rays  Thermionic Emission  F i g u r e 4.15 S c h e m a t i c D i a g r a m o f I n t e r a c t i o n s Between a n Beam a n d a n I r r a d i a t e d S u r f a c e I n d i c a t i n g E n e r g y Loss Mechanisms.  80 impinging  beam a s shown i n f i g u r e s 4.16 a n d 4.17. The e n e r g y  distribution  of these  backscattered  determined experimentally figure  electrons  and these  4.18. D o i n g an a p p r o p r i a t e in  can also  results are  i n t e g r a t i o n of t h e  distribution  results  backscattered  e l e c t r o n s a s shown i n f i g u r e 4.19.  the  power  The a p p l i c a t i o n o f t h e s e to  t h e power d i s t r i b u t i o n  straight is  forward.  independent  losses  experimental  m a t e r i a l , the surface of the been o b s e r v e d d u r i n g m e l t i n g  state  skull  of  to  observations  the b a c k s c a t t e r i n g of  the p h y s i c a l  in  energy  due  i n e l e c t r o n beam r e m e l t i n g  Although  of  shown  be  the  i s not  electrons irradiated  i s rarely plane.  I t has  uranium that the l i q u i d  surface  27 deforms, forming  a  depression  at the  seem u n l i k e l y  that the previous  the  situation.  melting  information the r e s u l t s In m e l t i n g surface  titanium  can  expect  backscattered reasonable beam e n e r g y normal t o  In  the  absence  lose  electrons.  In  of  the  incident  20%  of  hearth  on  be  the s k u l l  lost  because  i n hearth  losses  melting heavier  m a t e r i a l s such as niobium, tantalum  question.  on  plane  the order  t o 50% would n o t  i t  of  to is  incident  i s not  t i t a n i u m , beam  s u p e r a l l o y s beam l o s s e s up  be  used.  a  remelting,  t h e beam  surface. Therefore will  proper  i t s energy  t o assume t h a t an a d d i t i o n a l 10% o f t h e will  would  s u r f a c e h a v e been  n o r m a l beam to  It  r e l a t i o n s h i p s would apply t o  f o r a plane  a  beam s p o t .  always melting  30%.  In  and  the  be o u t o f  the  81  Figure  4.16 Number o f B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A t o m i c Number o f t h e I r r a d i a t e d M a t e r i a l f o r a N o r m a l Beam, A c c e l e r a t i n g V o l t a g e = 10 K V . 2  82  F i g u r e 4.17 Number o f B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A n g l e o f I n c i d e n c e o f t h e Beam, A c c e l e r a t i n g V o l t a g e = 10 K V . 2  83  F i g u r e 4.18 E n e r g y D i s t r i b u t i o n o f B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A t o m i c Number f o r a N o r m a l Beam, A c c e l e r a t i n g V o l t a g e = 10 K V . 2  84  Figure  4.19 Power L o s s e s Due t o B a c k s c a t t e r e d E l e c t r o n s a s a F u n c t i o n o f A t o m i c Number f o r a N o r m a l Beam, A c c e l e r a t i n g V o l t a g e = 10 K V . 2  85 4.3.1.2 M e l t R a t e  Adjustments  I n h e a r t h m e l t i n g t h e power r e q u i r e d t o m e l t material  i s i m p o r t a n t . I t i s advantageous  the  i n t h e steady  s t a t e m o d e l t o remove t h i s h e a t i n p u t f r o m t h e o v e r a l l  heat  b a l a n c e . T h i s i s b e s t done  heat  b a l a n c e on t h e s k u l l  (see figure  The o v e r a l l  IN  Q  +  Q  EB  Since  Q  I  Q  N  L  EB  Q  and  Q Q  U  T  +  by e x a m i n i n g  Q  4.20).  h e a t b a l a n c e i s g i v e n by :  IN  =  are  Q  OUT  +  Q  SURF  +  Q  HRTH*  terms  allows  (  4  2  )  a s s o c i a t e d w i t h t h e heat content  the m a t e r i a l being melted t h e d e f i n i t i o n these  theoverall  them  to  be  of  of the baseline f o r  calculated.  Using  room  temperature as the b a s e l i n e gives t h e f o l l o w i n g :  IN  Q  I  =  0  ...(4.3)  N  and • Q  OUT  • =  Q  MELT  where Q  M E L T  • +  Q  SH  i s the melting  (  power  and Q  g H  4  4  )  i s t h e power  86  QEB  Q,  Q'«EB  Q SURF  Q IN  Q OUT  V  F i g u r e 4.20 S c h e m a t i c Heat Beam S k u l l .  ^HRTH  Flow Diagram  of t h e E l e c t r o n  87  required t o superheat the  material. This  allows the  heat  b a l a n c e t o be w r i t t e n a s :  • Q  •  •  EB~ L Q  For  =  Q  •  •  EB MELT = Q  titanium  Q  + Q  M E L  H  amounts  balance and since  •  SH SURF HRTH*  T  kWhr/kg  °»43  S  results  obtained  t o approximately  and Q  from t h e 2% o f  i s 0.192  G H  temperature  the t o t a l  heat  i t i s t e m p e r a t u r e dependent r e s u l t i n g  increased numerical i n s t a b i l i t y approach w i l l  ,_ v  + Q  *  Whr/kg°C. U s i n g t h e model, Q g  • + Q  in  i t h a s been i g n o r e d .  g i v e t h e model a t e n d e n c y  to predict  This  slightly  h i g h e r t e m p e r a t u r e s a t t h e pour l i p o f t h e s k u l l . Finally  Q  EB  Q  L  Q  EB  L  Q  Q  t h e h e a t b a l a n c e c a n be w r i t t e n :  MELT  +  Q  SURF  +  Q  HRTH  ...(4.6)  or  Q  EB  Q  MELT  For  a  Q  EB  power  Q  SURF HRTH* + Q  distribution  ...(4.7)  i t i s inaccurate  s i m p l y remove t h e m e l t i n g power f r o m t h e o v e r a l l as t h e  melting  takes  place l o c a l l y  at  T h e r e f o r e t h e m e l t i n g power d i s t r i b u t i o n f r o m t h e a p p l i e d power  distribution.  power  t h e melt  must be  to  input basin.  subtracted  88  The t h e work  power d i s t r i b u t i o n s u s e d d u r i n g  h a v e been  derived  from  those  the course  typically  used  V i k i n g u s i n g the procedure d e s c r i b e d above. A t y p i c a l distribution  of by  power  i s shown i n f i g u r e 4.21.  4.3.2 R e s u l t s The  r e s u l t s of running  distribution  shown  in  f i g u r e 4.21  t r a n s f e r due t o l i q u i d m o t i o n 4.25.  (figure  produced e a r l i e r and  operating  hearth  no  enhanced  heat  produced and t h e p o o l  4.24) a r e i n a g r e e m e n t w i t h t h o s e using  condition  and  power  a r e shown i n f i g u r e s 4.22  The r e g u l a r p o o l p r o f i l e  indicated  t h e model u s i n g t h e  with  furnaces.  c a l c u l a t e d do n o t a g r e e w i t h t h o s e  boundary  observations  The  surface  depth results  the temperature d i s t r i b u t i o n  consequently  to  made  in  temperatures  experimentally  determined  27 by A. J o h n s o n . be  due  to  the  distribution in the  This discrepancy scale  down  i n surface  in  size  of  and t h e s i z e of t h e h e a r t h .  operating practices  c a u s e some d i s c r e p a n c i e s 4.3.2.1 E f f e c t o f Power The  effect  of V i k i n g  temperature both  the  may power  As w e l l d i f f e r e n c e s a n d A.  i n the r e s u l t i n g  Johnson  thermal  may  fields.  Density  o f an  increase  in  power d e n s i t y  was  d e t e r m i n e d by i n c r e a s i n g t h e t o t a l  power i n p u t t o t h e h e a r t h  by  increase  10%.  increase  As  expected,  t h e power  in  the  depth,  temperature  pool  i n the pool  (see  pool  volume,  results peak  in  an  surface  f i g u r e s 4.26 t o 4.29) a s  well  POWER DISTRIBUTION C  q  10  to  3; > n- »-3  c O0) 03 n no Q) o  POWER STAGE STAGE 1 E23 S T A G E 2 E3 S T A G E 3 fSI S T A G E 4 K3 S T A G E 5  o  H-"rJ  0  2 O  z Q rt cr c  a: O O o  w  o .  >-  rr O  3  G u> (D Oi  q I  3  - 2 . 0  —r— 3.0  13.0  8.0  X  COORD I NATE (CM)  18.0  23.0  28.  90  T—I ^~ CD  CD CM  O  Figure  4.22  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 °C, T o t a l Power = 33 KW, LTCMF = 1.  91 o  o  s" 11  F i g u r e 4.23  s:6  g/z.  S'S  S'E  SM  S'O-  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, T o t a l Power = 33 KW, LTCMF = 1.  Figure  4.24  Pool P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 33 KW, LTCMF = 1.  93 o  9"ll  9'6  S'L.  r wni  9"9  9'E  9'I  9"0-  ' nyoon 1  F i g u r e 4.25 S u r f a c e T e m p e r a t u r e D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 1.  Contouring 33 KW,  94 as  the  radiation  evaporation  and  loss  The  results  thermal c o n d u c t i v i t y  4.33  in  increases  o b t a i n e d from factor  the  liquid  in  the pool  to  power  liquid  distribution  shown i n f i g u r e s 4.30  through  4.4. I n g e n e r a l , e n h a n c i n g  using  forced  volume  convection  and p o o l  surface  temperatures a r e  about a c o r r e s p o n d i n g drop i n  depth as  well  environment  and a l l o y  reduced  t h e heat l o s t  heat  produces  o u t t h e t e m p e r a t u r e g r a d i e n t s . As a r e s u l t  l i q u i d motion,  4.3.3  due  increasing the  using the  i n f i g u r e 4.21 a r e  flattening  elements  Motion  and summarized i n t a b l e  flow  alloy  (see table 4.4).  4.3.2.2 I n f l u e n c e o f L i q u i d  indicated  of  as  of t h e  bringing  to the  furnace  element e v a p o r a t i o n .  The H e a t B a l a n c e As shown  previously the  heat balance  on t h e  EB  skull i s • Q  EB  Using  • Q  *>  •  L  Q  MELT  this  Q  EB  • Q  •  SURF  relationship  +  and  Q  HRTH*  the  d i s t r i b u t i o n model i t i s p o s s i b l e  ,, ...(4.7)  results  of  the  t o calculate the  power  thermal  95  S' I  Figure  4.26  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e L i q u i d P o o l , C o n t o u r i n g T e m p e r a t u r e = 1625 °C, T o t a l Power = 36 KW, LTCMF = 1.  96 o  CD  i S'll  Figure  4.27  i  i S'6  i  i S'£  (N3)  i  i S'S  i  i S'E  'QdOOG A  i  i S'l  i r S'O-  C o n t o u r Map S h o w i n g t h e B o u n d a r i e s o f t h e S o l i d S k u l l , C o n t o u r i n g T e m p e r a t u r e = 1595 °C, T o t a l Power = 36 KW, LTCMF = 1.  Figure  4.28  Pool P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 36 KW, LTCMF = 1.  98  F i g u r e 4.29  Surface Temperature D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 1.  Contouring 36 KW,  99  Beam Power  LTCMF  Surface Hearth H e a t L o s s Power KW  KW  KW  Peak Pool Pool S u r f a c e DepthVolume Temp. cm cm °C 3  Evaporation Rates kg/hr Ti Al  15.2  1  6.7  8.5  2260  30.0  1 26  0.1251  0.7714  16.6  1  7.7  8.9  2340  35.0  171  0.2622  1.2852  15.2  2  5.9  9.3  2085  37.5  1 70  0.0331  0.3273  15.2  5  5.1  1865  42.5  175  0.0062 0.1096  10.1  T a b l e 4.4 Summary o f H e a t B a l a n c e C a l c u l a t i o n s , P o o l D a t a a n d E v a p o r a t i o n R a t e s f o r Power D i s t r i b u t i o n Model.  Figure  4.30  P o o l P r o f i l e s at the C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 33 KW, LTCMF =2.  o  £• I  Figure  4.31  Surface Temperature D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 2.  Contouring 33 KW,  102  m CXI c n c\j  o  ' CM ' o . c n ' o , r -  o  o  in o  on o  0"SZ910"S£910"SZSl  "dW31 Figure  'jdns  i  i 0'6  i  i  O'L  i  i 0"S  i—i—i—i—r 0'£ 0"l 0"l-  (N3) 'QdOOD Z  4.32 P o o l P r o f i l e s a t t h e C e n t e r l i n e , C o n t o u r i n g I n t e r v a l = 100 °C, T o t a l Power = 33 KW, LTCMF = 5.  1 03 o  i  9'll  i  i  9'6  i  i  9"Z.  (N3)  i  i  9'9  i  i  9' £  'Gd003 k  i — i — i — r  9*1  9'0-  F i g u r e 4.33 S u r f a c e T e m p e r a t u r e D i s t r i b u t i o n , I n t e r v a l = 100 °C, T o t a l Power = LTCMF = 5 .  ,  Contouring 33 KW,  1 04  efficiency  o f t h e e l e c t r o n beam h e a r t h m e l t i n g p r o c e s s .  The  t h e r m a l e f f i c i e n c y c a n be d e f i n e d a s : Q  T?„ = "  MELT  +  Q  HRTH  Q EB  ...(4.8)  As s u m m a r i z e d of  i n table  4.5 t h e t h e r m a l  efficiency  t h e h e a r t h f u r n a c e i s on t h e o r d e r o f 5 0 % . The e f f i c i e n c y  decreases with  i n c r e a s i n g power  l e v e l and  increasing  liquid  movement.  efficiency  c a n be  a t t r i b u t e d t o the  temperature r e s u l t i n g  These  increases  variations variations  from i n c r e a s e d  with  in  thermal  in  surface  i n p u t power o r  liquid  motion. 4.4.  Beam S p o t  Temperature  S i n c e t h e power the  model have  been t i m e a v e r a g e d ,  that are calculated very  d i s t r i b u t i o n s used  are also  i n t e n s e moving p o i n t  transients area t o questions  i n the s k u l l ,  which as  the to  the temperature  t i m e a v e r a g e d . The  source should r e s u l t  for fields  use of  i n some  a  large  the l a r g e s t a t the s u r f a c e over the  power  the  as input  is  degree  applied. to  This  which  the  raises  some  calculated  temperatures would agree w i t h a c t u a l temperatures w i t h i n t h e hearth. In  order  to  produced underneath the  examine t h e  temperature  beam s p o t , an  transients  unsteady s t a t e  heat  1 05  LTCMF  Melting Power MLT  Surface Radiation SURF  KW  KW  1  8.5  6.7  1  8.5  2 5  Hearth Power Q  HRTH KW  Beam Losses Q  L  Total Input Q  EB  Thermal Efficiency V  KW  KW  %  8.5  10.2  33.9  50.1  7.7  8.9  10.8  35.9  48.5  8.5  5.9  9.2  10.2  33.9  52.2  8.5  5.1  10.0  10.2  33.9  54.6  T a b l e 4.5 T h e r m a l E f f i c i e n c y o f E l e c t r o n Beam M e l t i n g Various Conditions.  Under  106 transfer  model  Intuitively  has  been  developed  the time taken t o  should decrease  as t h e  (see Appendix  reach a s p e c i f i c  power  level  p r e d i c t s t h i s a s shown i n f i g u r e  1).  temperature  increases.  The  model  4.34. I n f i g u r e  4.35  power  h a s been a p p l i e d t o t h e m e l t s u r f a c e a t t h e c e n t e r l i n e  fora  p e r i o d o f 20 m i l l i s e c o n d s a n d t h e n t h e m e l t h a s been a l l o w e d to the  c o o l . T h i s i s a worst beam r a r e l y  remains  c a s e s c e n a r i o i n t h e EB h e a r t h in  t i m e . As t h e f i g u r e shows  any l o c a t i o n  the melt a t the c e n t e r l i n e  down t o t h e b u l k t e m p e r a t u r e a long time r e l a t i v e EB h e a r t h f u r n a c e  f o r this length  as of  cools  i n l e s s than 2 seconds. T h i s i s  t o t h e c y c l e t i m e s used  ( V i k i n g uses  in a  c y c l e times  industrial  o f 14  seconds  l o n g compared  to the  typically). If cycle  the  c o o l down  time then the  m o v i n g power s o u r c e the  time  averaged  time i s  temperature will  f l u c t u a t i o n s c a u s e d by  n o t be  temperature  significant. will  t h a n t h e peak t e m p e r a t u r e a t t a i n e d Observations of skull two  skull  In t h i s  case  only s l i g h t l y  less  t h e beam s p o t .  pool i n  In t h e Johnson  describe e l l i p t i c a l  i n between t h e m e l t end and t h e pour  eye  only very  slight  are  observed.  In  circular there  under  the molten  support t h i s c o n c l u s i o n .  o f t h e f o u r guns  be  other areas  and remains  is  an  changes i n  easily  stationary  the  p a t t e r n s over  discernable  bright  the  l i p . To t h e n a k e d  t h e beam  f o r long  Johnson  hearth furnace  brightness i n this  where  the  pattern  p e r i o d s of spot  region  and  is  time, large  107  Figure  4.34  U n s t e a d y S t a t e R e s p o n s e o f t h e T e m p e r a t u r e Under t h e Beam S p o t a s a F u n c t i o n o f Power I n p u t .  108  BEAM SPOT  TEMPERATURE  o  TIME  Figure  (SEC)  4.35 U n s t e a d y S t a t e R e s p o n s e o f t h e T e m p e r a t u r e Under t h e Beam S p o t A f t e r 20 m i l l i s e c o n d s a t 200 KW.  109 variations 4.5.  i n brightness.  Experimental Do t o  furnace skull  Verification the  at Viking, were  complexity  and  verification  not  conducted.  size  of  the  hearth  on t h e  Viking  experiments  were  experiments Instead,  c o n d u c t e d on s m a l l e r b l o c k s o f m a t e r i a l a t b o t h  Johnson  and  Viking. I n t h e V i k i n g e x p e r i m e n t a 28 cm ( 1 1 " x 19" x 6") placed  i n s i d e a 60  hand f u l l on  b l o c k of T i 6 A l 4 V  cm ( 2 4 " ) d i a m e t e r  of n a i l s ,  furnace  chamber.  An 80  diameter  circle  was t h e n  describing  centered  on  was s h u t  o f f and  a 10  the block  a  placed  inside  the  cm  (4")  face  and  40 m i n u t e s  the  the upper  t o remain t h e r e . A f t e r approximately  e l e c t r o n beam  was  c r u c i b l e mould and  b l o c k was t h e n p l a c e d KW beam  cm  u s e d . The b l o c k  f o r m a r k i n g t h e l i q u i d p o o l , were  t h e u p p e r s u r f a c e . The  allowed  was  x 48 cm x 15  was a l l o w e d  to  cool. In the experiment conducted same p r o c e d u r e  was  Ti6Al4V  number o f  and  a  c r u c i b l e mould.  carried out. A  The b l o c k  iron  a t J o h n s o n , much cylindrical  w a s h e r s were  was t h e n  heated  block  placed by  the  a  in  pattern f o r a period  time  a r r i v e a t steady  The  beam  cool.  was t h e n  to  e x t i n g u i s h e d and  the block  a  centered  e l e c t r o n beam d e s c r i b i n g a c i r c u l a r l o n g enough f o r t h e b l o c k  of  of  state.  allowed  to  1 10 In  each  case,  were s e c t i o n e d and p o o l . The  etched  resulting blocks  to  show t h e e x t e n t  photomacrographs showing  l i q u i d pools are Due  to  the  Viking t r i a l pool p r o f i l e  of the  liquid the  geometry  employed  at  c o u l d not  be d o n e .  The  and  4.37  r e c t a n g u l a r g e o m e t r y and  c o u l d be  material  of  cylindrical  the experiment  d i d use  of  the boundaries  shown i n f i g u r e s 4.36  Johnson, m o d e l l i n g  block  the  compared t o  the  a computer  resulting run of  the  model. An  initial  assuming t h a t  r u n u s i n g a beam l o s s f a c t o r of  t h e beam  was  t e m p e r a t u r e s on  the order  boundary  figure  (see  resemblence t o the made by s p r e a d i n g  of  that  shown i n  the  tight,  3000 °C and  4.38)  one out  very  actually  in  u n s a t i s f a c t o r y . The  results  power d i s t r i b u t i o n  input  actually  used  i n the  the  One p r e d i c t e d and  explanation actual  or  no  Further  runs  a l a r g e r area f i t  and to  4.39. profiles  experiment  f i t  the  experiment.  f a c t o r s c o n t r i b u t i n g to the pool  little  pool  approach the pool  are a l s o into  a  produced the best  shown i n f i g u r e  the model can  obtained  bears  and  surface  h i g h e r and  i n p u t power o v e r  the a c t u a l p o o l p r o f i l e as Although  produced  f i g u r e 4.36.  u s i n g i n c r e a s e d beam l o s s f a c t o r s  30%  the  poor c o n s i d e r i n g model  Clearly  and  is the  the  power  there are  other  profile.  f o r the  pool p r o f i l e s  discrepancy is  between  the  a difference in  the  Figure  4.36  Photomacrograph of t h e B l o c k Used D u r i n g the E x p e r i m e n t a t V i k i n g S h o w i n g t h e B o u n d a r y of t h e L i q u i d Pool. A - L i q u i d Pool, B - S o l i d Block, M a g n i f i c a t i o n : 0.375X , Beam Power : 80 KW.  Figure  4.37 P h o t o m a c r o g r a p h of t h e B l o c k Used D u r i n g the E x p e r i m e n t a t J o h n s o n Showing t h e Boundary of the L i q u i d P o o l . A - L i q u i d P o o l , B - S o l i d B l o c k , M a g n i f i c a t i o n : 0. 5 8 0 X .  ^ to  Figure  4.38  P o o l P r o f i l e s of the B l o c k Model U s i n g the C o n d i t i o n s o f t h e V i k i n g E x p e r i m e n t . LTCMF = Power = 56 KW, 5 cm i n d i a .  11 4  ©  'o  -R _ r— " a  •UJ  i  (  i  i  i  0 081 O K I  ,oix)  dW3i  i  i  i—i—i  i  i—i—i—i—i—i—i—i—i—i—i—r OL OS OE 0 ! 0 1 iwJ) - d a o o o z  009 O'Sl OEI a'II 0 6 jans  F i g u r e 4.39 P o o l P r o f i l e s o f t h e B l o c k M o d e l f o r t h e V i k i n g E x p e r i m e n t . LTCMF = 1. Power = 50 KW, G a u s s i a n D i s t . a = 10 cm.  115 assumed a n d a c t u a l  fluid  flow regimes.  For example, t h e very  h i g h t e m p e r a t u r e s u n d e r t h e beam s p o t a n d i n t h e c e n t e r circular  beam p a t t e r n  the x - y plane  c o u l d cause increased  with l i t t l e  d i r e c t i o n . The i m p o s i t i o n  o r no  of t h i s  c a u s e h e a t t o f l o w more r e a d i l y the p o o l t o s p r e a d .  fluid  of a  flow  in  increased flow i n the fluid  flow regime  i n the x - y plane  T h i s would a l s o  result  in a  z  would causing  shallower  pool. It  has  been f o u n d 31  welding  process  '  that  indicate  tension effects natural  the wider  into  that the f l u i d  may be  convection.  velocities  surface  to the  These i n v e s t i g a t i o n s  weld pool  modelling  the  effects  can  32  contribute significantly welding.  by r e s e a r c h e r s  as h i g h Obviously  tension  fluid  flow  regime  the f l u i d  regime i n the  v e l o c i t y due t o as twice these  during  those  surface  caused  increased  by  fluid  on t h e s u r f a c e o f t h e l i q u i d p o o l may a c c o u n t f o r pool observed  during the experimental  work.  116 CHAPTER 5 Hearth  5.1.  Design  and O p e r a t i o n  Factors A f f e c t i n g Hearth The  hearth  design  furnace  These i n c l u d e fluid  is  and  dissolution  regimes t h a t develop  separtion  compounds. The i m p o r t a n c e p r i m a r i l y dependent  o r no  concern  paramount importance The  heat  of  the thermal  considerations. elements, and t h e  inter  and  the  ability  non-metallic  end use  of t h e  i n g o t and  instance, evaporation  when  beam  r e a c t i o n volume f o r t h e  is the  rates are of  m e l t i n g CP t i t a n i u m  but a r e of  when m e l t i n g a n y a l l o y m a t e r i a l .  transfer  m o d e l c a n be  the e f f e c t s of the mixing parameters on  electron  of any o f t h e s e c o n s i d e r a t i o n s  on t h e  s t a r t i n g m a t e r i a l . For little  an  alloying  t o p r o v i d e adequate or  of  number o f  evaporation of  f l o w and t h e r m a l  and O p e r a t i o n  operation  a f f e c t e d by a  the  of t h e h e a r t h  Design  regime, the  used t o  (in a qualitative  e f f e c t of thermal  1 16  evaluate  conditions  way) at  1 17  the  s u r f a c e of the s k u l l  addition melt  and t h e i n f l u e n c e of geometry.  In  t h e e f f e c t of v a r i o u s o p e r a t i n g c o n d i t i o n s such  as  r a t e a n d power d i s t r i b u t i o n  on t h e t h e r m a l  evaporation  r a t e s c a n be e x a m i n e d .  5.2.  Design  Hearth  Design trial  and  o f an e l e c t r o n beam  error  exercise.  hearth  Although  regimes  i s currently  i t is  possible  estimate c e r t a i n parameters r e q u i r e d (such as h e a r t h from e x p e r i e n c e ,  i tisdifficult  parameters such as e v a p o r a t i o n  t o assess  of t h e t h e r m a l  regime developed  f l o w v a r i a b l e s and e v a p o r a t i o n obtained  from the thermal Using  terms mass  r a t e parameters can then  be  easy t o  evaluate  boundary c o n d i t i o n s  and e v a p o r a t i o n  on  rates.  Geometry  shape of t h e h e a r t h . the hearth  into  The e f f e c t  isdifficult  d i s t r i b u t i o n s would  to  in  d u r i n g o p e r a t i o n . The  thermal  balance  Geometry t a k e s  sizes  length)  profile.  the e f f e c t s of geometry and  5.2.1  to  mathematical  t o be e v a l u a t e d  t h e model i t i s r e l a t i v e l y  the p o o l volume, heat  a  some o f t h e o t h e r  r a t e s . Using a  model a l l o w s a g i v e n h e a r t h d e s i g n  and  of  t o determine  be r e q u i r e d  i n order t o maintain  t h e pour l i p .  account  both  changing  the  size  the length  because d i f f e r e n t  f o r the  l i q u i d metal  and  different  from t h e melt  of  power hearth basin  118 The profile  i n f l u e n c e of  skull  thickness  on t h e  thermal  i s e a s i l y d e t e r m i n e d s i n c e t h e power d i s t r i b u t i o n i s  independent of the s k u l l The  effect  determined using  thickness.  of  altering  a rectangular  power d i s t r i b u t i o n  shown i n  skull  skull  40  thickness cm x  f i g u r e 5.1  was  10 cm.  was u s e d  The  and  the  t h i c k n e s s v a r i e d f r o m 3 cm t o 11 cm. The is  variation  of p o o l volume w i t h s k u l l  shown i n f i g u r e 5.2 a n d i s s u m m a r i z e d  readily skull  seen, the  p o o l volume i n  t h i c k n e s s . As w e l l t h e  variation with skull a given  i n t a b l e 5.1 As  unaffected  by c h a n g e s  maximum p o o l d e p t h h a s  t h i c k n e s s . T h i s would  power d i s t r i b u t i o n ,  thickness  any s k u l l  is in  little  indicate that f o r  thickness larger  that  t h e p r e d i c t e d p o o l d e p t h w o u l d be a d e q u a t e . From t h i s o r no  advantage  represents bigger The  a  to  result,  t h e r e would appear  t o be  little  thicker skull.  A thick  skull  using a  larger, bulkier  and b e t t e r h a n d l i n g  advantage  piece  of  equipment  material  requiring  and a b i g g e r  enclosure.  to using a t h i c k e r s k u l l  the heat f l u x  t o the water c o o l e d  The  heat  average  flux  f i g u r e 5.3). Therefore use o f  a tube  hearth.  This  and c a s t  copper h e a r t h  through  decreases d r a m a t i c a l l y with  becomes a p p a r e n t when  any  particular  increasing skull  using a thick  h e a t f l u x e s due t o t h e p o o r t h e r m a l  surface  thickness  skull allows  construction in  type of c o n s t r u c t i o n  i s examined.  f o r the  the water  i n unable t o handle contact  (see  cooled high  between t h e c a s t  \\9  o. «0  1 20  Pool Volume vs. Skull Thickness 90  65 -f 2  1 4  1  1  1  !  6  3  10  12  T h i c k n e s s (cm)  Figure  5.2  The V a r i a t i o n o f P o o l Volume w i t h Thickness.  Skull  121  Skull Thickness (cm)  Pool Volume (cm ) 3  Melt Efficiency  Evaporation Rates  (%) Ti  (kg/hr)  Total Average Heat F l u x (w cm" ) 2  Al  3.0  89.4  56.2  0.0185  0.2099  20.65  5.0  81.6  56.2  0.0161  0.1894  16.26  7.0  77.1  56.1  0.0151  0.1797  13.42  9.0  71.3  56.1  0.0139  0.1687  11.42  11.0  68.1  56.1  0.0132  0.1623  9.94  T a b l e 5.1 R e s u l t s o f C a l c u l a t i o n s on t h e EB S k u l l Function of Thickness.  as a  d n> cn  Average HeaL F l u x  CO  H < n  t--  7T PJ CD  t-"-  in o cn 3  > < 0)  0)  0)  Legend A m e l t b a s i n  »-• c  x l e n g t h e d g e  rt  •  p o u rl i p  G3  bottom  ffi o v e r a l l  cn Thickness  (cm)  123 c o p p e r and  the  cheaper than channels  23 network.  tube  the  u s u a l method  i n a copper  t h e power d i s t r i b u t i o n (see  figure  p o o l volume  is  also  of  drilling  the  coolling  the  s k u l l without  changing  block.  A l t e r i n g the width  volume  has  5.4).  are o f f s e t  of  the The  o p p o s i t e e f f e c t on b e n e f i t s of  somewhat by  temperatures that r e s u l t  i n higher  the  specific  but no  melting  i n c r e a s e due  to  loss in energy).  The  a combination  of  the  also results  surface  surface heat l o s s e s . a loss in  evaporation  The  thermal (or  rates  the also  greater surface area  Increasing  the width  and  of  the  i n a d e c r e a s e of the a v e r a g e heat f l u x  e f f e c t s of  s k u l l width  are  readily  when t h e p o o l p a r a m e t e r s a r e p l o t t e d a g a i n s t a skull  width  ( d e f i n e d as  distribution parameters heat  and  the  r a t i o of t h e w i d t h  the a c t u a l  s k u l l w i d t h ) . The  ( i . e . pool volume, flux) a l l  p a r a m e t e r as  vary  the  evaporation  linearly  shown i n f i g u r e  From  to  should  be  also  depend on  the  of t h e various  r a t e of  power pool  aluminum  dimensionless  5.5.  point  used. O b v i o u s l y  apparent  dimensionless  with this  of  view  c o n s t r u c t i o n c o s t s , some c o m b i n a t i o n  will  in  mould. The  and  pool  increase  the m e l t i n g e f f i c i e n c y  increased surface temperature. skull  the  the  increased  increased surface temperature represents "efficiency"  . significantly  It  hearth  of a t h i c k , wide  the c h o i c e size  of  of s k u l l  of c o n t a i n m e n t  mould skull  dimensions vessel,  the  • q^PTM l i n n s  M4T  sumiOA Tjood  R T  uoi^eijeA  30  t*9 SJHBTJ  Pool Volume (cm**3) co o  o  o o  ro  o  _L_  > 13 O O  CO  c  O 13 a  o ^  3 ro  C5 •  o  o  c  c  ro  c  CO  ro  o  Mass Flux Al ( K g / h r ) CO  CO  ro  Average Heat Flux (W/cm *2) +  cn  1 25  Skull Width (cm)  Pool Volume (cm ) 3  Melt Efficiency  Evaporation Rates  (%) Ti  (kg/hr)  Total Average Heat F l u x (W cm" ) 2  Al  5.0  77.13  49.9  0.0151  0.1797  13.42  7.0  98.38  49.8  0.0193  0.2193  10.64  10.0  103.38  49.8  0.0204  0.2292  9.33  12.0  106.38  49.7  0.0212  0.2358  8.33  15.0  108.38  49.6  0.0218  0.2409  7.17  T a b l e 5.2 R e s u l t s o f C a l c u l a t i o n s on t h e EB S k u l l F u n c t i o n of Width  as a  127 e q u i p m e n t r e q u i r e d t o remove t h e s k u l l and  the  Since  tolerance  f o r the evaporation  the changing of  hearth  from t h e h e a r t h of a l l o y  both the t h i c k n e s s  mould  elements.  and w i d t h  o n l y a m i n o r e f f e c t on p o o l v o l u m e , i t i s  of  the  reasonable  to conclude that the q u a n t i t y of l i q u i d metal i s c o n t r o l l e d by  the  length  of t h e  hearth  and the  power  distribution  applied to the s k u l l . 5.2.2 The E f f e c t o f t h e H e a r t h M o u l d Using one  limiting  around the s k u l l  represents  c a s e f o r h e a t t r a n s f e r d u r i n g EBCHR. The  extreme i s t o freely  a h e a r t h mould  allow a l l  surfaces of the  t o the containment v e s s e l . Although  skull  to  other radiate  the hearth  f u n c t i o n s as  protection against  l i q u i d metal,  a s a pour l i p a t t h e c r u c i b l e end of t h e s k u l l  and  as support  heat flow  used  stock  overflow  of  i t also increases the  from the s k u l l . The  similar  f o r unmelted feed  b r e a k o u t and  mould  computer  model  to the Viking  earlier  (see  was r u n  a  g e o m e t r y w i t h t h e power figure  5.5).  t e m p e r a t u r e p r o f i l e s were o b t a i n e d geometry and u s i n g  using  t h e same  For  skull  and  distribution  comparison,  f o r a s k u l l of the  power d i s t r i b u t i o n  but  t h e w a t e r c o o l e d c o p p e r m o u l d . The r e s u l t s o f t h e s e  the same  without runs a r e  shown i n f i g u r e s 5.7 a n d 5.8 a n d s u m m a r i z e d i n t a b l e 5.3. It cooled  is  fairly  copper hearth  obvious  that  removing  from t h e a r o u n d t h e s k u l l  the  water  increases the  id C n  POWER DISTRIBUTION  cn  o rt)  rt  POWER STAGE CE S T A G E 1 eZl S T A G E 2 STAGE 3 STAGE 4 K 2 STAGE 5  rt>  p d-  in 0) rt CD  n  r| rr cr c 3 rt o o c  rt  I— 3  O  UJ  1  a  G ui  rt) a  o  a: O O a  rt O  n < 0> M c  0> rt (D rt  rt> n rt) o rt  q IN-2.0  3.0  8.0  13.0  X  18.0  23.0  28.0  COORD I NATE (CM) ro  CD  1 29  F i g u r e 5.7  P o o l P r o f i l e s f o r EB S k u l l U s i n g a Water Copper H e a r t h .  Cooled  1 30 o  F i g u r e 5.8  Pool P r o f i l e s C o o l e d Copper  f o r EB S k u l l Hearth.  Without  a Water  131  Mould  YES NO  ~  Table  Peak Surface Temperature (°C)  Pool Volume (cm )  (%)  2175  113.5  51.5  0.0754  0.5313  2275  252.9  43.5  0.3287  1.5373  3  Thermal Efficiency  Evaporation Rates Ti  (kg/hr)  5.3 E v a p o r a t i o n R a t e s , P o o l D a t a a n d H e a t B a l a n c e C a l c u l a t i o n s f o r an EB S k u l l W i t h a n d W i t h o u t a Hearth Mould.  Al  132 p o o l volume  by a p p r o x i m a t e l y  150%. T h i s  increase  v o l u m e i s a c c o m p a n i e d by a t h r e e f o l d i n c r e a s e evaporation efficiency an  rate  aluminum  of the furnace  increase  a hearth  of  free  skull  furnace  indicated  profiles  6/4.  The  thermal i s due  than  to of the  skull. p e r i l s of o p e r a t i n g  without  in  appreciably different  using  figure  an e l e c t r o n  a water cooled  5.8.  From  the  copper  diagram  i t i s apparent that l i q u i d metal w i l l  pour l i p basin.  i n t h e maximum  a l s o decreases but t h i s  i s not  One o f t h e  is  pool  i n s u r f a c e h e a t l o s s . The m e l t i n g e f f i c i e n c y  copper surrounded  melting  from  in  into the  ingot c r u c i b l e  and a l s o  hearth of  flow  pool  from t h e  from t h e  I t w o u l d be e a s y t o remedy t h i s s i t u a t i o n distribution  beam  choosing  a different  power  the  skull).  However, t h e r i s k s a s s o c i a t e d w i t h t h i s p r a c t i c e  are  likely  too great  to  (or increasing  by  melt  the size  g a i n wide acceptance i n  of  commercial  operations. 5.3. F u r n a c e  Operations  From t h e factor during  in  determining  EB  hearth  distribution. will  thermal the  standpoint, parameters  remelting  is  the s i n g l e of the  molten  pool  power  level  and  the  The i m p r o p e r s e l e c t i o n o f a power  s u r e l y cause the r e m e l t i n g o p e r a t i o n  of t h e  care  and  (such as h e a r t h  attention paid design).  biggest  t o other  distribution  to f a i l  regardless  pertinent  areas  133 Using the procedure described (see  section  4.3.1)  distribution applied  i t  i s possible  earlier to  in this  arrive  at  work power  t h a t c a n be u s e d i n t h e m o d e l s t a r t i n g w i t h  power d i s t r i b u t i o n a n d  making adjustments f o r  beam l o s s e s a n d m e l t r a t e .  Therefore the influence  r a t e on t h e t h e r m a l r e g i m e  c a n be i n v e s t i g a t e d .  common p r a c t i c e  melting  during  a  campaign  an both  of  melt  It is  also  have  some  to  d e g r e e o f o p e r a t o r c o n t r o l o v e r t h e power d i s t r i b u t i o n . T h i s control  is  attained  either  p h y s i c a l l y c o n t r o l one o r operator  access  controls  the  to  power  by a l l o w i n g  more o f t h e  the  the  guns, or g i v i n g  programmable  distribution.  p r a c t i c e s c a n be e x a m i n e d u s i n g  operator  The  to the  controller  which  effects  these  t h e t h e r m a l model  of  developed  here. An a d d i t i o n a l d e s i g n of a  application  power d i s t r i b u t i o n  that  temperature d i s t r i b u t i o n . Although has  not  been  considerable 5.3.1  demonstrated  expense  E f f e c t of Melt The e f f e c t  thermal  regime  was  d i s t r i b u t i o n s of the  in  5.4.  will  model produce a  possible this  is  this  work  due  melt  rate  the given  procedure to  the  involved. Rate of  altering  evaluated f o r m shown  l e v e l s and m e l t r a t e s a s s o c i a t e d table  of the  the  using  a  i f figure with  series  on  the  of  power  5.9. The  power  them a r e s u m m a r i z e d i n  Melt Rate (kg/hr)  Power Level (W cm" ) Stage 2 2  Stage 1  Stage  40.0  24.0  1 15.0  153.0  50.0  30.0  103.0  153.0  60.0  36.0  91.0  153.0  70.0  42.0  79.0  153.0  80.0  48.0  67.0  153.0  90.0  54.0  55.0  153.0  100.0  60.0  43.0  153.0  Table  5.4 Power L e v e l s U s e d a s a F u n c t i o n o f M e l t  3  Rate.  136 The table  r e s u l t s of  5.5.  For  this  the c a l c u l a t i o n s particular  g e o m e t r y and  power l e v e l ,  produce pool  depths greater  melt r a t e s greater insufficient  than the  of  skull  t h a t 40  kg/hr  skull thickness  kg/hr produce  s u p r i s i n g l y , both  and  l i q u i d pools  of  efficiency  and  the thermal  the m e l t i n g e f f i c i e n c y  r i s e as t h e  (see  increase  f i g u r e 5.10). T h i s a  decrease  maximum e v a p o r a t i o n figure  less  5.11).  This  in  rate  i n these  both of  melt rate i s  the pool  the  indicates  two  parameters  volume  alloying  t h a t the  increased  and  elements  higher  the  melting process the  s o l e aim  scrap or does not these volume  and  b e t t e r product  of the m e l t i n g  produce a  primary  i s achieved.  stage  (see  melting  c o n t a i n exhogenous p a r t i c l e s i n the  (a m e a s u r e o f t h e  residence  the  liquid  state) should  the  removal  of  the  between t h r o u g h p u t  also  be  offending and  particle  from m a t e r i a l ( i . e . WC  feed  stock  or T i N ) .  l a r g e enough t o a l l o w  s e p a r t i o n and  If pool  t i m e of t h e m a t e r i a l  some  CP  which  then the  s p e c i e s . Thus  the  i s true  i s to consolidate  electrode  defects are present  This  is the  r a t e t h e b e t t e r t h e m e t a l t h r o u g h p u t , t h e more e f f i c i e n t  if  in  volume.  Not  m a t c h e d by  outlined  configuration  melt r a t e s  t h a n 90  are  in for  balance  dissolution  r a t e s must be a t t a i n e d . 5.3.2  Power  Distribution  With the power d i s t r i b u t i o n  single has  exception  of  l i q u i d motion,  the g r e a t e s t e f f e c t  on  the  the  thermal  1 37  Melt Rate (kg/hr)  Pool Volume (cm ) 3  Specific Energy (KWH/kg)  Evaporation Rates (kg/hr) Ti  Al  60.0  1023.0  1.84  0.2196  1.9551  70.0  811.0  1.57  0.1093  1.1167  80.0  515.0  1.38  0.0645  0.6521  90.0  217.0  1.22  0.0446  0.3909  Table  5.5 P o o l D a t a , E v a p o r a t i o n R a t e s a n d H e a t B a l a n c e C a l c u l a t i o n s as a F u n c t i o n of Melt Rate.  138  T h e r m a l Efficiency As A F u n c t i o n of Melt Rate 60 •2.75 Specific Energy  50 A Thermal Efficiency  O CD  •2.25  40H h 1.75  • —4  O  o  30 H  CD 125  20  30  Figure  c O  Melt Efficiency  C£3  CD  50  70  Melt Rate (Kg/hr)  90  5.10 T h e r m a l E f f i c i e n c y a s a F u n c t i o n  no  of Melt  #  •0.75  Rate.  139  Pool Parameters As A Function of Melt Rate  40  50  60  70  80  90  100  Melt Rate ( K g / h r )  Figure  5.11 P o o l Volume a n d E v a p o r a t i o n Function of Melt Rate.  Rate of A l as a  140 r e g i m e i n e l e c t r o n beam r e m e l t i n g . The e f f e c t the  t o t a l power  shown  supplied to the  (see s e c . 4.3.2.1). I t  power l e v e l  results  s u r f a c e heat  beam power i n p u t size  of the  distribution  can a l s o  With  have a  large effect  on t h e  distribution  circle  effect  to of  was  of  i n the previous  t h i s power r a d i u s 5 cm  altering  d e m o n s t r a t e d by  o b t a i n e d by c o r r e c t i n g  same t o t a l  i s shown  size  of  figure  the  power  base  case  (see f i g u r e final  5.14)  case  was  used i n the  base  a r e a . The r e s u l t i n g  power  5.15.  The m o d e l was t h e n r u n The r e s u l t s  total  (see  shrinking the  t h e e l l i p s e shape  i n figure  i s the  representing a  p o w e r . The  case t o a rectangule of i d e n t i c a l  distributions.  the  used  section.  distribution  i n the width direction  maintaining the  distribution  thermal  chosen f o r t h e base case i s  o f 80 k g / h r u s e d  power d i s t r i b u t i o n and  power  skull.  alteration  The  electron  i n the  power o f 3.15 KW t o t h e b a s e power d i s t r i b u t i o n 5.13).  the  i n c r e a s e s i n p o o l volume,  5.12. I t i s t h e same power d i s t r i b u t i o n  for a melt r a t e  a d d i t i o n of a  increasing  a r e a . These changes  The power d i s t r i b u t i o n  The f i r s t  previously  evaporation rates.  power i n p u t  i n figure  s k u l l h a s been  i t i s a l s o p o s s i b l e t o change t h e shape and  r e g i m e i n t h e EB  shown  increasing  was f o u n d t h a t  in significant  l o s s e s and  of  u s i n g each of these  o f t h e r u n s a r e shown  5.16 t o 5.19 a n d s u m m a r i z e d i n t a b l e 5.6.  in  power figures  \0  POWER DISTRIBUTION id C  PONER LEVEL  cn  > T3  •D  Co O Q. C (-•• (0 rt- it  48.0 H/CN**2 W 67.0 H/CM*M2 • 134.0 W/CMx*2  201.0 W/CM**2 B l 335.0 W/CM*x2 268.0 W/CM**2 Z Z 67.0 W/CM**2 ES3 153.0 W/CM*x2  (-••  o a in  rr rt  0 ^ rt CT V C (D rt W O  01 3 01 ro —  (az —' o UJ  5 <° a  c e o o o  o 0) CO 01 • ro — • cn  o o n  i—•  n>  ' 1— -5.0  10.0  I  25.0  55.0  40.0  COORDINATE  70.0  85.0  [CM)  4^  ro  143  CO  c  at cn  CD X  no o o  o in  o rt  ar  ID W  01  -1 .0  in  3.0  i—i  9.0  r  15.0  21.0  1  27  0  X  1 33.0  1  I 39.0  n  I 45.0  COORDINATE  I  I 51 . 0  (CM)  i — 6i 3 . 0  57.0  r  69.0  75  i—r  0  81  o 0) in ft)  4^ cn  CTi  147  03 .m . c n  CO  .cn CO  in  in' _  CJ . in TLU  3  I—  oCE  or—•  O  CJ  . r-' CM  " o  ' rg " a . in o  ' cn a  ' on  i—i—r S 2 L I S ' U S ^ l - 016 0IX) 'dW31 JcjfWD) -  (  t  F i g u r e 5.18 P o o l P r o f i l e s  i—IT  .CO  0'6 0"C- I "QcJOOG Z  f o r Power D i s t r i b u t i o n  2  a  Figure  5.19  P o o l P r o f i l e s f o r Power  Distribution  1 49  Power Distribution  Pool Volume (cm ) 3  Thermal Efficiency (%)  Evaporation Rates (kg/hr) Ti  Al  BASE  515.0  50.0  0.0645  0.6521  1  671.0  48.8  0.2135  1.3206  2  739.0  48.1  0.4120  2.6667  3  445.0  50.4  0.0700  0.6505  Table  5.6 P o o l D a t a , H e a t B a l a n c e C a l c u l a t i o n s a n d E v a p o r a t i o n R a t e s f o r Power D i s t r i b u t i o n R u n s .  150 The an  increase  increase  a d d i t i o n of the e x t r a c i r c l e in  pool  i n pool  distribution  in  i n t e r m s of  In  an  case  of a l u m i n u m . F i n a l l y  variation  the  The  location  losses without t h e use  in  pool  direction  volume  43%  is  i n the e v a p o r a t i o n  provided  of  maximizing  little  as  large  the  be  rate  input  this  evaporation Finally  seems t o  power  be  particular  produce  distributions within  or c i r c u l a r  be  possible.  one  incraases  no  while  should  i n c r e a s i n g p o o l volume.  temperature  elliptical  can  volume  power  any  or  to  rates.  a s u r f a c e a r e a as  used as  appreciably  pool  o f r e c t a n g u l a r power d i s t r i b u t i o n s  although  of  significant conclusion  produce s l i g h t l y higher pool 5.4.  rate.  rates,  s l i g h t l y more u n i f o r m melt  i n c r e a s i n g the e v a p o r a t i o n  c o n c e n t r a t i n g power i n  s h o u l d not  Shrinking also  increase  data a  purpose  u n i f o r m l y over  p r a c t i c e of  100%.  is  distribution  minimizing evaporation spread  theoretical  t h e p o o l volume or e v a p o r a t i o n  From t h i s made. F o r  width  this  t h e change i n shape from e l l i p t i c a l  power  i n both  Accompanying  i n c r e a s e i n the  the  increase  a c c o m p a n i e d by a f o u r f o l d  a rectangular  ^30%.  r a t e of aluminum of over  disadvantageous this  of  v o l u m e i s an  maximum e v a p o r a t i o n t h e power  volume  o f power p r o d u c e s  the  distributions  volumes.  Summary Four a s p e c t s of h e a r t h  been e x a m i n e d a s t o t h e i r the thermal  r e g i m e . The  o p e r a t i o n and  e f f e c t s on  design  the pool parameters  c o n c l u s i o n s a r r i v e d at are  :  have and  151 p o o l volume i s  determined p r i m a r i l y  l e n g t h a n d power d i s t r i b u t i o n of  and i s l a r g e l y  (skull)  independent  s k u l l depth or width,  pool  volume  is  increased).by  increased  removing  from around the s k u l l evaporation  (or  the water - this  melt  rates  cooled  practice  can  copper  be  hearth  increases both the  r a t e s and t h e r i s k s ,  More e f f i c i e n t  use o f t h e i n p u t power i s made when m e t a l  throughput rates are increased power must be w e i g h e d a g a i n s t t h e most e f f i c i e n t low  by h e a r t h  evaporation  combination  - this refining  use  of  and  of h i g h p o o l  rate i s obtained  power o v e r a s l a r g e an a r e a  efficient  volume  by s p r e a d i n g  as p o s s i b l e .  the  with input  152 CHAPTER 6 Summary a n d R e c o m m e n d a t i o n s f o r F u t u r e Work 6.1. Summary I n t h i s work a t h r e e d i m e n s i o n a l transfer  model  of  the  developed.  The p r i n c i p a l  the model  to  volume)  predict  which  efficient  are  method o f  electron  beam  steady hearth  o b j e c t i v e of t h e certain  in  has  been  (such  determining  operation of the  heat  work was t o  parameters  important  state  use  as  pool  the  e l e c t r o n beam  most hearth  furnace. During crucial  t h e work  i t was  parameters f o r determining  hearth are the f l u i d applied  to  determine  the  flow  t h e two  the thermal  regime i n t h e  a n d t h e power  skull.  the size,  shown t h a t  These  shape and  two  distribution/level  factors  temperature  essentially  distribution  the l i q u i d p o o l . Other f a c t o r s  were c o n s i d e r e d a s t o  e f f e c t s on  These  factors the  the  liquid  (depth, w i d t h ) , the presence  i n f l u e n c e of melt  the thermal primary  pool.  to  theoretical  aluminum from a t i t a n i u m a l l o y melt  1 52  their  of t h e h e a r t h mould and factors also  f l o w a n d power  maximum  of  geometrical  a f a r lesser extent  c o n s i d e r a t i o n s of f l u i d The  included  r a t e . These o t h e r  regime but  most  affect  than  the  input.  evaporation  rate  was a l s o c a l c u l a t e d .  of In  153 general  i t was  found  that  parameters which produced (and and  therefore increased settling  rate. to  surface evaporation The  the  s k u l l . An  being  Increased  pool  rate  m e l t i n g was r e f l e c t e d  volume  employed  i n the  in  decrease i n the more e f f i c i e n t r a t e s used balance  effect  reducing  power d i s t r i b u t i o n  beam  applied  resulted i n less  a decrease  i n the  evaporation  rate  the  electron  an i n c r e a s e  to  energy in  melt  refining capability,  of a l l o y  elements and  i n commercial  operations  must a r r i v e  o f r e f i n i n g , power u s a g e a n d e v a p o r a t i o n  on  movement  a a  u s e o f t h e i n p u t e n e r g y f o r m e l t i n g . The m e l t  Skull pool  geometry volume  ( i . e . depth, and  d i m e n s i o n s do h a v e a s i g n i f i c a n t flux  during  increased melt r a t e  exception  liquid while  time  elements.  d e l i v e r e d to the s k u l l . Therefore  rate results  volume  evaporation  v e l o c i t y was t h e o n l y  r a t e of a l l o y i n g  melt  input  l i q u i d pool  increased the surface  generalization. increased  of  refining capacity, dissolution  times) also  significantly  combination  an i n c r e a s e d  Increasing the f l u i d this  any  effect  i n c r e a s e d , t h e average heat f l u x  a  small  rates.  Skull  on t h e a v e r a g e  o u t o f t h e s k u l l . As t h e d e p t h a n d w i d t h  some  rates.  width) has  evaporation  at  heat  dimensions are  through the s k u l l  surface  154  decreases. This allows the  f o r c o n s t r u c t i o n of hearth  mould  l e s s c o s t l y t u b e a n d c a s t method i f t h e s k u l l  were  by  large  enough. 6.2.  Recommendations f o r F u t u r e  6.2.1  Work  Verification At  the  present  u n v e r i f i e d . Although operation  time  the  i t predicts  model  trends  is  essentially  observed  during  - s u c h a s p o o l d e p t h s i n t h e r a n g e o f 2 - 4 cm a n d  evaporation  rates increasing with  the degree  to  unknown. O n l y  which the an  i n c r e a s i n g power  thermal  extensive  regime i s  program  of  predicted  modelling  s k u l l s a n d t h e i r a c c o m p a n y i n g m o u l d s f o l l o w e d by experiments  on  similar  equipment  input  can  is  smaller conducting  verify  the  model  completely. 6.2.2 Power D i s t r i b u t i o n a n d Power •The e x a c t surface  of  completely  a  d e t a i l s o f t h e d e l i v e r y o f power t o t h e  l i q u i d metal  known.  by  an e l e c t r o n  For instance,  s u r f a c e , t h e programmed a n g l e o f m a t e r i a l being due this  Level  irradiated a l l  the  I t would a l s o descriptions  of  power  shape o f  a r e not  the  liquid  t h e e l e c t r o n beam a n d  c o n t r i b u t e t o t h e power  to electron backscattering. Further type of i n f o r m a t i o n  beam  work i n  the lost  determining  i s required. be h e l p f u l t o  have a c c e s s t o  d i s t r i b u t i o n s used  in  good  industrial  155 hearth  furnaces.  e x a c t l y where skull,  this  programmed type of  Although  t h e beam location  an  is  operator  a c t i n g on  does  can  of  the  correspond  to  the  not always  l o c a t i o n . The i n c r e a s e i n t h e r e l i a b i l i t y  data  would  Evaporation  tell  the surface  require significant  financial  f r o m t h e p r i v a t e s e c t o r and i s n o t l i k e l y 6.2.3  always  t o be  of t h i s support  forthcoming.  Rates  F u r t h e r work on t h e m o d e l l i n g o f e v a p o r a t i o n d u r i n g e l e c t r o n beam r e m e l t i n g in  t h i s work i s a t  some i n d i c a t i o n  the  thermal  i s r e q u i r e d . The m o d e l  best p r i m i t i v e , attempting  of the expected  Investigation regime  into  and  rates  evaporation  the unsteady  the  fluid  used  only to  give  rates. state nature  flow  regime  of  should  accompany a n y p r o g r a m e x a m i n i n g  e v a p o r a t i o n r a t e s . Both  these  t h e amount o f e v a p o r a t i o n .  f a c t o r s can g r e a t l y e f f e c t  6.3. C o n c l u d i n g  Remarks  The m o d e l first  attempt  of  at  developed  in this  q u a n t i f y i n g the  r e g i m e s d u r i n g e l e c t r o n beam h e a r t h o f work r e m a i n s t o be done more e f f i c i e n t l y  thermal  represents and  flow  remelting. A great  deal  t o energy usage,  characteristics.  a  mass  to allow the process  with respect  r a t e s and r e f i n i n g  work  to  operate  evaporation  List  of References  1.  D. A p e l i a n , C.H. E n t r e k i n : I n t e r n a t i o n a l M e t a l s R e v i e w s , 3J_, No. 2, 1986, 7 7 - 8 9 .  2.  S. S c h i l l e r , U. H e i s i g , S. P a n z e r : " E l e c t r o n T e c h n o l o g y " , 1982, New Y o r k , W i l e y .  3.  H.R. S m i t h : " E l e c t r o n Beam P r o c e s s i n g " , 1972, D a y t o n , O h i o , U n i v e r s a l T e c h n o l o g y C o r p o r a t i o n , 1 a 1 - 1a66.  4.  K. Amboss : " E l e c t r o n a n d I o n Beam T e c h n o l o g y " , 1974, P r i n c e t o n , N J , The E l e c t r o c h e m i c a l S o c i e t y , 482 - 5 1 7 .  5.  H. R a n k e , V. B a u e r , J . H e i m e l : " E l e c t r o n Beam R e m e l t i n g a n d R e f i n i n g - S t a t e o f t h e A r t 1986",, P r o c . C o n f . , 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 , E n g l e w o o d , N J . , 98 - 96.  6.  A. M i t c h e l l , D. T r i p p : " I n t e r n a t i o n a l C o n f e r e n c e on T i t a n i u m P r o d u c t s a n d A p p l i c a t i o n s " , P r o c . C o n f . , 1987, T i t a n i u m D e v e l o p m e n t A s s o c . , D a y t o n , O h i o , 1011 - 1019.  7.  K. T a k a g i : M a s t e r ' s T h e s i s , U n i v e r s i t y C o l u m b i a , 1984.  8.  J . P . L a u g h l i n : " I n t e r n a t i o n a l C o n f e r e n c e on T i t a n i u m P r o d u c t s a n d A p p l i c a t i o n s " , P r o c . C o n f . , 1987, T i t a n i u m D e v e l o p m e n t A s s o c . , D a y t o n , O h i o , 879 - 8 8 3 .  9.  C.H. E n t r e k i n : " E l e c t r o n Beam R e m e l t i n g a n d R e f i n i n g S t a t e o f t h e A r t 1985 P t . I " , P r o c . C o n f . , 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 , E n g l e w o o d , N J . , 40 - 47.  10. ASM M e t a l s Handbook, V o l 3, A m e r i c a n M e t a l s P a r k , O h i o , 357.  156  Beam  of B r i t i s h  Society  for Metals,  157 11. E.E. B r o w n , R.W. H a t a l a : " E l e c t r o n Beam R e m e l t i n g a n d R e f i n i n g - S t a t e o f t h e A r t 1985 P t . I I " , P r o c . C o n f . , 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 , E n g l e w o o d , N J . , 103 - 117. 12. C. d'A H u n t , J.H.C. Lowe, S. K. H a r r i n g t o n : " E l e c t r o n Beam R e m e l t i n g a n d R e f i n i n g - S t a t e o f t h e A r t 1985 P t . I " , P r o c . C o n f . , 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 , E n g l e w o o d , N J . , 58 - 7 0 . 13. V. F o r s b e r g , W. Herman : " I n t e r n a t i o n a l C o n f e r e n c e on T i t a n i u m P r o d u c t s a n d A p p l i c a t i o n s " , P r o c . C o n f . , 1987, T i t a n i u m D e v e l o p m e n t A s s o c . , D a y t o n , O h i o , 904 - 9 1 7 . 14. A. M i t c h e l l , K. T a k a g i : " E l e c t r o n Beam R e m e l t i n g R e f i n i n g - S t a t e o f t h e A r t 1984", 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 , E n g l e w o o d , N J . , 88 - 9 9 . 15.  "Thermal P r o p e r t i e s of T i t a n i u m A l l o y s " , Defense M a t e r i a l s Information Center, B a t t e l l e Memorial I n s t i t u t e , Columbus, Ohio.  16. 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 C o l u m b i a , 1978. 17. H. F e n e c h , W.M. Rohsenow : T r a n s . T r a n s . 8 5 , 1963, 15. 18. W.B.  J_, 1 970,  : Met. T r a n s . ,  849.  2, 1971, 2087.  F. K r i e t h , W.Z. B l a c k : " B a s i c H e a t T r a n s f e r " , 1980, New Y o r k , H a r p e r a n d Row.  21. M. C h o u d h a r y , J . S z e k e l y 453. 22.  of B r i t i s h  ASME, J . o f H e a t  E i s e n , A. Campagna : M e t . T r a n s . ,  19. L . F . C a r v a j a l , G.E. G e i g e r 20.  and  : Met. Trans.,  _1_1_B, 1980, 439 -  S.V. J o s h i : 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 C o l u m b i a , 1971.  158 23. R.H. McKoon : " E l e c t r o n Beam R e m e l t i n g a n d R e f i n i n g S t a t e o f t h e A r t 1986", P r o c . C o n f . , 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 , E n g l e w o o d , N J . , 45 - 5 2 . 24. I.V. S a m a r a s e k e r a , J.K. Brimacombe Q u a r t e r l y , J J 8 , 1979, 251 - 2 6 6 .  : Can. Met.  25. L. Hageman, D. Young : " A p p l i e d I t e r a t i v e A c a d e m i c P r e s s , New Y o r k , 1 9 8 1 . 26. C.H. E n t r e k i n  : Unpublished  Methods",  work.  27. H.S. K h e s h g i , P.M. G r e s h o : " E l e c t r o n Beam R e m e l t i n g a n d R e f i n i n g - S t a t e o f t h e A r t 1986", P r o c . C o n f . , 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 , E n g l e w o o d , N J . , 68 - 97. 28. R. P a s k o  : Unpublished  work.  29. C E . H a r r i s o n : M a s t e r ' s 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 C o l u m b i a , 1984. 30. H.R. H a r k e r , C.H. E n t r e k i n , " I n t e r n a t i o n a l C o n f e r e n c e on T i t a n i u m P r o d u c t s a n d A p p l i c a t i o n s " , P r o c . C o n f . , 1987, T i t a n i u m D e v e l o p m e n t A s s o c . , D a y t o n , O h i o , 939 - 9 4 7 . 31. H e i p l e and Roper  : Weld J . , 6_1_, 1982, p p . 97,  32. O r e p e r e t . a l . : Weld J . , 6 2 , 1983, p p . 3 0 7 .  APPENDIX 1 Beam Spot Model Introduction The are time  power d i s t r i b u t i o n s  a v e r a g e d due  s t a t e heat  flow  t o the  used  i n the h e a r t h  r e s t r i c t i o n s of  equation. This  implies  temperatures r e p o r t e d are time averaged  model  the  steady  t h a t a l l of as w e l l .  In  the  actual  operation, temperature v a r i a t i o n s at  the s u r f a c e occur  due  to the  These  if  moving  significantly l o s s e s due  p o i n t power l a r g e , may  source.  be i m p o r t a n t  to radiation  and a l l o y  variations,  i n determining  element  losses  heat  due  to  evaporation. In temperature  order  to  evaluate  variations  and  the  magnitude  determine  b e t w e e n them and power d e n s i t y ,  the  d e v e l o p e d . The  Ballantyne  model i s  relationship other  s t a t e heat  16  after  the  beam sweep s p e e d a n d  time dependent v a r i a b l e s , a s i m p l e unsteady model p a t t e r n e d  of  flow  28  and  Pasko  c a p a b l e of p r e d i c t i n g  as a f u n c t i o n of time i n a c y l i n d r i c a l  has  been  temperatures  section.  Formulation S i n c e the i n c l u s i o n of time i n a heat represents the advantageous  inclusion  t o choose  of  an  extra  flow  dimension,  problem i t  a c o o r d i n a t e system which a l l o w s  159  is the  160 exclusion  of  one  of t h e  system which p r o v i d e s the  physical  symmetry  system  (see  heat  flow d i r e c t i o n s .  this considering i scylindrical  figure  A1.1).  assumption the unsteady  3 3T k3T 3 3T — (k— ) + + — ( k — ) 3x 3r r 3r 3z 3z  = pC  from a l l other e f f e c t s ,  surfaces  possible.  Therefore  3T k — = 0 where r = r 9r  axial  3T • k — = 0 where z = z 9z  m  a  a  x  m  The  the a f f e c t s of the conditions  the  perimeter  t o be a d i a b a t i c  and  ...(A1.2)  x  ...(A1.3)  assumption of a x i a l  symmetry i m p l i e s  c  t o p boundary  condition at  reflects  the three  the surface.  i s b e i n g d e l i v e r e d by t h e e l e c t r o n  i s being radiated  that  ...(A1.4)  transfer processes taking place energy  bottom  or :  S  surface  beam  were made a s  3T k — = 0 where r = 0 9r  The  symmetry  „  c  S  axial  equation i s  q  S  on  ...(A1.1)  t h e boundary  were c o n s i d e r e d  with  3T — P3t  I n an a t t e m p t t o i s o l a t e  simple as  the  s t a t e heat flow  only  the constraints  coordinates  Making  The  t o t h e f u r n a c e atmosphere  heat  At the top beam, h e a t  and m a t e r i a l  is  161  162 evaporating power and  requiring  radiation  t h a t used i n  the  latent  heat  of v a p o r i z a t i o n .  boundary c o n d i t i o n s  the h e a r t h  furnace  model  The  are equivalent ( e q u a t i o n 2.4)  to with  t h e e x c e p t i o n t h a t a t i m e a v e r a g e d power d i s t r i b u t i o n  i s not  requi red. The  e v a p o r a t i o n p o r t i o n of the t h e r m a l  t o p s u r f a c e i s a r r i v e d a t by m u l t i p l y i n g t h e e v a p o r a t i o n by melted.  The  equation  the  rate  ( e q u a t i o n 4.1).  from l i q u i d  with  r a t e of  of e v a p o r a t i o n  the r a t e l i m i t i n g  t h e gas  evaporation  t o gas  and  This equation  phases. Using  not d i f f u s i o n  3T 4 4 - k , , — = P ( r ) - o e ^ i T Z - T*) 9z S S A 0  S  S  Due  to  the  p  u s e d . The  equations  technique  are  implicit smaller  time  using  splitting  steps to reduce  non-linear nature  Langmuir providing  transformation the l i q u i d  under a good The  top  /2irRMT  or  vacuum boundary  ...(A1.5)  of  solution  a finite  generated  solved  technique  i s the  this.  complex n a t u r e  impossible. Therefore  the  p°C  b o u n d a r y c o n d i t i o n , an a n a l y t i c a l is  by  being  :  AH -  of  is valid  a pure m a t e r i a l  t h e n be w r i t t e n a s  l a t e n t heat  in either  i n f i n i t e pumping s p e e d i n s u r e s  c o n d i t i o n can  the  the m a t e r i a l  i s given  step for evaporation  l o a d on  the  top  to the  surface equations  d i f f e r e n c e approach  by  the  the each  finite  difference  alternating  direction  time  step  into  the e r r o r a s s o c i a t e d w i t h  of t h e r a d i a t i o n  was  boundary c o n d i t i o n .  four the  

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